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\section{Introduction}
The theory of the neutrino-pair radiation from nucleons in the $\pi ^{0}$-
condensed nucleon matter of neutron stars was developed in \cite%
{Voskresensky} under the basic simplification that the amplitude of the pion
field is small with respect to the pion mass, $\left| {\bf \varphi }\right|
^{2}\ll m_{\pi }^{2}$, so that the nucleon-pion interaction may be
considered as a perturbation, and the matrix element of the reaction is
given by the following diagrams,
\psfig{file=Fig1.eps} where the wavy line represents the external field of
the pion condensate, and the initial and final nucleon states are the plane
waves. The conclusion has been made that the corresponding neutrino energy
losses depend on temperature as $T^{6}$.
What we demonstrate in this letter is that the above result is valid only in
the more strong limit, $\left| {\bf \varphi }\right| ^{2}\ll T^{2}$, which
is never fulfilled in the case of developed condensate field. In the more
realistic case, $\left| {\bf \varphi }\right| ^{2}\gg T^{2}$, the process is
suppressed exponentially. The physical reason for this is a periodic
structure of the developed pion condensate. The repeated interactions of a
nucleon with the periodic pion field modify the nucleon spectrum, which is
split into two bands, and the energy gap between the bands is about a few $%
MeV$ or larger, even if the amplitude of the pion field is small compared
with the pion mass. Consequently, at temperatures less than a few $MeV$, at
which the above reactions has been thought to be important, the efficiency
of the process is much less than previously estimated. The analogous
mechanism of suppression of the neutrino-pair radiation was considered for
the electron bremsstrahlung in the crystalline crust of the neutron star %
\cite{Pethik}, and for neutrino emission from the bubble phase of the
stellar nuclear matter \cite{L1993}.
In the following we use the system of units $\hbar =c=1$.
\section{Nucleon states in the pion field}
By assuming a pseudovector nucleon interaction with the pion field ${\bf %
\varphi =}\left( \varphi _{1},\varphi _{2},\varphi _{3}\right) $ the
effective nucleon Hamiltonian can be written as
\begin{equation}
{\cal H}_{nr}=-\frac{1}{2M^{\ast }}{\bf \nabla }^{2}{\bf +}\frac{f}{m_{\pi }}%
\left( {\bf \sigma }\nabla \right) \left( {\bf \varphi \tau }\right)
\label{Hnr}
\end{equation}%
where the spin, $\sigma _{i}$, and isospin, $\tau _{i}$, operators are given
by the Pauli matrices; $M^{\ast }$ is the effective nucleon mass; and $%
f=0.988$ is the pion-nucleon coupling constant.
To examine the nucleon spectrum in the presence of the $\pi ^{0}$
condensate, let us consider a simple model with the condensed classical pion
field ${\bf \varphi =}\left( 0,0,\varphi _{0}\right) $ of the form of a
standing wave \cite{Migdal}
\begin{equation}
\varphi _{0}\left( {\bf r}\right) =\sqrt{\frac{2}{3}}a\left( \sin kx+\sin
ky+\sin kz\right) \label{fi}
\end{equation}%
with a real amplitude $a$, so that $a^{2}=\left\langle {\bf \varphi }%
^{2}\right\rangle .$ In this case we have%
\begin{equation}
{\cal H}_{nr}=-\frac{1}{2M^{\ast }}{\bf \nabla }^{2}{\bf +}\sqrt{\frac{2}{3}}%
f\frac{a}{m_{\pi }}k\left( {\bf \sigma R}\right) \tau _{3}
\end{equation}%
with%
\begin{equation}
{\bf R}\left( {\bf r}\right) =\left( \cos kx,\cos ky,\cos kz\right) .
\label{R}
\end{equation}%
Thus the nucleon quasi-particle moves in the three-dimensional periodic
potential%
\begin{equation}
U\left( {\bf r}\right) =\sqrt{\frac{2}{3}}f\frac{a}{m_{\pi }}k\left( {\bf %
\sigma R}\right) \tau _{3},
\end{equation}%
which can be recast as the sum over reciprocal lattice vectors%
\begin{equation}
U\left( {\bf r}\right) {\bf =}\sqrt{\frac{2}{3}}f\frac{a}{m_{\pi }}%
kt_{3}\sum_{{\bf K}}\left( {\bf \sigma R}_{{\bf K}}\right) \exp \left( i{\bf %
Kr}\right) . \label{Ur}
\end{equation}%
Here $t_{3}=\pm 1$ for protons and neutrons respectively. The matrix element
\begin{equation}
{\bf R}_{{\bf K}}=\frac{1}{V}\int_{V}d^{3}r{\bf R}\left( {\bf r}\right) \exp
\left( i{\bf Kr}\right)
\end{equation}%
is given by the integral over the volume of the elementary cell%
\begin{equation}
V=\left( \frac{2\pi }{k}\right) ^{3}.
\end{equation}%
One can easily find that there are only six non-zero reciprocal lattice
vectors for which the matrix element does not vanish, ${\bf R}_{{\bf K}}\neq
0$. The direct calculation yields%
\begin{equation}
{\bf R}_{{\bf K}}=\frac{1}{2}{\bf n}_{{\bf K}},
\end{equation}%
where the unit vector ${\bf n}_{{\bf K}}$ has the following components
\begin{eqnarray}
{\bf n}_{{\bf K}} &=&{\bf n}_{{\bf -K}}=\left( 1,0,0\right) \ \ \ \ \text{if
\ \ \ }{\bf K}=\left( \pm k,0,0\right) \nonumber \\
{\bf n}_{{\bf K}} &=&{\bf n}_{{\bf -K}}=\left( 1,0,0\right) \ \ \ \ \text{if
\ \ \ }{\bf K}=\left( \pm k,0,0\right) \label{UK} \\
{\bf n}_{{\bf K}} &=&{\bf n}_{{\bf -K}}=\left( 0,0,1\right) \ \ \ \ \text{if
\ \ \ }{\bf K}=\left( 0,0,\pm k\right) \nonumber
\end{eqnarray}%
Thus we can write%
\begin{equation}
U\left( {\bf r}\right) {\bf =}U_{k}t_{3}\sum_{{\bf K}}\left( {\bf \sigma n}_{%
{\bf K}}\right) \exp \left( i{\bf Kr}\right) ,
\end{equation}%
where
\begin{equation}
U_{k}=\frac{1}{2}\sqrt{\frac{2}{3}}fa\frac{k}{m_{\pi }} \label{Uk}
\end{equation}
By assuming that the periodic potential $U\left( {\bf r}\right) $ is small
with respect to the nucleon energy, near the Fermi surface, the nucleons can
be considered as almost free quasi-particles of the effective mass $M^{\ast
} $. Then the crystal potential has the most effect on the nucleon states
for which the free particle energies $\varepsilon _{{\bf p}}$ and $%
\varepsilon _{{\bf p-K}}$ are almost equal for some reciprocal lattice
vector ${\bf K}$, or, equivalentely, if the component of the quasi-particle
momentum in the direction of the reciprocal lattice vector is close to $%
p_{\parallel }\simeq K/2$. The latter condition means that, in the above
approximation, the summation over reciprocal lattice vectors can be
restricted by the values
$\left| {\bf K}\right| <2p_{F}\text{.}$
Typically the pion condensation appears at $k\simeq \left( 1.4\div
1.7\right) m_{\pi }$, and the condition (\ref{Kmax}) means the summation
over all of the six reciprocal lattice vectors as given by Eq. (\ref{UK}).
At $p_{\parallel }\simeq K/2$ one has $\varepsilon _{{\bf p}}\simeq
\varepsilon _{{\bf p-K}}$. In this case, up to the leading terms, the
equation of motion of the nucleon has the form%
\begin{equation}
\left[ E+\frac{1}{2M}{\bf \nabla }^{2}{\bf +}t_{3}U_{k}e^{i{\bf Kr}}\left(
{\bf \sigma n}_{{\bf K}}\right) \right] \psi =0,
\end{equation}%
and the nucleon wave function can be written as a superposition of two plane
waves%
\begin{equation}
\psi \left( {\bf r,\sigma }\right) =\left[ A\left( {\bf p,}\sigma \right)
\exp \left( i{\bf pr}\right) +B\left( {\bf p,}\sigma \right) \exp \left( i%
{\bf pr-}i{\bf Kr}\right) \right] ,
\end{equation}%
where the functions $A\left( {\bf p,}\sigma \right) $ and $B\left( {\bf p,}%
\sigma \right) $ obey the following set of equations%
\begin{equation}
\left( E-\varepsilon _{{\bf p}}\right) A\left( {\bf p,}\sigma \right) {\bf +}%
t_{3}\left( {\bf \hat{\sigma}n}_{{\bf K}}\right) U_{k}B\left( {\bf p,}\sigma
\right) =0, \label{A}
\end{equation}%
\begin{equation}
\left( E-\varepsilon _{{\bf p-K}}\right) B\left( {\bf p,}\sigma \right)
+t_{3}\left( {\bf \hat{\sigma}n}_{{\bf K}}\right) U_{k}A\left( {\bf p,}%
\sigma \right) =0, \label{B}
\end{equation}%
$\allowbreak $which has a non-trivial solution if
\begin{equation}
\left( E-\varepsilon _{{\bf p-K}}\right) \left( E-\varepsilon _{{\bf p}%
}\right) {\bf -}U_{k}^{2}=0.
\end{equation}%
As it follows from this dispersion equation, the energy eigenvalues, are
devided into two bands,
\begin{equation}
E_{{\bf p}}^{\pm }=\frac{\varepsilon _{{\bf p}}+\varepsilon _{{\bf p-K}}}{2}%
\pm \sqrt{\left( \frac{\varepsilon _{{\bf p}}-\varepsilon _{{\bf p-K}}}{2}%
\right) ^{2}+U_{k}^{2}}, \label{Epm}
\end{equation}%
shown in Fig. 1, where the upper and the lower band are denoted as ${\bf +}$
and $-$ respectively.
\psfig{file=spectr.eps}{Fig. 1 Red solid line represents the nucleon spectrum as given
by Eq, (10). Dashed line corresponds to a free nucleon.
The arrow shows a possible nucleon transition accompanied by the neutrino-pair
emission.} $\allowbreak $
Near $\varepsilon _{{\bf p}}\simeq \varepsilon _{{\bf p-K}}$, the nucleon
spin function can be taken as the eigenstate of ${\bf \hat{\sigma}n}_{{\bf K}%
}${\bf :}
\begin{equation}
\left( {\bf \hat{\sigma}n}_{{\bf K}}\right) \chi _{\lambda }=\lambda \chi
_{\lambda },\ \ \ \lambda =\pm 1,
\end{equation}%
Thus, from Eqs. (\ref{A}), (\ref{B}) we obtain the following normalized wave
function for the upper band%
\begin{equation}
\psi _{{\bf p,\lambda }}^{+}\left( {\bf r,\sigma }\right) =\left[ u_{{\bf p}%
}\exp \left( i{\bf pr}\right) {\bf -}t_{3}v_{{\bf p}}\left( {\bf \hat{\sigma}%
n}_{{\bf K}}\right) \exp \left( i{\bf pr-}i{\bf Kr}\right) \right] \chi
_{\lambda }\left( \sigma \right) , \label{p}
\end{equation}%
while for the lower band we have%
\begin{equation}
\psi _{{\bf p,\lambda }}^{-}\left( {\bf r,\sigma }\right) =\left[ v_{{\bf p}%
}\exp \left( i{\bf pr}\right) {\bf +}t_{3}u_{{\bf p}}\left( {\bf \hat{\sigma}%
n}_{{\bf K}}\right) \exp \left( i{\bf pr-}i{\bf Kr}\right) \right] \chi
_{\lambda }\left( \sigma \right) . \label{m}
\end{equation}%
The ''coherence factors'' are given by%
\begin{equation}
u_{{\bf p}}^{2}=\frac{1}{2}\left( 1+\frac{\xi _{{\bf p}}}{\zeta _{{\bf p}}}%
\right) ,\ \ \ \ v_{{\bf p}}^{2}=\frac{1}{2}\left( 1-\frac{\xi _{{\bf p}}}{%
\zeta _{{\bf p}}}\right) ,\ \ \ u_{{\bf p}}v_{{\bf p}}=\frac{\left|
U_{k}\right| }{2\zeta _{{\bf p}}},
\end{equation}%
with
\begin{equation}
\xi _{{\bf p}}=\frac{\varepsilon _{{\bf p}}-\varepsilon _{{\bf p-K}}}{2},\ \
\ \ \ \ \zeta _{{\bf p}}=\sqrt{\xi _{{\bf p}}^{2}+U_{k}^{2}}.
\end{equation}%
In contrast to the ordinary plane waves, in the nucleon wave functions (\ref%
{p}), (\ref{m}), the nucleon interaction with the condensed pion field is
''exactly'' taken into account\footnote{%
In our context we use term ''exactly'' to stress that the energy spectrum (%
\ref{Epm}) can be obtained only by summation of the perturbation series to
all orders.}. As it was expected the spectrum of the corresponding nucleon
states is of the band-like structure similar to that for electrons in a
cristalline metal. As given by Eq. (\ref{Epm}), the energy gap in the
nucleon spectrum is $\Delta =2U_{k}$. As will be shown in the next section,
even in the case of a small amplitude of the pion field, $T\ll a\ll m_{\pi }$%
, the energy gap, $U_{k}\gg T$, exponentially suppresses the neutrino-pair
bremsstrahlung from the nucleons, and the neutrino energy losses in this
process are much less than estimated before in \cite{Voskresensky}.
\section{Neutrino energy losses}
When, in the nucleon wave functions, the condensed pion field is ''exactly''
taken into account the neutrino-pair bremsstrahlung%
\begin{equation}
n\rightarrow n+\nu +\bar{\nu} \label{br}
\end{equation}
can be described by the following diagram,
\psfig{file=Fig2.eps} where the lepton pair is radiated by the nucleon
undergoing a transition between its eigen-states in the pion field. If the
initial and final nucleon states are in the same band, the radiation is
kinematically forbidden. Indeed, in this case the four-momentum $q=\left(
q_{0},{\bf q}\right) $ transferred from the nucleon is space-like\footnote{%
Since the group velocity of the nucleon is $\nabla _{{\bf p}}E<1$, the
energy difference between nucleon states $q_{0}=\left| {\bf q}\right| \nabla
_{{\bf p}}E$ is less than $\left| {\bf q}\right| $.}, $q_{0}<\left| {\bf q}%
\right| $, while the total momentum of the lepton pair is a time-like, $%
q_{0}>\left| {\bf q}\right| $. Consequently it is impossible simultaneously
to conserve the energy and momentum in the reaction. However, even for small
momentum transfers, there exists a finite energy difference between
different bands, so the neutrino radiation is kinematically allowed for the
nucleon transitions shown by the arrow in Fig. 1.
We now examine the process in which the neutron quasi-particle $\left(
t_{3}=-1\right) $ makes a transition from the upper band to the lower one.
The neutrino interaction with the non-relativistic neutrons is of the form%
\footnote{%
Since our goal is to prove the exponential suppression of the process, we
omit the correction to the weak vertex caused by NN correlations, which is
known to suppress the neutrino energy losses up to 100 times \cite%
{Voskresensky}.}%
\begin{equation}
\hat{H}=-\frac{G_{F}}{2\sqrt{2}}\,\left( \,J_{0}l_{0}-g_{A}\,{\bf J}{\bf l}%
\right) ,
\end{equation}%
where $G_{F}$ is the Fermi constant, and $g_{A}=1.26$. The vector current of
the non-relativistic neutron has only the time component, $J_{0}\equiv \hat{%
\Psi}^{\dagger }\hat{\Psi}$, while the axial-vector current is given by its
space components, ${\bf J}\equiv \hat{\Psi}^{\dagger }{\bf \sigma }\hat{\Psi}
$. The neutral weak current of neutrinos is of the standard form
\begin{equation}
l_{\mu }=\bar{\Psi}_{\nu }\gamma _{\mu }\left( 1-\gamma _{5}\right) \Psi
_{\nu },
\end{equation}%
We consider the total energy which is emitted into neutrino pairs per unit
volume and time. By Fermi's ''golden'' rule we have%
\begin{eqnarray*}
Q &=&\sum_{\nu }\int \frac{d^{3}pd^{3}p^{\prime }}{(2\pi )^{6}}\int \frac{%
d^{3}q_{1}}{2q_{1}^{0}(2\pi )^{3}}\frac{d^{3}q_{2}}{2q_{2}^{0}(2\pi )^{3}}%
2\pi \delta \left( E_{{\bf p}}^{+}-E_{{\bf p}^{\prime }}^{-}-q^{0}\right) \\
&&\;\times q^{0}\sum_{{\bf K}}\sum_{\lambda }\overline{\left| {\cal M}%
_{fi}\right| ^{2}}f\left( E_{{\bf p}}^{+}\right) \left( 1-f\left( E_{{\bf p}%
^{\prime }}^{-}\right) \right) ,
\end{eqnarray*}%
The integration goes over the phase volume of neutrinos and antineutrinos of
total energy $q^{0}=q_{1}^{0}+q_{2}^{0}$ and total momentum ${\bf q=q}_{1}+%
{\bf q}_{2}$. The symbol $\sum_{\nu }$\ \ indicates that summation over the
three neutrino types has to be performed. The square of the matrix element
of the reaction (\ref{br}) summed over spins of initial and final particles
has the following form%
\begin{equation}
\sum_{\lambda }\overline{\left| {\cal M}_{fi}\right| ^{2}}%
=G_{F}^{2}g_{A}^{2}u_{{\bf p}}^{2}v_{{\bf p}}^{2}\left( 2\pi \right)
^{3}\delta \left( {\bf p}-{\bf p}^{\prime }\right) \delta _{\mu i}\delta
_{\nu j}\left( \delta _{ij}-\delta _{i3}\delta _{j3}\right)
\mathop{\rm Tr}%
l_{\mu }l_{\nu }
\end{equation}%
Here we take into account that, in the degenerate case $T\ll \mu $, one can
neglect a small momentum of the neutrino par, $\left| {\bf q}\right| \sim T%
{\bf \ll }\left| {\bf p}_{F}\right| $, by making the following replacement $%
\delta \left( {\bf p}-{\bf p}^{\prime }-{\bf q}\right) \simeq \delta \left(
{\bf p}-{\bf p}^{\prime }\right) $. The distribution function of initial
nucleon as well as blocking of its final states are taken into account by
the Pauli blocking-factor $f\left( E^{+}\right) \left( 1-f\left(
E^{-}\right) \right) $, where $f$ is the Fermi distribution function%
\begin{equation}
f\left( E\right) =\left( \exp \left( \frac{E-\mu }{T}\right) +1\right) ^{-1}.
\end{equation}%
We assume that neutrinos can escape freely from the matter.
By inserting $\int d^{4}q\delta ^{\left( 4\right) }\left(
q-q_{1}-q_{2}\right) =1$ in this equation, and making use of the Lenard's
integral \
\begin{eqnarray}
&&\int \frac{d^{3}q_{1}}{2q_{1}^{0}}\frac{d^{3}q_{2}}{2q_{2}^{0}}\;\delta
^{\left( 4\right) }\left( q-q_{1}-q_{2}\right) {\rm Tr}\left( l_{\mu }l_{\nu
}^{\ast }\right) \nonumber \\
&=&\frac{4\pi }{3}\left( q_{\mu }q_{\nu }-q^{2}g_{\mu \nu }\right) \Theta
\left( q_{\mu }^{2}\right) \Theta \left( q^{0}\right) , \label{Lenard}
\end{eqnarray}%
where $\Theta (x)$ is the Heaviside step function, we can write%
\begin{eqnarray*}
Q &=&\sum_{\nu }\sum_{{\bf K}}\frac{4\pi }{3}G_{F}^{2}g_{A}^{2}\int d^{4}q\
\Theta \left( q_{\mu }^{2}\right) \Theta \left( q^{0}\right) q^{0}\int \frac{%
d^{3}p}{(2\pi )^{9}}2\pi \delta \left( E_{{\bf p}}^{+}-E_{{\bf p}%
}^{-}-q^{0}\right) \\
&&\;f\left( E_{{\bf p}}^{+}\right) \left( 1-f\left( E_{{\bf p}}^{-}\right)
\right) \left( q_{\mu }q_{\nu }-q_{\lambda }^{2}g_{\mu \nu }\right) u_{{\bf p%
}}^{2}v_{{\bf p}}^{2}\delta _{\mu i}\delta _{\nu j}\left( \delta
_{ij}-\delta _{i3}\delta _{j3}\right)
\end{eqnarray*}%
Then the direct calculation of the integrals gives
\begin{equation}
Q=\sum_{\nu }\sum_{{\bf K}}\frac{1}{60\pi ^{5}}G_{F}^{2}g_{A}^{2}M^{2}\frac{%
U_{k}^{2}}{K}\int_{2U_{K}}^{\infty }\frac{q_{0}^{6}}{\sqrt{%
q_{0}^{2}-4U_{k}^{2}}}\frac{dq_{0}}{e^{\frac{q_{0}}{T}}-1}.
\end{equation}%
Here the lower limit of integration corresponds to the minimal energy of the
neutrino pair radiated due to the nucleon transition from the upper to the
lower band.
According to Eq. (\ref{UK}), there are only six non-zero reciprocal lattice
vectors for which the matrix element ${\bf R}_{{\bf K}}$ does not vanish. By
performing summation over reciprocal lattice vectors and taking into account
three neutrino flavours, $\sum_{\nu }=3$, we obtain
\begin{equation}
Q=\frac{3}{10\pi ^{5}}G_{F}^{2}g_{A}^{2}M^{2}U_{k}^{2}\frac{1}{k}%
\int_{2U_{k}}^{\infty }\frac{q_{0}^{5}}{\sqrt{q_{0}^{2}-4U_{k}^{2}}}\frac{%
dq_{0}}{e^{\frac{q_{0}}{T}}-1}. \label{Q}
\end{equation}
The energy gap
\begin{equation}
\Delta =2U_{k}=\sqrt{\frac{2}{3}}f\frac{a}{m_{\pi }}k
\end{equation}%
between the nucleon energy bands rapidly increases along with increasing of
the amplitude of the condensed pion field. For a developed condensate one
has $2U_{k}\gg T.$ In this low-temperature limit, typical for neutron stars,
the neutrino energy losses are exponentially suppressed. Indeed, in the case
$2U_{k}/T\gg 1$, the main contribution to the integral, in Eq. (\ref{Q}),
comes from the vicinity of the lower limit where $x\simeq 2U_{k}/T$.
Therefore, in the case of developed pion condensate, we obtain the following
estimate
\begin{equation}
Q_{{\rm low-temp}}\simeq \frac{24\sqrt{2}}{5\pi ^{5}}G_{F}^{2}g_{A}^{2}M^{2}%
\frac{1}{k}TU_{k}^{7}e^{-4U_{k}/T}. \label{Qlt}
\end{equation}
In the limiting case of small amplitude of the pion field $2U_{K}\ll T$,
from Eq. (\ref{Q}) we have%
\begin{equation}
Q\simeq \frac{2\pi }{315}G_{F}^{2}g_{A}^{2}M^{2}f^{2}\frac{a^{2}}{m_{\pi
}^{2}}kT^{6}, \label{Qpert}
\end{equation}%
which reproduces the result obtained in \cite{Voskresensky} by the
perturbation theory. Notice that the energy losses (\ref{Qpert}) are
actually 3 times larger than that given in \cite{Voskresensky} due to
summation over 3 neutrino flavours.
\section{Conclusion}
To summarize, neutrino-pair bremsstrahlung from nucleons in the periodic field of the $\pi^0$ condensate is much less important than suggested by earlier estimates. The reason for this is that, at the temperatures typical for cooling neutron stars, the process is suppressed by band structure effects.
|
{
"timestamp": "2004-11-01T10:11:49",
"yymm": "0411",
"arxiv_id": "astro-ph/0411025",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411025"
}
|
\section{Introduction}
Very Long Baseline Interferometry at millimetre wavelengths (mm-VLBI) allows the detailed imaging of
compact galactic and extragalactic radio sources with angular resolutions unreached by any other
astronomical observing technique. It offers a unique possibility to image
the direct vicinity of the super-massive black holes (SMBH) thought to be located
in the centres of powerful radio galaxies and other Active Galactic Nuclei (AGN), including
the SMBH in our Galaxy (Sgr\,A*). The highest possible
angular and spatial resolution is also required to answer the still unsolved and fundamental question of how
the powerful radio jets of AGN are launched and how they are accelerated and collimated.
In radio interferometry the angular resolution can be improved either by increasing
the separation between the radio telescopes, or by observing at shorter wavelengths. The first
possibility leads to VLBI with orbiting radio antennas in space (e.g. VSOP, ARISE). The second
alternative leads to mm-VLBI, in which ground-based radio telescopes observe at frequencies
above $\sim 80$\,GHz. The resolution of a global mm-VLBI array at 230 GHz would be about $25-30$
micro-arcseconds (1 $\mu$as = $10^{-6}$ arcsec), similar to the resolution of future space-VLBI at
43 GHz.
Since the innermost region of an AGN is invisible at centimetre wavelengths
(due to intrinsic self-absorption), mm-VLBI offers the additional advantage to penetrate
this opacity barrier, opening the direct view onto the ``central engine''.
Here we report on recent developments in mm-VLBI, with particular emphasis on
VLBI experiments performed at the highest accessible VLBI frequencies of 86, 150 and 230 GHz.
We demonstrate that global VLBI at 230 GHz now is technically feasible and yields detections of AGN with an
angular resolution of $\sim 30 \mu$as. When combined with future
antennas like CARMA (USA), the SMA (Hawaii), the LMT (Mexico), and ALMA (Chile), the sensitivity
could be increased to a level so that detailed studies of galactic
and extragalactic (super-massive) black holes with a spatial resolution of only a few Schwarzschild radii
($R_{\rm S}$) will become possible.
\vspace{-0.5cm}
\section{Imaging the Base of the Jet in M\,87 with 20 ${\bf R_{\rm S}}$}
The Global 3\,mm-VLBI Array (GMVA) has been operational since 2003 (see http://www.mpifr-bonn.mpg.de/\-div/\-vlbi/\-globalmm).
At 86\,GHz, it combines the European antennas with the VLBA. In Europe, the following observatories
participate in the GMVA: Effelsberg, Onsala, Mets\"ahovi, Pico Veleta and Plateau de Bure. When
compared with the stand-alone VLBA, the GMVA is a factor of $3-4$ more sensitive. This is mainly due to
the participation of the two IRAM telescopes (the 30\,m telescope on Pico Veleta in Spain, and the 6x15\,m
inter\-fero\-meter on Plateau de Bure in France). The GMVA is open to the scientific community and
presently observes twice per year, in spring and autumn. In each session and for logistical reasons,
the observations are scheduled in time blocks of $3-6$ days duration, depending on proposal pressure.
For the near future (2005 onwards) it is planned to change the default recording mode from presently 256 Mbit/s
to 512 Mbit/s, giving better sensitivity. Since at the VLBA the tape consumption is
limited (to $\leq$ 2 tapes in 24 hrs), the duty cycle for the recording (= time used for recording / total time)
is at present only 0.21 (for 512 Mbit/s), and 0.43 (for 256 Mbit/s). The duty cycle and uv-coverage could
be increased, if in the future the VLBA were able to change the tapes more frequently or if the VLBA would record
on hard-disks. The latter has the additional advantage that recording rates of $> 512$ Mbit/s would become
possible also globally.
As an example for an image obtained with the Global 3\,mm-VLBI Array, we show in Figure 1 a new 86 GHz
VLBI image of the inner jet of M\,87 (= 3C\,274). The following stations contributed to this map:
Effelsberg (B), Onsala (S), Pico Veleta (V), Plateau de Bure (phased array) (P), and 8 VLBA stations (all except BR and SC).
The data were recorded at 256 Mbit/s using the MK5 disk recording in Europe and tape recording
at the VLBA. The source was detected on the B-V-P baselines with SNR $\leq 200$ and on the baselines to and within the VLBA
with SNR $\leq 50$. After initial fringe fitting at the Bonn correlator and narrowing of the search windows using
3C\,273 as fringe tracer (SNR $< 330$), the data were imported into AIPS with the new task
MK4IN (Alef \& Graham 2002). The final fringe fitting and amplitude calibration was done using the
standard procedures in AIPS. The final imaging was done using the Difmap package.
At a distance of 18.7 Mpc of M\,87, the angular resolution of 300 x 60 $\mu$as
corresponds to a spatial scale of 30 x 6 light days, or 100 x 20 Schwarzschild radii (assuming a
3 x $10^9$ $\rm{M}_\odot$ BH). Thus, the central engine and the inner jet can be studied
with a similar spatial resolution as the less massive, but closer SMBH in Sgr\,A*.
In fact, owing to its higher declination, M\,87 is easier to observe with VLBI and might be
even a better candidate than Sgr\,A* for the imaging of the event horizon around a SMBH.
One of the main differences between these two objects of course is, that the jet of M\,87 is related to a radio-loud
galaxy, whereas Sgr\,A* has a much lower radio-luminosity and shows no jet. The study
of both sources therefore should help to obtain a better understanding how jets are formed in general, and
how they are accelerated and collimated.
The fact that in M\,87 the jet can be traced down to scales of only a few ten Schwarzschild radii without
a large reduction of its brightness temperature is very noteworthy. This may
give new constraints to the theories of jet formation.
The comparison of the width of the jet at its origin with the expected size of the light cylinder
can help to discriminate between jet models, e.g. whether magnetic sling-shot models (e.g. Blandford \& Payne 1982) or
models with direct coupling to the BH spin (e.g. Blandford \& Znajek 1977) are more appropriate.
\begin{figure}
\includegraphics[bb=50 185 470 625,clip=,angle=-90,width=.5\textwidth]{krichbaum-fig1.ps}
\caption{
VLBI image of M\,87 (3C\,274, Virgo A) obtained in April 2003 at 86 GHz with the Global Millimetre VLBI Array (GMVA).
Contour levels are -0.5, 0.5, 1, 2, 4, 8, 16, 32, and 64 \% of the peak flux of 0.79 Jy/beam.
The beam size is 0.30 x 0.06 mas, pa=-6.3$^\circ$. The identification of the easternmost
jet component as VLBI core or as a counter-jet is still uncertain.}
\end{figure}
\vspace{-0.3cm}
\subsection{VLBI Observations of Sgr\,A* with European Antennas}
\begin{table}
\begin{tabular}{llclcc}
Station &Country &Diameter & T$_{\rm sys}^{\rm zenith}$ &Efficiency & SEFD \\
& &[m] & [K] & [\%] & [Jy] \\ \hline
Effelsberg &Germany & 100 & 130 & 8 & 950 \\
Pico Veleta &Spain & 30 & 120 & 55 & 860 \\
P. de Bure &France & 6x15 & 120 & 65 & 570 \\
Onsala &Sweden & 20 & 200 & 45 & 3900 \\
Mets\"ahovi &Finland & 14 & 200 & 30 & 12000 \\ \hline
Yebes &Spain & 40 & 150 & $\sim$40 & $\sim$830 \\
Noto &Italy & 32 & 150 & $\sim$30 & $\sim$2300 \\
SRT &Italy & 64 & 150 & $\sim$40 & $\sim$430 \\
\end{tabular}
\caption{Present and future telescopes, which in principle could enlarge the
European VLBI array at 3\,mm (80-90 GHz). The station names and locations are shown in columns 1 \& 2. Column 3
gives the antenna diameter, column 4 a typical system temperature (at zenith) of the receiver, column 5 the aperture
efficiency and column 6 the system equivalent flux density (SEFD) defined as the ratio of T$_{\rm sys}$ and
antenna gain (in [K/Jy]). The top 5 stations in the table participate regularly in global 3mm-VLBI observations
together with the VLBA. The stations listed below the intersecting line are future candidates.
}
\vspace{-0.5cm}
\end{table}
The compact radio source Sgr\,A*, which is located at the Centre of our Galaxy, most likely harbors
the nearest super-massive black hole. VLBI-images of Sgr\,A* at cm-wavelengths are heavily
affected by interstellar scatter broadening (e.g. Marcaide et al. 1999 and references therein), which blurs
the underlying source structure. Since the image broadening decreases
quadratically with increasing frequency, VLBI observations at mm-wavelengths allows us to
penetrate the scattering screen and image the source behind. Early VLBI observations
at 7\,mm (Krichbaum et al. 1993, Lo et al. 1998), 3\,mm (Krichbaum et al. 1998, Doeleman et al. 2001) and 1\,mm
(Krichbaum et al. 1998)
already indicated that the source appears slightly larger than the expected scattering size. New VLBA
observations at 7\,mm confirm this effect and now also suggest a possible variation of
the VLBI structure with time (Bower et al. 2004). Structural variability is not unexpected in view of the flux density
variations seen in the radio and near infrared bands (e.g. Zhao et al. 2001, Genzel et al. 2004),
and may provide an important building block for our understanding of the true nature of this enigmatic source.
Motivated by this, we simulate 3\,mm VLBI observations of Sgr\,A* with the existing European antennas and adding
new telescopes which may be able in the near future to participate in mm-VLBI. Here we argue that it is
a worthwhile effort to equip the following stations with 3\,mm receivers: the new 40\,m antenna built
by the Yebes group, the 32\,m antenna with its new adaptive reflector in Noto (Sicily), and the 64\,m Sardinia
radio telescope. In Table 1 we summarize the antenna characteristics, in Figure 2 we show uv-coverages for
Sgr\,A*, subsequently adding the new stations (note: Sgr\,A* is invisible from Onsala and
Mets\"ahovi).
With the VLBA, 86\,GHz VLBI observations of Sgr\,A* theoretically should yield an image with a maximum
resolution of 0.07\,mas (minor axis of beam). When the partially resolved source flux falls below
the baseline detection threshold, the resolution degrades. With a size of 0.18\,mas for Sgr\,A* at 86\,GHz
and 0.4-0.5\,Jy baseline sensitivity (assuming 512 Mbit/s, SEFD = 4300 Jy), the resulting maximum angular
resolution of the VLBA is 650\,M$\lambda$ or 0.16\,mas.
An European array with the telescopes listed in Table 1 could image the source with quite similar
resolution (500 \,M$\lambda$, 0.21\,mas) than the VLBA, but also with higher sensitivity. While the source would be
just marginally detected on the 500-600\,M$\lambda$ VLBA baseline with a SNR of $5-7$,
the sensitive European baselines (e.g. Effelsberg to IRAM) would see it with a SNR of $20-45$
(baseline detection thresholds: VLBA - VLBA : 480\,mJy, Pico - Effelsberg: 100\,mJy; Pico - PdBure: 76\,mJy).
When combined with other European mm-VLBI stations located in southern Europe (see in Fig. 2), very good
3\,mm-VLBI images of Sgr\,A* could be obtained. The high SNR of the measured visibilities and the good
uv-coverage would facilitate a more accurate determination of the source size and structure, with smaller
uncertainties than in previous VLBI observations. Small error bars on the source size, however, are
absolutely necessary for the clear detection of structural variability.
We conclude: the large and sensitive new radio telescopes being built in Spain and Italy, will
significantly extend the European VLBI baselines to the south. This will result in
better VLBI images of many radio sources, particularly for those with relatively low
declinations (e.g. M\,87). At short cm- and mm-wavelengths, where the
uv-coverage of the existing arrays (EVN: at 1.3 \& 7\,mm, GMVA: at 3\,mm) is still not very dense, the
participation of these new telescopes would lead to much better VLBI images.
If equipped with 3\,mm receivers, these stations could also play an important role in
the imaging of nearby SMBHs, as e.g. for Sgr\,A*.
\begin{figure}
\includegraphics[width=.22\textwidth]{krichbaum-fig2a.ps}
\includegraphics[width=.22\textwidth]{krichbaum-fig2b.ps}
\includegraphics[width=.22\textwidth]{krichbaum-fig2c.ps}
\includegraphics[width=.22\textwidth]{krichbaum-fig2d.ps}
\caption{Simulated uv-coverages for a VLBI observation of Sgr\,A* at 86\,GHz.
The simulations are done for the following telescopes (4 simulations arranged
from top left to bottom right): (a) Effelsberg, Pico Veleta, Plateau de Bure (present array),
(b) plus Yebes, (c) plus Noto, (d) plus SRT (Sardinia Radio Telescope).
}
\vspace{-0.5cm}
\end{figure}
\section{Towards Shorter Wavelengths - VLBI at 2\,mm and 1\,mm}
In order to demonstrate the technical feasibility of VLBI at wavelengths shorter than 3\,mm,
several VLBI pilot experiments were performed. At 2\,mm (147\,GHz) the following
telescopes were available: Pico Veleta (30\,m, Spain), Heinrich-Hertz Telescope (10\,m, Mt. Graham, Arizona), Kitt Peak
Telescope (12\,m, Kitt Peak, Arizona), Mets\"ahovi (14\,m, Finland) and SEST (15\,m, Chile). In two experiments performed
in 2001 and 2002, about one dozen mm-bright quasars were detected on the short continental baselines
in Europe (Pico-Metsa) and in the USA (HHT-KP) (Greve et al. 2002, Krichbaum et al. 2002).
A big success was the detection of 3 quasars also on the 4.2\,G$\lambda$ long transatlantic baseline between Pico Veleta
and the Heinrich-Hertz Telescope: NRAO150 (SNR=7), 1633+382 (SNR=23) and 3C279 (SNR=75).
In addition to continuum sources at 147\,GHz, also several SiO masers were observed (at 129\,GHz) and detected
on short baselines (Doeleman et al. 2002).
This success motivated another VLBI experiment one year later (April 2003);
this time at the shorter wavelength of 1.3\,mm (230\,GHz). In this experiment
the following stations participated: Pico Veleta, the 6x15\,m IRAM interferometer on
Plateau de Bure (as phased array), the Heinrich-Hertz Telescope and the 12\,m telescope on Kitt Peak.
Instead of recording on tapes, the new MK5 disk recording was chosen. The data were
recorded at a rate of 512 Mbit/s. In this observation,
the following sources were detected on the 880\,M$\lambda$ long baseline between Pico Veleta and
Plateau de Bure: NRAO\,150 (SNR=10.7), 3C\,120 (SNR=8.2), 0420-014 (SNR=24.9), 0736+017 (SNR=7.1), 0716+714
(SNR=6.8), OJ287 (SNR=10.4), 3C\,273 (SNR=8.2), 3C\,279 (SNR=9.6), and BL\,Lac (SNR=9.0).
Sensitivity limitations and some technical problems restricted the number of detected sources
on the 6.4\,G$\lambda$ long transatlantic baseline between Pico Veleta and HHT to
the quasar 3C\,454.3 (SNR=7.3). The BL\,Lac object 0716+714 was marginally detected (SNR=6.4).
No transatlantic fringes were seen to Plateau de Bure.
After the experiment and during correlation, it became obvious that the phase stability
of Plateau de Bure was not perfect and that some additional phase noise in the data degraded
the SNR of the detections on the baselines to this station by about a factor of 3-4. The problem is
under investigation and will be fixed soon.
Although the number of sources detected on the Pico Veleta - HHT baseline still is small,
the results demonstrate the technical feasibility of global 1\,mm VLBI. The detections also
mark a new record in angular resolution in astronomy (size $< 32 \mu$as) and indicate
the presence of ultra-compact emission regions in AGN, even at the highest frequencies.
For the quasar 3C\,454.3 (z=0.859, see also Pagels et al., this conference), the detection
was made at a rest frame frequency of 428 GHz.
At 2 and 1.3\,mm-wavelengths, the brightness temperatures of the detected AGN appears not to be
significantly lower than at cm-wavelengths. There are, however, indications that the source
compactness might be variable (for different sources and for a given source also with time). This is not
unexpected considering the known and often dramatic flux density and spectral variability in quasars,
which is much more pronounced at mm- than at cm-wavelengths.
\section{Future Outlook}
Micro-arsecond resolution imaging of compact radio sources with mm-VLBI is now possible, but still needs
further improvement. To obtain
an image fidelity comparable to present day cm-VLBI images, one needs a better
uv-coverage and a lower single baseline detection threshold, i.e. a higher array sensitivity.
The capabilities of global 3\,mm VLBI can be further improved by the addition of large telescopes
in Europe (Yebes, SRT), in the USA (GBT, CARMA) and in Central and South America (LMT, ALMA), even
if not all of these telescopes are optimized for 3\,mm-VLBI. When compared to the stand-alone VLBA, a sensitivity
improvement by at least a factor of $5-10$ appears possible.
At the shorter wavelengths (2\,mm, 1\,mm) several bright sources are already detected on long transatlantic
baselines. This demonstrates the feasibility of VLBI at these short wavelengths. However the number of
available antennas still is very small and the uv-coverage therefore correspondingly sparse. Thus, the
future success of VLBI at and below 2\,mm will depend critically on the availability of
a larger number of mm-antennas, which can observe at $\lambda \leq 2$\,mm. Major
steps towards better sensitivity and uv-coverage would be the addition of
the LMT (in Mexico), and the addition of the large millimetre interferometers CARMA (in California),
the SMA (in Hawaii), and ALMA (in Chile) to the mm-VLBI array. In combination with these very sensitives telescopes, the
smaller millimetre telescopes (APEX, KP-12m, JCMT, HHT) would efficiently contribute to the global uv-coverage.
One should also consider the ALMA prototype antennas, which are presently located in Socorro (New Mexico).
Their relocation to suitable places could fill existing gaps in the uv-plane. Only the combination of the
large with the smaller mm- and sub-mm antennas will lead to a global mm-VLBI array, which finally has a
high enough sensitivity to allow the imaging of those regions, where the coupling between
accretion disk, black hole and jet occurs.
In nearby objects and with a spatial resolution of only a few Schwarzschild radii, it should
be possible to reach the "event horizon" or at least the inner part of the accretion disk around
the central SMBH, if it radiates and is visible at mm-/sub-mm wavelengths.
Another important aspect and future goal is the VLBI polarimetry in the millimetre and sub-millimetre
domaine. At these short wavelengths, the jet base should become optically thin and the polarization should
be high. Sub-mm VLBI polarimetry therefore should allow observing the expected time-variable magnetic
field configuration in the BH-jet system. This will facilitate detailed tests of relativistic magneto-hydrodynamical
jet- and dynamo-models, which are presently proposed as a likely mechanism for jet creation.
The ongoing development of the VLBI recording systems towards higher sampling rates and larger bandwidths
(several Gbit/s) already points in the right direction and towards higher baseline sensitivities
(Graham et al., 2002, Whitney et al. 2003). At mm- and sub-mm
wavelengths, it will be also very important to correct instantaneously for the phase fluctuations introduced by the
Earth's atmosphere on short timescales (seconds to minutes). Simultaneous dual-frequency observations and/or
water vapor radiometry will help to extend the phase coherence and integration times and by this contribute to the
necessary sensitivity enhancement.
Thus one can hope that within less than a decade from now, the detailed imaging of the direct vicinity
of SMBHs and their `event horizon' will really become possible.
\begin{acknowledgements}
2\,mm and 1\,mm VLBI is a joint effort of the following observatories: MPIfR (Bonn), IRAM (France and Spain),
Mets\"ahovi Radio Observatory (Finland), MIT-Haystack Observatory (USA), Arizona Radio Observatory (USA),
and Onsala Radio Observatory (Sweden and Chile).
The results presented here would have been not possible without the help of many people at each of the
participating observatories. Thanks to them all!
\end{acknowledgements}
|
{
"timestamp": "2004-11-17T10:27:07",
"yymm": "0411",
"arxiv_id": "astro-ph/0411487",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411487"
}
|
\section{{\bf Introduction}}
V1143 Cygni (HD 185912; HR 7484; BD$+54^\circ 2193$;
$V_{max}=+5.86$; B-V=+0.46;
$\alpha=19^h 38^m 41^s.18$; $\delta=+54^{\circ} 58'25''.7$) is a
double-lined eclipsing binary, consists of a pair of F5V stars
with high orbital eccentricity (e=0.540) and a relatively long
period of 7.640 days (see Fig.1). According to Andersen et al.
(1987) who have determined the stellar and orbital properties of
this system accurately, the corresponding radii and masses are
$r_1=1.346\pm0.023R_{\odot}$, $r_2=1.323\pm0.023R_{\odot}$ and
$m_1=1.391\pm0.016M_{\odot}$, $m_2=1.347\pm0.013M_{\odot}$,
respectively.
\begin{figure}[h]
\epsfxsize=6cm
\centerline{
\epsffile{figure1.PS} } \caption{Relative orbit of V1143 Cygni drawn
in scale.}
\end{figure}
It is known that V1143 Cyg is one of the best examples of
eclipsing binaries with apsidal motion, in which the observed rate
of apsidal motion is grater than the value predicted by general
relativity and stellar evolutionary models.
The observed rate of apsidal motion is due to the contribution of
two terms; a classical term as well as the general-relativistic
term. In this sence, the observational apsidal motion rate is
\begin{equation}
\dot{\omega}_{obs}=\dot{\omega}_{cl}+\dot{\omega}_{GR},
\end{equation}\\
where $\dot{\omega}_{cl}$ denotes the classical or Newtonian term
and $\dot{\omega}_{GR}$ is the relativistic contribution which can
be determined using the formulas given by Gimenez (1985). In the
case of V1143 Cyg, the observed rate of apsidal motion is
$\dot{\omega}_{obs}=3^{\circ}.52/100^{yr}\pm
0^{\circ}.72/100^{yr}$ (Burns at al. 1996) while Andersen at al.
(1987) calculated a faster theoretical apsidal motion of
$\dot{\omega}_{theo}=4^{\circ}.25/100^{yr}\pm
0^{\circ}.72/100^{yr}$ in which the expected relativistic and
classical (Newtonian) contributions to apsidal motion are
$\dot{\omega}_{cl}=2^{\circ}.39/100^{yr}$ and
$\dot{\omega}_{GR}=1^{\circ}.86/100^{yr}$, respectively. It is
seen that the classical and relativistic contributions are of the
same order in this system.
\section{{\bf Observations}}
V1143 Cygni was observed during 24 nights from July to September
2000 at Biruni Observatory of Shiraz University
(Longitude: $52^{\circ}31'$ E, Latitude: $29^{\circ} 36'$ N).
Observations were made with a 51cm cassegrainian telescope
equipped with an uncooled RCA4509 multiplier phototube. Two stars
HD 184240 ($V_{max}=+6.30$) and HD 186239 ($V_{max}=+7.10$)
were selected as comparison and
check stars respectively. The integration time for all of the
observations were fixed to 10 seconds. The output signals of the
photomultiplier were fed to a computer after amplification, using
an A/D convertor. The measurements were made using B and V filters
of intermediate-bandpass blue $\lambda_{max}=4400 \AA$ and yellow
$\lambda_{max}=5530 \AA $ which are matched closely to the
Johnson's UBV system. Times were converted to Heliocentric Julian
Day Number (HJD). Data reduction and atmospheric corrections were
done to obtain the complete light-curves in two filters, using a
computer code developed by G. P. McCook. Figures 2 and 3,
represent the observed light curves in B and V filters
respectively. In order to enhance the accuracy of the present work
, two other minima were observed on July 16 and 18, 2002. This
time, the star HD 185978 (F8; $m_V=+7^m.8$) was selected as the
comparison star.
\begin{figure}[h]
\epsfxsize=8cm
\centerline{
\epsffile{figure2.PS} } \caption{Observed light curve of V1143 Cygni
in the B filter.}
\end{figure}
\begin{figure}[h]
\epsfxsize=8cm
\centerline{
\epsffile{figure3.PS} } \caption{Observed light curve of V1143 Cygni
in the V filter.}
\end{figure}
\section{{\bf Times of minima and light curve analysis}}
From the observed light curves, heliocentric times of minima (one primary and one secondary)
were computed by fitting a Lorentzian function to the observed minima data points
(see Figures 4 and 5).
This function can be expressed as
\begin{equation}
y={y_o}+{\frac{2A}{\pi}} {\frac{w}{4(x-{x_c})^{2}+w^2}},
\end{equation}
where $y_o$ is the baseline offset, A is total area under the
curve from baseline, $x_c$ is the center of the minimum and $w$ is
full width of the minimum at half height. The minima were
calculated according to the ephemeris given by Andersen et al.
(1987)
\begin{equation}
Min.I=HJD 2449234.6144+7^{d}.64075217 \times E,\\
\end{equation}
and are given in Tables 2 and 3.
The corresponding errors in primary and secondary minima are:
$\pm 0.00066$ and $\pm 0.00220$, respectively.
Meanwhile, in each filter, the depths of minima are as follows:\\ \ \\
Filter B: Min.I: $0^m.53\pm 0.02$, Min.II: $0^m.25\pm 0.02$\\ \ \\
Filter V: Min.I: $0^m.48\pm 0.02$, Min.II: $0^m.23\pm 0.02$\\
\begin{figure}[h]
\epsfxsize=8cm
\centerline{
\epsffile{figure4.PS} } \caption{A sample Lorentzian fit to the
primary minimum.}
\end{figure}
\begin{figure}[h]
\epsfxsize=8cm
\centerline{
\epsffile{figure5.PS} } \caption{A sample Lorentzian fit to the
secondary minimum.}
\end{figure}
The B and V light curves of V1143 Cyg have been analyzed
separately by using the Wilson code in order to derive
photometric elements of this system.
The program consists of two main FORTRAN programs {\bf LC}
(for generating light and radial velocity curves) and {\bf DC} (to
perform differential corrections and parameter adjustment of the
LC output). The model which upon the program is based on, has been
described and quantified in papers by Wilson (1979, 1990, 1993).
Since the system V1143 Cygni is detached with both components
residing well inside their respective Roche lobes (Burns et al.
1996), the solution was performed in {\bf mode 2}. In our
analysis, we assumed a value of zero for the third light
($l_3=0$). Also we fixed the ratio of the axial rotation rate to
the mean orbital rate for stars 1 and 2 ($F_1,F_2$). Before
running the {\bf LC} code, the position of the preastron was
estimated from the apsidal motion study as discussed in section
4. In order to optimize the observed parameters given in Table 1,
we wrote an auxiliary computer programm to compute the sum of
squares of the residuals in both filters by using the{\bf LC}
output. Figures 6 and 7 show
the sum of the squared residuals versus selected parameters for the two filters.
For example, Figure 6(a) represents the weighted sum of the
squared residuals $\sum\omega r^2$ (SSR) versus {\it eccentricity
(e)} and
{\it inclination angle (i)} for filter B.\\
\begin{figure}[h]
\epsfxsize=11cm
\centerline{
\epsffile{figure6.PS} } \caption{The weighted sum of squared residuals
$\sum\omega r^2$ (SSR) versus parameters using Wilson's {\bf LC}
code in B filter.}
\end{figure}
\begin{figure}[h]
\epsfxsize=11cm
\centerline{
\epsffile{figure7.PS} } \caption{The weighted sum of squared residuals
$\sum\omega r^2$ (SSR) versus parameters using Wilson's {\bf LC}
code in V filter.}
\end{figure}
Figures 8 and 9, show the optimized theoretical light curves
together with the observed light curves in B and V filters. The
theoretical curves correspond to the optimized parameters given in
Table 1.\newpage
\begin{center}
Table 1.\\ Optimized parameters of V1143 Cygni.
\end{center}
\begin{tabular}{lll}\hline
Parameter & Filter B & Filter V \\ \hline
$i$ & $87.3 \pm0.1$ & $ 87.1 \pm0.1$ \\
$e$ & $0.536 \pm0.005$ & $ 0.539\pm0.005$ \\
$\omega$ & $0^\circ.855 \pm0.004$ & $0^\circ.855 \pm0.004$ \\
$q(\frac{M_2}{M_1})$& $0.99 \pm0.01$ & $0.98\pm0.01 $\\
$\Omega_1$ & $17.4\pm0.5$ & $18.0\pm0.5 $\\
$\Omega_2$ & $19.6\pm0.5$ & $20.0\pm0.5$ \\
$\frac{L_2}{L_1}$ &$ 0.89 \pm0.05$ & $0.82\pm0.05$ \\ \hline
\end{tabular}
Finally, the observed times of minima on July 16 and 18, 2002 are
calculated according to the ephemeris \\
\begin{equation}
Min.I=HJD 2447087.5669+7^{d}.64075095 \times E, \\
\end{equation}
given by Burns et al. (1996)
and are tabulated in Tables 2 and 3:
\begin{figure}[h]
\epsfxsize=8cm
\centerline{
\epsffile{figure8.PS} } \caption{Theoretical light curve ({\bf LC}
output)
in filter B. Points are the observational data.}
\end{figure}
\begin{figure}[h]
\epsfxsize=8cm
\centerline{
\epsffile{figure9.PS} } \caption{Theoretical light curve ({\bf LC}
output)
in filter V. Points are the observational data.}
\end{figure}
\section{{\bf Apsidal motion}}
Due to the deep, narrow eclipses of V1143 Cyg and its high eccentricity, the rate of apsidal
motion can be determined exactly by analysis of primary and secondary eclipse timings.
In their paper, Guinan and Maloney (1985) described the procedure that must be
followed to determine the apsidal motion rate from the change in the displacement of the
secondary minimum from the half point (0.5 phase) according to
\begin{equation}
D=[(t_{2}-t_{1})-0.5{\times}\rm Period].
\end{equation}
D in turn, is related to $\omega$ the longitude of preastron by the formula given
by Sterne (1939):
\begin{equation}
D=\frac{P}{\pi}[\tan^{-1}( \frac{e\cos({\omega})}{{(1-e^2)}^{1/2}}) + \frac{e\cos({\omega})}
{1-{e^2}{\sin^2({\omega})}} {(1-e^2)}^{1/2}],
\end{equation}
where P is the period and e is eccentricity. The observed photoelectric times of primary
and secondary minima from 1969 to 2002
are collected in Tables 2 and 3. As you can see from Table 4, the
slow decrease in $D$ is due to the advance of the line of apsides
of the orbit. By computing the slope of a line fitted to all of
the secondary minima given in Table 3, we determined an observed
rate of apsidal motion of
$\dot{\omega}_{obs}=3^{\circ}.72/100^{yr}\pm 0.37/100^{yr}$ (see
Figure 10).
\newpage
\begin{center}
Table 2.\\
The photoelectric times of primary minima for V1143 Cyg. The O-Cs are computed according to equation 7.\\
\begin{tabular}{lcrl}\hline
H.JD. & O-C(day) & Epoch & Reference\\
(2400000.+) & & & \\ \hline
39339.616 & -0.22773 & -376 & Snowden and Koch (1969)$^1$ \\
39385.6881 & -0.00011 & -370 & Snowden and Koch (1969)\\
40837.4313 & 0.00017 & -180 & Battistini et al. (1973) \\
41135.4208 & 0.00033 & -141 & Battistini et al. (1973)\\
42212.7651 & -0.00142 & 0 & Koch (1977) \\
42617.727 & 0.00061 & 53 & Koch (1977)\\
43305.3943 & 0.00021 & 143 & Guinan et al. (1987) \\
45253.7858 & -0.00008 & 398 & Gimenez and Margrave (1985)\\
47087.5669 & 0.00049 & 638 & Burns at al. (1996)\\
48019.73800 & -0.00016 & 760 & Caton and Burns (1993) \\
49234.6144 & -0.00336 & 919 & Lacy and Fox (1994) \\
51771.34410 & -0.00338 & 1251 & Dariush et al. (2001) \\
52474.29443 & -0.00225 & 1343 & Dariush et al. (2003)\\
\hline
\end{tabular}
\end{center}
\begin{verbatim}
1. This minimum is observed spectroscopically and because of its
unusually high O-C, it is not shown in the O-C diagram.
2. This minimum was published as 2439385.6831 by Hamme and Wilson
(1984) which seems to be misprint.
\end{verbatim}
\newpage
\begin{center}
Table 3.\\
The photoelectric times of secondary minima for V1143 Cyg, together with\\
the computed values of $D$ and $\omega$, using equation 6. The O-Cs
are computed according to equation 8.
\begin{tabular}{lcrllll}\hline
H.JD. & O-C(day) & Epoch & D & $\omega$ & Reference\\
(2400000.+) & & & & & \\ \hline
38932.932 & -0.00431 & -430 & 1.8685 & 47.97 & Snowden and Koch (1969) \\
38978.7807 & 0.00000 & -424 & 1.8727 & 47.83 & Snowden and Koch (1969)\\
42615.75 & -0.01896 & 52 & 1.8439 & 48.82 & Koch (1977)$^1$\\
43066.575 & 0.00286 & 111 & 1.8646 & 48.11 & Koch (1977)\\
44487.7482 & -0.00002 & 297 & 1.8579 & 48.34 & Gimenez and Margrave (1985)\\
47085.5910 & -0.00598 & 637 & 1.8449 & 48.78 & Burns at al. (1996)\\
51792.28645 & -0.00122 & 1253 & 1.8370 & 49.05 & Dariush et al. (2001)\\
52472.30445 & -0.00834 & 1342 & 1.8281 & 49.35 & Dariush
et al. (2003) \\ \hline
\end{tabular}
\end{center}
\begin{verbatim}
1.Due to its unusually high O-C, it is not included in determination of
the observed rate of apsidal motion.
\end{verbatim}
\begin{figure}[h]
\epsfxsize=8cm
\centerline{
\epsffile{figure10.PS} } \caption{The displacement of the secondary
minima given in Table 3,
from the 0.5 phase versus epoch. The observed rate of apsidal motion ($\dot{\omega}_{obs}$) can be
calculated for the slope of this curve.}
\end{figure}
The residuals given in Tables 2 and 3 are calculated for each of
the minima according to the ephemeris given by Gimenez and
Margrave (1985)
\begin{equation}
Min.I =HJD 2442212.76652 + 7^{d}.64075217 \times E,
\end{equation}
and
\begin{equation}
Min.II=HJD 2442218.45092 + 7^{d}.64073165 \times E.
\end{equation}
The computed (O-C)s versus Julian day is plotted in Figure 11 for
both primary and secondary minima. This diagram shows no changes
in period within the 0.01 days.
\begin{figure}[h]
\epsfxsize=8cm
\centerline{
\epsffile{figure11.PS} } \caption{The O-C diagram in days for all the
primary and the secondary
minima tabulated in Tables 2 and 3.}
\end{figure}
\section{{\bf Results and discussion}}
Table 4 contains independent determinations of the observed
apsidal motion of V1143 Cyg together with the corresponding
period of apsidal revolution $U$. The relation between the apsidal
motion period $U$ and the observed rate of apsidal motion
($\dot{\omega}_{obs}$) has a simple form
\begin{equation}
U=\frac{360P}{\dot{\omega}_{obs}},
\end{equation}
where $\dot{\omega}_{obs}$ is expressed in degrees per cycle and
$P$ is the anomalistic period expressed in days. To determine a
more accurate value for $\dot{\omega}_{obs}$ we need more accurate
timings of secondary minima. From Table 4, it seems that expanding
our observational baseline, may decrease the discrepancy between
$\dot{\omega}_{obs}$ and $\dot{\omega}_{theo}$. Figure 12 shows
$D$ as a function of $\omega$ using equation 6 for V1143 Cygni and
DI Herculis. In the case of DI Herculis
(P=10.550 days, e=0.49)
the discrepancy is much larger than that of V1143 Cygni. For this system the observed
rate of apsidal motion is about one-third of the theoretical one which is due to
the classical and relativistic contribution (Claret, 1998; Dariush and Riazi, 2003).
The dots in Figure 12, represent our
observational period for the V1143 Cyg and DI Herculis apsidal
revolution. It is clear that up to now, our observational baseline
covers only a very small fraction of the period of apsidal
revolution $U$. In the case of DI Herculis, the presence of a
third body is a plausible explanation for the discrepancy
between $\dot{\omega}_{obs}$ and $\dot{\omega}_{theo}$, but until
no further observational evidence has been reported this
possibility. Our results for V1143 Cyg supporting $3^\circ.72\pm 0.37/100 yr$
is slightly closer to the theoretical apsidal motion rate
($4^\circ.25\pm 0.72/ 100yr$), computed with previous studies of
this system
\begin{figure}[h]
\epsfxsize=8cm
\centerline{
\epsffile{figure12.PS} } \caption{ $D$ as a function of $\omega$ using
equation 6 for V1143 Cygni and DI Herculis.}
\end{figure}
\begin{center}
Table 4.\\
Determined rate of apsidal motion for V1143 Cygni
\begin{tabular}{llrl}\hline
${\dot\omega_{obs}}^\circ$/100$^{yr}$ & $\pm$Error& U$^{yr}$ & Source \\ \hline
3.49 & $\pm 0.38$ & 10320 & Khaliullin (1983) \\
3.36 & $\pm 0.19$ & 10710 & Gimenez and Margrave (1985)\\
3.52 & $\pm 0.72$ & 10230 & Burns et al. (1996)\\
3.72 & $\pm 0.37$ & 9680 & Present study$^1$\\ \hline
\end{tabular}
\end{center}
\begin{verbatim}
1.This value is computed from all the secondary minima presented
in Table 4 except 2442615.75 of Koch (1977). Including this
minimum, it is changed to 3.45 degree/100yr +/- 0.65 degree/100yr.
\end{verbatim}
\begin{center}
{\bf Acknowledgments}\\
\end{center}
We would like to thanks Mr. Mehdi Nazem for his help during the
observations.
\newpage
\begin{center}
{\bf References:}\\
\end{center}
Andersen, J., Garcia, J. M., Gimenez, A., and Nordstr$\ddot o$m, B., 1987,
{\it Astron.Astrophys.}, {\bf 174}, 107.\\
Battistini, P., Bonifazi, A., and Guarnieri, A., 1973, {\it IBVS} {\bf 817}.\\
Burns, J. F., Guinan, E. F., and Marshall, J. J., 1996, {\it IBVS} {\bf 4363}.\\
Caton, D. B., and Burns, W. C., 1993, {\it IBVS} {\bf 3900}.\\
Claret, A. 1998, {\it Astron.Astrophys.}, {\bf 330}, 533.\\
Claret, A., and Willems, B., 2002, {\it Astron.Astrophys.}, {\bf 388}, 518.\\
Dariush, A., Afroozeh, A., Riazi, N., 2001, {\it IBVS} {\bf 5136}.\\
Dariush, A., and Riazi, N., 2003, {\it Astrophys.Space.Sci.}, {\bf 283}, 253.\\
Dariush, A., Zabihinpoor, S. M., Bagheri, M. R., Jafarzadeh, Sh.,
Mosleh, M., and Riazi, N., 2003, {\it IBVS} {\bf 5456}.\\
Gimenez, A., 1985, {\it Astrophys.J.}, {\bf 297}, 405.\\
Gimenez, A., and Margrave, T. E., 1985, {\it Astron.J.}, {\bf 90(2)}, 358.\\
Guinan, E.F. and Maloney, F.P., 1985,{\it Astron.J.}, {\bf 90}, 1519.\\
Guinan, E. F., Najafi, I., Zamani-Noor, F., and Boyd, P. T., 1987, {\it IBVS} {\bf 3070}.\\
Hamme, W. V., and Wilson, R. E., 1984, {\it Astron.Astrophys.}, {\bf 141}, 1.\\
Khaliullin, Kh. F., 1983, {\it Astron.Trisk}, No 1262.\\
Koch, R. H., 1977, {\it Astron.J.},{\bf 82}, 653.\\
Lacy, C.H. and Fox, G.W., 1994, {\it IBVS} {\bf 4009}.\\
Semeniuk, I., 1968, {\it Acta.Astron.}, {\bf 18}, 1.\\
Snowden, M. S., and Koch, R. H., 1969, {\it Astrophys.J.},{\bf 156}.\\
Sterne, T.E., 1939(a), {\it Mon.Not.R.Astron.Soc.}, {\bf 99}, 451.\\
Sterne, T.E., 1939(b), {\it Mon.Not.R.Astron.Soc.}, {\bf 99}, 662.\\
Wilson, R.E., 1979, {\it Astrophys.J.}, {\bf 234}, 1054.\\
Wilson, R.E., 1990, {\it Astrophys.J.}, {\bf 356}, 613.\\
Wilson, R.E., 1993, in {\it New Frontiers in Binary Star
Research}, ed. K. C. Leung and
Nha, I.S., A.S.P. Conf. Ser., {\bf 38}, 91.\\
\end{document}
|
{
"timestamp": "2004-11-30T12:13:35",
"yymm": "0411",
"arxiv_id": "astro-ph/0411788",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411788"
}
|
\section{Introduction}
Radio galaxies are excellent laboratories to disentangle the role of central
active nuclei in host galaxies evolution and the relation of intergalactic
environment and the central activity. The complex structure of radio galaxies
put severe troubles to carry out specific optical studies in this matter.
Fortunately, the recent technical and computational improvements on integral field spectroscopy (IFS hereafter) open new frontiers to multitude of astronomical projects.
In the particular case of radio galaxies, IFS data allow to separate the
different components and to perform an independent analysis of the various
subsystems and their evolution.
3C~120 is a Seyfert 1 galaxy at a redshift of $z=0.033$. It has been
studied extensively at many wide-range wavelengths, with special attention in
X-ray and radio frequencies \citep[e.g.][]{walk87,mole88,walk97,ogle04}.
Although 3C~120 was classified as an early-type galaxy based on a
visual inspection of ground-based images \citep{zwic71}, \cite{sarg67}
reported a faint spiral structure. However, its optical morphology is still
not clear showing an elliptical shape with some peculiarities \citep{hansen87}
and could be a merger remnant galaxy \citep{mole88}. Spectroscopic analysis
has shown that 3C~120 presents a rotation curve most likely indicating the
presence of an undetected disk \citep{mole88}. Based on this result, 3C~120 is
normally quoted as an early-type spiral or S0 galaxy \citep{mole88}.
Although Seyfert galaxies are generally considered as radio-quiet objects,
3C~120 is an active radio-loud source showing a superluminal
one-sided jet that extends $\sim$25$\arcsec$ out of the core
\citep{read79,walk88,walk97}.
Different authors \cite[e.g.][]{bald80,pere86,soub89,hua88} reported the
existence of several continuum and emission line dominated structures in this
object. Some of these structures have a clear relation with the radio-jet.
\cite{hjor95} detected a continuum dominated optical counterpart of the
radio-jet using deep broad-band ground based images. On the other hand, an
EELR (E$_1$) associated with the jet was found at $\sim$5$\arcsec$ west of
the nucleus \citep{hua88,axon89,sanc04c}. The nature of the remaining
structures is still unclear \citep[e.g.][]{pere86,mole88}.
This paper compiles data from IFS optical observations of 3C~120 using the
optical fibers system INTEGRAL covering a central region of
$\sim16\arcsec\times12\arcsec$ field-of-view. We combined our IFS data with
high-resolution Hubble Space Telescope (HST) images. We describe the optical
fibers instrument, observations, and data reduction in section \S2. In section
\S3, we describe the emission line profiles in the circumnuclear region of
3C~120 and we present 2D spectra diagrams for different emission lines
(appendix A). In the following sections, we present the stellar morphology and
the structures detected (section \S4.1), the average colors of the galaxy
and the colors of those structures (section \S4.2), and the emission-line
morphology and ionization structure (section \S4.3). In section \S5, we apply
a new technique to separate the spectra of different components coexisting in
3C~120 and analyze the properties of each component. We present the velocity
field of the central region of 3C~120 and the discussion of several
kinematic perturbations in section \S6. A distance of 198 Mpc is assumed for
3C~120 throughout the paper (H$_0$=50 km s$^{-1}$ Mpc$^{-1}$) which
corresponds to a scale of $\sim 1 $kpc/$\arcsec$.
\section{Experimental Set-Up, Observations, and Data Reductions}
\subsection{Integral Field Spectroscopy}
IFS optical observations with fibers are based on the idea of
connecting the focal plane of the telescope with the spectrograph slit using a
fiber bundle. In this way, when an extended object is observed, each fiber
receives light coming from a particular region of the object. Each individual
spectrum appears well separated on the detector and therefore, spatial and
spectral information are collected simultaneously. The wavelength limitations
essentially depend on the characteristics of the spectrograph itself. The
spatial resolution depends on the fiber sizes and the prevailing seeing
conditions during the observations. The spatial coverage of fiber systems is
relatively small \citep[see ][ for a counter example]{lefe03}, but they are
very useful when studying small size objects, such us the circumnuclear region
of nearby active galaxies \citep[e.g.][]{garcia01}, blue compact dwarf
galaxies \citep[e.g.][]{cairos02} or gravitational lenses
\citep[e.g.][]{motta02,wiso03}.
IFS of 3C~120 was obtained the 26th of February 2003 at the Observatorio del
Roque de los Muchachos (ORM) on the island of La Palma, Spain. The 4.2m
William Herschel Telescope (WHT) was used in combination with the fiber system
INTEGRAL \citep{arr98,arr99,medi98} and the WYFFOS spectrograph
\citep{Bing94}. The observations were carried out under photometric conditions
and an average seeing of 1$\farcs$2. The standard bundle 2 of INTEGRAL was
used during these observations. This bundle consists of 219 fibers, each one
with a diameter of 0$\farcs$9 projected on the sky. A central rectangle is
formed by 189 fibers covering a field-of-view of $16\arcsec\times12\arcsec.3$,
and the remaining 30 fibers form a ring with a diameter of $90\arcsec$. Figure
1 illustrates the actual distribution of the science fibers on the
focal plane.
The WYFFOS spectrograph was equipped with a 300 groove mm$^{-1}$ grating
centred on 5500 \AA\ (spectral coverage: 3500-9000 \AA ). A Tek6 CCD array of
$1124\times1124$ pixels of 24 $\mu$m size was used, giving a linear dispersion
of about 3 \AA \ pixels$^{-1}$. With this configuration, and pointing to the
central region of 3C~120, three exposures of 1800 seconds each were taken.
The data were reduced using IRAF standard routines \citep{tody86}. Although
the reduction of IFS data from fiber-based instruments does not differ
significantly from standard spectroscopic data reduction, we describe briefly
in this section the reduction procedure.
A master bias frame was built by averaging different bias frames taken along
the night. This bias frame was then subtracted from the science frames. In
observations with optical fibers, flats-fields are obtained illuminating the
focal plane uniformly, and obtaining spectra (the so-called flat-spectra).
Thus, for a particular wavelength, the differences in response among fibers
dominate the flat-spectra shape. These differences are due to their distinct
focal-ratio degradation, position at the entrance of the spectrograph, etc. In
this way, flat-spectra are used to homogenize the response of all the fibers.
Flat-spectra are also used to obtain the polynomial fits to define the fiber
path along the detector and extract the individual spectra from the whole
image. Each spectrum appears well separated on the detector, with a width of
approximately two pixels in the spatial direction (according to the core fiber
image size of the fiber bundle used for these observations). The trace and
extraction of individual spectra was performed using the standard routine
APALL of IRAF. After this operation, frames of $1124\times1124$ pixels were
reduced to $1040\times219$ pixel: each pixel in the spatial direction contains
the spectrum of each particular fiber, a total of 219 fibers.
The wavelength calibration was done using the IDENTIFY and REIDENTIFY routines
of IRAF. In order to carry out this procedure, we selected several isolated
and well-distributed arc lines. Although it might be difficult to determine
the actual uncertainty produced by the wavelength calibration, we have used
the sky lines in our spectra to estimate the final wavelength errors, being
smaller that 35.5 kms$^{-1}$ and 29.5 kms$^{-1}$ at blue and red wavelengths,
respectively.
The SP1045+378 flux standard star (Isaac Newton Group Database) was observed
the same night and under similar conditions as for 3C~120. The standard star
has been used to calibrate a flux ratio by comparing with the standard-star
flux tables of Stone (1977). We reduced the SP1045+378 frames in a similar way
than the object ones, correcting them for differential atmospheric refraction
\citep{fili82} using E3D \citep{sanc04}. Then, we extracted the observed
spectrum of the star by co-adding the spectra of the central 37 fibers, which
includes $>$99\% of the total flux. Comparing this observed spectrum with the
flux-calibrated spectrum of SP1045+378 we derived a sensitivity curve that we
used to calibrate the object frames.
We combine the IDA tool \citep{garcia02} and the Euro3D visualization package
\citep{sanc04} to analyze the data and generate two-dimensional maps of any
spectral feature (intensity, velocity, width, etc). Maps recovered from
spectra are images of 51$\times$37 pixels with a scale 0$\farcs$3/pixel. While
the spatial sampling of the used INTEGRAL configuration is 0$\farcs$9 (that
is, the fiber diameter of the bundle 2), the centroid of any peak in our maps
can be measured with an accuracy of around 1/5 of the fiber diameter, that is
$\sim 0\farcs 2$ \citep[e.g.][]{medi98}.
\subsection{HST imaging}
Wide-Field Planetary (WFPC) camera images of 3C~120 in different bands are
available on the archive of the Hubble Space Telescope (HST). We obtained
these images to study the morphology of this object. Table \ref{tab_hst}
summarizes the properties of these images. The data set comprises three
broadband images (F555W, F675W and F814W), which roughly corresponds to the
standard $V$, $R$ and $I$-bands, and a medium band image (F547M). These images
basically sample the continuum emission in 3C~120, since the equivalent width
of mostly all the emission lines is very small compared with the width of the
bands. However, there is a non neglectible contamination, dominated by the
[OIII] and H$\beta$ emission lines in the F555W-band image (and H$\alpha$ in
the F657W-band image). The F547M-band image can be used as an estimation of
the pure continuum emission, since the emission lines are at the edge of its
transmission curve. The data set is deep enough to study the different
structures present in this object \citep{soub89}.
\section{Atlas of Spectra}
Figure 2 shows the nuclear spectrum of 3C~120 in the full
wavelength range. This spectrum has been obtained by co adding the seven
spectra closest to the continuum peak. This is almost equivalent to a hexagonal
aperture of 1$\farcs$6 in radius centred at the optical nucleus of 3C~120.
We can easily recognize several emission lines and the characteristic broad
component of permitted emission lines of Seyfert 1 galaxies.
In the Appendix A, we present the individual spectra corresponding to each of
the different observed positions (fibers) in selected spectral intervals that
include the most important emission lines (spectra diagrams).
\subsection{Emission-Line Profiles}
The profiles of the emission lines in the nuclear spectrum of 3C~120 show a
considerable blending due to the wings of the broad component of permitted
lines (Fig. 2). The H$\delta$ profile appears to be
considerably broader than any other Balmer line, but this is not the case of
H$\delta$ as in previously found by other studies \citep{phil75,bald80}. The
HeI$\lambda5876$ shows also a considerable broadening. At this step, we cannot
discuss the origin of these features in terms of recombination models and
reddening effects because the nuclear spectrum in figure
2 includes the contribution of the nucleus and the
surrounding galaxy. In section \S5 we separate both contributions and we will
be in position to tackle this discussion. The low spectral resolution of the
current IFS data prevents the detection of double peaks reported by
\cite{axon89}. However, the emission lines show asymmetric profiles in some
locations outside the central region.
\section{Data analysis and results}
\subsection{Broad band distribution and morphological structures.}
\label{bb_hst}
Broad-band images of 3C~120 were recovered from our IFS data by coadding the
flux in spectral ranges that mimic the band-passes of the previously refereed
HST images and using an interpolation routine \citep{sanc04,garcia02}. The
specific wavelength ranges for each band were 3900-4900 \AA ($B$-band),
5000-6000 \AA ($V$-band), 5600-5700 \AA ($V'$-band), 6100-7100 \AA ($R$-band)
and 7550-8550 \AA ($I$-band). All images show a bright nucleus on top of
a weak host galaxy. Figure 3A presents the intensity map
corresponding to the $V$-band filter. The intensity contours have an
elliptical shape except for those at around 4 arcsec from the central peak,
showing an elongation toward the west. There are remarkable similarities
between the restored map (Fig. 3A) and the F555W-band image
from the HST (Fig. 3B), despite of the differences in
wavelength ranges and spatial sampling. The superior resolution of the HST
images allows to directly detecting several structures. These structures have been previously reported
subtracting a galaxy template to broad and narrow band ground base images
\citep[e.g.][]{bald80,pere86,hua88,soub89}. We have labeled, using the
nomenclature introduced by \cite{soub89}, the different detected structures
(figure 3B). The HST/F555W-band image allows to resolve them,
showing a richer level of structures. In particular, structures B, and C
are composed of a smoothed low surface brightness component and more luminous
clumpy substructures. Structure A seems to be more collimated, as previously
noticed by ground based imaging, which explains why it was initially confused
with the optical counterpart of the radio jet \citep[e.g.][]{pere86}. It shows
also a low surface brightness component and four clear clumps. We overplot in
figure 3B the radio-map at 4885 MHz (Walker 1997), showing that
the structure A is most likely related to the structure B rather than to the
radio-jet. Indeed, the HST images suggest that both structures belong to a
sequence of clumpy knots which are physically connected. Dust lines and shell structures
not detected from the ground are seen in the inner regions. Two shell arcs at
$\sim1\arcsec$ north and south of the central peak are clearly detected in the HST
images. Although previous HST images show extensions to the north-west and south-east in the inner region of 3C~120 \citep{zir98}, to our knowledge this is the first time that the shell structure is clearly visible. The north and south shells are named S$_N$ and S$_S$, respectively, hereafter. The S$_S$ seems to be connected to structure A by a faint tail.
A 2D modeling of the broad-band images was performed using GALFIT
\citep{peng02} to obtain a clear picture of the morphology of 3C~120. This
program has been extensively tested in the image decomposition of QSO/hosts
\citep{sanc04b}. The 2D model comprises a narrow Gaussian function (to model
the nucleus) and a galaxy template (to model the galaxy) both convolved with a
PSF. We performed the fit twice, first using a Sersic law \citep{sers68} to
characterize the galaxy and then using a de~Vaucouleurs law \citep{deva48}.
This method allows us to determine the morphological type of the host galaxy,
based on the obtained Sersic index, and to get a good determination of the
galaxy flux, based on the modeling the de~Vaucouleurs law \citep{sanc04b}.
The PSF was created using a field star for the analysis of the HST images, and a
calibration star for the analysis of the broad-band images recovered from the
IFS data.
We applied first the fitting technique to the HST images because of their
better spatial resolution.
The nucleus was saturated in the HST/F814W-band, which prevents us to do a
two-model fitting, and limits the reliability of any morphological
classification, which strongly depends on the shape of the profile in the
inner regions. To derive a rough estimation of the host magnitude we masked
the nucleus, and fixed the scale to the average of the values derived in the
other available HST/bands. Further checks, explained below, demonstrated that
this approach was valid. Table \ref{tab_galfit} summarizes the results from
this analysis. For each band it shows the derived Sersic index, the nucleus
and host magnitudes and the effective radius of the host galaxy. The derived
Sersic indices, all near or larger than the nominal value of 4 for an
early-type galaxy, confirm the morphological classification of the host galaxy
of 3C~120 as a bulge dominated galaxy.
We subtracted the object template (galaxy+nucleus) derived by the 2D modeling
from the original images, obtaining a residual image for each band. These
residual images were used to study the properties of the different structures,
once decontaminated from the smooth component. As an example, the residual
image of the $V$-band and the HST/F555W-band are shown in Figure
3C and 3D, respectively. The already quoted
structures are now clearly identified. The S$_S$ shell coincides with an
extended emission line region (EELR) previously detected in this object
\citep{hua88,soub89,sanc04c} and labeled as E$_2$ \citep{soub89}.
\subsection{Colors and gaseous distributions in 3C~120}
As already quoted all the broad-band images recovered from the IFS data show a
similar morphology than the $V$-band image shown in figure 3A.
Combining these recovered broad-band maps, we obtained color maps of 3C~120.
Figure 4A shows the $V-I$ color map derived from the IFS data. It
shows a blue circumnuclear region elongated to the southwest. The elongation
towards the west seen in the broad band images has also a counterpart in the
color maps showing bluer colors at a region $\sim4\farcs5$ west of the nucleus
than its surroundings. Figure 4B presents the $V-I$ color map
derived from the F555W and F814W-band HST images. Despite of their different
spatial resolution, figures 4A and 4B present remarkable
similarities. The different structures quoted in section \S4.1 show bluer
$V-I$ colors than the average color of the host galaxy, as already noticed
\citep[e.g.][]{fraix91}. Figure 4B
also shows evidence for two weaker shells farther from the central peak than S$_N$
and S$_S$ but also at the north and south. However, these ``secondary'' shells
are fainter, being at the detection limit.
Both the $V$-band and the F555W-band images are contaminated by the emission
from [OIII] and H$\beta$. A rough estimation of how strong that
contamination is can be obtained by subtracting a scaled continuum image, clean
of those contaminations. Figure 4C shows the residual image of the
INTEGRAL $V$-band image after the subtraction of an adequate continuum (the
$V'$-band image). A similar estimation was done for the HST images,
subtracting the continuum dominated F547M-band image from the F555W-band image
(Figure 4D). This image was smoothed using an 11$\times$11 pixels
median kernel to increase the signal-to-noise ratio. The resulting maps are a
rough estimation of the distribution of the [OIII]+H$\beta$ emission. The
position of two of the EELRs detected in this object (E$_1$ and E$_2$,
hereafter) is indicated \citep{soub89,sanc04c}. The residual images of the
continuum, after subtracting a model template (Fig. 3C and
3D), are overplotted for comparison purposes. The S$_N$ and
S$_S$ shells show a strong gaseous emission in the HST images, which in the
S$_S$ extends towards the E$_2$ region.
Figure 5 shows the $V-R$ colors of the host galaxy and the
different structures as a function of the $R-I$ colors. As already noticed
in the color image, the structures have bluer $V-R$ colors than the host
galaxy. This indicates most probably the presence of somewhat younger stellar
populations associated with them. However, they show a wider range of $R-I$
colors than that of $V-R$ colors. Indeed, the average $R-I$ color of the host
galaxy is within the range of $R-I$ colors of the different structures. The
color-to-color distribution derived from synthetic models has been included in
the figure for comparisons. The discontinuous line shows the colors of single
stellar populations calculated using the \cite{bruz03} models. We assumed a
solar metalicity and a \cite{chab03} IMF. The labels indicate the logarithm
of the stellar population ages in Gyrs. In general terms, the color-to-color
distribution does not match with that simple model, apart from the case of the
S$_N$ and S$_S$ shells. For the remaining structures, the $R-I$ colors
correspond on average to a young stellar population of $\sim$6$\times$10$^8$
Gyr, but the $V-R$ colors correspond to older stellar populations. This may
indicate that the real populations are a composite of populations. The dotted
line shows the colors of different galaxy types (E, Sab, Sbc, Scd and Irr)
\citep{fuku95}, which mainly correspond to different mixes of stellar
populations. The $R-I$ color of the host galaxy mainly corresponds to a
Sab-Scd galaxy. On average, the structures have $V-R$ colors that correspond
to the same kind of galaxies (Sab-Scd), but their $R-I$ colors expand over all
the range of possible colors.
So far we have ignored the effects of dust in the color-to-color distribution
shown in Figure 5. However, the dust content in 3C~120 is
rather high; an average value of A$_V\sim 4$ mags has been estimated
\citep{sanc04c}. Dust extinction redden both $V-R$ and $R-I$ colors in an
almost similar way \citep{fitz99}. The effects of dust in the color-to-color
distribution is illustrated with an arrow in Fig. 5. It is
clear that the dust is not homogeneously distributed in the galaxy: e.g.,
typical dust lanes are seen in figure 3B. A combined effect of
different stellar populations and a non-homogeneous distribution of dust could
explain the observed color-to-color distribution. In that case, the S$_N$ and
S$_S$ structures would be dust-free areas dominated by a single stellar
population.
However, results from broadband colors should be taken with care. The
contamination of broadband filters by strong emission lines can drastically
affect the morphology of color maps. In the case of 3C~120 the problem is even
worse because of the contribution and contamination from the broad component
of the Balmer emission lines in the central regions.
\subsection{Line-Intensity Maps and Ionization Structure}
We performed a line profile fitting and de-blending in order to study the
integrated high and low ionization gas distribution, their physical properties
and kinematics. We fit a single Gaussian to any of the bright emission lines
in the spectra. For the Balmer lines, we included a second broader Gaussian
to fit the wide line from the broad line region (BLR). In spite of our poor
spectral resolution, we found that several spectra have evidence of
substructures, showing asymmetric profiles with blue or red wings indicating
the presence of several gaseous systems in 3C~120. Indeed, previous authors
(Axon et al. 1989) reported the existence of several components in the emission lines of 3C~120, but to apply a line-profile decomposition to separate the
different gaseous components would be unrealistic because of the low spectral
resolution of the current IFS data. The broad component of the Balmer lines is
confined to the nucleus discarding an extended broad emission line region.
Figure 6 shows the intensity maps of [OIII]$\lambda5007$ (figure
6A) and H$\alpha$ (figure 6B) after the
deblending. Intensity contours of ionized gas are clearly elongated toward the
E$_2$ structure. Emission line maps present an elongation to the west, forming
a clear secondary peak at around 5 arscec from the nucleus in the [OIII]
intensity map. This secondary peak corresponds to the E$_1$ structure
previously quoted (section \S4.2). The ratio of the H$\alpha$ intensity of the
nucleus and E$_1$ is much smaller than that of [OIII]$\lambda5007$, suggesting
a high ionization nature of the latter. The nuclear region shows a low
ionization degree in the [OIII]$\lambda5007$/H$\beta$ map surrounded by a ring
of high ionization (figure 6C), with larger values at the south
of the nucleus, coincident with the location of E$_2$. A high ionisation
region expands from the ring to the west, increasing the ionisation degree
along this direction. The [NII]$\lambda6584$/H$\alpha$ line ratio map (figure
6D) also shows a ring structure, surrounding the nuclear
region. The E$_1$ is on the path of the 3C~120 radio-jet, as well as the
continuum dominated structure {\it A}.
The high [OIII]/H$\beta$ ratio of E$_1$ (figure 6C) discards a
relation between {\it A} and E$_1$, assuming that {\it A} is a star forming
region in a spiral arm or a tidal tail, and points, more likely, to a direct
connection between E$_1$ and the radio-jet \citep{sanc04b}. While the nucleus
and the {\it A} region present [OIII]$\lambda5007$/H$\beta$ and
[NII]$\lambda6584$/H$\alpha$ ratios at the limit of HII regions in a
diagnostic diagram \citep{veil87}, those ratios of E$_1$ and its west surroundings are placed on the high-ionization region .
The distribution of the H$\alpha$/H$\beta$ line ratio (figure
6E) shows a dust lane crossing 3C~120 along the
southeast-northwest direction. The high Balmer decrement at the west points to
a high dust obscuration around E$_1$. However, we cannot rule out that the
high H$\alpha$/H$\beta$ ratio may be related to the low signal to noise of the
narrow component of H$\beta$ in the circumnuclear region of the galaxy. The
electronic density derived from the [SII]$\lambda6716$/$\lambda6730$ presents
a patchy structure, showing an enlargement in the location of E$_1$. In the next
section, the ionization conditions of the different structures in the observed
region of 3C~120 are studied in detail.
\section{Decoupled spectra of the different components in 3C~120}
In the previous section, we used the traditional method (Gaussian fitting) to
deblend the different gaseous components. IFS optical observations record
spatial and spectral information simultaneously. Taking advantage of this
fact, we have developed a technique (3D-modeling hereafter) to disentangle
the spectra of the main components of 3C~120 (nucleus+host), and to obtain
cleaned spectra of the different structures described above. This technique
has been successfully applied recently \citep{sanc04c}. A brief summary of the
technique is described in the Appendix B, and will be explained in
detail in a separate article (S\'anchez et al., in prep.). In summary, the
technique provides us with a spectrum of each of the main components of the
object plus a residual data cube that can be used to derive cleaned spectra of
the different structures in the object and/or analyze their morphological and
kinematical properties. This is an extension of the 2D modeling of images
applied in section \S4.1 to each of the monochromatic images of the data cube
derived from IFS optical observations. We applied the 3D modeling to the
current data and we derived the 3C~120 nucleus spectrum, the mean host galaxy
spectrum and a residual data-cube of spectra.
\subsection{Morphological structures from the residual data cube of spectra}
To compare the results from the proposed technique to those of the traditional
method, figure 7A shows the greyscale of the residual from the 2D
modeling of the F555W/HST-band image obtained in section \S4.1 (figure
3D). We over plotted the contours of a narrow-band image
centred on the continuum adjacent to the [OIII] emission line (5204-5246 \AA)
extracted from the residual data cube of spectra obtained after the 3D
modeling. The main continuum dominated structures (A, B and C) are
detected $\sim$1$\arcsec$ from the nucleus. As expected, the shell
structure detected at $\sim$1$\arcsec$ north and south of the nucleus in the
HST images is not detected with the IFS data. A combination of the superior
resolution of the HST images, the effects of the seeing and the sampling of
our IFS data can explain it. In figure 7B we present the
greyscale of the F555W$-$F547M image (obtained in section \S4.2, figure
4D) and the contours of a narrow-band image centred on the
[OIII]$\lambda$5007 emission line (5170-5200\AA) extracted from the residual
data cube. As we quoted above (section \S4.2), the F555W-F547M image is a
rough estimation of the [OIII]-H$\beta$ emission map. Although both images
are in good agreement, integral field spectroscopic data are more adequate to
detect extended emission line regions (EELRs) that are diluted in broadband
filter images, like the F555W-band one. It is possible to identify the E$_1$
and E$_2$ structures, and a third region (E$_3$ hereafter) at the north-west
of the F555W-F547M image. \cite{soub89} already detected these emission line
regions, using narrow-band imaging centred on the [OIII] emission line
spectral region. E$_1$ and E$_2$ are completely coincident with their reported
positions. The peak of E$_3$ found by \cite{soub89} is not included within the field of view of INTEGRAL. We are confident of our detection, and most probably we are seeing a tail
previously not detected of E$_3$ towards the nucleus.
\subsection{Spectra of the nucleus and the host galaxy}
\label{hg_spec}
Figure 8 and 9 show the spectra of the different
deblended components of 3C~120. Figure 8A shows the spectrum of
the nucleus. Its continuum follows roughly a power-law with a slope of
$\alpha\sim-0.5$ ($F_\lambda\propto\lambda^\alpha$). It contains
broad-emission lines, without any trace of them in the spectrum of the host
galaxy (figure 8B), which indicates that the decoupling
technique has worked properly. The intensities of the narrow emission lines is
weaker in the nucleus than in the host galaxy spectrum. Indeed, many emission
lines are clearly detected in the host galaxy spectrum but not in the nucleus.
To derive the properties of the emission lines in the spectrum of the nucleus,
we fit Gaussian functions to the different emission lines. The adjacent
continuum was modelled with a low-degree polynomial function. To increase the
accuracy of our model, different nearby emission lines were fitted together,
defining line systems with the same systemic velocity and
full-width-half-maximum (FWHM). After several attempts, the best (and
simplest) modeling was obtained including two decoupled broad systems, with
$\sim$8796 km s$^{-1}$\ and $\sim$2188 km s$^{-1}$, and a narrow system, with $\sim$700 km s$^{-1}$
. Recently, \cite{ogle04} also reported two broad and one narrow component
for the H-Balmer lines in the nucleus of this galaxy with a similar velocity
dispersions than those derived here. Table \ref{elines1} lists the result
from the fit, including the wavelength, the flux and their 1$\sigma$ errors
for each detected line. The uncertainties in the determination of the flux of
the narrow emission lines affected by line blending are high as expected. The
H$\alpha$/H$\beta$ line ratios are in the range of $\sim$3.0-3.6 for the
different line systems, in agreement to the nominal values for case B
recombination \citep{oste89}. Therefore both the narrow and broad emission
line regions are not significantly affected by dust, as expected for a type 1
AGN nucleus. We estimated an effective temperature of $\sim$2.3$\times$10$^5$
K using the ([OIII]$\lambda$5007+[OIII]$\lambda$4959)/[OIII]$\lambda$4363 line
ratio and the relation between this ratio and temperature \citep{oste89}
typical of this kind of AGN.
Figure 8B shows the average spectrum of the host galaxy together
with the spectra of different synthetic models consisting on single stellar
populations of different ages. We used the \cite{bruz03} models, assuming a
solar metallicity and a \cite{chab03} initial mass function (IMF). The effects
of dust were included in the model using the extinction curve of \cite{fitz99}
for a dust absorption of $A_V\sim$4 mags (the average value in the host
galaxy, as we will show below). Two extreme cases were plotted, representing
an old ($\sim$16Gyr, red line) and a young stellar population ($\sim$0.01Gyr,
blue line). These extreme cases are a clear over simplification, but they can
be used just for qualitative comparisons. A single stellar population cannot
describe the observed spectrum: the optical slope between $\lambda\sim$5000
\AA \ and $\lambda\sim$6500 \AA \ matches well with that of an old stellar
population. However, the slope at larger wavelengths is flatter, more similar
to that of a young stellar population. We already noticed it (section \S4.2)
when we analyzed the broadband colors of the host galaxy (Figure
5). Only a mix of different stellar populations can explain the
observed spectrum.
We included in figure 8B the spectrum of a fifty-to-fifty mix of
both components (orange line). Even this simple model can describe quite well
the spectrum of the host galaxy for $\lambda>$4500\AA. At shorten wavelengths
there is an increase of the flux, which does not match with the mix model
(dashed line). It is well known that powerful radio galaxies show a
significant blue-UV excess \citep{lill84} most probably due to scattered light
from the nucleus than to young stellar populations \citep{mcca87}. This
scattered light can be well described by a power-law. Adding a power-law to the
mix model we can describe the average host galaxy spectrum for any wavelength
(orange solid line). Therefore, three components are needed to explain
qualitatively the spectrum of the host galaxy: (a) an underlying old stellar
population, (b) a young stellar population, and (c) a nebular continuum, most
probably scattered light from the nucleus. We did not perform a proper fitting
to determine the best fractions of the different components because it would
not change the qualitative statement, and, in any case, the specific fractions
will depend on the assumed synthetic models for the young and old populations
and we chose them arbitrary. The contribution of each component to the total
flux are $\sim$5\% (a), $\sim$55\% (b) and $\sim$40\% (c) at 4500 \AA \ and
$\sim$35\%(a), $\sim$64\% (b) and $\sim$2\% (c) at 6000 \AA . Similar results
have been found in the study of other powerful radio galaxies
\cite[e.g.][]{tadh96}.
Table \ref{elines1} lists the results derived from fitting a single Gaussian
to the emission lines in the host galaxy spectrum. As we quoted above, only
narrow emission lines corresponding to a dispersion velocity of FWHM$\sim$700
km s$^{-1}$ \ are detected. These emission lines are considerably stronger than the
narrow-emission line region in the spectrum of the nucleus, indicating the
presence of an extended narrow-emission line region which extends through most
of the field of view. Similar results have been found in spectroscopic studies
of radio-quite type 1 AGNs \citep[e.g.][]{jahn02,jahn04}. The H$\alpha$/H$\beta$\ line ratio
is $\sim$10, indicating a dust absorption of $A_V\sim$4 mags when comparing
with the nominal case B recombination value \citep{oste89} and using the
\cite{fitz99} extinction curve. In average the host galaxy contains a
rather high dust content, but the dust distribution is not uniform as we can
see in figure 6E. We estimated an effective temperature of
$\sim$14000 K using the
([OIII]$\lambda$5007+[OIII]$\lambda$4959)/[OIII]$\lambda$4363 line ratio and
an electron density of n$_e$$\sim$10 cm$^{-3}$ using the
[SII]$\lambda$6716/6731 line ratio. To derived these quantities, we used the
relations between these line ratios and the measured parameters
\citep{oste89}. The line ratios log([OIII]$\lambda$5007/H$\beta$)$\sim$1.2 and log([NII]$\lambda$6583/H$\alpha$)=$-$0.4 indicate
most probably a direct photo ionization by the UV flux from the AGN,
consistent with the result shown in Section \S4.3.
\subsection{Spectra of the EELRs: Origin of the ionization}
\label{EELRs_spec}
The integrated individual spectra of the structures E$_1$, E$_2$, E$_3$
described in section \S5.1 were extracted from the residual data cube. Figure
9A presents the spectra of each EELRs (E$_1$, E$_2$, E$_3$),
which are remarkable similar. They have a clear gaseous nature, without
appreciable continuum emission. This result confirms that the continuum
dominated structure $A$ is not related to E$_1$ as we pointed out in section
\S4.2. A visual inspection of the relative strength of the different lines
indicates that the dominant ionization source is the AGN or shocks, rather
than a star formation process. This was already noted in section \S4.3 and
figure 6C for E$_1$, and E$_2$. Table \ref{elines2} lists the
result of the Gaussian modeling of the emission lines in these spectra.
Different line ratios and parameters derived from them are listed in Table
\ref{param}. We included the [OIII]$\lambda$5007/H$\beta$, [NII]$\lambda$6583/H$\alpha$, [SII]$\lambda$6716/[SII]$\lambda$6731 \ line ratios, and the derived
dust absorption ($A_V$), electron density ($n_e$) and effective temperature
(T$_{eff}$). Those values of the nucleus and the host galaxy spectra were also
included for comparison purposes. As we already mentioned, the ionization
conditions are similar in the different EELRs, being also similar to those of
the nucleus and the host. Figure 10 shows the classical diagnostic
diagram of the [OIII]$\lambda$5007/H$\beta$ \ line ratio as a function of [NII]$\lambda$6583/H$\alpha$ \ for the different
components of 3C~120, including the division between AGNs and star forming
regions \citep{veil87}. The major difference is found in the position of the
nucleus in the diagram, which seems to lie in the location of star forming
regions. This is most probably due to the large uncertainties in the [NII]$\lambda$6583/H$\alpha$\ line
ratio derived for the spectrum of the nucleus, largely influenced by the
deblending process of the narrow and broad emission lines, as we quoted above.
The derived effective temperature (T$\sim1.4\times10^{4}$ K) for the host galaxy spectrum does not exclude the hot stars as a possible origin for the ionization but the high [SII]/H$\alpha$, [NII]/H$\alpha$, and [OIII]/H$\beta$ ratios
indicate that the average ionization for the host is the AGN or/and a shock
process. i.e., the observed spectra cannot be found in HII regions.
The [NII]$\lambda$6583/H$\alpha$ \ line ratio of the E$_3$ region is lower than that value for the rest
of the structures. E$_3$ is the faintest detected EELR and it was not
completely covered by our field-of-view, according to the [OIII] maps shown
by \cite{soub89}. Its spectrum is noisier and with some clear defects at
specific wavelengths produced by the modeling technique. A detailed
inspection of the spectrum shown in Fig. 9 shows that the
[NII]$\lambda$6583 line is distorted by one of these defects, affecting the
[NII]$\lambda$6583/H$\alpha$ \ line ratio, which error is clearly larger than the formal error plotted
in Fig. 10. Despite these caveats, it is clear that star formation
processes do not dominate the ionization of the different EELRs.
The dust content in the EELRs is lower than the mean obscuration of the host. In particular,
for the E$_2$ and E$_3$ regions the dust absorption is almost half of the
average in the host galaxy. On the other hand, the electron density in those
clouds is $\sim$10 times lower than the average. However, this density is
$\sim$10 times higher in the E$_1$ region, i.e. $\sim$100 times larger than in
the other clouds. This indicates most probably a different origin of those
clouds. In \cite{sanc04c} we discussed in detail the nature of the E$_1$
emission line region. It is most probably associated with the radio jet that
crosses this cloud, compressing it due to its lateral expansion, and splitting
it in two different kinematics regions \citep{axon89,sanc04c}. The
compression is reflected in the increase of the electron density. The nature
of the ionization is not clear in this cloud, since a post-shock zone can also
give rise to the observed line ratios. The similarities between the line
ratios for the different EELRs, and between them and the host galaxy ones, may
indicate a similar origin for the ionization. That is, direct photo ionization
from the UV-field of the AGN. In that case the effect of the jet over the
E$_1$ would be reduced to a split of the cloud and a compression that give
rise to a density enhancement.
\subsection{Spectral energy distribution of the continuum dominated condensations}
\label{cont_sed}
Figure 9B shows the spectral energy distribution (SED) of the
continuum dominated structures A, B and C detected in 3C~120 within the
field-of-view of our IFS data. The SEDs were obtained by an average of the
integrated spectra of the different structures extracted from the residual
data cube over spectral ranges of 300\AA\ width. The low intensity of the
continuum dominated structures and the subsequent low signal-to-noise prevents
us of using directly the extracted spectra for this analysis. The SEDs were
cut at 7500\AA\ since at larger wavelengths the imperfect subtraction of the
sky-lines and the noise enhancement strongly affect the reliability of the
derived SED. Synthetic spectra for three different stellar populations of
different ages were included for comparison purposes. These spectra are
similar to those shown in the figure 8B (described in \S5.2),
but without including the effects of dust. The SEDs are almost flat in the
plotted ranges, being roughly consistent with a mix of a young stellar
population with an underlying old stellar population. This result agrees with
the results based on the broadband colors of the structures, discussed in
\S4.2. Therefore, the continuum structures are experiencing a decrease of
the dust content, rather than an increase of the star formation. This seems to
be particularly valid for the A structure, which lies in a minimum of the dust
content, derived from the H$\alpha$/H$\beta$\ line ratio (See Fig. 6E).
Unfortunately, we cannot check it for the B and C structures, because the
gaseous emission in those regions is less luminous and therefore the H$\alpha$/H$\beta$\
ratio is uncertain.
So far only starlight was considered to explain the emission found in the
different condensations. However their SEDs could be also explained
considering other components, like scattered-light from the nucleus and/or
synchrotron emission associated with the radio-jet. \cite{hjor95} detected
polarization in the condensation $A$, which direction and magnitude were
consistent with those found by \cite{walk87}. This may indicate a connection
between that condensation and the radio jet. Indeed, it may indicate that
condensation $A$ consists, at least partially, of optical synchrotron
emission. However, as it was shown in \S4.1 \citep[and noticed by][]{hjor95},
$A$ does not follow the radio-jet into the core, being more likely associated
with $B$ and $C$. Furthermore, the measured polarization may be due to
unsubtracted scattered-light from the nucleus \citep{hjor95}, that we detected
in the average spectrum of the host (Fig. 8 and \S5.2). Indeed,
the SEDs of the three different condensations do not differ significantly (Fig.
9B), and, in particular, they show almost the same $V-R$ colors
(Fig. 5), which indicates, most probably, a similar origin for
all of them. Since most of the scattered-light has been removed
and the synchrotron radiation could also contaminate $A$, these condensations
are most probably dominated by starlight.
\section{3C~120 Velocity Field}
The central wavelengths of the Gaussians fitted to the individual spectra
(section \S4.3) give us the radial velocity associated to the ionized gas at
each position. Interpolating the individual emission line centroids, we
obtain the velocity field of different emission lines. Uncertainties due to
wavelength calibration ($<$ 35 km s$^{-1}$) and those related to the fitting
process ($\sim$ 15 km s$^{-1}$) are small enough to be irrelevant for the
following discussion.
Figure 11 shows the velocity field of the ionized gas derived
from [OIII]$\lambda5007$ and H$\alpha$ lines. While H$\alpha$ traces the
kinematics of low ionisation gas that describe the general pattern of the
galaxy, [OIII] draws the signatures of high ionization gas, characterizing the
most perturbed regions. Previous kinematics studies of 3C~120 found a rather
chaotic velocity field \citep{bald80} and some slight evidences of a
co-existing rotating system \citep{bald80,mole88}. Despite of the different
spatial resolution and coverage of the IFS data in this paper, our results are
in agreement with those of \cite{bald80} and \cite{mole88} pointing towards
highly distorted kinematics in this object.
The unclear morphology and highly distorted gas kinematics of 3C~120 makes
a simple interpretation of the velocity structure difficult. The ionized gas
velocity field presents a general regular pattern, with larger velocities at
the north-west and lower at the south-east that may be consistent with a disk
rotating around an axis along the north-east direction (figure
11A and 11B). Several kinematical perturbations
can be identify in the velocity maps as well as a regular rotation.
A velocity gradient is located at $\sim$5$\arcsec$ west of the nucleus,
aligned with the radio-jet (figure 3B), at the position of
E$_1$ (figure 11A). Using a better spectral
resolution, \cite{axon89} show that there are two different kinematics
components rather than a velocity gradient. The north component is receding
while the south component is approaching. The kinematical perturbation is most
probably due to the lateral expansion of the radio-jet \citep{axon89}: as
already quoted, the interaction of the radio-jet and the intergalactic gas
produces an enhancement of the gas density (section \S4.3), and perturbs the
kinematics \citep{sanc04}.
The velocity field derived from [OIII] is much more distorted to the east
(farther than 4 arcsec from the nucleus) than that one derived from H$\alpha$.
Although the signal to noise in the IFS spectra is low and uncertainties are
larger in that region, differences in the kinematics may be explained by the
different origin of these lines. The comparison of the continuum dominated
structures and the velocity field derived from the H$\alpha$ shows that $A$
and $B$ coincide with a velocity distortion located at the west of the nucleus
(figure 11B). This kinematical feature is similar to those
found in the velocity field of spiral galaxies because of the arms
\citep[e.g.][]{knap04}. This gives support to the idea of identifing the $A$
and $B$ regions as structures in an spiral arm in 3C~120 in spite of its
poor gas content and continuum emission dominated by starlight. We will
discuss in detail the origin of this arm-like structure in section \S7.
The comparison of the ionized gas structures E$_2$, and E$_3$ (figure
7B) and the velocity field gives clues to the origin of the EELRs
(figure 11C). The morphology of E$_3$ follows remarkably well an
east-west gradient in the velocity map. E$_2$ is in a close to constant velocity
region. Both clouds are in regions which present strong
kinematical perturbations from the canonical rotation. These perturbations are
more prominent in the velocity structure of the high ionization gas than that
of the low ionization gas.
Fitting a single Gaussian to the emission lines in the residual data cube of
spectra (section \S5), we have derived the velocity behavior of the E$_1$,
E$_2$, and E$_3$ residual structures. Figure 11D shows the
velocity map derived from the [OIII] lines in the residual spectra. We will
refer to this map as the velocity map of residuals hereafter. The kinematics
of the structures trace remarkable well the perturbations in the velocity
field, although their kinematical features are smoother in the latter. The
smoothing is a clear consequence of the blending of the host galaxy and the
structure spectra. The velocity map of the residuals represents much better
the kinematics of the gas clouds in E$_1$, E$_2$, and E$_3$. We found a clear
gradient of velocities in the region corresponding to the three structures
pointing to inflows/outflows of gas at different inclination angles and
outside of the galactic disk. While regions E$_1$ and E$_2$ present a
north-south velocity gradient, E$_3$ shows an east west gradient in good
agreement with the [OIII] velocity map (figure 11A).
Despite of the bulge dominated morphology found in the inner regions of
3C~120, the velocity field derived from H$\alpha$ indicates the presence of
a rotating disk along the north-west/south-east (figure 11B).
In that case, the receding velocities at the northwest suggest that southeast
is the face closer to the observer. The velocity gradient also traces the line
of zero velocities, but it is not clearly defined in the derived velocity field
because of the distortions in the center introduced by the Sy1
nucleus. With a larger spatial coverage, \cite{bald80} determined a
PA=72$^{\circ}\pm15^{\circ}$ for the minor kinematics axis. This angle agrees with
the semi-minor axis determined from the 2D modeling of the galaxy (section
\S4.1). A visual inspection of the velocity fields in figures
11A and 11B indicates that this PA may be a good
estimation of the minor kinematics axis, when considering the distortion at
the west. However, this distortion is most likely produced by the interaction of
the radio-jet with the E$_1$ cloud \citep{sanc04}, and most probably is not
related with the rotating disk. The isovelocity contours of the velocity field
derived from H$\alpha$ and the position of the kinematical center ($\sim0.8"$
north-west of the nucleus) suggest that the minor kinematical axis is most
likely along PA$\sim50^{\circ}$, which is not far from the photometric
determinations and within their uncertainties. Therefore, we will
consider hereafter a PA=$-40^{\circ}$ for the major kinematics axis of 3C~120.
Figure 12 shows the rotation curve of 3C~120 derived from
the velocity field of H$\alpha$, assuming a PA of $-40^{\circ}$ for the major
kinematics axis. This curve is clearly distorted at the south-east due to the
kinematical perturbations associated with E$_2$. However, the north-west
portion of the curve is remarkable similar to those of galactic rotating disks
\citep{binn98}. The effects of the perturbations in the south-east portion of
the curve must be removed prior to model the 2D distribution of the rotating
component of the galaxy. For doing so, it was assumed that the approaching
portion of the rotation curve (south-east) follows a symmetrical counterpart
of the receding portion (north-west). This {\it hypothetical} rotation curve
was included in Figure 12. This curve was then used to
derive a template of the velocity field by applying a simple rotational model
\citep{miha81}, for different inclinations varying from $\sim$5$^{\circ}$ to
$\sim$85$^{\circ}$ . Subtracting those templates from the H$\alpha$ velocity
field and minimizing the differences at the northwest region (where the
rotation curve really corresponds to the 3C~120 kinematical behavior), we
estimated an inclination angle of $\sim40^{\circ}$ for the disk component of
3C~120. The velocity field template corresponding to that inclination angle is
shown in figure 11E. This estimation of the inclination is in
agreement with the determination from the external isophote \citep{mole88}. As
quoted before, the internal isophotes are less elongated and distorted at the
west than the external ones. This suggests a different inclination angle,
smaller at the inner than at the outer regions. Indeed, the ellipticities
derived from the 2D modeling (section \S4.1) indicate that the inclination
may range from $\sim20^{\circ}$ to $\sim43^{\circ}$ in the center, in
agreement with the kinematical estimation. The differences in inclination
from the center to the outer regions, although small, suggest a slightly warped
disk component in 3C~120.
Figure 11F shows the residual of the [OIII] velocity map once
subtracted the template shown in Fig. 11E (residual velocities
hereafter). This residual map shows an almost flat area of 0$\pm20$ km s$^{-1}$ at
the north-west region, within the uncertainties of the velocity determination.
At the west, coincident with the path of the radio-jet and the location of
E$_1$, there is a north-south gradient of $\sim148$ km/s, in agreement with
the results by \cite{axon89}. At the location of E$_2$ there is a rather
chaotic residual velocity structure, with velocities ranging from $\sim$270 to
460 km/s. A slight east-west gradient of $\sim32$ km/s is found at the
location of E$_3$.
The residual velocities (Fig. 11F) and the velocity map of the
residuals (Fig. 11D) present a rather good qualitative agreement
in spite of the strong conceptual and practical differences of the two methods
used to derived them. This supports the idea of the presence of a rotational
component in 3C~120.
Due to the low spectral resolution of our data no attempt was done to
analyze the velocity dispersion maps.
\section{discussion}
In previous sections we presented several aspects of 3C~120 that describe a
puzzle environment. 3C 120 is morphologically speaking a bulge dominated
galaxy, which contains a rotating stellar disk and several continuum-
dominated structures and EELRs.
We analyzed in detail those structures in order to determine their nature. The $E_1$ is caused by the interaction of the radio-jet with the intergalactic
medium, as already discussed in \cite{sanc04c}.
The newly reported shells in the central region of 3C 120 is an
addition to the complex picture of this object.
Shells are common structures in elliptical and SO galaxies, although a few
number of spirals also show the presence of this kind of features
\citep{malin83,seitzer88}. Although models
considering internal shock waves were proposed to account for the
presence of shells in galaxies \citep[see e.g.][]{williams85}
the most accepted idea is that they are generated by merging processes \citep[see e.g.][and reference therein]{hern92}. Numerical simulations indicate that the number and
sharpness of shells in merger remnants depend on the mass ratio and the
absence (or presence) of a central buldge in the progenitors \citep{gonz04}.
The radial distribution
of shells depends on the potential of the host galaxy and their morphology
can appear aligned or randomly distributed around the galaxy \citep{prieur90}.
3C 120 shows two well defined shells, labelled as S$_N$ and S$_S$, and
some
other shell-like faint features which are aligned along an axis of
PA$\sim-25^{\circ}$. When shells are well aligned along a certain axis,
this axis is always close to the major axis of the galaxy \citep{prieur90}, which is out case.
Shells can be found in a wide range of radii from the nucleus, from only a
few to hundreds of Kpc, such us in the case of NGC 3923 \citep{prieur88}. The
innest shell detected in 3C 120 is at around 1 arcsec north ($\sim 1 $ kpc)
of the nucleus. Shells found close to the nucleus
implies that a dissipative process has played an important role in the
formation of the shells \citep{prieur90}. Therefore, the presence of the detected
circumnuclear shells (section \S4.1) strongly suggests the idea of
considering 3C 120 as a late stage merger.
The arm-like clumpy structures A and B (section \S4.1) show kinematics that
resembles those of a spiral arm as explained in section \S6. However, spiral
arms are active star forming regions, and no star formation was detected in
either of the continuum-dominated structures. In that case, these structures
could be also the remnant of a merging process. Indeed, the regions B, A and
the S$_s$ seem to be connected in our residual and color maps (Fig 3C and 4B),
and S$_s$ is associated with the E$_2$ EELR. A kinematics feature (Fig. 11B)
is coincident with these structures. Numerical simulations indicate that these
kind of structures are found as remnants of merging processes
\citep[e.g.][]{howa93,heyl96,miho96} being difficult to be explained by any other mechanism. Assuming that scenario, the presence of
an over density of gas in the south shell may indicate an inflow, where the
warmer gas has dropped faster into the inner regions. This picture is in good
agreement with the kinematics features at the south shell (E$_2$ clouds
kinematics) and its residuals. The regular velocity in this region suggests
that this inflow may occur in a plane outside the galactic disk and close to
the line of sight. Although the kinematics of the merger remnants
strongly depends on the viewpoint, the velocity and velocity dispersion
measured for E$_2$ is in good agreement with merger simulations
\citep{heyl96}. Theoretical models indicate
that inflows channeling gas
from the outer to the nuclear regions may appear in the late stages of a
merging process \citep[e.g.][]{miho96}. These inflows have been already
observed in different merging galaxies \citep[e.g.][]{arr02,arr03}.
The discussion above suggests that the structures A, B, and the E$_2$ EELR
can be identified with a tidal tail driving material to the central region most probably due to a past merger event in 3C 120. Most of the analyzed continuum dominated
structures shows a rather young stellar population, indicating a recent
(but not ongoing) star formation rate. This result agrees with the picture
of a post-merging event. Similar results have been found in the study of
the optical colors of host galaxies AGNs \citep{jahnke04,sanc04b} .
The poor spatial coverage of the third identified EELR, E$_3$, prevents us of
drawing a clear picture of its origin. In fact, this region corresponds (see
section \S5.1) to a tail extension of a larger structure already reported in
this object \citep{soub89}. Although we cannot be conclusive with our current
data, the kinematical results most probably indicate that E$_3$ is associated
with an inflow/outflow of gas to/from the circumnuclear region.
It has long been suggested that strong interactions and galaxy mergings may
(re-)ignite nuclear activity \citep[e.g.][]{sand88}. Galaxy interactions can
produce the loss of momentum required to allow the infall of gas towards the
nuclear regions, gas that would feed the AGN. Many authors found that AGN
hosts show distorted morphologies, reminiscent of past merging events
\citep[e.g.][]{mcle94a,mcle94b,bahc97,sanc03b}. They are found in environments
with high probability of experiencing interactions \cite[e.g.][]{sanc03a}. And
their hosts, mostly early-type galaxies, present anomalous blue colors
\cite[e.g.][]{sanc04c}. All together it may indicate that, if not all, at
least a family of AGNs is generated by the merging/interaction between
galaxies \citep{cana01}. A merging event alone id not enough to
generate an AGN. The presence of a massive black-hole in the progenitor galaxy
is a basic requerement. Massive black-holes are only found in bulge-dominated
massive galaxies, due to the black-hole/bulge mass relation
\citep[e.g.][]{mago98,korm01}. Indeed, recent results (S\'anchez et al., in
prep.) show that the fraction of AGNs increases in galaxy mergers between two
large galaxies or a large with an small galaxy.
Our current results agree with this sceneario. 3C~120 is a bulge dominated
galaxy which has, most probably, experienced a merging event with a less
massive galaxy. That galaxy was completely disrupted in the merging process,
falling in parts which produce, most probably, many of the observed structures
\citep{mole88}. This may explain the different stellar populations of the
structures and the average stellar population in the object (\S4.2) as
it has been previously reported for other galaxies \citep{prieur90}. A
subtantial fraction of its gas has been channelled towards the inner regions,
following the detected arm-like structure, and concentrating in the E$_2$
region.
\section{Summary}
We obtained integral field optical spectroscopy of the Seyfert 1 radio galaxy
3C~120. The homogeneous data, excellent spatial and spectral coverage, and
good spatial and spectral resolution make this atlas a useful tool for
studying 3C~120 in the optical. These IFS optical data were combined with high
resolution HST imaging. The analysis of these data suggests that a Seyfert 1
nucleus at the center, an early type galaxy, and several structures formed
as a consequence of a merging process in the past history constitute the
radio galaxy 3 120. At least one of these
structures is identified with an inflow, which is feeding the central engine.
A radio-jet is escaping from the center, perturbing the gas on its path.
The main results from the analysis of this dataset were the following:
\begin{itemize}
\item[1.] Several continuum-dominated structures were detected in 3C 120,
which do not follow the mean distribution (bulge dominated) of the stellar
component. Some of these structures were shells in the central Kpc of the
object, which may indicate a past merging process. The colors of the
different components were not compatible with a single stellar population in
this galaxy.
\item[2.] Three emission line structures were identified (E$_1$, E$_2$ and
E$_3$) which were not associated with the general behavior of the galaxy.
These gaseous structures presented a high level of ionization. The origin of
structure E$_1$ was the interaction between the intergalactic medium and the
radio-jet emerging from the nucleus.
\item[2.] The spectra of the nucleus and the host galaxy were decoupled,
obtaining a data cube of residuals. From that residual data cube, we
extracted and analyzed the spectra of the different structures in 3C 120.
\item[3.] The velocity field indicated a rotational component plus several
kinematical perturbations associated with the identified emission
structures.
\item[4.] The continuum-dominated structures A, B, the S$_s$ shell and the
EELR E$_2$ seem to be physically associated, belonging to the inner-most
part of a tidal-tail, remanent of a past merging event. This tail has
channeled gas into the inner regions, in an inflow, that has generated
E$_2$.
\end{itemize}
The 4.2-m William Herschel Telescope is operated by the Isaac Newton Group at
the Observatorio de Roque de los Muchachos of the Instituto de Astrof\'\i sica
de Canarias. We thank all the staff at the Observatory for their kind support.
This project is part of the Euro3D RTN on IFS, funded by the EC under contract
No. HPRN-CT-2002-00305. This project has used images obtained from the HST
archive, using the ESO archiving facilities. We would like to thank Dr.Walker
that has kindly provided us with the radio maps of 3C~120.
\clearpage
|
{
"timestamp": "2004-11-11T17:40:12",
"yymm": "0411",
"arxiv_id": "astro-ph/0411298",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411298"
}
|
\section{Introduction} \label{s1}
The Bergman kernel for complex projective
manifolds is the smooth kernel of the orthogonal projection
from the space of smooth sections of
a positive line bundle $L$ on the space of holomorphic sections of $L$,
or, equivalently, on the kernel of the Kodaira-Laplacian
$\Box^L= \overline{\partial} ^L\overline{\partial} ^{L*}
+ \overline{\partial} ^{L*}\overline{\partial} ^L$ on $L$.
It is studied in \cite{Tian,Ru,Zelditch,Catlin,BSZ,Lu,Wang1, KS01},
in various generalities,
establishing the diagonal asymptotic expansion
for high powers of $L$. Moreover, the
coefficients in the diagonal asymptotic expansion encode geometric
information
about the underlying complex projective manifolds.
This diagonal asymptotic expansion
plays a crucial role in the recent work of Donaldson \cite{D} where the
existence of K\"ahler metrics with constant scalar curvature is
shown to be closely related to Chow-Mumford stability.
In \cite{DLM}, Dai, Liu and Ma
studied the asymptotic expansion of the Bergman kernel
of the spin$^c$ Dirac operator associated to a positive line bundle
on a compact symplectic manifold,
and related it to that of the corresponding heat kernel. As a by
product, they gave a new proof of the above results.
This approach is inspired by Local Index Theory,
especially by the analytic localization
techniques of Bismut-Lebeau \cite[\S 11]{BL}.
Another natural generalization of the operator $\Box^L$
in symplectic geometry was initiated by Guillemin and Uribe
\cite{GU}. In this very interesting short paper,
they introduce a renormalized Bochner--Laplacian (cf.\,\eqref{laplace})
which is exactly $2\Box ^L$ in the K\"ahler case.
The asymptotic of the
spectrum of the renormalized Bochner--Laplacian on $L^p$ when $p\to \infty$
is studied in various generalities in
\cite{BU,Bra1,GU} by applying
the analysis of Toeplitz structures
of Boutet de Monvel--Guillemin \cite{BoG},
and in \cite{MM} as a direct application of Lichnerowicz formula.
Of course, there exists also a replacement of the $\db$--operator and of the
notion of holomorphic section based on a construction of Boutet de
Monvel--Guillemin \cite{BoG} of a first order pseudodifferential operator
$D_b$ which mimic the $\db_b$ operator on the circle bundle associated
to $L$. However, $D_b$ is neither canonically defined nor unique. This
point of view was adopted in a series of papers
\cite{BU1,SZ02,BSZ1}.
In this paper, we will study the asymptotic expansion of the
generalized Bergman kernel of the renormalized Bochner-Laplacian,
namely the smooth kernel of the projection on its bound states
as $p\to \infty$. The advantage of this approach is that the renormalized
Bochner-Laplacian has geometric meaning and is canonically defined.
Moreover, it does not require the passage to the associated circle bundle
as we can work directly on the base manifold.
Let's explain our results in detail.
Let $(X,\omega} \newcommand{\g}{\Gamma)$ be a compact symplectic manifold of real dimension $2n$.
Assume that there exists a Hermitian line bundle $L$ over $X$ endowed with
a Hermitian connection $\nabla^L$ with the property that
$\frac{\sqrt{-1}}{2\pi}R^L=\omega$, where $R^L=(\nabla^L)^2$
is the curvature of $(L,\nabla^L)$. Let $(E,h^E)$ be a Hermitian vector
bundle on $X$ with Hermitian connection $\nabla^E$ and curvature $R^E$.
Let $g^{TX}$ be a Riemannian metric on $X$
and ${\bf J}:TX\longrightarrow TX$ be the skew--adjoint
linear map which satisfies the relation
\begin{equation} \label{0.1}
\omega} \newcommand{\g}{\Gamma(u,v)=g^{TX}({\bf J}u,v)\quad\text{for}\quad u,v \in TX.
\end{equation}
Let $J$ be an almost complex structure
such that $g^{TX}(J\cdot, J\cdot)=g^{TX}(\cdot, \cdot)$,
$\omega} \newcommand{\g}{\Gamma(J\cdot, J\cdot)=\omega} \newcommand{\g}{\Gamma(\cdot, \cdot)$
and that $\omega} \newcommand{\g}{\Gamma(\cdot, J\cdot)$ defines a metric on $TX$.
Then $J$ commutes with ${\bf J}$, and $-J{\bf J}\in \End(TX)$ is positive,
thus $J= {\bf J} (-{\bf J}^2)^{-1/2}$.
We introduce the Levi-Civita connection $\nabla^{TX}$ on $(TX, g^{TX})$
with its curvature $R^{TX}$ and scalar curvature $r^X$.
Let $\nabla ^XJ \in T^*X \otimes \End(TX)$ be the covariant
derivative of $J$ induced by $\nabla ^{TX}$. Let $\Delta ^{L^p\otimes E}$
be the induced Bochner-Laplacian acting on $\cali{C}} \newcommand{\cA}{\cali{A}^\infty (X,L^p\otimes E)$.
We fix a smooth Hermitian section $\Phi$ of $\End (E)$ on $X$.
Let $\{e_i\}_i$ be an orthonormal frame of $(TX, g^{TX})$. Set
\begin{align}\label{0.2}
&\tau(x)=-\pi \tr_{|TX} [J{\bf J}]= \frac{\sqrt{-1}}{2} R^L (e_j, J e_j) >0,\\
&\mu_0=\displaystyle\inf_{{u\in T_x X,\,x\in X}}
\sqrt{-1} R^L_x(u, Ju)/|u|^2_{g^{TX}} >0 , \label{0.21}\\
& \Delta_{p, \Phi} = \Delta ^{L^p\otimes E} -p\tau + \Phi.\label{laplace}
\end{align}
By \cite[Cor. 1.2]{MM} (also cf. \cite{BU,GU,Bra1,BiV}),
there exists $C_L>0$ (which can be estimated precisely by using the $\cali{C}} \newcommand{\cA}{\cali{A}^0$-norms of
$R^{TX}$, $R^E$, $R^L$, $\nabla ^XJ$ and $\Phi$ cf. \cite[p.656\,--\,658]{MM})
independent of $p$ such that
\begin{equation}\label{0.3}
\spec\,\Delta_{p, \Phi} \subset [-C_L, C_L] \cup [2p\mu_0 -C_L, + \infty[\,.
\end{equation}
where we denote by $\spec(A)$ the spectrum of any operator $A$.
Let $\mH_p$ be the eigenspace of $\Delta_{p,\Phi}$ with the eigenvalues in
$[-C_L, C_L]$.
Then for $p$ large enough,
again by \cite[Cor. 1.2]{MM} (also cf. \cite{BU,GU} when $E$ is
trivial and ${\bf J}=J$)
\begin{align}\label{0.0}
&\dim \mH_p =d_p = \int_X \ch(L^p\otimes E)\td(TX)\\
&\hspace*{3mm}
={\rm rk} (E) \int_X \frac{c_1(L)^n}{n!} p^n
+ \int_X \Big(c_1(E) + \frac{{\rm rk} (E)}{2} c_1(TX)\Big)
\frac{c_1(L)^{n-1}}{(n-1)!} p^{n-1} + \cali{O}} \newcommand{\cE}{\cali{E}(p^{n-2}), \nonumber
\end{align}
where $\ch(\cdot), c_1(\cdot), \td(\cdot)$ are the Chern character,
the first Chern class and the Todd class of the corresponding complex
vector bundles ($TX$ is a complex vector bundle with complex structure $J$).
Let $\{S^p_i\}_{i=1}^{d_p}$ be any orthonormal basis of
$\mH_p$ with respect to the inner
product \eqref{0c2} such that $\Delta_{p, \Phi} S^p_i = \lambda_{i,p} S^p_i$.
For $q\in \field{N}$, we define $B_{q,p}\in \cali{C}} \newcommand{\cA}{\cali{A} ^\infty (X, \End (E))$ as follows,
\begin{align} \label{0.4}
B_{q,p}(x) &= \sum_{i=1}^{d_p} \lambda_{i,p}^q S^p_i (x) \otimes (S^p_i(x))^*,
\end{align}
here we denote by $\lambda_{i,p}^0=1$.
Clearly, $B_{q,p}(x)$ does not depend on the choice of $\{S^p_i\}$.
Let $\det {\bf J}$ be the determinant function of ${\bf J}_x\in \End(T_xX)$.
A corollary of Theorem \ref{t3.8} is one of our main results:
\begin{thm}\label{t0.1} There exist smooth coefficients
$b_{q,r}(x)\in \End (E)_x$ which are polynomials in $R^{TX}$,
$R^E$ {\rm(}and $R^L$, $\Phi${\rm)}, their derivatives of order
$\leqslant 2(r+q)-1$ {\rm(}resp. $2(r+q)${\rm)}, and reciprocals
of linear combinations of eigenvalues of ${\bf J}$ at $x$, with
\begin{equation}\label{0.5}
b_{0,0}=(\det {\bf J})^{1/2} \Id_E,
\end{equation}
such that for any $k,l\in \field{N}$, there exists
$C_{k,\,l}>0$ such that for any $x\in X$, $p\in \field{N}$,
\begin{align}\label{0.6}
&\Big |\frac{1}{p^n}B_{q,p}(x)
- \sum_{r=0}^{k} b_{q,r}(x) p^{-r} \Big |_{\cali{C}} \newcommand{\cA}{\cali{A} ^l} \leqslant C_{k,\,l}\: p^{-k-1}.
\end{align}
Moreover, the expansion is uniform in the following sense{\rm:} for any fixed $k,l\in \field{N}$,
assume that the derivatives of $g^{TX}$, $h^L$, $\nabla ^L$, $h^E$, $\nabla ^E$, $J$ and $\Phi$
with order $\leqslant 2n+2k+2q+l+4$ run over a set bounded in the $\cali{C}} \newcommand{\cA}{\cali{A}^l$--\,norm
taken with respect to the parameter $x\in X$ and, moreover, $g^{TX}$ runs over a set
bounded below. Then the constant $C_{k,\,l}$ is independent of $g^{TX}${\rm;}
and the $\cali{C}} \newcommand{\cA}{\cali{A} ^l$-norm in \eqref{0.6} includes also the
derivatives on the parameters.
\end{thm}
\noindent
By derivatives with respect to the parameters we mean directional derivatives in the spaces
of all appropriate $g^{TX}$, $h^L$, $\nabla ^L$, $h^E$, $\nabla ^E$, $J$ and $\Phi$ (on which
$B_{q,p}$ and $b_{q,r}$ implicitly depend).
We calculate further
the coefficients $b_{0,1}$ and $b_{q,0}$\,, $q\geqslant1$ as follows\footnote{Here
$|\nabla ^X J |^2= \sum_{i j}|(\nabla_{e_i} ^X J)e_j |^2$
which is two times the corresponding $|\nabla ^X J |^2$ from \cite{MM04a}.}.
\begin{thm}\label{t0.2} If $J={\bf J}$, then for $q\geqslant1$,
\begin{align}\label{0.8}
&b_{0,1}= \frac{1}{8\pi}\Big[r^X + \frac{1}{4} |\nabla ^X J |^2
+ 2\sqrt{-1} R^E (e_j,Je_j)\Big],\\
&b_{q,0}=\Big (\frac{1}{24} |\nabla ^{X}J| ^2
+\frac{\sqrt{-1}}{2} R^E (e_j,Je_j) + \Phi\Big )^q.\label{0.7}
\end{align}
\end{thm}
Let us check our formulas with the help of
the Atiyah-Singer formula \eqref{0.0}.
Let $T^{(1,0)}X = \{ v\in TX\otimes_\field{R} \field{C}; Jv=\sqrt{-1}v\}$ be the almost
complex tangent bundle on $X$ and let $P^{1,0}= \frac{1}{2} (1-\sqrt{-1}J)$
be the natural projection from $TX\otimes_\field{R} \field{C}$ onto $T^{(1,0)}X$.
Then $\nabla ^{1,0}= P^{1,0}\nabla ^{TX} P^{1,0}$ is a
Hermitian connection on $T^{(1,0)}X$, and
the Chern-Weil representative of $c_1(TX)$ is $c_1(T^{(1,0)}X, \nabla ^{1,0})
= \frac{\sqrt{-1}}{2 \pi} \tr_{|T^{(1,0)}X} (\nabla^{1,0})^2$.
By (\ref{g11}),
\begin{align}\label{0.10}
(\nabla^{1,0})^2 = P^{1,0} \Big[R^{TX}
-\frac{1}{4} (\nabla ^XJ)\wedge (\nabla ^XJ)\Big]P^{1,0}.
\end{align}
Thus if $J={\bf J}$,
then by (\ref{0.10}), (\ref{g29}), (\ref{g30}), (\ref{g36}) and (\ref{g38}),
\begin{align}\label{0.11}
\left\langle c_1(T^{(1,0)}X, \nabla ^{1,0}), \omega\right\rangle
&= \frac{1}{4\pi} \Big(r^X + \frac{1}{4} |\nabla ^X J |^2\Big).
\end{align}
Therefore, by integrating over $X$ the expansion \eqref{0.6} for $k=1$ we obtain \eqref{0.0}, so
\eqref{0.8} is compatible with \eqref{0.0}.
Theorem \ref{t0.1} for $q=0$ and \eqref{0.8} generalize the results of
\cite{Catlin,Zelditch,Lu} and \cite{Wang1}
to the symplectic case.
The term $r^X + \frac{1}{4} |\nabla ^X J |^2$
in \eqref{0.8} is called the Hermitian scalar curvature in the literature
\cite[Chap.\,10]{Ga04}
and is a natural substitute for the
Riemannian scalar curvature in the almost-K\"ahler case.
It was used by Donaldson \cite{D99} to define the moment map on the space of
compatible almost-complex structures.
We can view \eqref{0.7} as an extension and refinement of
the results of \cite{BU2}, \cite[\S 5]{GU} about the density of
states function of $\Delta_{p, \Phi}$ (cf. Remark \ref{t5.2} for
the details).
In \cite{DLM}, Dai, Liu and Ma also focused on the {\em full off-diagonal}
asymptotic expansion
(cf. \cite[Theorem 4.18]{DLM}) which is needed to study the Bergman kernel
on orbifolds, and the only small eigenvalue of the operator is $0$
when $p\to\infty$, thus they had the key equation \cite[(4.89)]{DLM}.
In the current situation, we have small eigenvalues (cf. \eqref{0.3})
and we are interested to prove Theorem \ref{t3.8}, that is, the
{\em near diagonal} expansion of the generalized Bergman kernels.
This result is enough for most of applications.
At first, the spectral gap \eqref{0.3} and the finite propagation speed of
solutions of hyperbolic equations allow to localize the problem.
Then we will combine the Sobolev norm estimates as in \cite{DLM}
and a formal power series trick to obtain Theorem \ref{t3.8},
and in this way, we get a method to compute the coefficients
(cf. \eqref{c90}, \eqref{1c54}) which is new also in the case of \cite{DLM}.
In a forthcoming paper \cite{MM04c}, we will find the {\em full off-diagonal}
asymptotic expansion of the generalized Bergman kernels by combining
the results here and in \cite{DLM}, and as a direct application,
we will study the Toeplitz operators on symplectic manifolds and
Donaldson Theorem \cite{D96} for the Kodaira map $\Phi_{p}$ \eqref{sz0}.
Let us provide a short road-map of the paper.
In Section \ref{s3}, we prove Theorem \ref{t0.1}.
In Section \ref{s4}, we compute the coefficients $b_{q,r}$,
and thus establish Theorem \ref{t0.2}.
In Section \ref{s5}, we explain some applications of our results.
Among others, we give a symplectic version
of the convergence of the induced Fubini-Study metric \cite{Tian}, and
we show how to handle the first-order pseudo-differential operator $D_b$
of Boutet de Monvel and Guillemin \cite{BoG}, which was studied
extensively by Shiffman and Zelditch \cite{SZ02}, and the operator
$\overline{\partial}+\overline{\partial}^*$ when $X$ is K\"ahler but ${\bf J}\neq J$.
We include also generalizations for non-compact or singular manifolds
and as a consequence we obtain an unified treatment
of the convergence of the induced Fubini--Study metric,
the holomorphic Morse inequalities
and the characterization of Moishezon spaces.
Some results of this paper have been announced in \cite{MM04a}.
We refer also the readers our recent book \cite{MM05b} for our approach.
\section{Generalized Bergman kernels }\label{s3}
As pointed out in Introduction, we will apply the strategy of the proof
in \cite{DLM}. However, we have small eigenvalues when $p\to \infty$
(cf. (\ref{0.3})), thus we cannot use directly the key equation
\cite[(4.89)]{DLM} to get a {\em full off-diagonal} asymptotic expansion
of the generalized Bergman kernels.
After localizing the problem, we will adapt the Sobolev norm estimates
developed in \cite{DLM} to our problem in Section \ref{s3.3}.
To complete the proof of Theorem \ref{t0.1},
we need to prove the vanishing of the coefficients $F_{q,r}$ ($r<2q$) in the expansion
\eqref{0ue45}.
We will introduce a formal power series trick to overcome this difficulty
and give a method to compute the coefficients in (\ref{0.6}).
The ideas used here are inspired by the technique of Local Index Theory,
especially by \cite[\S 10, 11]{BL}.
This Section is organized as follows. In Section \ref{s3.1}, we
explain that the asymptotic expansion of the generalized Bergman kernel
$P_{q,p}(x,x')$ is local on $X$ by using the spectral gap (\ref{0.3})
and the finite propagation speed of solutions of hyperbolic equations.
In Section \ref{s3.2}, we obtain an asymptotic expansion of
$\Delta_{p,\Phi}$ in normal coordinates. In Section \ref{s3.3}, we
study the uniform estimate of the generalized Bergman kernels of the
renormalized Bochner-Laplacian $\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t$. In Section \ref{s3.4}, we
study the Bergman kernel of the limit operator $\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0$. In Section
\ref{s3.5}, we compute some coefficients $F_{q,r}(r\leqslant 2q)$ in
the asymptotic expansion in Theorem \ref{tue14}. Finally, in
Section \ref{s3.6}, we prove Theorem \ref{t0.1}.
\subsection{Localization of the problem}\label{s3.1}
Let $a^X$ be the injectivity radius of $(X, g^{TX})$.
We fix $\varepsilon\in (0,a^X/4)$.
We denote by $B^{X}(x,\varepsilon)$ and $B^{T_xX}(0,\varepsilon)$ the open balls
in $X$ and $T_x X$ with center $x$ and radius $\varepsilon$, respectively.
Then the map $ T_x X\ni Z \to \exp^X_x(Z)\in X$ is a
diffeomorphism from $B^{T_xX} (0,\varepsilon)$ on $B^{X} (x,\varepsilon)$ for
$\varepsilon\leqslant a^X$. From now on, we identify $B^{T_xX}(0,\varepsilon)$
with $B^{X}(x,\varepsilon)$ for $\varepsilon \leqslant a^X$.
Let $\langle\,,\,\rangle_{L^p\otimes E}$ be the metric on
$L^p\otimes E$ induced by $h^L$ and $h^E$
and $dv_X$ be the Riemannian volume form of $(TX, g^{TX})$.
The $L^2$--scalar product on $\cali{C}} \newcommand{\cA}{\cali{A}^\infty (X,L^p\otimes E)$,
the space of smooth sections of $L^p\otimes E$, is given by
\begin{equation}\label{0c2}
\langle s_1,s_2 \rangle =\int_X\langle s_1(x),
s_2(x)\rangle_{L^p\otimes E}\,dv_X(x)\,.
\end{equation}
We denote the corresponding norm with $\norm{\,\cdot\,}_{L^2}$.
Let $\nabla^{TX}$ be the Levi--Civita connection
of the metric $g^{TX}$ and $\nabla^{L^p\otimes E}$ be the
connection on $L^p\otimes E$ induced by $\nabla^L$ and $\nabla^E$.
Let $\{e_i\}_i$ be an orthonormal frame of $TX$.
Then the Bochner-Laplacian on $L^p\otimes E$ is given by
\begin{equation}\label{0c1}
\Delta ^{L^p\otimes E} =- \sum_i \Big [ (\nabla ^{L^p\otimes E}_{e_i} )^2 -
\nabla ^{L^p\otimes E}_{\nabla ^{TX}_{e_i}{e_i}}\Big ].
\end{equation}
Let $P_{\mH_p}$ be the orthonormal projection from
$\cali{C}} \newcommand{\cA}{\cali{A} ^\infty (X,L^p\otimes E)$ onto $\mH_p$, the span of eigensections of
$\Delta_{p,\Phi}= \Delta ^{L^p\otimes E} -p \tau +\Phi$ corresponding to
eigenvalues in $[-C_L, C_L]$.
\begin{defn} \label{d3.0}
The smooth kernel of $(\Delta_{p,\Phi})^qP_{\mH_p}\,,\,q\geqslant 0$ (where
$(\Delta_{p,\Phi})^0=1$), with respect to $dv_X(x')$ is
denoted $P_{q,p}(x,x')$ and is called a {\em generalized Bergman kernel\/}
of $\Delta_{p,\Phi}$\,.
\end{defn}
The kernel $P_{q,p}(x,x')$ is a section of
$\pi^*_1(L^p\otimes E)\otimes\pi^*_2(L^p\otimes E)^*$ over $X\times X$,
where $\pi_1$ and $\pi_2$ are the projections of $X\times X$ on the first
and second factor. Using the notations of \eqref{0.4} we can write
\begin{equation}\label{0.42}
P_{q,p}(x,x')=\sum^{d_p}_{i=1}\lambda^q_{i,p}S^p_i(x)\otimes(S^p_i(x'))^*
\in(L^p\otimes E)_x\otimes(L^p\otimes E)^*_{x'} .
\end{equation}
Since $L^p_x\otimes(L^p_x)^*$ is canonically isomorphic to $\field{C}$, the
restriction of $P_{q,p}$ to the diagonal $\{(x,x):x\in X\}$ can be
identified to $B_{q,p}\in\cali{C}} \newcommand{\cA}{\cali{A}^\infty(X,E\otimes E^*)=\cali{C}} \newcommand{\cA}{\cali{A}^\infty(X,\End(E))$.
Let $f : \field{R} \to [0,1]$ be a smooth even function such that
$f(v)=1$ for $|v| \leqslant \varepsilon/2$, and $f(v) = 0$ for $|v| \geqslant \varepsilon$.
Set
\begin{align} \label{0c3}
F(a)= \Big(\int_{-\infty}^{+\infty}f(v) dv\Big)^{-1}
\int_{-\infty}^{+\infty} e ^{i v a} f(v) dv.
\end{align}
Then $F(a)$ is an even function and
lies in the Schwartz space $\mathcal{S} (\field{R})$ and $F(0)=1$.
Let $\widetilde{F}$ be the holomorphic function on
$\field{C}$ such that $\widetilde{F}(a ^2) =F(a)$.
The restriction of $\widetilde{F}$
to $\field{R}$ lies in the Schwartz space $\mS (\field{R})$. Then there exists
$\{c_j\}_{j=1}^{\infty}$ such that for any $k\in \field{N}$, the function
\begin{align} \label{0c5}
F_k(a)= \widetilde{F}(a) - \sum_{j=1}^k c_j a ^j \widetilde{F}(a),
\end{align}
verifies
\begin{align} \label{0c6}
F_k^{(i)}(0)= 0 \quad \mbox{\rm for any} \ 0< i\leqslant k.
\end{align}
\begin{prop}\label{0t3.0}
For any $k,m\in \field{N}$, there exists $C_{k,m}>0$ such that for $p\geqslant1$
\begin{align}\label{0c7}
\left|F_k\big(\tfrac{1}{\sqrt{p}}\Delta_{p,\Phi}\big)(x,x')
- P_{0,p}(x,x')\right|_{\cali{C}} \newcommand{\cA}{\cali{A} ^m(X\times X)}
\leqslant C_{k,m} p^{-\frac{k}{2} +2(2m+2n+1)}.
\end{align}
Here the $\cali{C}} \newcommand{\cA}{\cali{A} ^m$ norm is induced by $\nabla^L$ and $\nabla^E$.
\end{prop}
\begin{proof}
By \eqref{0c3}, for any $m\in \field{N}$, there exists $C'_{k,m}>0$ such that
\begin{equation}\label{c9}
\sup_{a\in \field{R}} |a|^m |F_k(a) | \leqslant C'_{k,m}.
\end{equation}
Set
\begin{align} \label{0c8}
G_{k,p}(a)= 1_{[\sqrt{p}\mu_0, +\infty[} (a) F_k(a),
\quad H_{k,p}(a)= 1_{[0, \frac{C_L}{\sqrt{p}}]} (|a|) F_k(a),
\end{align}
By (\ref{0.3}), for $p$ big enough,
\begin{equation} \label{0c9}
F_k\big(\tfrac{1}{\sqrt{p}}\Delta_{p,\Phi}\big)
= G_{k,p}\big(\tfrac{1}{\sqrt{p}}\Delta_{p,\Phi}\big)
+ H_{k,p}\big(\tfrac{1}{\sqrt{p}}\Delta_{p,\Phi}\big).
\end{equation}
As $X$ is compact, there exist $\{x_i\}_{i=1}^r$ such that
$\{U_i = B^X(x_i,\varepsilon)\}_{i=1}^r$ is a covering of $X$.
We identify $B^{T_{x_i}X}(0,\varepsilon)$ with $B^{X}(x_i,\varepsilon)$
by the exponential map as above. We identify
$ (L^p\otimes E)_Z$ for $Z\in B^{T_{x_i}X}(0,\varepsilon)$
to $ (L^p\otimes E)_{x_i}$ by parallel transport with respect
to the connection $\nabla^{L^p\otimes E}$ along the curve
$\gamma_Z: [0,1]\ni u \to \exp^X_{x_i} (uZ)$. Let $\{e_i\}_i$
be an orthonormal basis of $T_{x_i}X$. Let $\widetilde{e}_i (Z)$ be the parallel
transport of ${e}_i$ with respect to $\nabla^{TX}$ along the above curve.
Let $\Gamma ^E, \Gamma ^L$
be the corresponding connection forms of $\nabla^E$, $\nabla^L $
with respect to any fixed frame for
$E,L$ which is parallel along the curve $\gamma_Z$
under the trivialization on $U_i$.
Denote by $\nabla_U$ is the ordinary differentiation
operator on $T_{x_i}X$ in the direction $U$. Then
\begin{equation}\label{c10}
\nabla ^{L^p\otimes E}_{e_j}
= \nabla _{e_j}+ p \Gamma ^L(e_j) + \Gamma ^E(e_j).
\end{equation}
Let $\varphi_i$ be a partition function associated to $\{U_i\}$. We define
an Sobolev norm on the $l$-th Sobolev space $H^l(X,L^p\otimes E)$ by
\begin{equation}\label{c11}
\| s\| _{H^l_p}^2 = \sum_i \sum_{k=0}^l \sum_{i_1 \cdots i_k=1} ^{2n}
\|\nabla_{e_{i_1}}\cdots \nabla_{e_{i_k}}(\varphi _i s)\|_{L^2}^2
\end{equation}
Then by \eqref{0.2}, \eqref{0c1}, \eqref{c10},
there exists $C>0$ such that for $p\geqslant1$, $s\in H^2(X, L^p\otimes E)$,
\begin{equation}\label{c12}
\|s\|_{H^2_p} \leqslant C(\|\Delta_{p,\Phi} s\|_{L^2} + p^2\|s\|_{L^2}).
\end{equation}
Let $Q$ be a differential operator of order $m\in \field{N}$ with
scalar principal symbol and with compact support
in $U_i$, then by (\ref{c12}) and $[\Delta_{p,\Phi},Q]$
is a differential operator of order $m+1$, we get
\begin{equation}\label{c13}
\begin{split}
\|Qs\|_{H^2_p}
&\leqslant C(\|\Delta_{p,\Phi}Q s\|_{L^2} + p^2 \|Qs\|_{L^2})\\
&\leqslant C(\|Q \Delta_{p,\Phi} s\|_{L^2} + p^2 \|s\|_{H^{m+1}_p}
+ p^2\|Qs\|_{L^2}).
\end{split}
\end{equation}
This means
\begin{equation}\label{c17}
\|s\|_{H^{2m+2}_p}
\leqslant C_m p^{4m} \sum_{j=0}^{m} \|\Delta_{p,\Phi}^j s\|_{L^2}.
\end{equation}
Moreover for ${\bf G}_{k,p}= G_{k,p}$ or $H_{k,p}$,
$\langle \Delta_{p,\Phi}^{m'}
{\bf G}_{k,p}(\frac{1}{\sqrt{p}}\Delta_{p,\Phi})Q s,s'\rangle
=\langle s,Q^ * {\bf G}_{k,p}(\frac{1}{\sqrt{p}}\Delta_{p,\Phi})
\Delta_{p,\Phi}^{m'} s'\rangle$, so
from \eqref{0c6}, \eqref{c9}, we know that for $l,m'\in \field{N}$,
there exist $C, C'>0$ such that for $p>1$,
\begin{equation}\label{c18}
\begin{split}
&\Big\|\Delta_{p,\Phi}^{m'} G_{k,p}\big(\tfrac{1}{\sqrt{p}}\Delta_{p,\Phi}\big)Qs\Big\|_{L^2}
\leqslant C p^{-l}\|s\|_{L^2},\\
& \Big \|\Delta_{p,\Phi}^{m'}
\Big (H_{k,p}\big(\tfrac{1}{\sqrt{p}}\Delta_{p,\Phi}\big)-P_{\mH_p}\Big )Qs
\Big \|_{L^2} \leqslant C' p^{2m-\frac{k}{2}}\|s\|_{L^2}.
\end{split}
\end{equation}
We deduce from \eqref{c17} and \eqref{c18} that if $P,Q$
are differential operators with compact support in
$U_i$, $U_j$ respectively, then for any $l\in \field{N}$, there exists
$ C>0$ such that for $p>1$,
\begin{equation}\label{c19}
\begin{split}
&\Big\|P G_{k,p}\big(\tfrac{1}{\sqrt{p}}\Delta_{p,\Phi}\big) Q s\Big\|_{L^2}
\leqslant C p^{-l} \|s\|_{L^2},\\
&\Big \|P \Big (H_{k,p}\big(\tfrac{1}{\sqrt{p}}\Delta_{p,\Phi}\big)
-P_{\mH_p}\Big ) Q s\Big \|_{L^2}
\leqslant C p^{2(m+m')-\frac{k}{2}} \|s\|_{L^2}
\end{split}
\end{equation}
On $U_i\times U_j$, we use Sobolev inequality, we know for any $l\in \field{N}$,
there exists $ C>0$ such that for $p>1$,
\begin{equation}\label{c20}
\begin{split}
&\Big |G_{k,p}\big(\tfrac{1}{\sqrt{p}}\Delta_{p,\Phi}\big)(x,x')\Big|_{\cali{C}} \newcommand{\cA}{\cali{A} ^m}
\leqslant C_{l,m} p^{-l},\\
&\Big | \Big (H_{k,p}\big(\tfrac{1}{\sqrt{p}}\Delta_{p,\Phi}\big)
-P_{0,p}\Big )(x,x')\Big|_{\cali{C}} \newcommand{\cA}{\cali{A} ^m}
\leqslant C p^{2(2m+2n+1)-\frac{k}{2}}.
\end{split}
\end{equation}
By \eqref{0c9} and \eqref{c20}, we get our Proposition \ref{0t3.0}.
\end{proof}
Using \eqref{0c3}, \eqref{0c5} and the finite propagation speed
\cite[\S 7.8]{CP}, \cite[\S 4.4]{T1}, it is clear that for $x,x'\in X$,
$F_k\big(\frac{1}{\sqrt{p}}\Delta_{p,\Phi}\big)(x,\cdot)$ only depends on
the restriction of $\Delta_{p,\Phi}$ to $B^X(x,\varepsilon p^{-\frac{1}{4}})$,
and $F_k\big(\frac{1}{\sqrt{p}}\Delta_{p,\Phi}\big)(x,x')= 0$,
if $d(x, x') \geqslant\varepsilon p^{-\frac{1}{4}}$.
This means that the asymptotic of $\Delta_{p,\Phi}^q P_{\mH_p}(x,\cdot)$
when $p\to +\infty$,
modulo $\cali{O}} \newcommand{\cE}{\cali{E}(p^{-\infty})$ (i.e. terms whose $\cali{C}} \newcommand{\cA}{\cali{A}^m$ norm is
$\cali{O}} \newcommand{\cE}{\cali{E}(p^{-l})$ for any $l,m\in\field{N}$),
only depends on the restriction of $\Delta_{p,\Phi}$ to
$B^X(x,\varepsilon p^{-\frac{1}{4}})$.
\subsection{Rescaling and a Taylor expansion of the
operator $\Delta_{p,\Phi}$}\label{s3.2}
We fix $x_0\in X$. From now on, we identify $B^{T_{x_0}X}(0,\varepsilon)$
with $B^{X} (x_0,\varepsilon)$.
For $Z\in B^{T_{x_0}X}(0,\varepsilon)$ we identify $L_Z, E_Z$ and $(L^p\otimes E)_Z$
to $L_{x_0}, E_{x_0}$ and $(L^p\otimes E)_{x_0}$
by parallel transport with respect to the connections
$\nabla ^L$, $\nabla ^E$ and $\nabla^{L^p\otimes E}$ along the curve
$\gamma_Z :[0,1]\ni u \to \exp^X_{x_0} (uZ)$.
Let $\{e_i\}_i$ be an oriented orthonormal
basis of $T_{x_0}X$, and let $\{e ^i\}_i$ be its dual basis.
For $\varepsilon >0$ small enough, we will extend the geometric
objects from $B^{T_{x_0}X}(0,\varepsilon)$ to $\field{R}^{2n} \simeq T_{x_0}X$
where the identification is given by
\begin{equation}\label{0c11}
(Z_1,\cdots, Z_{2n}) \in \field{R}^{2n} \longrightarrow \sum_i
Z_i e_i\in T_{x_0}X
\end{equation}
such that $\Delta_{p,\Phi}$ is the
restriction of a renormalized Bochner-Laplacian on $\field{R}^{2n}$ associated
to a Hermitian line bundle with positive curvature. In this way,
we replace $X$ by $\field{R}^{2n}$.
At first, we denote by $L_0$, $E_0$ the trivial bundles with fiber $L_{x_0},
E_{x_0}$ on $X_0= \field{R}^{2n}$. We still denote by $\nabla ^L,
\nabla ^E$, $h^L$ etc. the connections and metrics on $L_0$,
$E_0$ on $B^{T_{x_i}X}(0,4\varepsilon)$ induced by the above
identification. Then $h^L$, $h^E$ is identified to the constant
metrics $h^{L_0}=h^{L_{x_0}}$, $h^{E_0}=h^{E_{x_0}}$.
Let $\rho: \field{R}\to [0,1]$ be a smooth even function such that
\begin{align}\label{1c14}
\rho (v)=1 \ \ {\rm if} \ \ |v|<2;
\quad \rho (v)=0 \ \ {\rm if} \ |v|>4.
\end{align}
Let $\varphi_\varepsilon : \field{R}^{2n} \to \field{R}^{2n}$ is the map defined by
$\varphi_\varepsilon(Z)= \rho(|Z|/\varepsilon) Z$.
Then $\Phi_0=\Phi\circ \varphi_\varepsilon$ is a smooth self-adjoint
section of $\End(E_0)$ on $X_0$.
Let $g^{TX_0}(Z)= g^{TX}(\varphi_\varepsilon(Z))$, $J_0(Z)= J(\varphi_\varepsilon(Z))$
be the metric and complex structure on $X_0$.
Set $\nabla ^{E_0}= \varphi_\varepsilon ^* \nabla ^{E}$. Then $\nabla ^{E_0}$
is the extension of $ \nabla ^{E}$ on $B^{T_{x_0}X}(0,\varepsilon)$.
If $\mR =
\sum_i Z_i e_i=Z$ denotes radial vector field on $\field{R}^{2n}$,
we define the Hermitian connection $\nabla ^{L_0}$ on $(L_0, h^{L_0})$ by
\begin{align}\label{1c15}
&\nabla ^{L_0}|_Z = \varphi_\varepsilon ^* \nabla ^{L} +\frac{1}{2}
(1-\rho ^2(|Z|/\varepsilon) ) R^{L}_{x_0} (\mR,\cdot).
\end{align}
Then we calculate easily that its curvature $R^{L_0}= (\nabla ^{L_0})^2$ is
\begin{equation}\label{1c16}
\begin{split}
R^{L_0}(Z) &= \varphi_\varepsilon ^* R^L + \frac{1}{2}d \Big((1-\rho ^2(|Z|/\varepsilon))
R^{L}_{x_0} (\mR,\cdot)\Big)\\
&= \Big(1-\rho ^2(|Z|/\varepsilon)\Big) R^{L}_{x_0}
+ \rho ^2(|Z|/\varepsilon) R^L_{\varphi_\varepsilon(Z)}\\
&- (\rho \rho')(|Z|/\varepsilon) \frac{Z_i e ^i}{\varepsilon |Z|}\wedge \big[
R^{L}_{x_0} (\mR,\cdot)- R^{L}_{\varphi_\varepsilon(Z)} (\mR,\cdot)\big].
\end{split}
\end{equation}
Thus $R^{L_0}$ is positive in the sense of (\ref{0.2}) for $\varepsilon$ small enough,
and the corresponding constant $\mu_0$ for $R^{L_0}$ is bigger
than $\frac{4}{5}\mu_0$. From now on, we fix $\varepsilon$ as above.
Let $\Delta_{p,\Phi_0}^{X_0}=\Delta^{L^p_0\otimes E_0}-p\tau_0-\Phi_0$
be the renormalized Bochner-Laplacian on $X_0$
associated to the above data, as in \eqref{laplace}. Observe that
$R^{L_0}$ is uniformly positive on $\field{R}^{2n}$,
so by the relations (3.2), (3.11) and (3.12) in \cite[p. 656\,--\,658]{MM}, we know that \eqref{0.3}
still holds for $\Delta_{p,\Phi_0}^{X_0}$.
Especially, there exists $C_{L_0}>0$ such that
\begin{align}\label{1c17}
&\spec \Delta_{p,\Phi_0}^{X_0}
\subset [-C_{L_0},C_{L_0}]\cup [\frac{8}{5} p\mu_0 -C_{L_0},+\infty[\,.
\end{align}
We note that $\Delta_{p,\Phi_0}^{X_0}$ has not necessarily discrete spectrum.
Let $S_L$ be an unit vector of $L_{x_0}$. Using $S_L$ and the above
discussion, we get an isometry $E_0\otimes L_0^p \simeq E_{x_0}$.
Let $P_{0,\mH_p}$ be the spectral projection of $\Delta_{p,\Phi_0}^{X_0}$
from $\cali{C}} \newcommand{\cA}{\cali{A}^\infty (X_0,L^p_0\otimes E_0)\simeq \cali{C}} \newcommand{\cA}{\cali{A}^\infty (X_0,E_{x_0})$
corresponding to the interval
$[-C_{L_0},C_{L_0}]$, and let $P_{0,q,p}(x,x')$
$(q\geqslant0)$ be the smooth kernels of
$P_{0,q,p}=(\Delta_{p,\Phi_0}^{X_0})^q P_{0,\mH_p}$
(we set $(\Delta_{p,\Phi_0}^{X_0})^0=1$)
with respect to the volume form $dv_{X_0}(x')$.
The following Proposition shows that $P_{q,p}$ and
$P_{0,q,p}$ are asymptotically close on $B^{T_{x_0}X}(0,\varepsilon)$ in the $\cali{C}} \newcommand{\cA}{\cali{A}^\infty$--\,topology, as $p\to\infty$.
\begin{prop} \label{p3.2} For any $l,m\in \field{N}$, there exists $C_{l,m}>0$
such that for $x,x' \in B^{T_{x_0}X}(0,\varepsilon)$,
\begin{equation}\label{1c19}
\left |(P_{0,q,p} -P_{q,p})(x,x')\right |_{\cali{C}} \newcommand{\cA}{\cali{A} ^m}
\leqslant C_{l,m} p^{-l}.
\end{equation}
\end{prop}
\begin{proof} Using \eqref{0c3} and \eqref{1c17}, we know that
for $x,x' \in B^{T_{x_0}X}(0,\varepsilon)$,
\begin{align}\label{1c20}
\left|F_k\big(\tfrac{1}{\sqrt{p}}\Delta_{p,\Phi}\big)(x,x')
- P_{0,0,p}(x,x')\right|_{\cali{C}} \newcommand{\cA}{\cali{A} ^m} \leqslant C_{k,m} p^{-\frac{k}{2} +2(m+n+1)}.
\end{align}
Thus from \eqref{0c7} and \eqref{1c20} for $k$ big enough,
we infer \eqref{1c19} for $q=0$; Now from the definition of
$P_{0,q,p}$ and $P_{q,p}$, we get \eqref{1c19} from \eqref{c10}
and \eqref{1c19} for $q=0$.
\end{proof}
It suffices therefore to study the kernel $P_{0,q,p}$ and for this purpose we
rescale the operator $\Delta_{p,\Phi_0}^{X_0}$.
Let $dv_{TX}$ be the Riemannian volume form of
$(T_{x_0}X, g^{T_{x_0}X})$.
Let $\kappa (Z)$ be the smooth positive function defined by the equation
\begin{equation}\label{c22}
dv_{X_0}(Z) = \kappa (Z) dv_{TX}(Z),
\end{equation}
with $\kappa(0)=1$.
Denote by $\nabla_U$ the ordinary differentiation
operator on $T_{x_0}X$ in the direction $U$,
and set $\partial_i=\nabla _{e_i}$.
If $\alpha = (\alpha_1,\cdots, \alpha_{2n})$ is a multi-index,
set $Z^\alpha = Z_1^{\alpha_1}\cdots Z_{2n}^{\alpha_{2n}}$.
We also denote by
$(\partial ^\alpha R^L)_{x_0}$ the tensor $(\partial^\alpha R^L)_{x_0}(e_i,e_j)
=\partial ^\alpha( R^L(e_i,e_j))_{x_0}$.
Denote by $t=\frac{1}{\sqrt{p}}$.
For $s \in \cali{C}} \newcommand{\cA}{\cali{A} ^{\infty}(\field{R}^{2n}, E_{x_0})$ and $Z\in \field{R}^{2n}$, set
\begin{equation}\label{c27}
\begin{split}
(S_{t} s ) (Z) = & s (Z/t),
\quad \nabla_{t} = tS_t^{-1}\kappa ^{\frac{1}{2}}
\nabla ^{L^p_0\otimes E_0}\kappa ^{-\frac{1}{2}}S_t,\\
& \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t= S_t^{-1} \,\tfrac{1}{p}\,\kappa ^{\frac{1}{2}}\,
\Delta_{p,\Phi_0}^{X_0} \,\kappa ^{-\frac{1}{2}}\,S_t.
\end{split}
\end{equation}
The operator $\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t$ is the rescaled operator, which we now develop in Taylor series.
\begin{thm}\label{t3.3} There exist polynomials $\mA_{i,j,r}$
{\rm(} resp. $\mB_{i,r}$, $\mC_{r}${\rm)}
{\rm(}$r\in \field{N}, i,j\in \{1,\cdots, 2n\}${\rm)}
in $Z$ with the following properties:
-- their coefficients are polynomials in
$R^{TX}$ {\rm(}resp. $R^{TX}$, $R^L$, $R^{E}$, $\Phi${\rm)}
and their derivatives at $x_0$ up to order $r-2$ {\rm(}resp. $r-1$, $r$,
$r-1$, $r${\rm)}\,,
-- $\mA_{i,j,r}$ is a monomial in $Z$ of degree $r$,
the degree in $Z$ of $\mB_{i,r}$ {\rm(}resp. $\mC_{r}${\rm)} has the same parity
with $r-1$ {\rm(}resp. $r${\rm)}\,,
-- if we denote by
\begin{align}\label{0c35}
\mO_{r} = \mA_{i,j,r}\nabla_{e_i}\nabla_{e_j}
+ \mB_{i,r}\nabla_{e_i}+ \mC_{r},
\end{align}
then
\begin{align}\label{c30}
\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t= \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0+ \sum_{r=1}^m t^r \mO_r + \cali{O}} \newcommand{\cE}{\cali{E}(t^{m+1}).
\end{align}
and there exists $m'\in \field{N}$ such that for any $k\in \field{N}$, $t\leqslant 1$
the derivatives of order $\leqslant k$ of the coefficients of the operator
$\cali{O}} \newcommand{\cE}{\cali{E}(t^{m+1})$ are dominated by $C t^{m+1} (1+|Z|)^{m'}$. Moreover
\begin{align}\label{c31}
&\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0 = -\sum_j \Big(\nabla_{e_j}+\frac{1}{2} R^L_{x_0}(Z, e_j) \Big)^2
-\tau_{x_0},\\
&\mO_1(Z)= -\frac{2}{3} ( \partial_j R^L)_{x_0} (\mR,e_i)Z_j
\Big(\nabla _{e_i}+ \frac{1}{2} R^L_{x_0}(\mR, e_i)\Big)
-\frac{1}{3} (\partial_i R^L)_{x_0} (\mR,
e_i)
- (\nabla _\mR \tau)_{x_0},\nonumber\\
&\mO_2(Z)= \frac{1}{3} \left \langle R^{TX}_{x_0} (\mR,e_i)
\mR, e_j\right \rangle_{x_0} \Big( \nabla_{e_i} + \frac{1}{2}
R^L_{x_0}(\mR,e_i)\Big)\Big( \nabla_{e_j}
+ \frac{1}{2} R^L_{x_0}(\mR,e_j)\Big)\nonumber\\
&\hspace*{3mm} +\Big [\frac{2}{3} \left \langle R^{TX}_{x_0}
(\mR, e_j) e_j,e_i\right \rangle_{x_0}
- \Big(\frac{1}{2}\sum_{|\alpha|=2}(\partial ^{\alpha}R^L)_{x_0}
\frac{Z^\alpha}{\alpha !} + R^E_{x_0} \Big)(\mR,e_i)\Big ]
\Big(\nabla_{e_i} + \frac{1}{2} R^L_{x_0}(\mR,e_i)\Big)\nonumber\\
&\hspace*{3mm} -\frac{1}{4} \nabla_{e_i}\Big(\sum_{|\alpha|=2}
(\partial ^{\alpha}R^L)_{x_0} \frac{Z^\alpha}{\alpha !}(\mR,e_i)\Big)
-\frac{1}{9}\sum_i \Big[\sum_j (\partial_j R^L)_{x_0} (\mR,e_i)Z_j\Big]^2
\nonumber\\
&\hspace*{3mm}-\frac{1}{12}\Big[\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0, \left \langle R^{TX}_{x_0} (\mR,e_i)
\mR, e_i\right \rangle_{x_0} \Big]
-\sum_{|\alpha|=2}(\partial ^{\alpha}\tau)_{x_0}
\frac{Z^\alpha}{\alpha !}+ \Phi_{x_0} .\nonumber
\end{align}
\end{thm}
\begin{proof}
Set $g_{ij}(Z)= g^{TX}(e_i,e_j)(Z) = \langle e_i,e_j\rangle_Z$
and let $(g^{ij}(Z))$ be the inverse of
the matrix $(g_{ij}(Z))$.
By \cite[Proposition 1.28]{BeGeV}, the Taylor expansion of $g_{ij}(Z)$
with respect to the basis $\{e_i\}$ to order $r$ is a polynomial of
the Taylor expansion of $R^{TX}$
to order $r-2$, moreover
\begin{equation}\label{0c30}
\begin{split}
&g_{ij}(Z) =
\delta_{ij} + \frac{1}{3}
\left \langle R^{TX}_{x_0} (\mR,e_i) \mR, e_j\right \rangle_{x_0}
+ \cali{O}} \newcommand{\cE}{\cali{E} (|Z|^3),\\
&\kappa(Z)= |\det (g_{ij}(Z))|^{1/2} = 1 +
\frac{1}{6} \left \langle R^{TX}_{x_0} (\mR,e_i) \mR, e_i\right \rangle_{x_0}
+ \cali{O}} \newcommand{\cE}{\cali{E} (|Z|^3).
\end{split}
\end{equation}
If $\Gamma _{ij}^l$ is the connection form of $\nabla ^{TX}$
with respect to the basis $\{e_i\}$, we have $(\nabla
^{TX}_{e_i}e_j)(Z) = \Gamma _{ij}^l (Z) e_l$. Owing to \eqref{0c30},
\begin{equation}\label{0c31}
\begin{split}
\Gamma _{ij}^l (Z)& = \frac{1}{2} g^{lk} (\partial_i g_{jk}
+ \partial_j g_{ik}-\partial_k g_{ij})(Z)\\
&= \frac{1}{3}\Big [
\left \langle R^{TX}_{x_0} (\mR, e_j) e_i, e_l\right \rangle _{x_0}
+ \left \langle R^{TX}_{x_0} (\mR, e_i) e_j, e_l\right \rangle_{x_0}\Big ]
+ \cali{O}} \newcommand{\cE}{\cali{E}(|Z|^2).
\end{split}
\end{equation}
Now by \eqref{0c1},
\begin{align}\label{c32}
\Delta_{p,\Phi} = - g^{ij} ( \nabla ^{L^p\otimes E}_{e_i}
\nabla ^{L^p\otimes E}_{e_j}- \nabla ^{L^p\otimes E}_{\nabla ^{TX}_{e_i}e_j} )
-p \tau +\Phi.
\end{align}
so from \eqref{c27} and \eqref{c32} we infer the expression
\begin{align}\label{0c37}
\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t=-g^{ij} (tZ) \Big [ \nabla_{t, e_i}\nabla_{t, e_j} - t
\Gamma _{ij}^l \nabla_{t, e_l} \Big ](tZ) - \tau (tZ)
+t^2\Phi(tZ).
\end{align}
Let $\Gamma ^E$, $\Gamma ^L$
be the connection forms of $\nabla^E$ and $\nabla^L$
with respect to any fixed frames for $E$, $L$
which are parallel along the curve $\gamma_Z$
under our trivializations on $B^{T_{x_0}X}(0,\varepsilon)$.
\eqref{c27} yields on $B^{T_{x_0}X}(0, \varepsilon/t)$
\begin{align}\label{0c36}
&\nabla_{t, e_i}|_{Z} = \kappa ^{\frac{1}{2}}(tZ)\Big(\nabla_{e_i}
+ \frac{1}{t} \Gamma ^L (e_i)(tZ) + t \Gamma ^E (e_i) (tZ)\Big)
\kappa ^{-\frac{1}{2}}(tZ).
\end{align}
Let $\Gamma ^\bullet= \Gamma ^E, \Gamma ^L$ and
$R^\bullet= R^E, R^L$, respectively.
By \cite[Proposition 1.18]{BeGeV}
the Taylor coefficients of $\Gamma ^\bullet (e_j) (Z)$ at $x_0$
to order $r$ are only determined by those of $R^\bullet$ to order $r-1$, and
\begin{align}\label{0c39}
\sum_{|\alpha|=r} (\partial^\alpha
\Gamma ^\bullet ) _{x_0} (e_j) \frac{Z^\alpha}{\alpha !}
=\frac{1}{r+1} \sum_{|\alpha|=r-1}
(\partial^\alpha R^\bullet ) _{x_0}(\mR, e_j)
\frac{Z^\alpha}{\alpha !}.
\end{align}
Owing to \eqref{0c30}, \eqref{0c39}
\begin{align}\label{0c41}
& \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t = - \Big ( \delta_{ij} - \frac{t^2}{3}
\left \langle R^{TX}_{x_0} (\mR,e_i) \mR, e_j\right \rangle
+ \cali{O}} \newcommand{\cE}{\cali{E} (t^3)\Big ) \kappa ^{\frac{1}{2}}(tZ) \Big \{\\
&\Big [\nabla_{e_i}+ \Big(\frac{1}{2} R^L_{x_0}
+ \frac{t}{3} (\partial_k R^L)_{x_0} Z_k +\frac{t^2}{4}\sum_{|\alpha|=2}
(\partial ^{\alpha}R^L)_{x_0} \frac{Z^\alpha}{\alpha !}
+ \frac{t^2}{2} R^E_{x_0}\Big ) (\mR,e_i)+ \cali{O}} \newcommand{\cE}{\cali{E} (t^3)\Big] \nonumber \\
&\Big [\nabla_{e_j}+\Big (\frac{1}{2} R^L_{x_0} + \frac{t}{3}
(\partial_k R^L)_{x_0} Z_k +\frac{t^2}{4}\sum_{|\alpha|=2}
(\partial ^{\alpha}R^L)_{x_0} \frac{Z^\alpha}{\alpha !}
+ \frac{t^2}{2} R^E _{x_0} \Big )(\mR,e_j) + \cali{O}} \newcommand{\cE}{\cali{E} (t^3)\Big] \nonumber \\
&\left.- t \Gamma ^l_{ij}(tZ) \Big (\nabla_{e_l}+
\frac{1}{2} R^L_{x_0}(\mR,e_l)
+ \cali{O}} \newcommand{\cE}{\cali{E} (t)\Big ) \right \} \kappa ^{-\frac{1}{2}}(tZ) \nonumber \\
&-\tau_{x_0} -t (\nabla_{\mR} \tau)_{x_0}
-t^2\sum_{|\alpha|=2}(\partial ^{\alpha}\tau)_{x_0}
\frac{Z^\alpha}{\alpha !}+ t^2 \Phi_{x_0} + \cali{O}} \newcommand{\cE}{\cali{E} (t^3). \nonumber
\end{align}
Relations \eqref{0c30} and \eqref{0c37}\,--\,\eqref{0c41} settle
our Theorem.
\end{proof}
\subsection{Uniform estimate of the generalized Bergman kernels}
\label{s3.3}
We shall estimate the Sobolev norm of the resolvent of $\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t$ so we introduce
the following norms.
We denote by $\left \langle\,\cdot\,,\cdot\,\right\rangle_{0,L^2}$ and $\norm{\,\cdot\,}_{0,L^2}$
the scalar product and the $L^2$ norm on $\cali{C}} \newcommand{\cA}{\cali{A} ^\infty (X_0, E_{x_0})$
induced by $g^{TX_0}, h^{E_0}$ as in \eqref{0c2}.
For $s\in \cali{C}} \newcommand{\cA}{\cali{A} ^{\infty}(X_0, E_{x_0}) $, set
\begin{align}\label{u0}
&\|s\|_{t,0}^2= \|s\|_{0}^2
= \int_{\field{R}^{2n}} |s(Z)|^2_{h^{E_{x_0}}}dv_{TX}(Z),\\
&\| s \|_{t,m}^2 = \sum_{l=0}^m \sum_{i_1,\cdots, i_l=1}^{2n}
\| \nabla_{t,e_{i_1}} \cdots \nabla_{t,e_{i_l}} s\|_{t,0}^2. \nonumber
\end{align}
We denote by $\left \langle s ', s \right\rangle_{t,0}$
the inner product on $\cali{C}} \newcommand{\cA}{\cali{A}^\infty (X_0, E_{x_0})$
corresponding to $\norm{\,\cdot\,}^2_{t,0}$\,.
Let $H^m_t$ be the Sobolev space of order $m$ with norm $\norm{\,\cdot\,}_{t,m}$.
Let $H^{-1}_t$ be the Sobolev space of order $-1$ and
let $\norm{\,\cdot\,}_{t,-1}$ be the norm on $H^{-1}_t$ defined by
$\|s\|_{t,-1} = \sup_{0\neq s'\in H^1_t }$
$|\left \langle s,s'\right \rangle_{t,0}|/\|s'\|_{t,1}$.
If $A\in \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R} (H^{m}, H^{m'})$ $(m,m' \in \field{Z})$, we denote by
$\norm{A}^{m,m'}_t$ the norm of $A$ with respect to the norms
$\norm{\,\cdot\,}_{t,m}$ and $\norm{\,\cdot\,}_{t,m'}$.
\begin{rem}\label{0t3.3} Note that $\Delta_{p,\Phi_0}^{X_0} $ is
self-adjoint with respect to $\norm{\,\cdot\,}_{0}$, thus
by \eqref{c22}, \eqref{c27}, \eqref{u0},
$\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t$ is a formal self adjoint elliptic operator with respect to
$\|\quad\|_{0}$,
and is a smooth family of operators with the parameter $x_0\in X$.
Thus $\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0$ and $\mO_r$ are also formal self-adjoint with respect to
$\norm{\,\cdot\,}_{0}$. This will simplify the computation of the coefficients
$b_{0,1}$ in (\ref{0.7}) (cf. \S \ref{s4.3})
and explains why we prefer to conjugate with $\kappa ^{1/2}$
comparing to \cite[(3.38)]{DLM}.
\end{rem}
\begin{thm} \label{tu1} There exist constants $ C_1, C_2, C_3>0$
such that for $t\in ]0,1]$ and any
$s,s'\in \cali{C}} \newcommand{\cA}{\cali{A} ^{\infty}_0(\field{R}^{2n}, E_{x_0})$,
\begin{equation}\label{u1}
\begin{split}
& \left \langle \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t s,s\right \rangle_{t,0}
\geqslant C_1\|s\|_{t,1}^2 -C_2 \|s\|_{t,0}^2 , \\
& |\left \langle \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t s, s'\right \rangle_{t,0}|
\leqslant C_3 \|s\|_{t,1}\|s'\|_{t,1}.
\end{split}
\end{equation}
\end{thm}
\begin{proof}
Relations \eqref{laplace} and \eqref{0c1} yield
\begin{align}\label{1u1}
\left \langle \Delta_{p,\Phi}s,s\right \rangle_{0,L^2} =
\|\nabla ^{ L^p_0\otimes E_0}s\|_{0,L^2}^2 - \left \langle \left (
p\tau-\Phi\right )s,s\right \rangle_{0,L^2}.
\end{align}
Thus from (\ref{c27}), (\ref{u0}) and (\ref{1u1}) we get
\begin{align}\label{1u2}
\left \langle \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t s,s\right \rangle_{t,0} = \|\nabla_ts\|_{t,0}^2
- \left \langle\left (
S_t^{-1}\tau -t^2\Phi\right )s,s\right \rangle_{t,0}.
\end{align}
which implies \eqref{u1}.
\end{proof}
Let $\delta$ be the counterclockwise oriented circle in $\field{C}$
of center $0$ and radius $\mu_0/4$.
\begin{thm}\label{tu4} The resolvent $(\lambda- \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1}$ exists for
$\lambda \in \delta$, and there exists $C>0$ such that for $t\in ]0,1]$,
$\lambda \in \delta$, and $x_0\in X$,
\begin{align}\label{ue2}
& \| (\lambda- \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1}\|^{0,0}_t \leqslant C,
& \| (\lambda- \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1}\|^{-1,1}_t
\leqslant C .
\end{align}
\end{thm}
\begin{proof} By \eqref{1c17}, \eqref{c27}, for $t$ small enough,
\begin{align}\label{1u3}
\spec \, \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t\subset
\big[-C_{L_0} t^2, C_{L_0} t^2\big] \cup \Big[\, \,\mu_0,+\infty\Big[.
\end{align}
Thus the resolvent $(\lambda- \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1}$ exists for $\lambda \in \delta$,
and we get the first inequality of (\ref{ue2}).
By \eqref{u1}
$(\lambda_0- \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1}$ exists for $\lambda_0\in\field{R}$,
$\lambda_0\leqslant -2C_2$, and
$\|(\lambda_0- \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1}\|^{-1,1}_t \leqslant \frac{1}{C_1}$. Now,
\begin{equation}\label{ue7}
(\lambda- \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1}= (\lambda_0- \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1}
- (\lambda-\lambda_0) (\lambda- \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1}(\lambda_0- \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1}.
\end{equation}
Thus for $\lambda\in \delta$, from (\ref{ue7}), we get
\begin{equation}\label{ue8}
\|(\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1}\|^{-1,0}_t \leq
\frac{1}{C_1} \left(1+\frac{4}{\mu_0}|\lambda-\lambda_0|\right).
\end{equation}
Changing the last two factors in \eqref{ue7} and applying \eqref{ue8}
we get
\begin{align}\label{ue9}
\|(\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1}\|^{-1,1}_t \leqslant \frac{1}{C_1}+
\frac{|\lambda-\lambda_0|}{{C_1}^2}
\left(1+\frac{4}{\mu_0}|\lambda-\lambda_0|\right)\leqslant C.
\end{align}
The proof of our Theorem is complete.
\end{proof}
\begin{prop} \label{tu5} Take $m \in \field{N}^*$. There exists $C_m>0$ such that
for $t\in ]0,1]$, $Q_1, \cdots, Q_m$
$\in \{ \nabla_{t,e_i}, Z_i\}_{i=1}^{2n}$ and $s,s'\in C
^{\infty}_{0}(X_0, E_{x_0})$, \begin{equation}\label{ue11} \left |\left \langle
[Q_1, [Q_2,\ldots, [Q_m, \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t]] \ldots ]s, s'\right
\rangle_{t,0} \right | \leqslant C_m \|s\|_{t,1} \|s'\|_{t,1}. \end{equation}
\end{prop}
\begin{proof}
Note that $[\nabla_{t,e_i}, Z_j]=\delta_{ij}$, hence
\eqref{0c37} implies that $[Z_j, \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t]$ verifies (\ref{ue11}).
On the other hand, we obtain from \eqref{c27}
\begin{align}\label{1ue2}
[\nabla_{t,e_i},\nabla_{t,e_j}]= \left(R^{L_0}(tZ)
+ t^2 R^{E_0}(tZ)\right)(e_i,e_j).
\end{align}
Thus from (\ref{0c37}) and (\ref{1ue2}),
we know that $[\nabla_{t,e_k}, \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t]$
has the same structure as $\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t$ for $t \in ]0,1]$,
i.e. $[\nabla_{t,e_k}, \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t]$ has the same type as
\begin{align}\label{1ue3}
\sum_{ij} a_{ij}(t,tZ) \nabla_{t,e_i}\nabla_{t,e_j}
+\sum_{i}b_{i}(t,tZ) \nabla_{t,e_i} + c(t,tZ),
\end{align}
and $a_{ij}(t,Z),b_{i}(t,Z), c(t,Z)$ and their derivatives in $Z$
are uniformly bounded for $Z\in \field{R} ^{2n}, t\in [0,1]$.
Moreover they are polynomials in $t$.
If $(\nabla_{t,e_i})^*$ is the adjoint of $\nabla_{t,e_i}$ with respect to
$\left \langle \,\cdot\, ,\,\cdot\,\right \rangle_{t,0}$\,, \eqref{u0} yields
\begin{align}\label{1ue4}
(\nabla_{t,e_i})^* =- \nabla_{t,e_i}- t (\kappa ^{-1}(e_i \kappa))(tZ).
\end{align}
Thus by (\ref{1ue3}) and (\ref{1ue4}), (\ref{ue11}) is verified for $m=1$.
By recurrence, it transpires that
$[Q_1,[Q_2,\ldots, [Q_m, \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t]] \ldots ]$
has the same structure \eqref{1ue3} as $\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t$\,, so from \eqref{1ue4}
we get the required assertion
\end{proof}
\begin{thm}\label{tu6} For any $t\in ]0,1]$, $\lambda \in \delta$,
$m \in \field{N}$, the resolvent $(\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1}$ maps $H^m_t$
into $H^{m+1}_t$. Moreover for any $\alpha\in \field{Z}^{2n}$,
there exists $C_{\alpha, m}>0$
such that for $t\in ]0,1]$, $\lambda \in \delta$,
$s\in \cali{C}} \newcommand{\cA}{\cali{A} ^\infty (X_0,E_{x_0})$,
\begin{equation}\label{ue12}
\|Z^\alpha (\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1} s\|_{t,m+1}
\leqslant C_{\alpha, m} \sum_{\alpha' \leqslant \alpha}
\|Z^{\alpha'} s\|_{t,m} .
\end{equation}
\end{thm}
\begin{proof} For $Q_1, \cdots, Q_m\in \{\nabla_{t,e_i}\}_{i=1}^{2n}$,
$Q_{m+1},\cdots, Q_{m+|\alpha|}\in\{Z_i\}_{i=1}^{2n}$, we can
express $Q_1\cdots$ $Q_{m+|\alpha|}(\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1}$ as a
linear combination of operators of the type \begin{equation}\label{ue13} [Q_1 [
Q_2,\ldots [Q_{m'},(\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1}]]\ldots ]Q_{m'+1} \cdots
Q_{m+|\alpha|}, \quad m'\leqslant m+|\alpha|. \end{equation} Let $\cali{R}} \newcommand{\Ha}{\cali{H}_{t}$ be the
family of operators $\cali{R}} \newcommand{\Ha}{\cali{H}_{t} = \{ [ Q_{j_1}[Q_{j_2},\ldots
[Q_{j_l},\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t]]\ldots ] \}$. Clearly, any commutator $[Q_1 [
Q_2,\ldots [Q_{m'},(\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1}]]\ldots ]$ is a linear
combination of operators of the form \begin{equation}\label{ue14}
(\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1}R_1(\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1}R_2 \cdots
R_{m'}(\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1} \end{equation} with $R_1, \cdots, R_{m'} \in
\cali{R}} \newcommand{\Ha}{\cali{H}_{t}$.
From Proposition \ref{tu5} we deduce that the norm $\norm{\,\cdot\,}_t^{1,-1}$ of the
operators $R_j\in \cali{R}} \newcommand{\Ha}{\cali{H}_{t}$ is
uniformly bounded by $C$.
By Theorem \ref{tu4} there exists $C>0$,
such that the norm $\norm{\,\cdot\,}_t^{0,1}$
of operators \eqref{ue14} is dominated by
$C$.
\end{proof}
The next step is to convert the estimates for the resolvent into estimates for
the spectral projection
${\mathcal{P}}}\def\mJ{{\mathcal{J}}}\def\bE{{\mathbf E}_{0,t}:(\cali{C}} \newcommand{\cA}{\cali{A}^{\infty}(X_0,E_{x_0}), \norm{\,\cdot\,}_0)\to
(\cali{C}} \newcommand{\cA}{\cali{A}^{\infty}(X_0,E_{x_0}), \norm{\,\cdot\,}_0)$
of $\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t$ corresponding to the interval $[-C_{L_0} t^2, C_{L_0}t^2]$.
Let ${\mathcal{P}}}\def\mJ{{\mathcal{J}}}\def\bE{{\mathbf E}_{q,t}(Z,Z')={\mathcal{P}}}\def\mJ{{\mathcal{J}}}\def\bE{{\mathbf E}_{q,t,x_0}(Z,Z')$, (with $Z,Z'\in X_0$, $q\geqslant0$) be the
smooth kernel of ${\mathcal{P}}}\def\mJ{{\mathcal{J}}}\def\bE{{\mathbf E}_{q,t}=(\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^q {\mathcal{P}}}\def\mJ{{\mathcal{J}}}\def\bE{{\mathbf E}_{0,t}$
(we set $(\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^0=1$) with respect to $dv_{TX}(Z')$. Note that
$\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t$ is a family of differential operators on $T_{x_0}X$ with coefficients
in $\End (E)_{x_0}$. Let $\pi : TX\times_{X} TX \to X$ be the
natural projection from the fiberwise product of $TX$ on $X$. Then
we can view ${\mathcal{P}}}\def\mJ{{\mathcal{J}}}\def\bE{{\mathbf E}_{q,t}(Z,Z')$ as a smooth section of $\pi ^* (\End
(E))$ over $TX\times_{X} TX$
by identifying a section $S\in \cali{C}} \newcommand{\cA}{\cali{A}^\infty (TX\times_{X}TX,\pi ^* \End (E))$
with the family $(S_x)_{x\in X}$, where
$S_x=S|_{\pi^{-1}(x)}$.
Let $\nabla ^{\End (E)}$ be the connection on $\End (E)$ induced by
$\nabla ^E$. Then $\nabla^{\pi^*\End (E)}$
induces naturally a $\cali{C}} \newcommand{\cA}{\cali{A} ^m$--\,norm of $S$ for the parameter $x_0\in X$.
\begin{thm}\label{tue8}
For any $m,m'\in\field{N}$, $\sigma>0$, there exists $C>0$,
such that for $t\in ]0,1]$, $Z,Z'\in T_{x_0}X$, $|Z|,|Z'|\leqslant \sigma$,
\begin{align}\label{ue15}
&\sup_{|\alpha|,|\alpha'|, r\leqslant m} \Big
|\frac{\partial^{|\alpha|+|\alpha'|}} {\partial Z^{\alpha}
{\partial Z'}^{\alpha'}} \frac{\partial^{r}}{\partial t^{r}}
{\mathcal{P}}}\def\mJ{{\mathcal{J}}}\def\bE{{\mathbf E}_{q,t}\left (Z, Z'\right )\Big |_{\cali{C}} \newcommand{\cA}{\cali{A} ^{m'}(X)} \leqslant C.
\end{align}
Here $\cali{C}} \newcommand{\cA}{\cali{A} ^{m'}(X)$ is the $\cali{C}} \newcommand{\cA}{\cali{A} ^{m'}$ norm for the parameter $x_0\in X$.
\end{thm}
\begin{proof} By (\ref{1u3}), for any $k\in \field{N}^*$, $q\geqslant0$,
\begin{align}\label{1ue15}
&{\mathcal{P}}}\def\mJ{{\mathcal{J}}}\def\bE{{\mathbf E}_{q,t}=(\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^q {\mathcal{P}}}\def\mJ{{\mathcal{J}}}\def\bE{{\mathbf E}_{0,t}= \frac{1}{2\pi i} \begin{pmatrix}
q+k-1 \\ k-1 \end{pmatrix} ^{-1} \int_{\delta} \lambda ^{q+k-1}
(\lambda - \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-k} d \lambda.
\end{align}
For $m\in \field{N}$, let $\mQ ^m$ be the set of operators
$\{\nabla_{t,e_{i_1}}\cdots \nabla_{t,e_{i_j}}\}_{j\leqslant m}$.
From Theorem \ref{tu6}, we deduce that
if $Q\in \mQ ^{m}$, there is $C_m>0$ such that
\begin{equation}\label{ue18}
\| Q (\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-m}\|_t^{0,0}
\leqslant C_m\,,\quad\text{for all $\lambda\in\delta$ }\, .
\end{equation}
Observe that $\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t$ is self--adjoint with respect to $\norm{\,\cdot\,}_{t,0}$,
so after taking the adjoint of (\ref{ue18}), we have
\begin{equation}\label{ue20}
\| (\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-m}Q\|_t^{0,0}
\leqslant C_m\,.
\end{equation}
From (\ref{1ue15}), (\ref{ue18}) and (\ref{ue20}), we obtain
\begin{align}\label{ue21}
&\|Q {\mathcal{P}}}\def\mJ{{\mathcal{J}}}\def\bE{{\mathbf E}_{q,t}Q' \|^{0,0}_t \leqslant C_{m}\,,\quad\text{for $Q,Q' \in \mQ ^{m}$}\,.
\end{align}
Let $|\,\cdot\,|_{(\sigma),m}$ be the usual Sobolev norm on
$\cali{C}} \newcommand{\cA}{\cali{A}^\infty(B^{T_{x_0}X}(0,\sigma+1),
E_{x_0})$ induced by $h^{E_{x_0}}$ and
the volume form $dv_{TX}(Z)$ as in (\ref{u0}).
Let $\|A\|_{(\sigma),m}$ be the operator norm of $A$
with respect to $|\,\cdot\,|_{(\sigma),m}$. Observe that by
(\ref{0c36}), (\ref{u0}), for $m>0$,
there exists $C_\sigma>0$ such that for $s\in
\cali{C}} \newcommand{\cA}{\cali{A}^\infty (X_0, E_{x_0})$, $\supp s \subset B^{T_{x_0}X}(0,\sigma +1)$,
\begin{align}\label{1ue21}
&\frac{1}{C_\sigma} \|s\|_{t,m}\leqslant |s|_{(\sigma),m}
\leqslant C_\sigma \|s\|_{t,m}.
\end{align}
Now \eqref{ue21} and \eqref{1ue21} together with Sobolev's inequalities
imply
\begin{equation}\label{ue23}
\sup_{|Z|,|Z'|\leqslant \sigma}| Q_Z Q'_{Z'} {\mathcal{P}}}\def\mJ{{\mathcal{J}}}\def\bE{{\mathbf E}_{q,t}(Z,Z') | \leqslant C\,,\quad\text{for $Q,Q' \in \mQ ^{m}$}\,.
\end{equation}
Thanks to \eqref{0c36} and \eqref{ue23} estimate \eqref{ue15} holds for $r=m'=0$.
\noindent
To obtain (\ref{ue15}) for $r \geqslant1$ and $m'=0$, note that from (\ref{1ue15}),
\begin{align}\label{ue28}
\frac{\partial^{r}}{\partial t^{r}} {\mathcal{P}}}\def\mJ{{\mathcal{J}}}\def\bE{{\mathbf E}_{q,t}=& \frac{1}{2\pi i}
\begin{pmatrix} q+k-1 \\ k-1 \end{pmatrix} ^{-1} \int_{\delta}
\lambda ^{q+k-1} \frac{\partial^{r}}{\partial
t^{r}}(\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-k} d \lambda\,,\quad\text{for $k\geqslant1$}\,.
\end{align}
Set
\begin{equation}\label{ue29}
I_{k,r} = \Big \{ ({\mathbf k},{\mathbf r})=(k_i,r_i)|
\sum_{i=0}^j k_i =k+j, \sum_{i=1}^j r_i =r,\, \, k_i, r_i \in \field{N}^*\Big \}.
\end{equation}
Then there exist $a ^{{\mathbf k}}_{{\mathbf r}} \in \field{R}$ such that
\begin{equation}\label{ue30}
\begin{split}
& A^{{\mathbf k}}_{{\mathbf r}} (\lambda,t) = (\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-k_0}
\frac{\partial^{r_1}\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t}{\partial t^{r_1}} (\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-k_1}
\cdots\frac{\partial^{r_j}\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t}{\partial t^{r_j}} (\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-k_j},\\
& \frac{\partial^{r}}{\partial t^{r}}
(\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-k}=
\sum_{({\mathbf k},{\mathbf r})\in I_{k,r} }
a ^{{\mathbf k}}_{{\mathbf r}} A ^{{\mathbf k}}_{{\mathbf r}} (\lambda,t).
\end{split}
\end{equation}
We claim that $A ^{{\mathbf k}}_{{\mathbf r}}(\lambda,t)$ is well defined and
for any $m\in \field{N}$, $k>2(m+r+1)$, $Q,Q'\in \mQ^m$,
there exist $C>0$, $N\in \field{N}$ such that for $\lambda\in \delta$,
\begin{align}\label{1ue30}
\|Q A ^{{\mathbf k}}_{{\mathbf r}}(\lambda,t)Q' s\|_{t,0}
\leq C \sum_{|\beta|\leq 2r} \|Z^\beta s\|_{t,0}.
\end{align}
In fact, by (\ref{0c37}), $\frac{\partial^{r}}{\partial t^{r}}\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t$ is
combination of $\frac{\partial^{r_1}}{\partial t^{r_1}}(g^{ij}(tZ))$
$(\frac{\partial^{r_2}}{\partial t^{r_2}}\nabla_{t,e_i})$
$(\frac{\partial^{r_3}}{\partial t^{r_3}}\nabla_{t,e_j})$,
$\frac{\partial^{r_1}}{\partial t^{r_1}}(d(tZ))$,
$\frac{\partial^{r_1}}{\partial t^{r_1}}(d_{i}(tZ))$
$(\frac{\partial^{r_2}}{\partial t^{r_2}}\nabla_{t,e_i})$.
Now $\frac{\partial^{r_1}}{\partial t^{r_1}}(d(tZ))$
(resp. $\frac{\partial^{r_1}}{\partial t^{r_1}}\nabla_{t,e_i}$)
($r_1\geq 1$),
are functions of the type as
$d'(tZ)Z^\beta$, $|\beta|\leq r_1$ (resp. $r_1+1$)
and $d'(Z)$ and its derivatives on $Z$ are bounded smooth functions on $Z$.
Let $\cali{R}} \newcommand{\Ha}{\cali{H}'_t$ be the family of operators of the type
$$\cali{R}} \newcommand{\Ha}{\cali{H}'_{t} = \{ [f_{j_1} Q_{j_1},
[f_{j_2} Q_{j_2},\ldots [f_{j_l} Q_{j_l},\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t]]\ldots ] \}$$
with $f_{j_i}$ smooth bounded (with its derivatives) functions
and $Q_{j_i}\in \{\nabla_{t,e_l}\}_{l=1}^{2n}$.
Now for the operator $A ^{{\mathbf k}}_{{\mathbf r}}(\lambda,t)Q'$,
we will move first all the term $Z^\beta$ in $d'(tZ)Z^\beta$
as above to the right hand side of this operator, to do so,
we always use the commutator trick, i.e., each time,
we consider only the commutation for $Z_i$,
not for $Z^\beta$ with $|\beta|>1$.
Then $A ^{{\mathbf k}}_{{\mathbf r}}(\lambda,t)Q'$ is as the form
$\sum_{|\beta|\leq 2r} L^t_\beta Q''_\beta Z^\beta$, and $Q''_\beta$ is obtained from
$Q'$ and its commutation with $Z^\beta$.
Now we move all
the terms $\nabla_{t,e_i}$ in
$\frac{\partial^{r_j} \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t}{\partial t^{r_j}}$
to the right hand side of the operator $L^t_\beta$.
Then as in the proof of Theorem \ref{tu6},
we get finally that $Q A ^{{\mathbf k}}_{{\mathbf r}}(\lambda,t)Q'$
is as the form $\sum_{|\beta|\leq 2r} \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}^t_\beta Z^\beta$ where $\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}^t_\beta$
is a linear combination of operators of the form
\begin{align*}
Q (\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-k'_0}R_1(\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-k'_1}R_2
\cdots R_{l'}(\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-k'_{l'}} Q''' Q'',
\end{align*}
with $R_1, \cdots, R_{l'} \in \cali{R}} \newcommand{\Ha}{\cali{H}'_{t}$, $Q'''\in \mQ^l$,
$Q''\in \mQ^m$, $|\beta|\leq 2 r$,
and $Q''$ is obtained from $Q'$ and its commutation with $Z^\beta$.
By the argument as in \eqref{ue18} and \eqref{ue20}, as $k>2(m+r+1)$,
we can split the above operator to two parts
\begin{align*}
&Q (\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-k'_0}R_1(\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-k'_1}R_2
\cdots R_{i}(\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-k''_{i}};\\
&(\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-(k'_{i}-k''_{i}) }\cdots
R_{l'}(\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-k'_{l'}} Q''' Q'',
\end{align*}
and the $\|\quad \|^{0,0}_t$-norm of each part is bounded
by $C$ for $\lambda\in \delta$. Thus
the proof of (\ref{1ue30}) is complete.
\comment{
We claim that $A ^{{\mathbf k}}_{{\mathbf r}}(\lambda,t)$ is well defined and
for $\lambda \in \delta$, $l\in \field{N}$,
\begin{equation}\label{1ue30}
\|A ^{{\mathbf k}}_{{\mathbf r}}(\lambda,t) s\|_{t,l}
\leqslant C \sum_{|\alpha|\leqslant 2r} \|Z^\alpha s\|_{t,l+2r-k}\,.
\end{equation}
In fact, by \eqref{0c37}, $\frac{\partial^{r}}{\partial t^{r}}\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t$ is
combination of $\frac{\partial^{r_1}}{\partial t^{r_1}}(g^{TX_0}_{ij}(tZ))
(\frac{\partial^{r_2}}{\partial t^{r_2}}\nabla_{t,e_i})
(\frac{\partial^{r_3}}{\partial t^{r_3}}\nabla_{t,e_j})$,
$\frac{\partial^{r_1}}{\partial t^{r_1}}(b(tZ))$,
$\frac{\partial^{r_1}}{\partial t^{r_1}}(a_{i}(tZ))
(\frac{\partial^{r_2}}{\partial t^{r_2}}\nabla_{t,e_i})$.
Now $\frac{\partial^{r_1}}{\partial t^{r_1}}(b(tZ))$
(resp. $\frac{\partial^{r_1}}{\partial t^{r_1}}\nabla_{t,e_i}$),
for $r_1\geqslant1$, are functions of the type $b'(tZ)Z^\beta$,
$|\beta|\leqslant r_1$ (resp. $r_1+1$) and $b'(Z)$ and its derivatives
with respect to $Z$ are bounded smooth functions.
In view \eqref{ue12}, we get \eqref{1ue30}.
}
\noindent
By \eqref{ue28}, \eqref{ue30} and the above argument, we get the
estimate \eqref{ue15} with $m'=0$.
Finally, for any vector $U$ on $X$,
\begin{align}\label{0ue30}
& \nabla ^{\pi ^* \End (E)}_U {\mathcal{P}}}\def\mJ{{\mathcal{J}}}\def\bE{{\mathbf E}_{q,t} =\frac{1}{2\pi i}
\begin{pmatrix} q+k-1 \\ k-1 \end{pmatrix} ^{-1} \int_{\delta}
\lambda ^{q+k-1} \nabla ^{\pi ^* \End (E)}_U (\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-k}
d \lambda .
\end{align}
Now we use a similar formula as \eqref{ue30} for
$\nabla ^{\pi ^* \End (E)} _U(\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-k}$ by replacing
$\frac{\partial^{r_1}\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t}{\partial t^{r_1}}$ by
$\nabla ^{\pi ^* \End (E)}_U \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t$, and remark that
$\nabla ^{\pi ^* \End (E)}_U \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t$ is a differential operator
on $T_{x_0} X$ with the same structure as $\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t$.
Then by the above argument, we conclude that \eqref{ue15} holds for $m'\geqslant1$.
\end{proof}
For $k$ big enough, set
\begin{equation}\label{ue31}
\begin{split}
& F_{q,r}= \frac{1}{2\pi i \, r! }
\begin{pmatrix} q+k-1 \\ k-1 \end{pmatrix}^{-1} \int_{\delta}
\lambda ^{q+k-1} \sum_{({\mathbf k},{\mathbf r})\in I_{k,r} }
a ^{{\mathbf k}}_{{\mathbf r}} A ^{{\mathbf k}}_{{\mathbf r}} (\lambda,0)d \lambda ,\\
&F_{q,r,t} = \frac{1}{r!}\frac{\partial^{r}}{\partial t^{r}}
{\mathcal{P}}}\def\mJ{{\mathcal{J}}}\def\bE{{\mathbf E}_{q,t}- F_{q,r}.
\end{split}
\end{equation}
Let $F_{q,r}(Z,Z')$ $(Z,Z'\in T_{x_0}X)$ be the
smooth kernel of $F_{q,r}$ with respect to $dv_{TX}(Z')$.
Then $F_{q,r}\in \cali{C}} \newcommand{\cA}{\cali{A}^\infty (TX\times_{X}TX,\pi ^* \End (E))$.
Certainly, as $t\to0$, the limit of $\|\quad\|_{t,m}$ exists,
and we denote it by $\|\quad\|_{0,m}$.
\begin{thm} \label{tue9} For any $r\geq 0$, $k>0$,
there exists $C>0$ such that for
$t \in [0,1], \lambda \in \delta$,
\begin{align}\label{ue32}
& \left \|\Big(\frac{\partial^{r}\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t}{\partial t^{r}} -
\frac{\partial^{r}\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t}{\partial t^{r}} |_{t=0}\Big )s \right \|_{t,-1}
\leqslant Ct \sum_{|\alpha|\leqslant r+3} \|Z^\alpha s\|_{0,1},\\
& \Big \|\Big (\frac{\partial^{r}}{\partial t^{r}} (\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-k}
-\sum_{({\mathbf k},{\mathbf r})\in I_{k,r} }
a ^{{\mathbf k}}_{{\mathbf r}} A ^{{\mathbf k}}_{{\mathbf r}} (\lambda,0)\Big )s\Big \|_{0,0}
\leqslant C t \sum_{|\alpha|\leqslant 4r+3} \|Z^\alpha s\|_{0,0}.\nonumber
\end{align}
\end{thm}
\begin{proof} Note that by (\ref{0c36}), (\ref{u0}), for $t\in [0,1]$, $k\geqslant1$,
\begin{align}\label{1ue35}
\|s\|_{t,0}= \|s \|_{0,0},\quad
\|s\|_{t,k}\leqslant C \sum_{|\alpha|\leqslant k} \|Z^\alpha s\|_{0,k}.
\end{align}
An application of Taylor expansion for
(\ref{0c37}) leads to the following estimate for
compactly supported $s,s'$:
\begin{equation}\label{ue33}
\Big | \left \langle \Big (\frac{\partial^{r}\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t}{\partial t^{r}} -
\frac{\partial^{r}\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t}{\partial t^{r}} |_{t=0} \Big )s,
s'\right \rangle_{0,0}\Big |
\leqslant C t \|s'\|_{t,1}\sum_{|\alpha|\leqslant r+3} \|Z^\alpha s\|_{0,1}.
\end{equation}
Thus we get the first inequality of (\ref{ue32}). Note that
\begin{align}\label{ue34}
& (\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1}- (\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0)^{-1}
=(\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1}(\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0) (\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0)^{-1}.
\end{align}
Now from (\ref{ue33}) and (\ref{ue34}),
\begin{align}\label{0ue34}
& \left \|\left ((\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1}- (\lambda-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0)^{-1} \right)s
\right \|_{0,0}
\leqslant Ct \sum_{|\alpha|\leqslant 3} \|Z^\alpha s\|_{0,1}.
\end{align}
After taking the limit, we know that Theorems \ref{tu4}, \ref{tu5} still hold
for $t=0$. Note that
$\nabla_{0,e_j} = \nabla_{e_j}+\frac{1}{2} R^L_{x_0}(\mR, e_j)$
by (\ref{0c36}). If we denote by $\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_{\lambda,t}=\lambda -\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_{t}$, then
\begin{multline}\label{0ue35}
A^{{\mathbf k}}_{{\mathbf r}} (\lambda,t)- A^{{\mathbf k}}_{{\mathbf r}} (\lambda,0)
= \sum_{i=1}^j \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_{\lambda,t}^{-k_0}\cdots
\left ( \frac{\partial^{r_i}\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t}{\partial t^{r_i}}
- \frac{\partial^{r_i}\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t}{\partial t^{r_i}} |_{t=0}\right )
\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_{\lambda,0}^{-k_{i}}\cdots \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_{\lambda,0}^{-k_j}\\
+ \sum_{i=0}^j \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_{\lambda,t}^{-k_0}\cdots
\left (\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_{\lambda,t}^{-k_i}- \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_{\lambda,0}^{-k_i} \right)
\left (\frac{\partial^{r_{i+1}}\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t}{\partial t^{r_{i+1}}}|_{t=0}\right)
\cdots \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_{\lambda,0}^{-k_j}.
\end{multline}
From the discussion after (\ref{1ue30}),
formulas (\ref{ue2}), (\ref{ue30})
and (\ref{0ue34}), we get (\ref{ue32}).
\end{proof}
\begin{thm} \label{tue12}
For $\sigma >0$, there exists $C>0$
such that for $t \in ]0,1]$,
$Z,Z'\in T_{x_0}X$, $|Z|, |Z'|\leqslant \sigma $,
\begin{align}\label{ue42}
\Big |F_{q,r,t}(Z,Z')\Big |\leqslant & C t^{1/2(2n+1)}.
\end{align}
\end{thm}
\begin{proof}
By (\ref{ue28}), (\ref{ue31}) and (\ref{ue32}), there exists $C>0$
such that for $t \in ]0,1]$,
\begin{align}\label{ue39}
& \|F_{q,r,t}\|_{(\sigma),0 }
\leqslant C t .
\end{align}
Let $\phi : \field{R}\to [0,1]$ be a smooth function with compact
support, equal $1$ near $0$, such that
$\int_{T_{x_0}X} \phi (Z) dv_{TX}(Z)=1$. Take $\nu \in ]0,1]$.
By the proof of Theorem \ref{tue8} and (\ref{ue31}),
$F_{q,r}$ verifies the similar inequality as in (\ref{ue15})
with $r=0$. Thus by (\ref{ue15}), there exists $C>0$ such that if
$|Z|,|Z'|\leqslant \sigma$, $U,U'\in E_{x_0}$,
\begin{multline}\label{ue43}
\Big | \left \langle
F_{q,r,t} (Z,Z') U,U' \right \rangle
-\int_{T_{x_0}X \times T_{x_0}X} \left \langle
F_{q,r,t}(Z-W,Z'-W') U,U' \right \rangle\\
\frac{1}{\nu ^{4n}} \phi (W/\nu) \phi (W'/\nu) dv_{TX}(W)dv_{TX}(W')\Big |
\leqslant C \nu |U||U'|.
\end{multline}
On the other hand, by (\ref{ue39}),
\begin{multline}\label{ue44}
\Big |\int_{T_{x_0}X \times T_{x_0}X} \left \langle
F_{q,r,t} (Z-W,Z'-W') U,U' \right \rangle\\
\frac{1}{\nu ^{4n}} \phi (W/\nu) \phi (W'/\nu) dv_{TX}(W)dv_{TX}(W')\Big |
\leqslant C t\frac{1}{\nu ^{2n}} |U||U'|.
\end{multline}
By taking $\nu = t^{1/2(2n+1)}$, we obtain (\ref{ue42}).
\end{proof}
Finally, we obtain the following off-diagonal estimate for the kernel of
${\mathcal{P}}}\def\mJ{{\mathcal{J}}}\def\bE{{\mathbf E}_{q,t}$.
\begin{thm} \label{tue14} For $k,m,m'\in \field{N}$, $\sigma>0$, there exists
$C>0$ such that if
$t\in ]0,1]$, $Z,Z'\in T_{x_0}X$, $|Z|,|Z'| \leqslant \sigma$,
\begin{align}\label{0ue45}
&\sup_{|\alpha|,|\alpha'|\leqslant m} \Big
|\frac{\partial^{|\alpha|+|\alpha'|}} {\partial Z^{\alpha}
{\partial Z'}^{\alpha'}}\Big ({\mathcal{P}}}\def\mJ{{\mathcal{J}}}\def\bE{{\mathbf E}_{q,t} - \sum_{r=0}^k
F_{q,r}t^r\Big ) (Z,Z') \Big |_{\cali{C}} \newcommand{\cA}{\cali{A} ^{m'}(X)} \leqslant C t^{k+1}.
\end{align}
\end{thm}
\begin{proof} By (\ref{ue31}), and (\ref{ue42}),
\begin{align}\label{0ue47}
&\frac{1}{r!} \frac{\partial^{r}}{\partial t^{r}} {\mathcal{P}}}\def\mJ{{\mathcal{J}}}\def\bE{{\mathbf E}_{q,t}
|_{t=0} = F_{q,r}.
\end{align}
Now by Theorem \ref{tue8} and (\ref{ue31}), $F_{q,r}$ has the same
estimate as $\frac{1}{r!}\frac{\partial^{r}}{\partial
t^{r}}{\mathcal{P}}}\def\mJ{{\mathcal{J}}}\def\bE{{\mathbf E}_{q,t}$, in (\ref{ue15}). Again from (\ref{ue15}), (\ref{ue31}),
and the Taylor expansion
\begin{align}\label{0ue46}
G(t)- \sum_{r=0}^k \frac{1}{r !} \frac{\partial ^r G}{\partial t^r}(0) t^r
= \frac{1}{k!}\int_0^t (t-t_0)^k
\frac{\partial ^{k+1} G}{\partial t^{k+1} }(t_0) dt_0,
\end{align}
we have (\ref{0ue45}).
\end{proof}
\subsection{Bergman kernel of $\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0$}
\label{s3.4}
The almost complex structure $J$ induces a splitting
$T_\field{R} X\otimes_\field{R} \field{C}=T^{(1,0)}X\oplus T^{(0,1)}X$,
where $T^{(1,0)}X$ and $T^{(0,1)}X$ are the eigenbundles of $J$
corresponding to the eigenvalues $\sqrt{-1}$ and $-\sqrt{-1}$ respectively.
We denote by $\det_{\field{C}}$ the determinant function on the complex
bundle $T^{(1,0)}X$.
Set
\begin{align}\label{0ue62}
\mathcal{J}= -2\pi \sqrt{-1} {\bf J}.
\end{align}
By \eqref{0.1}, $\mathcal{J}\in \End (T^{(1,0)}X)$
is positive, and $\mathcal{J}$ acting on $TX$ is skew-adjoint.
For any tensor $\psi$ on $X$, we denote by $\nabla ^{X}\psi$
the covariant derivative of $\psi$ induced by $\nabla ^{TX}$.
Thus $\nabla ^{X} \mJ, \nabla ^{X} J \in
T^*X \otimes \End(TX)$, $\nabla ^{X} \nabla ^{X} \mJ\in T^*X
\otimes T^*X \otimes \End(TX)$. We also adopt
the convention that all tensors will be evaluated at the base point
$x_0\in X$, and most of the time, we will omit the subscript
$x_0$.
Let $P^N$ be the orthogonal projection from
$(L^2 (\field{R}^{2n}, E_{x_0}), \|\quad\|_0=\|\quad\|_{t,0})$ onto $N=\Ker \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0$,
and let $P^N(Z,Z')$ be the smooth kernel of $P^N$ with respect
to $dv_{T_{x_0}X}(Z)$. Then $P^N(Z,Z')$ is the
the Bergman kernel of $\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0$. For $Z,Z'\in T_{x_0}X$, we have
\begin{align}\label{ue62}
P^N(Z,Z') =\frac{\det_{\field{C}}{\mathcal{J}_{x_0}}}{(2\pi)^n}
\exp\Big (- \frac{1}{4}
\left \langle (\mathcal{J}^2_{x_0})^{1/2}(Z-Z'),(Z-Z') \right \rangle
+\frac{1}{2} \left \langle \mathcal{J}_{x_0} Z,Z' \right \rangle\Big ).
\end{align}
Now we discuss the eigenvalues and eigenfunctions of $\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0$
in more precise way. We choose $\{ w_i\}_{i=1}^n$
an orthonormal basis of $T^{(1,0)}_{x_0} X$, such that
\begin{align}\label{0ue52}
\mathcal{J}_{x_0}= {\rm diag} (a_1 , \cdots, a_n)\in {\rm End}
(T^{(1,0)}_{x_0} X),
\end{align}
with $0< a_1\leqslant a_2\leqslant \cdots \leqslant a_n$, and let
$\{w^j\}_{j=1}^n$ be its dual basis. Then
$e_{2j-1}=\tfrac{1}{\sqrt{2}}(w_j+\overline{w}_j)$ and
$e_{2j}=\tfrac{\sqrt{-1}}{\sqrt{2}}(w_j-\overline{w}_j)\,,
j=1,\dotsc,n\, $
forms an orthonormal basis of $T_{x_0}X$.
We use the coordinates on $T_{x_0}X\simeq\field{R}^{2n}$ induced by $\{ e_i\}$ as in \eqref{0c11}
and in what follows we also introduce the complex coordinates $z=(z_1,\cdots,z_n)$
on $\field{C}^n\simeq\field{R}^{2n}$. Thus
$Z=z+\overline{z}$, and $w_i=\sqrt{2}\tfrac{\partial}{\partial z_i}$,
$\overline{w}_i=\sqrt{2}\tfrac{\partial}{\partial\overline{z}_i}$.
We will also identify $z$ to $\sum_i z_i\tfrac{\partial}{\partial z_i}$
and $\overline{z}$ to
$\sum_i\overline{z}_i\tfrac{\partial}{\partial\overline{z}_i}$ when
we consider $z$ and $\overline{z}$ as vector fields. Remark that
\begin{equation}\label{g0}
\Big\lvert\tfrac{\partial}{\partial z_i}\Big\rvert^2=
\Big\lvert\tfrac{\partial}{\partial\overline{z}_i}\Big\rvert^2
=\dfrac{1}{2}\,,
\quad\text{so that $|z|^2=|\overline{z}|^2=\dfrac{1}{2} |Z|^2$\,.}
\end{equation}
It is very useful to rewrite $\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0$ by using
the creation and annihilation operators. Set
\begin{equation}\label{0g1}
\nabla_{0,\cdot}= \nabla_{\cdot} + \frac{1}{2} R^L_{x_0}(\mR, \cdot),
\quad b_i=-2\nabla_{0,\tfrac{\partial}{\partial z_i}},\quad
b^{+}_i=2\nabla_{0,\tfrac{\partial}{\partial \overline{z}_i}},
\quad b=(b_1,\cdots,b_n)\,.
\end{equation}
Then by \eqref{0ue62}, and \eqref{0ue52}, we have
\begin{equation}\label{g1}
b_i=-2{\tfrac{\partial}{\partial z
_i}}+\frac{1}{2}a_i\overline{z}_i\,,\quad
b^{+}_i=2{\tfrac{\partial}{\partial\overline{z}_i}}+\frac{1}{2}a_i
z_i,
\end{equation}
and for any polynomial $g(z,\overline{z})$ on $z$ and $\overline{z}$,
\begin{align}\label{g2}
&[b_i,b^{+}_j]=b_i b^{+}_j-b^{+}_j b_i =-2a_i \delta_{i\,j},\\
&[b_i,b_j]=[b^{+}_i,b^{+}_j]=0\, ,\nonumber\\
& [g(z,\overline{z}),b_j]= 2 \tfrac{\partial}{\partial z_j}g(z,\overline{z}),
\quad [g(z,\overline{z}),b_j^+]
= - 2\tfrac{\partial}{\partial \overline{z}_j}g(z,\overline{z})\,. \nonumber
\end{align}
By \eqref{0.2}, \eqref{0ue52}, $\tau_{x_0}= \sum_i a_i$.
Thus from \eqref{c31}, \eqref{0ue52}, \eqref{0g1}-\eqref{g2},
\begin{equation}\label{g3}
\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0=\sum_i b_i b^{+}_i.
\end{equation}
\begin{rem} \label{r3.4} Let $L=\field{C}$ be the trivial holomorphic line bundle
on $\field{C}^n$ with the canonical section $1$. Let $h^L$ be the metric on $L$
defined by $|1|_{h^L}(z) = e ^{-\frac{1}{4}\sum_{j=1}^n a_{j}|z_{j}|^2}$
for $z\in \field{C}^n$. Let $g^{T\field{C} ^n}$ be the canonical metric on $\field{C}^n$.
Then $\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_{0}$ is twice the corresponding Kodaira-Laplacian
$\overline{\partial}^{L*}\overline{\partial}^{L}$ under the trivialization
of $L$ by using the unit section
$e ^{\frac{1}{4}\sum_{j=1}^n a_{j}|z_{j}|^2} 1$.
\end{rem}
\begin{thm}\label{t3.4}
The spectrum of the restriction of $\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0$ on $L^2(\field{R}^{2n})$ is given by
\begin{equation}\label{g4}
\spec{{\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0}_{\upharpoonright\,{L^2(\field{R}^{2n})}}}=
\Big\lbrace2\sum_{i=1}^n\alpha_i a_i\,:\,
\alpha =(\alpha_1,\cdots,\alpha_n)\in\field{N} ^n\Big\rbrace
\end{equation}
and an orthogonal basis of the eigenspace of $2\sum_{i=1}^n\alpha_i a_i$
is given by
\begin{equation}\label{g5}
b^{\alpha}\big(z^{\beta}\exp\big({- \frac{1}{4}\sum_i
a_i|z_i|^2}\big)\big)\,,\quad\text{with $\beta\in\field{N}^n$}\,.
\end{equation}
\end{thm}
\begin{proof} At first $z^{\beta}\exp\big({- \frac{1}{4}\sum_i
a_i|z_i|^2}\big)$, $\beta\in\field{N}^n$ are annihilated by $b^{+}_i$
$(1\leq i\leq n)$, thus they are in the kernel of
${\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0}_{\upharpoonright\,{L^2(\field{R}^{2n})}}$.
Now, by (\ref{g2}), \eqref{g5} are eigenfunctions of
${\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0}_{\upharpoonright\,{L^2(\field{R}^{2n})}}$ with eigenvalue
$2\sum_{i=1}^n\alpha_i a_i$.
But the span of functions \eqref{g5} includes
all the rescaled Hermite polynomials
multiplied by $\exp\big({- \frac{1}{4}\sum_i a_i|z_i|^2}\big)$,
which is an orthogonal basis of $L^2(\field{R}^{2n})$ by \cite[\S 6]{T2}.
Thus the eigenfunctions in \eqref{g5}
are all the eigenfunctions of ${\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0}_{\upharpoonright\,{L^2(\field{R}^{2n})}}$.
The proof of Theorem \ref{t3.4} is complete.
\end{proof}
Especially an orthonormal basis of
$\Ker \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0\upharpoonright_{L^2(\field{R}^{2n})}$ is
\begin{align}\label{g6}
\Big(\frac{a ^\beta}{(2\pi)^n 2 ^{|\beta|} \beta!}
\prod_{i=1}^n a_i\Big)^{1/2}z^\beta
\exp\Big (-\frac{1}{4} \sum_{j=1}^n a_j |z_j|^2\Big )\,,\quad \beta\in\field{N}^n\,.
\end{align}
From \eqref{g6}, we recover \eqref{ue62}:
\begin{equation}\label{g7}
\begin{split}
P^N(Z,Z') &=\frac{1}{(2\pi)^n}\prod_{i=1}^n
a_i\:\:\exp\Big(-\frac{1}{4}\sum_i
a_i\big(|z_i|^2+|z^{\prime}_i|^2 -2z_i\overline{z}_i'\big)\Big).
\end{split}
\end{equation}
Recall that the operators $\mO_1,\mO_2$ were defined in (\ref{c31}).
Theorem \ref{t3.5} below is crucial in proving the vanishing result of
$F_{q,r}$ (cf. Theorem \ref{0t3.6}).
\begin{thm}\label{t3.5}
We have the relation
\begin{equation}\label{g8} P^N\mO_1 P^N =0.\end{equation}
\end{thm}
\begin{proof} From \eqref{0.1}, for $U,V,W\in TX$,
$\langle(\nabla ^{X}_U{\bf J})V,W\rangle=(\nabla ^{X}_U\omega)(V,W)$, thus
\begin{equation} \label{g9}
\langle(\nabla ^{X}_U{\bf J})V,W\rangle
+\langle(\nabla ^{X}_V{\bf J})W,U\rangle+\langle
(\nabla ^{X}_W{\bf J})U,V\rangle=d\omega(U,V,W)=0.
\end{equation}
By \eqref{0.1} and \eqref{0.2},
\begin{equation}\label{g10}
\begin{split}
&R^L(U,V)=\langle \mJ U,V \rangle, \\
& (\nabla ^{X}_U R^L) (V,W)
= \langle (\nabla ^{X}_U\mJ) V,W \rangle,\\
&\nabla _U\tau =
- \frac{\sqrt{-1}}{2}\tr|_{TX}[\nabla ^{X}_U(J\mJ)].
\end{split}
\end{equation}
As $J$, $\mathcal{J}\in \End (TX)$ are
skew-adjoint and commute, $\nabla ^{X}_{U}J$, $\nabla ^{X}_{U}\mathcal{J}$
are skew-adjoint and $\nabla ^{X}_{U}(J\mathcal{J})$ is symmetric.
From $J^2=-\Id$, we know that
\begin{align}\label{g11}
J(\nabla ^{X} J)+(\nabla ^{X} J)J=0,
\end{align}
thus $\nabla ^{X}_{U}J$ exchanges $T^{(1,0)}X$ and $T^{(0,1)}X$.
From \eqref{g9} and \eqref{g10}, we have
\begin{equation}\label{g12}
\begin{split}
&(\nabla _\mR\tau)_{x_0}=-2\sqrt{-1} \left\langle
(\nabla ^{X}_{\mR}(J\mJ))\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial\overline{z}_i}\right\rangle
=2\left\langle(\nabla ^{X}_{\mR}\mJ)\tfrac{\partial}{\partial
z_i},
\tfrac{\partial}{\partial\overline{z}_i}\right\rangle,\\
&(\partial_i R^L)_{x_0}(\mR,e_i)
=2\Big\langle(\nabla ^{X}_{\tfrac{\partial}{\partial
z_i}}\mJ) \mR,\tfrac{\partial}{\partial\overline{z}_i}\Big\rangle
+ 2 \Big\langle(\nabla ^{X}_{\tfrac{\partial}{\partial\overline{z}_i}}\mJ)
\mR,\tfrac{\partial}{\partial z_i}\Big\rangle \\
&\hspace*{20mm}= 4\Big\langle(\nabla ^{X}_{\tfrac{\partial}{\partial
z_i}}\mJ) \mR,\tfrac{\partial}{\partial\overline{z}_i}\Big\rangle
-2 \left\langle(\nabla ^{X}_{\mR}\mJ) \tfrac{\partial}{\partial
z_i},\tfrac{\partial}{\partial\overline{z}_i}\right\rangle .
\end{split}
\end{equation}
From \eqref{c31}, \eqref{g2}, \eqref{g10} and \eqref{g12}, we infer
\begin{equation} \label{g13}
\begin{split}
\mO_1&=-\frac{2}{3}\Big[\left\langle(\nabla ^{X}_\mR\mJ)\mR,
\tfrac{\partial} {\partial z_i}\right\rangle b^+_i
- \left\langle(\nabla ^{X}_\mR\mJ)\mR,\tfrac{\partial}
{\partial\overline{z}_i}\right\rangle b_i\\
&\hspace*{10mm}+2\Big\langle(\nabla ^{X}_{\tfrac{\partial}{\partial
z_i}}\mJ)\mR,\tfrac{\partial}{\partial\overline{z}_i}\Big\rangle+
2\left\langle(\nabla ^{X}_\mR\mJ)\tfrac{\partial}{\partial
z_i},\tfrac{\partial} {\partial\overline{z}_i}\right\rangle\Big]\\
&=-\frac{2}{3}\Big[\left\langle(\nabla ^{X}_\mR\mJ)\mR,
\tfrac{\partial} {\partial z_i}\right\rangle b^+_i
-b_i \left\langle(\nabla ^{X}_\mR\mJ)\mR,\tfrac{\partial}
{\partial\overline{z}_i}\right\rangle\Big].
\end{split}
\end{equation}
Note that by (\ref{g1}), (\ref{g7}),
\begin{equation} \label{g14}
(b^+_iP^N)(Z,Z^{\prime}) =0\,,\quad \,
(b_iP^N)(Z,Z^{\prime})=a_i(\overline{z}_i-\overline{z}^{\prime}_i)P^N(Z,Z^{\prime}).
\end{equation}
We learn from \eqref{g14} that for any polynomial $g(z,\overline{z})$ in $z,\overline{z}$
we can write $g(z,\overline{z})P^N(Z,Z^{\prime})$ as
sums of $b^\beta g_{\beta} (z,\overline{z}')P^N(Z,Z^{\prime})$ with
$g_{\beta} (z,\overline{z}')$ polynomials in $z,\overline{z}'$.
By Theorem \ref{t3.4},
\begin{equation}\label{g15}
P^N b^\alpha g(z,\overline{z})P^N=0\,,\quad\text{for $|\alpha|>0$},
\end{equation}
and relations \eqref{g13}\,--\,\eqref{g15} yield the desired relation \eqref{g8}.
\end{proof}
\subsection{Evaluation of $F_{q,r}$}
\label{s3.5}
For $s\in \field{R}$, let $[s]$ denote the greatest integer
which is less than or equal to $s$.
Let $f(\lambda, t)$ be a formal power series with values in $
\End(L^2(\field{R}^{2n},E_{x_0}))$ \begin{equation}\label{c54} f(\lambda, t)=
\sum_{r=0}^\infty t^r f_r(\lambda), \quad f_r(\lambda)\in
\End(L^2(\field{R}^{2n}, E_{x_0})). \end{equation}
By (\ref{c30}), consider
the equation of formal power series for $\lambda\in \delta$,
\begin{align}\label{c55}
(-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0 +\lambda - \sum_{r=1}^\infty t^r \mO_r) f(\lambda, t) =
\Id_{L^2(\field{R}^{2n}, E_{x_0})}.
\end{align}
Let $N^\bot$ be the orthogonal space of $N$ in
$L^2(\field{R}^{2n},E_{x_0})$, and $P^{N^\bot}$ be the orthogonal
projection from $L^2(\field{R}^{2n},E_{x_0})$ to $N^\bot$. We decompose
$f(\lambda, t)$ according the splitting $L^2(\field{R}^{2n}, E_{x_0})=
N\oplus N^\bot$,
\begin{align}\label{c56}
g_r(\lambda) = P^{N}f_r(\lambda),\quad
f^\bot_r(\lambda)=P^{N^\bot}f_r(\lambda).
\end{align}
Using (\ref{c56}) and identifying the powers of $t$ in
(\ref{c55}), we find that
\begin{equation}\label{c57}
\begin{split}
&g_0(\lambda) = \frac{1}{\lambda}P^{N},
\quad f^\bot_0(\lambda)= (\lambda -\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0)^{-1}P^{N^\bot} ,\\
&f^\bot_r(\lambda)=(\lambda -\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0)^{-1}
\sum_{j=1}^{r} P^{N^\bot}\mO_j f_{r-j}(\lambda),\\
& g_r(\lambda) = \frac{1}{\lambda } \sum_{j=1}^{r} P^{N}\mO_j
f_{r-j}(\lambda).
\end{split}
\end{equation}
\begin{lemma}\label{t3.6} For $r\in \field{N}$,
$\lambda ^{[\frac{r}{2}] +1} g_r(\lambda)$, $\lambda
^{[\frac{r+1}{2}] }f^\bot_r(\lambda)$ are holomorphic functions
for $|\lambda|\leqslant \mu_0/4$ and
\begin{equation}\label{c81}
(\lambda ^{r+1} g_{2r})(0) =( P^N\mO_2 P^N
- P^N \mO_1 \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1} P^{N^\bot} \mO_1 P^N)^r P^N.
\end{equation}
\end{lemma}
\begin{proof}
By \eqref{c57} we know that Lemma \ref{t3.6} is true for $r=0$.
Assume that Lemma \ref{t3.6} is true for $r\leqslant m$. Now, by
Theorem \ref{t3.4}, (\ref{c57}) and the recurrence assumption, it follows that $\lambda
^{[\frac{m}{2}]+1 }f^\bot_{m+1}(\lambda)$ is holomorphic for
$|\lambda|\leqslant \mu_0/4$, and
\begin{equation}\label{c82}
\lambda ^{[\frac{m+1}{2}] +1} g_{m+1}(\lambda)
= \lambda ^{[\frac{m+1}{2}]} \sum_{i=1}^{m+1} P^N \mO_i\Big [
g_{m+1-i}(\lambda) + f^\bot_{m+1-i}(\lambda) \Big ] .
\end{equation}
By our recurrence, $\lambda ^{[\frac{m+1}{2}]-1}
f^\bot_{m+1-j}(\lambda)$, $\lambda ^{[\frac{m+1}{2}]-1}
g_{m-j}(\lambda)$, $\lambda ^{[\frac{m+1}{2}]}
f^\bot_{m}(\lambda)$ are
holomorphic for $|\lambda|\leqslant \mu_0/4$, $j\geqslant2$.
Thus by Theorem \ref{t3.5}, and \eqref{c57} and \eqref{c82},
$\lambda ^{[\frac{m+1}{2}] +1} g_{m+1}(\lambda)$
is also holomorphic for $|\lambda|\leqslant \mu_0/4$, and
\begin{multline}\label{c84}
(\lambda ^{[\frac{m+1}{2}] +1} g_{m+1})(0)
=\Big( \lambda ^{[\frac{m+1}{2}]}
(P^N \mO_1 f^\bot_{m} +P^N\mO_2 g_{m-1} ) \Big)(0)\\
= -P^N \mO_1 \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1} P^{N^\bot} \mO_1
\Big( \lambda ^{[\frac{m+1}{2}]}( g_{m-1}
+ f^\bot_{m-1})\Big)(0)\
+( \lambda ^{[\frac{m+1}{2}]} P^N\mO_2 g_{m-1})(0) .
\end{multline}
If $m$ is odd, then by (\ref{c84}) and recurrence assumption,
\begin{multline}\label{c85}
(\lambda ^{[\frac{m+1}{2}] +1} g_{m+1})(0)
=P^N (-\mO_1 \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1} P^{N^\bot} \mO_1 + \mO_2
)P^N(\lambda ^{[\frac{m-1}{2}] +1} g_{m-1})(0) \\
= ( P^N\mO_2
P^N- P^N \mO_1 \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1} P^{N^\bot} \mO_1 P^N)^{[\frac{m+1}{2}]}P^N .
\end{multline}
The proof of Lemma \ref{t3.6} is complete.
\end{proof}
\begin{thm} \label{0t3.6} There exist $J_{q,r}(Z,Z')$ polynomials
in $Z,Z'$ with the same parity as $r$ and $\deg J_{q,r}(Z,Z')\leq 3r$,
whose coefficients
are polynomials in $R^{TX}$, $R^E$ {\rm(}\,and $R^L$, $\Phi${\rm)}
and their derivatives of order $\leqslant r-1$ {\rm(}\,resp. $r${\rm)},
and reciprocals of linear combinations of eigenvalues of ${\bf J}$
at $x_0$\,, such that
\begin{align}\label{c86}
&F_{q,r}(Z,Z')= J_{q,r}(Z,Z')P^N(Z,Z').
\end{align}
Moreover,
\begin{equation}\label{1c52}
\begin{split}
&F_{0,0}=P^N,\\
&F_{q,r} =0, \quad\text{for $q>0, r< 2q$}\,, \\
&F_{q,2q} =\big(P^N\mO_2P^N
- P^N \mO_1 \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1} P^{N^\bot} \mO_1 P^N\big)^q\,
P^N\quad\text{for $q>0$}\,.
\end{split}
\end{equation}
\end{thm}
\begin{proof} Recall that ${\mathcal{P}}}\def\mJ{{\mathcal{J}}}\def\bE{{\mathbf E}_{q,t}= (\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^q {\mathcal{P}}}\def\mJ{{\mathcal{J}}}\def\bE{{\mathbf E}_{0,t}$.
By (\ref{1ue15}),
${\mathcal{P}}}\def\mJ{{\mathcal{J}}}\def\bE{{\mathbf E}_{q,t} = \frac{1}{2 \pi i} \int_{\delta}\lambda ^q (\lambda
-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t)^{-1}d\lambda$ .
Thus by (\ref{ue28}), (\ref{ue31}) and (\ref{c56}),
\begin{equation}\label{c90}
F_{q,r}= \frac{1}{2\pi i} \int_{\delta}\lambda ^q g_r (\lambda)d
\lambda + \frac{1}{2\pi i} \int_{\delta}\lambda ^q f_r^\bot
(\lambda)d \lambda.
\end{equation}
From Lemma \ref{t3.6} and (\ref{c90}), we get (\ref{1c52}).
Generally, from Theorems \ref{t3.3}, \ref{t3.4}, Remark \ref{0t3.3},
\eqref{g7}, \eqref{c57}, \eqref{c90} and the residue formula, we conclude that
$F_{q,r}$ has the form \eqref{c86}.
\end{proof}
From Theorems \ref{t3.4}, \ref{t3.5}, (\ref{c57}), (\ref{c90})
and the residue formula,
we can get $F_{q,r}$ by using the operators $\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1}$,
$P^N$, $P^{N^\bot}$, $\mO_k (k\leq r)$.
This give us a direct method to compute $F_{q,r}$
in view of Theorem \ref{t3.4}. In particular, we get
\footnote{The formula $F_{0,2}$ in \cite[(20)]{MM04a} missed the last
two terms here which are zero at $(0,0)$ if ${\bf J}=J$,
cf. Section \ref{s4.3}.}
\begin{equation}\label{g51}
\begin{split}
F_{0,1}=& -P^N\mO_1\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1}P^{N^\bot}-P^{N^\bot}\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1}\mO_1P^N,\\
F_{0,2}=& \frac{1}{2 \pi i} \int_{\delta}
\Big[(\lambda -\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0)^{-1} P^{N^\bot} (\mO_1 f_1 + \mO_2 f_0)(\lambda)
+ \frac{1}{\lambda} P^N (\mO_1 f_1+ \mO_2 f_0)(\lambda)\Big]d\lambda \\
=&\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1}P^{N^\bot}\mO_1 \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1}P^{N^\bot}\mO_1 P^N
- \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1}P^{N^\bot}\mO_2 P^N\\
&+ P^N\mO_1\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1}P^{N^\bot}\mO_1\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1}P^{N^\bot}
- P^N \mO_2\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1}P^{N^\bot}\\
&+P^{N^\bot}\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1}\mO_1 P^N\mO_1\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1}P^{N^\bot}
-P^N\mO_1 \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-2} P^{N^\bot}\mO_1 P^N.
\end{split}
\end{equation}
\subsection{Proof of Theorem \ref{t0.1}}
\label{s3.6}
Recall that $P_{0,q,p}= (\Delta_{p,\Phi_0}^{X_0})^q P_{0,\mH_p}$.
By (\ref{c22}), (\ref{c27}), for $Z,Z'\in
\field{R}^{2n}$, \begin{equation}\label{0c53} P_{0,q,p}(Z,Z') =
t^{-2n-2q}\kappa ^{-\frac{1}{2}}(Z) {\mathcal{P}}}\def\mJ{{\mathcal{J}}}\def\bE{{\mathbf E}_{q,t}(Z/t, Z'/t) \kappa ^{-\frac{1}{2}}(Z').
\end{equation}
By \eqref{1c19}, \eqref{0c53}, Proposition \ref{p3.2},
Theorems \ref{tue14} and \ref{0t3.6},
we get the following main technical result of this paper, the near off-diagonal expansion
of the generalized Bergman kernels:
\begin{thm} \label{t3.8} For $k,m,m'\in \field{N}$, $k\geqslant2q$, $\sigma>0$,
there exists
$C>0$ such that if $p\geqslant1$, $Z,Z'\in T_{x_0}X$, $|Z|,|Z'| \leq
\sigma/\sqrt{p}$,
\begin{multline}\label{1c53}
\sup_{|\alpha|+|\alpha'|\leqslant m} \Big|
\frac{\partial^{|\alpha|+|\alpha'|}} {\partial Z^{\alpha}
{\partial Z'}^{\alpha'}}\Big ( \frac{1}{p^n}P_{q,p} (Z,Z')\\
- \sum_{r=2q}^k F_{q,r} (\sqrt{p}Z,\sqrt{p}Z')
\kappa ^{-\frac{1}{2}}(Z)\kappa^{-\frac{1}{2}}(Z')
p^{-\frac{r}{2}+q}\Big ) \Big |_{\cali{C}} \newcommand{\cA}{\cali{A} ^{m'}(X)}
\leqslant C p^{-\frac{k-m}{2}+q}.
\end{multline}
\end{thm}
\noindent
Set now $Z=Z'=0$ in \eqref{1c53}. By Theorem \ref{0t3.6},
we obtain \eqref{0.6} and
\begin{align}\label{1c54}
b_{q,r}(x_0)= F_{q,2r+2q} (0,0).
\end{align}
Hence \eqref{0.5} follows from \eqref{ue62} and \eqref{1c54}.
The statement about the structure of $b_{q,r}$
follows from Theorems \ref{t3.4} and \ref{0t3.6}.
To prove the uniformity part of Theorem \ref{t0.1}, we notice that
in the proof of Theorem \ref{tue8}, we only use the derivatives of
the coefficients of $\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_t$ with order $\leqslant 2n+2m+m'+2$.
Thus by \eqref{0ue46},
the constants in Theorems \ref{tue8}, \ref{tue12} and \ref{tue14}
are uniformly bounded, if with respect to a fixed metric $g^{TX}_0$,
the $\cali{C}} \newcommand{\cA}{\cali{A}^{2n+2m+m'+4}$\,--\,norms on $X$ of the data
{\rm(}$g^{TX}$, $h^L$, $\nabla ^L$, $h^E$, $\nabla ^E$, $J$ and $\Phi${\rm)}
are bounded, and $g^{TX}$ is bounded below.
Moreover, taking derivatives with respect to the parameters we obtain
a similar equation as \eqref{0ue30}, where
$x_0\in X$ plays now a role of a parameter. Thus
the $\cali{C}} \newcommand{\cA}{\cali{A}^{m'}$--\,norm in \eqref{1c53} can also include the parameters
if the $C^{m'}$--\,norms (with respect to the parameter $x_0\in X$) of the
derivatives of above data with order $\leqslant 2n+2m+2k+m'+4$ are bounded.
Thus we can take $C_{k,\, l}$ in \eqref{0.6} independent of $g^{TX}$
under our condition.
This achieves the proof of Theorem \ref{t0.1}.
\section{Computing the coefficients $b_{q,r}$}
\label{s4}
In principle, Theorem \ref{t3.4}, the equations \eqref{c57}, \eqref{c90}
and the residue formula give us a direct method to
calculate $b_{q,r}$ by recurrence. Actually, it is computable for
the first few terms $b_{q,r}$ in (\ref{0.6}) in this way.
This Section is organized as follows.
In Section \ref{s4.1}, we will give a simplified formula
for $\mO_2P^N$ without the assumption ${\bf J}=J$. In Sections \ref{s4.2},
\ref{s4.3},
we will compute $b_{q,0}$ and $b_{0,1}$ under the assumption ${\bf J}=J$,
thus proving Theorem \ref{t0.2}.
In this Section, we use the notation in Section \ref{s3.4},
and all tensors will be evaluated at the base point $x_0\in X$.
Recall that the operators $\mO_1,\mO_2$ were defined in (\ref{c31}).
\subsection{A formula for $\mO_2P^N$}\label{s4.1}
We will use the following Lemma to evaluate $b_{q,r}$ in (\ref{0.6}).
\begin{lemma}\label{t4.1}
The following relation holds:
\begin{equation}\label{g17}
\begin{split}
&\mO_2P^N = \Big\{\frac{1}{3}b_ib_j
\left \langle R^{TX} (\mR,\tfrac{\partial}{\partial\overline{z}_i})
\mR, \tfrac{\partial}{\partial\overline{z}_j}\right \rangle
+ \frac{1}{2} b_i \sum_{|\alpha|=2}(\partial ^{\alpha}R^L)_{x_0}
(\mR,\tfrac{\partial}{\partial \overline{z}_i})
\frac{Z^\alpha}{\alpha !} \\
&+ \frac{4}{3} b_j \left[ \left \langle R^{TX}
(\tfrac{\partial}{\partial z_i},\tfrac{\partial}{\partial\overline{z}_i})
\mR, \tfrac{\partial}{\partial\overline{z}_j}\right \rangle
- \left \langle R^{TX} (\mR,\tfrac{\partial}{\partial z_i})
\tfrac{\partial}{\partial \overline{z}_i},
\tfrac{\partial}{\partial\overline{z}_j}\right \rangle \right]
+R^E (\mR,\tfrac{\partial}{\partial \overline{z}_i})b_i \\
& +
\left \langle (\nabla ^{X} \nabla ^{X}\mJ)_{(\mR,\mR)}
\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial \overline{z}_i} \right \rangle
+ 4 \left \langle R^{TX} (\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial z_j}) \tfrac{\partial}{\partial \overline{z}_i},
\tfrac{\partial}{\partial \overline{z}_j}\right \rangle \Big\} P^N\\
&+ \Big(-\frac{1}{3}\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0 \left \langle R^{TX}
(\mR,\tfrac{\partial}{\partial z_j})
\mR, \tfrac{\partial}{\partial \overline{z}_j}\right \rangle
+\frac{1}{9} |(\nabla_\mR^X \mJ) \mR |^2
-\sum_{|\alpha|=2}(\partial ^{\alpha}\tau)_{x_0}
\frac{Z^\alpha}{\alpha !}+ \Phi \Big) P^N.
\end{split}
\end{equation}
\end{lemma}
\begin{proof}
Set
\begin{equation}\label{g18}
\begin{split}
I_1& = \frac{1}{2} \sum_{|\alpha|=2}(\partial ^{\alpha}R^L)_{x_0}
(\mR,\tfrac{\partial}{\partial \overline{z}_i})
\frac{Z^\alpha}{\alpha !}b_i\\
&- \frac{1}{2} {\tfrac{\partial}{\partial \overline{z}_i}}
\Big(\sum_{|\alpha|=2}(\partial ^{\alpha}R^L)_{x_0}
(\mR,\tfrac{\partial}{\partial z_i})
\frac{Z^\alpha}{\alpha !}\Big)
-\frac{1}{2}{\tfrac{\partial}{\partial z_i}}
\Big(\sum_{|\alpha|=2}(\partial ^{\alpha}R^L)_{x_0}
(\mR,\tfrac{\partial}{\partial \overline{z}_i})
\frac{Z^\alpha}{\alpha !}\Big), \\
I_2&=\frac{1}{3} \left \langle R^{TX}_{x_0}
(\mR,\tfrac{\partial}{\partial\overline{z}_i})
\mR, \tfrac{\partial}{\partial\overline{z}_j}\right \rangle b_ib_j\\
&-\frac{4}{3} \left[\left \langle R^{TX}_{x_0}
(\mR,\tfrac{\partial}{\partial \overline{z}_i} )
\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial \overline{z}_j}\right \rangle
+ \left \langle R^{TX}_{x_0} (\mR,\tfrac{\partial}{\partial z_i} )
\tfrac{\partial}{\partial \overline{z}_i},
\tfrac{\partial}{\partial \overline{z}_j}\right \rangle\right] b_j.
\end{split}
\end{equation}
From \eqref{c31}, \eqref{0ue62}, \eqref{g1}, \eqref{g2}, \eqref{g10},
\eqref{g18}, and since $\mJ$ is purely imaginary,
\begin{equation}\label{g19}
\begin{split}
\mO_2 &= I_1 +I_2-\frac{1}{3}\Big [\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0, \left \langle R^{TX}
(\mR,\tfrac{\partial}{\partial z_j})
\mR, \tfrac{\partial}{\partial \overline{z}_j}\right \rangle\Big]
+ R^E (\mR,\tfrac{\partial}{\partial \overline{z}_i})b_i\\
&+ \frac{1}{3} \left[
\left \langle R^{TX} (\mR,\tfrac{\partial}{\partial z_i})
\mR, \tfrac{\partial}{\partial \overline{z}_j}\right \rangle
(-2b_jb_i^+ -2a_i \delta_{ij})
+ \left \langle R^{TX} (\mR,\tfrac{\partial}{\partial z_i})
\mR, \tfrac{\partial}{\partial z_j}\right \rangle b_i^+b_j^+\right]\\
&+\Big( \frac{2}{3} \left \langle R^{TX}
(\mR, e_j) e_j,\tfrac{\partial}{\partial z_i}\right \rangle
- \Big(\frac{1}{2}\sum_{|\alpha|=2}(\partial ^{\alpha}R^L)_{x_0}
\frac{Z^\alpha}{\alpha !}
+R^E\Big) (\mR,\tfrac{\partial}{\partial z_i})\Big) b_i^+\\
&+\frac{1}{9} |(\nabla_\mR^X \mJ) \mR |^2
-\sum_{|\alpha|=2}(\partial ^{\alpha}\tau)_{x_0}
\frac{Z^\alpha}{\alpha !}+ \Phi.
\end{split}
\end{equation}
In normal coordinates, $(\nabla^{TX}_{e_i}e_j)_{x_0}=0$,
so from \eqref{0c31}, at $x_{0}$,
\begin{equation}\label{g20}
\begin{split}
\nabla _{e_j} \nabla _{e_i}\left \langle \mJ e_k, e_l \right
\rangle =& \left \langle (\nabla ^{X}_{e_j} \nabla
^{X}_{e_i} \mJ) e_k
+ \mJ(\nabla ^{TX}_{e_j} \nabla ^{TX}_{e_i} e_k), e_l \right\rangle
+ \left \langle \mJ e_k,
\nabla ^{TX}_{e_j} \nabla ^{TX}_{e_i}e_l \right \rangle \\
=& \left \langle (\nabla ^{X}_{e_j} \nabla ^{X}_{e_i} \mJ) e_k,
e_l \right \rangle
-\frac{1}{3} \left \langle R^{TX}(e_j,e_i)e_k+ R^{TX}(e_j,e_k)e_i,
\mJ e_l \right \rangle \\
&+\frac{1}{3} \left \langle R^{TX}(e_j,e_i)e_l+
R^{TX}(e_j,e_l)e_i, \mJ e_k \right \rangle .
\end{split}
\end{equation}
From \eqref{g10}, \eqref{g20},
\begin{equation}\label{g21}
\begin{split}
&\sum_{|\alpha|=2}(\partial ^{\alpha}R^L)_{x_0} (e_k,e_l)
\frac{Z^\alpha}{\alpha !}
= \frac{1}{2} (\nabla _{e_j} \nabla _{e_i}
\left \langle \mJ e_k, e_l \right \rangle )_{x_0} Z_iZ_j \\
&= \frac{1}{2} \left \langle (\nabla ^{X} \nabla ^{X}\mJ)_{(\mR,\mR)} e_k,
e_l \right \rangle + \frac{1}{6} \left[ \left \langle R^{TX}
(\mR, e_l)\mR, \mJ e_k \right \rangle - \left \langle R^{TX}
(\mR, e_k)\mR, \mJ e_l \right \rangle \right].
\end{split}
\end{equation}
Thus
\begin{align}\label{g22}
\sum_{|\alpha|=2}(\partial ^{\alpha}R^L)_{x_0}
(\mR, e_l) \frac{Z^\alpha}{\alpha !}
= \frac{1}{2} \left \langle (\nabla ^{X} \nabla ^{X}\mJ)_{(\mR,\mR)} \mR,
e_l \right \rangle
+ \frac{1}{6}\left \langle R^{TX} (\mR, \mJ \mR)\mR, e_l \right
\rangle .
\end{align}
From (\ref{g2}), (\ref{g7}), (\ref{g18}) and (\ref{g22}), we know that
\begin{equation}\label{g23}
\begin{split}
I_1 =\,&\frac{1}{2}\, b_i \sum_{|\alpha|=2}(\partial ^{\alpha}R^L)_{x_0}
(\mR,\tfrac{\partial}{\partial \overline{z}_i})
\frac{Z^\alpha}{\alpha !} \\
&+\frac{1}{12}\left[{\tfrac{\partial}{\partial z_i}}
\left \langle R^{TX}_{x_0} (\mR, \mJ \mR)\mR,
\tfrac{\partial}{\partial \overline{z}_i} \right\rangle
- {\tfrac{\partial}{\partial \overline{z}_i}}
\left \langle R^{TX}_{x_0} (\mR, \mJ \mR)\mR,
\tfrac{\partial}{\partial z_i} \right\rangle \right] \\
&+ \frac{1}{4} \left[ {\tfrac{\partial}{\partial z_i}}
\left \langle (\nabla ^{X} \nabla ^{X}\mJ)_{(\mR,\mR)} \mR,
\tfrac{\partial}{\partial \overline{z}_i} \right \rangle
- {\tfrac{\partial}{\partial \overline{z}_i}}
\left \langle (\nabla ^{X} \nabla ^{X}\mJ)_{(\mR,\mR)} \mR,
\tfrac{\partial}{\partial z_i} \right \rangle \right] .
\end{split}
\end{equation}
The definition of $\nabla ^{X}\nabla ^{X}\mJ$, $R^{TX}$ and \eqref{g9} imply,
for $U,V,W,Y\in TX$,
\begin{equation}\label{g24}
\begin{split}
&\left \langle R^{TX}(U,V)W,Y \right \rangle
=\left \langle R^{TX}(W,Y) U,V\right \rangle,\\
&R^{TX}(U,V)W+ R^{TX}(V,W)U+ R^{TX}(W,U)V=0, \\
& (\nabla ^{X}\nabla ^{X}\mJ)_{(U,V)}- (\nabla ^{X}\nabla ^{X}\mJ)_{(V,U)}
=[R^{TX}(U,V),\mJ],\\
&\left \langle (\nabla ^{X}\nabla ^{X}\mJ)_{(Y,U)}V,W \right \rangle
+ \left \langle (\nabla ^{X}\nabla ^{X}\mJ)_{(Y,V)}W,U \right \rangle
+\left \langle (\nabla ^{X}\nabla ^{X}\mJ)_{(Y,W)}U,V \right \rangle =0.
\end{split}
\end{equation}
Note that $\mJ,(\nabla ^{X}\nabla ^{X}\mJ)_{(Y,U)}$ are skew-adjoint,
by \eqref{0ue52} and \eqref{g24},
\begin{equation}\label{g25}
\begin{split}
{\tfrac{\partial}{\partial z_i}}&
\left \langle (\nabla ^{X} \nabla ^{X}\mJ)_{(\mR,\mR)} \mR,
\tfrac{\partial}{\partial \overline{z}_i} \right \rangle
- {\tfrac{\partial}{\partial \overline{z}_i}}
\left \langle (\nabla ^{X} \nabla ^{X}\mJ)_{(\mR,\mR)} \mR,
\tfrac{\partial}{\partial z_i} \right \rangle \\
=& \left \langle 2 (\nabla ^{X} \nabla ^{X}\mJ)_{(\mR,\mR)}
\tfrac{\partial}{\partial z_i}
+ 2(\nabla ^{X} \nabla ^{X}\mJ)_{(\mR,\tfrac{\partial}{\partial z_i})} \mR
+\Big[R^{TX} (\tfrac{\partial}{\partial z_i}, \mR), \mJ \Big] \mR,
\tfrac{\partial}{\partial \overline{z}_i} \right \rangle \\
-& \left \langle
2(\nabla ^{X} \nabla ^{X}\mJ)_{(\mR,\tfrac{\partial}{\partial \overline{z}_i})} \mR
+ \Big[R^{TX} (\tfrac{\partial}{\partial \overline{z}_i}, \mR),
\mJ \Big] \mR,
\tfrac{\partial}{\partial z_i} \right \rangle \\
=&4 \left \langle (\nabla ^{X} \nabla ^{X}\mJ)_{(\mR,\mR)}
\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial \overline{z}_i} \right \rangle
+ \left \langle 2 a_i R^{TX} (\mR, \tfrac{\partial}{\partial z_i}) \mR
- R^{TX} (\mR,\mJ \mR) \tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial \overline{z}_i} \right \rangle,
\end{split}
\end{equation}
\begin{equation*}
\begin{split}
{\tfrac{\partial}{\partial z_i}}&\left \langle R^{TX} (\mR, \mJ \mR)\mR,
\tfrac{\partial}{\partial \overline{z}_i} \right\rangle
- {\tfrac{\partial}{\partial \overline{z}_i}}
\left \langle R^{TX} (\mR, \mJ \mR)\mR,
\tfrac{\partial}{\partial z_i} \right\rangle \\
&= 2\left \langle R^{TX} (\mR, \mJ \mR)\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial \overline{z}_i} \right\rangle
+2a_i \left \langle R^{TX} (\mR, \tfrac{\partial}{\partial \overline{z}_i})\mR,
\tfrac{\partial}{\partial z_i} \right\rangle \\
&+\left \langle R^{TX} (\tfrac{\partial}{\partial z_i}, \mJ \mR)\mR,
\tfrac{\partial}{\partial \overline{z}_i} \right\rangle
-\left \langle R^{TX} (\tfrac{\partial}{\partial \overline{z}_i}, \mJ \mR)\mR,
\tfrac{\partial}{\partial z_i} \right\rangle \\
&=3\left \langle R^{TX} (\mR, \mJ \mR)\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial \overline{z}_i} \right\rangle
+2a_i \left \langle R^{TX} (\mR, \tfrac{\partial}{\partial z_i})\mR,
\tfrac{\partial}{\partial \overline{z}_i} \right\rangle .
\end{split}
\end{equation*}
Thus by (\ref{g23})-(\ref{g25}),
\begin{multline}\label{g26}
I_1 = \frac{1}{2} b_i \sum_{|\alpha|=2}(\partial ^{\alpha}R^L)_{x_0}
(\mR,\tfrac{\partial}{\partial \overline{z}_i})\frac{Z^\alpha}{\alpha !}
+\left \langle (\nabla ^{X} \nabla ^{X}\mJ)_{(\mR,\mR)}
\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial \overline{z}_i} \right \rangle \\
+ \frac{2}{3}a_i \left \langle R^{TX}
(\mR, \tfrac{\partial}{\partial z_i})\mR,
\tfrac{\partial}{\partial \overline{z}_i} \right\rangle.
\end{multline}
Now by (\ref{g2}), (\ref{g18}) and (\ref{g24}),
\begin{multline}\label{g27}
I_2 =\,\frac{4}{3}\, b_j \left[ \left \langle R^{TX}
(\tfrac{\partial}{\partial z_i},\tfrac{\partial}{\partial\overline{z}_i})
\mR, \tfrac{\partial}{\partial\overline{z}_j}\right \rangle
- \left \langle R^{TX} (\mR,\tfrac{\partial}{\partial z_i} )
\tfrac{\partial}{\partial \overline{z}_i},
\tfrac{\partial}{\partial \overline{z}_j}\right \rangle\right] \\
+ \frac{1}{3}\,b_ib_j
\left \langle R^{TX} (\mR,\tfrac{\partial}{\partial\overline{z}_i})
\mR, \tfrac{\partial}{\partial\overline{z}_j}\right \rangle
- \frac{8}{3} \left \langle R^{TX}
(\tfrac{\partial}{\partial z_j},\tfrac{\partial}{\partial z_i})
\tfrac{\partial}{\partial \overline{z}_i},
\tfrac{\partial}{\partial\overline{z}_j}\right \rangle \\
+ \frac{4}{3}\,\left[ \left \langle R^{TX}
(\tfrac{\partial}{\partial z_i},\tfrac{\partial}{\partial\overline{z}_i})
\tfrac{\partial}{\partial z_j},
\tfrac{\partial}{\partial\overline{z}_j}\right \rangle
- \left \langle R^{TX} (\tfrac{\partial}{\partial z_j},
\tfrac{\partial}{\partial\overline{z}_i}) \tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial\overline{z}_j}\right \rangle \right]\\
= \frac{4}{3}\, b_j \left[ \left \langle R^{TX}
(\tfrac{\partial}{\partial z_i},\tfrac{\partial}{\partial\overline{z}_i})
\mR, \tfrac{\partial}{\partial\overline{z}_j}\right \rangle
- \left \langle R^{TX} (\mR,\tfrac{\partial}{\partial z_i})
\tfrac{\partial}{\partial \overline{z}_i},
\tfrac{\partial}{\partial\overline{z}_j}\right \rangle \right] \\
+\frac{1}{3}\,b_ib_j
\left \langle R^{TX} (\mR,\tfrac{\partial}{\partial\overline{z}_i})
\mR, \tfrac{\partial}{\partial\overline{z}_j}\right \rangle
+ 4 \left \langle R^{TX} (\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial z_j}) \tfrac{\partial}{\partial \overline{z}_i},
\tfrac{\partial}{\partial \overline{z}_j}\right \rangle .
\end{multline}
Finally (\ref{g3}), (\ref{g14}), (\ref{g19}), (\ref{g26}) and (\ref{g27}),
yield (\ref{g17}).
\end{proof}
\noindent
From (\ref{g10}), (\ref{g13}), (\ref{g15}), (\ref{1c52}) and (\ref{g17}),
follows
\begin{multline}\label{0g27}
F_{1,2}= J_{1,2}P^N = \Big(
2R^E (\tfrac{\partial}{\partial z_i},\tfrac{\partial}{\partial \overline{z}_i})
+4 \left \langle R^{TX} (\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial z_j}) \tfrac{\partial}{\partial \overline{z}_i},
\tfrac{\partial}{\partial \overline{z}_j}\right \rangle + \Phi \Big) P^N\\
+ P^N \Big(\left \langle (\nabla ^{X} \nabla ^{X}\mJ)_{(\mR,\mR)}
\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial \overline{z}_i} \right \rangle
+\frac{\sqrt{-1}}{4}
\tr_{|TX} \Big(\nabla ^{X} \nabla ^{X}(J\mJ)\Big)_{(\mR,\mR)} \\
+ \frac{1}{9} |(\nabla_\mR^X \mJ) \mR |^2
+ \frac{4}{9} \left\langle(\nabla ^{X}_\mR\mJ)\mR,
\tfrac{\partial} {\partial z_i}\right\rangle b^+_i \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1}b_i
\left\langle(\nabla ^{X}_\mR\mJ)\mR,\tfrac{\partial}
{\partial\overline{z}_i}\right\rangle \Big) P^N.
\end{multline}
\subsection{The coefficients $b_{q,0}$\,}\label{s4.2}
In the rest of this Section we assume that ${\bf J}=J$.
A very useful observation is that
\eqref{g9}, \eqref{g11} imply
\begin{equation}\label{g29}
\begin{split}
&\text{$\mJ=-2\pi \sqrt{-1} J$ and $a_i=2\pi$ in (\ref{0ue52}), $\tau=2\pi n$.
$\nabla ^{X}_U J$ is skew-adjoint} \\
&\text{and the tensor $\left\langle(\nabla ^{X}_\cdot J)\cdot,
\cdot \right\rangle$
is of the type $(T^{*(1,0)}X)^{\otimes 3}\oplus (T^{*(0,1)}X)^{\otimes 3}$.}
\end{split}
\end{equation}
Before computing $b_{q,0}$, we establish the relation between
the scalar curvature $r^X$ and $|\nabla ^{X}J| ^2$.
\begin{lemma}
\begin{align}\label{g28}
r^X= 8 \left \langle R^{TX} (\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial \overline{z}_j})\tfrac{\partial}{\partial z_j},
\tfrac{\partial}{\partial \overline{z}_i} \right \rangle
- \frac{1}{4} |\nabla ^{X}J|^2.
\end{align}
\end{lemma}
\begin{proof}
By (\ref{g29}),
\begin{align}\label{g30}
|\nabla ^{X}J| ^2
=4 \Big \langle (\nabla ^{X}_{\tfrac{\partial}{\partial z_i}}J) e_j,
(\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_i}}J) e_j\Big \rangle
=8 \Big \langle (\nabla ^{X}_{\tfrac{\partial}{\partial z_i}} J)
\tfrac{\partial}{\partial z_j} ,
(\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_i}}J)
\tfrac{\partial}{\partial \overline{z}_j}\Big \rangle .
\end{align}
By (\ref{g0}), (\ref{g9}) and (\ref{g29}),
\begin{multline}\label{g31}
\Big \langle(\nabla ^{X}_{\tfrac{\partial}{\partial z_j}}J)
\tfrac{\partial}{\partial z_i},
(\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_i}}J)
\tfrac{\partial}{\partial \overline{z}_j}\Big\rangle
= 2\Big \langle(\nabla ^{X}_{\tfrac{\partial}{\partial z_j}}J)
\tfrac{\partial}{\partial z_i},\tfrac{\partial}{\partial z_k}\Big\rangle
\Big \langle (\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_i}}J)
\tfrac{\partial}{\partial \overline{z}_j}, \tfrac{\partial}{\partial \overline{z}_k}
\Big\rangle \\
= 2\Big \langle(\nabla ^{X}_{\tfrac{\partial}{\partial z_i}}J)
\tfrac{\partial}{\partial z_k}
- (\nabla ^{X}_{\tfrac{\partial}{\partial z_k}}J)
\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial z_j}\Big\rangle
\Big \langle (\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_i}}J)
\tfrac{\partial}{\partial \overline{z}_k}, \tfrac{\partial}{\partial \overline{z}_j}
\Big\rangle \\
=\Big\langle (\nabla ^{X}_{\tfrac{\partial}{\partial z_i}} J)
\tfrac{\partial}{\partial z_k} ,
(\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_i}}J)
\tfrac{\partial}{\partial \overline{z}_k}\Big\rangle
-\Big \langle(\nabla ^{X}_{\tfrac{\partial}{\partial z_k}}J)
\tfrac{\partial}{\partial z_i},
(\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_i}}J)
\tfrac{\partial}{\partial \overline{z}_k}\Big\rangle .
\end{multline}
By \eqref{g30} and \eqref{g31},
\begin{align}\label{g32}
\Big\langle(\nabla ^{X}_{\tfrac{\partial}{\partial z_j}}J)
\tfrac{\partial}{\partial z_i},
(\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_i}}J)
\tfrac{\partial}{\partial \overline{z}_j}\Big\rangle
= \frac{1}{16}|\nabla ^{X}J| ^2.
\end{align}
Now, from (\ref{g11}), we get
\begin{align}\label{g34}
(\nabla ^{X}\nabla ^{X}J)_{(U,V)}J +(\nabla ^{X}_UJ)\circ (\nabla ^{X}_VJ)
+ (\nabla ^{X}_VJ)\circ (\nabla ^{X}_UJ)
+J (\nabla ^{X}\nabla ^{X}J)_{(U,V)}=0.
\end{align}
thus from (\ref{g24}), (\ref{g29}) and (\ref{g34}),
for $u_1,u_2, u_{3}\in T^{(1,0)}X$, $\overline{v}_1,\overline{v}_2\in T^{(0,1)}X$,
\begin{equation}\label{g35}
\begin{split}
&(\nabla ^{X}\nabla ^{X}J)_{(u_1,u_2)}u_{3},\,
(\nabla ^{X}\nabla ^{X}J)_{(\overline{v}_1,\overline{v}_2)}u_{3}\in T^{(0,1)}X,
\quad (\nabla ^{X}\nabla ^{X}J)_{(u_1,\overline{v}_2)}u_{3}\in T^{(1,0)}X, \\
&2\sqrt{-1} \left\langle (\nabla ^{X}\nabla ^{X}J)_{(u_1,\overline{v}_1)}
u_2 , \overline{v}_2\right\rangle
= \left\langle (\nabla ^{X}_{u_1}J) u_2,
(\nabla ^{X}_{\overline{v}_1}J)\overline{v}_2\right\rangle \, .
\end{split}
\end{equation}
Formulas \eqref{g24} and \eqref{g35} yield
\begin{multline}\label{0g35}
\left\langle (\nabla ^{X}\nabla ^{X}J)_{(u_1,u_2)}\overline{v}_1,
\overline{v}_2\right\rangle
= -\left\langle (\nabla ^{X}\nabla ^{X}J)_{(u_1,\overline{v}_1)}\overline{v}_2,
u_2\right\rangle
- \left\langle (\nabla ^{X}\nabla ^{X}J)_{(u_1,\overline{v}_2)}u_2,
\overline{v}_1\right\rangle\\
= \frac{1}{2\sqrt{-1}} \left\langle (\nabla ^{X}_{u_1}J)u_2,
(\nabla ^X_{\overline{v}_1} J)\overline{v}_2
- (\nabla ^X_{\overline{v}_2} J)\overline{v}_1\right\rangle.
\end{multline}
From (\ref{g24}), (\ref{g32}) and (\ref{0g35}), we deduce
\begin{multline}\label{g36}
\left \langle R^{TX} (\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial z_j}) \tfrac{\partial}{\partial \overline{z}_i},
\tfrac{\partial}{\partial \overline{z}_j}\right \rangle
=\frac{\sqrt{-1}}{2} \left \langle [R^{TX} (\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial z_j}), J] \tfrac{\partial}{\partial \overline{z}_i},
\tfrac{\partial}{\partial \overline{z}_j}\right \rangle\\
=\frac{\sqrt{-1}}{2} \left \langle
\Big( (\nabla ^{X}\nabla ^{X}J)_{(\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial z_j})}
- (\nabla ^{X}\nabla ^{X}J)_{(\tfrac{\partial}{\partial z_j},
\tfrac{\partial}{\partial z_i})} \Big)
\tfrac{\partial}{\partial \overline{z}_i},
\tfrac{\partial}{\partial \overline{z}_j}\right \rangle\\
= \frac{1}{4} \Big\langle(\nabla ^{X}_{\tfrac{\partial}{\partial z_i}}J)
\tfrac{\partial}{\partial z_j} ,
(\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_i}}J)
\tfrac{\partial}{\partial \overline{z}_j}\Big\rangle
= \frac{1}{32}|\nabla ^{X}J|^2.
\end{multline}
The scalar curvature $r^X$ of $(X,g^{TX})$ is
\begin{equation}\label{g38}
\begin{split}
r^X =& -\left \langle R^{TX} (e_i,e_j)e_i,e_j \right \rangle
= -4 \left \langle R^{TX} (\tfrac{\partial}{\partial z_i},e_j)
\tfrac{\partial}{\partial \overline{z}_i},e_j \right \rangle\\
=&-8 \left \langle R^{TX} (\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial z_j})\tfrac{\partial}{\partial \overline{z}_i},
\tfrac{\partial}{\partial \overline{z}_j} \right \rangle
- 8 \left \langle R^{TX} (\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial \overline{z}_j})\tfrac{\partial}{\partial \overline{z}_i},
\tfrac{\partial}{\partial z_j} \right \rangle .
\end{split}
\end{equation}
In conclusion, relations \eqref{g36} and \eqref{g38} imply \eqref{g28}.
\end{proof}
\noindent
From \eqref{g13} and \eqref{g29} we know
\begin{align}\label{g41}
\mO_1=\frac{2}{3}b_i \left\langle(\nabla ^{X}_{\overline z}\mJ)\overline z,
\tfrac{\partial}{\partial\overline{z}_i}\right\rangle
- \frac{2}{3} \left\langle(\nabla ^{X}_z\mJ)z,
\tfrac{\partial}{\partial z_i} \right\rangle b_i^+.
\end{align}
Hence by (\ref{g2}), (\ref{g14}), (\ref{g29}) and (\ref{g41}),
\begin{equation}\label{g42}
\begin{split}
(\mO_1P^N )(Z,Z') =& \frac{2}{3} \Big( b_i
\left\langle(\nabla ^{X}_{\overline z}\mJ)\overline z,\tfrac{\partial}
{\partial\overline{z}_i}\right\rangle P^N\Big) (Z,Z')\\
=& \frac{2}{3} \Big\{\Big( \frac{b_ib_j}{2\pi}
\Big\langle (\nabla ^{X}_{\tfrac{\partial}
{\partial\overline{z}_j}}\mJ)\overline{z}',\tfrac{\partial}
{\partial\overline{z}_i} \Big\rangle
+ b_i \left\langle(\nabla ^{X}_{\overline{z}'}\mJ)\overline{z}',
\tfrac{\partial}{\partial\overline{z}_i}\right\rangle \Big) P^N\Big\} (Z,Z').
\end{split}
\end{equation}
By Theorem \ref{t3.4}, \eqref{g15} and \eqref{g42}, we have
\begin{align}\label{0g42}
&(\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1} P^{N^\bot} \mO_1 P^N)(Z,Z')\\
&\hspace*{10mm}= \frac{2}{3}\Big\{ \Big( \frac{b_ib_j}{16\pi ^2} \Big\langle
(\nabla ^{X}_{\tfrac{\partial}{\partial\overline{z}_j}}\mJ)\overline{z}',
\tfrac{\partial}{\partial\overline{z}_i}\Big\rangle
+ \frac{b_i}{4\pi} \left\langle(\nabla ^{X}_{\overline{z}'}\mJ)\overline{z}',
\tfrac{\partial}{\partial\overline{z}_i}\right\rangle \Big) P^N\Big\}(Z,Z'),
\nonumber\\
&P^N \mO_1 \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1} P^{N^\bot} \mO_1 P^N
=- \frac{2}{3} P^N \left\langle(\nabla ^{X}_z\mJ)z,
\tfrac{\partial}{\partial z_k} \right\rangle b_k^+ \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1}
P^{N^\bot} \mO_1P^N .\nonumber
\end{align}
(\ref{g0}), (\ref{g2}), (\ref{g14}), \eqref{g41} and (\ref{0g42}) imply
\begin{multline}\label{g43}
\frac{2}{3} \Big(\left\langle(\nabla ^{X}_z\mJ)z,
\tfrac{\partial}{\partial z_k} \right\rangle b_k^+ \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1}
P^{N^\bot} \mO_1P^N \Big )(Z,Z')
= \frac{4}{9} \left\{\left\langle(\nabla ^{X}_z\mJ)z,
\tfrac{\partial}{\partial z_k} \right\rangle \right. \\
\left. \times \left(- \frac{b_i}{4\pi} \Big\langle
(\nabla ^{X}_{\tfrac{\partial}{\partial\overline{z}_i}} \mJ)
\tfrac{\partial}{\partial\overline{z}_k}
+(\nabla ^{X}_{\tfrac{\partial}{\partial\overline{z}_k}} \mJ)
\tfrac{\partial}{\partial\overline{z}_i} ,\overline{z}' \Big\rangle
+ \left\langle(\nabla ^{X}_{\overline{z}'}\mJ)\overline{z}',
\tfrac{\partial}{\partial\overline{z}_k}\right\rangle \right) P^N\right\} (Z,Z')\\
= \left\{\left[ -\frac{b_i}{9\pi} \left\langle(\nabla ^{X}_z\mJ)z,
\tfrac{\partial}{\partial z_k} \right\rangle
\left\langle (\nabla ^{X}_{\tfrac{\partial}{\partial\overline{z}_i}} \mJ)
\tfrac{\partial}{\partial\overline{z}_k}
+(\nabla ^{X}_{\tfrac{\partial}{\partial\overline{z}_k}} \mJ)
\tfrac{\partial}{\partial\overline{z}_i} ,\overline{z}' \right\rangle \right. \right.\\
-\frac{2}{9\pi} \left\langle(\nabla ^{X}_{z}\mJ)
\tfrac{\partial}{\partial z_i}
+(\nabla ^{X}_{\tfrac{\partial}{\partial z_i}}\mJ) z,
\tfrac{\partial}{\partial z_k} \right\rangle
\left\langle (\nabla ^{X}_{\tfrac{\partial}{\partial\overline{z}_i}} \mJ)
\tfrac{\partial}{\partial\overline{z}_k}
+(\nabla ^{X}_{\tfrac{\partial}{\partial\overline{z}_k}} \mJ)
\tfrac{\partial}{\partial\overline{z}_i} ,\overline{z}' \right\rangle \\
\left. \left. + \frac{2}{9}\left\langle(\nabla ^{X}_{z}\mJ)z,
(\nabla ^{X}_{\overline{z}'}\mJ)\overline{z}'\right\rangle\right] P^N \right\} (Z,Z').
\end{multline}
Thanks to \eqref{g11}, (\ref{g14}), (\ref{g29}), (\ref{g30}) and (\ref{g31}) we obtain
\begin{multline}\label{g44}
\frac{1}{9} \Big |(\nabla_\mR^X \mJ) \mR \Big |^2P^N(Z,Z')=
\frac{8\pi ^2}{9} \left\langle(\nabla ^{X}_{z}J)z,
(\nabla ^{X}_{\overline{z}}J)\overline{z} \right\rangle P^N(Z,Z') \\
=\frac{8\pi ^2}{9} \left\{\left\langle(\nabla ^{X}_{z}J)z,
\frac{b_ib_j}{4\pi ^2} (\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_i}}J)
\tfrac{\partial}{\partial \overline{z}_j}
+ \frac{b_i}{2 \pi} \Big[
(\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_i}}J)
\overline{z}'+(\nabla ^{X}_{\overline{z}'}J)
\tfrac{\partial}{\partial \overline{z}_i} \Big]
+ (\nabla ^{X}_{\overline{z}'}J)\overline{z}' \right\rangle P^N\right\}(Z,Z')\\
= \frac{8\pi ^2}{9} \left\{ \left[\left\langle \frac{b_ib_j}{4\pi ^2}
(\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_i}}J)
\tfrac{\partial}{\partial \overline{z}_j}
+\frac{b_i}{2 \pi}\Big (
(\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_i}}J)
\overline{z}'+(\nabla ^{X}_{\overline{z}'}J)
\tfrac{\partial}{\partial \overline{z}_i}\Big) ,
(\nabla ^{X}_{z}J)z \right\rangle\right.\right.\\
+\frac{b_i}{2\pi ^2} \left\langle
(\nabla ^{X}_{\tfrac{\partial}{\partial z_j}}J)z
+ (\nabla ^{X}_{z}J)\tfrac{\partial}{\partial z_j},
(\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_i}}J)
\tfrac{\partial}{\partial \overline{z}_j}
+(\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_j}}J)
\tfrac{\partial}{\partial \overline{z}_i} \right\rangle \\
+ \frac{1}{\pi} \left\langle
(\nabla ^{X}_{\tfrac{\partial}{\partial z_i}}J)z
+(\nabla ^{X}_{z}J)\tfrac{\partial}{\partial z_i},
(\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_i}}J)\overline{z}'
+(\nabla ^{X}_{\overline{z}' }J)\tfrac{\partial}{\partial \overline{z}_i}
\right\rangle\\
\left.\left. + \left\langle (\nabla ^{X}_{z}J)z,
(\nabla ^{X}_{\overline{z}'}J)\overline{z}'\right\rangle
+ \frac{3}{16 \pi ^2} |\nabla ^{X}J|^2
\right] P^N\right\}(Z,Z').
\end{multline}
Taking into account (\ref{g24}), (\ref{g35}) and $\left\langle [R^{TX}(\overline{z},z), J]
\tfrac{\partial}{\partial z_i} ,
\tfrac{\partial}{\partial \overline{z}_i}\right\rangle=0$, we get
\begin{multline}\label{g45}
\left\langle (\nabla ^{X}\nabla ^{X}J)_{(\mR,\mR)}
\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial \overline{z}_i}\right\rangle
= \left\langle (\nabla ^{X}\nabla ^{X}J)_{(z,\overline{z})}
\tfrac{\partial}{\partial z_i} +(\nabla ^{X}\nabla ^{X}J)_{(\overline{z},z)}
\tfrac{\partial}{\partial z_i} ,
\tfrac{\partial}{\partial \overline{z}_i}\right\rangle\\
=-\sqrt{-1} \left\langle (\nabla ^{X}_zJ)
\tfrac{\partial}{\partial z_i} ,
(\nabla ^{X}_{\overline{z}}J)\tfrac{\partial}{\partial \overline{z}_i}\right\rangle.
\end{multline}
From (\ref{g14}), (\ref{g30}) and (\ref{g45}),
\begin{multline}\label{g46}
\left\langle (\nabla ^{X}\nabla ^{X}\mJ)_{(\mR,\mR)}
\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial \overline{z}_i}\right\rangle P^N (Z,Z')
= - 2 \pi
\left\langle (\nabla ^{X}_zJ)
\tfrac{\partial}{\partial z_i} ,
(\nabla ^{X}_{\overline{z}}J)\tfrac{\partial}{\partial \overline{z}_i}\right\rangle
P^N(Z,Z')\\
=- 2 \pi \Big\{ \Big\langle (\nabla ^{X}_zJ)
\tfrac{\partial}{\partial z_i} , \frac{b_j}{2 \pi}
(\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_j}}J)
\tfrac{\partial}{\partial \overline{z}_i}+ (\nabla ^{X}_{\overline{z}'}J)
\tfrac{\partial}{\partial \overline{z}_i} \Big\rangle P^N\Big\}(Z,Z')\\
=- \Big\{\Big [\Big\langle b_j
(\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_j}}J)
\tfrac{\partial}{\partial \overline{z}_i}
+ 2\pi (\nabla ^{X}_{\overline{z}'}J)\tfrac{\partial}{\partial \overline{z}_i}
, (\nabla ^{X}_zJ)\tfrac{\partial}{\partial z_i} \Big\rangle
+\frac{1}{4} |\nabla ^{X}J|^2 \Big ] P^N\Big\}(Z,Z').
\end{multline}
Recall that the polynomial $J_{q,2q} (Z,Z')$ was defined in (\ref{c86}).
From $J\mJ=2\pi \sqrt{-1}$, (\ref{g15}), (\ref{0g27}),
(\ref{g36}) and (\ref{g41})-(\ref{g46}),
$J_{1,2}(Z,Z')$ is a polynomial on $z,\overline{z}'$, and each monomial of $J_{1,2}$
has the same degree in $z$ and $\overline{z}'$; moreover
\begin{align}\label{g48}
J_{1,2}(0,0)
= \frac{1}{24} |\nabla ^{X}J| ^2_{x_0} +2 R^E_{x_0}
(\tfrac{\partial}{\partial z_i},\tfrac{\partial}{\partial \overline{z}_i})
+ \Phi_{x_0}.
\end{align}
Using (\ref{g7}), (\ref{1c52}) and the recurrence,
we infer that each monomial of $J_{q,2q}$
has the same degree in $z$ and $\overline{z}'$, and
\begin{align}\label{g49}
J_{q,2q}(0,0) = (J_{1,2}(0,0))^q.
\end{align}
In view of (\ref{g7}), (\ref{c86}), (\ref{1c54}), (\ref{g48}) and
(\ref{g49}) we obtain (\ref{0.7}).
\comment{\begin{rem}\label{t4.2} If ${\bf J}\neq J$, it seems that the
monomial of $J_{1,2}$ will involve the term $z^\alpha$ and ${\overline{z}'}^\beta$,
thus (\ref{g49}) will not hold in general.
\end{rem}}
\subsection{The coefficient $b_{0,1}$ }\label{s4.3}
By (\ref{1c54}), we need to compute $F_{0,2} (0,0)$.
By \eqref{g42} and \eqref{0g42}, we know that
\begin{equation}\label{0g52}
(\mO_1 P^N)(Z,0)=0, \quad (\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1}P^{N^\bot} \mO_1 P^N)(0,Z^\prime)=0.
\end{equation}
Thus the first and last two terms in \eqref{g51} are zero at $(0,0)$.
Thus we only need to compute $-(\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1}P^{N^\bot} \mO_2 P^N)(0,0)$,
since the third and fourth terms in \eqref{g51}
are adjoint of the first two terms by Remark \ref{0t3.3}.
Let $h_i(z)$ and $f_{ij}(z)$, $(i,j=1,\cdots, n)$ be arbitrary
polynomials in $z$.
By Theorem \ref{t3.4}, (\ref{g2}), (\ref{g7}) and (\ref{g14}), we have
\begin{equation}\label{g52}
\begin{split}
&(b_ih_iP^N)(0,0) = -2 \frac{\partial h_i}{\partial z_i}(0),\, \quad
(b_ib_jf_{ij}P^N)(0,0) =
4 \frac{\partial ^2 f_{ij}}{\partial z_i\partial z_j}(0),\\
&(\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1} b_if_{ij}b_jP^N)(0,0)=
-\frac{1}{2\pi}\frac{\partial ^2 f_{ij}}{\partial z_i\partial z_j}(0).
\end{split}
\end{equation}
Owing to Theorem \ref{t3.4}, (\ref{g30}), (\ref{g32}), (\ref{g44})
and (\ref{g52}),
\begin{multline}\label{g53}
-\frac{1}{9} \Big( \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1} P^{N^\bot}
\Big |(\nabla_\mR^X \mJ) \mR \Big |^2P^N\Big)(0,0)=- \frac{8}{9} \left\{
\Big[
\frac{b_ib_j}{32\pi } \left\langle(\nabla ^{X}_{z}J)z,
(\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_i}}J)
\tfrac{\partial}{\partial \overline{z}_j} \right\rangle \right.\\
\left. +\frac{b_i}{8\pi} \Big\langle
(\nabla ^{X}_{\tfrac{\partial}{\partial z_j}}J)z
+ (\nabla ^{X}_{z}J)\tfrac{\partial}{\partial z_j},
(\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_i}}J)
\tfrac{\partial}{\partial \overline{z}_j}
+(\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_j}}J)
\tfrac{\partial}{\partial \overline{z}_i} \Big\rangle \Big] P^N\right\}(0,0)\\
= \frac{1}{9 \pi}
\Big \langle(\nabla ^{X}_{\tfrac{\partial}{\partial z_i}}J)
\tfrac{\partial}{\partial z_j} +
(\nabla ^{X}_{\tfrac{\partial}{\partial z_j}}J)
\tfrac{\partial}{\partial z_i},
(\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_i}}J)
\tfrac{\partial}{\partial \overline{z}_j}
+2 (\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_i}}J)
\tfrac{\partial}{\partial \overline{z}_j}\Big \rangle
= \frac{1}{16\pi } |\nabla ^X J |^2\,,
\end{multline}
and by Theorem \ref{t3.4}, (\ref{g46}) and (\ref{g52}),
\begin{multline}\label{g54}
-\Big( \mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1} P^{N^\bot} \left\langle (\nabla ^{X}\nabla ^{X}\mJ)_{(\mR,\mR)}
\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial \overline{z}_i}\right\rangle P^N\Big) (0,0)\\
= \Big(\frac{b_j}{4\pi} \Big\langle (\nabla ^{X}_z J)
\tfrac{\partial}{\partial z_i} ,
(\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_j}}J)
\tfrac{\partial}{\partial \overline{z}_i} \Big\rangle P^N\Big) (0,0)
= -\frac{1}{16\pi}|\nabla ^X J |^2.
\end{multline}
Observe that by (\ref{g14}), for a polynomial $g(z)$ in $z$,
the constant term of $\frac{1}{P^N}\frac{b^\alpha}{2^{|\alpha|}} g(z)P^N$ is
the constant term of $(\tfrac{\partial}{\partial z})^\alpha g$.
Thus in the term $-\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1} P^{N^\bot}\mO_2 P^N$,
by \eqref{g14}, the contribution of $\frac{1}{2}b_i
\sum_{|\alpha|=2}(\partial ^{\alpha}R^L)_{x_0}
(\mR,\tfrac{\partial}{\partial \overline{z}_i})\frac{Z^\alpha}{\alpha !}$
in $\mO_2$ consists of the terms whose total degree of $b_i$ and $\overline{z}_j$
is same as the degree of $z$. Hence we only need to consider
the contribution from the terms where the degree of $z$ is $2$.
By (\ref{g22}), (\ref{g29}), (\ref{g35}), (\ref{0g35})
and $\left\langle [ R^{TX}(\overline{z}, z), \mJ]z,
\tfrac{\partial}{\partial \overline{z}_i}\right\rangle =0$, this term is
\begin{multline}\label{g55}
I_3= \frac{1}{4} b_i \Big [
\left\langle (\nabla ^{X}\nabla ^{X}\mJ)_{(z,z)}\overline{z}
+(\nabla ^{X}\nabla ^{X}\mJ)_{(z,\overline{z})}z
+(\nabla ^{X}\nabla ^{X}\mJ)_{(\overline{z},z)}z,
\tfrac{\partial}{\partial \overline{z}_i}\right\rangle\\
+\frac{1}{3} \left\langle R^{TX}(\overline{z},\mJ z)z + R^{TX}(z,\mJ \overline{z})z,
\tfrac{\partial}{\partial \overline{z}_i} \right\rangle \Big ]\\
=-\frac{\pi}{4} b_i \Big [ \Big\langle (\nabla ^{X}_zJ)z,
3 (\nabla ^{X}_{\overline{z}}J)\tfrac{\partial}{\partial \overline{z}_i}
- (\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_i}}J)\overline{z}\Big\rangle
+\frac{4}{3}\left\langle R^{TX}(z,\overline{z})z,
\tfrac{\partial}{\partial \overline{z}_i} \right\rangle \Big ].
\end{multline}
Therefore, from (\ref{g14}), (\ref{g24}), (\ref{g32}), (\ref{g36}), (\ref{g52})
and (\ref{g55}), we get
\begin{multline}\label{g58}
- \Big(\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1} P^{N^\bot}I_3P^N\Big) (0,0)
= \frac{\pi}{4} \Big\{\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1} b_i \Big [\frac{4}{3}
\left\langle R^{TX}(z, \tfrac{\partial}{\partial \overline{z}_j})z,
\tfrac{\partial}{\partial \overline{z}_i} \right\rangle\\
+ \Big\langle (\nabla ^{X}_zJ)z,
3 (\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_j} }J)
\tfrac{\partial}{\partial \overline{z}_i}
- (\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_i}}J)
\tfrac{\partial}{\partial \overline{z}_j}\Big\rangle \Big ]
\frac{b_j}{2\pi}P^N\Big\} (0,0) \\
= - \frac{1}{16\pi} \Big[ \frac{4}{3} \left\langle
R^{TX} (\tfrac{\partial}{\partial z_j},\tfrac{\partial}{\partial \overline{z}_j})
\tfrac{\partial}{\partial z_i}
+ R^{TX}(\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial \overline{z}_j})\tfrac{\partial}{\partial z_j},
\tfrac{\partial}{\partial \overline{z}_i}\right\rangle\\
+\Big\langle
(\nabla ^{X}_{\tfrac{\partial}{\partial z_i}} J)\tfrac{\partial}{\partial z_j}
+ (\nabla ^{X}_{\tfrac{\partial}{\partial z_j}} J)
\tfrac{\partial}{\partial z_i},
3(\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_i}}J)
\tfrac{\partial}{\partial \overline{z}_j}
-(\nabla ^{X}_{\tfrac{\partial}{\partial \overline{z}_j}}J)
\tfrac{\partial}{\partial \overline{z}_i}\Big\rangle \Big ] \\
= - \frac{5}{192\pi } |\nabla ^X J |^2 - \frac{1}{6\pi}\left\langle
R^{TX} (\tfrac{\partial}{\partial z_i},\tfrac{\partial}{\partial \overline{z}_j})
\tfrac{\partial}{\partial z_j},
\tfrac{\partial}{\partial \overline{z}_i}\right\rangle.
\end{multline}
Thanks to (\ref{g14}), (\ref{g36}) and (\ref{g52}) we have
\begin{multline}\label{g59}
\frac{1}{3} \Big(P^{N^\bot} \left \langle R^{TX}
(\mR,\tfrac{\partial}{\partial z_i})
\mR, \tfrac{\partial}{\partial\overline{z}_i} \right \rangle P^N\Big) (0,0)\\
= \frac{1}{3} \Big(P^{N^\bot}
\left \langle R^{TX} (z,\tfrac{\partial}{\partial z_i})
\tfrac{\partial}{\partial \overline{z}_j}
+ R^{TX}(\tfrac{\partial}{\partial \overline{z}_j},
\tfrac{\partial}{\partial z_i})z,
\tfrac{\partial}{\partial\overline{z}_i}\right \rangle
\frac{b_j}{2\pi} P^N\Big) (0,0)\\
= -\frac{1}{3\pi} \Big [ \left \langle R^{TX}
(\tfrac{\partial}{\partial z_j},\tfrac{\partial}{\partial z_i})
\tfrac{\partial}{\partial \overline{z}_j}
+ R^{TX}(\tfrac{\partial}{\partial \overline{z}_j},
\tfrac{\partial}{\partial z_i})\tfrac{\partial}{\partial z_j},
\tfrac{\partial}{\partial\overline{z}_i}\right \rangle \Big ] \\
= - \frac{1}{96\pi } |\nabla ^X J |^2 + \frac{1}{3\pi}\left\langle
R^{TX} (\tfrac{\partial}{\partial z_i},\tfrac{\partial}{\partial \overline{z}_j})
\tfrac{\partial}{\partial z_j},
\tfrac{\partial}{\partial \overline{z}_i}\right\rangle.
\end{multline}
By (\ref{g29}), (\ref{g52}), (\ref{g53}), (\ref{g54}),
(\ref{g58}), (\ref{g59}), and the discussion above \eqref{g55},
\begin{multline}\label{g60}
-\Big(\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1} P^{N^\bot} \mO_2 P^N\Big) (0,0)=
- \Big\{ \Big[\frac{b_ib_j }{24\pi}
\left \langle R^{TX} (z,\tfrac{\partial}{\partial\overline{z}_i})
z, \tfrac{\partial}{\partial\overline{z}_j}\right \rangle
+\frac{b_i}{4\pi}R^E (z,\tfrac{\partial}{\partial \overline{z}_i})\\
+ \frac{b_j}{3\pi} \left( \left \langle R^{TX}
(\tfrac{\partial}{\partial z_i},\tfrac{\partial}{\partial\overline{z}_i})
z, \tfrac{\partial}{\partial\overline{z}_j}\right \rangle
- \left \langle R^{TX} (z,\tfrac{\partial}{\partial z_i})
\tfrac{\partial}{\partial \overline{z}_i},
\tfrac{\partial}{\partial\overline{z}_j}\right \rangle \right)
\Big]P^N \Big\} (0,0) \\
+\frac{1}{3} \Big( P^{N^\bot} \left \langle R^{TX}
(\mR,\tfrac{\partial}{\partial z_i})
\mR, \tfrac{\partial}{\partial\overline{z}_i} \right \rangle P^N \Big) (0,0)
- \Big(\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1} P^{N^\bot} I_3 P^N \Big) (0,0) \\
= -\frac{1}{6 \pi} \left \langle R^{TX}
(\tfrac{\partial}{\partial z_i},\tfrac{\partial}{\partial\overline{z}_i})
\tfrac{\partial}{\partial z_j}
+ R^{TX} (\tfrac{\partial}{\partial z_j},\tfrac{\partial}{\partial\overline{z}_i})
\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial\overline{z}_j}\right \rangle
+ \frac{1}{2\pi}R^E (\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial \overline{z}_i}) \\
+\frac{2}{3\pi} \left[ \left \langle R^{TX}
(\tfrac{\partial}{\partial z_i},\tfrac{\partial}{\partial\overline{z}_i})
\tfrac{\partial}{\partial z_j},
\tfrac{\partial}{\partial\overline{z}_j}\right \rangle
- \left \langle R^{TX} (\tfrac{\partial}{\partial z_j},
\tfrac{\partial}{\partial z_i}) \tfrac{\partial}{\partial \overline{z}_i},
\tfrac{\partial}{\partial\overline{z}_j}\right \rangle \right]\\
- \frac{7}{192\pi } |\nabla ^X J |^2 + \frac{1}{6\pi}\left\langle
R^{TX} (\tfrac{\partial}{\partial z_i},\tfrac{\partial}{\partial \overline{z}_j})
\tfrac{\partial}{\partial z_j},
\tfrac{\partial}{\partial \overline{z}_i}\right\rangle\\
= \frac{1}{2\pi} \left\langle
R^{TX} (\tfrac{\partial}{\partial z_i},\tfrac{\partial}{\partial \overline{z}_j})
\tfrac{\partial}{\partial z_j},
\tfrac{\partial}{\partial \overline{z}_i}\right\rangle
+\frac{1}{2\pi}R^E (\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial \overline{z}_i}).
\end{multline}
Formulas (\ref{g28}), (\ref{g60}) and the discussion
at the beginning of Section \ref{s4.3} yield finally
\begin{equation}\label{g61}
\begin{split}
b_{0,1}(x_0)=&F_{0,2}(0,0)
=- \Big(\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1} P^{N^\bot} \mO_2 P^N \Big) (0,0)
- \Big ( \Big(\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0^{-1} P^{N^\bot} \mO_2 P^N \Big) (0,0)\Big)^*\\
=& \,\frac{1}{8\pi}\Big[r^X_{x_0}+ \frac{1}{4} |\nabla ^X J |^2_{x_0}
+ 2\sqrt{-1} R^E_{x_0} (e_j,Je_j)\Big].
\end{split}
\end{equation}
The proof of Theorem \ref{t0.2} is complete.
\begin{rem} In the K\"ahler case, i.e. $J$ is integrable and $L,E$
are holomorphic, then $\mO_1=0$, and the above computation
simplifies a lot.
\end{rem}
\section{Applications} \label{s5}
In this Section, we discuss various applications of our results.
In Section \ref{s5.1},
we study the density of states
function of $\Delta_{p,\Phi}$.
In Section \ref{s5.3}, we explain how to handle
the first-order pseudo-differential operator $D_b$
of Boutet de Monvel and Guillemin \cite{BoG} which was studied
extensively by Shiffman and Zelditch \cite{SZ02}.
In Section \ref{s5.31}, we prove a symplectic version of
the convergence of the Fubini-Study metric of
an ample line bundle \cite{Tian}.
In Section \ref{s5.4}, we show how to handle the operator
$\overline{\partial}+\overline{\partial}^*$ when $X$ is K\" ahler but ${\bf J}\neq J$.
Finally, in Sections \ref{s5.5}, \ref{s5.6}, we establish some generalizations
for non-compact or singular manifolds.
\subsection{Density of states function}\label{s5.1}
Let $(X,\omega} \newcommand{\g}{\Gamma)$ be a compact symplectic manifold of real dimension $2n$
and $(L,\nabla ^L, h^L)$
is a pre-quantum line bundle as in Section \ref{s1}.
Assume that $E$ is the trivial bundle $\field{C}$,
$\Phi=0$ and $\mathbf{J}=J$. The latter
means, by \eqref{0.1}, that $g^{TX}$ is the Riemannian metric
associated to $\omega$ and $J$. We denote by
$\vol (X) =\int_X \frac{\omega ^n}{n!}$ the Riemannian volume of $(X,g^{TX})$.
Recall that $d_p$ is defined in (\ref{0.0}).
Our aim is to describe the asymptotic distribution of the energies
of the bound states as $p$ tends to infinity.
We define the spectrum counting function of $\Delta_{p}:=\Delta_{p,0}$ by
$N_{p}(\lambda)=\#\left\{i\,:\,\lambda_{i,p}\leqslant\lambda\right\}$
and the spectral density measure on $[-C_L,C_L]$ by
\begin{gather}
\nu_p=\frac{1}{d_p}\,\frac{d}{d\lambda}N_{p}(\lambda)\,,\quad
\lambda\in[-C_L,C_L]\,.\label{f2}
\end{gather}
Clearly, $\nu_p$ is a sum of Dirac measures supported on
$\spec\Delta_{p}\cap[-C_L,C_L]$. Set
\begin{equation}\label{f3}
\varrho:X\longrightarrow\field{R}\,,\quad\varrho(x)
=\frac{1}{24}\abs{\nabla^{X}J}^2\,.
\end{equation}
\begin{thm}\label{spectral-density}
The weak limit of the sequence $\{\nu_p\}_{p\geqslant1}$ is the direct image measure
$\varrho_{\ast}\Big(\dfrac{1}{\vol (X)}\dfrac{\omega} \newcommand{\g}{\Gamma^n}{n!}\Big)$, that is,
for any continuous function $f\in\cali{C}} \newcommand{\cA}{\cali{A}([-C_L,C_L])$, we have
\begin{equation}\label{f4}
\lim_{p\to\infty}\int_{-C_L}^{C_L}f\,d\nu_p
=\frac{1}{\vol (X)}\int_{X}(f\circ\varrho)\,\frac{\omega} \newcommand{\g}{\Gamma^n}{n!}\:\:.
\end{equation}
\end{thm}
\begin{proof}
By \eqref{0.4}, we have for $q\geqslant1$ (now $E$ is trivial):
$B_{q,p}(x)=\sum_{i=1}^{d_p} \lambda_{i,p}^q\,\abs{S^{p}_{i}(x)}^2$,
which yields by integration over $X$,
\begin{gather}
\frac{1}{d_p}\int_{X} B_{q,\,p}\,dv_{X}=
\frac{1}{d_p}\sum_{i=1}^{d_p} \lambda_{i,p}^q=
\int_{-C_L}^{C_L}\lambda^q\,d\nu_p(\lambda)\,,\label{f5}
\end{gather}
since $S^{p}_{i}$ have unit $L^2$ norm. On the other hand,
\eqref{0.0}, \eqref{0.6} entail for $p\to \infty$,
\begin{align}\label{f6}
\frac{1}{d_p}\int_{X} B_{q,\,p}\,dv_{X}&=
\frac{p^n}{d_p}\int_{X} b_{q,0}\,dv_{X}+\frac{\cali{O}} \newcommand{\cE}{\cali{E}(p^{n-1})}{d_p}\\
&= \frac{1}{\vol (X)}\int_{X}\varrho^q\,dv_{X} + \cali{O}} \newcommand{\cE}{\cali{E}(p^{-1}).\nonumber
\end{align}
We infer from \eqref{f5} and \eqref{f6} that \eqref{f4} holds for
$f(\lambda)=\lambda^q$, $q\geqslant1$. Since this is obviously true for
$f(\lambda)\equiv1$, too, we deduce it holds for all polynomials.
Upon invoking the Weierstrass approximation theorem, we get \eqref{f4}
for all continuous functions on $[-C_L,C_L]$. This achieves the proof.
\end{proof}
\begin{rem} \label{t5.2} A function $\varrho$ satisfying \eqref{f4} is called spectral
density function. Its existence and uniqueness were demonstrated
by Guillemin-Uribe \cite{GU}.
As for the explicit formula of $\varrho$, the paper \cite{BU2}
is dedicated to its computation. Our formula (\ref{f3}) is different from
\cite[Theorem 1.2]{BU2}
\footnote{In \cite[(3.7)]{BU2}, the leading term of
$G_{0j}$ should be $\kappa ^{-1/2} b^{(1)}_j$ which was missed therein,
as the principal terms of $\frac{\partial}{\partial s}$,
$\frac{\partial}{\partial y^j}$ are $\partial_0$, $T^l_j \partial_l$ by
\cite[equation after (3.11)]{BU2}. Now, from \cite[(3.5)]{BU2},
$ b^{(1)}_j$ is $\frac{1}{2}\langle Jz,T^l_j \partial_l\rangle$.
Thus $\mL_0$ in \cite[(3.8)]{BU2} is incorrect.
\cite[Theorem 1.2]{BU2} is $\varrho(x)=
-\frac{5}{24}\abs{\nabla^{X}J}^2$. }.
\end{rem}
An interesting corollary of (\ref{f3}) and (\ref{f4})
is the following result which was first stated in \cite[Cor. 1.3]{BU2}.
\begin{cor}\label{t5.3}
The spectral density function is identically zero if and only if
$(X,\omega,J)$ is K{\"a}hler.
\end{cor}
\begin{rem}\label{t5.4}
Theorem \ref{spectral-density} can be slightly generalized. Assume namely
that ${\bf J}=J$ and
$E$ is a Hermitian vector bundle as in Section \ref{s1} such that
$R^E=\eta\otimes\Id_E$, $\Phi=\varphi\Id_E$,
where $\eta$ is a $2$-form and $\varphi$ a real function on $X$.
Then there exists a spectrum density function satisfying \eqref{f4}
given by
\begin{equation}\label{f8}
\varrho:X\longrightarrow\field{R}\,,\quad\varrho(x)=
\frac{1}{24}\abs{\nabla^{X}J}^2+\frac{\sqrt{-1}}{2}\eta(e_j,Je_j) +
\varphi\,.
\end{equation}
The proof is similar to the previous one, as
$\tr_{E_x}B_{q,p}(x)=\sum_{i=1}^{d_p} \lambda_{i,p}^q\,\abs{S^{p}_{i}(x)}^2$.
\end{rem}
\subsection{Almost-holomorphic Szeg{\" o} kernels}\label{s5.3}
We use the notations and assumptions from Section \ref{s5.1},
especially, ${\bf J}=J$. Then $\tau=2\pi n$.
Let $Y= \{u\in L^*, |u|_{h^{L^*}}=1\}$ be the unit circle bundle in $L^*$.
Then the smooth sections of $L^p$ can be identified to the smooth functions
$\cali{C}} \newcommand{\cA}{\cali{A}^\infty (Y)_p= \{ f\in \cali{C}} \newcommand{\cA}{\cali{A}^\infty(Y,\field{C});
f(ye ^{i\theta})$ $= e ^{ip\theta} f(y)\, \, {\rm for}\, \,
e ^{i\theta}\in S^1, y\in Y \}$,
here $ye ^{i\theta}$ is the $S^1$ action on $Y$.
The connection $\nabla ^L$ on $L$ induces a connection
on the $S^1$-principal bundle $\pi : Y\to X$,
and let $T^H Y \subset TY$ be the corresponding horizontal bundle.
Let $g^{TY}= \pi ^ * g^{TX} \oplus d\theta ^2$ be the metric on
$TY= T^HY\oplus TS ^1$, with $d\theta ^2$
the standard metric on $S^1= \field{R}/2\pi \field{Z}$.
Let $\Delta_Y$ be the Bochner-Laplacian on $(Y,g^{TY})$, then by construction,
it commutes with the generator $\partial_\theta$ of the circle action,
and so it commutes with the horizontal Laplacian
\begin{align}\label{f20}
\Delta_h= \Delta_Y+ \partial_\theta ^2,
\end{align}
then $\Delta_h$ on $\cali{C}} \newcommand{\cA}{\cali{A}^\infty (Y)_p$ is identical with $\Delta ^{L^p}$
on $\cali{C}} \newcommand{\cA}{\cali{A}^\infty (X, L^p)$ (cf. \cite[\S 2.1]{BU1}).
In \cite[Lemma 14.11, Theorem A 5.9]{BoG}, \cite{BoS},
\cite[(3.13)]{GU}, they construct
a self-adjoint second-order pseudodifferential operator $Q$ on $Y$
such that
\begin{align}\label{f21}
V= \Delta_h +\sqrt{-1} \tau \partial_\theta - Q
\end{align}
is a self-adjoint pseudodifferential operator of order zero on $Y$,
and $V,Q$ commute with the $S^1$-action.
The orthogonal projection $\Pi$ onto the kernel of $Q$ is called
the {\em Szeg\"o projector} associated with the almost CR manifold $Y$.
In fact, the Szeg\"o
projector is not unique or canonically defined, but the above construction
defines a canonical choice of $\Pi$ modulo smoothing operators.
In the complex case, the construction produces the usual Szeg\"o projector
$\Pi$.
We denote the operators on $\cali{C}} \newcommand{\cA}{\cali{A}^\infty (X, L^p)$ corresponding to $Q$, $V$,
$\Pi$ by $Q_p,V_p$, $\Pi_p$, especially,
$V_p(x,y)= \frac{1}{2\pi} \int_0^{2\pi} e ^{-ip\theta}
V(x e ^{i\theta},y)d\theta$. Then by (\ref{f21}),
\begin{align}\label{f22}
Q_p= \Delta ^{L^p}- p \tau -V_p.
\end{align}
By \cite[\S 4]{GU}, there exists $\mu_1>0$ such that for $p$ large,
\begin{align}\label{f23}
\spec Q_p \subset \{0\}\cup [\mu_1 p,+\infty[.
\end{align}
Since the operator $V_p$ is uniformly bounded in $p$,
naturally, from (\ref{0.3}), (\ref{0.0}), we get
\begin{align}\label{f24}
\dim \Ker Q_p=d_p =\int_X \td(TX)\ch(L^p).
\end{align}
Now we explain how to study the Szeg\"o
projector $\Pi_p$ \footnote{As Professor Sj\"ostrand pointed out to us,
in general, $\Pi_p-P_{0,p}$ is not $\cali{O}} \newcommand{\cE}{\cali{E}(p^{-\infty})$
as $p\to \infty$, where $P_{0,p}$ is the smooth kernel
of the operator $\Delta_{0,p}$ (Definition \ref{d3.0}).
This can also be seen from the presence of a contribution coming from $\Phi$ in the
expression \eqref{0.6} of the coefficient $b_{0,2}$.}. This can
be done from our point of view.
Recall $\widetilde{F}$ is the function defined after (\ref{0c3}).
Let $\Pi_p(x,x')$, $\widetilde{F}(Q_p)(x,x')$ be the smooth kernels
of $\Pi_p$, $\widetilde{F}(Q_p)$ with respect to the volume form $dv_X(x')$.
Note that $V_p$ is a 0-order pseudodifferential operator on $X$ induced from
a 0-order pseudodifferential operator on $Y$.
Thus from (\ref{f22}), (\ref{f23}), we have the analogue of
\cite[Proposition 3.1]{DLM} (cf. Proposition \ref{0t3.0}):
for any $l,m\in \field{N}$, there exists $C_{l,m}>0$ such that for $p\geqslant1$,
\begin{align}\label{f25}
|\widetilde{F}(Q_p)(x,x') -\Pi_p (x,x') |_{\cali{C}} \newcommand{\cA}{\cali{A}^m(X\times X)}\leqslant C_{l,m} p^{-l}.
\end{align}
By finite propagation speed \cite[\S 4.4]{T1}, we know that
$\widetilde{F}(Q_p)(x, x')$ only depends on the restriction of $Q_p$ to
$B^X(x,\varepsilon)$, and is zero if $d(x, x') \geqslant\varepsilon$. It transpires
that the asymptotic of $\Pi_p(x,x')$ as $p\to \infty$ is localized on
a neighborhood of $x$.
Thus we can translate our analysis from $X$ to the manifold $\field{R}^{2n} \simeq
T_{x_0}X=:X_{0}$ as in Section \ref{s3.2}, especially, we extend
$\nabla ^L$ to a
Hermitian connection $\nabla ^{L_{0}}$ on
$(L_0,h^{L_0})=(X_0\times L_{x_{0}},h^{L_{x_{0}}} )$
on $T_{x_0}X$ in such a way so that
we still have positive curvature $R ^{L_{0}}$; in addition
$R^{L_{0}}=R ^{L}_{x_{0}}$ outside a compact set.
Now, by using a micro-local partition of unity, one can still construct
the operator $Q^{X_0}$ as in
\cite[Lemma 14.11, Theorem A 5.9]{BoG}, \cite{BoS}, \cite[(3.13)]{GU},
such that
$V^{X_0}$ differs from $V$ by a smooth operator in a neighborhood of $0$.
On $X_0$, and $Q^{X_0}$ still verifies (\ref{f23}).
Thus we can work on $\cali{C}} \newcommand{\cA}{\cali{A}^\infty (X_0,\field{C})$ as in Section \ref{s3.3}.
We rescale then the coordinates as in (\ref{c27})
and use the norm (\ref{u0}). The $V^{X_0}_p$ is a
0-order pseudodifferential operator on $X_0$ induced from
a 0-order pseudodifferential operator on $Y_0$. This guarantees that the operator
rescaled from $V^{X_0}_p$ will have the similar expansion as \eqref{c30}
with leading term $t^2 R_2$ in the sense of pseudo-differential operators.
From (\ref{f24}) and \cite[(3.89)]{DLM},
similar to the argument in \cite[Theorem 3.18]{DLM}, we can also get the full
off diagonal expansion for $\Pi_p$,
which is an extension of \cite[Theorem 1]{SZ02}, where the authors obtain
(\ref{f26}) for $|Z|, |Z'|\leqslant C/\sqrt{p}$ with $C>0$ fixed.
More precisely, recalling that $P^N$ is the Bergman kernel of $\mathscr{L}} \def\cC{\mathscr{C}} \def\cR{\mathscr{R}_0$ as in \eqref{ue62}, \eqref{g7}
we have:
\begin{thm} \label{tue17}
There exist polynomials ${\bf j}_r(Z,Z')$ $(r\geqslant0)$ of $Z,Z'$
with the same parity with $r$,
and ${\bf j}_0=1$, $C''>0$ such that
for any $k,m,m'\in \field{N}$, there exist $N\in \field{N}, C>0$ such that for
$\alpha,\alpha'\in \field{Z}^{2n}$, $|\alpha|+|\alpha'|\leqslant m$,
$Z,Z'\in T_{x_0}X$, $|Z|, |Z'|\leqslant \varepsilon$, $x_0\in X$, $p>1$,
\begin{multline}\label{f26}
\left |\frac{\partial^{|\alpha|+|\alpha'|}}
{\partial Z^{\alpha} {\partial Z'}^{\alpha'}}
\left (\frac{1}{p^n} \Pi_p(Z,Z')
-\sum_{r=0}^k ({\bf j}_r P^N)(\sqrt{p} Z,\sqrt{p} Z') \kappa ^{-\frac{1}{2}}(Z)
\kappa ^{-\frac{1}{2}}(Z')
p^{-r/2}\right )\right |_{\cali{C}} \newcommand{\cA}{\cali{A} ^{m'}(X)}\\
\leqslant C p^{-(k+1-m)/2} (1+|\sqrt{p} Z|+|\sqrt{p} Z'|)^N
\exp (- \sqrt{C''\mu_1 } \sqrt{p} |Z-Z'|)+ \cali{O}} \newcommand{\cE}{\cali{E}(p^{-\infty}).
\end{multline}
\end{thm}
The term $\kappa ^{-\frac{1}{2}}$ in (\ref{f26})
comes from the conjugation of the operators as in (\ref{1c53}),
$\cali{C}} \newcommand{\cA}{\cali{A} ^{m'}(X)$ is the $\cali{C}} \newcommand{\cA}{\cali{A} ^{m'}$ -norm for the parameter $x_0\in X$,
and we use the trivializations from Section \ref{s3.2},
the term $\cali{O}} \newcommand{\cE}{\cali{E}(p^{-\infty})$ means that for any $l,l_1\in \field{N}$,
there exists $C_{l,l_1}>0$ such that its $\cali{C}} \newcommand{\cA}{\cali{A}^{l_1}$-norm is dominated
by $C_{l,l_1} p^{-l}$.
We leave the details to the interested reader.
\subsection{Symplectic version of Kodaira Embedding Theorem}\label{s5.31}
Let $(X,\omega)$ be a compact symplectic manifold of real dimension
$2n$ and let $(L,\nabla^L,h^L)$ be a pre-quantum line bundle
and let $g^{TX}$ be a Riemannian metric on $X$
as in Section \ref{s1}.
Recall that $\mH_p\subset\cali{C}} \newcommand{\cA}{\cali{A}^\infty(X,L^p)$ is the span of those
eigensections of $\Delta_p=\Delta ^{L^p}-\tau p$ corresponding
to eigenvalues from $[-C_L,C_L]$.
We denote by $\field{P}\mH^*_p$ the projective space associated to the dual of
$\mH_p$ and we identify $\field{P}\mH^*_p$ with the Grassmannian of hyperplanes in
$\mH_p$. The {\em base locus} of $\mH_p$ is the
set $\operatorname{Bl}(\mH_p)=\{x\in X:s(x)=0\,\,
\text{for all}\,s\in\mH_p\,\}$.
As in algebraic geometry, we define the Kodaira map
\begin{equation}\label{sz0}
\begin{split}
&\Phi_p:X\smallsetminus\operatorname{Bl}(\mH_p)\longrightarrow\field{P}\mH^*_p\\
&\Phi_p(x)=\{s\in\mH_p:s(x)=0\}
\end{split}
\end{equation}
which sends $x\in X\smallsetminus\operatorname{Bl}(\mH_p)$
to the hyperplane of sections vanishing at $x$.
Note that $\mH_p$ is endowed with the induced $L^2$ product \eqref{0c2}
so there is a well--defined Fubini--Study metric
$g_{FS}$ on $\field{P}\mH^*_p$ with the associated form $\omega_{FS}$.
\begin{thm} \label{sym-Kodaira}
Let $(L,\nabla^L)$ be a pre--quantum line bundle over a compact symplectic
manifold $(X,\omega)$. The following assertions hold true{\rm:}
{\rm (i)} For large $p$, the Kodaira maps $\Phi_p:X\longrightarrow\field{P}
\mH^*_p$ are well defined.
{\rm (ii)} The induced Fubini--Study metric $\frac{1}{p}\Phi^*_p
(\omega_{FS})$ converges in the $\cali{C}} \newcommand{\cA}{\cali{A}^\infty$ topology
to $\omega$\,{\rm;} for any
$l\geqslant 0$ there exists $C_l>0$ such that
\begin{equation} \label{sz1}
\Big|\frac{1}{p}\,\,\Phi^*_p(\omega_{FS})
-\omega\Big|_{\cali{C}} \newcommand{\cA}{\cali{A}^l}\leqslant\frac{C_l}{p} .
\end{equation}
{\rm (iii)} For large $p$ the Kodaira maps $\Phi_p$ are embeddings.
\end{thm}
\begin{rem}
1) Assume that $X$ is K\" ahler and $L$ is a holomorphic bundle.
Then $\Delta_p$ is the twice the Kodaira-Laplacian
and $\mH_p$ coincides with the
space $H^0(X,L^p)$ of holomorphic sections of $L^p$. Then (i) and (iii) are
simply the Kodaira embedding theorem. Assertion (ii) is due to Tian
\cite[Theorem A]{Tian} as an answer to a conjecture of Yau.
In \cite{Tian} the case
$l=2$ is considered and the left--hand side of \eqref{sz1} is estimated by
$C_l/\sqrt p$. Ruan \cite{Ru} proved the $\cali{C}} \newcommand{\cA}{\cali{A}^\infty$ convergence and
improved the bound to $C_l/p$. Both papers use the peak section method,
based on $L^2$--estimates for ${\overline\partial}$.
A proof for $l=0$ using the heat kernel
appeared in Bouche \cite{Bou}. Finally, Zelditch deduced (ii) from the
asymptotic expansion of the Szeg\" o kernel \cite{Zelditch}.
2) Borthwick and Uribe \cite[Theorem 1.1]{BU1},
Shiffman and Zelditch \cite[Theorems\,2, \,3]{SZ02} prove a different
symplectic version of \cite[Theorem A]{Tian} when ${\bf J}=J$.
Instead of $\mH_p$, they use the space
$H^0_J(X,L^p):={\rm Im}(\Pi_p)$ (cf. \cite[p.601]{BU1}, \cite[\S 2.3]{SZ02},
\S \ref{s5.3})
of `almost holomorphic sections' proposed by Boutet de Monvel
and Guillemin \cite{BoG}, \cite{BoS}.
\end{rem}
\begin{proof}
Let us first give an alternate description of the map $\Phi_p$ which relates
it to the Bergman kernel. Let $\{S^p_i\}^{d_p}_{i=1}$ be any orthonormal
basis of $\mH_p$ with respect to the inner product \eqref{0c2}. Once we
have fixed a basis, we obtain an identification $\mH_p\cong\mH^*_p\cong
\field{C}^{d_p}$ and $\field{P}\mH^*_p\cong\field{C}\field{P}^{d_p-1}$. Consider the commutative
diagram.
\begin{equation} \label{sz1.1}
\begin{CD}
X\smallsetminus\operatorname{Bl}(\mH_p)@>\Phi_p>>\field{P}\mH^*_p\\
@VV{\Id}V @VV{\cong}V\\
X\smallsetminus\operatorname{Bl}(\mH_p)@>\widetilde\Phi_p>>\field{C}\field{P}^{d_p-1}
\end{CD}
\end{equation}
Then
\begin{equation} \label{sz2}
\Phi^*_p(\omega_{FS})=\widetilde\Phi^*_p\Big(\frac{\sqrt{-1}}{2\pi}
\partial\overline\partial\log\sum^{d_p}_{j=1}|w_j|^2\Big),
\end{equation}
where $[w_1,\ldots,w_{d_p}]$ are homogeneous coordinates
in $\field{C}\field{P}^{d_p-1}$.
To describe $\widetilde\Phi_p$ in a neighborhood of a point
$x_0\in X\smallsetminus\operatorname{Bl}(\mH_p)$, we choose
a local frame
$e_L$ of $L$ and write $S^p_i=f^p_ie^{\otimes p}_L$
for some smooth functions $f^p_i$. Then
\begin{equation} \label{sz3}
\widetilde\Phi_p(x)=[f^p_1(x);\ldots;f^p_{d_p}(x)],
\end{equation}
and this does not depend on the choice of the frame $e_L$.
(i) Let us choose an unit frame $e_L$ of $L$. Then $|S^p_i|^2=
|f^p_i|^2|e_L|^{2p}=|f^p_i|^2$, hence
\begin{equation*}
B_{0,p}=\sum^{d_p}_{i=1}|S^p_i|^2=\sum^{d_p}_{i=1}|f^p_i|^2 .
\end{equation*}
Since $b_{0,0}>0$, the asymptotic expansion \eqref{0.6} shows that $B_{0,p}$
does not vanish on $X$ for $p$ large enough, so the sections $\{S^p_i\}^
{d_p}_{i=1}$ have no common zeroes. Therefore $\Phi_p$ and $\widetilde
\Phi_p$ are defined on all $X$.
(ii) Let us fix $x_0\in X$. We identify a small geodesic ball
$B^X(x_0,\varepsilon)$
to $B^{T_{x_0}X}(0,\varepsilon)$ by means of the exponential map and consider
a trivialization of $L$ as in Section \ref{s3.2}, i.e. we trivialize $L$
by using an unit frame $e_L(Z)$ which is parallel with respect
to $\nabla ^L$ along $[0,1]\ni u\to uZ$
for $Z\in B^{T_{x_0}X}(0,\varepsilon)$.
We can express the Fubini--Study metric as
\begin{equation*}
\frac{\sqrt{-1}}{2\pi}\partial\overline\partial
\log\Big(\sum^{d_p}_{j=1}|w_j|^2\Big)=\frac{\sqrt{-1}}
{2\pi}\left[\frac{1}{|w|^2}\sum^{d_p}_{j=1}dw_j\wedge d\overline w_j
-\frac{1}{|w|^4}\sum^{d_p}_{j,k=1}\overline w_jw_k\,dw_j\wedge d\overline
w_k\right] ,
\end{equation*}
and therefore, from \eqref{sz3},
\begin{multline} \label{sz4}
\Phi^*_p(\omega_{FS})(x_0)=\frac{\sqrt{-1}}{2\pi}
\left[\frac{1}{|f^p|^2}\sum^{d_p}_
{j=1}df^p_j\wedge d\overline{f^p_j}
-\frac{1}{|f^p|^4}\sum^{d_p}_{j,k=1}\overline{f^p_j}f^p_k\,df^p_j\wedge d
\overline{f^p_k}\right](x_0)\\
=\frac{\sqrt{-1}}{2\pi}\big[f^p(x_0,x_0)^{-1}d_xd_yf^p(x,y)
-f^p(x_0,x_0)^{-2}d_xf^p(x,y)\wedge d_yf^p(x,y)\big]\vert_{x=y=x_0},
\end{multline}
where $f^p(x,y)=\sum^{d_p}_{i=1}f^p_i(x)\overline{f^p_i}(y)$ and
$|f^p(x)|^2=f^p(x,x)$.
Since
\begin{equation} \label{0sz4}
P_{0,p}(x,y)=f^p(x,y)e^p_L(x)\otimes e^{p}_L(y)^*,
\end{equation}
thus $P_{0,p}(x,y)$ is $f^p(x,y)$ under our trivialization of $L$.
By \eqref{0c30}, Theorem \eqref{0t3.6}, and \eqref{1c53}, we
obtain
\begin{multline} \label{0sz5}
\frac{1}{p}\,\,\Phi^*_p(\omega_{FS})(x_0)
=\frac{\sqrt{-1}}{2\pi}\Big[\,\frac{1}{F_{0,0}}d_xd_yF_{0,0}
- \frac{1}{F_{0,0}^2}d_xF_{0,0}\wedge d_yF_{0,0}\Big](0,0)\\
-\frac{\sqrt{-1}}{2\pi}\frac{1}{\sqrt{p}}\Big[\,\frac{1}{F_{0,0}^2}
(d_xF_{0,1}\wedge d_yF_{0,0}+ d_xF_{0,0}\wedge d_yF_{0,1})\Big](0,0)
+\cali{O}} \newcommand{\cE}{\cali{E}\big(1/p\big).
\end{multline}
Using again \eqref{g7}, \eqref{1c52}, we obtain
\begin{equation} \label{0sz6}
\frac{1}{p}\,\,\Phi^*_p(\omega_{FS})(x_0)
=\frac{\sqrt{-1}}{4\pi}\sum^n_{j=1} a_jdz_j\wedge
d \overline z_j|_{x_0}+\cali{O}} \newcommand{\cE}{\cali{E}\big(1/p\big)=\omega(x_0)+\cali{O}} \newcommand{\cE}{\cali{E}\big(1/p\big),
\end{equation}
and the convergence takes place in the $\cali{C}} \newcommand{\cA}{\cali{A}^\infty$ topology with respect to
$x_0\in X$.
(iii) Since $X$ is compact, we have to prove two things for $p$ sufficiently
large: (a)~$\Phi_p$ are immersions and (b) $\Phi_p$ are injective.
We note that (a) follows immediately from \eqref{sz1}.
To prove (b) let us assume the contrary, namely that there exists
a sequence of distinct points $x_p\neq y_p$ such that
$\Phi_p(x_p)=\Phi_p(y_p)$. Relation \eqref{sz1.1} implies that
$\widetilde\Phi_p(x_p)=\widetilde\Phi_p(y_p)$,
where $\widetilde\Phi_p$ is defined by any particular choice of basis.
The key observation is that Theorem \ref{t3.8} ensures the existence of
a sequence of {\em peak sections\/} at each point of $X$.
The construction goes like follows.
Let $x_0\in X$ be fixed. Since $\Phi_p$ is point base free for large $p$,
we can consider the hyperplane $\Phi_p(x_0)$ of all sections of $\mH_p$
vanishing at $x_0$. We construct then an orthonormal basis $\{S^p_i\}^{d_p}_
{i=1}$ of $\mH_p$ such that the first $d_p-1$ elements belong to $\Phi_
p(x_0)$. Then $S^p_{d_p}$ is a unit norm generator of the orthogonal
complement of $\Phi_p(x_0)$, and will be denoted by $S^p_{x_0}$.
This is a peak section at $x_0$. We note first that
$|S^p_{x_0}(x_0)|^2=B_{0,p}(x_0)$ and $P_{0,p}(x,x_0)=
S^p_{x_0}(x)\otimes S^p_{x_0}(x_0)^*$ and therefore
\begin{equation}\label{sz4.1}
S^p_{x_0}(x)=\frac{1}{B_{0,p}(x_0)}P_{0,p}(x,x_0)\cdot S^p_{x_0}(x_0).
\end{equation}
From \eqref{1c53} we deduce that for a sequence $\{r_p\}$ with $r_p\to0$ and
$r_p\sqrt{p}\to\infty$,
\begin{equation} \label{sz5}
\int_{B(x_0,r_p)}|S^p_{x_0}(x)|^2\,dv_X(x)=1-\cali{O}} \newcommand{\cE}{\cali{E}(1/p)\,,
\quad \text{for $p\to\infty$}.
\end{equation}
Relation \eqref{sz5} explains the term `peak section': when $p$ grows, the
mass of $S^p_{x_0}$ concentrates near $x_0$.
Since $\Phi_p(x_p)=\Phi_p(y_p)$
we can construct as before the peak section $S^p_{x_p}=S^p_{y_p}$ as the
unit norm generator of the orthogonal
complement of $\Phi_p(x_p)=\Phi_p(y_p)$. We fix in the sequel such a section
which peaks at both $x_p$ and $y_p$.
We consider the distance $d(x_p,y_p)$ between the two points $x_p$ and $y_p$.
By passing to a subsequence we have two possibilities: either
$\sqrt{p}d(x_p,y_p)\to\infty$ as $p\to\infty$
or there exists a constant $C>0$ such that $d(x_p,y_p)\leqslant C/\sqrt{p}$ for all $p$.
Assume that the first possibility is true.
For large $p$, we learn from relation \eqref{sz5}
that the mass of $S^p_{x_p}=S^p_{y_p}$
(which is $1$) concentrates both in neighborhoods $B(x_p,r_p)$ and $B(y_p,r_p)$ with
$r_p=d(x_p,y_p)/2$ and approaches
therefore 2 if $p\to\infty$. This is a contradiction which
rules out the first possibility.
To exclude the second possibility we follow \cite{SZ02}. We identify as usual
$B^X(x_p,\varepsilon)$ to $B^{T_{x_p}X}(0,\varepsilon)$ so the point
$y_p$ gets identified to $Z_p/\sqrt{p}$ where $Z_p\in B^{T_{x_p}X}(0,C)$.
We define then
\begin{equation}\label{sz6}
f_p:[0,1]\longrightarrow\field{R}\;,\quad f_p(t)
=\frac{|S_{x_p}^p(tZ_p/\sqrt{p})|^2}{B_{0,p}(tZ_p/\sqrt{p})}\;.
\end{equation}
We have $f_p(0)=f_p(1)=1$ (again because $S^p_{x_p}=S^p_{y_p}$)
and $f_p(t)\leqslant1$ by the definition of the generalized Bergman kernel.
We deduce the existence of a point $t_p\in(0,1)$ such that $f''_p(t_p)=0$.
Equations \eqref{1c53}, \eqref{sz4.1}, \eqref{sz6} imply the estimate
\begin{equation}\label{sz7}
f_p(t)=e^{-\frac{t^2}{4}\sum_j a_j|z_{p,j}|^2}\big(1+g_p(tZ_p)/\sqrt{p}\big)
\end{equation}
and the $\cali{C}} \newcommand{\cA}{\cali{A}^2$ norm of $g_p$ over $B^{T_{x_p}X}(0,C)$ is
uniformly bounded in $p$.
From \eqref{sz7}, we infer that
$|Z_p|^2_0:=\frac{1}{4}\sum_j a_j|z_{p,j}|^2=\cali{O}} \newcommand{\cE}{\cali{E}(1/\sqrt{p})$.
Using a limited expansion $e^x=1+x+x^2\varphi(x)$ for $x=t^2|Z_p|^2_0$
in \eqref{sz7} and taking derivatives, we obtain
$f''_p(t)=-2|Z_p|^2_0+\cali{O}} \newcommand{\cE}{\cali{E}(|Z_p|^4_0)+\cali{O}} \newcommand{\cE}{\cali{E}(|Z_p|^2_0/\sqrt{p})
=(-2+\cali{O}} \newcommand{\cE}{\cali{E}(1/\sqrt{p}))|Z_p|^2_0$.
Evaluating at $t_p$ we get $0=f''_p(t_p)=(-2+\cali{O}} \newcommand{\cE}{\cali{E}(1/\sqrt{p}))|Z_p|^2_0$,
which is a contradiction since by assumption $Z_p\neq0$.
This finishes the proof of (iii).
\end{proof}
\begin{rem}
Let us point out complementary results which are analogues of
\cite[(1.3)--(1.5)]{BU1} for the spaces $\mH_p$.
Computing as in \eqref{sz4}
the pull-back $\Phi_p^*h_{FS}$ of the Hermitian metric
$h_{FS}=g_{FS}-\sqrt{-1}\,\omega_{FS}$
on $\field{P} \mH^*_p$, we get the similar inequality
to (\ref{sz1}) for $g_{FS}$ and $\omega} \newcommand{\g}{\Gamma(\cdot,J\cdot)$.
Thus, $\Phi_p$ are asymptotically
symplectic and isometric. Moreover, arguing as in \cite[Proposition\,4.4]{BU1}
we can show that $\Phi_p$ are `nearly holomorphic'\,:
\begin{equation}\label{sz9}
\frac{1}{p}\,\lVert\partial\Phi_p\rVert\geqslant C\,,\quad
\frac{1}{p}\,\lVert\overline\partial\Phi_p\rVert=\cali{O}} \newcommand{\cE}{\cali{E}(1/p)\,,\quad\text{for some $C>0$}\,,
\end{equation}
uniformly on $X$, where $\Vert\,\cdot\,\Vert$ is the operator norm.
\end{rem}
\subsection{Holomorphic case revisited}\label{s5.4}
In this Section, we assume that $(X,\omega, J)$ is K\"ahler
and the vector bundles $E,L$ are holomorphic on $X$,
and $\nabla ^E,\nabla ^L$ are the holomorphic Hermitian
connections on $(E,h^E)$, $(L,h^L)$. As usual,
$\frac{\sqrt{-1}}{2 \pi} R^L=\omega} \newcommand{\g}{\Gamma$.
But we will work with an arbitrary ({\em non-K\"ahler})
Riemannian metric $g^{TX}$ on $TX$ compatible with $J$.
That is, in general ${\bf J}\neq J$ in \eqref{0.1}.
The use of non-K\"ahler metrics is useful for example in Section \ref{s5.6}.
Set
\begin{align} \label{f11}
\Theta(X,Y)= g^{TX}(JX,Y).
\end{align}
Then the 2-form $\Theta$ need not to be closed
(the convention here is different to
\cite[(2.1)]{B} by a factor $-1$). We denote by $T^{(1,0)}X$, $T^{(0,1)}X$
the holomorphic and anti-holomorphic tangent bundles as in Section \ref{s3.4}.
Let $\{e_i\}$ be an orthonormal frame of $(TX, g^{TX})$.
Let $g^{TX}_{\omega} \newcommand{\g}{\Gamma}(\cdot,\cdot):= \omega} \newcommand{\g}{\Gamma(\cdot,J\cdot)$ be the metric on $TX$
induced by $\omega} \newcommand{\g}{\Gamma, J$.
We will use a subscript $\omega} \newcommand{\g}{\Gamma$ to indicate the objects corresponding to
$g^{TX}_{\omega} \newcommand{\g}{\Gamma}$, especially $r^{X}_{\omega} \newcommand{\g}{\Gamma}$ is the scalar curvature of
$(TX,g^{TX}_{\omega} \newcommand{\g}{\Gamma})$, and $\Delta_\omega} \newcommand{\g}{\Gamma$ is the Bochner-Laplace
operator as in \eqref{0c1} associated to $g^{TX}_{\omega} \newcommand{\g}{\Gamma}$.
Let $\overline{\partial} ^{L^p\otimes E,*}$ be the formal adjoint of
the Dolbeault operator $\overline{\partial} ^{L^p\otimes E}$
on the Dolbeault complex
$\Omega ^{0,\bullet}(X, L^p\otimes E)$ with the scalar product
induced by $g^{TX}$, $h^L$, $h^E$ as in (\ref{0c2}).
Set $D_p = \sqrt{2}( \overline{\partial} ^{L^p\otimes E}
+ \overline{\partial} ^{L^p\otimes E,*})$.
Then $D_p^2=
2( \overline{\partial} ^{L^p\otimes E}\overline{\partial} ^{L^p\otimes E,*}
+\overline{\partial} ^{L^p\otimes E,*}\overline{\partial} ^{L^p\otimes E})$
preserves the $\field{Z}$-grading of $\Omega ^{0,\bullet}(X, L^p\otimes E)$.
Then for $p$ large enough,
\begin{equation} \label{f12}
\Ker D_p =\Ker D_p^2 = H^0 (X,L^p\otimes E).
\end{equation}
Here $D_{p}$ is not a spin$^c$ Dirac operator on
$\Omega ^{0,\bullet}(X, L^p\otimes E)$, and $D^2_p$ is not
a renormalized Bochner--Laplacian as in (\ref{laplace}).
Let $P_p(x,x')$ $(x,x'\in X)$ be the smooth kernel of the orthogonal
projection from $\cali{C}} \newcommand{\cA}{\cali{A} ^\infty(X, L^p\otimes E)$ on $\Ker D_p^2$
with respect to the Riemannian volume form $dv_X(x')$ for $p$ large enough.
Recall that we denote by $\det_{\field{C}} $ the determinant function on the complex
bundle $T^{(1,0)}X$. We denote by $|{\bf J}|= (-{\bf J}^2)^{-1/2}$,
then $\det_{\field{C}}|{\bf J}|= (2\pi)^{-n}\Pi_i\, a_i$
under the notation in \eqref{0ue52}.
Now we explain how to put it in the frame of our work.
\begin{thm}\label{nonkahler}
The smooth kernel $P_p(x,x')$
has a full off--diagonal asymptotic expansion analogous to \eqref{f26}
with ${\bf j}_0=\det_\field{C} |{\bf J}|$ as $p\to\infty$\,.
The corresponding term $b_{0,1}$ in the expansion \eqref{0.6} of $B_{0,p}(x):=P_p(x,x)$
is given by
\begin{equation} \label{af12}
b_{0,1}= \frac{\det_\field{C} |{\bf J}|}{8\pi}\Big[r^X_\omega} \newcommand{\g}{\Gamma
-2 \Delta_\omega} \newcommand{\g}{\Gamma \Big(\log({\det}_\field{C} |{\bf J}|)\Big)
+ 4 R^E (w_{\omega} \newcommand{\g}{\Gamma,j},\overline{w}_{\omega} \newcommand{\g}{\Gamma,j})\Big].
\end{equation}
here $\{w_{\omega} \newcommand{\g}{\Gamma,j}\}$ is an orthonormal basis of $(T^{(1,0)}X, g^{TX}_\omega} \newcommand{\g}{\Gamma)$.
\end{thm}
\begin{proof}
As pointed out in \cite[Remark 3.1]{MM}, by \cite[Theorem 1]{BiV},
there exist $\mu_0, C_L>0$ such that for any $p\in \field{N}$ and
any $s\in\Omega^{>0}(X,L^p\otimes E):
=\bigoplus_{q\geqslant 1}\Omega^{0,q}(X,L^p\otimes E)$,
\begin{equation}\label{main1}
\norm{D_{p}s}^2_{L^ 2}\geqslant(2p\mu_0-C_L)\norm{s}^2_{L^ 2}.
\end{equation}
Moreover $\spec D^2_p \subset \{0\}\cup [2p\mu_0 -C_L,+\infty[$.
Let $S^{-B}$ denote the 1-form with values in antisymmetric
elements of $\End(TX)$ which is such that if $U,V,W \in TX$,
\begin{align}\label{f13}
\langle S^{-B}(U)V,W \rangle = - \frac{\sqrt{-1}}{2}
\Big( (\partial- \overline{\partial} )\Theta\Big)(U,V,W).
\end{align}
The Bismut connection $\nabla ^{-B}$ on $TX$ is defined by
\begin{align}\label{f14}
\nabla ^{-B} = \nabla ^{TX} + S^{-B}.
\end{align}
Then by \cite[Prop. 2.5]{B89}, $\nabla ^{-B}$ preserves the metric $g^{TX}$
and the complex structure of $TX$.
Let $\nabla ^{\det}$ be the holomorphic Hermitian connection on
$\det (T^{(1,0)}X)$ with its curvature $R^{\det}$.
Then these two connections induce naturally
an unique connection on $\Lambda (T^{*(0,1)}X)$
which preserves its $\field{Z}$-grading, and with the connections
$\nabla ^L, \nabla ^E$, we get a connection $\nabla ^{-B,E_p}$ on
$\Lambda (T^{*(0,1)}X)\otimes L^p\otimes E$. Let $\Delta ^{-B,E_p}$
be the Laplacian on $\Lambda (T^{*(0,1)}X)\otimes L^p\otimes E$
induced by $\nabla ^{-B,E_p}$ as in (\ref{0c1}).
For any $v\in TX$ with decomposition $v=v_{1,0}+v_{0,1}
\in T^{(1,0)}X\oplus T^{(0,1)}X$, let ${\overline v^\ast_{1,0}}\in T^{*(0,1)}X$
be the metric dual of $v_{1,0}$.
Then $c(v)=\sqrt{2}({\overline v^\ast_{1,0}}\wedge-i_{v_{\,0,1}})$
defines the Clifford action of $v$ on $\Lambda (T^{*(0,1)}X)$, where $\wedge$ and $i$
denote the exterior and interior product respectively.
We define a map $^c: \Lambda (T^*X) \to C(TX)$, the Clifford bundle
of $TX$, by sending
$e ^{i_1}\wedge\cdots \wedge e ^{i_j}$ to $c(e_{i_1})\cdots c(e_{i_j})$
for $i_1< \cdots < i_j$. For $B\in \Lambda^3(T^*X)$, set
$|B|^2= \sum_{i<j<k}|B(e_i,e_j,e_k)|^2$.
Then we can formulate \cite[Theorem 2.3]{B89} as following,
\begin{align}\label{f16}
D_p^2= \Delta ^{-B,E_p} + \frac{r^X}{4} + {^c(R^E +pR^L
+ \frac{1}{2} R^{\det})}
+ \frac{\sqrt{-1}}{2} {^c(\overline{\partial}\partial \Theta) }
- \frac{1}{8} |( \partial- \overline{\partial} )\Theta|^2.
\end{align}
We use now the connection $\nabla ^{-B,E_p}$ instead of $\nabla ^{E_p}$
in \cite[\S 2]{DLM}. Then by \eqref{main1}, \eqref{f16}, everything
goes through perfectly well and as in \cite[Theorem 3.18]{DLM},
so we can directly apply the result in \cite{DLM} to get the
{\em full off-diagonal} asymptotic expansion of the Bergman kernel.
As the above construction preserves the $\field{Z}$-grading on
$\Omega ^{0,\bullet}(X, L^p\otimes E)$, we can also directly work
on $\cali{C}} \newcommand{\cA}{\cali{A} ^{\infty} (X, L^p\otimes E)$.
Now, we need to compute the corresponding $b_{0,1}$.
Now $h^E_\omega} \newcommand{\g}{\Gamma:= ({\det}_{\field{C}}|{\bf J}|)^{-1} h^E$ defines a metric on $E$,
let $R^E_\omega} \newcommand{\g}{\Gamma$ be the curvature associated to the holomorphic Hermitian connection of $(E, h^E_\omega} \newcommand{\g}{\Gamma)$,
then
\begin{equation}\label{af17}
R^E_\omega} \newcommand{\g}{\Gamma =R^E -\overline{\partial}\partial \log ({\det}_{\field{C}}|{\bf J}|).
\end{equation}
Thus
\begin{equation}\label{af18}
\sqrt{-1}R^E_\omega} \newcommand{\g}{\Gamma(e_{\omega} \newcommand{\g}{\Gamma,j},Je_{\omega} \newcommand{\g}{\Gamma,j})
= 2 R^E_\omega} \newcommand{\g}{\Gamma(w_{\omega} \newcommand{\g}{\Gamma,j},\overline{w}_{\omega} \newcommand{\g}{\Gamma,j})
=\sqrt{-1}R^E( e_{\omega} \newcommand{\g}{\Gamma,j},Je_{\omega} \newcommand{\g}{\Gamma,j}) -\Delta_\omega} \newcommand{\g}{\Gamma \log ({\det}_{\field{C}}|{\bf J})\,.
\end{equation}
Let $\left\langle\,\cdot,\cdot \right \rangle_\omega} \newcommand{\g}{\Gamma$ be the Hermitian product on
$\cali{C}} \newcommand{\cA}{\cali{A}^\infty (X, L^p\otimes E)$
induced by $g^{TX}_\omega} \newcommand{\g}{\Gamma, h^L, h^E_\omega} \newcommand{\g}{\Gamma$. Then
\begin{equation}\label{af19}
(\cali{C}} \newcommand{\cA}{\cali{A}^\infty (X, L^p\otimes E), \left\langle \quad \right \rangle_\omega} \newcommand{\g}{\Gamma)
= (\cali{C}} \newcommand{\cA}{\cali{A}^\infty (X, L^p\otimes E), \left\langle \quad \right \rangle)\,,
\quad dv_{X,\omega} \newcommand{\g}{\Gamma} =({\det}_{\field{C}}|{\bf J}|) dv_{X}.
\end{equation}
Observe that $H^0(X, L^p\otimes E)$ does not depend on $g^{TX}$, $h^L$ or $h^E$.
If $P_{\omega} \newcommand{\g}{\Gamma,p}(x,x')$, ($x,x'\in X$) denotes the smooth kernel of the orthogonal projection from
$(\cali{C}} \newcommand{\cA}{\cali{A}^\infty (X, L^p\otimes E), \left\langle\,\cdot,\cdot \right \rangle_\omega} \newcommand{\g}{\Gamma)$
onto $H^0(X, L^p\otimes E)$ with respect to $dv_{X,\omega} \newcommand{\g}{\Gamma}(x)$,
we have
\begin{equation}\label{af20}
P_{p}(x,x')= ({\det}_{\field{C}}|{\bf J}|(x')) P_{\omega} \newcommand{\g}{\Gamma,p}(x,x').
\end{equation}
Now for the kernel $ P_{\omega} \newcommand{\g}{\Gamma,p}(x,x')$, we can apply Theorem \ref{t0.1} (or \cite[Theorem 1.3]{DLM})
since $g^{TX}_\omega} \newcommand{\g}{\Gamma(\cdot, \cdot)= \omega} \newcommand{\g}{\Gamma(\cdot, J\cdot)$ is a K\"ahler metric on $TX$,
and \eqref{af12} follows from \eqref{0.5} and \eqref{af18}.
\end{proof}
\begin{rem} Certainly, the argument in this Subsection goes through
the orbifold case as in \cite[\S 4.2]{DLM}.
\end{rem}
\subsection{Generalizations to non-compact manifolds} \label{s5.5}
Let $(X,\Theta)$ be a K\"ahler manifold and $(L,h^L)$
be a holomorphic Hermitian line bundle over $X$.
As in Section \ref{s5.4}, let $R^L, R^{\det}$ be the curvatures
of the holomorphic Hermitian connections $\nabla ^L,\nabla^{\det}$
on $L$, $\det (T^{(1,0)}X)$, and let $J^L\in \End(TX)$ such that
$\frac{\sqrt{-1}}{2\pi}R^L(\cdot,\cdot)=\Theta (J^L\cdot,\cdot)$. The space
of holomorphic sections of $L^p$ which are $L^2$ with respect to the norm
given by \eqref{0c2} is denoted by $H^0_{(2)}(X,L^p)$.
Let $P_p(x,x')$, $(x,x'\in X)$ be the Schwartz kernel of
the orthogonal projection $P_p$ from the $L^2$ section of $L^p$
onto $H^0_{(2)}(X,L^p)$ with respect to the Riemannian volume form $dv_X(x')$
associated to $(X,\Theta)$.
Then by the ellipticity of the Kodaira-Laplacian and Schwartz kernel theorem,
we know $P_p(x,x')$ is $\cali{C}} \newcommand{\cA}{\cali{A}^\infty$.
Choose an orthonormal basis $(S^p_i)_{i\geqslant 1}$
of $H^0_{(2)}(X,L^p)$. For each local holomorphic frame $e_L$ we
have $S^p_i=f^p_i e^{\otimes p}_L$ for some local holomorphic functions
$f^p_i$.
Then $B_p(x):=P_p(x,x)=\sum_{i\geqslant 1}|S^p_i(x)|^2
=\sum_{i\geqslant 1}|f^p_i(x)|^2|e^{\otimes p}_L|^2$ is a smooth function.
We have the following generalization of Theorem \ref{t0.1}.
\begin{thm} \label{noncompact}
Assume that $(X,\Theta)$ is a complete K\"ahler manifold.
Suppose that there exist $\varepsilon>0\,,\,C>0$ such that
one of the following assumptions
holds true\,{\rm:}
\begin{gather}
\text{ $\sqrt{-1}R^L>\varepsilon\Theta\,,\,\sqrt{-1}R^{\det}>-C\Theta$. }\label{i}\\
\text{$L=\det (T^{*(1,0)}X)$, $h^L$ is induced by $\Theta$ and
$\sqrt{-1}R^{\det}<-\varepsilon\Theta$. }\label{ii}
\end{gather}
The kernel $P_p(x,x')$
has a full off--diagonal asymptotic expansion analogous to \eqref{f26}
with ${\bf j}_0=\det_\field{C} |J^L|$ as $p\to\infty$,
uniformly for any $x,x'\in K$, a compact set of $X$.
Especially there exist coefficients $b_r\in\cali{C}} \newcommand{\cA}{\cali{A}^\infty(X)\,,\,r\in\field{N}$,
such that for any compact set $K\subset X$, any $k,l\in\field{N}$, there exists
$C_ {k,l,K}>0$ such that for $p\in \field{N}$,
\begin{equation} \label{ell1}
\Big|\frac{1}{p^n}B_p(x)-\sum^k_{r=0}b_r(x)p^{-r}\Big|_{\cali{C}} \newcommand{\cA}{\cali{A}^l(K)}\leqslant
C_{k,l,K}\,p^{-k-1}.
\end{equation}
Moreover, $b_0= (\det J^L)^{1/2}$ and $b_1$ equals $b_{0,1}$ from \eqref{af12}.
\end{thm}
\begin{proof} By the argument in Section \ref{s3.1}, if
the Kodaira--Laplacian $\Box^{L^p}=\frac{1}{2}\Delta_p:=\frac{1}{2}\Delta_{p,0}$
acting on sections of $L^p$ has a spectral gap as in \eqref{0.3},
then we can localize the problem, and we get directly (\ref{ell1})
from Section \ref{s3.3}.
Observe that $D^2_p|_{\Omega^{0,\bullet}}= \Delta_p$.
In general, on a non-compact manifold, we define a
self-adjoint extension of $D^2_p$ by
\begin{equation*}
\begin{split}
\Dom\,D^2_p=\big\lbrace u\in\Dom\,&\db^{L^p}\cap\,\Dom\,\db^{L^p*} \,:\,
\db^{L^p}u\in\Dom\,\db^{L^p*}\,,\:\db^{L^p*} u\in \Dom\db^{L^p} \big\rbrace\,,\\
D^2_p\,u&=2(\db^{L^p}\db^{L^p*} +\db^{L^p*}\db^{L^p})\,u\,,\quad \text{for $u\in\Dom D^2_p$}\,.
\end{split}
\end{equation*}
The quadratic form associated to $D^2_p$ is the form $Q_p$ given by
\begin{equation}\label{ell2}
\begin{split}
&\Dom\,Q_p:=\Dom\, \db^{L^p}\cap\;\Dom\, \db^{L^p*}\\
Q_p(u,v)=&2 \big\langle \db^{L^p}u\,,\db^{L^p}v\big\rangle
+ 2 \big\langle \db^{L^p*}u\,,\db^{L^p*}v
\big\rangle\,,\quad u,v\in \Dom\,Q_p\,.
\end{split}
\end{equation}
In the previous formulas $\db^{L^p}$ is the weak maximal extension
of $\db^{L^p}$ to $L^2$ forms and $\db^{L^p*}$ is its Hilbert space adjoint.
We denote by $\Omega^{0,\bullet}_0(X,L^p)$ the space of smooth compactly
supported forms and by $L^{0,\bullet}_2(X,L^p)$
the corresponding $L^2$-completion.
Under one of the hypotheses \eqref{i} or \eqref{ii} there
exists $\mu>0$ such that for $p$ large enough
\begin{equation}\label{ell4}
Q_p(u)\geqslant \mu p\norm{u}^2\,,\quad u\in\Dom Q_p\cap L^{0,q}_2(X,L^p)\,
\,{\rm for} \, \, q>0.
\end{equation}
Indeed, the estimate holds for $u\in\Omega^{0,q}_0(X,L^p)$ since
the Bochner-Kodaira formula \cite[Prop. 3.71]{BeGeV} reduces to
$Q_p(u)\geqslant 2 \big\langle\,(p R^L+ R^{\det})(w_i,\overline{w}_j)
\overline{w}^j\wedge i_{\overline{w}_i} u\,,u\big\rangle\, $,
for $u\in\Omega^{0,q}_0(X,L^p)$, where $\{w_i\}$ is an orthonormal frame of
$T^{(1,0)}X$.
But this implies \eqref{ell4} in general, since $\Omega^{0,\bullet}_0(X,L^p)$
is dense in $\Dom Q_p$ with respect to the graph norm, as the metric is
complete.
Next, consider $f\in\Dom\,\Delta_p\cap L^{0,0}_2(X,L^p)$ and set $u=\db^{L^p}f$.
It follows from the definition of the Laplacian and \eqref{ell4} that
\begin{equation}\label{ell5}
\lVert\Delta_p f\rVert^2= 2\big\langle\, \db^{L^p*}u\,,\db^{L^p*}u\big\rangle
=Q_p(u)\geqslant \mu p\norm{u}^2=\mu p\big\langle\,\Delta_p f\,,f\big\rangle\,.
\end{equation}
This clearly implies $\spec(\Delta_p)\subset\{0\}\cup[p\mu,\infty[$ for large $p$.
\end{proof}
Theorem \ref{noncompact} permits an immediate generalization of Tian's
convergence theorem. Tian \cite[Theorem 4.1]{Tian} already proved
a non--compact version for convergence in the $\cali{C}} \newcommand{\cA}{\cali{A}^2$ topology
and convergence rate $1/\sqrt{p}$\,.
Another easy consequence are
holomorphic Morse inequalities for the space $H^0_{(2)}(X,L^p)$.
Observe that the quantity $\sum_{i\geqslant 1}|f^p_i(x)|^2$
is not globally defined, but the current
\begin{equation}\label{ell0}
\omega_p=\frac{\sqrt{-1}}{2\pi}\,\partial\overline\partial\log
\Big(\sum_{i\geqslant 1}|f^p_i(x)|^2\Big)
\end{equation}
is well defined globally on $X$. Indeed, since
$R^L=-\partial\overline\partial\log|e_L|_{h^L}^2$
we have
\begin{equation}\label{ell0.1}
\frac{1}{p}\omega_p-\frac{\sqrt{-1}}{2\pi}R^L=\frac{\sqrt{-1}}{2\pi p}\,
\partial\overline\partial\log B_p \, .
\end{equation}
If $\dim H^0_{(2)}(X,L^p)<\infty$ we have by \eqref{sz0} that
$\omega_p=\Phi^*_p(\omega_{FS})$ where $\Phi_p$ is defined as in
\eqref{sz0} with $\cali{H}} \newcommand{\cD}{\cali{D}_p$ replaced by $H^0_{(2)}(X,L^p)$.
\begin{cor} Assume one of the hypotheses \eqref{i} or \eqref{ii} holds.
Then\,{\rm:}
{\rm(a)}
for any compact set $K\subset X$ the restriction $\omega_p|_{K}$ is a smooth
$(1,1)$-form for sufficiently large $p${\rm;}
moreover, for any $l\in\field{N}$ there exists a
constant $C_{l,K}$ such that
\begin{equation*}
\Big|\frac{1}{p}\omega_p-\frac{\sqrt{-1}}{2\pi} R^L\Big|_{\cali{C}} \newcommand{\cA}{\cali{A}^l(K)}
\leqslant\frac{C_{l,K}}{p}\;;
\end{equation*}
{\rm(b)} the Morse inequalities hold in bidegree $(0,0)$\,{\rm:}
\begin{equation} \label{ell6}
\liminf_{p\longrightarrow\infty}p^{-n}\dim H^0_{(2)}(X,L^p)\geqslant
\frac{1}{n!}\int_X\Big(\frac{\sqrt{-1}}{2\pi}R^L\Big)^n .
\end{equation}
In particular, if $\dim H^0_{(2)}(X,L^p)<\infty$, the manifold $(X,\Theta)$ has
finite volume.
\end{cor}
\begin{proof}
Due to \eqref{ell1}, $B_p$ doesn't vanish on any given compact set $K$
for $p$ sufficiently large. Thus, (a) is a consequence of \eqref{ell1}
and \eqref{ell0.1}.
Part (b) follows from Fatou's lemma, applied on $X$ with the measure $\Theta^n/n!$
to the sequence $p^{-n}B_p$ which converges pointwise to
$(\det J^L)^{1/2}=
\big(\frac{\sqrt{-1}}{2\pi}R^L\big)^n/\Theta ^n$ on $X$.
\end{proof}
\begin{rem}
Under the hypothesis \eqref{ii}, the inequality \eqref{ell6} is
\cite[Theorem 1.1]{NT} of Nadel--Tsuji, where Demailly's
holomorphic inequalities on compact sets $K\subset X$ were used.
The volume estimate is essential in their compactification theorem of complete K\"ahler
manifolds with negative Ricci curvature (a generalization of the fact that arithmetic
varieties can be complex--analytically compactified). The Morse inequalities
\eqref{ell6} were also used by Napier--Ramachandran \cite{Na-R} to show that
some quotients of the unit ball in $\field{C}^n$ ($n>2$) having a strongly pseudoconvex end
have finite topological type (for the compactification of such quotients see also
\cite{MY}).
\end{rem}
\begin{rem}\label{non-comp and non-kahler}
The statement of Theorem \ref{noncompact} still holds true for a {\em non-K\"ahler complete} metric
$\Theta$ satisfying \eqref{i} and having bounded torsion
$T=[i(\Theta),\partial\Theta]$, i.e.
$\abs{T}\leqslant C$, where $\abs{T}$ is the norm with respect to
$\Theta$ (that is, Theorem \ref{nonkahler} has a non-compact version analogous
to Theorem \ref{noncompact}).
As in the proof of Theorem \ref{noncompact}, the localization argument in Section \ref{s3.1}
goes through provided we can prove the existence of the
spectral gap of the Kodaira-Laplacian $\Box^{L^p}$\,.
This follows by applying the generalized Bochner-Kodaira-Nakano formula of
Demailly \cite[Theorem 0.3]{dem2} with torsion term as in \cite[Theorem 1]{BiV}.
\end{rem}
Another generalization is a version of Theorem \ref{t0.1}
for covering manifolds.
Let $\tx$ be a paracompact smooth manifold, such that there is a discrete group
$\g$ acting freely on $\tx$ with a compact quotient $X=\tx/\g$.
Let $\pi_{\Gamma}:\tx\longrightarrow X$ be the projection.
Assume that there exists a $\g$--invariant pre--quantum line bundle
$\til{L}} \newcommand{\te}{\til{E}$ on $\tx$ and a $\g$--invariant connection $\nabla^{\til{L}} \newcommand{\te}{\til{E}}$ such that
$\tom=\frac{\sqrt{-1}}{2\pi}(\nabla^{\til{L}} \newcommand{\te}{\til{E}})^2$ is non--degenerate.
We endow $\tx$ with a $\g$--invariant Riemannian metric $g^{T\tx}$.
Let $\tj$ be an $\g$-invariant almost complex structure on $T\tx$ which is separately
compatible with $\tom$ and $g^{T \tx}$.
Then $\til{\mathbf J}$, $g^{T \tx}$, $\tom$, $\tj$, $\til{L}} \newcommand{\te}{\til{E}$, $\te$ are the pull-back of the corresponding
objects in Section \ref{s1} by the projection $\pi_{\Gamma}:\tx \to X$.
Let $\Phi$ be a smooth Hermitian section of $\End(E)$,
and $\til{\Phi}=\Phi\circ \pi_{\Gamma}$.
Then the renormalized Bochner-Laplacian $\tdel$ is
\begin{equation*}
\tdel=\Delta^{\til{L}} \newcommand{\te}{\til{E}^p\otimes\te}-p\,(\tau\circ \pi_{\Gamma})+\til{\Phi}
\end{equation*}
which is an essentially self--adjoint operator. It is shown in \cite[Corollary 4.7]{MM} that
\begin{equation}\label{0ell6}
\spec\tdel\subset [-C_L,C_L]\cup[2p\mu_0-C_L,+\infty[\,,
\end{equation}
where $C_L$ is the same constant as in Section \ref{s1} and $\mu_0$ is introduced in
\eqref{0.21}.
Let $\widetilde{\mH}_p$ be the eigenspace of $\tdel$ with the eigenvalues in
$[-C_L, C_L]$:
\begin{equation}\label{0.31}
\widetilde{\mH}_p=\operatorname{Range}E\big([-C_L, C_L],\tdel\big)\,,
\end{equation}
where $E(\,\cdot\,,\tdel)$ is the spectral measure of $\tdel$.
From \cite[Corollary 4.7]{MM},
the von Neumann dimension of $\widetilde{\mH}_p$ equals $d_p=\dim \mH_p$\/.
Finally, we define the generalized Bergman kernel $\widetilde{P}_{q,p}$
of $\tdel$ as in Definition \ref{d3.0}.
Unlike most of the objects on $\tx$,
$\widetilde{P}_{q,p}$ is not $\g$--invariant.
By \eqref{0ell6} and the proof of Proposition \ref{0t3.0},
the analogue of \eqref{0c7} still holds
on any compact set $K\subset \tx$. By the finite propagation speed
as the end of Section \ref{s3.1}, we have:
\begin{thm}\label{t0.11} We fix
$0<\varepsilon_0 < \inf_{x\in X}\{\text{injectivity radius of $x$}\}$.
For any compact set $K\subset\tx$ and $k,l\in \field{N}$, there exists
$C_{k,\,l,\,K}>0$ such that for $x,x'\in K$, $p\in \field{N}$,
\begin{equation}\label{0.61}
\begin{split}
&\Big |\widetilde{P}_{q,p}(x,x')
- P_{q,p}(\pi_{\Gamma}(x),\pi_{\Gamma}(x'))\Big |_{\cali{C}} \newcommand{\cA}{\cali{A} ^l(K\times K)}
\leqslant C_{k,\,l,\,K}\: p^{-k-1}\, ,\quad {\rm if}\,\, d(x,x')< \varepsilon_0,\\
&\Big |\widetilde{P}_{q,p}(x,x')\Big |_{\cali{C}} \newcommand{\cA}{\cali{A} ^l(K\times K)}
\leqslant C_{k,\,l,\,K}\: p^{-k-1}\, ,\quad
{\rm if}\,\, d(x,x')\geqslant\varepsilon_0
\end{split}
\end{equation}
Especially, $\widetilde{P}_{q,p}(x,x)$ has the same asymptotic expansion as
$B_{q,p}(\pi_{\Gamma} (x))$ in Theorem \ref{t0.1} on any compact set $K\subset\tx$.
\end{thm}
\begin{rem}
Theorem \ref{t0.11} works well for coverings of non-compact manifolds.
Let $(X,\Theta)$ be a complete K\"ahler manifold, $(L,h^L)$ be a
holomorphic line bundle on $X$ and let $\pi_{\Gamma}:\tx\to X$ be a Galois
covering of $X=\tx/\g$. Let $\widetilde{\Theta}$ and $(\widetilde{L},h^{\widetilde{L}})$ be the
inverse images of $\Theta$ and $(L,h^L)$ through $\pi_{\Gamma}$.
If $(X,\Theta)$ and $(L,h^L)$ satisfy one of the
conditions \eqref{i} or \eqref{ii}, $(\tx,\widetilde{\Theta})$ and
$(\widetilde{L},h^{\widetilde{L}})$ have the same properties. We obtain therefore as
in \eqref{ell6} (by integrating over a fundamental domain):
\begin{equation}
\liminf_{p\longrightarrow\infty}p^{-n}\dim_{\g} H^0_{(2)}(\tx,\til{L}} \newcommand{\te}{\til{E}^p)\geqslant
\frac{1}{n!}\int_X\Big(\frac{\sqrt{-1}}{2\pi}R^L\Big)^n .
\end{equation}
where $\dim_{\g}$ is the von Neumann dimension of the $\g$--module
$H^0_{(2)}(X,L^p)$. Such type of inequalities imply as in \cite{TCM:01}
weak Lefschetz theorems \`a la Nori.
\end{rem}
\subsection{Singular polarizations} \label{s5.6}
Let $X$ be a compact complex manifold. A {\em singular K\"ahler metric} on
$X$ is a closed, strictly positive $(1,1)$-current $\omega$.
This means there exist locally strictly plurisubharmonic functions
$\varphi\in L^1_{loc}$ such that $\sqrt{-1}\partial\db\varphi=\omega$.
If the cohomology class of $\omega$ in $H^2(X,\field{R})$ is integral,
there exists a holomorphic line bundle $(L,h^L)$, endowed with a
singular Hermitian metric, such that $\frac{\sqrt{-1}}{2\pi}R^L=\omega$
in the sense of currents. We call $(L,h^L)$ a {\em singular polarization}
of $\omega$. If we change the metric $h^L$, the curvature of the new metric
will be in the same cohomology class as $\omega$. In this case we speak of a
polarization of $[\omega]\in H^2(X,\field{R})$.
Our purpose is to define an appropriate notion of polarized
section of $L^p$, possibly by changing the metric of $L$,
and study the associated Bergman kernel.
First recall that a Hermitian metric $h^L$ is called {\em singular }
if it is given in local
trivialization by functions $e^{-\varphi}$ with
$\varphi\in L^1_{\mathrm{loc}}$.
The curvature current $R^L$ of $h^L$ is well defined and
given locally by the currents $\partial\db\varphi$.
By the approximation theorem of Demailly \cite[Theorem 1.1]{dem},
we can assume that $h^L$ is smooth outside a proper analytic set
$\Sigma\subset X$. Using this fundamental fact, we will introduce
in the sequel the {\em generalized Poincar\'e metric} on $X\smallsetminus\Sigma$.
Let $\pi:\widetilde{X}\longrightarrow X$ be a resolution of singularities such that $\pi:
\widetilde{X}\smallsetminus\pi^{-1}(\Sigma)\longrightarrow
X\smallsetminus \Sigma$ is biholomorphic and $\pi^{-1}(\Sigma)$
is a divisor with only simple normal crossings.
Let $g^{T\widetilde{X}}_0$ be an arbitrary smooth $J$-invariant metric
on $\widetilde X$ and $\Theta '(\cdot,\cdot)=g^{T\widetilde{X}}_0(J\cdot,\cdot)$
the corresponding $(1,1)$-from. The generalized Poincar\'e metric on
$X\smallsetminus \Sigma=\widetilde X \smallsetminus\pi^{-1}(\Sigma)$
is defined in \cite[\S 3]{Zu:79} by
\begin{equation}\label{poin}
\Theta_{\varepsilon_0}=\Theta '
-\varepsilon_0\sqrt{-1}\sum_i\partial\db\log(-\log\|\sigma_i\|^2_i)^2\,,
\quad \text{$0<\varepsilon_0\ll 1$ fixed},
\end{equation}
where $\pi^{-1}(\Sigma)= \cup_i \Sigma_i$ is the decomposition into
irreducible components $\Sigma_i$ of $\pi^{-1}(\Sigma)$
and each $\Sigma_i$ is non-singular; $\sigma_i$ are sections
of the associated holomorphic line bundle $[\Sigma_i]$ which vanish to
first order on $\Sigma_i$, and $\|\sigma_i\|_i$ is the norm for a smooth
Hermitian metric on $[\Sigma_i]$ such that $\|\sigma_i\|_i<1$.
\begin{lemma}\label{lem-poin}
{\rm(i)} The generalized Poincar\'e metric \eqref{poin} is a complete Hermitian metric
of finite volume. It has bounded torsion and Ricci curvature.
{\rm(ii)} If $(E,h^E)$ is a holomorphic bundle over $X$ with smooth Hermitian metric
and $H_{(2)}^0(X\smallsetminus\Sigma,E)
=\big\lbrace u\in L^{0,0}_2(X\smallsetminus \Sigma, E\,,\,\Theta_{\varepsilon_0}\,,h^E):
\db^{E}u=0\big\rbrace$ then $H_{(2)}^0(X\smallsetminus\Sigma,E)=H^0(X,E)$.
\end{lemma}
\begin{proof}
To describe the metric more precisely we denote by
$\field{D}$ the unit disc in $\field{C}$ and by $\field{D}^*=\field{D}\smallsetminus\{0\}$.
On the product $(\field{D}^*)^l\times\field{D}^{n-l}$ we
introduce the metric
\begin{equation} \label{compl12,16}
\omega_P=\frac{\sqrt{-1}}{2}\sum^l_{k=1}\frac{dz_k\wedge d\overline z_k}
{|z_k|^2(\log|z_k|^2)^2}+\frac{\sqrt{-1}}{2}\sum^n_{k=l+1}dz_k\wedge d
\overline z_k.
\end{equation}
For any point $p\in\pi^{-1}(\Sigma)$ there exists
a coordinate neighbourhood $U$ of $p$ isomorphic to $\field{D}^n$ in which
$(X\smallsetminus\pi^{-1}(\Sigma))\cap U=\{z=(z_1,\ldots,z_n):z_1\neq 0,\ldots,z_l\neq 0
\}$. Such coordinates are called special. We endow $(X\smallsetminus\pi^{-1}(\Sigma))
\cap U\cong(\field{D}^*)^l\times\field{D}^{n-l}$ with the metric \eqref{compl12,16}.
Now, a calculation in special coordinates as in \cite[Prop.\,3.4]{Zu:79}
show that the metrics \eqref{poin} and \eqref{compl12,16} are equivalent.
From this the first assertion of (i) follows.
We wish to show that there
exist a constant $C>0$ such that
\begin{equation}\label{zar-ell9}
\sqrt{-1}R^{\det}>-C\Theta_{\varepsilon_0}\,,\; |T_{\varepsilon_0}|<C\,.
\end{equation}
where $T_{\varepsilon_0}=[\Theta_{\varepsilon_0},\partial\Theta_{\varepsilon_0}]$
is the torsion operator of $\Theta_{\varepsilon_0}$ and $|T_{\varepsilon_0}|$ is its norm with
respect to $\Theta_{\varepsilon_0}$.
Now $\partial\Theta_{\varepsilon_0}=\partial\Theta'$ by \eqref{poin}, so
it extends smoothly over $\tx$, and thus we get the second relation of
\eqref{zar-ell9}.
We turn now to the first condition of \eqref{zar-ell9}. We have
\begin{equation}\label{zar-ell10}
\Theta_{\varepsilon_0}=\Theta '+ 2\sqrt{-1} \varepsilon_0
\sum_i \Big(\frac{R^{[\Sigma_i]}}{\log\|\sigma_i\|^2_i}
+\frac{\partial\log\|\sigma_i\|^2_i\wedge
\overline\partial\log\|\sigma_i\|^2_i}{(\log\|\sigma_i\|^2_i)^2}\Big )\, .
\end{equation}
The terms $R^{[\Sigma_i]}/\log\|\sigma_i\|^2_i$ tend to zero as we approach $\Sigma$
so they can be absorbed in $\Theta'$ and do not contribute to the singularity of
$\Theta_{\varepsilon_0}$ near $\Sigma$\,.
To examine the last term let us localize to a point $x_0\in\Sigma$\,.
We choose special coordinates in a neighborhood $U$ of $x_0$ in which
$\Sigma_j$ has the equation $z_j=0$ for $j=1,\dots,k$ and $\Sigma_j$, $j>k$,
do not meet $U$.
Then for $1\leq i\leq k$, $\|\sigma_i\|^2_i=u_i|z_i|^2$ for some positive smooth function $u_i$
on $U$ and
\begin{equation}\label{zar-ell11}
\frac{\partial\log\|\sigma_i\|^2_i\wedge\overline\partial\log\|\sigma_i\|^2_i}{(\log\|\sigma_i\|^2_i)^2}=
\frac{dz_i\wedge d\overline{z}_i+v_i}{|z_i|^2(\log\|\sigma_i\|^2_i)^2}
\end{equation}
where $v_i$ is a smooth $(1,1)$--\,form on $U$. Without loss of generality we may assume that
$\Theta'$ is the Euclidean metric on $U$ so that $\Theta'^n$ is the Euclidean
volume element. Then there exists a smooth function $\beta$ such that
\begin{equation}\label{zar-ell12}
\Theta_{\varepsilon_0}^n=
\left(1+\frac{1+\beta(z)}{\prod_i|z_i|^2(\log\|\sigma_i\|^2_i)^2}\right)\,
\Theta'^n\,=:\gamma(z)\Theta'^n\,.
\end{equation}
and consequently
\begin{multline}\label{ell13}
\sqrt{-1}\,R^{\det}=-\sqrt{-1}\,\partial\overline\partial\log\gamma(z)=
-\sqrt{-1}\,\Big(\frac{\partial\overline\partial\gamma(z)}{\gamma(z)}-
\frac{\partial\gamma(z)\wedge\overline\partial\gamma(z)}{\gamma(z)^2}\Big)\\
\geqslant-\sqrt{-1}\,\frac{\partial\overline\partial\gamma(z)}{\gamma(z)}\;.
\end{multline}
A brute force calculation of $-\sqrt{-1}\,\partial\overline\partial\gamma(z)/\gamma(z)$
and comparison to the singularities of $\Theta_{\varepsilon_0}$ given by \eqref{zar-ell11}
show that
$\sqrt{-1}\,R^{\det}>-C\Theta_{\varepsilon_0}$ for some positive constant $C$\,.
This achieves the proof of \eqref{zar-ell9}.
Let us prove (ii). First observe that $\Theta_{\varepsilon_0}$ dominates
the euclidian metric in special coordinates near $\pi^{-1}(\Sigma)$,
being equivalent with \eqref{compl12,16}. Therefore it dominates some positive multiple
of any smooth Hermitian metric on $\tx$. We deduce that, given a smooth Hermitian
metric $\Theta''$ on $X$, there exists a constant $c>0$ such that
$\Theta_{\varepsilon_0}\geqslant c\Theta''$ on $X\smallsetminus\Sigma$.
It follows that elements of $H_{(2)}^0(X\smallsetminus\Sigma,E)$ are $L^2$
integrable with respect to the smooth metrics $\Theta''$ and $h^E$ over $X$, which entails
they extend holomorphically to sections of $H^0(X,E)$ by \cite[Lemme\,6.9]{De:82}.
We have therefore $H_{(2)}^0(X\smallsetminus\Sigma,E)\subset H^0(X,E)$. The reverse inclusion
follows from the finiteness of the volume of $X\smallsetminus\Sigma$ in the Poincare metric.
\end{proof}
We can construct as in \cite[\S 4]{Ta:94}
a singular Hermitian line bundle
$(\widetilde{L},h^{\widetilde L})$ on $\widetilde{X}$ which is strictly
positive and $\widetilde{L}|_{\widetilde{X}\smallsetminus\pi^{-1}(\Sigma)}\cong
\pi^*(L^{p_0})$, for some $p_0\in\field{N}$.
We introduce on $L|_{X\smallsetminus\Sigma}$ the metric
$(h^{\widetilde L})^{1/p_0}$
whose curvature extends to a strictly positive $(1,1)$--current on $\widetilde X$. Set
\begin{subequations}
\begin{align}\label{ell7}
&h^L_{\varepsilon}=(h^{\widetilde L})^{1/p_0}\,
\prod_i(-\log\|\sigma_i\|^2_i))^\varepsilon\,, \quad 0<\varepsilon\ll 1\,,\\
&\label{ell8}
H^0_{(2)}(X\smallsetminus \Sigma,L^p)
=\big\lbrace u\in L^{0,0}_2(X\smallsetminus \Sigma,
L^p\,,\,\Theta_{\varepsilon_0}\,,h^L_{\varepsilon}):\db^{L^p}u=0\big\rbrace .
\end{align}
\end{subequations}
The space $H^0_{(2)}(X\smallsetminus \Sigma,L^p)$ is the space of
$L^2$-holomorphic sections relative to the metrics $\Theta_{\varepsilon_0}$ on
$X\smallsetminus \Sigma$ and
$h^L_\varepsilon$ on $L|_{X\smallsetminus \Sigma}$.
Since $(h^{\widetilde L})^{1/p_0}$ is bounded away from zero (having
plurisubharmonic weights),
the elements of this space are $L^2$ integrable with respect to the Poincar\'e metric
and a smooth metric $h^L_{*}$ of $L$ over whole $X$. By Lemma \ref{lem-poin} (ii)
we have $H^0_{(2)}(X\smallsetminus \Sigma,L^p)\subset H^0(X,L^p)$.
(Here we cannot infer the other inclusion since $h^L$ might be infinity on $\Sigma$.)
The space $H^0_{(2)}(X\smallsetminus \Sigma,L^p)$ is our space of polarized sections of $L^p$.
\begin{cor}\label{moi}
Let $(X,\omega)$ be a compact complex manifold with a singular K\"ahler
metric with integral cohomology class. Let $(L,h^L)$ be a singular
polarization of $[\omega]$ with strictly positive curvature current having
singular support along a proper analytic set $\Sigma$\,.
Then
the Bergman kernel of the space of polarized sections \eqref{ell8}
has the asymptotic expansion as in Theorem \ref{noncompact}
for $X\smallsetminus\Sigma$.
\end{cor}
\begin{proof}
We will apply Remark \ref{non-comp and non-kahler} to the
non--K\"ahler Hermitian manifold
$(X\smallsetminus \Sigma,\Theta_{\varepsilon_0})$ equipped with the Hermitian bundle
$(L|_{X\smallsetminus \Sigma}, h^L_\varepsilon)$.
Thus we have to show that there
exist constants $\eta>0$, $C>0$ such that
\begin{align}\label{ell9}
&\sqrt{-1}R^{(L|_{X\smallsetminus \Sigma},\,
h^L_\varepsilon)}>\eta\Theta_{\varepsilon_0}\,,
\;\sqrt{-1}R^{\det}>-C\Theta_{\varepsilon_0}\,,\; |T_{\varepsilon_0}|<C\,.
\end{align}
where $T_{\varepsilon_0}=[i(\Theta_{\varepsilon_0}),\partial\Theta_{\varepsilon_0}]$
is the torsion operator of $\Theta_{\varepsilon_0}$ and $|T_{\varepsilon_0}|$ is its norm with
respect to $\Theta_{\varepsilon_0}$.
The first one results for all $\varepsilon$ small enough from \eqref{poin}, \eqref{ell7}
and the fact that the curvature of $(h^{\widetilde L})^{1/p_0}$
extends to a strictly positive $(1,1)$--\,current on $\widetilde X$
(dominating a small positive multiple of $\Theta'$ on $\widetilde X$).
The second and third relations were proved in \eqref{zar-ell9}.
This achieves the proof of Corollary \ref{moi}.
\end{proof}
\begin{rem}
(a) Corollary \ref{moi} gives an alternative proof of the characterization
of Moishezon manifolds given by Ji--Shiffman \cite{JS:93}, Bonavero
\cite{Bo:93} and Takayama \cite{Ta:94}. Indeed,
any Moishezon manifold possesses a strictly positive singular polarization
$(L,h^L)$.
Conversely, suppose $X$ has such a polarization. Then as in \eqref{ell6}, we
have $\dim H^0_{(2)}(X\smallsetminus \Sigma,L^p)\geqslant C p^n$ for some $C>0$
and $p$ large enough.
Since $H^0_{(2)}
(X\smallsetminus \Sigma,L^p)\subset H^0(X,L^p)$, it follows that $L$ is big and
$X$ is Moishezon.
(b) By \cite[Proposition 6.6. (f)]{De:96}, or \cite{Ts:92}, any big line bundle $L$ on a
projective manifold carries a singular Hermitian metric
having strictly positive curvature current with singularities along a proper analytic set.
(c) The results of this section hold also for reduced compact complex spaces $X$
possessing a holomorphic line bundle $L$ with singular Hermitian metric $h^L$
having positive curvature current (see \cite{Ta:94} for definitions). This is just a matter of
desingularizing $X$. As space of polarized sections we obtain $H^0_{(2)}(X\smallsetminus \Sigma,L^p)$
where $\Sigma$ is an analytic set containing the singular set of $X$.
\end{rem}
\subsection*{Acknowledgments}
We express our hearty thanks to Professors Jean-Michel Bismut, Jean-Michel Bony
and Johannes Sj\"ostrand for useful conversations. It's a pleasure to
acknowledge our intellectual debt to Xianzhe Dai and Kefeng Liu.
We are grateful Professor Paul Gauduchon for the reference \cite{Ga04}
on the Hermitian scalar curvature.
Our collaboration was partially supported by the European Commission
through the Research Training Network ``Geometric Analysis''.
\providecommand{\href}[2]{#2}
|
{
"timestamp": "2005-11-15T22:37:26",
"yymm": "0411",
"arxiv_id": "math/0411559",
"language": "en",
"url": "https://arxiv.org/abs/math/0411559"
}
|
\section{Introduction}
Polyakov have shown that there is no deconfinement phase for the pure
2+1D compact U(1) gauge theory\cite{POLYAKOV77}. In the
confinement phase instantons proliferate and the gauge field acquires a mass
gap. After the seminal work\cite{POLYAKOV77} a good deal of theoretical
efforts have been devoted to the question of how the presence of matter field
with fundamental charge modifies the dynamics of the U(1) gauge
field\cite{EINHORN,FRADKIN79,NAGAOSA93,MUDRY,NAYAK,NAGAOSA00,ICHINOSE,WEN02,RWspin,HERBUT,KLEINERT,HERMELE}.
The dynamics of the U(1) gauge field crucially depends on the number and the
dynamics of the matter fields\cite{IOFFE,MURTHY,WEN02,RWspin,SENTHIL04,HERMELE}.
Theoretical analysis is most feasible if there are a large number of matter
fields. One loop calculations show that the gauge coupling is renormalized to
be $g^2 \sim \frac{\Lambda}{N}$ with $N$, the number of matter fields and
$\Lambda$, the mass of the matter fields\cite{MURTHY}. Consequently the
instanton acquires a large scaling dimension ($\sim N$) and becomes irrelevant
at the critical point in the limit $\Lambda \to 0$
\cite{IOFFE,WEN02,RWspin,SENTHIL04,HERMELE}.
Then it is interesting to ask how a change in the dynamics of matter field
affects the dynamics of the U(1) gauge field.
The self-interaction of massive matter fields was shown to qualitatively modify the short distance
potential between test charge in the non-compact 2+1D quantum
electrodynamics\cite{GHOSH,ABREU}. An alternative way of modifying the
dynamics of matter fields is to put the matter fields under a strong
additional gauge interaction.
In this paper, we are going to consider a system of 2+1D $U(1)$ gauge
theory coupled with matter fields in 2+1D where the matter fields in turn
interact strongly with a $SU(N)$ gauge field in p+1D.
(Here the 2+1D space-time is a subspace of the p+1D space-time with $p=2, 4, 6$.)
When $p = 4, 6$ ($p = 2$) the SU(N) gauge coupling becomes weak at low (high) energy.
In this regime the theory reduces to the aforementioned
2+1D U(1) gauge theory coupled with matter fields.
Then how will the dynamics of the 2+1D U(1) gauge field be modified
at high (low) energy for $p = 4, 6$ ($p = 2$) where the SU(N) gauge coupling becomes strong ?
Usual perturbative picture is not suitable to describe the strong coupling effect.
The aim of the present paper is to examine the non-perturbative effect of
the strong SU(N) gauge coupling on the 2+1D U(1) gauge field.
For some strongly coupled gauge theories, including the one under
consideration, it is advantageous to use dual string
theory\cite{AHARONY}. The exact duality between gauge and string theories has
been anticipated from the observation that the Wilson loop in gauge theory
satisfies a loop equation of string\cite{POLYAKOV98}.
The first concrete example for
this idea was conjectured as a duality between the type IIB string theory in
the anti-de Sitter space and ${\cal N}=4$ supersymmetric SU(N) gauge theory in
3+1D\cite{MALDACENA,GUBSER,WITTEN}. The duality has opened a
variety of possibilities for a new understanding on many strong coupling
phenomena of gauge theories\cite{AHARONY}. From the dual gravity description
the confining nature of the 2+1D SU(N) gauge theory has been
confirmed\cite{WITTEN2}. Recently the idea has been applied to construct
QCD-like gauge theory including fundamental matter
fields\cite{KARCH,KRUCZENSKI,BABINGTON,CHERKIS,NUNEZ,ERDMENGER}.
Most recently dual gravity backgrounds have been found for an infinite family of quiver gauge theories\cite{BENVENUTI}.
The field theory of our interest is a nonsupersymmetric theory.
It contains a p+1D SU(N) gauge theory with matter fields in
the adjoint representation of the SU(N) gauge group,
and a U(1) gauge theory that lives on a 2+1D subspace.
It also contain matter fields on the 2+1D subspace that carry fundamental
charges for both U(1) and SU(N) gauge fields. To understand the dynamics of
the $U(1)$ gauge field, we would like to integrate out the SU(N) gauge field
and the matter fields to obtain an effective theory of the U(1) gauge field.
However, this is not easy to do in the strong coupling limit. In this paper, we
like to show that, in the large $N$ limit, we can obtain the effective action
using a duality relation between the above field theory and
D-brane in superstring theory.
The above 2+1D/p+1D U(1)/SU(N) gauge theory
has a dual description in terms of superstring theory where we consider a probe
D2-brane lying parallel to a large number of Dp-branes in type IIA superstring
theory. However the full field theory describing the brane system is larger
than the field theory of our interest. Fortunately, it is possible to study a
reduced field theory from the brane configuration in the probe limit, as will
be explained below. We first identify the full degrees of freedom in the field
theory for the brane configuration, then explain how we obtain the reduced
field theory of our interest.
The low energy field theory on the D2-brane is the 2+1D U(1) gauge theory. The
U(1) gauge field comes from open string with its two ends on the D2-brane. The
open strings connecting different Dp-branes give rise to p+1D SU(N) gauge
fields on the Dp-brane. The matter fields, carrying fundamental charges for
both U(1) and SU(N) gauge fields, come from the open strings that connect the
D2- and Dp-branes. There are also U(1)/SU(N) neutral scalars coming from the
open strings with its two ends on the D2-branes. They describe fluctuations in
the relative position of the D2 and Dp-branes. In the supersymmetric case
($p=2,6$) there are also fermionic partners to all of the bosonic modes. These
are the degrees of freedom of the full field theory for the brane
configuration.
We consider the probe action of the D2-brane in the gravity background
dual to the $N$ Dp-branes.
In the probe limit, the back reaction of the D2-brane to metric is not included.
More specifically, the fluctuations of the neutral scalars are frozen
by fixing the position and the flat shape of the probe brane. We treat the
separation between the D2 and Dp branes as a non-dynamical parameter ignoring
the fluctuations. The fermionic modes on the D2-brane do not have geometrical
meaning like the position of brane because they can not have vacuum expectation
values. Thus we just ignore the fluctuations of those modes in the effective
action. Certainly, we also ignore the fluctuations of 2+1D U(1) gauge field.
We see that in the probe limit, the probe action only includes the effect of
integrating out all the p+1D fields including the SU(N) gauge field,
and the fundamental matter fields,
but not the fluctuations of the 2+1D U(1) gauge field, neutral scalars, and
their fermionic partners on the D2-brane. Thus one can regard the probe action
with background U(1) gauge field as an effective action for the U(1) gauge
field which is obtained by integrating out the SU(N) gauge field
along with other p+1D fields and the fundamental matter field.
This, in turn, can be interpreted as the effective action obtained
from the reduced field theory which includes all the degrees of freedom
of the full field theory except for the neutral
scalars and fermions coming from the strings
with their two ends attached to the D2-brane.
In this approach, the effective coupling strength of the SU(N) gauge
interaction and the mass of the matter fields can be tuned independently by the
separation between the branes and the string coupling constant. Using the
resulting 2+1D U(1) effective action, we can examine how the U(1) gauge
coupling and the mass gap of the U(1) gauge theory change as the energy scale
(set by the separation between branes) varies. The mass gap of the U(1) gauge
theory is generated by the proliferation of the U(1) instanton.
It should be emphasized that the U(1) effective action is not meant to describe
the full field theory of the branes. We use the brane configuration as a tool
to integrate out the strongly coupled matter fields and the SU(N) gauge field.
In the full field theory of the brane, the neutral scalars and the fermionic
modes should be allowed to fluctuate along with the U(1) gauge field. This
makes significant differences in the dynamics of instanton. First, in the full
field theory the neutral scalars acquire space-time dependent expectation value
in the presence of U(1) instanton. In the brane picture, the probe brane is
bent near the instanton, which, in turn, modify the interaction between
instantons. Second, the presence of the fermionic zero modes on the D2-brane
associated with the underlying supersymmetry for $p=2, 6$ will suppress the
multi-instanton effects in the full field theory. As a result, the U(1) photon
(equivalently, the scalar dual to the photon in 2+1D) remains massless with
$16$ sypercharges\cite{SEIBERG}. However there are multi-instanton effects in
the reduced non-supersymmetric field theory of our interest. This is because
there are no such neutral scalars and fermions in our field theory model.
There are unbroken supersymmetries in the D2/Dp system for $p=2$ and $6$.
The full field theory includes not only the (p+1)D super SU(N) gauge theory
but also the 2+1D super U(1) gauge theory.
With the suppression of fluctuations of neutral scalars and the U(1) gaugino on the D2-brane,
the reduced field theory becomes a non-supersymmetric 2+1D U(1) gauge theory.
However the U(1) gauge theory is still coupled to the supersymmetric (p+1)D SU(N) gauge theory with the fundamental matter fields.
We will examine how the dynamics of the U(1) gauge field is affected by the fundamental matters which are strongly coupled to the super SU(N) gauge field.
For $p=4$ there is no supersymmetry. The absence of supersymmetry makes the
D2/D4 system unstable. Eventually the D2-brane will collapse to the D4-branes
and will be dissolved into flux in the D4-branes. In the full unstable field
theory there are tachyons describing the transverse fluctuations of the
D2-brane. For the purpose of exploring the full non-perturbative structure of
string theory it is essential to consider the tachyon condensation\cite{ASEN}.
Here we freeze the unstable fluctuations and study the field theory describing
the fluctuations along the stable direction. Conceptually this is
similar to the Gliozzi-Scherk-Olive(GSO) projection in constructing
tachyon-free string theories out of full string spectrum including tachyons.
However the identification of stable field theory is less clear in the present
case because the tachyonic modes are coupled to other stable modes while there
is no such coupling in the GSO projection. Thus it is not clear {\it in
priori} whether the D2/D4 brane with fixed distance describes the 2+1D/4+1D
U(1)/SU(N) gauge theory. In this paper we present a clue that it may be really
the case.
\section{Effective action of the U(1) gauge theory from dual gravity approach}
First we consider a general configuration involving D2 and Dp branes in
10-dimensional type IIA string theory with $p$ an even integer.
Then we will discuss $p=2, 6, 4$ cases in the order that the number of supercharges is lowered.
Consider N coincident Dp-branes and one D2-brane where the Dp-branes are
extended in $0,1,...,p$ directions and the D2-brane, in $0,1,2$ directions.
In the field theory limit\cite{MALDACENA,ITZHAKI} the low energy theory
consists of two decoupled theories : 1) 9+1D gravitational
theory, and 2) 2+1D U(1) gauge theory on the D2 brane and
p+1D SU(N) gauge theory on the Dp branes. The U(1) and SU(N)
gauge theories are coupled with each other through matter fields. The matter
fields come from stretching strings between the D2-brane and the Dp-branes,
and thus they are extended only in the 0,1,2 directions. They carry
fundamental charges for both U(1) and SU(N) gauge fields. Since there are $N$
different possibilities of the string's ending on the Dp-branes, the number of
matter fields is proportional to $N$. For each one of $N$ there are a few
light string modes and a tower of infinitely massive modes.
We replace the N Dp-branes with a gravitational background. The Euclidean
metric for the N Dp-branes which are located at $x^{p+1} = x^{p+2} = ... = x^9
= 0$ is (in string frame)\cite{AHARONY}
\begin{equation}
ds^2 =
\frac{1}{\sqrt{H_p(r)}} \left[ \sum_{i=0}^{p} (dx^i)^2 \right]
+ \sqrt{H_p(r)} \left[ \sum_{j=p+1}^{9} (dx^j)^2 \right],
\label{metric}
\end{equation}
where $H_p(r) = 1 + \left( \frac{L_p}{r} \right)^{7-p}$ with $r = \sqrt{
\sum_{j=p+1}^{9} (x^j)^2 }$ and $L_p^{7-p} = \frac{ (2\pi)^{7-p} g_s l_s^{7-p}
N}{ (7-p) \Omega_{8-p} }$. $g_s$ is the string coupling in the asymptotic
region ($r \rightarrow \infty$), $l_s$, the string length scale and
$\Omega_{d}$, the volume of the unit d-dimensional sphere. The classical
gravity approximation is reliable if the curvature is small in the unit of
string length and the local string coupling is small,
\begin{equation}
| {\cal R} | l_s^2 << 1, \mbox{\hspace{0.5cm}} e^\phi << 1,
\label{smallRg}
\end{equation}
where ${\cal R}$ is the scalar curvature of the metric (\ref{metric}) and
$e^\phi = g_s H_p^{\frac{(3-p)}{4}}$ with $\phi$, the dilaton field.
In the probe limit ($N >> 1$) the back reaction of the D2-brane on the metric
is negligible. The effective action of the U(1) gauge field on the D2-brane
is given by the Dirac-Born-Infeld action,
\begin{eqnarray}
\Gamma & = & \frac{1}{(2\pi)^2 l_s^3} \int dx^3 e^{-\phi}
\sqrt{det(G^{ind}_{\mu \nu} + 2\pi l_s^2 F_{\mu\nu})},
\label{Gamma1}
\end{eqnarray}
where $G^{ind}_{\mu \nu}$ is the induced metric on the probe D2-brane.
Here we suppress the transverse fluctuations of the D2-brane and treat $r$ as a parameter.
We take the field theory limit\cite{MALDACENA,ITZHAKI},
\begin{eqnarray}
l_s & \rightarrow & 0, \nonumber \\
g_{YM}^2 & = & (2 \pi)^{p-2} g_s l_s^{p-3} = {\mbox {\rm fixed}}, \nonumber \\
\Lambda & = & \frac{r}{l_s^2} = {\mbox {\rm fixed}}.
\label{limit}
\end{eqnarray}
Here $g_{YM}^2$ is the coupling constant for the p+1D SU(N) gauge
theory on the Dp-branes. $\Lambda$ is the mass scale associated with the
tension of string stretching between the D2 and Dp-branes.
The resulting field theory contains an $SU(N)$ gauge field in $p+1$
dimensional space-time and a $U(1)$ gauge field in $2+1$ dimensional space-time
which is a subspace of the $p+1$ dimensional space-time. The field theory
also contain bosonic and fermionic matter fields in $2+1$ dimension
that carry both the SU(N) gauge charge and the U(1) gauge charge.
The mass scale of the matter field and the high energy cut-off scale of the field theory is of order $\Lambda$.
In this paper, we like to understand the dynamics of the U(1) gauge field in $2+1$ dimensions.
The Dp/Dq ($p>q$) brane systems have been considered in order to add fundamental matters to the q+1D gauge theories\cite{KARCH,KRUCZENSKI,BABINGTON,CHERKIS,NUNEZ,ERDMENGER}.
In the previous studies\cite{KARCH,KRUCZENSKI,BABINGTON,CHERKIS,NUNEZ,ERDMENGER} the gauge coupling in the light (Dq) brane was taken to be finite in the field theory limit.
In this limit the gauge coupling in the heavy (Dp) brane vanishes.
The gauge symmetry in the heavy brane becomes a global flavor symmetry.
In our case we do the opposite in order to study the effect of strong coupling in the heavy brane.
We take the gauge coupling in the heavy brane finite.
Then the gauge coupling in the light brane becomes infinite.
It is noted that the bare 2+1D gauge coupling is $g_s l_s^{-1} \rightarrow \infty$ in the field theory limit (\ref{limit}) for $p=4$ and $6$.
Then how do we obtain a finite 2+1D gauge coupling ?
This is possible because the fundamental matter fields renormalize the gauge coupling to a finite value.
In other words there is no bare kinetic energy term for the 2+1D gauge field but it is generated by the fluctuations of the fundamental matter fields.
In the strong coupling limit of the SU(N), we take the 't Hooft limit where
the effective Yang-Mills coupling $g_{eff}^2 = g_{YM}^2 N \Lambda^{p-3}$ is
fixed in the large N limit\cite{THOOFT}.
We like to obtain
the low energy effective action for the U(1) gauge field
in this limit by integrating out the SU(N) gauge fields and the
matter fields.
Instead of directly integrating out the SU(N) gauge fields and the
matter fields, we go back to the string theory and integrate out all the
string modes to obtain the
the low energy effective action for the U(1) gauge field:
\begin{eqnarray}
\Gamma & = &
\frac{M^2}{g^2} \int dx^3 \sqrt{ F^2 + M^4 },
\label{Gamma}
\end{eqnarray}
where
\begin{eqnarray}
g^2 & = & [ (7-p) \Omega_{8-p} ]^{\frac{(p-2)}{4}}
(2\pi)^{\frac{(p-2)(2p-13)}{4}} g_{YM}^{\frac{(6-p)}{2}} N^{\frac{(2-p)}{4}}
\Lambda^{\frac{(7-p)(p-2)}{4}}, \nonumber \\
M^4 & = & (2 \pi)^{2p-11} (7-p)
\Omega_{8-p} N^{-1} g_{YM}^{-2} \Lambda^{7-p},
\label{obs}
\end{eqnarray}
and $F^2 \equiv \sum_{\mu > \nu} F_{\mu \nu} F_{\mu \nu}$, the square of the U(1) gauge field strength.
$\Gamma$ corresponds to the effective action generated by vacuum fluctuations of strings in the background of the U(1) gauge field on the probe D2-brane and the gravitational field dual to the Dp-branes\cite{DBI}.
In the weak string coupling limit ($e^{\phi} <<1$) the leading contributions come from the disk diagrams of string world sheet.
In the field theory side the weak string coupling limit corresponds to the 't Hooft limit, and the disk diagrams to the planar diagrams (see Fig. 1(a)).
Higher order diagrams (e.g., cylinder diagrams) in string theory corresponds to non-planar diagrams in field theory (see Fig. 1(b)).
The disk diagram is order of $e^{-\phi} \sim N$ and the cylinder diagram, $e^{0} \sim 1$.
Thus the Dirac-Born-Infeld action which is order of $e^{-\phi}$ captures the fluctuations of matter fields and the SU(N) gauge fields in the leading order of $N$ in the field theory side\cite{AHARONY}.
The effects of matter fields and the SU(N) gauge field are encoded in the nontrivial metric background of Eq.(\ref{Gamma1}) and in $g^2$ and $M$ of Eq.(\ref{Gamma}).
Note that it is possible to regard the brane position as a non-dynamical parameter
because the fluctuations of strings with their two ends on the D2-brane are negligible
by factor of $1/N$ in the probe limit.
The derivative terms such as $(\partial F)^n$ are ignored in this action.
If the action is expanded in $F^2$, the coefficient of the quadratic term becomes $\frac{1}{2 g^2}$.
Thus $g^2$ is identified as the gauge coupling of the 2+1D U(1) gauge theory.
$M$ is the mass scale above which higher order terms become important.
$M$ can be also identified as the size of non BPS instanton as will be discussed later.
The conditions for the small curvature and small string coupling
(\ref{smallRg}) become\cite{ITZHAKI}
\begin{equation}
1 << g_{eff}^2 << N^{\frac{4}{7-p}}.
\label{the_range0}
\end{equation}
Now we determine the mass gap of the 2+1D compact U(1) gauge
field as a function of $\Lambda$.
We have to consider instantons because of the compactness of the gauge field.
It is emphasized again that the instanton we consider here is non-supersymmetric even though the background is supersymmetric for $p=2, 6$.
This is because the excitations of scalar fields are suppressed.
The instanton is an event localized in space-time where the
U(1) flux changes by $2\pi$\cite{POLYAKOV77}. Using the dual field strength
$b_\mu = \frac{1}{2} \epsilon_{\mu \nu \lambda} F_{\nu \lambda}$, we divide
$b$ into the longitudinal part and the transverse part,
\begin{equation}
b_\mu = b_\mu^{in} + (\partial \times a)_\mu,
\end{equation}
where the longitudinal part $b_\mu^{in}$ is contributed from the instantons
and satisfies
\begin{equation}
\partial \cdot b^{in} = 2 \pi \rho,
\label{inst}
\end{equation}
with $\rho$, the instanton density. We consider one instanton of charge $q$
at the origin with $q$, an integer. The instanton action is obtained by
minimizing the effective action Eq.(\ref{Gamma}) with respect to the
transverse field $a$. The resulting equation of motion for $a$,
\begin{equation}
\partial \times \frac{ b^{in} + \partial \times a }{\sqrt{ (b^{in} + \partial \times a)^2 + M^4 }} = 0
\end{equation}
is solved by introducing a dual scalar field $\xi$,
\begin{equation}
\frac{ b^{in} + \partial \times a }{\sqrt{ (b^{in} + \partial \times a)^2 + M^4 }} = \partial \xi.
\end{equation}
From Eq.(\ref{inst}) $\xi$ satisfies
\begin{equation}
\partial \cdot \left(
\frac{ \partial \xi }{ \sqrt{ 1 - (\partial \xi)^2 } }
\right)
= 2 \pi q \frac{\delta^{(3)}(x)}{M^2},
\label{flux}
\end{equation}
with $\delta^{(3)}(x)$, the three-dimensional delta function resulting in
\begin{equation}
(\partial_r \xi)^2 = \frac{1}{ (\sqrt{2/q} M r)^4 + 1 }.
\end{equation}
For $r << M^{-1}$ the dual scalar field increases linearly with distance.
On the other hand for $r >> M^{-1}$ we obtain $\xi \sim 1/r$.
Thus we identify the length scale $M^{-1}$ as the core size of instanton.
From Eq.(\ref{Gamma}) the
instanton action is readily obtained to be
\begin{eqnarray}
I_{c}(q) & = &
\frac{M^4}{g^2} \int d^3 x \left[ \frac{1}{\sqrt{ 1 - (\partial \xi)^2 } } - 1 \right] \nonumber \\
& = & \frac{2 \pi M q^{\frac{3}{2}}}{g^2} \int_{0}^\infty dy [ \sqrt{ 4y^4 + 1} - 2y^2 ]
\approx 5.5 \frac{q^{\frac{3}{2}} M}{g^2}.
\label{core}
\end{eqnarray}
It is noted that the action of instanton is finite without short distance
divergence and that the action is proportional to the charge $q$ with a fractional power $\frac{3}{2}$.
Both of these features are due to the higher
order terms of field strength in the effective action (\ref{Gamma}) which
become important near the center of the instanton.
It is noted that the energy scale associated with the instanton core is smaller than the cut-off scale, that is, $M \sim \frac{\Lambda}{ g_{eff}^{1/2} } << \Lambda$.
Thus the core structure of the instanton can be reliably studied from the effective action (\ref{Gamma}) as far as $\Lambda^{-1} << r$.
Now we consider many instantons. Eq.(\ref{flux}) is modified as
\begin{equation}
\partial \cdot \left(
\frac{ \partial \xi }{ \sqrt{ 1 - (\partial \xi)^2 } }
\right)
= \frac{ 2 \pi}{M^2} \sum_a q_a \delta^{(3)}(x-x_a).
\label{flux2}
\end{equation}
If the distance between instantons is much larger than $M^{-1}$ the dual
scalar field becomes
\begin{equation}
\xi(x) \approx \frac{1}{2M^2} \sum_a \frac{q_a}{|x-x_a|}
\end{equation}
leading to the Coulomb interaction between instantons\cite{POLYAKOV77},
\begin{equation}
\Gamma = \sum_a I_c(q_a) + \frac{\pi}{g^2} \sum_{a>b} \frac{q_a q_b}{|x_a - x_b|}.
\end{equation}
Owing to the screening property of the 3D Coulomb gas the
$\frac{1}{x}$ potential is screened to be $\frac{e^{-m_c x}}{x}$ where $m_c$,
the mass gap of the U(1) gauge field\cite{POLYAKOV77}. With $M^{-1}$
identified as a cut-off length scale for instanton the mass gap is given by
$m_c^2 \sim \frac{M^3}{g^2} e^{-I_c}$ with $I_c$, the instanton action with
unit charge\cite{POLYAKOV77,GOPFERT}.
Using Eq.(\ref{obs}) the mass gap is obtained to be
\begin{equation}
m_c^2 \sim g_{YM}^{\frac{4}{3-p}} \lambda^{\frac{(p-5)(p-7)}{4}} e^{-c_p \lambda^{\frac{(p-3)(p-7)}{4}} },
\label{mass}
\end{equation}
where
$\lambda = \Lambda (g_{YM}^2)^\frac{1}{(p-3)} N^{\frac{1}{p-7}}$ and
$c_p = (2 \pi)^{\frac{(2p+3)(11-p)}{4}} [ (7-p) \Omega_{8-p} ]^{\frac{3-p}{4}} \int_0^\infty dy [ \sqrt{ 1+ 4y^4} - 2 y^2 ]$.
\subsection{$p=2$}
The full field theory is the SU(N+1) super Yang-Mills theory with $16$ supercharges where the gauge group is broken to $SU(N) \times U(1)$ for nonzero $\Lambda$.
The vector multiplet consists of the gauge field, 7 scalars and 8 Majorana spinors.
As discussed in the introduction we suppress the fluctuations of the scalars
and fermions in the U(1) sector of the vector multiplet in order to study
non-supersymmetric U(1) gauge theory. The resulting theory is a 2+1D field
theory with a $U(1)$ gauge field, a $SU(N)$ gauge field, and some
bosonic/fermionic matter fields in the fundamental representation of
$U(1)\times SU(N)$ and adjoint representation of $SU(N)$. The DBI action
(\ref{Gamma}) is the effective action for the U(1) gauge boson on the probe
D2-brane after the supermultiplets on the N D2-branes and the stretched string
modes are integrated out. This corresponds to integrating out the $SU(N)$ gauge
field and the matter fields in the field theory.
The mass of the stretched string modes is given by $\Lambda$.
This configuration is stable because the gravitational attraction is balanced by the coupling to the Ramond-Ramond field which we did not show in (\ref{Gamma}).
The conditions for the small curvature and small string coupling (\ref{smallRg})
becomes
\begin{equation}
g_{YM}^2 N^{\frac{1}{5}} << \Lambda << g_{YM}^2 N.
\label{the_range2}
\end{equation}
For $\Lambda > g_{YM}^2 N$ the curvature becomes large in string unit and gravity solution is not reliable.
Instead perturbative field theory is reliable in this UV limit.
For $\Lambda < g_{YM}^2 N^{\frac{1}{5}}$ the local string coupling becomes large and the 11-th dimension of the M-theory appears\cite{ITZHAKI}.
We will concentrate only on the IIA gravity description in the range (\ref{the_range2}).
The U(1) gauge coupling and the inverse size of the instanton is given by
\begin{equation}
g^2 = g_{YM}^2 , \mbox{\hspace{0.5cm}} M = \left( \frac{\Lambda^5}{24 \pi^4 g_{YM}^2 N } \right)^{1/4}.
\label{obs2}
\end{equation}
The original Yang-Mills coupling $g_{YM}^2$ is restored for the U(1) sector of the SU(N+1) gauge theory
as expected.
There is no loop correction to the U(1) gauge coupling.
This is because the integrated SU(N) gauge field and the matter field have $16$ supercharges.
The flow of the U(1) gauge coupling is solely determined by the dimensional scaling as is shown in Fig. 2(a).
The loop correction is absent also in the regime of the perturbative SU(N) gauge theory.
Thus the dimensionless U(1) gauge coupling is likely to behave as $\Lambda^{-1}$
in the whole range of the energy scale including both the weak and strong (IIA gravity) coupling regimes (see Fig. 2(a)).
One can readily obtain the action of the instanton and the mass gap of the U(1) gauge field from (\ref{mass}).
The U(1) instanton considered here is different from the supersymmetric instanton of the full SU(N+1) gauge theory\cite{INSTANTON}.
We are considering a non-supersymmetric instanton which involves the excitation of only the U(1) gauge field on the probe brane.
The supersymmetric instanton\cite{INSTANTON} corresponds to Euclidean D0-brane stretched between the probe D2-branes and one of N D2-brane which involves the excitations of the gauge fields and scalar fields on both sides of the branes.
The mass gap caused by the U(1) instanton is displayed in Fig. 3(a).
It is interesting to note that the mass gap of the U(1) gauge theory increases
as the mass of the matter field decreases while $g_{MY}$ is kept fixed.
This is contrary to the U(1) gauge theory coupled with `free' matter fields where lighter matters would be more effective in screening gauge field.
The opposite trend in the present case is the strong coupling effect of the additional SU(N) gauge field.
Even though mass of matter fields decreases, the increasing trend of the effective coupling in the 2+1D SU(N) gauge theory makes it harder for the matters to be polarized at lower energy.
This is an example showing that change in the dynamics of matter fields can drastically change their screening behavior.
\subsection{$p=6$}
The parallel D6/D2 brane configuration preserve 8 supersymmetries\cite{CHERKIS,ERDMENGER}.
This configuration is also stable because it is a BPS state.
The 2+1D degrees of freedom consist of one vector multiplet, one neutral hyper multiplet
and $N$ fundamental hyper multiplets.
The scalars in the vector multiplet describes the transverse fluctuations of probe brane
in the directions $x^7$, $x^8$ and $x^9$ and
the scalars in the neutral hypermultiplet, in the directions $x^3$, $x^4$, $x^5$ and $x^6$.
The $N$ fundamental hyper multiplets are stretched string modes.
The neutral hyper multiplet, and the fermions and the scalars in the vector multiplet are suppressed in the effective action Eq.(\ref{Gamma}).
The mass of the matter field is again given by $\Lambda$.
The conditions for the small curvature and small string coupling (\ref{smallRg})
becomes
\begin{equation}
\left( \frac{1}{g_{YM}^2 N} \right)^{1/3} << \Lambda << \frac{N}{(g_{YM}^2)^{1/3}}.
\label{the_range6}
\end{equation}
Note that the Yang-Mills coupling $g_{YM}^2$ has a dimension of $(length)^3$ in (6+1)-dimension.
The lower bound of $\Lambda$ is the threshold between the strong coupling regime at high energy and the weak coupling regime at low energy.
The U(1) gauge coupling and the inverse size of the instanton becomes
\begin{equation}
g^2 = \frac{ 2 \Lambda }{ N },
\mbox{\hspace{0.5cm}}
M = \left( \frac{ 8 \pi^2 \Lambda}{g_{YM}^2 N } \right)^{\frac{1}{4}}.
\label{obs6}
\end{equation}
The dimensionless gauge coupling $g^2 \Lambda^{-1}$ does not flow with $\Lambda$.
Moreover the renormalized gauge coupling is independent of the Yang-Mills gauge coupling even in the strong coupling regime of the 6+1D Yang-Mills theory, that is, $g_{eff}^2 >> 1$.
This is consistent with the one-loop result in the weak coupling regime $g^2 \Lambda^{-1} \sim 1/N$.
Thus it is likely that the dimensionless U(1) gauge coupling
does not flow in the whole range of energy scale including both the strong and
weak coupling regimes of the SU(N) gauge theory as is shown in Fig. 2(b).
If the U(1) gauge coupling has different value at high energy
it will be quickly renormalized to $g^2 \Lambda^{-1} \sim 1/N$ at low energy, which
is represented as dotted lines in Fig. 2(b).
The mass gap as a function of the normalized energy scale (mass of the matter fields) is displayed in Fig. 3(b).
At lower energies the 6+1 gauge coupling becomes weaker resulting in the smaller mass gap.
This is opposite to the $p=2$ case.
\subsection{$p=4$}
The D4/D2-brane system breaks all
supersymmetries and the gravitational attraction renders this system unstable.
This can be seen from the $\Lambda$ dependence of the effective action in
Eq.(\ref{Gamma}) with $F$ set to be $0$.
However here we are interested in the dynamics of the U(1) gauge field on the
D2-brane at a fixed position.
For this we suppress the transverse fluctuations of the D2-brane and treat $\Lambda$ as a
parameter.
We will see a clue that the neglect of the transverse fluctuations in the gravity
description corresponds to the neglect of all unstable modes in the
full unstable field theory thus defining a well defined field theory problem.
If we ignore the tachyonic modes, $\Lambda$ can be
regarded as bare mass of the non-tachyonic matter fields which comes from the
stretching strings. However in the non-supersymmetric case ($p=4$) the actual
mass of the matter field may be different from $\Lambda$ owing to the coupling
with the SU(N) gauge field. This is especially true if the matter fields are
strongly coupled with the SU(N) gauge theory.
The conditions for the small curvature and small string coupling
(\ref{smallRg}) become
\begin{equation}
\frac{1}{g_{YM}^2 N} << \Lambda << \frac{N^{\frac{1}{3}}}{g_{YM}^2}
\label{the_range1}
\end{equation}
and the U(1) gauge coupling $g^2$ and the mass scale $M$ in the effective
action (\ref{Gamma}),
\begin{equation}
g^2 = \left( \frac{ g_{YM}^2 \Lambda^3 }{ 4 \pi^3 N } \right)^{\frac{1}{2}},
\mbox{\hspace{0.5cm}}
M = \left( \frac{\Lambda^3}{\pi g_{YM}^2 N } \right)^{\frac{1}{4}}.
\label{obs4}
\end{equation}
The 4+1D gauge coupling has a dimension of length and becomes weaker as energy is lowered.
The lower bound of $\Lambda$ in (\ref{the_range1}) is the threshold energy
$\Lambda_c \sim \frac{1}{N g_{YM}^2}$ which divides the strong and weak coupling regimes of the 4+1D SU(N) gauge theory.
The gravity solution is valid only in the strong coupling regime ($\Lambda >> \Lambda_c$).
In this region the dimensionless gauge coupling scales as
$g^2 \Lambda^{-1} \sim \left( \frac{\Lambda g_{YM}^2}{N} \right)^{\frac{1}{2}}$.
The flow of the U(1) gauge coupling is shown in Fig. 2(c).
The mass gap of the compact U(1) gauge theory is displayed in Fig. 3(c).
As in the case of $p=6$ the mass gap decreases with decreasing energy scale.
It is instructive to compare to
the case where the 4+1D SU(N) gauge theory decouples and
$N$ species of matter fields with mass $\Lambda$ are coupled
only with the U(1) gauge field.
In this case the one-loop effect renormalizes the U(1) gauge coupling to
$g^2 \sim \frac{\Lambda}{N}$\cite{IOFFE,MURTHY,WEN02,SENTHIL04,HERMELE} and the dimensionless gauge coupling at
the energy scale $\Lambda$ does not flow with the energy scale.
Theories with different gauge couplings flow to the fixed point at low energy.
This is displayed in Fig. 4 which is essentially the same as the one in $p=6$ case (Fig. 2(b)).
The reason why the U(1) gauge coupling decreases with decreasing energy in the presence
of SU(N) gauge field can be explained in the following way. At lower energy
the SU(N) gauge coupling becomes weaker. As a result the matter fields become
more polarizable and more effective in screening the U(1) gauge field leading
to the decreasing behavior of the gauge coupling.
In the non-supersymmetric background ($p=4$) there is no cancellation between bosonic and fermionic fields.
In this case the SU(N) gauge field play an important role in determining the U(1) gauge coupling in the probe brane.
This is contrary to the supersymmetric $p=6$ case where there is no flow of dimensionless gauge coupling even in the strong coupling regime of the SU(N) gauge theory.
It is interesting to note that the gravity solution in (\ref{obs4}) predicts the U(1) gauge coupling at the threshold energy $\Lambda_c$ to be $g^2 \sim \frac{\Lambda}{N}$ which is consistent with the prediction of the weakly coupled SU(N) gauge theory.
This is a nontrivial consistent check to our earlier assumption
that the gravity solution with the neglect of the unstable mode
describes the 2+1D/4+1D U(1)/SU(N) theory with fundamental matters
in the strong coupling regime of the SU(N) theory.
Even though the gravity solution begins to loose its validity around the
threshold the qualitative feature is captured.
It is reminded that $\Lambda$ is not necessarily the same as the mass of the
matter field in the strong coupling regime because there is no supersymmetry for the D2/D4 case.
Therefore it is hard to directly
interpret the scaling dimension of $g^2$ and $M$ in Eq.(\ref{obs4}).
However the ratio between the scaling dimension is meaningful,
\begin{equation}
\frac{d \ln(M \Lambda^{-1})}{d \ln(g^2 \Lambda^{-1})} = -\frac{1}{2}
\end{equation}
because the ratio is independent of definition of $\Lambda$. This exponent
$-1/2$ shows how the mass scale associated with the instanton size scales
relative to the gauge coupling.
\section{conclusion}
In summary, we studied how a change in the dynamics of fundamental matter fields
caused by strong coupling to SU(N) gauge field changes their screening property
in the 2+1D compact U(1) gauge theory.
For this, we considered the probe action of a D2-brane in the gravity
background dual to a large number of coincident Dp-branes by treating the
separation between the branes as a parameter.
We studied the effects of the
SU(N) gauge field of the Dp-branes on the dynamics of the 2+1D compact compact
U(1) gauge field of the D2-brane as the effective coupling strength of the
SU(N) gauge theory is tuned by the separation. We determined the gauge
coupling, the size of instanton and the mass gap of the
non-supersymmetric compact U(1) gauge theory as a function of the separation.
The results are interpreted in terms of the 2+1D U(1) gauge theory and the
p+1D SU(N) gauge theory which are coupled with each other through
a large number of matter fields in fundamental representation of both U(1) and
SU(N) gauge groups.
It is found that the strong coupling of the matter fields to the SU(N) gauge field
can drastically modify the dynamics of the U(1) gauge field.
In the supersymmetric D6/D2 brane system
the renormalized U(1) gauge coupling is shown to be
independent of the (6+1)-dimensional
Yang-Mills coupling even in the strong coupling regime.
For D4/D2 case it is shown that the dimensionless
U(1) gauge coupling decreases with decreasing separation in the strong
coupling regime for the SU(N) gauge theory and that it is continuously
connected with the value in the weak coupling regime.
\section*{ACKNOWLEDGEMENTS}
We would like to thank P. A. Lee, A. Hanany and T. Senthil for helpful discussions.
We are especially grateful to H. Liu for various advices at an early stage of this work.
S.L. was supported by the Post-doctoral
Fellowship Program of Korea Science $\&$ Engineering Foundation (KOSEF) and
NSF grant DMR-0201069.
X.G.W. was supported by NSF Grant No. DMR--01--23156,
NSF-MRSEC Grant No. DMR--02--13282, and NFSC No. 10228408.
|
{
"timestamp": "2005-09-01T22:59:28",
"yymm": "0411",
"arxiv_id": "hep-th/0411239",
"language": "en",
"url": "https://arxiv.org/abs/hep-th/0411239"
}
|
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\begin{document}
\rightline{UG-FT-175/04}
\rightline{CAFPE-45/04}
\rightline{FFUOV-04/17}
\rightline{QMUL-PH-04-10}
\rightline{hep-th/0411181}
\rightline{January 2005}
\vspace{1truecm}
\centerline{\LARGE \bf Giant Gravitons and Fuzzy $CP^2$}
\vspace{1.3truecm}
\centerline{
{\large \bf Bert Janssen${}^{a,}$}\footnote{E-mail address:
{\tt bjanssen@ugr.es}},
{\large \bf Yolanda Lozano${}^{b,}$}\footnote{E-mail address:
{\tt yolanda@string1.ciencias.uniovi.es}}
{\bf and}
{\large \bf Diego Rodr\'{\i}guez-G\'omez${}^{b,c,}$}\footnote{E-mail address:
{\tt diego@fisi35.ciencias.uniovi.es}}
}
\vspace{.4cm}
\centerline{{\it ${}^a$ Departamento de F\'{\i}sica Te\'orica y del Cosmos and}}
\centerline{{\it Centro Andaluz de F\'{\i}sica de Part\'{\i}culas Elementales}}
\centerline{{\it Universidad de Granada, 18071 Granada, Spain}}
\vspace{.4cm}
\centerline{{\it ${}^b$Departamento de F{\'\i}sica, Universidad de Oviedo,}}
\centerline{{\it Avda.~Calvo Sotelo 18, 33007 Oviedo, Spain}}
\vspace{.4cm}
\centerline{{\it ${}^c$Queen Mary, University of London,}}
\centerline{{\it Mile End Road, London E1 4NS, UK}}
\vspace{2truecm}
\centerline{\bf ABSTRACT}
\vspace{.5truecm}
\noindent In this article we describe the giant graviton configurations in $AdS_m\times S^n$
backgrounds
that involve 5-spheres,
namely, the giant graviton in $AdS_4\times S^7$ and the dual giant graviton in
$AdS_7\times S^4$, in terms of dielectric gravitational waves. Thus, we conclude
the programme initiated in hep-th/0207199 and pursued in hep-th/0303183 and
hep-th/0406148 towards the microscopical description of giant gravitons in
$AdS_m\times S^n$ spacetimes.
In our construction the
gravitational waves expand due to Myers dielectric effect onto
``fuzzy 5-spheres'' which
are described as $S^1$ bundles over fuzzy $CP^2$. These fuzzy manifolds
appear as solutions of the matrix model that comes up as the action for M-theory
gravitational waves.
The validity of our
description is checked by confirming the agreement with the Abelian
description in terms of a spherical M5-brane
when the number of waves goes to
infinity.
\newpage
\section{Introduction}
It is by now well-known that giant gravitons can be described microscopically in terms
of dielectric \cite{myers}
gravitational waves \cite{DTV,BMN,JL2,JLR,JLR2}. In
$AdS_m\times S^n$
spacetimes the gravitational waves expand into a fuzzy $S^{(n-2)}$ brane included in
$S^n$, for the genuine giant graviton,
or into a fuzzy $S^{(m-2)}$ brane included in $AdS_m$, for the dual giant graviton, both
carrying angular momentum on the spherical part of the geometry.
For the giant graviton in
$AdS_7\times S^4$ and the dual giant graviton in $AdS_4\times S^7$ these fuzzy spheres
are ordinary non-commutative $S^2$ \cite{JL2}, whereas for both the giant and dual giant
gravitons in $AdS_5\times S^5$ the corresponding ``fuzzy 3-spheres'' are defined as
$S^1$ bundles over non-commutative $S^2$ base manifolds \cite{JLR}.
In all cases
perfect agreement is found, when the number of waves goes to infinity,
with the (Abelian) description in \cite{GST,GMT,HHI} in
terms of spherical test branes.
This agreement provides the strongest support to the
(non-Abelian) dielectric constructions of \cite{JL2,JLR}.
The key point in the non-Abelian description of giant gravitons is the construction of the action for the
system of coincident waves, and the identification of the dielectric and magnetic moment
couplings responsible of their expansion. Given that
$AdS_m\times S^n$ is not weakly curved one cannot use linearised Matrix
theory results as those in \cite{TVR1,TVR2}.
The action appropriate to describe coincident gravitational waves in non-weakly
curved M-theory backgrounds was constructed in \cite{JL2}. This action contains
dielectric and magnetic moment couplings to the 3-form potential of eleven dimensional
supergravity, which are responsible for the expansion of the waves into a dielectric
or magnetic moment M2-brane with the topology of a fuzzy $S^2$, which constitute,
respectively, the
dual giant and giant graviton configurations in the $AdS_4\times S^7$ and $AdS_7\times S^4$
backgrounds. Using this action one can also describe coincident Type IIB gravitational
waves in non-weakly curved $AdS$ type backgrounds, after reduction and T-duality \cite{JLR}.
In the Type IIB action the T-duality direction occurs as an special isometric direction,
and this turns out to
be essential in the construction of the fuzzy manifold associated to both giant and dual
giant graviton configurations in $AdS_5\times S^5$. Describing the ``fuzzy 3-sphere'' as
an $S^1$ bundle over
a fuzzy 2-sphere, the isometric direction is precisely the coordinate along the fibre.
This same action was
later used in \cite{JLR2} to describe the giant graviton solutions in the
$AdS_3\times S^3\times T^4$ Type IIB background
in terms of expanding waves. In this case
the waves expand into a fuzzy cylinder whose basis is contained either in $S^3$ or
in $AdS_3$. In all these cases the agreement between these descriptions and the Abelian
descriptions of \cite{GST,GMT,HHI} for large number of waves provides the strongest
support for the validity of the non-Abelian actions for coincident
waves\footnote{Of course, together with the
fact that the action for
M-theory waves reduces to Myers action for D0-branes when they propagate along the eleventh
direction. This was in fact the key ingredient in \cite{JL2} in order to extend the
linearised Matrix theory result to more general backgrounds. See this reference
for more details.}.
In this paper we would like to complete the programme initiated in \cite{JL2}, and
pursued in \cite{JLR} and \cite{JLR2}, with the microscopical description of the giant
graviton in $AdS_4\times S^7$ and the dual giant graviton in $AdS_7\times S^4$.
One expects that microscopically these gravitons will be described in terms of
a magnetic moment or dielectric M5-brane with the topology of a fuzzy 5-sphere.
Fuzzy $S^n$ with $n>2$ are however quite complicated technically. The general
strategy is to identify them as subspaces of suitable spaces (spaces which admit a
symplectic structure) and to introduce conditions to restrict the functions to be
on the sphere \cite{Ram1,Ram2} \cite{S-J}.
We will see that in our construction the ``fuzzy $S^5$'' is simply defined as an
$S^1$ bundle over a fuzzy $CP^2$, in very much the same way the ``fuzzy $S^3$'' in
\cite{JLR} was defined as an $S^1$ bundle over a fuzzy $S^2$. Again the
coordinate along the $S^1$ fibre is, as in the fuzzy $S^3$ of \cite{JLR}, an
isometric direction present in the action describing the waves. The existence of this
direction is, moreover, crucial in order to construct the right dielectric
couplings that will cause the expansion of the waves. As we will discuss
the expanded 5-brane will be a longitudinal brane wrapped on this direction.
The fuzzy $CP^2$ has been extensively studied in the literature
(see for instance \cite{NR}-\cite{KNR}).
In the context of Myers dielectric effect it was studied in \cite{TV} \cite{ABS}.
$CP^2$ is the coset manifold $SU(3)/U(2)$. $G/H$ coset manifolds
can be described as fuzzy surfaces if $H$ is the isotropy group of the lowest weight state
of a given irreducible representation of $G$ \cite{Madore,TV}.
When $G=SU(2)$ all lowest weight vectors have isotropy group $U(1)$, and therefore one describes
a fuzzy $SU(2)/U(1)$, i.e. a fuzzy $S^2$ for any choice of irreducible representation.
In the case of $SU(3)$ irreducible representations can be parametrised by two integers
$(n,m)$, corresponding to the number of fundamental and anti-fundamental indices.
The lowest weight vector in the representations $(n,0)$ or $(0,n)$ has isotropy group $U(2)$,
whereas for any other irreducible representation it has isotropy group $U(1)\times U(1)$.
Therefore, choosing a $(n,0)$ or a $(0,n)$ irreducible representation one describes a
fuzzy $CP^2$.
We will use this result to describe ``the fuzzy $S^5$'', associated to the giant graviton in
$AdS_4\times S^7$ and the dual giant graviton in $AdS_7\times S^4$, as an $S^1$ bundle over
a fuzzy $CP^2$ base manifold. We will then use the action for coincident M-theory
gravitational waves to find the corresponding ground state configuration.
A key ingredient in this construction is the identification in the action of the
direction along the $S^1$ bundle. For this purpose we will start in Section 2 by
recalling some properties of the action for coincident M-theory gravitational waves
constructed in \cite{JL2}. Then in Section 3 we will use this action to describe
microscopically the giant graviton in $AdS_4\times S^7$. We will see that the
corresponding macroscopical description is in terms of a longitudinal M5-brane with
$S^5$ topology.
In Section 4 we present the
analogous description for the dual giant graviton in $AdS_7\times S^4$. In both cases we
show the explicit agreement with the macroscopical description in \cite{GST,GMT} for
large number of gravitons. In Section 5 we present our Conclusions, where we discuss the
connection between our solution and other 5-brane solutions to Matrix
theory actions previously found in the literature, as well as the supersymmetry properties
of our configurations.
\sect{The action for M-theory gravitational waves}
The action for coincident
M-theory gravitational waves constructed in \cite{JL2} is given by:
\begin{equation}
\label{Mwaves}
S=S^{\rm BI}+S^{\rm CS}
\end{equation}
with BI action given by
\begin{equation}
S^{\rm BI}=-T_0 \int d\tau {\rm STr} \{ k^{-1}\sqrt{-P[E_{00}+E_{0i}
(Q^{-1}-\delta)^i_k E^{kj}E_{j0}]{\rm det Q}}\}\, ,
\end{equation}
where
\begin{equation}
\label{mred}
E_{\mu\nu}={\cal G}_{\mu\nu}+k^{-1}(\mbox{i}_k C^{(3)})_{\mu\nu}\, , \qquad
{\cal G}_{\mu\nu}=g_{\mu\nu}-\frac{k_\mu k_\nu}{k^2}
\end{equation}
and
\begin{equation}
Q^i_j=\delta^i_j+ik[X^i,X^k]E_{kj}\, ,
\end{equation}
and CS action given by
\begin{equation}
\label{di}
S^{\rm CS}=T_0 \int d\tau {\rm STr} \{ -P[k^{-2} k^{(1)}]+iP[(\mbox{i}_X \mbox{i}_X)C^{(3)}] +
\frac12 P[(\mbox{i}_X\mbox{i}_X)^2\mbox{i}_k C^{(6)}] -\frac{i}{6} P[(\mbox{i}_X\mbox{i}_X)^3
\mbox{i}_k N^{(8)}]\}
\end{equation}
where $\mbox{i}_k N^{(8)}$ denotes the Kaluza-Klein monopole potential \cite{BEL}.
In this action
$k^\mu$ is an Abelian Killing vector that points on the direction of propagation of the
waves. This direction is isometric, because the background fields are
either contracted with the Killing vector, so that any component along the isometric
direction of the contracted field vanishes, or pulled back in the worldvolume with
covariant derivatives relative to the isometry (see \cite{JL2} for their explicit
definition)\footnote{The reduced metric ${\cal G}_{\mu\nu}$ appearing in (\ref{mred})
is in fact defined such that its pull-back with ordinary derivatives equals the pull-back
of $g_{\mu\nu}$ with these covariant derivatives.}.
To understand why this is so we need to recall the construction of this action.
Expression (\ref{Mwaves}) was obtained by uplifting to eleven dimensions the action for Type
IIA gravitational waves derived in \cite{JL1} using Matrix String theory in a weakly curved background,
and then going beyond the weakly curved background approximation
by demanding agreement with Myers action for D0-branes
when the waves propagate along the eleventh direction.
In the action for Type IIA waves the
circle in which one Matrix theory is compactified in order to construct Matrix String theory
cannot be
decompactified in the non-Abelian case \cite{JL1}. In fact, the action exhibits a
$U(1)$ isometry associated to translations along this direction, which by construction is
also the direction on which the waves propagate.
A simple way to see this is to recall that the
last operation in the 9-11 flip involved in the construction of Matrix String theory
is a T-duality from fundamental strings wound around the
9th direction. Accordingly, in the action we find a minimal coupling to
$g_{\mu 9}/g_{99}$, which is the momentum operator $k^{-2}k_\mu$
in adapted coordinates. Therefore, by
construction, the action (\ref{Mwaves}) is designed to describe BPS waves with
momentum charge along the compact isometric direction.
It is important to mention that in the Abelian limit, when all dielectric couplings and $U(N)$ covariant
derivatives\footnote{Which are of course implicit in the pull-backs of the non-Abelian
action (\ref{Mwaves}).}
disappear, (\ref{Mwaves}) can be Legendre transformed into an action in
which the dependence on the isometric direction has been restored. This action is
precisely the usual action for a massless particle written in terms of an auxiliary $\gamma$
metric (see \cite{JL2} and \cite{JL1} for the details), where no information remains about the
momentum charge carried by the particle.
Let us now look at the couplings to the 3-form potential of eleven dimensional supergravity. We clearly find a dipole coupling in the
CS part of the action and a
magnetic moment coupling
\begin{equation}
\label{mm}
[X^i,X^k](\mbox{i}_k C^{(3)})_{kj}
\end{equation}
in the BI part\footnote{Let us stress that through its contraction with the Killing vector the 3-form potential acquires the necessary rank to couple in the BI action.}.
These couplings play a crucial role
in the microscopical description of the $AdS_4\times S^7$ dual giant graviton and the $AdS_7\times S^4$ giant graviton, respectively.
Let us parametrise
the $AdS_m\times S^n$ background as
\begin{equation}
\label{ads}
ds^2=-(1+\frac{r^2}{{\tilde L}^2})dt^2+\frac{dr^2}{1+\frac{r^2}{{\tilde L}^2}}+r^2d\Omega_{m-2}^2
+L^2(d\theta^2+\cos^2{\theta}d\phi^2+\sin^2{\theta}d\Omega_{n-2}^2)
\end{equation}
\begin{equation}
C^{(m-1)}_{t\alpha_1\dots\alpha_{m-2}}=-\frac{r^{m-1}}{{\tilde L}}\sqrt{g_\alpha}\, ,\qquad
C^{(n-1)}_{\phi\beta_1\dots\beta_{n-2}}=a_n L^{n-1}\sin^{n-1}{\theta}\sqrt{g_\beta}
\end{equation}
where $a_4=a_5=1$, $a_7=-1$, $\alpha_i$ ($\beta_i$) parametrise the
$S^{m-2}$ ($S^{n-2}$) contained in $AdS_m$ ($S^n$) as
\begin{equation}
d\Omega^2_{m-2}=d\alpha_1^2+\sin^2{\alpha_1}(d\alpha_2^2+\sin^2{\alpha_2}
(\dots +\sin^2{\alpha_{m-3}} d\alpha^2_{m-2}))
\end{equation}
(similarly for $\beta_i$) and $\sqrt{g_\alpha}$ ($\sqrt{g_\beta}$)
denotes the volume element on the unit $S^{m-2}$ ($S^{n-2}$).
Consider the $AdS_7\times S^4$ background and take $r=0$, $\theta={\rm constant}$
and $\phi$ time-dependent, i.e. the ansatz for the giant graviton configuration. In this case there is a non-vanishing 3-form potential
\begin{equation}
C^{(3)}_{\phi\beta_1\beta_2}=L^3\sin^3{\theta}\sqrt{g_\beta}
\end{equation}
which (when rewritten in terms of Cartesian coordinates) clearly couples as in (\ref{mm}), given that the gravitons carry $P_\phi$
angular momentum, which identifies $\phi$ with the isometric direction in the action (so that $k^\mu=\delta^\mu_\phi$).
On the other hand, taking the $AdS_4\times S^7$ background and the dual giant graviton
ansatz, $\theta=0$, $r={\rm constant}$ and $\phi$ time-dependent, the non-vanishing
3-form potential is
\begin{equation}
C^{(3)}_{t\alpha_1\alpha_2}=-\frac{r^3}{{\tilde L}}\sqrt{g_\alpha}
\end{equation}
which couples through (\ref{di}).
The detailed computations of the potentials associated to these configurations were performed in \cite{JL2}, and perfect agreement was found for large number of gravitons with the
macroscopical calculations in \cite{GST,GMT}.
Consider now the backgrounds in which the gravitons expand into M5-branes.
Taking the giant graviton ansatz in the $AdS_4\times S^7$ background one finds a
non-vanishing 6-form potential
\begin{equation}
\label{6mag}
C^{(6)}_{\phi\beta_1\dots\beta_5}=-L^6\sin^6{\theta}\sqrt{g_\beta}\, .
\end{equation}
Clearly this potential does not couple in the action for the system of waves if we identify $\phi$ with the isometric direction, given that in the pull-back involved in
\begin{equation}
\label{C6}
\int d\tau P[(\mbox{i}_X\mbox{i}_X)^2 \mbox{i}_k C^{(6)}]
\end{equation}
only $\phi$ is time-dependent, and this component is already taken through the interior product with $k^\mu$.
Similarly, the dual giant graviton
ansatz in $AdS_7\times S^4$ yields
\begin{equation}
\label{6di}
C^{(6)}_{t\alpha_1\dots\alpha_5}=-\frac{r^6}{{\tilde L}}\sqrt{g_\alpha}
\end{equation}
which again does not couple in the action, this time because the quadrupolar coupling (\ref{C6})
has $\phi$ component and therefore is not of the form (\ref{6di}).
A puzzle then arises regarding the microscopical description of these giant graviton configurations
in terms of dielectric gravitational waves.
By analogy with other expanded configurations one would expect the gravitons to expand into fuzzy 5-spheres due to quadrupolar electric or magnetic moment couplings to the 6-form potential. Since a 5-sphere has 5 relative dimensions with respect to a point-like object, the 6-form potential has to be contracted as well with the Killing direction in order to be able to couple to a one dimensional
worldvolume.
However, we have seen that if this Killing direction is the direction of propagation the only coupling of this form present in the action for M-theory waves (\ref{Mwaves}) vanishes for the $AdS$ backgrounds that we want to study.
\vspace{1cm}
In order to find a possible solution to this puzzle let us recall the microscopical description
of the giant graviton configurations in the $AdS_5\times S^5$ background \cite{JLR}.
These configurations correspond microscopically
to Type IIB gravitational waves expanding into $S^3$ D3-branes due to their
dielectric or magnetic moment interaction with the 4-form
RR potential of the background. This potential is
\begin{equation}
\label{c4}
C^{(4)}_{\phi\beta_1\beta_2\beta_3}=L^4\sin^4{\theta}\sqrt{g_\beta}\, ,
\end{equation}
for the giant graviton, and
\begin{equation}
\label{c4d}
C^{(4)}_{t\alpha_1\alpha_2\alpha_3}=-\frac{r^4}{L}\sqrt{g_\alpha}
\end{equation}
for the dual giant graviton.
Consider now that the action for Type IIB waves contained the coupling that one would
naturally expect to involve the 4-form potential:
\begin{equation}
\int d\tau P[(\mbox{i}_X\mbox{i}_X)i_k C^{(4)}]\, .
\end{equation}
By the same arguments above one easily checks that this coupling vanishes
both for the giant and dual giant graviton potentials.
The action describing Type IIB waves constructed in \cite{JLR}
contains however a second isometric direction, with Killing vector
$l^\mu$.
This direction is the direction in which one performs the T-duality
transformation that has to be made in order to obtain the action for Type IIB waves
from the (reduction of the) action for M-theory waves.
In the Abelian limit one can interpret the resulting action as a dimensional
reduction over the T-duality direction, as one usually does, but this cannot be done in
the non-Abelian
case, in part due to the presence of non-trivial dielectric couplings. One finds, in particular,
the following coupling in the CS part of the action
\begin{equation}
\label{C4di}
\int d\tau P[(\mbox{i}_X\mbox{i}_X)i_l C^{(4)}]\, ,
\end{equation}
together with a magnetic moment coupling to the 4-form potential in the
BI part of the action,
\begin{equation}
\label{C4mag}
[X^i,X^k] (i_k i_l C^{(4)})_{kj}\, .
\end{equation}
This isometric action for Type IIB waves
is therefore valid to describe waves propagating in
backgrounds which contain a $U(1)$ isometric
direction. This is the case for the
$AdS_5\times S^5$ background, where the $U(1)$ isometry is that corresponding
to the translations along the
$S^1$-fibre in the description of the 3-sphere (contained in
$S^5$ (for the giant graviton) or $AdS_5$ (for the dual giant graviton)) as an
$S^1$-fibre over an $S^2$ base manifold.
In fact rewriting the potentials (\ref{c4}) and (\ref{c4d})
in adapted coordinates to this isometry it is easy to see that the coupling (\ref{C4di})
is non-vanishing for the 4-form potential associated to the dual giant graviton
and the coupling (\ref{C4mag}) for the one associated to the giant graviton. Indeed the
detailed computation of the corresponding non-Abelian potentials shows perfect agreement with
the macroscopical calculations in \cite{GST,HHI} for large number of gravitons.
Let us stress however that the right dielectric couplings that cause the expansion
of the gravitons can only be constructed in spacetimes with a $U(1)$ isometry.
\vspace{1cm}
The discussion above suggests that something similar can be happening
for the gravitons expanding into 5-spheres that we are considering in this
article.
The $S^5$ can similarly be described as a $U(1)$ bundle, in this case
over the two dimensional complex projective space, $CP^2$. Therefore, there is
a $U(1)$ isometry in the background that could allow the
construction of further dielectric couplings.
However, the action
that we know for M-theory waves contains only one isometric direction, that we naturally
identified
with the direction of propagation of the waves, since, as we discussed,
they are minimally coupled to the
momentum operator in this direction.
Consider instead that we identified this direction
with the $U(1)$ fibre of the $U(1)$-decomposition of the 5-sphere.
In this case one can see that the coupling (\ref{C6}) is non-vanishing both for the giant and dual giant graviton potentials. We will see this in detail in the next sections, when we write $S^5$ as an $S^1$ fibre over $CP^2$ in adapted coordinates. Denoting $\chi$ the coordinate adapted to the isometry, (\ref{C6}) becomes
\begin{equation}
\label{C6bis}
\int d\tau X^l X^k X^j X^i \Bigl[C^{(6)}_{\chi ijkl 0}+C^{(6)}_{\chi ijkl \phi}\dot{\phi}\Bigr]\, ,
\end{equation}
and one easily finds that the first term is non-vanishing for the dual giant graviton whereas the second one is non-vanishing for the giant graviton.
We then propose to use the action (\ref{Mwaves}) to describe gravitons propagating in a
spacetime with a compact isometric direction, $\chi$, with, by construction,
non-vanishing momentum-charge along that direction, $P_\chi$, and with a non-zero velocity
along a different transverse direction, $\dot{\phi}$, as implied by the giant
and dual giant gravitons ans\"atze.
Clearly, in order to describe giant graviton configurations, which only carry momentum $P_\phi$,
we will have to set the momentum charge $P_\chi$ to zero at the end of the calculation. We will see that this calculation matches exactly the macroscopical calculation in \cite{GST,GMT} for large number of gravitons, which will provide the strongest check for the validity of our proposal.
Of course, a more direct microscopical description of giant gravitons with momentum
$P_\phi$ living in a spacetime with an isometric direction $\chi$ would be in terms
of an effective action containing both $\chi$ and $\phi$ as isometric directions,
and with momentum charge only with respect to the second one.
As we have seen this type of action exists in the Type IIB theory. However the construction
of a similar type of action for M-theory waves cannot be made based on duality arguments.
One possibility would be to start from the two isometric action for Type IIB waves, T-dualize and
uplift to eleven dimensions. The detailed calculation shows that in the non-Abelian case
both the T-duality direction and the eleventh dimension become isometric directions in
the M-theory action. Therefore, the resulting action is adequate for the study of M-theory
gravitational waves propagating in a spacetime with three $U(1)$ directions (the other
isometric direction is the direction of propagation of the waves). These isometries are
however not present in the M-theory backgrounds that we want to study.
Let us finally remark that an action for M-theory gravitational waves with two isometric
directions would have to be highly non-perturbative. This would
be in the same spirit of \cite{BMN}, where it is argued that the 5-brane cannot appear
as a classical solution to the pp-wave Matrix model because the scaling of its radius
with the coupling constant is more non-perturbative than
the one corresponding to a classical solution.
Coming back to our action, if the direction of propagation is also isometric,
we cannot have a magnetic coupling to
the 6-form potential like the second one in (\ref{C6bis}). Therefore the
only way to find such a magnetic coupling is in the BI action, through something like
\begin{equation}
Q^i_j=\delta^i_j +\dots +[X^i,X^k][X^l,X^m](\mbox{i}_k\mbox{i}_l C^{(6)})_{klmj}\, .
\end{equation}
However, quadratic couplings of this sort are by no means predicted
by T-duality (plus the uplift to M-theory)\footnote{This is not the case for the Type IIB waves, where the corresponding coupling is a dipole coupling
$$Q^i_j=\delta^i_j+\dots-i[X^i,X^k](\mbox{i}_k\mbox{i}_l C^{(4)})_{kj}$$
neatly predicted by T-duality from the action for Type IIA waves (see \cite{JLR} for the details).}.
Clearly this is due to the fact that an action containing this kind of couplings must be highly
non-perturbative. Therefore it could only be derived \`a la Myers from a non-perturbative action
for coincident D-branes. Although such a non-perturbative action is known for a single brane
(it is well-known that worldvolume duality of the BI vector yields the action which is valid
in the strong coupling regime) it is not known for coincident branes.
Therefore, we seem to be stuck with some fundamental problem in D-brane actions.
In the next two sections we show how the action (\ref{Mwaves}) can be used to correctly
describe microscopically the giant graviton in $AdS_4\times S^7$ and the dual giant
graviton in $AdS_7\times S^4$. We will compare in both cases to the corresponding
macroscopical descriptions and see that there is perfect agreement for large number of gravitons.
\sect{The giant graviton in $AdS_4 \times S^7$}
The giant graviton solution of \cite{GST} in this spacetime is in terms of an
M5-brane with the topology of a 5-sphere contained in $S^7$,
carrying angular momentum along the $\phi$-direction and magnetic moment with respect
to the 6-form potential of the background:
\begin{equation}
C^{(6)}_{\phi\beta_1\dots\beta_5}=-L^6\sin^6{\theta}\sqrt{g_\beta}
\end{equation}
where $\beta_i$ are the angles parametrising the unit 5-sphere in $S^7$:
\begin{equation}
d\Omega_5^2=d\beta_1^2+\sin^2{\beta_1}(d\beta_2^2+\sin^2{\beta_2}(d\beta_3^2+
\sin^2{\beta_3}(d\beta_4^2+\sin^2{\beta_4}d\beta_5^2)))
\end{equation}
Taking the ansatz $r=0$, $\theta=$ constant, $\phi=\phi(\tau)$ in (\ref{ads}) we expect to find the gravitons
expanding into a non-conmutative $S^5$ with radius $L\sin{\theta}$. As we have mentioned in the previous section, describing the $S^5$
as a $U(1)$ bundle over the two dimensional complex projective space, $CP^2$, it is natural to identify
the isometry of the action for the coincident gravitons with the $U(1)$ coordinate.
\subsection{$S^5$ as a $U(1)$-bundle over $CP^2$}
It is well-known that the 5-sphere can be described as a $U(1)$ bundle over the two
dimensional complex projective space. Here we will review some details of this
construction and introduce a convenient set of adapted coordinates to the $U(1)$
isometry. We will mainly follow references \cite{GP,pope}. The reader is referred to
those references for more details.
The unit $S^5$ can be represented as a submanifold of
$\ensuremath{\mathbb{C}}^3$ with coordinates $(z_0,z_1,z_2)$ satisfying
$\bar{z}_0 z_0+{\bar z}_1 z_1 +\bar{z}_2 z_2=1$.
This is invariant under $z_i\rightarrow z_i e^{i\alpha}$, and $CP^2$ is the space of
orbits under the action of this circle group. The projection of points in $S^5$
onto these orbits is the $U(1)$-fibration of $S^5$. Setting:
\begin{equation}
\xi_1=z_1/z_0\, , \qquad \xi_2=z_2/z_0\, , \qquad z_0=|z_0|e^{i\chi}\, ,
\end{equation}
and defining:
\begin{equation}
\label{Acon}
A=\frac{i}{2}(1+|\xi_1|^2+|\xi_2|^2)^{-1}[\bar{\xi}_1 d\xi_1+\bar{\xi}_2 d\xi_2
-{\rm c.c.}]\, ,
\end{equation}
the metric on the $S^5$ may be written as
\begin{eqnarray}
\label{5sph}
d\Omega_5^2&=&(d\chi-A)^2+(1+\sum_k |\xi_k|^2)^{-1}\sum_i |d\xi_i|^2- \nonumber\\
&&-(1+\sum_k |\xi_k|^2)^{-2}\sum_{i,j}\xi_i\bar{\xi}_j d\bar{\xi}_id\xi_j \nonumber\\
&=&(d\chi-A)^2+ds^2_{CP^2}\, ,
\end{eqnarray}
since $CP^2$ is the projection orthogonal to the vector $\partial /\partial\chi$.
$ds^2_{CP^2}$ is the Fubini-Study metric for $CP^2$:
\begin{eqnarray}
ds^2_{CP^2}&=&(1+\sum_k |\xi_k|^2)^{-1}\sum_i |d\xi_i|^2
-(1+\sum_k |\xi_k|^2)^{-2}\sum_{i,j}\xi_i\bar{\xi}_j d\bar{\xi}_id\xi_j= \nonumber\\
&=&\frac{\partial^2 K}{\partial\xi^i\partial\bar{\xi}^{\bar{j}}}d\xi^i
d\bar{\xi}^{\bar{j}}\, ;
\end{eqnarray}
with $K=\log{(1+|\xi^1|^2+|\xi^2|^2)}$. Therefore $CP^2$ has a K\"ahler structure with
K\"ahler form $J=i\partial\bar{\partial}K$. The field strength $F=dA=2J$ is a
solution of Maxwell's equations, the so-called ``electromagnetic instanton'' of
\cite{Traut}. It is self-dual and satisfies that its integral $\int F\wedge F$
associated with the
second Chern class is equal to $4\pi^2$. This solution to Maxwell's equations will in
fact play a role in the macroscopical description of giant gravitons
of sections 3.4 and 4.1.
One can obtain a real four dimensional metric on $CP^2$ by defining coordinates
$(\varphi_1,\varphi_2,\psi,\varphi_3)$, $0\leq\varphi_1\leq \pi/2$,
$0\leq\varphi_2\leq\pi$, $0\leq\psi\leq 4\pi$, $0\leq \varphi_3\leq 2\pi$, as
\cite{pope}:
\begin{eqnarray}
\xi_1&=&\tan{\varphi_1}\cos{\frac{\varphi_2}{2}}e^{i(\psi+\varphi_3)/2} \nonumber\\
\xi_2&=&\tan{\varphi_1}\sin{\frac{\varphi_2}{2}}e^{i(\psi-\varphi_3)/2}
\end{eqnarray}
to give
\begin{equation}
\label{phis}
ds^2_{CP^2}=d\varphi_1^2+\frac14 \sin^2{\varphi_1}\Bigl[\cos^2{\varphi_1}
(d\psi+\cos{\varphi_2}d\varphi_3)^2+d\varphi_2^2+
\sin^2{\varphi_2}d\varphi_3^2\Bigr]\, .
\end{equation}
In these coordinates the connection $A$ defined in (\ref{Acon}) is given by
\begin{equation}
\label{Acon2}
A=-\frac12 \sin^2{\varphi_1}(d\psi+\cos{\varphi_2} d\varphi_3)
\end{equation}
and the 6-form potential of the background reads:
\begin{equation}
C^{(6)}_{\phi\chi\varphi_1\varphi_2\psi\varphi_3}=-\frac18 L^6\sin^6{\theta}
\sin^3{\varphi_1}\sin{\varphi_2}\cos{\varphi_1}\, .
\end{equation}
Clearly, the background is isometric in the $\chi$-direction, and this is the direction that we are going to identify with the isometric direction in the action (\ref{Mwaves}), i.e. $k^\mu=\delta^{\mu}_\chi$.
Let us now make the non-commutative ansatz for the 5-sphere.
Inspired by the results of \cite{JLR} for $AdS_5\times S^5$, where it was found that
the non-commutative manifold on which the giant (and dual giant) graviton expands is
defined as an $S^1$ bundle over a non-commutative $S^2$,
we make the ansatz that the non-commutative manifold onto
which the giant graviton in $AdS_4\times S^7$
expands is defined as an $S^1$ bundle over a non-commutative
$CP^2$.
\subsection{The fuzzy $CP^2$}
In this subsection we review some basic properties about the fuzzy $CP^2$. The
fuzzy $CP^2$ has been extensively studied in the literature (see for instance
\cite{TV},\cite{NR}-\cite{KNR}). In this section we will mainly follow the notation in \cite{ABIY}.
$CP^2$ is the coset manifold $SU(3)/U(2)$, and can be defined as the submanifold of
$\ensuremath{\mathbb{R}}^8$ determined by the constraints:
\begin{eqnarray}
\label{condi}
&&\sum_{i=1}^8 x^i x^i=1 \nonumber\\
&&d^{ijk}x^j x^k =\frac{1}{\sqrt{3}}x^i
\end{eqnarray}
where $d^{ijk}$ are the components of the totally symmetric
$SU(3)$-invariant tensor defined
by
\begin{equation}
\lambda^i \lambda^j=\frac23 \delta^{ij}+(d^{ijk}+if^{ijk})\lambda^k\, ,
\end{equation}
where $\lambda^i$, $i=1,\dots,8$ are the Gell-Mann matrices.
In this set of constraints only four are independent (the first one, for instance,
is a consequence of the rest), therefore they define a four dimensional manifold.
A matrix level definition of the fuzzy $CP^2$
can be obtained by impossing the conditions (\ref{condi}) at the level of matrices.
Defining a set of coordinates $X^i$, $i=1,\dots,8$ as
\begin{equation}
\label{defX}
X^i=\frac{1}{\sqrt{C_N}} T^i
\end{equation}
with $T^i$ the generators of $SU(3)$ in an $N$ dimensional irreducible representation and $C_N$ the
quadratic Casimir of $SU(3)$ in this representation, the first
constraint in (\ref{condi}) is trivially satisfied through the quadratic Casimir of the group
\begin{equation}
\sum_{i=1}^8 X^i X^i=\frac{1}{C_N}\sum_{i=1}^8 T^i T^i =\frac{1}{C_N}C_N=
\unity\, ,
\end{equation}
whereas the rest of the constraints are satisfied for any $n$
(see Appendix A in \cite{ABIY})
if the $X^i$'s are taken
in the $(n,0)$ or
$(0,n)$ representations of $SU(3)$, parametrising the irreducible representations of
$SU(3)$ by two integers
$(n,m)$ corresponding to the number of fundamental and anti-fundamental indices.
They become at the level of matrices
\begin{equation}
\label{const2}
d^{ijk}X^jX^k=\frac{\frac{n}{3}+\frac12}{\sqrt{\frac13 n^2+n}}X^i\, .
\end{equation}
Therefore, to describe the fuzzy $CP^2$ the non-commuting coordinates $X^i$ have to be
taken in the $(n,0)$ or $(0,n)$ irreducible representations of $SU(3)$.
This is in agreement with the fact that $G/H$ cosets can be made fuzzy if $H$ is the
isotropy group of the lowest weight state of a given irreducible representation of $G$
\cite{Madore,TV}. Therefore,
different irreducible representations, having associated different isotropy
subgroups, can give rise to different cosets $G/H$. $CP^2$ has $G=SU(3)$ and $H=U(2)$, and
this is precisely the isotropy subgroup of the $SU(3)$ irreducible representations $(n,0)$ and $(0,n)$.
Any other choice of $(n,m)$ has isotropy group $U(1)\times U(1)$, and therefore yields
to a different coset, $SU(3)/(U(1)\times U(1))$.
It will be useful later to know
that the irreducible representations $(n,0)$, $(0,n)$ have dimension N
given by
\begin{equation}
N=\frac12 (n+1)(n+2)
\end{equation}
and quadratic Casimir
\begin{equation}
C_N=\frac13 n^2+n\, .
\end{equation}
$CP^2$ can be embedded in $\ensuremath{\mathbb{R}}^8$ using coherent state techniques \cite{perelo}
(see also Appendix B in \cite{TV}). In our coordinates (\ref{phis}) we have:
\begin{eqnarray}
X^1&=&\frac12 \sin{2\varphi_1}\cos{\frac{\varphi_2}{2}}
\cos{\frac{\psi+\varphi_3}{2}} \nonumber\\
X^2&=&-\frac12 \sin{2\varphi_1}\cos{\frac{\varphi_2}{2}}
\sin{\frac{\psi+\varphi_3}{2}} \nonumber\\
X^3&=&\frac12[\sin^2{\varphi_1}(1+\cos^2{\frac{\varphi_2}{2}})-1]\nonumber\\
X^4&=&\frac12 \sin{2\varphi_1}\sin{\frac{\varphi_2}{2}}
\cos{\frac{\psi-\varphi_3}{2}} \nonumber\\
X^5&=&-\frac12 \sin{2\varphi_1}\sin{\frac{\varphi_2}{2}}
\sin{\frac{\psi-\varphi_3}{2}} \nonumber\\
X^6&=&\frac12 \sin^2{\varphi_1}\sin{\varphi_2}\cos{\varphi_3} \nonumber\\
X^7&=&-\frac12 \sin^2{\varphi_1}\sin{\varphi_2}\sin{\varphi_3} \nonumber\\
X^8&=&\frac{1}{2\sqrt{3}}(3\sin^2{\varphi_1}\sin^2{\frac{\varphi_2}{2}}-1)\, ,
\end{eqnarray}
for which
\begin{equation}
\sum_{i=1}^8 (dX^i)^2=d\varphi_1^2+\frac14 \sin^2{\varphi_1}\Bigl[\cos^2{\varphi_1}
(d\psi+\cos{\varphi_2}d\varphi_3)^2+d\varphi_2^2+
\sin^2{\varphi_2}d\varphi_3^2\Bigr]=ds^2_{CP^2}\, .
\end{equation}
We also have that
\begin{equation}
\sum_{i=1}^8 (X^i)^2=\frac13\, ,
\end{equation}
so that in order to fulfill this constraint
we will have to slightly modify
our definition (\ref{defX}). We then get, in the $(n,0)$ or $(0,n)$ representations
\begin{equation}
X^i=\frac{1}{\sqrt{3}\sqrt{C_N}}T^i=\frac{1}{\sqrt{n^2+3n}}T^i\, , \qquad
i=1,\dots,8\, .
\end{equation}
With this normalisation the commutation relations of the $X^i$ become
\begin{equation}
[X^i,X^j]=\frac{i}{\sqrt{n^2+3n}}f^{ijk}X^k\, ,
\end{equation}
with $f^{ijk}$ the structure constants of $SU(3)$ in the notation
\begin{equation}
[\lambda^i,\lambda^j]=2i f^{ijk} \lambda^k\, .
\end{equation}
\subsection{The microscopical description}
Let us now take the giant graviton ansatz,
$r=0$, $\theta={\rm constant}$, $\phi=\phi(\tau)$ in the
$AdS_4\times S^7$ background. We find, in Cartesian coordinates
\begin{equation}
ds^2=-dt^2+L^2\cos^2{\theta}d\phi^2+L^2\sin^2{\theta}\Bigl[(d\chi-A)^2
+(dX^1)^2+\dots+(dX^8)^2\Bigr]
\end{equation}
and
\begin{equation}
C^{(6)}_{\phi\chi ijkl}=2L^6\sin^6{\theta} f^{[ijm}f^{kl]n}X^mX^n\, .
\end{equation}
Taking now $k^\mu=\delta^\mu_\chi$ in the action (\ref{Mwaves}) we have
that
\begin{eqnarray}
&&k=L\sin{\theta}\, , \qquad E_{00}=-1+L^2\cos^2{\theta}{\dot{\phi}}^2\, , \nonumber\\
&&Q^i_j=\delta^i_j-\frac{L^3\sin^3{\theta}}{\sqrt{n^2+3n}}f^{ijk} X^k\, ,
\qquad i,j=1,\dots,8\, ,
\end{eqnarray}
and substituting in the action we find
\begin{equation}
S^{\rm BI}=-T_0\int d\tau {\rm STr}\Bigl\{\frac{1}{L\sin{\theta}}\sqrt{1-L^2\cos^2{\theta}
{\dot{\phi}}^2}
\sqrt{\unity+\frac32 \frac{L^6\sin^6{\theta}}{n^2+3n}X^2+\frac{9}{16}
\frac{L^{12}\sin^{12}{\theta}}{(n^2+3n)^2}X^2X^2+\dots}\Bigr\}\, .
\end{equation}
Here we have dropped those contributions to ${\rm det}Q$ that will vanish when taking
the symmetrised trace, and ignored higher powers of $n^2+3n$ which will vanish
in the large $N$ ($\Leftrightarrow n\rightarrow \infty$) limit.
These terms on the other hand cannot be nicely arranged into higher powers of the
quadratic Casimir without explicit use of the constraints (\ref{const2}).
Up to order $n^{-4}$ we have that
\begin{equation}
\sqrt{\unity+\frac32 \frac{L^6\sin^6{\theta}}{n^2+3n}X^2+\frac{9}{16}
\frac{L^{12}\sin^{12}{\theta}}{(n^2+3n)^2}X^2X^2+\dots}=\unity+\frac34
\frac{L^6\sin^6{\theta}}{n^2+3n}X^2\, .
\end{equation}
We also have for the CS part of the action:
\begin{equation}
S^{\rm CS}=\frac{T_0}{2}\int d\tau {\rm STr}\Bigl\{P[(\mbox{i}_X\mbox{i}_X)^2
i_k C^{(6)}]\Bigr\}=\int d\tau \frac{NT_0}{4}\frac{L^6\sin^6{\theta}}{n^2+3n}\dot{\phi}
\end{equation}
It is then easy to compute the symmetrised trace to finally
arrive, in Hamiltonian
formalism, to
\begin{equation}
\label{Hgiant}
H=\frac{P_\phi}{L}\sqrt{1+\tan^2{\theta}\Bigl(1-\frac{NT_0}{4(n^2+3n)P_\phi}
L^6\sin^4{\theta}\Bigr)^2+\frac{N^2T_0^2}{P_\phi^2\sin^2{\theta}}\Bigl(1+\frac12
\frac{L^6\sin^6{\theta}}{n^2+3n}\Bigr)}
\end{equation}
where $P_\phi$ is a conserved quantity given that $\phi$ is cyclic in the Lagrangian.
We should stress that this expression is an approximated expression in which higher
powers of $n^2+3n$ vanishing in the large $n$ limit have been omitted.
In order to describe giant graviton configurations we must set to zero the momentum
along the $\chi$-direction, $P_\chi$. Recall however that, by construction, the $N$
gravitons carry momentum $P_\chi=NT_0$.
The difference between $P_\chi$ being zero or not is merely
a coordinate transformation, a boost in $\chi$. However, how to perform coordinate
transformations in non-Abelian actions is an open problem \cite{DS, Hassan, DSW, Dom}.
In order to clarify how the limit $P_\chi\rightarrow 0$ must be taken we study
in the next subsection the macroscopical description of this microscopical
configuration, in terms of a spherical M5-brane carrying both $P_\phi$ and
$P_\chi$ charges. The agreement between the two descriptions in the large $N$ limit
will make
clear that in the limit $P_\chi\rightarrow 0$
the last term in the Hamiltonian should vanish. On the other hand, the term
\begin{equation}
\label{nv}
\frac{NT_0}{4(n^2+3n)P_\phi}\, ,
\end{equation}
whose presence is in fact crucial in
order to find the giant graviton configuration, remains finite, since
$N=(n+1)(n+2)/2$ for $(n,0)$ and $(0,n)$ irreps, so that
numerator and denominator in (\ref{nv}) scale with the same power of $n$.
In fact, in the $N\rightarrow \infty$ ($\Leftrightarrow n\rightarrow\infty$) limit both terms are compensated, and the finite result
$T_0/8P_\phi$ is reached, in perfect agreement with the macroscopical calculation.
This compensation does not however take place in the last term in the Hamiltonian.
Therefore, in order to describe giant graviton configurations we minimize
\begin{equation}
\label{Hmic}
H_{\rm mic}=\frac{P_\phi}{L}\sqrt{1+\tan^2{\theta}\Bigl(1-\frac{NT_0}{4(n^2+3n)P_\phi}
L^6\sin^4{\theta}\Bigr)^2}
\end{equation}
with respect to $\theta$. We then find two solutions with energy $P_\phi/L$: $\sin{\theta}=0$,
which corresponds to the point-like
graviton, and
\begin{equation}
\sin{\theta}=\Bigl(\frac{4(n^2+3n)P_\phi}{NT_0L^6}\Bigr)^{1/4}
\end{equation}
which corresponds to the giant graviton, with radius
\begin{equation}
R=\Bigl(\frac{4(n^2+3n)P_\phi}{NT_0L^2}\Bigr)^{1/4}\, .
\end{equation}
Clearly, for the giant graviton
\begin{equation}
P_\phi\leq\frac{NT_0L^6}{4(n^2+3n)}
\end{equation}
which provides the microscopical bound to the angular momentum predicted by the
stringy exclusion principle. When the number of gravitons is large we find that
\begin{equation}
P_\phi\leq\frac{T_0L^6}{8}={\tilde N}
\end{equation}
with ${\tilde N}$ the units of 6-form flux on the 5-sphere in the macroscopical
description of \cite{GST,GMT}, given by
${\tilde N}=A_5 T_5 L^6$ with $A_5$ the area of the unit 5-sphere. We therefore
find perfect agreement with the bound found in \cite{GST}. The same holds true for the
radius of the configuration, which for large number of gravitons is given by
\begin{equation}
R=\Bigl(\frac{8P_\phi}{T_0L^2}\Bigr)^{1/4}=L
\Bigl(\frac{P_\phi}{\tilde N}\Bigr)^{1/4}
\end{equation}
exactly as in \cite{GST,GMT}.
In this section we have achieved the right microscopical description of the giant graviton in $AdS_4\times S^7$ in
terms of gravitons expanding into a ``fuzzy 5-sphere'', defined as a $U(1)$ bundle over the
fuzzy $CP^2$. We have seen that the coordinate along the fibre must be isometric in the action,
and this has forced our choice of $k^\mu$ pointing on that direction, which has in turn
introduced, by construction, a non-vanishing momentum $P_\chi$ on the configuration. The
macroscopical description of such a configuration should then be in terms of a spherical
M5-brane carrying both
$P_\phi$ and $P_\chi$ charges. We will perform the detailed macroscopical description in these
terms in the next subsection. The agreement between the two descriptions will provide
further justification to the limit taken to arrive at expression (\ref{Hmic}) as the
right microscopical Hamiltonian describing gravitons with angular momentum $P_\phi$.
\subsection{The macroscopical calculation}
The simplest way to describe an M5-brane living in a spacetime with a
$U(1)$ isometry and carrying momentum along that direction is by uplifting
a D4-brane to M-theory keeping the eleventh direction compact. The resulting
brane is therefore a longitudinal 5-brane. Doing this we
obtain the action adequate to describe an M5-brane whose worldvolume
contains an isometric direction. Clearly this applies to the M5-brane with the
topology of an $S^5$. In this case the isometric direction is the coordinate along the
fibre in the decomposition of the $S^5$ as a $U(1)$ fibre over $CP^2$.
The worldvolume of such a brane is therefore effectively four-dimensional,
and locally that of a $CP^2$.
For the non-vanishing eleven dimensional fields involved in our
$AdS$ backgrounds we find the following action for a wrapped M5-brane:
\begin{equation}
\label{M5w}
S=-T_4 \int d^5\xi \Bigl\{ k \sqrt{{\rm det}({\cal G}+k^{-1}F)}-P[i_k C^{(6)}]
-\frac12 P[k^{-2} k^{(1)}]\wedge F\wedge F\Bigr\}\, .
\end{equation}
Here $F$ is the field strength associated to M2-branes wrapped on the isometric direction ending on the
M5-brane, since the uplifting of the BI field strength of the D4-brane, $F+B^{(2)}$,
to M-theory gives
$F+i_k C^{(3)}$ \footnote{In the action (\ref{M5w}) we have set $C^{(3)}$ to zero,
which is valid for our particular backgrounds.}. ${\cal G}$ is the reduced metric defined in
(\ref{mred}) and
we have denoted $T_4$ the tension of the brane to explicitly take into account that
its spatial worldvolume is 4-dimensional.
As we discussed in Section 2 $k^{-2}k^{(1)}$ is identified with the momentum operator
along the isometric direction. Therefore we can switch on momentum charge
on the M5-brane by choosing a non-vanishing field strength such that
\begin{equation}
\label{FF}
\int_{CP^2}F\wedge F=8\pi^2 N\, ,
\end{equation}
since then
\begin{equation}
\frac{T_4}{2}\int d\tau d^4\sigma P[k^{-2}k^{(1)}]\wedge F\wedge F=
NT_0\int d\tau P[k^{-2}k^{(1)}]\, .
\end{equation}
With $F$ satisfying (\ref{FF}) we are therefore dissolving $N$ gravitons, propagating
along the isometric direction, in the worldvolume of the 5-brane.
The field satisfying this condition is in fact the electromagnetic instanton that we
discussed in section 3.1, which now must have instanton number
equal to twice the number of gravitons. This gives
\begin{equation}
F=\sqrt{\frac{N}{2}}\Bigl(-\sin{2\varphi_1}d\varphi_1\wedge d\psi-
\sin{2\varphi_1}\cos{\varphi_2}d\varphi_1\wedge d\varphi_3+
\sin^2{\varphi_1}\sin{\varphi_2}d\varphi_2\wedge d\varphi_3\Bigr)\, .
\end{equation}
Identifying the isometric direction in the action (\ref{M5w}) with $\chi$ in (\ref{5sph})
and integrating over the spatial worldvolume of the M5-brane we arrive at
\begin{equation}
S=-4\pi^2T_4\int d\tau \Bigl\{ L\sin{\theta}\sqrt{1-L^2\cos^2{\theta}{\dot{\phi}}^2}
\Bigl(\frac{L^4\sin^4{\theta}}{8}+\frac{N}{L^2\sin^2{\theta}}\Bigr)-\frac18 L^6\sin^6{\theta}
\dot{\phi}\Bigr\}
\end{equation}
and, in Hamiltonian formalism, to
\begin{equation}
\label{Hgiantm}
H=\frac{P_\phi}{L}\sqrt{1+\tan^2{\theta}\Bigl(1-\frac{\pi^2T_4L^6\sin^4{\theta}}
{2P_\phi}\Bigr)^2+\frac{16\pi^4T_4^2N^2}{P_\phi^2\sin^2{\theta}}
\Bigl(1+\frac{L^6\sin^6{\theta}}
{4N}\Bigr)}\, .
\end{equation}
Comparing with the Hamiltonian constructed in \cite{GST,GMT}, which describes a
spherical 5-brane with momentum $P_\phi$, we see that a non-vanishing momentum
along the $\chi$-direction translates into an additional piece depending on $N$
inside the squared root. Comparing
this expression with the microscopical Hamiltonian (\ref{Hgiant}) we find that they exactly agree in the large $N$ limit,
when $N\sim n^2/2$. In the macroscopical Hamiltonian (\ref{Hgiantm}) it is clearer however that the
limit of zero momentum along the fibre direction is reached when there are zero gravitons
propagating in the $\chi$ direction
dissolved in the worldvolume of the 5-brane, which neatly sets to zero the last term in the
Hamiltonian. This further justifies the elimination of the corresponding term in
the microscopical Hamiltonian (\ref{Hgiant}) in section 3.3.
\section{The dual giant graviton in $AdS_7\times S^4$}
\subsection{The microscopical description}
In this section we briefly describe the microscopical description of the
dual giant graviton in $AdS_7\times S^4$. One expects that microscopically the
gravitons expand into a 5-sphere with radius $r$ due to the coupling to the
6-form potential $C^{(6)}_{t\alpha_1\dots\alpha_5}$. As before, we describe the
``fuzzy 5-sphere'' as a $U(1)$ bundle over the fuzzy $CP^2$, and we embed the $CP^2$ in
$\ensuremath{\mathbb{R}}^8$. We then have for the dual giant graviton ansatz:
\begin{eqnarray}
&&ds^2=-(1+\frac{r^2}{{\tilde L}^2})dt^2+L^2d\phi^2+r^2\Bigl[(d\chi-A)^2+(dX^1)^2+\dots
(dX^8)^2\Bigr]\nonumber\\
&&C^{(6)}_{0\chi ijkl}=2\frac{r^6}{{\tilde L}}f^{[ijm}f^{kl]n}X^mX^n
\end{eqnarray}
Substituting in the action (\ref{Mwaves}) we have that
\begin{eqnarray}
&&k=r\, ,\qquad E_{00}=-(1+\frac{r^2}{{\tilde L}^2})+L^2{\dot{\phi}}^2\, ,\nonumber\\
&&Q^i_j=\delta^i_j-\frac{r^3}{\sqrt{n^2+3n}}f^{ijk}X^k\, , \qquad i,j=1,\dots,8
\end{eqnarray}
and
\begin{eqnarray}
S&=&-T_0\int d\tau {\rm STr}\Bigl\{\frac{1}{r}\sqrt{1+\frac{r^2}{{\tilde L}^2}-L^2{\dot{\phi}}^2}
\sqrt{\unity+\frac32 \frac{r^6}{n^2+3n}X^2+\frac{9}{16}\frac{r^{12}}{(n^2+3n)^2}X^2X^2
+\dots}\nonumber\\
&&-\frac{r^6}{{\tilde L}}\frac{N}{4(n^2+3n)}\Bigr\}\, .
\end{eqnarray}
Up to order $n^{-4}$ we arrive at the Hamiltonian
\begin{equation}
\label{Hgiantd}
H=\sqrt{\Bigl(1+\frac{r^2}{{\tilde L}^2}\Bigr)\Bigl(\frac{{P_\phi}^2}{L^2}
+\frac{N^2T_0^2r^{10}}{16(n^2+3n)^2}+\frac{N^2T_0^2}{r^2}
(1+\frac12 \frac{r^6}{n^2+3n})\Bigr)}-\frac{NT_0}{4(n^2+3n)}\frac{r^6}{{\tilde L}}
\end{equation}
Now we have to take the limit $P_\chi=0$ which amounts to setting to zero the last
term in the squared-root, as we discussed in detail when we studied the giant
graviton in $AdS_4\times S^7$. We are then left with
\begin{equation}
\label{Hmicd}
H_{\rm mic}=\sqrt{\Bigl(1+\frac{r^2}{{\tilde L}^2}\Bigr)\Bigl(\frac{{P_\phi}^2}{L^2}
+\frac{N^2T_0^2r^{10}}{16(n^2+3n)^2}\Bigr)}
-\frac{NT_0}{4(n^2+3n)}\frac{r^6}{{\tilde L}}\, ,
\end{equation}
which again is an approximated expression in which higher order powers of
$n^2+3n$ vanishing in the large $n$ limit have been omitted.
Minimizing with respect to $r$ we find two solutions with energy $P_\phi/L$:
$r=0$, which corresponds to the point-like graviton, and
\begin{equation}
r=\Bigl(\frac{4(n^2+3n)P_\phi}{NT_0L{\tilde L}}\Bigr)^{1/4}
\end{equation}
which corresponds to the dual giant graviton. When $N\rightarrow\infty$
\begin{equation}
r\rightarrow \Bigl(\frac{8P_\phi}{T_0L{\tilde L}}\Bigr)^{1/4}=
{\tilde L} \Bigl(\frac{P_\phi}{{\tilde N}}\Bigr)^{1/4}
\end{equation}
in agreement with the result in \cite{GMT}.
\subsection{The macroscopical calculation}
Using the action (\ref{M5w}) we can easily describe the previous configurations
in terms of an M5-brane wrapped on the $\chi$-direction and carrying $P_\phi$ and
$P_\chi$ momentum charges. $P_\chi$ is switched on by dissolving $N$ gravitons
with momentum on this direction in the worldvolume of the M5-brane.
We find
\begin{equation}
S=-4\pi^2 T_4 \int d\tau \Bigl[r\Bigl(\frac{r^4}{8}+\frac{N}{r^2}\Bigr)
\sqrt{1+\frac{r^2}{{\tilde L}^2}
-L^2{\dot{\phi}}^2}-\frac18 \frac{r^6}{{\tilde L}}\Bigr]\, ,
\end{equation}
and, in Hamiltonian formalism
\begin{equation}
\label{Hmacd}
H=\sqrt{1+\frac{r^2}{{\tilde L}^2}}\sqrt{\frac{P_\phi^2}{L^2}+16\pi^4 T_4^2 r^2
\Bigl(\frac{r^4}{8}+\frac{N}{r^2}\Bigr)^2}-\frac{\pi^2 T_4}{2}
\frac{r^6}{{\tilde L}}\, .
\end{equation}
Clearly the condition $P_\chi=0$ is met for $N=0$ gravitons dissolved in the
worldvolume, which amounts to setting to zero the $N$-term in the squared-root.
Comparing to (\ref{Hgiantd}) this further justifies the limit taken in that expression,
yielding to the microscopical potential (\ref{Hmicd}). In fact, when
$N\rightarrow\infty$ ($\Leftrightarrow n\rightarrow \infty$) we find perfect agreement
between (\ref{Hmacd}) and the microscopical potential (\ref{Hmicd}).
\section{Conclusions}
We have shown that the giant graviton configurations in $AdS_m\times S^n$
backgrounds that
involve 5-spheres are described microscopically in terms of gravitational waves
expanding into ``fuzzy 5-spheres'' which are defined as $S^1$ bundles over fuzzy
$CP^2$. The explicit construction can be done due to the fact that the action used
to describe the system of coincident waves contains a special $U(1)$ isometric
direction that can be identified with the $U(1)$ fibre.
In this description
the gravitons expand into a longitudinal M5-brane which has four manifest dimensions
and one wrapped on the $U(1)$ direction.
This brane carries quadrupolar magnetic moment with respect to the 6-form potential of
the background in the $AdS_4\times S^7$ case, or quadrupolar electric moment in the
$AdS_7\times S^4$ case. In both cases it has a non-vanishing angular momentum along
the spherical part of the geometry, $P_\phi$.
The details of the construction show that there is as
well a non-vanishing momentum along the $U(1)$ direction, which has to be set to zero
to find the right point-like graviton and giant graviton configurations in the background.
We have seen that
in that case not only the radii of the giant gravitons but also the Hamiltonian
that they minimize
agree with the macroscopical results in \cite{GST,GMT} for large number of gravitons.
Gravitational waves
propagating both along the spherical part of the geometry (the $\phi$ direction) and
the compact $U(1)$ direction can be described macroscopically in terms of
a longitudinal M5-brane with velocity $\dot{\phi}$
wrapped on the isometric direction. The action associated to this brane
can be easily constructed by just
uplifting the action of the D4-brane to M-theory, while maintaining
the eleventh direction compact.
The
$\int C^{(1)}F\wedge F$ term in the CS part of the D4-brane action is then
uplifted to $\int k^{-2}k^{(1)} F\wedge F$, with $F$ now associated to wrapped M2-branes
ending on the M5-brane. Therefore a momentum charge along the compact direction is
simply switched on by
taking $F$ with non-vanishing instanton number. The comparison between this
description and our microscopical description shows exact agreement for large
number of gravitons. Moreover, this comparison can be used
to clarify the right way to set to zero the momentum
along the compact direction to finally obtain the correct giant graviton configurations.
Our action for M-theory waves, therefore, provides an explicit Matrix action which is
solved by some sort of non-commutative 5-sphere. Moreover, although we have not checked the
supersymmetry properties of our configurations, the agreement with the macroscopical
description of \cite{GST,GMT} suggests that they should occur as BPS solutions
preserving the same half of the supersymmetries as the point-like graviton \cite{GMT}.
To our knowledge this would be the first example of a physical matrix model,
coming up as the action
for coincident M-theory gravitational waves, admitting some fuzzy 5-sphere as a
supersymmetry preserving solution.
Let us stress that the ``fuzzy 5-sphere'' that we have constructed
is defined as an $S^1$ bundle
over the fuzzy $CP^2$, and is therefore different from previous fuzzy 5-spheres
discussed in the literature \cite{Ram1,Ram2,S-J}.
In particular, our solution does not
show $SO(6)$ covariance, this invariance being broken down to $SU(3)\times U(1)$, whereas
this is the case for the fuzzy 5-sphere in \cite{Ram1,Ram2,S-J}.
The $SO(6)$ invariance might still be present in a non-manifest way, after all
the $SO(6)$ covariance of the classical 5-sphere is also not explicit when
it is described as an $S^1$-bundle over $CP^2$.
Another difference is that our solution approaches neatly the classical $S^5$ in the large
$N$ limit, where all the non-commutativity disappears. This is not the case for the
fuzzy 5-sphere in \cite{Ram1,Ram2,S-J}. Indeed, the right dependence of the radius of
the 5-sphere giant graviton with $P_\phi$, $R\sim P_\phi^{1/4}$, is only achieved
within the $S^1$ bundle over $CP^2$ description, the corresponding dependence of the
fuzzy 5-sphere of \cite{Ram1,Ram2,S-J} being given by $R\sim P_\phi^{1/5}$.
Another difference is that our ``fuzzy $S^5$'' inherits its symplectic
structure from the K\"ahler form of the fuzzy $CP^2$, whereas in the construction in
\cite{Ram1,Ram2} the bundle structure corresponds to a $CP^3$ base and a
$CP^2$ fibre. Therefore, there are clear differences between the two constructions.
Non-supersymmetric longitudinal M5-branes with $CP^2\times S^1$ topology have
been obtained as
explicit solutions of Matrix theory in \cite{NR}.
Our longitudinal M5-branes, although similar in the explicit construction,
have $S^5$ topology, once the necessary twist in the fibre is taken into
account. This twist should provide the global extension of the local residual
supersymmetry found in \cite{NR}, in terms of spinors charged under the gauge
potential whose field strength is the K\"ahler form (see \cite{DLP,pope}).
Longitudinal 5-branes with other topologies have also been shown to arise as
solutions to Matrix theory in \cite{CLT,Ho,K1,K2} (see also the fuzzy funnel
solution in \cite{CMT}).
In general, to find these solutions it is
necessary to include additional Chern-Simons terms or mass terms. Very recently \cite{N}
there has been some speculation on how the fuzzy 5-sphere of \cite{Ram1,Ram2}
might appear as a solution to the pp-wave Matrix model of \cite{BMN}. It would be interesting
to elucidate the relation between the new Chern-Simons coupling conjectured in this reference
and the dielectric couplings constructed in this paper once the Penrose limit is
taken. This would help clarifying if indeed the resulting Matrix action would allow
for transverse 5-brane solutions \cite{MSR}.
\subsubsection*{Acknowledgements}
We would like to thank J.M. Figueroa-O'Farrill and S. Ramgoolam for useful discussions.
The work of B.J. is done as part of the program ``Ramon y Cajal'' of the
M.E.C. (Spain). He was also partially supported by the M.E.C. under contract FIS
2004-06823 and by the Junta de Andaluc\'{\i}a group FQM 101.
The work of Y.L. and D.R-G.
has been
partially supported by CICYT grant BFM2003-00313 (Spain).
D.R-G. was supported in part by a F.P.U.
Fellowship from M.E.C. He would like to thank the String Theory group
at Queen Mary College (London) for its hospitality while part of this work was done.
|
{
"timestamp": "2005-01-26T12:54:40",
"yymm": "0411",
"arxiv_id": "hep-th/0411181",
"language": "en",
"url": "https://arxiv.org/abs/hep-th/0411181"
}
|
\section*{ Acknowledgments}
We thank C. Blume for his help with the NA49 data.
We acknowledge the support of the German Bundesministerium f\"ur Bildung und
Forschung (BMBF), the Polish State Committee for Scientific
Research (KBN) grant 2P03 (06925), the
National Research Foundation (NRF, Pretoria) and the URC of the
University of Cape Town.
|
{
"timestamp": "2005-02-24T13:18:22",
"yymm": "0411",
"arxiv_id": "hep-ph/0411187",
"language": "en",
"url": "https://arxiv.org/abs/hep-ph/0411187"
}
|
\section{introduction}
Recently, studies making use of the operator product expansion
(OPE) have provided evidence for the importance of the condensate
$<A_\mu^aA_a^\mu>$ [1-3]. (There is a suggestion that such a
condensate may be related to the presence of instantons in the
vacuum [4].) The importance of that condensate raises the question
of gauge invariance and there are now a large number of papers
that address that and related issues [5-19]. We will not attempt
to review that large body of literature, but will consider how the
presence of an $<A_\mu^aA_a^\mu>$ condensate affects the form of
the gluon propagator. We may mention the work of Kondo [7] who was
responsible for introducing a BRST-invariant condensate of
dimension two, \begin{eqnarray} \mathcal{Q}=\frac{1}{\Omega}<\int
d^4x\,\mbox{Tr}\left(\frac{1}{2}A_\mu(x)A_\mu(x)-\alpha ic(x)\cdot
\bar{c}(x)\right)>, \end{eqnarray} where $c(x)$ and $\bar{c}(x)$ are
Faddeev-Popov ghosts, $\alpha$ is the gauge-fixing parameter and
$\Omega$ is the integration volume. Kondo points out that $\Omega$
reduces to $A_{min}^2$ in the Landau gauge, $\alpha=0$. The
minimum value of the integrated squared potential is $A^2_{min}$,
which has a definite physical meaning [7].
In a recent work we considered the Minkowski-space gluon
propagator in the presence of an $A^2$ condensate [20]. We found
that the propagator has no on-mass-shell poles, so that the gluon
was a nonpropagating mode in the presence of the vacuum condensate
[21]. The form we obtained for the propagator was \begin{eqnarray}
D^{\mu\nu}(k)=\left(g^{\mu\nu}-\frac{k^\mu k^\nu}{k^2}\right)D(k)
\end{eqnarray} with \begin{eqnarray}
D(k)=\frac{Z_1}{k^2-m^2+\frac{4}{3}\frac{k^2m^2}{k^2-m^2}}.\end{eqnarray}
Here $Z_1$ is a normalization parameter which we put equal to 3.82
so that we may obtain a continuous representation as we pass from
Minkowski to Euclidean space. In Fig. 1 we show $D(k)$ with
$m^2=0.25$ GeV$^2$. (We remark that $D(k)=0$ when $k^2=m^2$,
$D(k)=-Z_1/m^2$ at $k^2=0$, and $D(k)\rightarrow Z_1/k^2$ for
large $k^2$.) If we choce $Z_1=15.28m^2=3.82$ our result for the
propagator will be continuous at $k^2=0$ when we consider both the
Euclidean-space and Minkowski-space propagators.
The organization of our work is as follows. In Section II we
discuss the Euclidean-space gluon propagator as obtained in a
lattice simulation of QCD in the Landau gauge and in the absence
of quark degrees of freedom [22]. (We also record in the Appendix
a number of semi-phenomenological analytic forms which are meant
to represent the Euclidean-space propagator.) In Section III we
summarize the results of our analysis and provide some additional
discussion.
\section{QCD lattice calculations and phenomenological forms for the Euclidean-space gluon propagator}
Results for the gluon propagator obtained in a lattice simulation
of QCD are given in Ref. [22]. In that work the authors also
record several phenomenological forms. We reproduce these forms in
the Appendix for ease of reference. Of these various forms we will
make use of model A of Ref. [22] which has the form \begin{eqnarray}
D^L(k^2)=Z\left[\frac{AM^{2\alpha}}{(k^2+M^2)^{1+\alpha}}+\frac{1}{k^2+M^2}L(k^2,M)\right],\end{eqnarray}
with \begin{eqnarray} L(k^2,M)\equiv
\left[\frac{1}{2}\ln(k^2+M^2)(k^{-2}+M^{-2})\right]^{-d_D},\end{eqnarray} and
$d_D=13/22$. The parameters used in Ref. [22] to provide a very
good fit to the QCD lattice data are \begin{eqnarray} Z=2.01^{+4}_{-5},\end{eqnarray} \begin{eqnarray}
A=9.84^{+10}_{-86},\end{eqnarray} \begin{eqnarray} M=0.54^{+5}_{-5},\end{eqnarray} and \begin{eqnarray}
\alpha=2.17^{+4}_{-19}.\end{eqnarray} Note that $M$ in GeV units is $1.018$
GeV. Rather than work with the lattice data we will use Eqs.
(2.1)-(2.6) when we compare our results with the lattice data. In
Fig. 2 we show $k^2D^L(k)$ of Eq. (2.1) and in Fig. 3 we show
$D^L(k)$. These functions are represented by the solid lines in
Figs. 2 and 3. Note that Eq. (1.3) may be written in Euclidean
space as \begin{eqnarray}
D_E(k)=-\frac{Z_1}{k^2_E+m^2-\frac{4}{3}\frac{k^2_Em^2}{k^2_E+m^2}}.\end{eqnarray}
This form is useful for $k_E^2<1$ GeV$^2$ and we therefore
consider various phenomenological forms which may be used to
extend Eq. (2.7) so that we may attempt to fit the lattice result
over a broader momentum range. To that end, we make use of Ref.
[23]. The authors of that work define the Landau gauge gluon
propagator as \begin{eqnarray}
<A^a_\mu(k)A^a_\nu(k')>=V\delta(k+k')\delta^{ab}\left(\delta_{\mu\nu}-\frac{k_\mu
k_\nu}{k^2}\right)\frac{Z(k^2)}{k^2}, \end{eqnarray} with \begin{eqnarray}
Z(k^2)=\omega\left(\frac{k^2}{\Lambda^2_{QCD}+k^2}\right)^{2\kappa}(\alpha(k^2))^{-\gamma},\end{eqnarray}
and $\gamma=-13/22$. (We do not ascribe any particular
significance to Eq. (2.9). We use Eq. (2.9) as a phenomenological
form which could be replaced by a form which provides a better fit
to the data within the context of our model at some future time.
We believe Eq. (2.9) is useful, since it is a simple matter to
remove the first term of that equation and introduce a propagator
that has the small $k^2$ behavior of our model.)
The authors of Ref. [23] introduce two choices for $\alpha(k^2)$
of Eq. (2.9). We use their form for $\alpha_2(k^2)$: \begin{eqnarray}
\alpha_2(k^2)=\frac{\alpha(0)}{\ln\left[e+a_1\left(\frac{k^2}{\Lambda^2_{QCD}}\right)^{a_2}\right]}.\end{eqnarray}
In their analysis they put $\kappa=0.5314$, $\Lambda_{QCD}=354$
MeV, $\alpha(0)=2.74$, $a_1=0.0065$ and $a_2=2.40$. (Here, we have
not recorded the uncertainties in these values which are given in
Table 2 of Ref. [23].) As we proceed, we will change these values
somewhat. As a first step we remove the first factor in Eq. (2.9)
and write \begin{eqnarray} Z(k^2)=Z_2(\alpha_2(k^2))^{-\gamma}.\end{eqnarray} We now use
$a_1=0.0080$ and $a_2=2.10$ rather than the values given above. In
Fig. 4 we show $(\alpha_2(k))^{13/22}$ as a function of $k$, using
our modified values of $a_1$ and $a_2$.
We now define \begin{eqnarray}
D_E(k_E)=-\frac{Z_2(\alpha(k^2))^{-\gamma}}{k_E^2+m^2-\frac{4}{3}\frac{k_E^2m^2}{k_E^2+m^2}}.\end{eqnarray}
The function $-k^2D_E(k_E)$ is shown in Fig. 2 as a dotted line.
In this calculation we have put $Z_2=2.11$. We find a good
representation of the lattice result for $k_E<2$ GeV,
In Fig. 3 we compare $D_E(k_E)$ with the result of the lattice
calculation which is represented by the solid line. In Fig. 5 we
combine our results in Minkowski and Euclidean space and show the
values of $k^2D(k^2)$ for both positive and negative $k^2$ values.
For positive $k^2$ we use $D(k)$ of Eq. (1.3) and for negative
values of $k^2$ we use $D_E(k_E^2)$ of Eq. (2.12). Equality of
these functions at $k^2=0$ implies $Z_1=Z_2(\alpha(0))^{13/22}$,
or $Z_1=1.81Z_2$. (In our work we have used $Z_1=3.82$ and
$Z_2=2.11$. See Eqs. (1.3) and (2.12).) In Fig. 6 we show $D(k^2)$
rather than $k^2D(k^2)$, which was shown in Fig. 5.
\begin{figure}
\includegraphics[bb=40 25 200 200, angle=0, scale=1]{fig1.eps}%
\caption{The function $D(k^2)$ of Eq. (1.3) is shown in Minkowski
space. The value for large $k^2$ is given by $Z_1/k^2$ with
$Z_1=3.87$. Here $m=0.50$ GeV.}
\end{figure}
\begin{figure}
\includegraphics[bb=40 25 240 240, angle=0, scale=1]{fig2.eps}%
\caption{The function $-k_E^2D_E(k)$ is shown. The solid line
represents the QCD lattice data, while the dotted line represents
$-k^2_ED_E(k)$ in the case that $D_E(k)$ is given in Eq. (2.12).}
\end{figure}
\begin{figure}
\includegraphics[bb=40 25 240 240, angle=0, scale=1]{fig3.eps}%
\caption{The function $-D_E(k)$ is shown. The solid line
represents the QCD lattice data, while the dotted line represents
$-D_E(k)$ of Eq. (2.12). [See Fig. 2.]}
\end{figure}
\begin{figure}
\includegraphics[bb=40 25 240 240, angle=0, scale=1]{fig4.eps}%
\caption{The function $(\alpha_2(k))^{13/22}$ is shown. [See Eq.
(2.10).] Note that $(\alpha_2(0))^{13/22}=1.81$.}
\end{figure}
\begin{figure}
\includegraphics[bb=40 25 240 240, angle=0, scale=1]{fig5.eps}%
\caption{For $k^2>0$ the solid line represents $k^2D(k^2)$ with
$D(k^2)$ given by Eq. (1.3). Here, $Z_1=3.82$. For $k^2<0$ we show
$k^2D_E(k^2)$, where $D_E(k^2_E)$ is given by Eq. (2.12) with
$Z_2=2.11$.}
\end{figure}
\begin{figure}
\includegraphics[bb=40 25 240 240, angle=0, scale=1]{fig6.eps}%
\caption{Same as Fig. 5 except that $D(k^2)$ is shown.}
\end{figure}
\section{Discussion and Conclusions}
In this work we have provided a representation of the gluon
propagator in both Euclidean and Minkowski space. The
Minkowski-space propagator has only complex poles and that implies
that the gluon is a nonpropagating mode in the QCD vacuum. Our
analysis takes into account the important condensate $<A_a^\mu
A_\mu^a>$ which is responsible for mass generation for the gluon.
Our work has some relation to that of Cornwall [24] who obtained a
gluon mass of $500\pm 200$ MeV in his analysis. Cornwall also
suggested that ``quark confinement arises from a vertex condensate
supported by a mass gap."
In recent work Gracey obtained a pole mass of the gluon of
$2.13\Lambda_{\overline{MS}}$ in a two-loop renormalization scheme
[25]. If we put $\Lambda_{\overline{MS}}=250$ MeV, the mass
obtained at two-loop order in Ref. [25] is $532$ MeV, which is
close to the value of $500$ MeV used in the present work. (We
remark that in Ref. [21] we obtained a gluon mass of $530$ MeV, if
we made use of Eq. (3.18) of that reference, which includes the
effect of including various exchange terms in our analysis of the
relevant matrix elements.)
|
{
"timestamp": "2004-11-18T15:00:06",
"yymm": "0411",
"arxiv_id": "hep-ph/0411234",
"language": "en",
"url": "https://arxiv.org/abs/hep-ph/0411234"
}
|
\section{Introduction}
Quantum transport in two-dimensional (2D) disordered systems
in inhomogeneous magnetic fields has been
studied extensively, in connection with the fractional
quantum Hall effect (FQHE).\cite{HLR}
In particular, the Anderson localization\cite{LR} in
a 2D system in random magnetic fields(RMF) with zero mean,
which arises in the mean field theory of the fractional
quantum Hall effect at filling factor $\nu=1/2$, has
been studied by many authors.\cite{PZ,KZ,SN,AHK,YG,Verges,SYN,Yakubo,KO}
From the theoretical point of view, whether such a system
has a metallic phase or not has been an important issue.
Systems in random magnetic fields belong
to the 2D unitary universality class which
is, in general, expected to have no metallic phase. If the
system has a metallic phase, it throws a question on the
validity of the conventional classification of the universality classes
of the Anderson transition.
Although many numerical as well as analytical attempts has been made to
clarify this point, the conclusions are so far still controversial.
On the other hand, the singular behavior of the conductance
fluctuations\cite{OSO} and the density of states at the band center $(E=0)$
has been investigated and argued that these singularities are governed by
the chiral unitary universality class.\cite{Furusaki,MBF}
Apart from such a theoretical interest,
the transport properties of 2D electron systems
in random magnetic fields are important to understand
the experiments in the quantum Hall systems.
In fact, the structure observed in the magnetoresistance
at the filling factor $1/2$ has been analyzed using
models with random magnetic fields.\cite{EMPW}
On the other hand, recently,
two-dimensional electron systems
in random magnetic fields have been realized experimentally
by attaching small magnets on the layer parallel to
the 2D electron gas in a semiconductor heterostructure.\cite{Ando}
In such a system, one can also control the strength of the
random magnetic fields.
The magnetotransport has been measured and interestingly, it is found
that the magnetoresistance
in systems having random magnetic fields exhibits similar structures to that
in the fractional quantum Hall system at the filling factor $\nu=1/2$.
\cite{Ando}
Since the origins of the magnetic field fluctuations in these
two systems are quite different,
it is important to clarify the origin of this similarity
in the magnetoresistance between these two systems.
One of the significant features of the magnetoresistance observed in
the recent experiment\cite{Ando}
is the increase of the magnetoresistance in two different scales of the
magnetic field.
One is the dip structure around zero field whose width is on
the order of the fluctuation of the random field, and the other is
the positive
magnetoresistance in a larger scale of the magnetic field
followed by the Shubnikov-de Haas oscillations.
In recent years, the semi-classical theory for the magnetotransport
in a smoothly varying random magnetic field has been developed,\cite{MWEPW}
where the snake state near zero magnetic field lines plays an important
role.
In the case of a weak disorder,
a pronounced positive magnetoresistance coming from
a classical origin has been reported for small magnetic fields.\cite{MWEPW}
Although the semi-classical approach provides qualitatively consistent
results with the experiments,
it is also important to examine the transport
property by the quantum mechanical calculations for its further
understanding.
In the present paper, we study the magnetoconductance in the
random magnetic field in two dimensions.
We consider a tight-binding model in random fluxes having no
spatial correlation, which has been analyzed by many authors
for the study of the Anderson localization in random magnetic fields.
In two dimensions, it has been shown that such a model has
the large localization length
near the band center and the diffusive behavior
is observed in that energy regime for numerically accessible length
scales.\cite{KO}
Since there is no spatial correlation
in random magnetic fields, the random magnetic field is not
smooth and hence the validity of
the semi-classical concepts is not obvious for the present model.
In our model, we find that the conductivity
is strongly suppressed by adding a uniform field and also
that it shows a saturation when the uniform field becomes on the order
of the fluctuation of the random magnetic field.
We adopt the equation of motion method in order to
examine numerically the diffusion of an electron in
inhomogeneous magnetic fields.
Adopting this method, we have an advantage that very large systems
compared with other numerical methods can be considered.
However, within the present method,
we are able to estimate the longitudinal conductivity $(\sigma_{xx})$
only, and not the Hall conductivity $(\sigma_{xy})$. Due to this,
we are not able to discuss the magnetoresistance directly.
As mentioned above, the 2D random magnetic field system has been shown
to have very large localization length near the band center.
It is then natural to assume that, in the present model, this
regime is responsible for the metallic
behavior observed in experiments. We therefore
confine ourselves to the case that the Fermi energy lies near
the band center.
First, we discuss the transport in the random magnetic fields with zero mean
and next, we
examine the effect of additional uniform magnetic fields.
The strength of the additional
uniform magnetic field considered in the present study
is assumed to be on the same order as or
smaller than that of the random magnetic field.
Implications from our numerical results
for the magnetoconductance are discussed
in comparison with the recent experimental results.
\section{Model and Method}
In order to describe the two-dimensional system in random
magnetic fields, we consider the following Hamiltonian
\begin{equation}
H = \sum_{<i,j>} V \exp({\rm i} \theta_{i,j}) C_i^{\dagger}
C_j + \sum_{i} \varepsilon_i C_i^{\dagger} C_i
\end{equation}
on the square lattice.
Here $C_i^{\dagger}(C_i)$ denotes the creation(annihilation) operator
of an electron on the site $i$ and $\{ \varepsilon_i \}$ denote the
random potential distributed independently in the
range $[ -W/2, W/2]$.
The phases $\{ \theta_{i,j} \}$ are related to the magnetic fluxes
$\{ \phi_i \}$ through the plaquette
$(i,i+\hat{x},i+\hat{x}+\hat{y},i+\hat{y})$ as
\begin{equation}
\theta_{i,i+\hat{x}} + \theta_{i+\hat{x},i+\hat{x}+\hat{y}}+
\theta_{i+\hat{x}+\hat{y},i+\hat{y}} + \theta_{i+\hat{y},i}
= -2\pi \phi_i / \phi_0
\end{equation}
where $\phi_0 = h/|e|$ stands for the unit flux.
The fluxes $\{ \phi_i \}$ are also assumed to be distributed independently
in each plaquette. The probability distribution $P(\phi)$ of the flux $\phi$
is given by
\begin{equation}
P( \phi ) = \left\{ \begin{array}{ll}
1/h_{\rm rf} & {\rm for }\quad |\phi /\phi_0|
\leq h_{\rm rf}/2 \\
0 & {\rm otherwise}
\end{array} \right. . \label{dist1}
\end{equation}
The variance of the distribution is accordingly given by
\begin{equation}
\langle \phi_i \phi_j \rangle = \frac{h_{\rm rf}^2}{12}\phi_0^2
\delta_{i,j} .
\end{equation}
In order to solve the time-dependent Schr\"{o}dinger equation
numerically, we employ the decomposition formula for
exponential operators.\cite{Suzuki}
The basic formula used in the present paper
is the forth order formula
\begin{equation}
\exp ( x[A_1 + \cdots + A_n]) = S(xp)^2 S(x(1-4p))S(xp)^2 + O(x^5),
\end{equation}
where
\begin{equation}
S(x) = {\rm e}^{xA_1/2}\cdots {\rm e}^{xA_{n-1}/2}{\rm e}^{xA_n}
{\rm e}^{xA_{n-1}/2}\cdots {\rm e}^{xA_1/2}.
\end{equation}
The parameter $p$ is given by $p=(4-4^{1/3})^{-1}$ and $A_1, \ldots , A_n$
are arbitrary operators.
We divide the Hamiltonian into five parts as in the
previous paper\cite{KO} so that each part is represented
as the direct product of $2\times 2$ matrices.
By applying this formula to the time evolution operator $U(t)\equiv
\exp(-{\rm i}
H t/\hbar)$, we obtain
\begin{eqnarray}
U(\delta t) &=& U_2(-{\rm i} p \delta t/\hbar)^2
U_2(-{\rm i}(1-4p)\delta t/\hbar)
U_2(-{\rm i} p \delta t/\hbar)^2 \\
& &+O(\delta t^5)
\end{eqnarray}
with
\begin{equation}
U_2(x) = {\rm e}^{xH_1/2}\cdots {\rm e}^{xH_4/2}{\rm e}^{xH_5}
{\rm e}^{xH_4/2}\cdots {\rm e}^{xH_1/2} ,
\end{equation}
where $H=H_1 + \cdots + H_5$. It is to be noted that the $U_2$ can
be expressed in an analytical form while the original evolution operator
$U$ can not be evaluated exactly without performing the
exact diagonalization of the whole system.
We are thus able to
consider larger system-sizes than other numerical methods, such as
exact diagonalization and the recursion method based on
the Landauer formula, which is one of the advantages of the
present method.
This method has already
been successfully applied to the case of
$W=0$ and $h_{\rm rf}=1$\cite{KO} as well as
to the 2D symplectic class.\cite{KO2}
The system we consider is the square lattice of the size $999 \times 999$
with the fixed boundary condition. All the length scales are measured in
units of the lattice constant $a$.
To prepare the initial wave packet with energy $E$, we numerically
diagonalize the subsystem ($21 \times 21$)
located at the center of the whole system and
take the eigenstate whose eigenvalue is the closest to $E$ as the initial
wave function.
In the following, we set the energy $E$ to be $E/V = -0.5$ which is
close to the band center.
The single time step $\delta t$
is set to be $\delta t = 0.02 \hbar/V$.
With this time step, the fluctuations of the expectation value of
the Hamiltonian is safely neglected throughout the present
simulation $(t \leq 200 \hbar/V)$.
We observe the second moment defined by
\begin{equation}
\langle \mathbf{r}^2 (t)\rangle_c \equiv \langle \mathbf{r}^2 (t)\rangle -
\langle \mathbf{r} (t)\rangle^2
\end{equation}
with
\begin{equation}
\langle \mathbf{r}^n(t) \rangle =
\sum_{\mathbf{r}} \mathbf{r}^n |\psi(\mathbf{r},t)|^2, \qquad (n=1,2)
\end{equation}
where $\psi(\mathbf{r},t)$ denotes the wave function at time $t$.
In the diffusive regime, the second moment is expected to
grow in proportion to $t$
\begin{equation}
\langle r^2 \rangle_c = 2dDt ,
\end{equation}
where the diffusion coefficient is denoted by $D$ and $d$ is
the dimensionality of the system. The
diffusion coefficient $D$ is related to the conductivity
by the Einstein relation $\sigma = e^2 D \rho$.\cite{KTH} Here
$\rho$ denotes the density of states. In Fig. 1, the
second moment is shown as a function of time for the case of
$W=0$. For each value of $h_{\rm rf}$,
five realizations of random magnetic fields are considered.
It is clearly seen that for $t \geq 50 \hbar/V$
the second moment increases in proportion to $t$, which means that
the system is in the diffusive regime. For $t \leq 50 \hbar/V$, we see
the ballistic behavior, where the second moment grows as
$\propto t^2$. By examining the behavior of the second moment,
we can clearly distinguish whether the system is in the diffusive regime
or in the ballistic regime.
To estimate the diffusion coefficient we discard the
data in the ballistic regime. We also do not consider the
time scale where the wave packet reaches to the edge of
the whole system.
Within the time scale considered in the present paper $t\leq 200 \hbar/V$,
we do not observe any sign of the saturation of the second moment
due to the finiteness of the system.
To obtain the conductivity, we need the
density of states too. We estimate it by the Green function
method\cite{SKM} for strips the width of which is $12 \leq M \leq 30$.
\begin{figure}
\includegraphics[scale=0.6]{fig1.eps}%
\caption{The second moment as a function of time
for $h_{\rm rf}=0.2, 0.25,
0.3$, and $0.4$. Five realizations of random magnetic fields are
considered for each $h_{\rm rf}$. \label{fig1}}
\end{figure}
\section{Numerical Result}
Let us first discuss how the diffusion
of an electron in random magnetic fields depends on
the strength of the fluctuation of random magnetic fields.
We estimate the diffusion coefficients from the behavior
of the second moment shown in Fig. 1. It is clearly seen that
the diffusion coefficient becomes smaller as the fluctuation
of the random magnetic fields increases.
The density of states at $E=-0.5V$
is estimated to be $\rho \approx 0.178$, $0.181$,
$0.179$, and $0.179 (1/a^2V)$ for $h_{\rm rf}=0.2$, $0.25$,
$0.3$ and $0.4$, respectively.
It depends on the strength of the
RMF very weakly near the band center.
On the other hand, the diffusion coefficient $D$ depends
strongly on the strength of the RMF which is
estimated to be $4D=85.22\pm 0.03$, $54.44\pm 0.04$, $37.40\pm0.03$
and $21.30\pm 0.02 (a^2V/\hbar)$ for $h_{\rm rf}=0.2$, $0.25$,
$0.3$ and $0.4$, respectively.
In Fig. 2, the conductivity, evaluated from these diffusion coefficients
and the density of states, is plotted for small values of $h_{\rm rf}$.
Here we clearly see that
the conductivity is inversely proportional to the square of $h_{\rm rf}$,
namely, $\sigma \propto 1/h_{\rm rf}^2$. This fact indicates that
the present regime would be well described by the
Born approximation.\cite{EMPW}
In the recent experiment,\cite{Ando} it is observed that the
relative change of the resistivity by the random magnetic fields
is in proportion to the square of the fluctuation
of the random magnetic fields, which is qualitatively
consistent with the present results (Fig. 2).
\begin{figure}
\includegraphics[scale=0.65]{fig2.eps}%
\caption{Conductivity in units of $e^2/h$ for $h_{\rm rf}=0.2$, $0.25$,
$0.3$ and $0.4$. \label{fig2}}
\end{figure}
Next, we apply an additional uniform magnetic field and
examine its effect on the electron diffusion. The uniform
magnetic field through
each plaquette of the square lattice is denoted by $\phi_{\rm uni}$.
The total magnetic field per plaquette is then $\phi_{\rm uni}+\phi$,
where $\phi_{\rm uni}$ is common to all the plaquette and $\phi$ is
distributed independently as (\ref{dist1}).
The second moments for $h_{\rm rf}=0.2$ under various values of
the uniform magnetic fields are shown in Fig. 3.
Here it is clear that the diffusion of electrons is
strongly suppressed by the additional uniform magnetic field.
Note that the strength of the uniform field considered is smaller than the
fluctuation of the random field $\sqrt{\langle \phi^2/\phi_0^2
\rangle} = h_{\rm rf}/\sqrt{12}= 0.0577\ldots $.
The diffusion coefficients and the density of states
estimated for various values of the uniform fields
are summarized in Table 1.
Here we see again that the density of states is not
sensitive to the uniform magnetic field.
We also perform numerical calculations for $h_{\rm rf}=0.3$ and
$0.4$. In Fig. 4,
the estimated conductivity is plotted as a function of the
uniform field scaled by $\sqrt{\langle \phi^2
\rangle}$.
For these values of
the random magnetic field, it is
commonly observed that the negative magnetoconductance
occurs when the uniform field is weaker than the
fluctuation of the random field.
It is also observed that at $\phi_{\rm uni} \approx
\sqrt{\langle \phi^2 \rangle}$, the conductivity takes a value
on the order of $e^2/h$ and is, interestingly, insensitive to the
magnitude of the random magnetic field.
\begin{table}
\caption{The diffusion coefficients $D$ and the density of states
$\rho$ for
$h_{\rm rf}=0.2$ \label{}}
\begin{ruledtabular}
\begin{tabular}{ccc}
$\phi_{\rm uni}/\phi_0$ & $4D$ & $\rho$\\
0 & 85.22 $\pm$ 0.03 & 0.178 \\
0.002 & 83.25 $\pm$ 0.03 & 0.179\\
0.005 & 76.37 $\pm$ 0.04 & 0.179 \\
0.01 & 59.54 $\pm$ 0.02 & 0.178\\
0.02 & 34.61 $\pm$ 0.04 & 0.179\\
0.04 & 14.65 $\pm$ 0.02 & 0.183 \\
0.06 & 8.69 $\pm$ 0.02 & 0.189\\
0.08 & 6.69 $\pm$ 0.02 & 0.177\\
0.1 & 5.31 $\pm$ 0.02 & 0.168\\
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{figure}
\includegraphics[scale=0.65]{fig3.eps}%
\caption{Electron diffusion for
$\phi_{\rm uni}/\phi_0=0$,
$0.01$, $0.02$, and $0.04$, where $\phi_{\rm uni}$ stands for
the strength of the uniform flux per plaquette. \label{fig3}}
\end{figure}
\begin{figure}
\includegraphics[scale=0.65]{fig4.eps}%
\caption{Magnetoconductance for
$h_{\rm rf}=0.2$, $0.3$ and $0.4$. The uniform magnetic
flux $\phi_{\rm uni}$ is scaled by the strength of the random
magnetic flux $\sqrt{\langle \phi^2 \rangle}$. Inset: The normalized
conductivity $\sigma /\sigma_0$ plotted as a function of
$\phi_{\rm uni}\cdot\sigma_0$. From the top, results for
$h_{\rm rf}=0.4$, $0.3$ and $0.2$ are presented.
The normalized data do not fall on the one curve, suggesting
the deviation from the Drude behavior. \label{fig4}}
\end{figure}
In order to see whether this peak structure around zero field can be
observed in the presence of the random potential $(W\neq 0)$, we
also examine the electron diffusion with the additional random potential.
We consider the case where $h_{\rm rf}=0.2$ and $W=0$, $1$ and $2$
and find that the peak structure
disappears at $W=2$ (Fig. 5). The random potential
$W=2$ on the square lattice
yields the mean free path on the order of 4 lattice
constants.\cite{ESZ} This result indicates that to observe
this enhancement of conductivity at zero uniform field
we need fairly clean samples.
\begin{figure}
\includegraphics[scale=0.7]{fig5.eps}%
\caption{Conductivity in units of $e^2/h$ for
$h_{\rm rf}=0.2$ and $W/V=0,1$ and
$2$. \label{fig5}}
\end{figure}
Estimation of the mean free path in the random magnetic
field, especially for $W=0$, is a subtle problem.\cite{AMW}
If we define the relaxation time for the transport\cite{AMW} by
$\tau_{\rm tr}=2D/v_{\rm F}^2$, where $v_{\rm F}$ denotes the Fermi velocity,
the mean free path $l_{\rm rf}$ in
the random magnetic field can be defined by $l_{\rm rf}= v_{\rm F}
\tau_{\rm tr}
= 2D/v_{\rm F}$. Near the band center, the average Fermi velocity
can be estimated to be on the order of
$2aV/\hbar$. The mean free path $l_{\rm rf}$
is then evaluated to be about $21a$, $9a$, and $5a$ for
$h_{\rm rf}=0.2$, $0.3$ and $0.4$, respectively.
\section{Discussion}
We have demonstrated
that in the absence of the uniform magnetic field,
the conductivity is inversely proportional to the square of the magnitude
of the random field, and hence is very sensitive to the strength of the
random magnetic field.
In contrast, we have also shown that
the conductivity takes a value on the order of $e^2/h$ and is
insensitive to the strength of the random magnetic field if the uniform
field is set to be
$\phi_{\rm uni}\approx \sqrt{\langle \phi^2 \rangle}$ (Fig. 4).
These two properties yield the peak structure at zero field, especially,
in the case of the weak random magnetic fields.
It may be useful to
discuss our results in view of the Drude formula $\sigma = \sigma_0 /
(1+\omega_c^2 \tau^2)$, where $\omega_c$ and $\tau$ are the
cyclotron frequency and the relaxation time, respectively.
Our results indicate that
$\sigma_0 \propto \tau \propto \langle\phi^2\rangle^{-1}$(Fig. 2).
The Drude theory then yields the form $\sigma /\sigma_0 =
1/(1+A(\phi_{\rm uni}\cdot \sigma_0)^2)$, where $A$ is a constant
fixed by the density of electrons and
independent of $\phi_{\rm uni}$ and $\phi$. We have found that
although
the data for small magnetic fields $ \phi_{\rm uni} \ll \sqrt{\langle \phi^2
\rangle }$ seem to be
consistent with this scaling form, it is unlikely that
the Drude formula accounts for the behavior around $\phi_{\rm uni} \approx
\sqrt{\langle \phi^2 \rangle }$.
The deviation from the
Drude theory, in which the resistivity is independent of the magnetic field,
comes from the fact that the conductivity is
insensitive to the magnitude of the random magnetic field at
$\phi_{\rm uni}\approx \sqrt{\langle \phi^2 \rangle}$ (Fig.4).
Let us consider this insensitivity of
the conductivity to the magnitude of the
random magnetic field at
$\phi_{\rm uni}\approx \sqrt{\langle \phi^2 \rangle}$
in more detail. With this condition
$\phi_{\rm uni}\approx \sqrt{\langle \phi^2 \rangle}$, the system is
almost equivalent to a system having
the random magnetic fields distributed in the range $0 \leq \phi /
\phi_0 \leq h$ and having
no uniform magnetic field $\phi_{\rm uni} = 0$.
We have then evaluated the conductivity
in such a system for
$0.2$, $0.3$, $0.4$ and $0.5$ and, indeed,
found that for all these values of $h$ the
conductivity is insensitive to the value of $h$ and
falls in the range $1.4 \sim 1.9 (e^2/h)$.
This would be one of the significant transport properties
specific to the random magnetic fields.
In order to consider a possible relationship to the
experiments, it may be useful to identify
the correlation length of the random fields in
experiments with the lattice constant of the present model.
Our calculation then implies that the sample having
mean free paths longer than 4 correlation
lengths of the random fields is very sensitive to the
application of uniform magnetic fields. This implication seems to be
consistent with the
experiments\cite{Ando} where the dip structure is observed
in samples having a mean free path larger than the correlation
length of the random magnetic field.
In summary, we have investigated the electron diffusion
in the random magnetic field with additional uniform
magnetic fields by the equation of motion
method. We have found that a sharp peak at zero uniform field
appears in magnetoconductance in the
absence of the spatial correlation of
random magnetic fields, where the semi-classical
theory can not be applied. The width of the peak turns out to be
on the order of the fluctuation of the random magnetic field.
The conductivity at $\phi_{\rm uni}\approx
\sqrt{\langle \phi^2 \rangle}$ is found to be insensitive to
the magnitude of the random magnetic field.
This peak structure disappears when the mean free path
becomes shorter by introducing the
random potential.
Although the present method enables us to simulate very large
systems, we can obtain the longitudinal conductivity $\sigma_{xx}$ only.
Detailed analysis of $\sigma_{xx}$ in experiments is
required to examine the relevance of the present results.
\begin{acknowledgments}
The authors thank M. Ando, Y.Ono, B.Kramer and S. Kettemann for valuable
discussions. Numerical calculations were performed by the facilities
of the Supercomputer Center, Institute for Solid State Physics,
University of Tokyo. This work is partly supported by the
Grants-in-aid No.13740244 from Japan Society for the Promotion of Science.
\end{acknowledgments}
|
{
"timestamp": "2004-11-10T05:13:49",
"yymm": "0411",
"arxiv_id": "cond-mat/0411248",
"language": "en",
"url": "https://arxiv.org/abs/cond-mat/0411248"
}
|
\section{Introduction}
Our Galaxy contains over two hundred known Supernova Remnants (SNRs), which are
an important source of energy and heavy element release into the interstellar
medium (ISM), and are also thought to be the sites of the acceleration of
cosmic rays. Over the last twenty years I have produced several versions of a
catalogue of Galactic SNRs, the most recent revised in 2004 January (see
Appendix~\ref{s:appendix}). Since the first version of the catalogue was
published in Green (1984), the number of identified Galactic SNRs has increased
considerably, from 145 to 231, and here I review some of the statistical
properties of Galactic remnants based on the most recent version of the
catalogue. In Section~\ref{s:catalogue} the catalogue is described, while the
selection effects applicable to the identification of Galactic SNRs are
discussed in Section~\ref{s:selection}. Some simple statistical properties of
the remnants are presented in Section~\ref{s:simple}, with more detailed
discussions of distance-dependant statistical studies of Galactic SNRs
(including a brief discussion of some aspects of extragalactic remnants) and
the Galactic distribution of SNRs given in Sections~\ref{s:distance} and
\ref{s:distribution} respectively. The summary parameters of the 231 remnants
from the 2004 January version of the catalogue of Galactic SNRs are presented
in Appendix~\ref{s:appendix}.
\section{The Catalogue}\label{s:catalogue}
The catalogue of Galactic SNRs contains: (i) basic parameters (Galactic and
equatorial coordinates, size, type, radio flux density, spectral index, and
other names); (ii) short descriptions of the observed structure at radio, X-ray
and optical wavelengths, as applicable; (iii) other notes on distance
determinations, pulsars or point sources nearby; and (iv) references.
Appendix~\ref{s:appendix} gives the basic parameters of all 231 remnants in the
2004 January version of the catalogue, and describes these parameters in more
details. The detailed version of the catalogue is available on the
World-Wide-Web from:
\\[6pt] \centerline{\tt http://www.mrao.cam.ac.uk/surveys/snrs/} \\[6pt]
which includes the descriptions, additional notes and references. The detailed
version is available as postscript or pdf for downloading and printing, or as
HTML web pages for each individual remnant. The web pages include links to the
`NASA Astrophysics Data System' for each of the nearly one thousand references.
Notes both on those objects no longer thought to be SNRs, and on many possible
and probable remnants that have been reported, are also included in the
detailed version of the catalogue. In addition to the observational selection
effects that are discussed further in Section~\ref{s:selection}, it should be
noted that the catalogue is far from homogeneous. Is is particularly difficult
to be uniform in terms of which objects are considered as definite remnants,
and are included in the catalogue, rather than listed as possible or probable
remnants which require further observations to clarify their nature. Although
many remnants, or possible remnants, were first identified from wide area
surveys, many others have been observed with a far from uniform set of
observational parameters, making uniform criteria for inclusion in the main
catalogue difficult. Also, some of the parameters included in the catalogue are
themselves of quite variable quality. For example, the radio flux density of
each remnant at 1~GHz. This is generally of good quality, being obtained from
several radio observations over a range of frequencies, both above and below
1~GHz. However, for a small number of remnants -- often those which have been
identified at other than radio wavelengths -- no reliable radio flux density,
or only a limit is available (which applies to 14 remnants in the current
catalogue).
\begin{figure
\centerline{\includegraphics[width=11.0cm]{s-sigmal}}
\caption{The distribution of surface brightness against Galactic longitude for
all 217 Galactic SNRs with defined surface brightnesses.\label{f:sigmal}}
\end{figure
\begin{figure
\centerline{\includegraphics[width=11.0cm]{s-sigmab}}
\caption{The distribution of surface brightness against Galactic latitude for
212 Galactic SNRs. The surface brightnesses of the five remnants with
$|b|>7^\circ$ are indicated by arrows at the left and right edges of the
plot.\label{f:sigmab}}
\end{figure
\section{Selection Effects}\label{s:selection}
Although several Galactic SNRs have been identified at other than radio
wavelengths, in practice the dominant selection effects are those that are
applicable at radio wavelengths. Simplistically, two selection effects
apply to the identification of Galactic SNRs (e.g.\ Green 1991), due to the
difficulty in identifying (i) faint remnants and (ii) small angular size
remnants. (In the case of extragalactic SNRs, the selection effects are
different, and these are discussed briefly -- particularly in the context of
SNRs identified in M82 -- in Section~\ref{s:extragalactic}.)
\subsection{Surface Brightness}\label{s:surface}
Clearly, SNRs need to have a high enough surface brightness for them to be
distinguished from the background Galactic emission. This selection effect is
{\em not} uniform across the sky, both because the Galactic background varies
with position, and because the sensitivities of available wide area surveys
covering different portions of the Galactic plane vary. The most recent
large-scale radio surveys that have covered much of the Galactic plane are: (i)
the Effelsberg survey at 2.7~GHz (Reich et al.\ 1990; F\"urst et al.\ 1990),
which covered $358^\circ < l < 240^\circ$ and $|b| < 5^\circ$; and (ii) the
MOST survey at 843~MHz (Whiteoak \& Green 1996; Green et al.\ 1999), which
covered $245^\circ < l < 355^\circ$, but only to $|b| < 1\fdg5$.
Figs.~\ref{f:sigmal} and \ref{f:sigmab} show the distribution of surface
brightness for known SNRs against Galactic longitude and latitude. These show
that in the anti-centre and away from $b=0^\circ$, where the Galactic
background is lower, fainter remnants are relatively more common than brighter
remnants, as expected. Also, there are fewer faint remnants identified in the
4th quadrant, which is due to the narrower range of the latitude coverage of
the MOST survey compared with that of the Effelsberg survey. (There are similar
numbers of remnants with surface brightnesses less than $10^{-21}$ {W m$^{-2}$\,Hz$^{-1}$\,sr$^{-1}$}
with $|b| \lesssim 1\fdg5$, in the 1st and 4th quadrant -- 9 and 7 respectively
-- but at higher Galactic latitude many more SNRs have been identified in the
1st quadrant than in the 4th -- 10 compared to 5 -- due to the wider latitude
coverage of the Effelsberg survey.)
\begin{figure
\centerline{\includegraphics[width=11.0cm]{h-post92}}
\caption{Histogram of the surface brightness of Galactic SNRs in the region
$358^\circ < l < 240^\circ$, $|b| < 5^\circ$ (i.e.\ the region covered by the
Effelsberg 2.7-GHz survey, Reich et al.\ 1990; F\"urst et al.\ 1990) identified
since 1991 (cf.\ Fig.~\ref{f:sigma} for the distribution for all Galactic
remnants).\label{f:post92}}
\end{figure
The Effelsberg survey detected new SNRs down to surface brightnesses
corresponding to $\approx 2 \times 10^{-22}$ {W m$^{-2}$\,Hz$^{-1}$\,sr$^{-1}$} at
1~GHz (Reich et al.\ 1988),
although the completeness limit for regions of brighter Galactic emission is
higher. This is not only because of the difficulty in identifying remnants in
the presence of extended Galactic emission, but is also due to confusion with
bright {{H\,{\sc ii}}} regions (which is relatively more of a problem at higher
frequencies). Since the new SNRs identified from the Effelsberg survey were
included in the version of the SNR catalogue published in Green (1991),
consideration of the surface brightness of other remnants in the survey region
that have subsequently been identified is useful for estimating the
completeness limit for this survey. Since 1991 an additional 24 remnants within
$358^\circ < l < 240^\circ$ and $|b| < 5^\circ$ have been included in the
catalogue, most in the first quadrant. These remnants have been identified from
a variety of observations, usually covering small regions of the Galactic
plane, rather than from large area surveys. Of these, five do not have good radio
observations available, and a histogram of the surface brightnesses of the
remaining 19 is shown in Fig.~\ref{f:post92}. Of these remnants, three have
surface brightnesses above $10^{-20}$ {W m$^{-2}$\,Hz$^{-1}$\,sr$^{-1}$}, two of which
(\SNR(0.3)+(0.0) and \SNR(1.0)-(0.1)) are close to the Galactic Centre, where
the background is particularly bright. Regarding these three bright remnants as
somewhat special cases, the surface brightnesses of the other more recently
identified SNRs suggest a completeness limit of $\approx 10^{-20}$
{W m$^{-2}$\,Hz$^{-1}$\,sr$^{-1}$} for the Effelsberg survey.
It is difficult to estimate the completeness limit of the MOST survey in a
similar way, as only three new remnants have been identified in the MOST survey
region since the remnants identified in this survey were included in the 1996
version of the catalogue. This is due to the limited number of telescopes able
to observe this part of the Galactic plane. Of these more recently identified,
only two have surface brightnesses (of $\approx 5 \times 10^{-22}$ and $4
\times 10^{-21}$ {W m$^{-2}$\,Hz$^{-1}$\,sr$^{-1}$} at 1~GHz). However, a comparison of the distribution of
the brighter SNRs in Galactic longitude suggests that the completeness limit in
the MOST survey region is not very different from that in the Effelsberg survey
region. There are 32 remnants in the 1st quadrant (i.e.\ covered by the
Effelsberg survey) with surface brightnesses above $10^{-20}$ {W m$^{-2}$\,Hz$^{-1}$\,sr$^{-1}$} and
27 in the 4th quadrant (i.e.\ covered by the MOST survey). Although the
Molonglo survey covers a smaller range in Galactic latitude than the Effelsberg
survey, this difference is not important, as only one of the bright remnants in
the 1st quadrant has $|b| > 1\fdg5$,
\begin{figure
\centerline{\includegraphics[width=11.0cm]{h-sigma}}
\caption{Distribution in surface brightness at 1~GHz of 217 Galactic SNRs. The
dashed line indicates the surface brightness completeness limit discussed in
Section~\ref{s:surface}.\label{f:sigma}}
\end{figure
So, the surface brightness limit for completeness of the current catalogue of
Galactic SNRs is approximately $10^{-20}$ {W m$^{-2}$\,Hz$^{-1}$\,sr$^{-1}$}. Fig.~\ref{f:sigma}
shows a histogram of the surface brightnesses of the 217 Galactic SNRs, of
which 64 are above this nominal surface brightness limit. As noted above, many
SNRs with surface brightnesses below this limit have been identified, both from
surveys and from other observations. These fainter remnants are predominantly
in regions of the Galaxy where the background is low, i.e.\ in the 2nd and 3rd
quadrants, and away from $b=0^\circ$, as shown in Fig.~\ref{f:lb}.
It is noticeable in Fig.~\ref{f:sigmal} that there are more remnants in the 2nd
quadrant than the 3rd (21 compared with 11). It seems likely that this is due
to the fact that the 2nd quadrant is more accessible to the wider range of
northern hemisphere radio telescopes than is the 3rd quadrant. Above the
nominal surface brightness limit given above, there are only very few remnants
in the 2nd and 3rd quadrants (3 and 2 respectively), i.e.\ for these bright
remnants there is no indication of any obvious deficit of remnants in the 3rd
quadrant. At first sight, Fig.~\ref{f:sigmab} appears to show that there are
more remnants identified away from the Galactic plane at positive latitudes
than at negative latitudes. There is an asymmetry in the number of SNRs at high
Galactic latitudes. There are 11 remnants with $|b| \ge +5^\circ$, but only 4
remnants with $|b| \le -5^\circ$. Most of these high positive latitude remnants
are in the 1st and 2nd quadrants, which suggests this asymmetry is related to
Gould's Belt (e.g.\ Stothers \& Frogel 1974), which is predominantly at
positive latitudes in these quadrants. However, it is not clear that these high
latitude remnants are close enough to be associated with Gould's Belt. There is
no evidence for any asymmetry in the number of remnants at low Galactic
latitudes; there are 49 with $b \ge +1^\circ$ and 44 with $b \le -1^\circ$,
which are not statistically different.
\begin{figure
\centerline{\includegraphics[angle=270,width=13.5cm]{s-lb-all}}
\quad\\
\centerline{\includegraphics[angle=270,width=13.5cm]{s-lb-bright}}
\caption{Galactic distribution of (top) all Galactic SNR and (bottom) those
SNRs with a surface brightness at 1~GHz greater than $10^{-20}$ {W m$^{-2}$\,Hz$^{-1}$\,sr$^{-1}$}.
(Note that the latitude and longitude axes are not to scale.)\label{f:lb}}
\end{figure
Ongoing and future observations will no doubt continue to detect more Galactic
SNRs, although it seems very likely that most of these objects will be faint,
and hence difficult to study in detail. Currently there are several large scale
radio surveys underway that will cover much of the Galactic plane
including\footnote{Also see: {\tt http://www.ras.ucalgary.ca/IGPS/} for further
information on the first three of these surveys.}: (i) the Canadian Galactic
Plane Survey (CGPS, see: Taylor et al.\ 2003) which covers much of the northern
Galactic plane, from $l \approx 55^\circ$ to $l \approx 195^\circ$; (ii) the
VLA Galactic Plane Survey (VGPS, see: Lockman \& Stil 2004) which covers
$l=18^\circ$ to $l=67^\circ$, (iii) the Southern Galactic Plane Survey (SGPS,
see McClure-Griffiths et al.\ 2001; McClure-Griffiths 2002), and (iv) the
second-epoch Molonglo Galactic Plane Survey (MGPS2, see Green 2002), which
covers $240^\circ \le l \le 5^\circ$, $|b| \le 10^\circ$. Examples of recently
identified SNRs include two faint remnants with surface brightnesses at 1~GHz
less than a few time $10^{-22}$ {W m$^{-2}$\,Hz$^{-1}$\,sr$^{-1}$}, which were found from CGPS
observations by Kothes et al.\ (2001). However, confusion with the Galactic
background -- particularly in regions near the Galactic Centre and near
$b=0^\circ$ -- will continue to be a limiting factor in the identification of
even moderately bright remnants. Comparison of radio observations over a wide
range of frequencies (so spectral index information can be used), or
observations at other than radio frequencies, may help to avoid some of the
limits caused by this confusion. Recent discoveries include three
new\footnote{These remnants are not included in the catalogue presented in
Appendix~\ref{s:appendix}, as their identification was published after that
catalogue was updated in 2004 January.}, faint SNRs near $l=11^\circ$ from
Brogan et al.\ (2004), which have surface brightness at 1~GHz of $(2{-}6)
\times 10^{-21}$ {W m$^{-2}$\,Hz$^{-1}$\,sr$^{-1}$}. These new remnants were identified because of
the wide range of frequencies, and the relatively high resolution of the radio
observations.
\begin{figure
\centerline{\includegraphics[width=11.0cm]{h-theta}}
\caption{Histogram of the angular size of 219 Galactic SNRs (12 remnants larger
than 100 arcmin are not included).\label{f:theta}}
\end{figure
As discussed below in Section~\ref{s:sigmad}, there is a general trend that
fainter remnants tend to be larger, and hence on average older, than brighter
remnants. However, because of the wide range of properties of Galactic SNRs
with known distances, the surface brightness selection effect applies not just
to old remnants, but also to young remnants. In particular, note that the
remnant of the SNR of {{\sc ad}} 1006 (see Table~\ref{t:historical}, below) is
fainter than the surface brightness completeness limit discussed above.
It should be noted that the study of Galactic SNRs by Filipovi\'c et al.\
(2002), which used the PMN Southern Survey images to extract flux densities of
SNRs at 5~GHz, is seriously limited by observational constraints. Since the PMN
observations were not processed to image extended objects (see Condon, Griffith
\& Wright 1993), the derived flux densities of many SNRs are in serious error.
\subsection{Angular Size}
Small angular size SNRs are likely to be missing from current catalogues. If
they are too small, then their structure is not well resolved by the available
Galactic plane surveys, and they would not be recognised as likely SNRs.
Fig.~\ref{f:theta} is shows the histogram of the angular sizes of known
remnants, which peaks at around 10~arcmin. (Note that for elongated remnants,
which have angular sizes given as $n \times m$ arcmin$^2$ in the catalogue, a
single diameter of $\sqrt{nm}$ has been used in this histogram, and in other
figures in this paper concerned with angular size.) The limiting angular size
varies for the different available wide area surveys. As discussed above, the
radio survey that covers most of the Galactic plane is the Effelsberg 2.7-GHz
survey, which has a resolution of $\approx 4.3$~arcmin. So, for this survey, any
remnants less than about 13~arcmin in diameter (i.e.\ 3 beamwidths) are not
likely to be recognised from their structures (although, as discussed in
Section~\ref{s:missing}, some searches have been made for small remnants among
the unresolved and barely resolved sources in the Effelsberg 2.7-GHz survey).
The MOST 843-MHz survey has a much better resolution, $\approx 0.7$~arcmin,
which implies that in the region of the Galactic plane covered by this survey
only remnants smaller than about $\approx 2$ arcmin (i.e.\ 3 beamwidths) might
be expected to be missed. However, although the MOST survey
detected 18 new SNRs
(Whiteoak \& Green 1996), the smallest new remnant is \SNR(345.7)-(0.2), which is $7
\times 5$ arcmin$^2$ in extent, i.e.\ several times larger than the nominal
limit of $\approx 2$ arcmin. Thus it is difficult to quote a single angular
size selection limit for current SNR catalogues, although it is clear that it
is difficult to identify small angular size remnants from existing wide area
surveys.
This selection effect is likely to be more important for filled-centre type
remnants than for shell type remnants. Even if filled-centre remnants are large
enough to be resolved in a survey at the level of several beamwidths, their
centrally brightened structures may not be striking enough to be able to
recognise them as filled-centre remnants. Using only radio continuum
observations, it is also easy to confuse the flat-spectrum synchrotron emission
from filled-centre remnants with thermal emission. Additional observations --
e.g.\ radio polarisation (as used to identify the small angular size
filled-centre remnant \SNR(54.1)-(0.3), see Reich et al.\ 1985), radio
recombination line non-detections (see, for example, Misanovic, Cram \& Green
2002), relatively low infra-red to radio ratios (e.g.\ Cohen \& Green 2001),
X-ray emission (e.g.\ Schaudel et al.\ 2002) -- are useful to distinguish
filled-centre remnants from thermal sources.
\subsection{Missing Young but Distant SNRs}\label{s:missing}
The lack of small angular size remnants -- i.e.\ young but distant remnants --
is particularly clear when the remnants of known `historical' Galactic
supernovae (see Stephenson \& Green 2002) are considered. These remnants are
relatively close-by -- as is expected, since their parent SNe were seen
historically -- and therefore sample only a small fraction of the Galactic
disc. Consequently we expect many more similar, but more distant remnants in
our Galaxy (e.g.\ Green 1985), but these are not present in current catalogues.
\begin{table}
\caption{Parameters of known historical SNRs, plus Cas~A.\label{t:historical}}
\begin{center}
\small\tabcolsep 2pt
\smallskip
\begin{tabular}{cccdcdccdccd}\hline
& & & \multicolumn{3}{c}{as observed} & & \multicolumn{2}{c}{if at 8.5~kpc} & & \multicolumn{2}{c}{if at 17~kpc} \\ \cline{4-6}\cline{8-9}\cline{11-12}
date & name or & distance & \dhead{size} & ${\mathit\Sigma}_{\rm 1~GHz}$ & \dhead{$S_{\rm 1~GHz}$}& & size & \dhead{$S_{\rm 1~GHz}$} & & size & \dhead{$S_{\rm 1~GHz}$} \\
& remnant & /kpc & \dhead{/arcmin} & {\footnotesize /{W m$^{-2}$\,Hz$^{-1}$\,sr$^{-1}$}} & \dhead{/Jy} & & /arcmin & \dhead{/Jy} & & /arcmin & \dhead{/Jy} \\ \hline
-- & Cas A & 3.4 & 5 & $1.6 \times 10^{-17}$ & 2720 & & 2.0 & 435 & & 1.0 & 109 \\
{{\sc ad}} 1604 & Kepler's & 2.9 & 3 & $3.2 \times 10^{-19}$ & 19 & & 1.0 & 2.2 & & 0.5 & 0.55 \\
{{\sc ad}} 1572 & Tycho's & 2.3 & 8 & $1.3 \times 10^{-19}$ & 56 & & 2.3 & 4.1 & & 1.1 & 1.0 \\
{{\sc ad}} 1181 & 3C58 & 3.2 & 7 & $1.0 \times 10^{-19}$ & 33 & & 2.6 & 4.7 & & 1.3 & 1.2 \\
{{\sc ad}} 1054 & {\footnotesize Crab nebula} & 1.9 & 6 & $4.4 \times 10^{-18}$ & 1040 & & 1.4 & 52 & & 0.7 & 13 \\
{{\sc ad}} 1006 & {\footnotesize \SNR(327.6)+(14.6)} & 2.2 & 30 & $3.2 \times 10^{-21}$ & 19 & & 7.7 & 1.3 & & 3.9 & 0.31 \\ \hline
\end{tabular}
\end{center}
\end{table}
Table~\ref{t:historical} gives the distances, angular sizes, flux
densities and surface brightnesses at 1~GHz, for the remnants of known
historical supernovae from the last thousand years, plus Cas~A (which although
its progenitor was not seen -- so it is not strictly a historical remnant -- is
known to be only about 300 years old). The distances used for these remnants
are those given in Section~\ref{s:distances}. This table also lists the
parameters of these remnants when scaled to larger distances of 8.5 and 17~kpc,
i.e.\ to represent how they would appear if they if they were at the other
side of the Galaxy (from the Galactic Centre, to the far point on the Solar
Circle). The number of other young (i.e.\ less than a thousand year old) SNRs
expected in the Galaxy can be estimated in two simple ways:
(i) from the expected supernova rate of one every 45 to 70 years (Cappellaro
2003), 15 to 22 young remnants are expected in total;
(ii) considering the fraction of the Galactic disc sampled by the historical
supernovae (6 in a thousand years), which are within $\approx 4$~kpc, i.e.\
about 16 per cent of the Galactic disc modelled as being uniform and having a
radius of $\approx 10$~kpc, implies there should be $\approx 40$ young remnants
(see also the discussions in Strom 1994). Of these, $\approx 80\%$ (see
Cappellaro 2003) are expected to be the remnants of massive supernovae -- i.e.\
from type Ib/Ic/II SNe -- and therefore be close to the Galactic plane.
From Table~\ref{t:historical}, any young SNRs in the Galaxy similar to the
known historical remnants, but in the far half of the Galaxy, would generally
be expected to have angular sizes less than a few arcmin, high surface
brightness, greater than $\approx 10^{-19}$ {W m$^{-2}$\,Hz$^{-1}$\,sr$^{-1}$} (although remnants
similar to the remnant of the SN of {{\sc ad}} 1006 would be much fainter). These
remnants would also be expected to lie close to the Galactic plane, with $|b|
\lesssim 1^\circ$. Although the above estimates are rather uncertain due to
intrinsic Poisson uncertainties from the small numbers of known historical
remnants, they imply that about a dozen or more young but distant remnants
might be expected. However, there are very few such remnants in the current Galactic
SNR catalogue. In fact there are only 3 known remnants with angular sizes of 2
arcmin or less: \SNR(1.9)+(0.3), \SNR(54.1)+(0.3) and \SNR(337.0)-(0.1).
Of these, \SNR(1.9)+(0.3) is the smallest, with an angular size of only
1.2~arcmin. No distance measurement is available for this remnant -- which
being close to $l=0^\circ$ makes kinematic methods unreliable -- but even if it
were at the far side of the Galaxy, at say 17~kpc, its physical size would only
be 6~pc. This is comparable to the sizes of the known historical remnants in
Table~\ref{t:historical} (which have physical diameters of 5, 3, 5, 7, 3 and 19
pc respectively). Another indication that this is indeed a young remnant is
that it shows a circularly symmetric limb-brightened shell of radio emission
(see Fig.~\ref{f:g1.9} for a previously unpublished image of this remnant at
1.5~GHz, from 1985 observations made with the NRAO's VLA\footnote{The National
Radio Astronomy Observatory is a facility of the National Science Foundation
operated under cooperative agreement by Associated Universities, Inc.}). The
known young shell remnants all show highly circular structures, whereas older
remnants tend to be less circular, as is expected as they expand into differing
regions of the interstellar medium. The distance to the filled-centre remnant
\SNR(54.1)+(0.3) is uncertain (e.g.\ Camilo et al.\ 2002), although its small
angular size of 1.5~arcmin again suggests it is physically small, and therefore
young, even if it were to be situated at the edge of the Galaxy. A reasonably
accurate distance estimate is available for \SNR(337.0)-(0.1) (see
Section~\ref{s:distances}), and this is $\approx 11$~kpc, which is consistent
with this being a young but distant SNR (with a diameter of $\approx 5$~pc,
from its angular size of 1.5 arcmin). The deficit of small remnants is also
illustrated in Fig.~\ref{f:sigmatheta}, which shows there are very few known
Galactic remnants with high surface brightnesses and small angular sizes.
\begin{figure
\centerline{\includegraphics[width=6cm]{g19+03-log}}
\caption{A VLA image of \SNR(1.9)+(0.3) at 1.5~GHz, with a resolution of $9.4
\times 7.2$ arcsec$^2$ at a position angle of $3^\circ$. The contour levels are
at $-1, 1 \sqrt{2^n}$ mJy beam$^{-1}$ for $n=0, 1, 2\dots$ (with the negative
contour dashed). The coordinates are J2000.0.\label{f:g1.9}}
\end{figure
\begin{figure
\centerline{\includegraphics[width=11.0cm]{s-sigmatheta}}
\caption{Distribution in surface brightness at 1~GHz against angular size for
known Galactic SNRs of angular size $\le 8$ arcmin. The five historical
remnants from Table~\ref{t:historical} included in this plot are indicated by
additional crosses. (For all but the smallest few remnants in the catalogue,
the angular size is given to the nearest arcmin.)\label{f:sigmatheta}}
\end{figure
This deficit of young but distant remnants has long been recognised, but
searches for remnants of this type (e.g.\
Green
\& Gull 1984; Helfand et al.\ 1984; Green 1985, 1989; Sramek et al.\ 1992; Misanovic et al.\
2002; see also Saikia et al.\ 2004) have had only limited success (identifying
the small remnants \SNR(1.9)+(0.3) and \SNR(54.1)+(0.3) noted above). Since the
missing young but distant remnants are expected to have angular sizes of a few
arcmin or less, they will not have been resolved sufficiently by single-dish
radio surveys with resolutions of several arcminutes, so will not have been
recognised as SNRs. Searching for these remnants is not easy because there are
very many candidate sources in single-dish surveys to choose from (e.g.\ in the
1st quadrant, there are over a thousand compact sources in the Effelsberg
survey with $|b| \le 1^\circ$), and only a fraction of these have been observed
with high enough resolution to be able to identify them if they were small
angular size SNRs. Moreover, the missing young but distant remnants are likely
to be in the most complex, and hence most confused, regions of the Galactic plane,
as being distant they will be close to $b=0^\circ$. The use of additional
observational indicators (e.g.\ radio spectral index, infra-red to radio ratios) has yet not proved efficient for improving such searches, nor have the
MOST survey (see Whiteoak \& Green 1996) and the NVSS (see Condon et al.\
1998), which cover the Galactic plane with slightly higher resolution than
available single-dish surveys. (Note, however, that the NVSS does contain
previously unrecognised SNRs, for example \SNR(353.9)-(2.0), with an angular
size of 13 arcmin, see Green 2001a.) The fact that such missing, small remnants
are likely to be in complex regions of the plane may mean that confusion is a
very significant problem, and not just at radio wavelengths. Further searches
for these missing young but distant remnants are required.
It should be noted that there are unlikely to be other luminous remnants in the
Galaxy like Cas A and the Crab nebula. Any such remnants, even on the far side
of the Galaxy, would have relatively high flux densities, and the nature of
all such sources in the Galactic plane is known. On the other hand, the remnant
of the SN of {{\sc ad}} 1006 is faint -- possibly because it is far from the
Galactic plane, in a low density region -- and distant remnants similar to this
would be particularly difficult to detect, as they would have both a small
angular size and low surface brightness. However, any remnants similar to the
other three historical remnants are detectable.
\section{Some Simple SNR Statistics}\label{s:simple}
In the current version of the catalogue, 77\% of remnants are classed as shell
(or possible shell), 12\% are composite (or possible composite), and 4\% are
filled-centre (or possible filled centre) remnants. The remaining 7\% have not
yet been observed well enough to be sure of their type, or else are objects
which are conventionally regarded as SNRs although they do not fit well into
any of the conventional types (e.g.\ CTB80 ($=$\SNR(69.0)+(2.7)), MSH
17$-$3{\em 9} ($=$\SNR(357.7)-(0.1))). Since the 1991 version of the catalogue
(Green 1991), the proportion of shell remnants in the catalogue has stayed very
similar, with the proportion of composite remnants increasing from 8\%, and the
proportion of filled centre remnants has decreasing from 7\%. The increase in
the proportion of composite remnants is because more recent, improved
observations have continued to identify many more shell remnants, but have also
identified faint, pulsar powered nebulae in what until then had been identified
as pure shell remnants (e.g.\ W44 ($=$\SNR(24.7)-(0.4))), and also that faint
shells have been detected around some filled-centre remnants (e.g.\
\SNR(21.5)-(0.9)).
\begin{figure
\centerline{\includegraphics[width=11.0cm]{s-stheta}}
\caption{Distribution of flux density at 1~GHz against angular size for known
Galactic SNRs with diameters $\le 30$~arcmin.\label{f:stheta}}
\end{figure
There are 14 Galactic SNRs that are either not detected at radio wavelengths,
or are poorly defined by current radio observations, so that their flux
density at 1~GHz cannot be determined with any confidence: i.e.\ 94\% have a
flux density at 1~GHz included in the catalogue. Of the catalogued remnants,
36\% are detected in X-ray, and 23\% in the optical. At both these wavelengths,
Galactic absorption hampers the detection of distant remnants.
Some of the properties of the Galactic SNRs in the catalogue, which are not
shown in other sections of this paper, are shown in Fig.~\ref{f:stheta}. This
shows the flux density at 1~GHz versus angular size for SNRs less than 30
arcmin in extent. This is of interest in terms of which remnants may appear as
bright, relatively compact Galactic plane sources (e.g.\ in future Planck
surveys). The most prominent sources are, not surprisingly, the very bright
SNRs Cas~A and the Crab nebula (see Table~\ref{t:historical}), which have
similar angular sizes, and very high flux densities. Cas~A has the higher flux
density at 1~GHz by a factor of about 2.6, but because the Crab nebula has a
much flatter spectrum (with $\alpha \approx 0.30$ compared with $\approx 0.77$
for Cas~A, e.g.\ Baars et al.\ 1977), the Crab nebula has the higher flux
density at frequencies above about 8~GHz. The statistics of the radio spectral
indices of Galactic SNRs are not discussed here, although there is a short
discussion of these in Green (2001b).
\section{Distance Dependant SNR Statistics}\label{s:distance}
\subsection{Distances to SNRs}\label{s:distances}
Many studies of Galactic SNRs require knowledge of the distances to remnants
(or equivalently their physical sizes, since their angular sizes are known).
However, accurate distances are not available for many known SNRs. The
distances that are available are obtained from a wide variety of methods --
e.g.\ optical expansion and proper motion studies, 21-cm {{H\,{\sc i}}} absorption
spectra, {{H\,{\sc i}}} column density (see Foster \& Routledge 2003), association with
{{H\,{\sc i}}} or CO features in the surrounding interstellar medium, or association
with other objects -- each of which is subject to their own uncertainties, and
some of which are subjective. Table~\ref{t:distances} presents distances for 47
Galactic SNRs available in the literature.
(In a few cases distances estimates to SNRs are also available from the
distances to associated pulsar, derived from the observed pulsar dispersion
measure and a model of the Galactic electron density distribution. However,
these have not been used in Table~\ref{t:distances}.)
In several cases the distances derived from {{H\,{\sc i}}} absorption measurements have
been recalculated using a modern `flat' rotation curve with a Galactocentric
radius of 8.5~kpc and a constant rotation speed of 220 km s$^{-1}$.
Additionally, in a few cases the distances given depend on re-interpretation of
the published observations. For \SNR(11.2)-(0.3) and \SNR(21.5)-(0.9), the
distances given correspond to the near distances of the last strong {{H\,{\sc i}}}
absorption seen (see also Safi-Harb et al.\ 2001 for \SNR(21.5)-(0.9)).
\begin{table
\caption{Galactic SNRs with distance measurements or
estimates.\label{t:distances}}
\begin{center}
\smallskip
\footnotesize
\begin{tabular}{rdllr}\hline
remnant & \dhead{distance} & method & reference & notes \\
& \dhead{/kpc} & & & \\ \hline
\SNR(4.5)+(6.8) & 2.9 & optical proper motion/velocity & Blair et al.\ (1991) & \\
\SNR(6.4)-(0.1) & 1.9 & {{H\,{\sc i}}} absorption & Vel{\'a}zquez et al.\ (2002) & \\
\SNR(11.2)-(0.3) & 4.4 & {{H\,{\sc i}}} absorption & Becker et al.\ (1985) & a,b \\
\SNR(18.8)+(0.3) & 14.0 & association with CO & Dubner et al.\ (2004) & \\
\SNR(21.5)-(0.9) & 4.6 & {{H\,{\sc i}}} absorption & Davelaar et al.\ (1986) & a,b \\[3pt]
\SNR(27.4)+(0.0) & 6.8 & {{H\,{\sc i}}} absorption & Sanbonmatsu \& Helfand (1992) & \\
\SNR(33.6)+(0.1) & 7.8 & {{H\,{\sc i}}} absorption & Frail \& Clifton (1989) & a \\
\SNR(34.7)-(0.4) & 2.8 & {{H\,{\sc i}}} absorption & Caswell et al.\ (1975) & a \\
\SNR(39.7)-(2.0) & 3.0 & association with {{H\,{\sc i}}} & Dubner et al.\ (1998) & \\
\SNR(43.3)-(0.2) & 10.0 & association with {{H\,{\sc i}}} & Lockhart \& Goss (1978) & a \\[3pt]
\SNR(49.2)-(0.7) & 6.0 & association with CO & Koo et al.\ (1995) & \\
\SNR(53.6)-(2.2) & 2.8 & association with {{H\,{\sc i}}} & Giacani et al.\ (1998) & \\
\SNR(55.0)+(0.3) & 14.0 & association with {{H\,{\sc i}}} & Matthews et al.\ (1998) & \\
\SNR(74.0)-(8.5) & 0.4 & optical proper motion/velocity & Blair et al.\ (1999) & \\
\SNR(74.9)+(1.2) & 6.1 & {{H\,{\sc i}}} column density & Kothes et al.\ (2003) & \\[3pt]
\SNR(84.2)-(0.8) & 4.5 & association with CO & Feldt \& Green (1993) & \\
\SNR(89.0)+(4.7) & 0.8 & association with CO and {{H\,{\sc i}}} & Tatematsu et al.\ (1990) & \\
\SNR(93.3)+(6.9) & 2.2 & {{H\,{\sc i}}} column density & Foster \& Routledge (2003) & \\
\SNR(93.7)-(0.2) & 1.5 & association with {{H\,{\sc i}}} & Uyan{\i}ker et al.\ (2002) & \\
\SNR(109.1)-(1.0) & 3.0 & association with {{H\,{\sc ii}}} region & Kothes et al.\ (2002) & \\[3pt]
\SNR(111.7)-(2.1) & 3.4 & optical proper motion/velocity & Reed et al.\ (1995) & \\
\SNR(114.3)+(0.3) & 0.7 & association with {{H\,{\sc i}}} & Yar-Uyan{\i}ker et al (2004) & \\
\SNR(116.5)+(1.1) & 1.6 & association with {{H\,{\sc i}}} & Yar-Uyan{\i}ker et al (2004) & \\
\SNR(116.9)+(0.2) & 1.6 & association with {{H\,{\sc i}}} & Yar-Uyan{\i}ker et al (2004) & \\
\SNR(119.5)+(10.2) & 1.4 & association with {{H\,{\sc i}}} & Pineault et al.\ (1993) & \\[3pt]
\SNR(120.1)+(1.4) & 2.3 & optical proper motion/velocity & Chevalier et al.\ (1980) & \\
\SNR(130.7)+(3.1) & 3.2 & {{H\,{\sc i}}} absorption & Roberts et al.\ (1993) & \\
\SNR(132.7)+(1.3) & 2.2 & association with CO & Routledge et al.\ (1991) & \\
\SNR(166.0)+(4.3) & 4.5 & association with {{H\,{\sc i}}} & Landecker et al.\ (1989) & \\
\SNR(166.2)+(2.5) & 8.0 & association with {{H\,{\sc i}}} & Routledge et al.\ (1986) & \\[3pt]
\SNR(184.6)-(5.8) & 1.9 & various & Trimble (1973) & \\
\SNR(189.1)+(3.0) & 1.5 & optical absorption & Welsh \& Sallmen (2003) & \\
\SNR(205.5)+(0.5) & 1.6 & various & Odegard (1986) & \\
\SNR(260.4)-(3.4) & 2.2 & association with {{H\,{\sc i}}} & Reynoso et al.\ (1995) & \\
\SNR(263.9)-(3.3) & 0.3 & pulsar parallax & Caraveo et al.\ (2001) & \\[3pt]
\SNR(292.0)+(1.8) & 6.0 & various & Gaensler \& Wallace (2003) & \\
\SNR(292.2)-(0.5) & 8.4 & {{H\,{\sc i}}} absorption & Caswell et al.\ (2004) & \\
\SNR(296.8)-(0.3) & 9.6 & association with {{H\,{\sc i}}} & Gaensler et al.\ (1998a) & \\
\SNR(315.4)-(2.3) & 2.3 & optical velocity & Sollerman et al.\ (2003) & \\
\SNR(320.4)-(1.2) & 5.2 & {{H\,{\sc i}}} absorption & Gaensler et al.\ (1999) & \\[3pt]
\SNR(327.4)+(0.4) & 4.8 & {{H\,{\sc i}}} absorption & McClure-Griffiths et al.\ (2001) & \\
\SNR(327.6)+(14.6) & 2.2 & optical proper motion/velocity & Winkler et al.\ (2003) & \\
\SNR(332.4)-(0.4) & 3.1 & {{H\,{\sc i}}} absorption & Caswell et al.\ (1975) & a \\
\SNR(337.0)-(0.1) & 11.0 & various & Sarma et al.\ (1997) & \\
\SNR(348.5)+(0.1) & 8.0 & {{H\,{\sc i}}} absorption & Caswell et al.\ (1975) & a \\[3pt]
\SNR(348.7)+(0.4) & 8.0 & {{H\,{\sc i}}} absorption & Caswell et al.\ (1975) & a \\
\SNR(349.7)+(0.2) & 14.8 & {{H\,{\sc i}}} absorption & Caswell et al.\ (1975) & a \\ \hline
\end{tabular}\\
\noindent Notes: a) distance recalculated using modern rotation curve; b) see
text for further discussion.
\end{center}
\end{table
\begin{figure
\centerline{\includegraphics[width=11.0cm]{h-withd}}
\caption{Distribution in surface brightness at 1~GHz of 47 Galactic SNRs with
known distances (see Section~\ref{s:distances}). The dashed line indicates the
surface brightness completeness limit discussed in
Section~\ref{s:surface}.\label{f:withd}}
\end{figure
The uncertainties in these distances are far from uniform. For kinematic
distances -- which are a large majority of the distances given in
Table~\ref{t:distances} -- there are always some uncertainties in deriving
distances from observed velocities, due to deviations from circular motion
(especially an issue for nearby remnants, and for those near $l=0^\circ$ and
$180^\circ$ where the observed velocity does not depend strongly on distance)
and ambiguities inside the Solar Circle. Generally, the published errors in
kinematic distances are less than $\approx 25\%$, although there are additional
possible larger uncertainties depending on whether the (often subjective)
association of a particular feature with a remnant is actually correct. A
potential bias for statistical studies is that many distance methods are more
easily applied to brighter remnants than to fainter ones. This is particularly the
case for 21-cm {{H\,{\sc i}}} absorption studies, which depend on a radio continuum from
the remnant being bright, otherwise any absorption could not be studied in
reasonable detail. But this also applies to some of the other methods, e.g.\
association with other {{H\,{\sc i}}} or CO features in the ISM, where fainter remnants
will not be well defined, so that clear morphological association with other
features will be more difficult. Indeed, Fig.~\ref{f:withd} shows a histogram
of the surface brightness of Galactic SNRs with known distances, which shows
these tend to be the brighter Galactic SNRs overall (see Fig.~\ref{f:sigma}). Thus,
it is likely that, at a given diameter, SNRs with known distances are biased to
brighter remnants. The number of SNRs with available distances is sufficiently
large that statistical studies -- see Section~\ref{s:sigmad} -- show that the
range of intrinsic luminosities of Galactic SNRs is large.
\begin{figure
\centerline{\includegraphics[angle=270,width=9cm]{s-sigmad}}
\caption{The surface brightness/diameter (${\mathit\Sigma}{-}D$) relation for 47
Galactic SNRs with known distances (see Table~\ref{t:distances}), shown as
filled circles. The open circle shows the parameters of RX
J$0852{\cdot}0{-}4622$ ($=$\SNR(266.2)-(1.2)), if it is at a distance of 200~pc
(see text for discussion). Note that the lower left part of this diagram is
likely to be seriously affected by selection effects.\label{f:sigmad}}
\end{figure
\begin{figure
\centerline{\includegraphics[angle=270,width=9cm]{s-ld}}
\caption{The luminosity/diameter ($L{-}D$) relation for 47 Galactic SNRs
with known distances (see Table~\ref{t:distances}).\label{f:ld}}
\end{figure
\subsection{The ${\mathit\Sigma}{-}D$ and $L{-}D$ Relations}\label{s:sigmad}
Since distances are not available for all SNRs, many statistical studies of
Galactic SNRs have relied on the surface-brightness/diameter, or
`${\mathit\Sigma}{-}D$' relation to derive distances for individual SNRs from their
observed flux densities and angular sizes. For remnants with known distances
($d$), and hence known diameters ($D$), physically large SNRs are fainter
(i.e.\ they have a lower surface brightness) than small remnants. Using this
correlation between ${\mathit\Sigma}$ and $D$ for remnants with known distances, a
physical diameter is deduced from the distance-independent {\em observed}
surface brightness of any remnant. Then a distance to the remnant can be
deduced from this diameter and the observed angular size of the remnant.
The ${\mathit\Sigma}{-}D$ relation for Galactic SNRs with known distances is shown in
Fig.~\ref{f:sigmad}. As discussed above (Section~\ref{s:distances}), the
distances to individual remnants are not homogeneous in quality, and many
depend on subjective interpretation of data.
Of the SNRs included in this figure, three are
`filled-centre' remnants (the Crab nebula ($=$\SNR(184.6)-(5.8)), 3C58
($=$\SNR(130.7)+(3.1)) and \SNR(74.9)+(1.2)). Nevertheless, Fig.~\ref{f:sigmad}
clearly shows a wide range of diameters for a given surface brightness, which
is a severe limitation in the usefulness of the ${\mathit\Sigma}{-}D$ relation for
deriving the diameters, and hence distances, to individual remnants. For a
particular surface brightness, the diameters of SNRs vary by up to about an
order of magnitude, or conversely, for a particular diameter, the range of
observed surface brightnesses seen varies by more than two orders of magnitude.
The correlation shown between surface brightness and diameter in
Fig.~\ref{f:sigmad} is, however, largely a consequence of the fact that it is a
plot of surface-brightness -- rather than luminosity -- against diameter, $D$.
Surface brightness is plotted, because it is the distance-independent
observable that is available for (almost) all SNRs, including those for which
distances are not available. For remnants whose distances are known, we can
instead consider the radio luminosity of the remnants. Since ${\mathit\Sigma}$ and
luminosity, $L$, depends on the flux density $S$, angular size $\theta$,
distance $d$ and diameter $D$, as
$$
{\mathit\Sigma} \propto {S \over \theta^2} \quad{\rm and}\quad
L \propto S d^2
$$
then
$$
{\mathit\Sigma} \propto {L \over (\theta d)^2} \quad{\rm or}\quad
{\mathit\Sigma} \propto {L \over D^2}.
$$
Thus, much of the correlation shown in the ${\mathit\Sigma}{-}D$ relation in
Fig.~\ref{f:sigmad} is due to the $D^{-2}$ bias that is inherent when plotting
${\mathit\Sigma}$ against $D$, instead of $L$ against $D$. The $L{-}D$ relation for
Galactic SNRs with known distances in Fig.~\ref{f:ld} shows that there is wide
range of luminosities for SNRs of all diameters. Cas~A is the most luminous
Galactic SNR, but it appears to be at the edge of a wide distribution of
luminosities. The wide range of luminosities is perhaps not surprising, given
that the remnants are produced for a variety of types of supernovae, and that
they evolve in regions of ISM with a range of properties (e.g.\ density), which
may well effect the efficiency of the radio emission mechanism at work. For
example, some SNRs may initially evolve inside a low-density, wind-blown
cavity, and then collide with the much denser regions of the surrounding ISM.
Furthermore, the full range of intrinsic properties of SNRs may be even wider
than that shown in Figs~\ref{f:sigmad} and \ref{f:ld}, as the selection effects
discussed above mean that it is difficult to identify small and/or faint SNRs.
One specific example is the recently
identified SNR RX J$0852{\cdot}0{-}4622$ ($=$\SNR(266.2)-(1.2), see Aschenbach
1998), which may extend the range of properties of SNRs considerably (see
Duncan \& Green 2000). The surface brightness of RX J$0852{\cdot}0{-}4622$ at
1~GHz is $\approx 6 \times 10^{-22}$ {W m$^{-2}$\,Hz$^{-1}$\,sr$^{-1}$}, which places it among the
faintest 20 per cent of catalogued remnants. If it is at a distance as small as
200~pc, as suggested by Aschenbach (1998) -- see also Redman et al.\ (2002) --
then its diameter would be only 7~pc, see Fig.~\ref{f:sigmad}. On the other
hand, the deficit of bright, large SNRs in Fig.~\ref{f:sigmad}
cannot be explained by any selection
effect, and so represents some real limit in the luminosity of remnants at a
particular diameter, related to the physics of the underlying radio emission
mechanism at work. This upper bound in the ${\mathit\Sigma}{-}D$ plane can be used to
derive an {\em upper limit} for the diameter of any SNR from its observed
surface brightness, and hence an upper limit on its distance. Another upper
bound on the distance to any remnant -- which may be as useful -- is to assume
it lies within the Galactic disc.
Case \& Bhattacharya (1998) derived a ${\mathit\Sigma}{-}D$ relation, based on the
distances available for 36 remnants -- not including filled-centre remnants --
from the 1996 version of the catalogue. They argue that
it is useful for deriving
distances for individual SNRs, with a fractional error of only 0.33, which is
very much smaller than the wide range in diameters for a given surface
brightness shown in Fig.~\ref{f:sigmad}. This optimistic result was only
obtained after excluding Cas~A from the remnants with known distances used to
derived the ${\mathit\Sigma}{-}D$ relation (as it was deemed to be sufficiently
different from other shell SNRs), and also after excluding 7 remnants with high
$z$-values (as three of these showed the largest deviation from the best-fit
${\mathit\Sigma}{-}D$ relation). It is, however, difficult to decide {\em a priori}
whether a remnant is or is not to be included in the subset of remnants for
which Case \& Bhattacharya derived distances with relatively small
uncertainties.
Also, it is not clear that any best-fit ${\mathit\Sigma}{-}D$ relation -- not
withstanding selection effect problems -- actually represents the evolutionary
track of individual SNRs (see Berkhuijsen 1986). The distribution of SNRs with
known distances is a snapshot in time of a population of remnants, and
individual remnants may evolve in the ${\mathit\Sigma}{-}D$ plane in directions quite
different from the overall power law fitted to the overall distribution of SNRs
(or to the upper limit of the distribution). As a simple example, consider the
situation where SNRs have a range of intrinsic luminosities, expand with a
constant luminosity up to some particular diameter -- which varies for
different SNRs depending on their environment (e.g.\ the surrounding ISM
density, which influences their expansion speed, which may affect the
efficiency of radio emission mechanism) -- after which their radio luminosity
fades rapidly. In this case, the locus of the upper bound to the highest
surface brightness remnants for a particular diameter is related to where the
luminosities of different SNRs begin to decrease, and does not represent the
evolutionary track of any individual remnant. The current direction of the
evolutionary track of only one Galactic SNR, Cas~A, can be estimated from
available observations. The flux density of Cas~A is decreasing at
approximately 0.8\% year$^{-1}$ (Baars et al.\ 1977; Rees 1990), and its bulk
expansion is 0.22\% year$^{-1}$ (Ag\"ueros \& Green 1999; but see DeLaney
et al.\ 2004 for alternative expansion timescales). These observations suggest
Cas~A is following a track with $\Sigma \propto D^{-5.6}$, although the
uncertainty in this slope is large (nominally $\pm 1.2$, if the uncertainty in
the secular flux density decrease is taken as 0.2\% year$^{-1}$, and the expansion
timescale of the remnant is between 400 to 500 years).
\subsection{An Aside: Extragalactic Selection Effects}\label{s:extragalactic}
As noted above, a major problem with statistical studies of Galactic SNRs is
the difficulty of obtaining reliable distances for remnants. Studies of samples
of remnants in external galaxies are more straightforward in this respect, as
all the remnants are at a very similar distance. Given this, then it is
arguably more appropriate to consider the $L{-}D$ relation, rather than the
${\mathit\Sigma}{-}D$, for extragalactic SNRs -- as the latter is not needed to
determine distances for individual remnants -- with the appropriate selection
effects. Then, for unresolved sources, in most radio studies the dominant
selection effect is a flux density limit, which corresponds to a fixed
luminosity limit for a particular galaxy.
In a recent study of the statistical properties of extragalactic SNRs in
several galaxies -- in addition to those in the Milky Way -- Arbutina et al.\
(2004) concluded that only in the case of M82 was there a good $L{-}D$
correlation, and hence a useful ${\mathit\Sigma}{-}D$ relation also. In other cases
there was a poor correlation between the luminosity and diameter of identified
SNRs (as noted in Section~\ref{s:sigmad} above for Galactic SNRs). However, in
their study Arbutina et al.\ have not correctly appreciated the observational
selection effects applicable to the sample of SNRs in M82. This sample of SNRs
was identified by Huang et al.\ (1994) from observations at 8.4~GHz. In Fig.~1
of Arbutina et al.\ a sensitivity limit corresponding to a luminosity of
$\approx 7 \times 10^{24}$ erg s$^{-1}$ Hz$^{-1}$ is plotted, which although
appropriate at 8.4~GHz, is not appropriate for the luminosities of the M82
sample of SNRs plotted in this figure, which are at 1~GHz. The true luminosity
limit from Huang et al.'s observations should be moved upwards on Arbutina et
al.'s Fig.~1 by $\approx 3$ (i.e.\ $\approx 8.4^{0.5}$ for a typical spectral
index of 0.5 to correct from 8.4 to 1~GHz). Moreover, the actual observational
selection effects provide a more stringent limit on the detectable luminosities
of the larger SNRs in the sample. Any remnants with a diameter of larger than
$\approx 3$~pc were resolved by Huang et al.'s observations, and a surface
brightness limit (equivalent to luminosity scaling as $D^2$), not a constant
luminosity limit is appropriate (see Fig.~2 of Huang et al., who show these
limits on a ${\mathit\Sigma}{-}D$ rather than a $L{-}D$ plot). Consequently, the
apparent range of luminosities of SNRs in M82 is strongly limited by selection
effects, particularly for the larger remnants. Indeed, it is noticeable that
the range of luminosities shown for the larger remnants in M82 appears smaller
than that of the smaller remnants, which seems unlikely to be real. Thus the
range of luminosities of SNRs in M82 is likely to extend to lower values, but
these objects have not been identified due to selection effects. In this case
the correlation between luminosity and diameter for remnants here is not
strong, as is the case in other galaxies, including our own. Consequently, the
${\mathit\Sigma}{-}D$ relation for M82 is also affected by selection effects, and is
therefore of limited use.
\section{Galactic SNR Distribution}\label{s:distribution}
The distribution of SNRs in the Galaxy is of interest for many astrophysical
studies, particularly in relation to their energy input into the ISM and for
comparison with the distributions of possible progenitor populations. Such
studies are, however, not straightforward, due to observational selection
effects and the lack of reliable distance estimates available for most identified
remnants. In particular, all SNRs in the anti-centre (i.e.\ 2nd and 3rd
Galactic quadrants) are outside the Solar Circle, at large Galactocentric
radii, in regions where the background Galactic emission is low, so that low
surface brightness remnants are relatively easy to identify (see
Section~\ref{s:selection}). Without taking selection effects into account, the
larger number of fainter SNRs in the anti-centre leads to an apparently broad
distribution of Galactic SNRs in Galactocentric radius (e.g.\ the very broad
distribution of SNRs derived by Li et al.\ (1991), who included all SNRs in
their analyses). A more complicated method to derive the radial distribution of
Galactic SNRs is that used by Case \& Bhattacharya (1996, 1998), following a
method used by Narayan (1987) for pulsars. This relies on (i) assuming
catalogues of Galactic SNRs are complete for SNRs within a distance of 3~kpc;
(ii) using ${\mathit\Sigma}{-}D$ derived distances for the SNRs, and (iii) attempts to
correct for observational selection effects using a scaling factor that varies
in many bins across the disc of the Galaxy. However, this is difficult given
the uncertainties in the usefulness of ${\mathit\Sigma}{-}D$ relation discussed above,
and the necessity of deconvolving selection effects from the observed
distribution of SNRs. An alternative approach is to investigate the
distribution of SNRs in Galactic coordinates, restricting the studies to
relatively bright remnants, for which current catalogues are thought to be
complete. van den Bergh (1988a,b) discussed the distribution of observed SNRs
and noted that high surface brightness remnants (in this case taken to be
${\mathit\Sigma}_{\rm 1~GHz} > 3 \times 10^{-21}$ W m$^{-2}$\,Hz$^{-1}$\,sr$^{-1}$) are concentrated in a
thin nuclear disc when plotted in Galactic coordinates. As noted by F\"urst's
comments to van den Bergh (1988b), this conclusion is strengthened by a more
realistic surface brightness completeness limit.
\begin{figure}
\centerline{\includegraphics[width=8cm,clip=]{hist-l-all}}
\quad\\
\centerline{\includegraphics[width=8cm,clip=]{hist-l-bright}}
\caption{The distribution in Galactic longitude of (top) all 231 Galactic SNRs,
and (bottom) the 64 high surface brightness SNRs with ${\mathit\Sigma}_{\rm 1~GHz} >
10^{-20}$ {W m$^{-2}$\,Hz$^{-1}$\,sr$^{-1}$}. Each plot shows as a solid line a histogram of the
observed longitudes of the remnants (left scale), and as a dotted line for
cumulative fraction (right scale).\label{f:all-bright}}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=8cm,clip=]{model-l-050}}
\quad\\
\centerline{\includegraphics[width=8cm,clip=]{model-l-065}}
\quad\\
\centerline{\includegraphics[width=8cm,clip=]{model-l-080}}
\caption{Model distribution in Galactic longitude of Gaussian distributions of
SNRs with three different Galactocentric radius scale lengths (cf.\ the
observed distribution in Fig.~\ref{f:all-bright}).\label{f:model}}
\end{figure}
More quantitatively -- following the method of Li et al., but using an
appropriate selection brightness cut-off -- the observed distribution of bright
SNRs with Galactic longitude can be compared with that expected from various
models. A major advantage of this method is that it avoids the problem that we
lack accurate distances to individual SNRs, although on the other hand it uses
only a sub-set of the known Galactic SNRs. Fig.~\ref{f:all-bright} shows the
observed distributions with Galactic longitude of all Galactic SNRs, and of the
64 remnants which have ${\mathit\Sigma}_{\rm 1~GHz} > 10^{-20}$ {W m$^{-2}$\,Hz$^{-1}$\,sr$^{-1}$} (this is
a similar number to the 71 remnants used in a similar study presented in Green
1996a, which used a slightly lower surface brightness cut-off applied to the
SNR catalogue of Green 1996b). By applying the surface brightness cut-off, so
that the surface brightness selection effect is not important, it is clear that
the distribution of Galactic remnants is actually much more concentrated
towards $l=0^\circ$ than if all remnants are considered (cf.\ Fig.~\ref{f:lb}).
Fig.~\ref{f:all-bright} shows evidence for a deficit of SNRs near
$l=350^\circ$, which may be a true deficit if there is a decrease in the space
density of SN progenitors towards the Galactic centre. However, it may also be,
in part at least, due to the difficulty of finding remnants in this region of
the Galactic plane, due to the very complex background emission and confusing
Galactic sources (e.g.\ {{H\,{\sc ii}}} regions). Any remaining incompleteness in
current catalogues, both for the surface brightness and angular diameter
selection effects, are expected to be worse closer to $b=0^\circ$ (because of
the increased confusion in the case of the surface brightness selection effect,
and the longer line-of-sight through the Galaxy for missing small, i.e.\ young
but distant remnants). Thus, the true distribution in $l$ is likely to be
somewhat {\sl narrower} than is indicated in Fig.~\ref{f:all-bright}. For
comparison with the observed distributions in Galactic longitude, simple Monte
Carlo models of the distribution of SNRs in the disc of the Galaxy were
constructed assuming a simple, circularly symmetric, Gaussian distribution,
where the probability distribution varies with Galactocentric radius, $R$, as
$$
\propto {\rm e}^{-(R/\sigma)^2},
$$
(where $\sigma$ is the Gaussian Galactocentric scale length, assuming the
distance to the Galactic Centre is 8.5~kpc). Fig.~\ref{f:model} shows plots of
the expected distribution of SNRs in Galactic longitude of three such models
for different scale lengths. As noted above, the true distribution is likely to
be somewhat narrower than that derived from the observations, due to residual
selection effects, so that this scale length is an upper limit. A $\chi^2$
comparison of the observed and model cumulative distributions indicates that
for this simple model, a scale length of $\approx 6.5$~kpc best matches the
observed distribution of high brightness SNRs.
The model distribution of SNRs derived above should, however, be interpreted
cautiously, as not only is it a simplistic model without spiral arm structures,
but also it is a model of the distribution of {\em observed} remnants. It is
far from clear what factors affect the brightness and lifetime of appreciable
radio emission from SNRs -- i.e.\ their observability at radio wavelengths --
and hence how close the distribution of observed SNRs is to the parent
supernovae distribution. The distribution of SNRs could reflect the
distribution of, for example, the density of the ISM, or the Galactic magnetic
field, if these are important factors in determining the brightness and lifetime
of radio emission from SNRs.
\section{Conclusions}
Here I have presented a recent catalogue of 231 Galactic SNRs, and have
discussed the selection effects that apply to the identification of remnants.
Both surface brightness and angular size selection effects are important, and
these need to be borne in mind when statistical studies of Galactic SNRs are
made. One consequence of the current angular size selection effect is that few
young but distant remnants have yet been identified in the Galaxy. These
objects are likely to be in complex regions of the Galactic plane, and further
observations -- using a wide range of radio wavelengths and/or the combination
of radio and other wavelengths -- are required to identify these missing
objects. For remnants with known distances, the intrinsic range of luminosity
of Galactic SNRs is large, which combined with selection effects, means that
the ${\mathit\Sigma}{-}D$ relation is of limited use for determining distances to
individual remnants, or for statistical studies.
\section*{Acknowledgements}
I am grateful to many colleagues for numerous comments on, and corrections to,
the various versions of the Galactic SNR catalogue, and for comments on early
versions of this work. I also thank the referee, Chris Salter, for his useful
and detailed report. This paper was finalised during an extended visit to the
National Radio Astronomy Observatory, Socorro, NM, USA in 2004 September, and I
am very grateful for their hospitality at that time. This research has made use
of NASA's Astrophysics Data System Bibliographic Services.
|
{
"timestamp": "2004-11-04T11:28:37",
"yymm": "0411",
"arxiv_id": "astro-ph/0411083",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411083"
}
|
\section{Introduction} \label{I}
Despite being discovered more than 30 years ago
\citep{Kleb-Stro-Olso:73}, and the fact that bursts have been detected
daily since the advent of the Compton Gamma-Ray Observatory in 1991,
there still does not exist a satisfactory theoretical model which
encompasses the entire $\gamma$-ray burst event. Nonetheless, in
spite of their strong heterogeneity, the abundance of observations has
led to a well described burst phenomenology, and is briefly summarised
below.
Burst durations are bimodal, with long bursts lasting $\sim100\,\s$
while short bursts last $\sim1\,\s$ \citep{Kouv-etal:93}. Long bursts typically
have inferred isotropic luminosities on the order of $\sim
10^{51-52}\,\erg/\s$ \citep[see, \eg,][and references
therein]{Piran:04}, although achromatic breaks in the afterglow
light curves suggest strong collimation \citep[with opening angles
$\sim5-10^\circ$,][]{Frai-etal:01} which, in turn, implies actual
luminosities on the order of $\sim 10^{48-49}\,\erg/\s$. The majority
of the prompt emission is in the form of apparently nonthermal
$\gamma$-rays, well fit by a broken power law, typically peaking near
$100$--$1000\,\keV$, and softening throughout the burst. The prompt
emission is composed of a large number of subpulses with typical
widths on the order of a second in which lower energy emission lag
behind, and are wider than, the higher energy emission.
Long bursts are followed by optical and radio afterglows thought to be
generated by a hot fireball interacting with the interstellar medium
\citep[see, \eg,][]{Mesz:02,Li-Chev:03}. These can
last many weeks and have in at least two cases (\mbox{GRB 980425} and
\mbox{GRB 030329}) included type Ib,c supernova (\mbox{SN 1998bw} and
\mbox{SN 2003dh}, respectively) light curves $7$--$10\,{\rm days}$
after the prompt emission. The total bolometric energy in these
bursts has been estimated from late time radio observations to be
$\sim 10^{51}\,\erg$ and roughly constant among bursts
\citep{Berg-Kulk-Frai:04}. To date, there have been no instances of
afterglow or supernova being associated with short bursts.
Due to similarities in their temporal structure, duration, and
spectra, X-ray flashes appear to be low luminosity cousins of
$\gamma$-ray bursts. Recently it has been shown that their bolometric
luminosity is indeed comparable to normal bursts \citep{Sode-etal:04}.
Despite this, their inferred isotropic energy is substantially less
($\sim10^{49}\,\erg$) placing X-ray flashes upon the peak--inferred
isotropic energy relation discovered by \citet{Amat-etal:02},
and supporting a unified interpretation of X-ray flashes and
$\gamma$-ray bursts \citep{Saka-etal:04}.
A number of mechanisms have been suggested for $\gamma$-ray bursts.
Frequently, these address either the central engine and emission
separately. Central engine models include black hole and/or neutron star
collisions
\citep{Eich-Livi-Pira-Schr:89,Pacz:91,Nara-Pacz-Pira:92},
magnetar birth \citep{Usov:92,Thom:94,Thom-Chan-Quat:04}, black hole
birth \citep{Viet-Stel:98}, and collapsar
\citep[see, \eg,][]{MacF-Woos-Hege:01} models. In most of these it is
not clear how the spectral and temporal structure of the burst is
produced. In contrast, there are also a number of models which focus
upon the emission. These include the cannonball
\citep[][and references therein]{Dado-Dar-DeRu:02},
shot gun \citep{Hein-Bege:99}, Comptonised jet
\citep{Thom:94,Ghis-Lazz-Celo-Rees:00}, and internal shock
\citep{Piran:04} models. Despite their ability to reproduce many of
the observed spectral and temporal features of the bursts, few of
these address directly the power source of the burst. As a result,
currently there are few models which can explain both the properties
of the observed emission and the prodigious energy of observed bursts
in a satisfactory manner.
Here a model involving supernovae in helium star--black hole and
helium star--neutron star binaries is presented. In this scenario the
supernova results in a transient period of rapid accretion onto the compact object, extracting
via magnetic torques its rotational energy at highly super-Eddington
luminosities in the form of a narrowly beamed, strongly
electromagnetically dominated jet. The prompt emission is produced by
Compton scattering supernova photons advected within the ejecta and
photons created at shocks driven into the ejecta by the jet. The
duration of the burst is limited by the rate of Compton cooling of the
jet, eventually creating an optically thick, moderately
relativistically expanding fireball which can produce the afterglow
emission. In comparison with the previous models this has two
advantages: firstly it provides a unified model for the central engine
and the prompt emission, and secondly, since helium star--neutron star
binaries are widely believed to be the progenitors of the observed
double pulsars, it is assured that supernova in these systems will
occur in sufficient quantity (see section \ref{PE:PS}). For
convenience, typical values for the pertinent quantities which
describe the model are collected in Table \ref{quantities} and a
schematic of the process is shown in Figure \ref{cartoon}.
The formation and energetics of the jet are discussed in Section \ref{JF}. Sections \ref{PE} and \ref{Ag}
describe the mechanisms by which the kinetic energy of the jet is
converted in the the observed prompt emission and subsequent
afterglow, including the limits this places upon the jet
dynamics. Expected observational implications of this model are
discussed in Section \ref{MI}. Finally, concluding remarks are
contained in Section \ref{C}.
\begin{table}
\begin{center}
\begin{tabular}{lcc}
\hline
Helium Star Mass & $M_{\rm He}$ & $2\,\Ms$\\
Helium Star Radius & $R_{\rm He}$ & $2\times10^{11}\,\cm$\\
Orbital Separation & $a$ & $4\times10^{11}\,\cm$\\
Supernova Ejecta Density & $\rho_{\rm sn}$ & $10^{-2}\,\g/\cm^3$\\
Supernova Ejecta Velocity & $v_{\rm sn}$ & $10^3\,\km/\s$\\
\hline
Neutron Star Mass & $M_{\rm NS}$ & $1.4\,\Ms$\\
Neutron Star Radius & $R_{\rm NS}$ & $10^6\,\cm$\\
Black Hole Mass & $M_{\rm BH}$ & $10\,\Ms$\\
\hline
Jet Lorentz Factor & $\Gamma$ & $20$\\
Jet Plasma $\sigma$ & $\sigma$ & $10^4$\\
\hline
\end{tabular}
\caption{A number of model parameters are listed together with some of
the jet characteristics obtained in the following sections.}
\label{quantities}
\end{center}
\end{table}
\begin{figure*}
\begin{center}
\includegraphics[width=1.5\columnwidth]{fig1.eps}
\caption{A cartoon of the mechanism is shown (not to scale) with the
principle features, supernova ejecta, accretion driven jets,
up-scattered entrained and shock produced photons from the ejecta,
and the regions where the jet is axially optically thin
($r\sim10^{11}\,\cm$) and becoming nonrelativistic.}
\label{cartoon}
\end{center}
\end{figure*}
\section{Jet Formation} \label{JF}
A high power jet (or highly collimated relativistic outflow) is a
central feature of most $\gamma$-ray burst models.
Observational constraints require that the luminosity of a jet with
opening angle $\sim \Gamma^{-1}$ (where $\Gamma$ is the bulk Lorentz
factor) be on the order of
\begin{equation}
L_{\rm jet} \simeq \frac{1}{4\Gamma^2} L_{\rm iso}
\simeq 6\times10^{47} \left(\frac{\Gamma}{20}\right)^{-2}
\left(\frac{L_{\rm iso}}{10^{51}\,\erg/\s}\right)
\,\erg/\s\,,
\label{JF:jet_luminosity}
\end{equation}
where $L_{\rm iso}$ is the isotropic equivalent luminosity of the
burst. For a burst of duration $T$, this corresponds to a total
energy budget of
\begin{equation}
E_{\rm tot} \simeq 6\times10^{49} \left(\frac{\Gamma}{20}\right)^{-2}
\left(\frac{L_{\rm iso}}{10^{51}\,\erg/\s}\right)
\left(\frac{T}{100\,{\rm s}}\right)
\,\erg \,.
\label{JF:burst_energy}
\end{equation}
The observational features of the model presented here depend only
upon the existence of a suitably luminous and electromagnetically pure
jet (see, \eg, section \ref{PE:IfJT}) in the vicinity of the
supernova, and are otherwise independent of the mechanism by which
such a jet is produced. Nonetheless, for concretness, here the jet is
presumed to be the result of accretion driven electromagnetic torques
upon a central compact object. Therefore, possible mechanisms for the
production of such a jet are discussed below in the cases where the
compact object is a black hole or a neutron star. However, this is in no means
meant to be an exhaustive discussion of the problem of jet formation,
which is beyond the scope of this paper.
\subsection{Black Hole Binaries}
A considerable literature exists regarding the production of
jets via the electromagnetic extraction of angular momentum from
accreting black holes
\citep[see, \eg,][and references therein]{Blan-Znaj:77,McKi-Gamm:04,Komi:04,Levi:05}.
The maximum energy that may be extracted by this method is determined by the
irreducible mass, $M_{\rm ir}$:
\begin{align}
E_{\rm tot}
&= M_{\rm BH} c^2 \left( 1 - \frac{M_{\rm ir}}{M_{\rm BH}} \right) \nonumber\\
&\simeq \frac{1}{8} j^2 M_{\rm BH} c^2\nonumber\\
&\simeq 10^{55} j^2\,\erg\,.
\end{align}
where $j$ is the dimensionless angular momentum of the black hole, and
the expansion is appropriate for $j\ll1$
\citep[see, \eg,][]{Misn-Thor-Whee:73}. Comparing this to the energy
required to power a burst (\cf equation \ref{JF:burst_energy}) requires
$j\gtrsim0.01$, and thus puts a rather weak limit upon the spin of the
black hole.
In the model presented here, the torque upon the black hole is due to
a substantially super-Eddington accretion flow resulting from the
accretion of the supernova ejecta. An estimate for the mass accretion
rate in this scenario is simply given by the Bondi-Hoyle rate,
\begin{align}
\dot{M}
&\simeq
\pi r_{\cal BH}^2 \rho_{\rm sn} v_{\rm sn}
\simeq
4 \pi \rho_{\rm sn} \left(\frac{G M_{\rm BH}}{c^2}\right)^2 \left(\frac{v_{\rm sn}}{c}\right)^{-3} c \nonumber \\
&\simeq
2\times10^{29}\,\g/\s\,,
\label{JF:BH:mass_accretion_rate}
\end{align}
where the Bondi-Hoyle radius is given by
\begin{equation}
r_{\cal BH}
=
\frac{2 G M_{\rm BH}}{v_{\rm sn}^2 + c_s^2}\,,
\end{equation}
in which the sound speed ($c_s$) may typically be ignored compared to the
ejecta bulk velocity. The resulting density in the vicinity of the
horizon is then
\begin{equation}
\rho
=
\frac{\dot{M}}{4 \pi r_h^2 c}
\simeq
\left(\frac{v_{\rm sn}}{c}\right)^{-3} \rho_{\rm sn}\,,
\end{equation}
where $r_h$ is the horizon radius. This implies a magnetic field
strength on the order of
\begin{align}
B_{\rm BH} &\simeq \sqrt{\rho_{\rm sn} c^2} \left(\frac{v_{\rm sn}}{c}\right)^{-3/2}\nonumber\\
&\simeq 2\times10^{13}\,\G\,,
\end{align}
presumably created via the magneto-rotational instability (MRI)
\citep{Hawl-Balb:95}.
The rotational energy may then be electromagnetically tapped by, \eg,
the Blandford-Znajek process \citep{Blan-Znaj:77}. Recent numerical
simulations of accreting black holes have produced Poynting flux
dominated jets with typical luminosities of
\begin{align}
L_{\rm BH}
&\simeq \Omega_{\rm BH}^4 B_{\rm BH}^2 \frac{1}{c^3}
\left(\frac{GM_{\rm BH}}{c^2}\right)^6\nonumber\\
&\simeq 2\times10^{48} j^4\,\erg/\s\,,
\end{align}
where $\Omega_{\rm BH} = j c / 2 r_h$ is a measure of the angular
velocity of the black hole \citep{McKi:05}. Therefore, rapidly
rotating black holes \cite[$j\gtrsim0.8$ for the black hole parameters
given here, \cf][]{Gamm-Shap-McKi:04} are easily capable of
producing a Poynting dominated jet of sufficient luminosity.
This scenario is similar to the central engine models for the
collapsar scheme \cite[see, \eg,][]{Poph-Woos-Frye:98}, the primary
difference being the accretion rate. In both cases, the accretion
flow is neutrino cooled, and thus able to reach substantially
super-Eddington (\cf~with $10^{18}\,\g/\s$) accretion rates. However,
whereas the typical collapsar accretion rates are on the order of
$1 M_\odot\,\s^{-1}$, here they are a considerably lower
$10^{-4} M_\odot\,\s^{-1}$.
\subsection{Neutron Star Binaries}
A second, and perhaps more speculative, mechanism by which the jet might be formed
involves a rapidly rotating neutron star. As with the black hole, in
this scenario the energy is provided by the spin of the star. The
total rotational energy available is
\begin{align}
E_{\rm tot} &= \frac{1}{2} I_{\rm NS} \Omega_{\rm NS}^2
\simeq \frac{1}{5} M_{\rm NS} R_{\rm NS}^2 \Omega_{\rm NS}^2
= \frac{G M_{\rm NS}^2}{5 R_{\rm NS}} \omega^2\\ \nonumber
&\simeq 10^{53} \omega^2 \,\erg \,,
\end{align}
where $\omega$ is the angular velocity in units of the breakup
velocity. In order to be sufficient to power a burst this already
requires that $\omega \gtrsim 0.03$ and hence the neutron star must be
rapidly rotating ($P\lesssim 15\,\ms$) consistent with observations of
millisecond pulsars.
As with the black hole, the accretion of the supernova ejecta may
mediate the extraction of the rotational energy. In the neutron star
case, the Bondi-Hoyle accretion rate is given approximately by
\begin{equation}
\dot{M} \simeq 5\times10^{27}\,\g/\s\,,
\end{equation}
at which, as in the black hole case, neutrino cooling dominates
\citep{Chev:89,Frye-Benz-Hera:96}.
Near the neutron star, this results in a density on the order of
\begin{equation}
\rho
\simeq
\frac{\dot{M}}{4\piR_{\rm NS}^2 v_r}
\simeq
10^4 \beta_r^{-1} \,\g\,\cm^{-3}\,,
\end{equation}
where $v_r \equiv \beta_r c$ is the infall velocity near the neutron
star surface. This implies an accretion magnetic field strength on the
order of
\begin{equation}
B_{\rm NS} \simeq 4\times10^{12} \beta_r^{-1/2}\,\G\,.
\end{equation}
Note that this is a lower limit as the radial infall velocity may be
substantially less than the speed of light. This field will reconnect with
the native stellar field on time scales comparable to the infall time
scale \citep{Ghos-Lamb:79b}. Therefore, it may not be necessary for the
native field of the neutron star to initially be dynamically
significant for it to subsequently strongly couple the star to the disk.
The resulting spin-down torque has been studied primarily in the
context of propellor accretion flows
\citep[see, \eg,][]{Ghos-Lamb:79b,Eksi-Hern-Nara:05,Roma-Usty-Kold-Love:04}.
Unfortunately, the magnitude of the spin down-torque is not a settled
issue. Nonetheless, recent numerical and analytical efforts have
implied that
\begin{equation}
\mathcal{M} \simeq B_{\rm d}^2 R_{\rm d}^2 v_{\rm esc} \eta \left(1-\eta\right)\,,
\end{equation}
where $R_{\rm d}$ is the radius at which the magnetic field is
dominated by the inertia of the disk, $B_{\rm d}$ is the magnetic
field at this radius, $v_{\rm esc}$ is the escape velocity from this
radius, and $\eta$, the so-called fastness parameter, is the
rotation frequency of the neutron star in units of the Keplerian
velocity at the dissipation radius
\citep{Roma-Usty-Kold-Love:04,Eksi-Hern-Nara:05}. Clearly the minimum
dissipation radius is the radius of the neutron star, and thus the
maximum {\em possible} luminosity is
\begin{equation}
L_{\rm NS max}
\simeq B_{\rm NS}^2 R_{\rm NS}^2 c\,\omega \left(1-\omega\right)\,,
\end{equation}
which is sufficient to drive a burst provided that
$B\gtrsim10^{13}\,\G$ and $\omega$ is near unity ($P\sim1\,\ms$). The
former is satisfied if $\beta_r\simeq0.1$. The latter must be
satisfied for a propellor to operate if the dissipation radius is near
the surface, placing a more stringent constraint upon the spin of the
star. It should be noted that such a high spin may be expected in
neutron star--helium star binaries if the helium star's hydrogen
envelope was expelled due to accretion by the neutron star or common
envelope evolution, as is commonly thought to be the case in such
systems \citep{Beth-Brow:98}.
In the standard propeller theory the dissipation radius is assumed to
be approximately the Alfv\'en radius, where the pressure from the
neutron star's native field balances the accretion ram pressure.
However, in the hyperaccreting millisecond pulsar case
(considered here) the dominant magnetic field is the accretion flow
field. Therefore, this case may be more analogous to accreting black
holes (in which the dissipation radius is roughly that of the horizon)
than to accreting pulsars.
The manner in which the dissipated energy is transported out of the
system is not clear a priori. Numerical simulations of propellors
have found the development of a magnetic chimney, suggestive of the
development of a jet, suggesting that some fraction of the
energy leaves in a Poynting flux \citep{Roma-Usty-Kold-Love:04}.
This interpretation is supported by the obvious parallels with the
accreting black hole case, in which a significant fraction of the
spin-down energy is channeled into a Poynting jet. However, a
definitive answer will depend upon the details of the accretion flow
and the magnetic coupling to the star, and will likely have to await
detailed numerical simulations.
\subsection{Jet Emergence} \label{JF:JE}
In order to have observational consequences it is necessary for the
jet to emerge from the growing supernova. This is analogous to the
problem of jet emergence in the collapsar model
\citep[see, \eg,][]{MacF-Woos-Hege:01} with a
reduction of the external medium density by eight orders of magnitude.
The ram pressure of the jet,
\begin{align}
P_{\rm jet} &\simeq \frac{\Gamma^2 L_{\rm jet}}{3\pi r^2 c}
\simeq \frac{L_{\rm iso}}{12\pi r^2 c} \\ \nonumber
&\simeq 1\times10^{15}
\left(\frac{L_{\rm iso}}{10^{51}\,\erg/\s}\right)
\left(\frac{r}{10^{12}\,\cm}\right)^{-2}
\,\erg/\cm^3 \,,
\end{align}
can be compared to the typical pressures in the supernova ejecta,
\begin{equation}
P_{\rm sn} \simeq \frac{1}{6} \rho_{\rm sn} v_{\rm sn}^2
\simeq 2\times10^{13}\,\erg/\cm^3 \,,
\end{equation}
(the thermal photon pressure will be comparable at the orbital
separations considered here) and hence the jet will easily escape the
ejecta.
\section{Prompt Emission} \label{PE}
The presence of a jet alone is insufficient to produce a $\gamma$-ray
burst. Also required is a mechanism by which the considerable kinetic
energy flux of the jet can be converted into the observed prompt
emission. Many such mechanisms have been discussed in the
literature. However, for highly beamed emission, a minimum
requirement is that the jet be optically thin along the jet axis.
This may be accomplished in a number of ways, including clumpy jets.
However, in the context of the model considered here, in which a large
number of seed photons are available entrained in the supernova
ejecta, this immediately suggests inverse-Compton scattering as the prompt
emission mechanism. This has the considerable advantage over internal
shock scenarios of being capable of converting the kinetic energy of
the jet into the prompt emission at efficiencies approaching unity
\citep[see, \eg,][]{Lazz-Ghis-Celo:99}.
\subsection{Photon Collimation} \label{PE:PC}
Due to the high bulk Lorentz factor, the scattered seed photons will
naturally be collimated to within $\Gamma^{-1}$ of the jet axis
\citep[see, \eg,][]{Bege-Siko:87}. However, in addition, there are
optical depth effects which will also serve to beam the scattered
photons.
In the rest frame of the jet electrons, the Thomson depth is given by
\begin{equation}
\tau = \int \sigma_T n_e c \d t' \,.
\end{equation}
When transformed into the lab frame this gives
\begin{align}
\tau &= \int \sigma_T n_e c \Gamma \left(1-\beta\cos\theta\right) \d t
\\ \nonumber
&=\int \sigma_T n_e \Gamma \left(1-\beta\cos\theta\right) \d r \,,
\end{align}
where in both cases $n_e$ is the proper electron number density.
For the two limiting cases of across the jet ($\theta\gg\Gamma^{-1}$)
and along the jet ($\theta\lesssim\Gamma^{-1}$), the optical depth is
\begin{align}
\tau_\parallel &= \frac{1}{2\Gamma}\sigma_T n_e r && \theta=0 \nonumber\\
\tau_\perp &= 2\sigma_T n_e r && \theta=\frac{\pi}{2}
\,,
\end{align}
where $n_e \propto r^{-2}$ (changing the power law index changes
$\tau_\parallel$ by factors of order unity) was assumed and a jet
width of $2 r/\Gamma$ was used. Therefore, despite the considerable
difference in scale length, the optical depth along the jet is a
factor of $\Gamma$ less than that across the jet. As a result, it is
possible to have a jet which is optically thin to photons within $\sim
\Gamma^{-1}$ of the jet axis while being optically thick to all
others. This provides an additional collimating mechanism and
explains why it is possible to Compton scatter a large number of the
seed photons incident on the jet while allowing the scattered burst
photons to escape.
\subsection{Implications for Jet Type} \label{PE:IfJT}
The requirement that at interesting radii $\tau_\parallel<1$
limits the baryon loading of the jet. The power associated with ions
in the jet is
\begin{align}
L_{\rm ions} &\simeq \frac{\pi r^2}{\Gamma} n_e m_p c^3
\simeq \frac{2 \pi r}{\sigma_T} m_p c^3 \tau_\parallel
\\ \nonumber
&\simeq 10^{-4} \left(\frac{\Gamma}{20}\right)^2
\left(\frac{r}{10^{11}\,{\rm cm}}\right)
\tau_\parallel L_{\rm jet}\,.
\end{align}
Hence, the condition that the jet be optically thin along it's axis at
$r\sim10^{11}\,{\rm cm}$ requires that in the jet,
\begin{align}
\sigma \equiv \frac{L_{\rm jet}}{L_{\rm ions}} - 1 \gtrsim 10^{4}\,,
\end{align}
where $\sigma$ is the ratio the Poynting flux to the kinetic energy
flux of the baryons. Therefore, the jet must be extremely
electromagnetically dominated. It should be noted however that
observational precedent for such an electromagnetically dominated
outflow exists in the context of the Crab pulsar in which
$\sigma\sim10^6$ at these radii \citep[see, \eg,][]{Vlah:04,Komi-Lyub:04}.
The magnetic fields within the jet required to generate a Poynting
flux with the $\gamma$-ray burst luminosities will typically be of order
\begin{align}
B &\simeq \sqrt{2 \frac{L_{\rm iso}}{r^2 c}}\nonumber\\
&\simeq 3\times10^9
\left(\frac{L_{\rm iso}}{10^{51}\,\erg/\s}\right)^{1/2}
\left(\frac{r}{10^{11}\,\cm}\right)^{-1}\,\G \,,
\label{B_estimate}
\end{align}
for $r\gg R_{\rm NS}$\footnote{The conical structure of
the jet will not necessarily extrapolate down to the acceleration
region which is expected to occur on a scale of many gravitational
radii. If it were this would suggest magnetic field strengths on the
order of $10^{15}\,\G$, as is suggested in magnetar models of
$\gamma$-ray bursts as opposed to the far more conservative
${\rm few}\times10^{13}\,\G$ discussed here.}.
The resulting synchrotron cooling time for an electron with Lorentz
factor $\gamma$ in the frame of the jet is then
\begin{align}
t_{\rm syn} &\simeq \frac{\gamma m_e c^2}{P_{\rm syn}}
\simeq 6\pi\frac{m_e c^2}{\sigma_T c B^2} \gamma^{-1}
\\ \nonumber
&\simeq 10^{-10}
\left(\frac{L_{\rm iso}}{10^{51}\,\erg/\s}\right)^{-1}
\left(\frac{r}{10^{11}\,{\rm cm}}\right)^2
\gamma^{-1}\,\s\,,
\end{align}
hence these electrons may always be treated as cold in this frame.
\subsection{Compton Seed Photons}
From momentum conservation, the energy of a photon after a single
Compton scatter is given by
\begin{equation}
\epsilon_f = \frac{1-\beta\cos\theta_i}
{1-\beta\cos\theta_f+
\epsilon_i\left(1-\cos\Theta\right)/\Gamma m_e c^2}
\epsilon_i\,,
\end{equation}
where $\theta_{i,f}$ are the initial and final photon propagation angle
with respect to the jet axis, and $\Theta$ is the angle between the
initial and final photon propagation direction. For photons scattered
to within $\Gamma^{-1}$ (as expected from relativistic beaming and
optical depth collimation), the resulting energy is given
approximately by
\begin{equation}
\epsilon_f = \min\left(2 \Gamma^2 \epsilon_i,\Gamma m_e c^2\right) \,.
\end{equation}
Therefore, in order to generate a burst with an isotropic luminosity
of $L_{\rm iso}$, a luminosity of
$L_{\rm iso}/8\Gamma^4\simeq10^{45}(\Gamma/20)^{-4}\,\erg/\s$ of seed photons
are required to impinge upon the jet.
There are two sources of seed photons, the thermal photons from the
supernova itself and those produced by strong shocks driven into the
ejecta by the jet. In the first case, due to the ejecta's high
Thomson depth, these will be entrained in, and in thermal equilibrium
with, the ejecta. Thus, assuming the ejecta cools adiabatically as it
expands, the photon temperature will be
\begin{equation}
T_\gamma
\simeq \frac{m_p v_{\rm sn}^2}{3 k} \left(\frac{r}{R_{\rm He}}\right)^{-2}
\simeq 4 \left(\frac{r}{R_{\rm He}}\right)^{-2}\,\keV\,,
\end{equation}
and hence the typical photon energy is on the order of $1\,\keV$ for
an orbital separation of $\gtrsim 2 R_{\rm He}$ which would be
expected if the Helium star is filling its Roche lobe. The associated
energy density of photons is
\begin{equation}
u_T = a T_\gamma^4 \sim 2\times10^{16} \left(\frac{r}{R_{\rm He}}\right)^{-8}
\sim 10^{14}\,\erg/\cm^3\,,
\end{equation}
(where here $a$ is the Stefan-Boltzmann constant and not to be
confused with the orbital separation) and thus the luminosity of
thermal seed photons available to the jet is approximately
\begin{equation}
L_T \sim u_T v_{\rm sn} \frac{2 r^2}{\Gamma}
\sim 10^{45}\left(\frac{r}{10^{12}\,\cm}\right)^2
\left(\frac{\Gamma}{20}\right)^{-1}\,\erg/\s \,,
\end{equation}
where clearly this depends upon the alignment of the jet relative to
the orbital plane through the dependence of $u_T$ upon the jet
position. In the second case, the kinetic energy of the ejecta is
thermalised at the jet boundary providing a seed photon luminosity of
roughly
\begin{align}
L_S
&\sim \frac{1}{2} \rho_{\rm sn} v_{\rm sn}^3 \frac{2 r^2}{\Gamma}
\\ \nonumber
&\sim 10^{45} \left(\frac{r}{10^{12}\,\cm}\right)^2
\left(\frac{\Gamma}{20}\right)^{-1}\,\erg/\s \,,
\end{align}
which is comparable to $L_T$. Depending upon the shock structure and
location these may also be thermalised. Due to its weaker
dependence upon $r$ ($\rho_{\rm sn}\propto r^{-3}$) this contribution will
dominate at large radii. The total luminosity of seed photons entering
the jet is then simply
\begin{equation}
L_{\rm seed} = L_T + L_S \,,
\end{equation}
which is sufficient.
\subsection{Implications for Jet Lorentz Factor} \label{PE:IfJLF}
Recent simulations suggest that in vacuum it would be expected that
the jet would accelerate due to magnetic stresses until it was no
longer electromagnetically dominated, \ie, $\Gamma$ would approach
$\sigma\sim10^4$ \citep{Spit-Aron:04}\footnote{Note that this is in contrast to
many of the theoretical estimates in the literature in which
behaviour expected by one-dimensional models is
$\Gamma\rightarrow\sigma^{1/3}$ \citep[see, \eg,][]{Mich:69,Gold-Juli:70,Bege-Lee:94,Besk-Kuzn-Rafi:98}.
\citet{Spit-Aron:04} claim that the discrepancy is largely due to
the various artificial assumptions that these early efforts
necessarily adopted, including spherical symmetry, split-monopole
field configurations, and the region of applicability of the
force-free approximation.}. If the seed photons are indeed
from a thermal distribution with $T_\gamma \sim 1\,\keV$ as argued in
the previous sections, the up-scattered emission would peak at a few
$\GeV$, far higher than is observed. However, in the situation under
consideration here, the Compton scattering will induce a drag on the
jet, substantially reducing its velocity. This will manifest itself
differently depending upon whether or not the jet is optically thin
along its axis.
When it is optically thin, the rate at which energy is lost by the jet
to the up-scattered photons at a given radius is simply given by the
rate at which seed photons enter the jet and their subsequent
up-scattered energy, \ie,
\begin{equation}
\frac{\d L}{\d r} = -2\Gamma^2 \frac{\d L_{\rm seed}}{\d r} \,,
\end{equation}
where $L_{\rm seed}$ as a function of radius is defined by
\begin{equation}
L_{\rm seed}(r) = \int_{r_0}^r \left( u_T + u_S \right) v_{\rm sn}
\frac{2 r}{\Gamma} \d r \,,
\end{equation}
and $u_S$ is the energy density of seed photons produced in the shocks
(of order $\rho_{\rm sn} v_{\rm sn}^2/2$). Since $\d L/\d r$ is negative definite it
will generally produce a deceleration in the jet.
In contrast, when $\tau_\parallel > 1$ the up-scattered photons cannot
escape. They will consequently rescatter, introducing a second term
to the energy loss:
\begin{equation}
\frac{\d L}{\d r} = -2\Gamma^2 \frac{\d L_{\rm seed}}{\d r}
- 2 L_{\rm seed} \frac{\d \Gamma^2}{\d r}
= -\frac{\d}{\d r} 2 \Gamma^2 L_{\rm seed} \,.
\end{equation}
Unlike the optically thin case, for strongly decelerating jets this
can produce a positive accelerating force. This provides an effective
mass loading of the jet, and since $\sigma$ is so high, will determine
the maximum $\Gamma$ reached, \ie,
\begin{align}
\Gamma &\simeq \frac{L_{\rm jet}}
{2 \Gamma^2 L_{\rm seed}\vert_{\tau_\parallel=1}}
\simeq \frac{L_{\rm iso}}{8 \Gamma^3 \left(u_T + u_S\right) v_{\rm sn} r^2}
\nonumber\\
&\longrightarrow \Gamma \simeq
30
\left(\frac{L_{\rm iso}}{10^{51}\,\erg/\s}\right)^{1/4}
\left(\frac{r}{10^{11}\,{\rm cm}}\right)^{-1/2} \,.
\label{Lorentz_limit}
\end{align}
Hence if the jet becomes optically thin near
$r=2\times10^{11}\,\cm$, $\Gamma\simeq20$ as assumed thus far.
After $\tau_\parallel$ drops below unity the jet will continue to
decelerate until it exits the ejecta or stalls.
The Lorentz factors considered here are substantially lower than those
implied by the typical resolution to the ``compactness problem''
\citep[which are $\gtrsim {\rm few}\times10^2$, see, \eg,][]{Lith-Sari:01}.
The runaway pair production associated with the ``compactness problem'' is
not present here as a result of the low energy of the seed photons
themselves. In the jet frame, the seed photons have
energy $\sim \Gamma \epsilon_{\rm sn}$, where $\epsilon_{\rm sn}$ is the
typical seed photon energy in the lab frame. Since in the jet frame
the electrons are cold, and thus all of the photon scattering is elastic,
the maximum energy in any photon-photon collision is
$2\Gamma\epsilon_{\rm sn}$, which for $\Gamma\sim20$ and
$\epsilon_{\rm sn}\sim1\,\keV$ is insufficient to pair produce.
Nonetheless, for a thermal seed photon distribution a high energy tail
will exist, including photons with $\epsilon > m_e c^2/\Gamma$ and
hence will be able to pair produce. The number density of such
photons accumulated in the jet frame by the end of the burst is
roughly given by
\begin{equation}
n_\epsilon \simeq g \Gamma \frac{u_T}{k T_\gamma} \frac{v_{\rm sn}}{c}
\left(\frac{m_e c^2}{\Gamma k T_\gamma}\right)^2
\exp\left(-\frac{m_e c^2}{\Gamma k T_\gamma}\right)\,,
\end{equation}
where the factor of $v_{\rm sn}/c$ accounts for the fact that in the jet the
photons are free streaming and $g\simeq0.15$ is a normalisation
factor. The resulting optical depth to pair production is then
$\sigma_T n_\epsilon r/\Gamma$, where $r$ is the radius in the lab
frame \citep[see, \eg,][]{Lith-Sari:01}. Therefore, the lepton
density at the end of the burst as a result of photon annihilation is
$n_{e^{\pm}} \simeq \sigma_T n_\epsilon^2 r/\Gamma$, with an associated
optical depth along the jet of
\begin{align}
\tau
&\simeq
\frac{1}{2\Gamma} \sigma_T n_{e^\pm} r \nonumber\\
&\simeq
\frac{1}{2} \left(\sigma_T n_\epsilon \frac{r}{\Gamma}\right)^2
\nonumber\\
&\simeq
3\times10^{-3}\,,
\end{align}
for $r\simeq10^{12}\,\cm$. Note that since pair annihilation has been
ignored, this is only an upper limit. Thus, it may be safely
concluded that pair production is insignificant during the the burst.
In general, both the jet and the seed photon density and temperature
will be expected to have radial structure. In the jet this is due to
competition between the magnetic stresses and the Compton drag. In
the seed photons this is due to the adiabatic cooling of the supernova
ejecta. A direct result of this structure is that, despite beginning
with thermal seed photons, the time integrated spectra can have the
observed broken power law shape where the break energy could
indeed be interpreted as the temperature of the up-scattered seed photons
when $\tau_\parallel\simeq1$, \ie, roughly
$2\Gamma^2\,\keV\sim{\rm few}\times10^2\,\keV$. For the case where
the seed photon temperature has a radial power-law dependence this has
already been explicitly shown to be the case by
\citet{Ghis-Lazz-Celo-Rees:00}. However, in this scenario,
since the density of seed photons depends upon distance from the
helium star, different orientations of the jet with respect to the
orbital plane will lead to substantial changes in the spectral slopes
of the integrated emission. Thus, because the compact object is
expected to have suffered a kick during its birth, the considerable
variation observed in burst spectral slopes would be expected.
\subsection{Population Statistics} \label{PE:PS}
For a given beaming angle
($\sim \Gamma^{-1}$) approximately $10^{-10} \Gamma^2$
bursts occur per galaxy per year \citep{Schm:01}. This implies that the formation rate
(${\cal R}_{\rm CO-He}$) for the progenitor systems satisfy
\begin{equation}
{\cal R}_{\rm CO-He} \gtrsim 4\times10^{-8} \left(\frac{\Gamma}{20}\right)^2
\,{\rm galaxy^{-1} yr^{-1}} \,.
\end{equation}
There is a considerable literature which addresses the formation rates
of neutron star--neutron star binaries and neutron star--black hole
binaries. Because in both cases it is believed that these are produced
by the evolution of compact object--Helium star binaries, the formation
rates of the former place lower limits upon the formation rate of the
latter. Therefore, the progenitor formation rate must be compared to
${\cal R}_{\rm NS-NS} \simeq 10^{-6}$ to $5\times10^{-4}\,{\rm
galaxy^{-1} yr^{-1}}$
\citep{Kalo-etal:04} and
${\cal R}_{\rm NS-BH} \gtrsim 10^{-4}\,{\rm galaxy^{-1} yr^{-1}}$
\citep{Beth-Brow:98}.
Hence, there are more than enough presumed compact object--Helium star
products to account for the number of $\gamma$-ray bursts observed.
It should also be noted that the formation rate of compact binaries is
expected to be considerably smaller than the formation rate of the
progenitor binaries due to the possibility of unbinding the
progenitor as a consequence of the ensuing supernova, and the fact
that the black hole--black hole binaries have been ignored.
\section{Afterglow} \label{Ag}
A consequence of electromagnetic domination in the jet is the absence
of internal shocks as long as the jet remains relativistic. However,
as the jet cools, the flow will become increasingly hydrodynamic.
When $\Gamma$ is of order unity, strong internal shocks may be
expected to develop. At this point the internal kinetic energy of
the jet can be thermalised, producing a moderately relativistic fireball,
which can then be analysed within the highly successful standard
fireball afterglow model (see, \eg, \citealt{Mesz:02};
\citealt{Koni-Gran:02} argue that many of the diverse afterglow
phenomena can be naturally produced if the afterglow is produced
inside of a pulsar-wind bubble). Due to the
velocity gradient at the end of the jet, when enough matter
accumulates a Compton thick head will develop, shutting off further
prompt emission. This can be expected to occur over the time scale for
the jet to proceed from the radius at which $\tau_\parallel\simeq1$
($\sim10^{11}\,\cm$) to the radius at which $\Gamma\sim1$
($\sim10^{12}\,\cm$), which for the scenario considered here is
approximately $30-100\,\s$. After this time, the jet will continue to
pump energy into the growing fireball at its head. Only when the
rotational energy of the compact object is sufficiently exhausted, or
accretion ceases, will the jet cease and the fireball expand under its
own pressure. For a neutron star this also can be expected to occur
over a comparable time scale as the prompt emission, while for a black
hole this can continue considerably longer (see section \ref{JF}).
\section{Model Implications} \label{MI}
This model has a number of direct observational implications discussed
below, many of which have already been detected.
\subsection{Burst Substructure}
The fact that the ejecta is Thomson thick and inhomogeneous leads to
considerable burst substructure. In this case seed photons will not
be continuously available, but rather will enter the jet in bunches
over timescales ($\delta t$) associated with the inhomogeneity length
scales ($\ell$), \ie,
\begin{equation}
\delta t \simeq \frac{\ell}{v_{\rm sn}} \sim 1\,\s\,,
\end{equation}
implying
\begin{equation}
\ell\sim10^3\,\km\sim10^{-3}R_{\rm He}\,.
\end{equation}
This will occur when an ejecta clump impacts the jet and is
subsequently either sheared apart, releasing the entrained photons, or
forms strong shocks and thus thermalising its bulk kinetic energy.
To lowest order, the rate of supplied seed photons may be treated as
uniform over $\delta t$ and vanishing otherwise. In this case, for a
single clump, the number of seed photons within the jet will evolve
according to
\begin{equation}
\frac{\d N_{\rm seed}}{\d t} =
\left\{
\begin{aligned}
\frac{\cal N}{\delta t} - \frac{\tau_b}{\delta t} N_{\rm seed} && 0<t<\delta t\\
- \frac{\tau_b}{\delta t} N_{\rm seed} && \delta t < t\\
\end{aligned}
\right.
\,,
\end{equation}
where ${\cal N}$ is the total number of seed photons in the clump and
\begin{equation}
\tau_b = \tau_\perp \frac{\Gamma c \delta t}{2 r}\,,
\end{equation}
which is expected to be $\sim 0.3$ at the point where most of the
emission occurs. The solution is trivially found to be
\begin{equation}
N_{\rm seed} =
{\cal N} {\rm e}^{\tau_b t/\delta t}
\left\{
\begin{aligned}
\frac{t}{\delta t} && 0<t<\delta t\\
1 && \delta t < t\\
\end{aligned}
\right.
\,.
\end{equation}
Since nearly every seed photon that enters the jet will be up-scattered
to $\gamma$-ray energies, this also provides the expected light curve
for a single subpulses
\begin{equation}
\frac{\d N_\gamma}{\d t} = f \frac{\tau_b}{\delta t} N_{\rm seed} \,.
\end{equation}
where $f$ is the fraction of singly scattered photons and is
typically of order unity. This has the fast rise--exponential decay
structure observed \citep[\cf][]{Norr-etal:96}. Furthermore, the
ratio of the exponential decay time to the (approximately) linear rise
time is $\tau_b^{-1} \sim 3$, as observed \citep{Norr-etal:96}. As
the jet Lorentz factor increases these subpulses would be expected to
be systematically more symmetric for a given inhomogeneity scale in
the supernova ejecta (which would not be expected to vary
considerably). Thus, more energetic bursts would be expected to
produce more symmetric subpulses, which has also been observed
\citep{Norr-etal:96}.
\subsection{Time Lags and Energy Dependent Subpulse Widths}
Photons at energies below $2\Gamma^2 T_\gamma$ will be due to both
the low energy tail of the seed photon distribution and multiple
scatters both within the jet and at the jet boundaries. The
high bulk Lorentz factor of the jet implies that the scattering angles
will typically be on the order of $\Gamma^{-1}$. Therefore,
for each encounter
\begin{equation}
\epsilon_f \simeq \frac{\epsilon_i}{1 + \Gamma \epsilon_i/m_e c^2} \,.
\end{equation}
Many encounters may be approximated by integrating the equation
\begin{equation}
\frac{\d\epsilon}{\d N_{\rm scat}} \simeq \frac{\epsilon}{1 + \Gamma \epsilon/m_e c^2} - \epsilon \,,
\end{equation}
to give
\begin{equation}
N_{\rm scat} \simeq \frac{m_e c^2}{\Gamma}
\left(\frac{1}{\epsilon}-\frac{1}{\epsilon_{\rm peak}}\right)
-\ln\left(\frac{\epsilon}{\epsilon_{\rm peak}}\right)
\sim \frac{m_e c^2}{\Gamma \epsilon}\,,
\end{equation}
and thus $\epsilon \sim N_{\rm scat}^{-1}$ in the limit of many scatters. Due to the additional
path length traversed by these photons, they will lag behind the
single scattered photons by a time $\propto N_{\rm scat}$. In addition, since
the scattered photons are performing a biased random walk, they will
spread in time $\propto N_{\rm scat}^{1/2}$. This gives the following scalings
\begin{equation}
\Delta t_{\rm lag}
\propto \Gamma^{-1} \epsilon^{-1}\,,
\end{equation}
and
\begin{equation}
\delta t \propto \Gamma^{-1/2} \epsilon^{-1/2} \,,
\end{equation}
where the former is consistent with the observed anti-correlation between the
time lags and the overall luminosity of the burst \citep{Band:97},
and the latter is in rough agreement with the observed relation $\partial\ln\delta
t/\partial\ln\epsilon \simeq -0.4$ \citep{Feni-etal:95}.
The relevant time scales for the time lag can be estimated directly by
noting that the jet width is roughly
\begin{equation}
\frac{r}{\Gamma} \sim 5\times10^9
\left(\frac{\Gamma}{20}\right)^{-1}
\left(\frac{r}{10^{11}\,\cm}\right) \,\cm \,,
\end{equation}
and hence the lower energy emission will lag by
$N_{\rm scat} (r/\Gamma c)\sim0.2N_{\rm scat}\,\s$. If a majority of the low energy
photons are produced deeper within the jet (where the Compton depth is
higher), this time scale can decrease by an order of magnitude. In
either case this is again consistent with observations of burst pulse time
lags \citep{Band:97}.
\subsection{Spectral Softening}
As the jet evolves, its axial optical depth will increase and the
position at which the majority of the emission arises from will move
outward towards regions of lower Lorentz factor. Near the end of the
burst the Lorentz factor at the $\gamma$-ray photosphere is expected
to be no more than a few. As a result, the emission can be expected
to soften considerably as the burst proceeds. The rate at which this
softening occurs depends upon the radial structure of the jet as
discussed in section \ref{PE:IfJLF} and would be expected to vary
considerably between bursts.
\subsection{Peak--Inferred Isotropic Energy Relation} \label{PIIER}
\citet{Amat-etal:02} \citep[and more recently][]{Amat:04} have
reported a correlation between the peak spectral energy and the
inferred isotropic energy of bursts, namely
$\epsilon_{\rm peak} \propto E_{\rm iso}^{1/2}$.
In the context of
the model presented here, this would be naturally expected if the
dominant factor in the variation between bursts was the maximum jet
Lorentz factor. This is not unexpected if the helium star supernovae
are similar, implying that scatter in the orbital
separation and compact object parameters, both of which enter most
significantly into the determination of $\Gamma$, produces the
variability amongst bursts. Then, in terms of the typical seed photon
energy ($\epsilon_{\rm sn}$), the peak $\gamma$-ray energy is
\begin{equation}
\epsilon_{\rm peak} \simeq 2 \Gamma^2 \epsilon_{\rm sn} \,.
\end{equation}
From equation (\ref{Lorentz_limit}) it is clear that
\begin{equation}
L_{\rm iso} \simeq 8 \Gamma^4 \left(u_T+u_S\right) v_{\rm sn} r^2
\propto \epsilon_{\rm peak}^2\,,
\end{equation}
where the last proportionality holds if the typical ejecta densities
and velocities and the radius at which the jet becomes optically thick
are similar amongst bursts. This may be trivially inverted to yield
the observed relation.
\subsection{X-ray Flash Characteristics}
For jet Lorentz factors $\sim\text{few}$, the peak of the emission
would occur in the X-rays, and thus this model provides a natural
explanation for X-ray flashes.
Note that this is neither a structured nor uniform jet model
\citep[\cf][]{Ross-Lazz-Rees:02,Lamb-Dona-Graz:04}. In the former,
viewing an azimuthally structured jet from different angles provides
the peak--inferred isotropic energy relation. In the latter, the
distribution is in the opening angle of the jet. In both it is
assumed that there is little scatter in the total energy of the burst.
In contrast, as seen in equation (\ref{Lorentz_limit}) with the same
assumptions made in the previous subsection, in the scenario presented
here the energy released in the form of $\gamma$-rays is expected to
scale as
\begin{equation}
L_{\rm jet} \propto \Gamma^{-2} \,.
\end{equation}
However, in this as well as in the normal bursts, this energy is
expected to be subdominant relative to the energy in the supernova
itself.
Since X-ray flashes share many of the temporal and spectral features
of $\gamma$-ray bursts, including lying upon the peak--inferred
isotropic luminosity relation of \citet{Amat:04}, it is
tempting to interpret them as simply subluminous bursts. Recently,
the bolometric energy of the X-ray flash \mbox{XRF 020903} has been measured
and was indeed found to be similar to typical $\gamma$-ray bursts
despite having a substantially lower inferred isotropic luminosity
\citep{Sode-etal:04}.
\subsection{Late Light Curve Variability}
Despite the large amount of energy released in the supernova, the
binary does not necessarily become unbound (indeed, it must not if
this is a viable formation scenario for double pulsars). Therefore,
in systems with a sizable amount of ejecta remaining in the vicinity
of a nascent compact remnant, variability due to the occasional
accretion by the companion compact object may be expected after the
supernova becomes optically thin, and thus late in the light curve.
This may explain the late time ($\sim50\,{\rm days}$ after the burst)
variability observed in the residual light curve of \mbox{SN 2003dh},
which has a rough time scale of a few days, implying an orbital
separation of $\sim10^{12}\,\cm$, in agreement with what would be
expected in this model \citep{Math-etal:03}.
\subsection{Polarisation}
\citet{Lazz-Ross-Ghis-Rees:04} have shown that inverse-Compton
scattering by jets can produce large degrees of linear polarisation
($\sim100\%$ in some cases).
This is a direct result of the relativistic aberration of the seed
photons in the jet frame and is a function of the viewing angle of
the jet, nearing unity for $\theta\sim\Gamma^{-1}$. Therefore,
jet models in which the prompt emission is due to inverse-Compton
scattering will generally exhibit wide variation in the degree of
linear polarisation, ranging from zero along the jet axis to unity
along the jet edge. As a consequence, the degree of linear polarisation
may be used to measure the jet viewing angle, breaking the degeneracy
between azimuthal jet structure and low luminosity in attempts to
utilise $\gamma$-ray bursts as cosmological standard candles.
Since, in the model presented here, the spectral evolution
of each subpulse to lower energy is due to multiple scatterings, the
polarisation fraction would be expected to decrease roughly
exponentially with the number of scatters and thus the polarisation
should be a strong function of energy as well.
Note that as long as the seed photon frequency in the jet frame is much
greater than the cyclotron frequency, as is expected to be the case
when the jet is optically thin axially (see equation
\ref{B_estimate}), the magnetic field will not significantly effect
the polarisation properties of the prompt emission.
Presently, the only measurement of the prompt emission polarisation
involved GRB 021206 and was found to be $80\%\pm20\%$ by
\citet{Cobu-Bogg:03} (though this claim continues to be
controversial). However, due to the large uncertainty this may be
consistent with both synchrotron self-Compton models and the model
presented here. If this is confirmed in future bursts, this
would provide strong evidence for Comptonised jet models.
\section{Conclusions} \label{C}
Rough estimates of the consequences of supernovae in compact
object--helium star binaries have been used to suggest that such an
event is a viable candidate for $\gamma$-ray bursts. Such events are
assumed to occur in the standard double pulsar evolution scenario.
Within this context a number of population calculations have been
performed, implying that indeed these events are likely to be frequent
enough to explain the number of bursts seen. This model is capable of
explaining a number of the burst characteristics, including the
subpulse light curves and energy dependence, late time spectral
softening, and the peak--inferred isotropic energy relation. In
addition, X-ray flashes and $\gamma$-ray bursts are naturally combined
into a single unified theory, as recently suggested by observations.
If the binary remains bound, as is necessarily the case for the double
pulsar formation scenarios, late time variability may be expected.
Such variability has been observed in the light curve of
\mbox{GRB 030329}/\mbox{SN 2003dh}. While no clear periodicity is
discernible, the typically time scales are consistent with an orbital
separation $\sim 10^{12}\,\cm$ as expected by this model. Future
observations of late time afterglow light curves can provide evidence
for the existence of binaries in $\gamma$-ray burst progenitors.
The prediction of significant degrees of polarisation in Compton
dragged jet models in general, and this model in particular, is
noteworthy. In the absence of strong structure within the jet, this
would necessarily lead to significant variation in the degree of
polarisation, ranging from zero to unity. Predictions for significant
degrees of polarisation which depend upon viewing angle has
significant implications for cosmological observations. Even in the
absence of a detailed understanding of the azimuthal jet structure,
polarisation provides a simple way in which to identify the viewing
angle, removing this degeneracy. Unfortunately, due to its
considerable uncertainty, the only measurement of prompt emission
polarisation is as yet unable to differentiate between
synchrotron self-Compton models and Comptonised jet models. Thus,
clearly future polarisation observations are required as well.
Many of the observational predictions of this model are generic to
Compton dragged jets. These include the spectrum
\citep{Ghis-Lazz-Celo-Rees:00}, polarisation
\citep{Lazz-Ross-Ghis-Rees:04}, subpulse lags, and subpulse energy
dependence \citep[\cf][]{Dado-Dar-DeRu:02}. To a lesser extent, the Amati
relation, and thus the X-ray flash characteristics, is also generally
expected as long as the primary difference among bursts is the jet
Lorentz factor.
This model is also very similar to the collapsar, with the obvious
difference that in this case the compact object is outside the helium
star (which need not be a Wolf-Rayet star here). As such, the most
immediate feature of the collapsar model, the association with
supernova, is present here as well. However, it is not possible to
simply subsume this model into the collapsar by placing the compact
object within the helium star; the primary difficulty being that the
solution to the compactness problem would no longer apply due to the
considerable supply of MeV photons in the central regions of the
supernova. As a result the jet would remain optically thick until
radii two orders of magnitude larger than those considered here.
Considering the likely fact that the supernova ejecta are travelling
radially from the compact object at this point, the mechanism proposed
here for producing the burst substructure will be unlikely to work.
Unique to this model is the manner in which the successes of
Comptonised jet and collapsar models are unified. Furthermore, it
provides a natural class of progenitors, in which supernovae are
already believed to occur as a result of independent observational
evidence. Thus, the results presented here may be regarded as both a
proposal for an new $\gamma$-ray burst mechanism and an investigation
into the observational consequences of what appears to be an
inevitable process given the currently observed double neutron star
systems.
\section*{Acknowledgements}
I would like to thank a number of people with whom I've had useful
discussions, including Jon McKinney, Ramesh Narayan, Andrew McFadyen,
Roger Blandford, Max Lyutikov, Yasser Rathore, Gerry Brown, and Ralph
Wijers. I would also like to thank the anonymous referee who's
suggestions substantially improved this manuscript. I would
especially like to thank Martin Rees for a number of insightful
conversations and generously hosting me during the time that much of
this work was completed. This research was supported by NASA-ATP
grant NAG5-12032 and NSF RTN grant AST-9900866.
\bibliographystyle{mn2e.bst}
|
{
"timestamp": "2005-08-01T18:50:52",
"yymm": "0411",
"arxiv_id": "astro-ph/0411778",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411778"
}
|
\section{Introduction}
\begin{quote}
``QCD nowadays has a split personality. It embodies hard
and soft physics, both being hard subjects and
the softer the
harder.''
\end{quote}
\centerline{Yuri Dokshitzer (2001) \cite{dokshitzer}}
\vspace{0.2cm}
Despite the great success of QCD as the field theory of hadronic
interactions, there still remains some open questions and
one of them is related to the hadron-hadron scattering at
\textit{high-energies} and
\textit{small momentum transfer} (soft diffraction).
The region of high energies is characterized by scattering
of particles with center of mass energy
$\sqrt s > 10$ GeV $\sim 10$ $m_p$ (the proton mass).
From the experimental point of view, diffractive processes
are associated with
a slow increase of the total cross sections,
the diffraction pattern in the differential cross section,
and rapidity gaps in the plots of pseudo rapidity {\it versus}
azimuthal angle. In the theoretical context, diffraction means
that
the initial and final states in the scattering process
have the same quantum numbers and, therefore, the
exchanged ``object" has the vacuum quantum numbers
(Pomeron). The soft diffractive processes are generally classified as double
diffraction dissociation, single diffraction dissociation and
elastic scattering. Introductory reviews on the area can be found in Refs.
\cite{bc,matthiae,engel,pred,ddln,menon02}.
High-energy elastic hadron scattering is the simplest soft
diffractive process and, at the same time a topical problem in
high-energy physics. Being associated with long distance
phenomena perturbative QCD can not
be applied. On the other hand,
the standard non-perturbative approach starts with the ground state
(vacuum), proceeds with bound states (mesons, barions) and eventually
reaches the scattering states. However, it is obvious that the vacuum
is a non-trivial
problem.
Moreover, even assuming some vacuum concept, to treat
only one gluon field
it is necessary to take into account more than 30 invariants, and
all that
becomes
a typical problem of statistical physics, with specific technical
approaches,
such as Monte Carlo simulation (lattice QCD). Although bound
states may be described,
the point is that,
presently, we do not know how
to calculate elastic scattering amplitudes from a pure nonperturbative
QCD formalism.
At this stage, phenomenology certainly plays an important role in the
search for
connections between experimental data, model descriptions, and the
possible development of new calculational schemes in the underlying
theory (QCD). Here, however, we are faced with another kind of
problem, namely, the wide variety of model descriptions, based on
different ideas and approaches, not always giving enough support
for the development of novel calculational schemes well founded on
QCD.
Based on the above facts,
our main strategy in the investigation
of the elastic sector
is to search for \textit{ model independent information}
that may
be extracted from the experimental data, through approaches that
have well established bases on the General Principles, theorems
and bounds from axiomatic quantum field theory
(the \textit{analytic approach}). Simultaneously,
we attempt to construct phenomenological models, in agreement
with the above Principles and connected,
in some way, with the underlying
dynamics of QCD.
In this review, it is presented some results we have obtained in the
area of elastic scattering in the last years, with focus on high-energy
proton-proton
($pp$) and antiproton-proton
($\bar{p}p$) elastic scattering.
The manuscript is organized as follows. In Sec. II we recall some
basic experimental and theoretical concepts,
defining also our notation. In Secs. III, IV, V, and VI we present the
main
results we have obtained throughout the analytic approach, model
independent analyses, eikonal models and
nonperturbative QCD, respectively. In Sec. VII we discuss
some perspectives in the area, from both experimental and theoretical
points of view.
A summary and some final remarks are the contents of Sec. VIII.
\section{Basic concepts}
In this section we recall the physical quantities that characterize
the elastic scattering and shortly review some principles, high-energy
theorems, and the main formulas associated with two basic pictures,
usually referred as s-channel (geometrical/optical picture) and t-channel
(exchange picture) \cite{matthiae,engel,pred,ddln,menon02}.
\subsection{Physical Quantities}
In elastic scattering, the connection between experimental data and theory
is done by means of the \textit{invariant scattering amplitude},
expressed in terms of
two Mandelstam variables, generally the
center-of-mass (c.m.) energy squared $s$ and the
four-momentum
transfer squared $t = -q^2$: $F = F(s,t)$.
It is expected that spin effects decrease as the energy increases
(for some recent results see \cite{mp}), and
neglecting spin, the physical quantities that characterize the
elastic scattering process are the
differential cross section,
\begin{eqnarray}
\frac{d\sigma}{dt}(s,t) = \frac{\pi}{k^2}|F(s,t)|^2,
\end{eqnarray}
where $k$ is the c.m. momentum, the elastic integrated cross section,
\begin{eqnarray}
\sigma_{el}(s) = \int_{-\infty}^{0} \frac{d\sigma}{dt}(s,t) dt, \nonumber
\end{eqnarray}
the total cross section (Optical Theorem),
\begin{eqnarray}
\sigma_{tot}(s) = \frac{4\pi}{k}\mathop{\mathrm{Im}} F(s,0),
\end{eqnarray}
the inelastic cross section
\begin{eqnarray}
\sigma_{inel}(s) = \sigma_{tot}(s) - \sigma_{el}(s), \nonumber
\end{eqnarray}
the $\rho$ parameter,
\begin{eqnarray}
\rho(s) = \frac{\mathop{\mathrm{Re}} F(s,0)}{\mathop{\mathrm{Im}} F(s,0)},
\end{eqnarray}
and the slope parameter,
\begin{eqnarray}
B(s) = \frac{d}{dt} \left[ \ln \frac{d\sigma}{dt}(s,t) \right]_{t=0}.
\end{eqnarray}
The corresponding experimental data have been analyzed and
compiled by the Particle Data Group and can be found in Ref.
\cite{pdg} and quoted references. In what follows we shall be
mainly interested in $pp$ and $\bar{p}p$ data in the regions:
$13.8$ GeV $\leq \sqrt s \leq 1.8$ TeV and
$0.01$ GeV$^{2} \leq q^{2} \leq 9.8$ GeV$^{2}$.
In a particular analysis we shall also use the
$pp$ data at $\sqrt s = 27.5$ GeV, in the
region $5.5$ GeV$^{2} \leq q^{2} \leq 14.2$ GeV$^{2}$.
Some treatment of cosmic-ray information on $pp$ total
cross sections at $\sqrt s =$ 6 - 40 TeV is also
presented.
In Figure 1 it is displayed the
experimental information available on $pp$ and $\bar{p}p$ total cross sections
from accelerators and cosmic-ray experiments. From that plot, it is clear that
the mathematical description of the increase of the total cross
sections at the highest energies is an open problem. As we shall
discuss, the
study of the effects of the discrepant points at the highest
energies is one of our goals.
Figure 2 shows the typical diffractive pattern that characterizes
the differential cross section. We note that the data cover the
region corresponding to 10 decades.
In Figure 3 it is displayed the slope parameter from
$pp$ and $\bar{p}p$ scattering as function of
the energy and determined in the region of small momentum transfer.
In what follows we shall refer to these three figures as
indicative of the empirical behavior of the quantities involved.
\begin{figure}
\begin{center}
\includegraphics[width=9.0cm,height=9.0cm]{menonf1.eps}
\caption{Total cross sections on $pp$ and $\bar{p}p$
from accelerator and cosmic-ray experiments (for complete list
of references and tables see \cite{alm03}).}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm,height=7.0cm]{menonf2.eps}
\caption{Differential cross section data and the diffractive pattern
from $pp$ elastic scattering at $\sqrt s =$ 52.8 GeV.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=6.cm,height=6cm]{menonf3.eps}
\caption{The slope parameter as function of the energy and determined
in the interval 0.01 $ < |t| < $ 0.20 GeV$^2$.}
\label{fig5}
\end{center}
\end{figure}
\subsection{Principles, theorems and high-energy bounds}
For our purposes, we recall some principles and theorems from
axiomatic quantum field theories \cite{teo}. The basic Principles are:
Lorentz Invariance, Unitarity (related with the conservation of
probability), Analyticity (related to causality) and Crossing
(connecting particle-particle and particle-antiparticle
interactions).
Analyticity and crossing allow the connections between real and imaginary
parts of the scattering amplitude by means of dispersion relations.
Several rigorous theorems and bounds may be deduced from the
basic Principles and axiomatic quantum field theory. Among them,
the Froissart-Martin bound concerns the increase of the total cross
section stating that
\begin{eqnarray}
\sigma_{tot} \leq C \log^2 \frac{s}{s_0}
\qquad {\rm as} \qquad
s \rightarrow \infty.
\end{eqnarray}
The Pomeranchuk Theorem treats the difference between
cross sections for particle-particle ($ab$) and particle-antiparticle
scattering ($a\bar{b}$). The original form was deduced when it was
believed that the cross section decreased to a constant value,
and in this case
$\sigma_{tot}^{ab} = \sigma_{tot}^{a\overline{b}}$
as $s \rightarrow \infty $. After the discovery of the rising of
the cross section, Grunberg and Truong obtained the
generalized or revised form of the Pomeranchuk Theorem, stating
that
\begin{eqnarray}
\frac{\sigma_{tot}^{ab} - \sigma_{tot}^{a\bar{b}}}
{\sigma_{tot}^{ab} + \sigma_{tot}^{a\bar{b}}} \rightarrow 0
\quad {\rm or} \quad
\frac{\sigma_{tot}^{ab}} {\sigma_{tot}^{a\overline{b}}} \rightarrow 1
\quad {\rm as} \quad
s \rightarrow \infty, \nonumber
\end{eqnarray}
and this means that, if the Froissart-Martin bound is reached, then
\begin{eqnarray}
\Delta \sigma \equiv \sigma_{tot}^{a\overline{b}} - \sigma_{tot}^{ab}
\leq C \frac{ \sigma_{tot}^{a\overline{b}} + \sigma_{tot}^{ab}}{\log s}
\leq C \log s.
\end{eqnarray}
By expressing the cross sections in terms of crossing even ($+$)
and odd ($-$) contributions,
\begin{eqnarray}
\sigma_{\pm}(s) = \frac{\sigma_{tot}^{ab} \pm
\sigma_{tot}^{a\bar{b}}}{2} , \nonumber
\end{eqnarray}
we have
$|\Delta \sigma| = | \sigma_{tot}^{a\overline{b}} -
\sigma_{tot}^{ab} | = 2\sigma_{-}$ .
Therefore,
$\Delta \sigma \equiv \sigma_{tot}^{a\overline{b}} - \sigma_{tot}^{ab}
\rightarrow 0$
if and only if $\sigma_{-}
\rightarrow 0$. This possible odd contribution is named Odderon
and the case of even dominance at asymptotic energies is
associated with the Pomeron.
\subsection{Basic pictures}
Nearly all the phenomenological models, able to describe
the experimental data on elastic hadron scattering, are based
on the Optical/Geometrical Picture (s-channel) and/or
the Exchange Picture (t-channel).
The corresponding formulas may be obtained from the
Partial Waves representation of the scattering amplitude,
\begin{eqnarray}
F(k,\theta) = \frac{i}{2k} \sum_{l=0}^{\infty} (2l + 1)
\left[ 1 - e^{2i\delta_l} \right] P_l(\cos \theta), \nonumber
\end{eqnarray}
where $\delta_l$ is the phase shift. In what follows, we outline
the main steps and formulas in both pictures.
\subsubsection{Optical/Geometrical Picture}
From the partial wave representation,
one considers the
high-energy limit and the semi-classical approximation, so that
the discrete angular momentum $l$ may be replaced by
the continuum impact parameter $b$,
\begin{eqnarray}
l = kb - \frac{1}{2}. \nonumber
\end{eqnarray}
In turn, the discrete
phase shifts $\delta_l$ are replaced by the continuum eikonal
function of $b$ and $s$, $\chi(s,b)$ and
\begin{eqnarray}
\sum_{l=0}^{\infty}...\ \ \rightarrow \ \ \int_{0}^{\infty} db...
\nonumber
\end{eqnarray}
The scattering amplitude
in this \textit{Eikonal
Representation}, with azimuthal symmetry assumed, reads
\begin{eqnarray}
F(s, q) = ik \int_{0}^{\infty} bdb J_{0}(qb)
[1 - e^{i\chi(s,b)}].
\end{eqnarray}
The quantity
\begin{eqnarray}
1 - e^{i\chi(s,b)} \equiv \Gamma (s,b)
\end{eqnarray}
is named Profile function. From Unitarity this function is related to
the probability that an inelastic event takes place at $b$ and $s$,
the Inelastic Overlap function:
\begin{eqnarray}
G_{inel}(s,b) = | \Gamma(s,b) |^2 - 2Re \Gamma(s,b).
\end{eqnarray}
Since in the Eikonal representation
\begin{eqnarray}
\ G_{inel}(s,b) = 1 - e^{-2 Im \chi(s,b)},
\end{eqnarray}
for
$Im\ \chi(s,b) \geq 0$ we have
$G_{inel}(s,b) \leq 1$, which implies in an automatically unitarized
representation.
\subsubsection{Exchange Picture}
In this picture, from the partial wave representation, one considers
the analytic continuation of the amplitude to complex angular momentum.
In the asymptotic limit ($s \rightarrow \infty$) and with symmetry
connecting the crossed channels
one arrives at the Watson-Sommerfeld-Gribov-Regge
representation for the scattering amplitude, expressed as a sum
over the poles of the amplitude (the Regge poles), as outlined in what
follows.
As it is known at high energies the number of partial waves is large,
and one way to circumvent that is to transform the sum of partial
waves into a complex integral, and then use the residues theorem
to obtain a new sum, but involving only the number of residues:
\begin{eqnarray}
\sum_{l=0}^{\infty} ...
\rightarrow \oint_{C} g(l) dl
\rightarrow \sum_{m=0}^{N} \left. Res\ g(l) \right|_{l =
l_{m}}. \nonumber
\end{eqnarray}
Detailed calculation allows one to obtain the following representation
for the scattering amplitude,
\begin{eqnarray}
F(k, \theta) = \sum_{i=1}^{N}
\frac{\beta_i(k)P_{\alpha_{i}(k)}(-\cos \theta)}{
\sin \pi \alpha_i(k)} + BI(k, \theta), \nonumber
\end{eqnarray}
where $BI(k,\theta)$ is called the Background integral.
By considering the high-energy limit (then $BI \rightarrow 0$) and crossing
(exchange four-momenta $p \rightarrow \quad
\Leftrightarrow \quad
\leftarrow -\bar{p}$)
we can replace the crossing channel variable
($\bar{\theta} \leftrightarrow s$)
\begin{eqnarray}
\cos \bar{\theta} = 1 - \frac{2s}{4m^2 - t} \quad
\rightarrow \quad \propto
- s \quad {\rm as} \quad s \rightarrow \infty, \nonumber
\end{eqnarray}
also,
\begin{eqnarray}
P_l(x) \rightarrow \left[\frac{2^l \Gamma(l + 1/2)}
{\sqrt \pi \Gamma(l + 1)}\right] x^l \quad {\rm for} \quad
x \rightarrow \pm \infty, \nonumber
\end{eqnarray}
and grouping all the $s$-independent quantities in a function
${\cal K}(t)$ we have
\begin{eqnarray}
P_{\alpha(t)}(-\cos \bar{\theta}) = {\cal K}(t)s^{\alpha(t)}
\quad {\rm for} \quad s \rightarrow \infty. \nonumber
\end{eqnarray}
Rearranging the terms we arrive at
a descending asymptotic series in powers of $s$, with leading
contribution:
\begin{eqnarray}
F(s,t) = \gamma(t) \xi(t) s^{\alpha(t)},
\end{eqnarray}
where $\gamma(t)$ is the residue function,
$\xi(t)$ the signature factor and
$\alpha(t) = \alpha(0) + \alpha' t$ the trajectory function.
This last function connects the spin and masses through the
Chew-Frautschi plot, as exemplified in Fig. 4.
In this picture the interaction of the colliding particles
is basically interpreted in terms of exchanges of Regge
poles (also Regge cuts) and the Pomeron (an \textit{ad hoc} trajectory
with intercept nearly above 1). We note that, as constructed,
the exchange picture is intended for asymptotic energies.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm,height=7.0cm]{menonf4.eps}
\caption{The Chew-Frautschi plot for some mesons and resonances.}
\end{center}
\end{figure}
\section{Analytic approach}
The Analytic Approach for elastic hadron-hadron scattering is based
on general principles and theorems from Quantum Field Theory.
It is
characterized by analytical parametrizations for
the imaginary part of the forward amplitude, together with the use of
dispersion relation
techniques. The central point is the simultaneous investigation
of the total cross section (imaginary part
of the scattering amplitude, Eq. (2)) and the
$\rho$ parameter (connected with the real part of the
amplitude, Eq. (3)).
For particle-particle and particle-antiparticle interactions,
dispersion relations are
consequences of the principles of Analyticity and Crossing.
In this context, they correlate real and imaginary
parts of crossing even ($+$) and odd ($-$) amplitudes,
which in turn are expressed in terms of the scattering amplitudes for
a given process and its crossed channel, for example, $ a + b $ and
$a + \bar{b} $:
\begin{equation}
F_{ab} = F_{+} + F_{-}, \qquad
F_{a\bar{b}} = F_{+} - F_{-}.
\end{equation}
At high energies, the standard singly subtracted integral dispersion
relations,
with poles removed, are given by
\begin{equation}
\mathop{\mathrm{Re}} F_{+}(s)= K + \frac{2s^{2}}{\pi}P\!\!\!\int_{s_{0}}^{+\infty}
\!\!\!\ d s'
\frac{1}{s'(s'^{2}-s^{2})}\mathop{\mathrm{Im}} F_{+}(s')
\label{eq:4}
\end{equation}
and
\begin{eqnarray}
\mathop{\mathrm{Re}} F_{-}(s)= \frac{2s}{\pi}P\!\!\!\int_{s_{0}}^{+\infty} \!\!\!
\ d s'
\frac{1}{(s'^{2}-s^{2})}\mathop{\mathrm{Im}} F_{-}(s'),
\label{eq:5}
\end{eqnarray}
where $K$ is the subtraction constant and, for $pp$ and $\bar{p}p$
scattering, $s_0=2m^2\sim 1.8$ GeV$^2$.
In this section we review some results obtained through this approach.
We start with the replacement of the above integral forms by derivative
operators (Derivative Dispersion Relations) and then we discuss
the use of analytic
models (Reggeons, Pomeron, Odderon) for parametrizations involving the total
cross section
and the $\rho$ parameter, the determination of bounds for
the soft Pomeron intercept, and the practical role of the subtraction constant.
In what follows we are mainly concerned with the $pp$ and $\bar{p}p$
elastic scattering,
since for particle and antiparticle interactions they correspond to
the highest energy
interval with available data and are the only set including the cosmic-ray
information on total cross sections ($pp$ scattering).
As commented before, the experimental data available on the
total
cross sections (Figure 1) are characterized by discrepant experimental
information at the highest energies, and one of our aims is to
investigate the effects of these discrepancies in the context
of the analytic models. This concern permeates all the
discussion in this Section.
\subsection{Derivative Dispertion Relations}
The use of dispersion relations in the investigation of scattering
amplitudes may be traced back to the end of fifties, when they were
introduced in the form of {\em Integral} Dispersion Relations (IDR).
Despite the
important results that have been obtained since then, one limitation of
the integral forms is their non-local character:
in order to obtain the real part of the amplitude, the imaginary part
must be known for all values of the energy. Moreover, the class of
functions that allows analytical integration is limited.
In the last years, we have investigated the applicability of {\em Derivative}
Dispersion Relations (DDR)
in place of integral forms \cite{mmp,alm01,almhadron02,alm03,lm03,am03}.
In Reference \cite{am03} we present a
recent review on different results
and statements
related to this replacement,
and a discussion
connecting these different aspects with the corresponding assumptions and
classes of functions considered in each case.
In particular, we have shown that for the class of functions which
are entire in the logarithm of the energy (as is the case of analytic
models at high energies)
it is possible to expand the integrand in the above formulas
and by considering a high-energy approximation, represented
by $s_0=2m^2 \rightarrow 0$, to integrate term by term.
In that case, as demonstrated in detail in
\cite{am03}, the derivative dispersion relations with
one subtraction reads
\begin{equation}
\frac{\mathop{\mathrm{Re}} F_+(s)}{s}= \frac{K}{s} +
\tan\left[\frac{\pi}{2}
\frac{\mathrm{d}}{\mathrm{d}\ln s}\right]\frac{\mathop{\mathrm{Im}} F_+(s)}{s},
\end{equation}
\begin{equation}
\frac{\mathop{\mathrm{Re}} F_-(s)}{s}=
\tan\left[\frac{\pi}{2}
\left(1 + \frac{\mathrm{d}}{\mathrm{d}\ln s}\right)\right]
\frac{\mathop{\mathrm{Im}} F_-(s)}{s},
\label{eq:11}
\end{equation}
where the series expansion is implicit in the tangent operator.
From this deduction one arrives to three formal results: (1) the
subtraction constant is preserved when the IDR are replaced by DDR and,
therefore, in principle, can not be disregarded in fit procedures;
(2) except for the subtraction constant, the DDR with entire functions
in the logarithm of the energy do not depend on any
additional free parameter; (3) the only approximation involved in the
replacement concerns the lower limit in the IDR (13-14), namely
$s_0 = 2m^2 \rightarrow 0$, which represents a high-energy
approximation.
In the next two subsections we discuss some uses of the DDR
with analytical models,
and in the third subsection we return to the replacement of IDR
by DDR, investigating the important role of the subtraction constant
from a practical point of view.
\subsection{Basic Models}
In this Subsection we make use of two basic and well
known parametrizations for the
total cross sections and investigate the effects
of the discrepancies in the experimental information
from cosmic-ray experiments.
\subsubsection{Ensembles}
In the cosmic-ray region, $6\ \textrm{TeV} < \sqrt s \leq 40$ TeV, the
discrepancies
on the total cross section information are
due to both experimental and theoretical uncertainties in the
determination of
$\sigma_{tot}^{pp}$ from p-air cross sections. The situation has been
recently
reviewed in detail in \cite{alm03}, where a complete list of references,
numerical
tables and discussions are presented.
From Fig. 1 we see that, despite the large error bars in
the cosmic-ray region, we can
identify two distinct sets of estimations: one corresponding to the results
by
the Fly's Eye Collaboration (Fly's Eye) together with those by the Akeno
Collaboration (Akeno); the other set associated with the results by Gaisser,
Sukhatme,
and Yodh
(GSY) together with with those by Nikolaev (Nikolaev).
Taken separately these two sets suggest different
scenarios for the increase of the total cross section,
as previously discussed in \cite{alm01,mm97,lm02}.
Based on these considerations, it is important to investigate the
behavior of the
total cross section by taking into account the discrepancies that
characterize the cosmic ray information.
To this end, in \cite{alm03} we have considered \textit{two ensembles
of data and
experimental information}, as follows:
\vspace{0.3cm}
$\bullet$ Ensemble I:
$\bar{p}p$ and $pp$ accelerator data + Akeno +
Fly's Eye;
\vspace{0.3cm}
$\bullet$ Ensemble II:
$\bar{p}p$ and $pp$ accelerator data + Nikolaev +
GSY.
\vspace{0.3cm}
To some extent, ensemble I represents a kind of high-energy standard
picture and ensemble II a nonstandard one.
\subsubsection{Analytic Models}
With analytical parametrizations for $pp$/$\bar{p}p$ total cross
sections, the connections with the $\rho$ parameter, Eq. (3),
are obtained by defining the associated crossing even and odd
quantities,
\begin{eqnarray}
\sigma_{\pm}(s) = \frac{\sigma_{tot}^{pp} \pm
\sigma_{tot}^{\bar{p}p}}{2},
\end{eqnarray}
using the high-energy normalization for the Optical Theorem,
\begin{eqnarray}
\sigma_{tot}(s)
\sim \frac{\mathop{\mathrm{Im}} F(s,0)}{s},
\end{eqnarray}
and the DDR given by Eqs. (15) and (16).
In \cite{alm03} we have considered two different parametrizations for
the total cross
sections, one introduced by Donnachie and Landshoff \cite{dl} and
other by
Kang and Nicolescu \cite{kn}.
The main difference concerns the asymptotic limits, which allow the
dominance of an even amplitude (Pomeron) or
the odd amplitude (Odderon), respectively. In this way, we may contrast
these possibilities with the standard and non-standard pictures
represented by Ensembles I and II.
The \textit{Donnachie-Landshoff} (DL) parametrization for the total
cross sections
is expressed by
\begin{eqnarray}
\sigma_{tot}^{pp} (s) = X s^{\epsilon} + Y s^{- \eta},
\qquad
\sigma_{tot}^{\bar{p}p} (s) = X s^{\epsilon} + Z s^{- \eta},
\end{eqnarray}
where the first contribution is associated with a single Pomeron
exchange (universal) and the second one with Reggeon exchange.
With the procedure explained above, we obtain the analytical
connections with the $\rho$ parameter for $pp$ and $\bar{p}p$
scattering:
\begin{eqnarray}
&&\rho ^{pp}(s) \sigma_{tot}^{pp}(s) =
\frac{K}{s} +
\left[X\tan\left(\frac{\pi\epsilon}{2}\right)\right]s^{\epsilon}
\nonumber \\
&+&
\left[\frac{(Y-Z)}{2}\cot\left(\frac{\pi\eta}{2}\right) -
\frac{(Y+Z)}{2}\tan\left(\frac{\pi\eta}{2}\right)\right]s^{- \eta},
\nonumber
\end{eqnarray}
\begin{eqnarray}
&&\rho ^{\overline{p}p}(s) \sigma_{tot}^{\overline{p}p}(s) =
\frac{K}{s} +
\left[X\tan\left(\frac{\pi\epsilon}{2}\right)\right]s^{\epsilon}
\nonumber \\
&+&
\left[\frac{(Z-Y)}{2}\cot\left(\frac{\pi\eta}{2}\right) -
\frac{(Y+Z)}{2}\tan\left(\frac{\pi\eta}{2}\right)\right]s^{- \eta}.
\nonumber
\end{eqnarray}
From the above formulas, since $\eta > 0$, this model predicts that,
asymptotically ($s \rightarrow \infty$),
\begin{eqnarray}
&\Delta \sigma& = \ \sigma^{\overline{p}p}_{tot}(s) -
\sigma^{pp}_{tot}(s)
\rightarrow 0, \nonumber \\
&\Delta \rho& = \ \rho^{\bar{p}p}(s) - \rho^{pp}(s) \rightarrow 0.
\nonumber
\end{eqnarray}
The parametrization for the total cross sections introduced by
\textit{Kang and Nicolescu} (KN), under the
hypothesis of the Odderon, is given by
\begin{eqnarray}
\sigma^{pp}_{tot}(s)= A_{1} + B_{1} \ln s +
k \ln^2 s, \nonumber
\end{eqnarray}
\begin{eqnarray}
\sigma^{\overline{p}p}_{tot}(s)= A_{2} + B_{2} \ln s
+ k \ln^2 s + \frac{2R}{s^{1/2}}, \nonumber
\end{eqnarray}
and the connections with $\rho$ read
\begin{eqnarray}
&&\rho ^{pp}(s) \sigma_{tot}^{pp}(s)=
\frac{K}{s} + \frac{\pi}{2}\left(\frac{B_1 + B_2}{2}\right)
\nonumber \\
&+&
\left(\pi
k + \frac{A_2 - A_1}{\pi}\right) \ln s + \left(\frac{B_2 - B_1}{2\pi}
\right)
\ln ^2 s - \frac{2R}{s^{1/2}}, \nonumber
\end{eqnarray}
\begin{eqnarray}
&&\rho ^{\overline{p}p}(s) \sigma_{tot}^{\overline{p}p}(s) =
\frac{K}{s} + \frac{\pi}{2}\left(\frac{B_1 +
B_2}{2}\right) \nonumber \\
&+& \left(\pi k - \frac{A_2 - A_1}{\pi}\right)\ln s -
\left(\frac{B_2 - B_1}{2\pi} \right) \ln ^2 s \nonumber.
\end{eqnarray}
Differently from the previous case, this model predicts that the difference
between the two
cross sections is given by
\begin{eqnarray}
&\Delta \sigma& =
(A_2 - A_1) + (B_2 - B_1)\ln s + 2Rs^{-1/2} \nonumber \\
&\rightarrow& \Delta A +
\Delta B \ln s \quad {\rm(asymptotically)}, \nonumber
\end{eqnarray}
so that, if $\Delta A
\not= 0$ and/or $\Delta B \not= 0$, the total cross section difference may
increase and
$\sigma_{tot}^{pp}$ may even become greater than $\sigma_{tot}^{\bar{p}p}$,
depending on the
values and signs of $\Delta A$ and $\Delta B$, which is formally in
agreement with
the theorems of Sec. II.B.
Moreover, if $\Delta A$ and $\Delta B$ are sufficiently small, so that
we may replace $\sigma_{tot}^{\bar{p}p} \approx \sigma_{tot}^{pp} \equiv
\sigma_{tot}(s)$, then,
asymptotically,
\begin{eqnarray}
\Delta \rho = \rho ^{\overline{p}p} - \rho ^{pp} \sim
- \frac{1}{\pi \sigma_{tot}(s)} \left\{ \Delta A \ln s + \Delta B \ln^2 s
\right\}. \nonumber
\end{eqnarray}
This means that, depending on the fit results, there may be a change of
sign in
$\Delta \rho$, with $\rho ^{pp}$ becoming greater than
$\rho ^{\overline{p}p}$ at some
finite energy. Therefore, the case of a crossing either in $\sigma_{tot}$
or $\rho$ is a sign of the odderon contribution in the imaginary or
real part of the amplitude, respectively.
\subsubsection{Fits and Results}
We have performed 16 different fits through the program CERN-MINUIT. In these
fits we have used both
ensembles I and II and both the DL and KN models.
For each of these four possibilities we have performed global and
individual
fits to $\sigma_{tot}$ and $\rho$ and, in each case, we either considered the
subtraction
constant $K$ as a free fit parameter, or assumed $K = 0$.
All the results are presented and discussed in detail in Ref. \cite{alm03}.
Our main conclusions are the following:
(1) Despite the small influence from different cosmic-ray estimations, the
results allow to extract an upper bound for the soft Pomeron
intercept: $1 + \epsilon = 1.094$;
(2) although global fits present good statistical results, in general, this
procedure constraints the rise of $\sigma_{tot}$;
(3) the subtraction constant as a free
parameter affects the fit results at both low and high energies;
(4) independently of the
cosmic-ray information used and the subtraction constant, global fits with
the Odderon
parametrization predict that, above $\sqrt s \approx 70$ GeV, $\rho_{pp}(s)$
becomes greater
than $\rho_{\bar{p}p}(s)$, and this result is in complete
agreement with all the data presently available. That result is displayed
in Fig. 5 and we can infer
$\rho_{pp} = 0.134\ \pm \ 0.005$ at $\sqrt s = 200$ GeV and $0.151\ \pm
\ 0.007$ at $500$ GeV (BNL RHIC energies).
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm,height=7.0cm]{menonf5a.eps}
\end{center}
\begin{center}
\includegraphics[width=8.0cm,height=7.0cm]{menonf5b.eps}
\caption{ Simultaneous fits to $\sigma_{tot}(s)$ and $\rho(s)$ through the KN
parametrization with $K=0$ and ensembles I (dotted curves for $pp$ and dashed
for $\bar{p}p$)
and II (solid curves for $pp$ and dot-dashed for $\bar{p}p$) \cite{alm03}.}
\end{center}
\end{figure}
\subsection{Non-degenerate Meson Trajectories}
The DL parametrization referred above
assumes degeneracies between the secondary reggeons, imposing a common
intercept for the $C=+1$ ($a_2, f_2$) and the $C=-1$ ($\omega,\rho$)
trajectories (see Fig. 4).
More recently, analysis treating global fits to
$\sigma_{tot}$ and $\rho$ have indicated that the best results are
obtained with
non-degenerate meson trajectories. In this case the forward
scattering amplitude is decomposed into three reggeon exchanges,
$
F(s) = F_{\tt I\!P}(s) + F_{a_2/f_2}(s) +
\tau F_{\omega/\rho}(s),
$
where the first term represents the exchange of a single soft Pomeron,
the other
two the secondary Reggeons and $\tau = + 1$ ($- 1$) for $pp$ ($\bar{p}p$)
amplitudes. Using the notation $\alpha_{\tt I\!P}(0)
= 1+\epsilon$, $\alpha_{+}(0) = 1 -\eta_{+}$ and $\alpha_{-}(0) =
1 -\eta_{-}$
for the intercepts of the Pomeron and the $C=+1$ and $C=-1$
trajectories, respectively, the total cross sections, Eq. (18), for $pp$
and $\bar{p}p$
interactions are
written as
\begin{eqnarray}
\sigma_{tot}(s) = X s^{\epsilon} + Y_{+}\, s^{-\eta_{+}} + \tau Y_{-}\,
s^{-\eta_{-}}
\end{eqnarray}
and the connection with the $\rho$ parameter by means of DDR is similar
to that
displayed in the last subsection.
Making use of this parametrization, in this section we present the
determination of
extrema bounds for the Pomeron intercept \cite{lm03} and a practical
analysis on the replacement of IDR by DDR together with a discussion
on the role of the subtraction constant \cite{am03}.
\subsubsection{Extrema Bounds for the Pomeron Intercept}
In order to analyze the extrema effects in the soft Pomeron
intercept due to discrepancies in
the experimental data, we performed a detailed analysis
including the highest and the lowest values of the total cross
section from both accelerators and cosmic-ray experiments.
As it is well known, in the accelerator region, the conflict concerns
the results
for $\sigma_{tot}^{\bar{p}p}$ at $\sqrt s = 1.8$ TeV reported by the
CDF Collaboration and those reported by the E710 and the
E811 Collaborations (Fig. 1).
In the cosmic-ray region, as we have discussed, the highest predictions for
$\sigma_{tot}^{pp}$
concern the result by Gaisser, Sukhatme, and Yodh together with those by
Nikolaev. In order to treat the lowest estimations in the cosmic-ray
region, we consider
the results obtained by Block, Halzen, and Stanev (BHS), by means of
a QCD-inspired model. As discussed in \cite{alm03}, the reason
for this choice is that, although the extracted
$\sigma_{tot}^{pp}(s)$ shows agreement with the
Akeno results, it is about $17$ mb below the Fly's Eye value at $30$ TeV
and therefore may be considered as a extreme lower estimate. All the
numerical tables and references can be found in \cite{alm03}.
In this case we have considered the following ensembles of experimental
information.
First we only consider
accelerator data in two ensembles with the following notation:
\vspace{0.3cm}
$\bullet$ Ensemble I: $\sigma^{pp}_{tot}$ and $\sigma^{\bar{p}p}_{tot}$
data ($10 \le \sqrt{s} \le
900$ GeV) + CDF datum ($\sqrt{s} = 1.8$ TeV);
\vspace{0.3cm}
$\bullet$ Ensemble II
: $\sigma^{pp}_{tot}$ and $\sigma^{\bar{p}p}_{tot}$
data ($10 \le \sqrt{s} \le
900$ GeV) + E710/E811 data ($\sqrt{s} = 1.8$ TeV).
\vspace{0.3cm}
Ensemble I represents the faster increase scenario for the rise of
$\sigma_{tot}$ from
accelerator data and ensemble II the slowest one. These ensembles
are then combined with
the highest and lowest estimations for $\sigma_{tot}^{pp}$ from
cosmic-ray experiments,
namely, the Nikolaev and the Gaisser, Sukhatme, and Yodh (NGSY)
results and
the Block, Halzen,
and Stanev (BHS) results, respectively. These new ensembles are
denoted by
\vspace{0.3cm}
$\bullet$ I + NGSY
\vspace{0.3cm}
$\bullet$ II + BHS
\vspace{0.3cm}
As in the previous analysis, we have considered both individual
fits to $\sigma_{tot}$,
and
simultaneous fits to $\sigma_{tot}$ and $\rho$, either in the case
where the subtraction
constant is considered as a free fit parameter or assuming $K = 0$.
From this analysis, in the case of only accelerator data,
we could infer the following upper and lower
values for the Pomeron intercept: $\alpha_{\tt I\!P}(0)=1.098\pm 0.004$
(global fits to ensemble I, with $K = 0$)
and $\alpha_{\tt I\!P}(0)=1.085\pm 0.004$
(individual fit to $\sigma_{tot}$ from ensemble II),
with bounds $1.102$ and $1.081$, respectively.
Adding the cosmic-ray information, we inferred
the following upper and lower values:
$\alpha_{\tt I\!P}(0)=1.104\pm 0.005$
(individual fit to $\sigma_{tot}$ from ensemble I + NGSY)
and $\alpha_{\tt I\!P}(0)=1.085\pm 0.003$
(global fits to ensemble II + BHS and $K$ as a free fit parameter
or individual fit to $\sigma_{tot}$ from this ensemble),
with bounds $1.109$ and $1.082$,
respectively. Therefore we may infer the following \textit{extrema}
bounds
for the soft Pomeron intercept:
\begin{eqnarray}
\alpha^{upper}_{\tt I\!P}(0) = 1.109,
\qquad
\alpha^{lower}_{\tt I\!P}(0) = 1.081. \nonumber
\end{eqnarray}
Figure 6 shows the total cross sections with parametrization
(20) and the above extrema bounds,
together with the experimental information available.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm,height=7cm]{menonf6.eps}
\end{center}
\caption{Fastest and
slowest increase scenarios for the rise of the total cross section
through parametrization (20) and
allowed by the experimental information available: fits to ensembles
I + NGSY (solid) and II (dashed) \cite{lmm03}.}
\end{figure}
Extensions of these \emph{extrema bounds} for the pomeron intercept to
meson-p, gamma-p and gamma-gamma scattering have been
discussed in \cite{lmm03}. By means of global fits to total cross section
data it is shown that these bounds are in
agreement
with the bulk of experimental data presently available, and
extrapolations
to higher energies indicate different behaviors for the rise of the
total cross sections.
We have also obtained new \emph{constrained bounds} for the Pomeron
intercept from spectroscopy data
(Chew-Frautschi plots) and have extended the analysis to baryon-p, meson-p,
baryon-n, meson-n,
gamma-p and gamma-gamma scattering \cite{lmm-npa04}. It is also
presented tests on factorization
and quark counting rules with both extrema and constrained bounds
(asymptotic energy region). In particular, at 14 TeV (CERN LHC)
the extrema and constrained bounds allow to infer
$\sigma_{tot} = 114 \pm 25$ mb and $105 \pm 10$ mb, respectively.
\cite{lmm-npa04}.
\subsubsection{IDR, DDR and the Subtraction Constant}
As commented before, we have shown in Ref. \cite{am03} that for entire
functions in the logarithm of the energy
the only approximation involved in the replacement of IDR by
DDR concerns the lower limit $s_0$ in the IDR: the high-energy
condition is reached by assuming that $s_0 = 2m^2 \rightarrow 0$
in Eqs. (13-14).
In that paper we have investigated the practical applicability of the
DDR and IDR in the
context of the Pomeron-reggeon parametrizations, with both degenerate
and
non-degenerate higher meson trajectories. By means of global fits to
$\sigma_{tot}(s)$ and $\rho(s)$ data from $pp$ and $\bar{p}p$ scattering,
we have tested all the 16 important variants that could affect the fit
results, namely the number of secondary reggeons, energy cutoff
(5 and 10 GeV), effects of
the high-energy approximation connected with the subtraction constant
and the
analytic
approach using both DDR and IDR with fixed $s_0$. Our results led to
the conclusion that
the high-energy approximation and the subtraction constant affect the
fit results at both low and high energies. This effect is a consequence
of the
fit procedure, associated with the strong correlation among the free
parameters.
A striking novel result concerns the practical role of the subtraction
constant. We have shown that, with the Pomeron-reggeon parametrizations,
once the subtraction constant is used as a free fit parameter, the
results
obtained with the DDR and with the IDR (with finite lower limit,
$s_0 = 2m^2$)
are the same up to 3 significant figures in the fit parameters and
$\chi^2/DOF$. This conclusion, as we have shown, is independent of the
number of secondary reggeons (DL or extended parametrization) or the
energy cutoff ($\sqrt s$ = 5 or 10 GeV). In Table I we display the
fit results with the extended parametrization and cutoff at 10 GeV.
\begin{table}[h]
\begin{center}
\caption{Simultaneous fits to $\sigma_{tot}$ and $\rho$
through the extended parametrization, $\sqrt s_{\mathrm{min}} =$
10 GeV
(154 data points), with $K$ as a free
parameter and using IDR with lower limit $s_0=2m^2$ and DDR
\cite{am03}.}
\label{tab:4}
\begin{tabular}{ccc}
\hline
& IDR with $s_0=2m^2$ & DDR \\
\hline
$X$ & 19.57 $\pm$ 0.79 & 19.58 $\pm$ 0.78 \\
$Y_+$ & 66.0 $\pm$ 6.7 & 66.0 $\pm$ 6.6 \\
$Y_-$ & -29.2 $\pm$ 4.0 & -29.2 $\pm$ 4.0 \\
$\epsilon$ & 0.0897 $\pm$ 0.0033 & 0.0897 $\pm$ 0.0033 \\
$\eta_+$ & 0.380 $\pm$ 0.033 & 0.380 $\pm$ 0.033 \\
$\eta_-$ & 0.520 $\pm$ 0.025 & 0.520 $\pm$ 0.024 \\
$K$ & -14 $\pm$ 48 & 104 $\pm$ 58 \\
$\chi^2/DOF$ & 1.10 & 1.10 \\
\hline
\end{tabular}
\end{center}
\end{table}
\section{ Model Independent Analysis}
This kind of analysis is characterized by model independent parametrizations
of the experimental data involved and the extraction of empirical or
semi-empirical information that can contribute with the development
of phenomenological models and the underlying theory. In this
section we review some results we have obtained in the investigation
of $pp$ and $\bar{p}p$ differential cross section data (unconstrained
and constrained fits, as will be explained) and the correlations
between the experimental data on total cross section and the slope
parameter (Eq. (4)).
\subsection{Differential Cross Section}
Several authors have investigated elastic hadron scattering by means
of parametrizations for the scattering amplitude and fits to the
differential
cross section data, Eq. (1). The extraction of the Profile, Eikonal and
Inelastic Overlap functions in the $b$-space (impact parameter) and,
in some special
cases, the Eikonal in the $q^2$-space, has led to important and novel
results related with geometrical aspects (radius, central
opacity), differences between charge distributions and hadronic matter
distributions, existence or not of eikonal zeros in the $q^2$-space and,
more recently, connections with pomerons, reggeons and nonperturbative
QCD aspects. In Ref. \cite{cmm03} we present a review and a critical
discussion
on the main results concerning this kind of analysis and also a wide
list of references to outstanding works.
The basic input in all these analyses is the parametrization of the
scattering amplitude as a sum of exponentials in $q^2$
(as empirically suggested by the diffractive pattern shown in
Fig. 2) and fits
to the differential cross section data. This parametrization allows
analytical expressions for the Fourier transform of the amplitude,
providing also analytical expressions for the quantities of interest
in the $b$-space.
In the next two subsections we review the results we have obtained
by means of unconstrained fits (fit parameters completely free, without
extracted dependences on the energy) \cite{cmm03,cm}, and discuss some
research in course related to constrained fits (including dependences
on the energy which are based on empirical information) \cite{acmmhadron04}.
\subsubsection{Unconstrained Fits and the Eikonal}
In the high energy region, $\sqrt s > $10 GeV,
differential cross section data are available at
$\sqrt s =$ 13.8, 19.5, 23.5, 30.7, 44.7, 52.8 and 62.5 GeV for $pp$
scattering and at $\sqrt s =$ 13.8, 19.4,
31, 53, 62, 546 and
1800 GeV for $\bar{p}p$ scattering.
Data from $pp$ scattering also exists at
$\sqrt s =$ 27.5 GeV
and 5.5 $\leq q^2 \leq$ 14.2 GeV$^2$ (but not on $\sigma_{tot}$
and $\rho$), and as we shall show, that
set plays a fundamental role in our analyse. See \cite{cmm03} for
a complete list of references.
As discussed in \cite{cmm03} two main problems are typical of
model independent analysis of the differential cross sections:
(1) Experimental data are available only over finite regions
of the momentum transfer (which in general are small,
$q^2 < $ 7 GeV$^2$) and the Fourier transform demands integration
from $q^2$ = 0 to infinity. This means that any fit is biased by
extrapolations and although some extrapolated
curves may look unphysical, they can not be excluded on
mathematical grounds.
(2) The exponential parametrization allows analytical determination
of the quantities in the $b$-space (profile, inelastic, eikonal
functions)
and also the statistical
uncertainties, by means of error propagation from the fit parameters.
However,
in this case, the translation of the eikonal
from b-space to the $q^2$-space can not be analytically
performed and neither the error propagation (through standard
procedures).
As a consequence, the unavoidable uncertainties from the fit
extrapolations
can not, in principle, be taken into account.
In what follows we review a model independent approach able to
minimize the above two problems.
- \leftline{\textit{Fit Procedure}}
In order to treat problem (1) we have used the following
procedure \cite{cm}.
Since it is known that for large $t$ the experimental data
do not depend on the energy at
$13.8$ GeV $\leq \sqrt s \leq 62$ GeV and that there exist
data at $\sqrt s = 27.5$ GeV in the region
$5.5$ GeV$^{2} \leq q^{2} \leq 14.2$ GeV$^{2}$, we have selected
two ensembles of
$pp$ and $\bar{p}p$ differential cross section data:
$\bullet$ Ensemble I: experimental data at each energy;
$\bullet$ Ensemble II: Ensemble I + data at $\sqrt s = 27.5$ GeV.
For the scattering amplitude we have introduced the following
model independent
analytical parametrization for both real and imaginary parts:
\begin{eqnarray}
F(s,q) =
\{ \mu \sum_{j=1}^{2} \alpha_{j} e^{-\beta_{j} q^{2}}
\} + i
\{ \sum_{j=1}^{n} \alpha_{j} e^{-\beta_{j} q^{2}} \},
\end{eqnarray}
\begin{eqnarray}
\mu = \frac{\rho(s)}{\alpha_{1} + \alpha_{2}} \sum_{j=1}^{n}
\alpha_{j}.
\end{eqnarray}
With the experimental $\rho$ value at each energy the
fits to the differential cross section data have been performed
through the CERN-MINUIT routine
and the validity or not of ensemble II is checked by means of
the MINUIT output and standard statistical
interpretation of the fit results (DOF, confidence levels).
For $pp$ scattering
we have found that the data at $\sqrt s =$ 13.8 GeV are
not compatible with ensemble II. In the case of $\bar{p}p$ scattering
none of the data sets are compatible with ensemble II.
Therefore, in what follows, ensemble II (data at
$\sqrt s =$ 27.5 GeV added) corresponds only to $pp$ scattering
at 6 energies: 19.5, 23.5, 30.7, 44.7, 52.8 and 62.5 GeV.
From the error matrix (variances and covariances),
$\chi^2/DOF$ and confidence intervals, we
infer the best values for the parameters and corresponding
errors $\Delta\alpha_{j}$, $\Delta\beta_{j}$.
By means of standard error propagation,
the uncertainties in the free parameters, $\Delta \alpha_{j}$,
$\Delta \beta_{j}$, ($j$ = 1, 2, ...) have been propagated to the
scattering
amplitude,
and then to the differential cross section, providing
\begin{eqnarray}
\frac{d\sigma}{dq^2} \pm \Delta \left(\frac{d\sigma}{dq^2}\right).
\end{eqnarray}
By adding and subtracting the corresponding uncertainties we may estimate
the confidence region associated with all the extrapolations, which cannot
be excluded
on statistical grounds. A typical result with ensembles I and II
is illustrated in Fig. 7, for $pp$ scattering at $\sqrt s$= 23.5 GeV.
We see that,
as expected, the effect
of adding the experimental data at $\sqrt s$= 27.5 GeV (when statistically
justified) is to reduce drastically the uncertainty region. That result will
be fundamental in the extraction of the empirical information on
the eikonal, as shown in what follows.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm,height=7.0cm]{menonf7.eps}
\caption{Regions of uncertainties (limited by the
solid lines) in fits to $pp$ differential
cross section data, Eq. (23), at $\sqrt s = 23.5$ GeV with ensembles
I (below) and II (above) \cite{cmm03}.}
\end{center}
\end{figure}
- \leftline{\textit{ Eikonal in the momentum transfer space}}
By means of the Fourier transform, Eqs.(7-8), the parametrization
(21-22) provides analytical expressions for the real and
imaginary parts of the Profile function, $\Gamma_{R}(s,b)$ and
$\Gamma_{I}(s,b)$, and also the associated uncertainties.
From the fit results,
together with error propagation, we have found that
\begin{eqnarray}
\frac{\Gamma_{I}^{2}(s,b)}{[1-\Gamma_{R}(s,b)]^{2}} \ll1,
\nonumber
\end{eqnarray}
and therefore, the imaginary part of the
eikonal may be approximated by
\begin{eqnarray}
\chi_{I}(s,b)\approx\ln{1\over1-\Gamma_{R}(s,b)}
\end{eqnarray}
and the uncertainty $\Delta\chi_{I}$ determined directly from
$\Delta\Gamma_{R}$
through propagation.
The next step is to go to the momentum transfer space
and concerns problem (2):
the Fourier transform can not be performed analytically
and therefore also the error propagation. For this reason
we used a semi-analytical method as follows.
Expanding the above equation, we express the remainder of the
series as
\begin{eqnarray}
R(s,b)=\ln[{1\over1-\Gamma_{R}(s,b)}]-\Gamma_{R}(s,b)
\end{eqnarray}
and then fit the numerical points (MINUIT) by a sum of Gaussians
in the impact parameter space:
\begin{eqnarray}
R_{\rm{fit}}(s,b)=\sum_{j=1}^{6} A_{j}e^{-B_{j}b^{2}}.
\end{eqnarray}
A typical result is displayed in Fig. 8.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm,height=7.0cm]{menonf8.eps}
\caption{Typical parametrization for the generated remainder $R(s,b)$
by means of Eq. (26) \cite{cmm03}.}
\end{center}
\end{figure}
With this, the errors
$\Delta A_{j}$ and $\Delta B_{j}$ may be propagated
determining $\Delta R_{fit}(s,q)$ and then
$\chi_{I}(s,q) \pm \Delta\chi_{I}(s,q)$. In order to
check the results and approximations, we performed also
numerical integration through the NAG routine.
\leftline{- \textit{Results}}
One of the main results extracted from this analysis is the
statistical evidence of eikonal zeros in the momentum transfer space,
first presented in \cite{cm}.
In order to investigate the position of the zeros and, mainly, to
determine the uncertainties in its values, we consider
the expected behavior of $\chi_{I}$ at large $q^2$, namely
$\chi_{I} \sim q^{-8}$.
In Fig. (9) we show a typical plot of the quantity $q^{8}\chi_{I}(s,q)$ as
function of $q^2$. The shaded areas correspond to the uncertainties
obtained from error propagation.
This example shows clearly the role and the effect of data at large values of
the momentum transfer. In fact, within the uncertainties, ensemble I
does not allow to infer a zero,
but with ensemble II, we find statistical evidence for the change of sign.
\begin{figure}
\begin{center}
\includegraphics[width=6.0cm,height=5.0cm]{menonf9a.eps}
\vspace{0.5cm}
\end{center}
\begin{center}
\includegraphics[width=6.0cm,height=5.0cm]{menonf9b.eps}
\caption{The eikonal in the transfer momentum space (multiplied
by $t^4$) for $pp$ at $\sqrt s = 30.7$ GeV with ensemble I
(above) and II (below) \cite{cmm03}.}
\end{center}
\end{figure}
From plots like that we can determine
the positions of the zeros and the associated errors from the extrems
of the uncertainty region (in general not symmetrical). The position
of the zero can also be obtained from the numerical method,
but without uncertainties.
In Figure 10 it is shown the position of the zeros as function
of the energy determined by means of both the semi-analytical
(with uncertainties) and numerical (without uncertainties) methods.
Despite the systematic difference on the values with these methods,
we may conclude that the position of the zero decreases as the
energy increases. Roughly, $q^2_0:\ 8.5 \rightarrow 6.0$ GeV$^2$
as $\sqrt s :\ 20 \rightarrow 60$ GeV.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm,height=7.0cm]{menonf10.eps}
\caption{Position of the eikonal zero in the momentum
transfer space as function of the energy \cite{cmm03}.}
\end{center}
\end{figure}
\leftline{- \textit{Discussion}}
As reviewed in \cite{cmm03}, there has been previous indication of eikonal
zeros in the momentum transfer space, but without associated
uncertainties. Our first statistical evidence, published in 1997,
indicated the position of the zero at $q^2_0 = 7 \pm 2$ GeV$^2$
\cite{cm}. In 2000, experiments performed at the Jefferson Laboratory,
on electron-proton
scattering, have indicated an unexpected decrease of the ratio
between the electric and magnetic proton form factors
as the momentum transfer increases from 0.5 to
5.6 GeV$^2$. Moreover, extrapolations from empirical
fits indicate a change of sign (zero) in this ratio,
just at $q^2 \approx $ 7.7 GeV$^2$. Since for
$pp$/$\bar{p}p$ scattering the eikonal is connected with the
hadronic matter form factor (see Sec. V, Eq. (38)), the above
results on the position of the
zeros suggest novel and important insights on possible correlations
between
hadronic and electromagnetic interactions. We discuss that subject in
\cite{mmmhadron04}, calling attention to the possibility of
\textit{hadronic form factors depending on the energy}.
We have also obtained the value of the imaginary part of
the Eikonal at zero momentum transfer, that is, the central
opacity. The results are displayed in Fig. 11. As discussed in
\cite{cmmm},
one naive way to test these results is with the Glauber
model for the scattering involving
hadrons $A$ and $B$, and the Optical Theorem at the elementary level.
In that case, the elementary cross section may be expressed
by
\begin{eqnarray}
\sigma_{elem}(s) = \frac{4\pi}{N_A N_B} \chi_{I}(s, q=0),
\nonumber
\end{eqnarray}
where
$N_A$ and $N_B$ are the number of constituents in hadrons $A$ and $B$,
respectively.
If we take $N_A = N_B = 3$ we obtain $\sigma_{elem} \sim 6$ mb
at the ISR region, a result in agreement with other estimations.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm,height=7.0cm]{menonf11.eps}
\caption{Imaginary part of the eikonal at zero momentum transfer,
as function of the energy, from analysis on $pp$ and $\bar{p}p$
scattering \cite{cmm03}.}
\end{center}
\end{figure}
In Ref. \cite{cmm03} we discuss the applicability of our results in the
phenomenological context,
outlining some connections with nonperturbative QCD and
presenting a critical review on the
main results concerning ``model-independent" analyses.
\subsubsection{Constrained Fits and Energy Dependence}
Despite the results obtained with the parametrization discussed
in the last subsection, due to the fit procedure, we do not have the
dependence on the energy of the free parameters $\alpha_i$, $\beta_i$.
Presently, we are investigating that subject and we review here some
preliminary results \cite{acmmhadron04}.
The energy dependence has been introduced according to some empirical
information: the increase of the total cross section
and of the slope parameter with $\ln^2 s$ and $\ln s$, respectively
(see Figs 1 and 3).
Let us consider the standard exponential parametrization
for the imaginary part of the amplitude, normalized as
\begin{eqnarray}
\frac{\mathop{\mathrm{Im}} F(s,q^2)}{s}={\sum_{i=1}^n}{\alpha}_i\exp[-\beta_iq^2].
\end{eqnarray}
At $q^2 = 0$, from the optical theorem, Eq. (18), we expect a
dependence of the parameters $\alpha_i$ with $\ln^2 s$, and the
slope represented by the parameters $\beta_i$ with $\ln s$.
These are the central choices in our approach. In order to treat
$pp$ and $\bar{p}p$ scatterings, in agreement with Analyticity
and Crossing, we introduce crossing even and odd amplitudes and
make use of the derivative dispersion relations,
Eqs. (15) and (16), to connect real
and imaginary parts of the amplitudes involved.
Specifically, for the imaginary part of the
scattering amplitude we consider the parametrizations
\begin{eqnarray}
\frac{\mathop{\mathrm{Im}} F_{pp}(s,q^2)}{s}={\sum_{i=1}^n}{\alpha}_i(s)
\exp[-\beta_i(s)q^2],
\end{eqnarray}
\begin{eqnarray}
\frac{\mathop{\mathrm{Im}} F_{\bar{p}p}(s,q^2)}{s}={\sum_{i=1}^n}{\bar{\alpha}}_i(s)
\exp[-\bar{\beta}_i(s)q^2],
\end{eqnarray}
and, based on the above arguments, we introduce the following general
dependences
on the energy
\begin{eqnarray}
\left\{\begin{array}{l@{\quad\quad}}
\label{6}\alpha_i(s)=A_i+B_i\ln(s)+C_i\ln^2(s)\\
\beta_i(s)=D_i+E_i\ln(s)
\end{array}\right.
\end{eqnarray}
for $pp$ scattering and
\begin{eqnarray}
\left\{\begin{array}{l@{\quad\quad}}
\label{7}\bar{\alpha}_i(s)=\bar{A}_i+\bar{B}_i\ln(s)+\bar{C}_i\ln^2(s)\\
\bar{\beta}_i(s)=\bar{D}_i+\bar{E}_i\ln(s)
\end{array}\right.
\end{eqnarray}
for $\bar{p}p$ scattering, where $i=1,2,...n$. Defining the
crossing even (+) and odd (-) amplitudes,
\begin{eqnarray}
\mathop{\mathrm{Im}} F_{+}(s,q^2)=\frac{\mathop{\mathrm{Im}} F_{pp}(s,q^2)+
\mathop{\mathrm{Im}} F_{\bar{p}p}(s,q^2)}{2},
\end{eqnarray}
\begin{eqnarray}
\mathop{\mathrm{Im}} F_{-}(s,q^2)=\frac{\mathop{\mathrm{Im}} F_{pp}(s,q^2)-
\mathop{\mathrm{Im}} F_{\bar{p}p}(s,q^2)}{2}.
\end{eqnarray}
the corresponding real parts can be determined by means of the
leading terms of the DDR, Eqs. (15-16), and so the corresponding real
parts of the
$pp$ and $\bar{p}p$ amplitudes. With these analytic amplitudes we
obtain the differential cross section:
\begin{eqnarray}
\frac{d{\sigma}}{dq^2}=\frac{1}{16\pi s^2}
|\mathop{\mathrm{Re}} F(s,q^2)+i\mathop{\mathrm{Im}} F(s,q^2)|^{2}.
\end{eqnarray}
In order to treat simultaneous fits to $pp$ and $\bar{p}p$ data
we have considered only sets available at nearly the same energy,
namely $\sqrt s \sim$ 19.5, 31, 53 and 62 GeV. As a preliminary test
we make use of data at the diffraction peak, outside the Coulomb-nuclear
interference region, 0.01 GeV$^2 < q^2 \leq $ 0.5 GeV$^2$, and
the data providing the optical point,
\begin{eqnarray}
\frac{d\sigma(s,q^2 = 0)}{dq^2} =
\frac{\sigma_{tot}(1+\rho^2)}{16\pi}.
\end{eqnarray}
We have performed simultaneous fits to the experimental data through the
MINUIT program. For this ensemble we used only two exponentials for
the imaginary part of the amplitude,
obtaining good reproduction of all the data
analyzed, as shown in Fig. 12. In Ref. \cite{acmmhadron04} we
also display the predictions for the differential cross sections at
the RHIC energies.
We are presently investigating the extension
of the analysis to the region of higher momentum transfer.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm,height=7.0cm]{menonf12a.eps}
\end{center}
\begin{center}
\includegraphics[width=8.0cm,height=7.0cm]{menonf12b.eps}
\caption{Differential cross section at the diffraction peak: fit results
and experimental data (displaced by a factor of 10) \cite{acmmhadron04}.}
\end{center}
\end{figure}
\subsection{Total Cross Sections and Slopes}
Another important quantity that characterizes the elastic hadron-hadron
scattering is the slope parameter, defined in Eq. (4). In practice it
may be determined by means of fits to the hadronic differential cross
section data in the region of small momentum
transfer, with the parametrization
\begin{eqnarray}
\frac{d\sigma}{dt} = \left[\frac{d\sigma}{dt}\right]_{t=0}
e^{-B|t|},
\end{eqnarray}
and, in general, it is connected with $\rho$ and $\sigma_{tot}$ through fits
in the region of Coulomb interference \cite{bc}.
The slope and the total cross section are also important quantities
in the determination of $\sigma_{tot}^{pp}$ from $\sigma^{p-air}$
(cosmic-ray experiments) but, as commented before, the procedure is
strongly model dependent. One reason is associated with the use of the
Glauber multiple diffraction formalism, in which $\sigma_{tot}(s)$
and $B(s)$ take part in the parametrization of the elastic amplitude,
\begin{eqnarray}
F^{pp}(s,t) \propto \sigma_{tot}^{pp}(s) \exp
\left\{ \frac{B(s) t}{2}\right\}.
\end{eqnarray}
As commented in \cite{alm03} and \cite{mmmbjp04}, different models
predict different relations between $\sigma_{tot}(s)$
and $B(s)$ and that is mirrored in the final value of the cross section,
contributing to the discrepancies already discussed.
Based on the above observations, we have investigated the
possibility to extract an empirical correlation between the
experimental data on $\sigma_{tot}(s)$
and $B(s)$, from $pp$ and $\bar{p}p$ scatterings.
For the slope parameter, we have selected the data above
the region of Coulomb-nuclear
interference and below
the ``break" in the hadronic slope at the diffraction
peak (localized at
$|t| \sim$ 0.2 GeV$^2$ at the ISR and Collider energies),
namely 0.01 $ < |t| < $ 0.20 GeV$^2$ (Fig. (3)). In this region, the
differential cross section data are well fitted by a single
exponential and therefore there is no change in the slope
associated with the $t$-dependence. For each energy we have compiled the
corresponding data on the total cross section.
Once more, the choice for a parametrization was based on the empirical
observation that at high energies $B(s)$ increases with the logarithm
of $s$. Since the Kang-Nicolescu parametrization for the
total cross section is expressed in terms of $\ln s$ (Sec. III.B.2),
we replaced this dependence by the slope parameter:
\begin{eqnarray}
\sigma_{tot}^{pp}(s) &=& c_1 + c_2B + c_3B^2,
\nonumber \\
\sigma_{tot}^{\overline{p}p}(s) &=& c_1 + c_2B +c_3B^2 +
c_4 e^{-B/2},
\label{KNeqs}
\end{eqnarray}
where $c_i$, $i = 1,2,3,4$ are free fit parameters.
That is a strictly mathematical choice, having
nothing to do with the physics or model concept behind the
Kang-Nicolescu parametrization.
Fits to the experimental data
have been performed with the
CERN-MINUIT program and the results are displayed in Fig. 13.
It is expected that extrapolations to cosmic-ray energies
may be useful in the determination of the $pp$ total cross
section from $p$-air cross section, allowing to connect
$\sigma_{tot} - B$ in an almost model independent way. We are
presently investigating this subject.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm,height=7.0cm]{menonf13.eps}
\caption{Total cross sections in terms of the slope, and
the parametrization (38) \cite{mmmbjp04}.}
\end{center}
\end{figure}
In Ref. \cite{mmmbjp04} we also made use of the Donnachie-Landshoff
parametrization, which predicts a faster increase of the total cross
section as function of the slope parameter. Moreover, in \cite{mmmbjp04}
we also present a critical discussion on the recent measurement of
the slope parameter at the BNL RHIC, at 200 GeV, by the pp2pp Collaboration.
We call attention to the fact that
the combination $B =$ 16.3 $\pm$ 1.8 GeV$^{-2}$
and $\sigma_{tot} =$ 51.6 mb, indicated by the pp2pp analysis,
is in disagreement with the general trend for the
behaviors of $\sigma_{tot}$ and $B$. If this ``peer" is
correct, new physics is necessary.
Using the above
$B$ value as input in our parametrizations, the corresponding
values of the total cross sections show agreement with
the $\sigma_{tot}$ versus $B$ data. However, these inferred
values for $\sigma_{tot}$ indicate new physics when plotted
as function of the energy. We conclude that
if this
measurement is correct and represents an hadronic quantity,
its high value may indicate a ``break" in the slope near
0.02 GeV$^2$, a phenomenon that was never observed in both
$pp$ and $\bar{p}p$ scattering,
at $\sqrt s \leq $ 62.5 GeV and $\sqrt s \leq $ 1.8 TeV,
respectively and therefore, once more,
new physics is necessary.
\section{Eikonal Models}
It is expected that the eikonal function in the
momentum transfer space, $\chi(s,q)$, may be connected with some
microscopic
aspects of the underlying field theory (elementary interactions,
form factors, structure functions) and, as
mentioned before, it corresponds to a unitarized scheme connected
with the experimental data.
Eikonal models are characterized by different choices for
$\chi(s,q)$. In what follows we discuss our
results and researches through two eikonal models
(geometrical and QCD-based)
\subsection{Geometrical Model - Inelastic Channel}
In this subsection we review the description of $\bar{p}p$
multiplicities
distributions (inelastic channel) from models for the elastic channel
in the context of the geometrical picture (contact interactions).
\leftline{- \textit{ Elastic and Inelastic Channels}}
Through the Unitarity and the Inelastic Overlap Function,
defined in Sec. II.C, we can connect elastic and
inelastic scattering. This is done by expressing
the topological cross section
for producting an even number $n$ of charged particles
at $s$ in terms of $G_{in}$:
\begin{eqnarray}
\sigma_{n}(s)= \int d^{2}{\bf b}\ \sigma_{n}(s,b) =
\int d^{2}{\bf b}\
G_{in}(s,b) \left[ \frac{\sigma_{n}(s,b)}{G_{in}(s,b)} \right].
\nonumber
\end{eqnarray}
If $n(s)$ and $<n>_{(s)}$ are the hadronic and averaged multiplicities,
respectively, by introducting the KNO variable
$Z=n(s)/<n>_{(s)}$,
the hadronic multiplicity distribution may be expressed by
\begin{eqnarray}
\Phi (s,Z)= <n>_{(s)} \frac{\sigma_{n}(s)}{\sigma_{in}(s)}
= \frac{\int d^{2} {\bf b} \frac{G_{in}(b,s)}{r(b,s)}
\varphi
(\frac{Z}{r(b,s)})}{ \int d^{2} {\bf b}\ G_{in} (b,s)},
\nonumber
\end{eqnarray}
where $\varphi$ is the elementary multiplicity distribution
and $r(b,s) = <n>_{(b,s)}/<n>_{(s)}$ the elementary multiplicity
function.
In Ref. \cite{bmv}, in the context of the
geometrical picture, the elementary contact interaction
process was based on $e^+e^-$ scattering data. In this
approach we express
\begin{eqnarray}
r(s,b)=\xi(s) \chi_{I}^{\gamma}(s,b), \nonumber
\end{eqnarray}
where
\begin{eqnarray}
\xi(s)= \frac{\int d^{2} b\ G_{in}(s,b)}{\int d^{2} b\
G_{in}(s,b) \chi_{I}^{\gamma}(s,b)} \nonumber
\end{eqnarray}
and the power $\gamma$
is determined by fitting the average multiplicity from
$e^+e^-$ scattering data through a power law parametrization:
\begin{eqnarray}
<n>_{e^+ e^-} =
A[\sqrt s]^{\gamma}. \nonumber
\end{eqnarray}
The elementary distribution
$\varphi(Z/r(b,s))$ is represented by a Gamma distribution
and determined also by fits to $e^+e^-$ data.
With inputs for $G_{in}(s,b)$ and/or $\chi_{I}(s,b)$, obtained from fits
to elastic scattering data,
we have no free parameter and
the hadronic multiplicity distribution as function
of $Z$ and $s$ may be inferred.
\leftline{- \textit{ Elastic-channel inputs and results}}
In Ref. \cite{bmv} we made use of three inputs from the elastic
sector. Two are based on the Multiple Diffraction Formalism,
in which the eikonal in the momentum transfer space is expressed by
\begin{eqnarray}
\chi (b,s)= C\int qdqJ_{0}(qb)G_{A}G_{B}f,
\end{eqnarray}
where $G_{A}$ and $G_{B}$ are the hadronic form
factors, $f$ the elementary (constituent - constituent) amplitude
and $C$ does not depend on the transferred momentum. In this case
we made use of the parametrizations used by Chou and Yang
and also by Menon and Pimentel. Both present good descriptions
of the experimental data in the elastic channel. The other input
corresponds to the Short Range Expansion of the inelastic overlap
function,
introduced by Henzi and Valin,
\begin{eqnarray}
G_{in}(b,s)=P(s)\exp \{-b^{2}/4B(s)\}k(x,s), \nonumber
\end{eqnarray}
with $k$ being expanded in terms of a short-range variable $x=b\
\exp \{-(\epsilon b)^{2}/4 B(s)\}$, i.e.
\begin{eqnarray}
k(x,s)=\sum_{n=0}^{N} \delta_{2n}(s) \left[ \frac{\epsilon\
\exp \{1/2\}}{\sqrt{2B(s)}}\ x \right]^{2n}. \nonumber
\end{eqnarray}
With particular parametrizations excellent agreement with experimental
data on $pp$ and
$\overline{p}p$ elastic scattering is achieved, allowing to infer the
black-edge-large (BEL) behavior.
In Ref. \cite{bmv} a detailed discussion is presented on several
variants from the elastic channel and parametrizations from
$e^-e^+$ scattering. In particular, the results for the
multiplicities distributions, with the BEL inelastic overlap
function, at
$\sqrt s =$ 52.6 and 546 GeV are displayed in Fig. 14,
together with the experimental data.
The prediction shows that
the violation of the KNO scaling is well described.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm,height=7.0cm]{menonf14a.eps}
\end{center}
\vspace{0.3cm}
\begin{center}
\includegraphics[width=8.0cm,height=7.0cm]{menonf14b.eps}
\vspace{0.3cm}
\caption{Scaled multiplicity distribution for inelastic $pp$ data at
$\sqrt s =$ 52.6 GeV and $\bar{p}p$ data at 546 GeV, compared with the
model predictions \cite{bmv}.}
\end{center}
\end{figure}
\subsection{QCD-inspired models}
In this section we outline some research in course with
the eikonal approach in connection with some QCD concepts.
As we shall discuss the main point concerns the
gluon-gluon contributions to the hadronic cross sections
which we have investigated either from a dynamical gluon mass
approach or by introducting the momentum scale in the gluonic distribution
functions.
After a review on the basic formalism we outline some aspects
of both approaches.
\subsubsection{Basic formalism}
The formalism was introduced by Afek, Leroy, Margolis and
Valin \cite{qcdb1} and developed by several authors, including (for
our purposes) Durand, Pi \cite{dp}, Block, Gregores, Halzen and Pancheri
\cite{qcdb2}.
Originally, the point was to separate contributions from
soft (S) and semi-hard (SH) inelastic processes by expressing
\begin{eqnarray}
G_{inel}(s,b) &=& 1 - \bar{P}_{S} \bar{P}_{SH} \nonumber \\
&=&
1 - e^{- 2Re\ \chi_{S}(s,b)} e^{- 2Re\ \chi_{SH}(s,b)},
\nonumber
\end{eqnarray}
where $\bar{P}_{S}$ is the probability of $NO$ soft inelastic
process and $\bar{P}_{SH}$ the probability of $NO$ semi-hard
inelastic process.
Therefore, that indicated an additive contribution in the eikonal:
$\chi(s,b) = \chi_{S}(s,b) + \chi_{SH}(s,b)$.
In the recent version by Block et al. \cite{qcdb2}
different elementary contributions from quarks and gluons
have been introduced:
the gluon-gluon contribution comes
from the parton model, the
quark-quark from regge parametrization and the quark-gluon
by phenomenological inputs. In what follows we shortly review the
main formulas.
The normalization for the eikonal reads
\begin{eqnarray}
F(s, q) = ik \int_{0}^{\infty} bdb J_{0}(qb)
\left[ 1 - e^{i\chi(s,b)} \right], \nonumber
\end{eqnarray}
\begin{eqnarray}
\chi(s,b) = \mathop{\mathrm{Re}} \chi(s,b) + i \mathop{\mathrm{Im}} \chi(s,b). \nonumber
\end{eqnarray}
For $pp$ and $\bar{p}p$ scattering the crossing even and odd
contributions are
expressed by
$\chi_{p\bar{p}} = \chi^+ + \chi^-$ and
$\chi_{pp} = \chi^+ - \chi^-$. The odd eikonal is assumed not
to contribute at
the asymptotic energies
and is parametrized by
\begin{eqnarray}
\chi^{-}(s,b) = C^{-} \frac{m_0}{\sqrt s}e^{i\pi/4}w(b,\mu_{odd}).
\nonumber
\end{eqnarray}
Analyticity (generation of real and imaginary parts) for the
even part is assumed as given by the prescription
\begin{eqnarray}
\chi^{+}(s,b) \quad &\Rightarrow& \quad
\chi^{+}(se^{-i\pi/2},b) = \nonumber \\
&=& Re\ \chi^{+}(s,b) + i Im\ \chi^{+}(s,b).
\nonumber
\end{eqnarray}
The even eikonal is expressed as a sum of three contributions, from
quark-quark ($qq$), quark-gluon ($qg$) and
gluon-gluon ($gg$) interactions,
\begin{eqnarray}
\chi^+(s,b) = \chi_{qq}(s,b) + \chi_{qg}(s,b) + \chi_{gg}(s,b), \nonumber
\end{eqnarray}
which individually factorize in $s$ and $b$,
\begin{eqnarray}
\chi_{ij}(s,b) = i \sigma_{ij}(s) w(b, \mu_{ij}), \nonumber
\end{eqnarray}
where $i,j = q,g$.
The {\it impact parameter distribution function} for each process comes from
convolution involving dipole form factors (Chou-Yang Model):
\begin{eqnarray}
w_{ii}(b, \mu_{ii}) = \int d^2\vec{b}' \rho_{i}(|\vec{b}'|
\rho_{i}(|\vec{b} - \vec{b}'|), \nonumber
\end{eqnarray}
\begin{eqnarray}
\rho(b) = <G(q)> = < \frac{1}{(1 + q^2/\mu^2)^2} >, \nonumber
\end{eqnarray}
where the angular brackets denote the symmetrical two-dimensional
Fourier transform. Therefore,
\begin{eqnarray}
w_{ii}(b) = \frac{1}{8} \frac{\mu_{ii}^2}{12\pi}[\mu_{ii} b]^3
K_3(\mu_{ii} b),
\nonumber
\end{eqnarray}
and for $i \not= j$ it is assumed that
\begin{eqnarray}
\mu_{ij} = \sqrt{\mu_{ii} \mu_{jj}}. \nonumber
\end{eqnarray}
The {\it elementary cross sections} for each process are introduced as
follows. The {\it quark-quark} contribution is parametrized as a
constant plus a
Regge
(even) term,
\begin{eqnarray}
\sigma_{qq}(s) = C + C_{R}^{+} \frac{m_0}{\sqrt s} \nonumber
\end{eqnarray}
and the {\it quark-gluon} term as
\begin{eqnarray}
\sigma_{qg}(s) = C_{qg} \log \frac{s}{s_0}. \nonumber
\end{eqnarray}
The {\it gluon-gluon} contribution is considered as the responsible for
the increase of the total cross section at the highest energies
and is
calculated through the parton model
approach,
\begin{eqnarray}
\sigma_{gg}(s) = c_{gg}\int_{0}^{1} d\tau F_{gg}(\tau) \hat\sigma_{gg}(\hat{s}),
\end{eqnarray}
with
\begin{eqnarray}
F_{gg} = \int_0^1 \int_0^1 dx_1 dx_2 f_g(x_1) f_g(x_2) \delta(\tau - x_1x_2),
\end{eqnarray}
where $f_g(x_i)$ is the gluon distribution function, $\tau = \hat{s}/s$,
and the symbol
$\hat{ }$
denotes the elementary process. In \cite{qcdb2} the elementary cross
section is given by
\begin{eqnarray}
\hat\sigma_{gg}(\hat{s}) =
\frac{9\pi \alpha_{s}^2}{m_{0}^2} \theta(\hat{s} - m_{0}^{2}),
\end{eqnarray}
implying a cutoff $m_0$ for the particle production threshold
and it is assumed the following simple parametrization for the gluon
distribution function
\begin{eqnarray}
f_g (x) &=& N_g \frac{(1 - x)^5}{x^{1 + \epsilon}}, \nonumber \\
N_g &=& \frac{1}{2}
\frac{(6 - \epsilon)(5 - \epsilon)...(1 - \epsilon)}{5!}.
\end{eqnarray}
The model has 6 fixed parameters, $m_0$, $\epsilon$, $\mu_{qq}$,
$\mu_{gg}$, $\mu_{odd}$, $\alpha_s$ and 6 free parameters, determined
from fits
to $pp$ and $p\bar{p}$ {\it forward} scattering data, namely
$\sigma_{tot}(s)$, $\rho(s)$ and $B(s)$ above 15 GeV \cite{qcdb2}.
We have shown that the model
applies only to forward and small momentum
transfer regions \cite{lmmhadron02}.
In the next two sections we shall discuss two researches in course
concerning the determination of the contribution from gluon-gluon
interactions, Eqs. (39-42).
\subsubsection{Dynamical gluon mass}
The possibility that the gluon propagator may be regularized
by a dynamically generated gluon mass \cite{cornwall} has
recently provided important phenomenological description
of several processes \cite{natale}. The approach allows to
calculate the contribution for the elementary $gg$ cross section
Eq. (42) and the main point is the association of the mass scale
with the dynamical gluon mass.
The basic ingredients are the expressions for the
dynamical gluon mass,
\begin{eqnarray}
M^2_g(\hat{s}) =m_g^2 \left[
\frac{ \ln [(\hat{s}+4{m_g}^2) / \Lambda_{\mathrm{QCD}} ^2]}{ \ln
[(4{m_g}^2) / \Lambda_{\mathrm{QCD}}^2] }\right]^{- 12/11},
\nonumber
\end{eqnarray}
and the associated running coupling constant
\begin{eqnarray}
\alpha_{s} (\hat{s})= \frac{4\pi}{\beta_0 \ln\left[
(\hat{s}+4M_g^2(\hat{s}))/\Lambda_{\mathrm{QCD}}^2 \right]},
\nonumber
\end{eqnarray}
where $\beta_0 = 11- \frac{2}{3}n_f$ and $n_f$ is the number of
flavors. Preliminary tests with these contributions, in the context
of the model described in the last subsection, have shown that the
experimental data on $\sigma_{tot}(s)$, $\rho(s)$ and $B(s)$ are well
described \cite{lmmmn}, as exemplified in Fig. (15). We are presently
investigating the contributions from the other elementary processes,
$qq$ and $qg$.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm,height=7.0cm]{menonf15.eps}
\caption{Description of the total cross sections through the QCD-based
model with dynamical gluon mass $m_0 =$ 500, 600 and 700 MeV \cite{lmmmn}.}
\end{center}
\end{figure}
\subsubsection{Momentum Scale}
Presently, we are attempting to improve
the descriptions of the QCD-inspired models by taking into account
the momentum transfer scale
in the gluon distribution functions. The point is to replace
the simple choice in Eq. (42), by distribution functions with
the $Q^2$ dependence, namely
\begin{eqnarray}
f_{g}(x_i) \rightarrow f_{g}(x_i, Q^2). \nonumber
\end{eqnarray}
That can be implemented, following the approach by Durand and Pi \cite{dp},
by introducing the differential cross section
\begin{eqnarray}
\hat\sigma_{gg}(\hat{s}) =
\int d|\hat{t}| \frac{d\hat\sigma_{gg}}{d|\hat{t}|}(\hat{s}, \hat{t}),
\nonumber
\end{eqnarray}
with
\begin{eqnarray}
\frac{d\hat\sigma_{gg}}{d|\hat{t}|}(\hat{s}, \hat{t}) &=&
\frac{9\pi\alpha_{s}^2}{2}
[\frac{3}{\hat{s}^2} + \frac{\hat{t}}{\hat{s}^3}
+ \frac{\hat{t}^2}{\hat{s}^4} + \frac{1}{\hat{t}^2} +
\frac{1}{\hat{s}\hat{t}} \nonumber \\
&-& \frac{\hat{t}}{\hat{s}(\hat{s} + \hat{t})^2}]
\nonumber
\end{eqnarray}
and by considering $Q^2 = |\hat{t}|$.
The novel input concerns the updated determinations of the gluon distribution
functions (CETEQ6), parametrized by means of
Chebyshev polynomials. Presently, the implementation in the QCD-inspired
approach is being
developed.
\section{Nonperturbative QCD}
As commented before the difficulties associated with high-energy
soft processes arise
from the fact that perturbative QCD can not be applied
and presently we do
not know how to calculate even the elastic hadron-hadron scattering
amplitudes from a pure nonperturbative
QCD formalism. However, progresses have been achieved through the approach
introduced by Landshoff and Nachtmann \cite{landotto}, developed by
Nachtmann \cite{otto1} and connected with the
Stochastic Vacuum Model (SVM) (introductory reviews may be
found in \cite{rev}).
In particular, through this
formalism and in some restricted kinematic conditions, it is possible to
connect the gluon two-point correlation function with elementary
(quark-quark) scattering amplitude.
In this section we review the
results we have obtained for these amplitudes with correlators determined
from lattice QCD and also in the context of the Constrained Instantons.
\subsection{Stochastic Vacuum Model}
The approach has its origins in the attempts by Landshoff and
Nachtmann to connect soft high-energy processes with nonperturbative
properties of the QCD vacuum, as for example, the {\it gluon condensate}
introduced by Shifman, Vainstein and Zakharov \cite{svz}. In the first
version \cite{landotto} quarks couple with Abelian gluons. The non-Abelian
version was developed by Nachtmann in the context of QCD and using the
eikonal method for high energy interactions \cite{otto1}.
The scattering amplitude is calculated by means of a functional integral
approach and is connected to a correlation function of two lightlike
Wegner-Wilson loops. These correlation functions can be evaluated through the
Stochastic Vacuum Model, in which the low frequency contributions
to the functional integral of QCD are described in terms of a stochastic
process by means of a cluster expansion \cite{svm}. The model incorporates
the gluon
condensate concept and assumes that the correlation of two field strengths
decreases rapidly with distance; due to an effective chromomagnetic monopole
condensate, the QCD vacuum acts as a dual superconductor.
In this formalism the low frequencies contributions in the functional
integral of QCD are described in terms of a stochastic process, by means of
a cluster expansion. The most general form
of the lowest cluster is the gauge invariant two-point field
strength correlator \cite{svm,dfk}
\begin{eqnarray}
&<&{\bf{F}}_{\mu \nu}^{\rm C}(x){\bf{F}}_{\rho
\sigma}^{\rm D}(y)>= \nonumber \\
&=&{\delta}^{\rm CD}g^{2}\frac{<FF>}{12(N_c^2-1)}\{(
{\delta}_{\mu\rho}{\delta}_{\nu \sigma}-{\delta}_{\mu \sigma}
{\delta}_{\nu\rho}){\kappa}D({z^2/a}^{2})+ \nonumber \\
&+&
\frac{1}{2}[{\partial}_{\mu}(z_{\rho}{\delta}_{\nu
\sigma}-z_{\sigma}{\delta}_{\nu
\rho}) + \nonumber \\
&+&{\partial}_{\nu}(z_{\sigma}{\delta}_{\mu
\rho}-z_{\rho}{\delta}_{\mu \sigma})](1-{\kappa})D_{1}({z^2/a}^{2})\},
\nonumber
\end{eqnarray}
where $z=x-y$ is the two-point distance, $a$ is a characteristic
correlation length, ${\kappa}$ a constant, $g^{2}<FF>$ the gluon
condensate and $N_c$ the number of colours
(${\rm C}, {\rm D}=1,...,N_c^2-1$).
The two scalar functions $D$ and $D_{1}$ describe the correlations
and they play a central role in the application
of the SVM to high energy scattering.
Once one has information
about $D$ and $D_{1}$, the SVM leads to the determination of the
elementary
quark-quark scattering amplitude, which constitutes important
input for models
aimed to construct hadronic amplitudes. The main formulas are as
follows.
The elementary amplitude $f$ in the momentum transfer space is expressed
in terms of the elementary profile $\gamma$
by
\begin{eqnarray}
f(q^2)=\int_0^{\infty}bdb J_0(qb)\gamma(b).
\end{eqnarray}
In the Nachtmann approach the no-colour exchange
parton-parton (loop-loop) amplitude can be written as
\begin{eqnarray}
\gamma&=&{\langle}Tr[{\cal{P}}e^{-ig{\int}_{loop 1}d{\sigma}_{\mu
\nu}F_{\mu \nu}(x;w)}-1] \nonumber \\
& & Tr[{\cal{P}}e^{-ig{\int}_{loop2}d{\sigma}_{\rho\sigma}
F_{\rho \sigma}(y;w)}-1]{\rangle}, \nonumber
\end{eqnarray}
where ${\langle}{\rangle}$ means the functional integration over the
gluon fields (the integrations are over the respective loop areas),
and $w$ is a common reference point from which the integrations are
performed. In the
SVM by taking the Wilson loops on the light-cone the {\it leading order}
contribution to the amplitude is given by
\begin{eqnarray}
\gamma(b)=\eta{\epsilon}^{2}(b),
\end{eqnarray}
where $\eta$ is a constant depending
on normalizations and
\begin{eqnarray}
\epsilon(b)=g^{2}{\int}{\int}d{\sigma}_{\mu
\nu}d{\sigma}_{\rho \sigma}Tr{\langle}F_{\mu \nu}(x;w)F_{\rho
\sigma}(y;w){\rangle}. \nonumber
\end{eqnarray}
After a two-dimensional integration, $\epsilon(b)$ can be
expressed in terms of the correlation functions
by
\begin{eqnarray}
\epsilon(b)=\epsilon_I(b)+\epsilon_{II}(b),
\end{eqnarray}
where
\begin{eqnarray}
\epsilon_I(b)={\kappa}{\langle}g^2FF{\rangle}
{\int}_{b}^{\infty}db'(b'-b){\cal{F}}_{2}^{-1}
[D(k^2)](b'),
\end{eqnarray}
\begin{eqnarray}
\epsilon_{II}(b)=({1-\kappa}){\langle}g^2FF{\rangle}
{\cal{F}}_{2}^{-1}[\frac{d}{dk^{2}}D_{1}(k^2)](b).
\end{eqnarray}
For
${\cal{D}}=D$ or ${\cal{D}}=D_{1}$ we have
${\cal{D}}(k^2)={\cal{F}}_4[{\cal{D}}(z^2)]$, where
${\cal{F}}_n$ denotes a n-dimensional Fourier transform.
With the above formalism, once one has inputs for the correlation
functions $D(z)$ and $D_1(z)$, the elementary amplitude in the
momentum transfer space, Eq. (43), may, in
principle, be evaluated through Eqs.
(44-47).
It is important to stress that, as constructed, the
formalism is intended for small momentum transfer ($q^2 \lesssim {\cal O}(1)$
GeV$^2$) and asymptotic energies $s \rightarrow \infty$.
Despite of these limitations, the investigation of soft high energy
scattering at the energies presently available has led to satisfactory
results
\cite{dfk,fp,berger}.
\subsection{Elementary Amplitudes}
In this section we review the results we have obtained from inputs
for the above correlators from lattice QCD
\cite{mmt,mm03}. We also comment the
research in course in the semi-classical context of Instantons
\cite{doro}.
\subsubsection{Correlators from Lattice QCD}
Numerical determinations of the above correlation
functions, in limited interval of physical distances, exist from lattice
QCD in both quenched approximation (absence of fermions)
and full QCD (dynamical fermions included) \cite{digi}.
With the procedure described above (see \cite{mmt} for all the
calculational details),
the elementary scattering amplitude in the momentum transfer
space can be
determined in numerical form.
In order to obtain analytical expressions, suitable for
investigating distinct contributions and also for phenomenological uses,
we have parametrized these numerical points through a sum of exponentials
in $q^2$:
\begin{eqnarray}
\frac{f(q^2)}{f(0)}=\sum_{i=1}^n\alpha_i e^{-\beta_i q^2}.
\end{eqnarray}
The results are displayed in Fig. (16) from both quenched
approximation and full QCD, together with the corresponding exponential
components.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm,height=7.0cm]{menonf16a.eps}
\end{center}
\begin{center}
\includegraphics[width=8.0cm,height=7.0cm]{menonf16b.eps}
\caption{Elementary amplitudes from quenched and full QCD and the
exponential components through parametrization (48) \cite{mm03}.}
\end{center}
\end{figure}
Our main conclusions are the following \cite{mm03}:
(1) the amplitudes decrease smoothly as the
momentum transfer increases and they do not present zeros; (2) the decreasing
is faster when going from quenched approximation to full QCD (with
decreasing quark masses), and this effect is associated with the increase
of the correlation lengths;
(3) the dynamical fermions generate two contributions in the
region of small momentum transfer, which are of the same order at
$q^2 \sim$ 1 GeV$^2$ (only one contribution is present in the
case of quenched approximation).
We understand that result (3) may suggest some kind of change in
the dynamics at the elementary level, near $q^2 \sim$ 1 GeV$^2$ and at
asymptotic energies. If that is true, some signal could be expected
at the hadronic level.
One possibility is that this effect can be associated with the position
of the
dip (or the beginning of the ``shoulder") in the hadronic (elastic) differential
cross section data. The asymptotic condition embodied in our
result indicates that $q^2 \sim$ 1 GeV$^2$ seems to be in agreement with
the limit of the shrinkage of the diffraction peak, empirically verified when
the energy increases in the region 23 GeV $ \leq \sqrt s \leq $ 1.8 TeV.
\subsubsection{ Correlators from the Instanton Approach}
By means of the stochastic vacuum formalism we also presently
investigate the elementary amplitudes using
correlators determined in the context of the \textit{constraint} instanton
approach, developed by A. Dorokhov and collaborators \cite{instanton}.
The basic picture is that of an instanton field dominating
at small distances and decreasing exponentially
at large distances in the physical vacuum.
In Ref. \cite{doro}, we make use of suitable parametrizations for the
correlators and investigate the effects of the contributions
from the short and long range correlations
in the determination of the full correlator.
Denoting those contributions
as $D_I(z)$ and $D_L(z)$, respectively, we introduce a
dimensionless
parameter $\alpha \equiv \eta_g \rho_c$ in terms of the {\it driven parameter}
$\eta_g$ and the {\it size parameter} $\rho_c$ \cite{doro}.
Since
$\eta_g$ is correlated with the relative contribution of each kind
of correlator, we consider two extreme cases: 1) equal contributions
(weights 0.5 and 0.5), corresponding to $\alpha = 1.0$; 2) almost pure
instanton contribution (weights 0.99 and 0.01), corresponding to
$\alpha = 0.1$.
In the lack of information on the long range component, and for our purpose,
we consider parametrization in a Gaussian form \cite{doro}
\begin{eqnarray}
D_L(z) = \exp \{-(2/2.5)^2 z^2\}. \nonumber
\label{independentlrc}
\end{eqnarray}
For the short range case, $\alpha = 0.1$, we introduce the parametrization,
\begin{eqnarray}
D_I(z)&=& 0.7119\exp\{-(2.403|z|)^2\} \nonumber \\
&+&0.2899\exp\{-(1.485|z|)^2\} \nonumber \\
&-& 9.456\times 10^{-3}\exp\{-1.277|z|\} . \nonumber
\label{ddzalfa0pt1}
\end{eqnarray}
The
full correlator is then determined by
\begin{eqnarray}
D(z) = 0.99 D_I(z) + 0.01 D_L(z).
\label{src}
\end{eqnarray}
For the long range case, $\alpha = 1.0$, the parametrization takes the form
\begin{eqnarray}
D_I(z)&=& 0.80084\exp\{-(2.3025|z|)^2\} \nonumber \\
&-&3.3846\times 10^{-2}\exp\{-(0.97119|z|)^2\} \nonumber \\
&+&0.24225\exp\{-2.7706|z|\}, \nonumber
\label{lrc}
\end{eqnarray}
and for the full correlator we have
\begin{eqnarray}
D(z) = 0.5 D_I(z) + 0.5 D_L(z).
\label{ddzwithlrc1pt0}
\end{eqnarray}
With the Eqs. (49) and (50) we can
calculate the elementary amplitude through the steps indicated
in Sec. VI.A. The results are displayed in Fig. (17).
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm,height=7.0cm]{menonf17.eps}
\caption{Elementary amplitudes from the constrained instanton approach.}
\end{center}
\end{figure}
A central result is that if the contribution from the long
range correlator is small, that is, an almost purely instanton case, the
corresponding elementary amplitude presents
a minimum. In terms of the associated differential cross section,
this implies a diffractive pattern in the momentum
transfer space, a result already indicated
in some phenomenological approaches \cite{menon,mmt}.
In the case of equal weights the amplitude decreases
monotonically with the momentum transfer.
We conclude that in the context of the instanton approach, the
balance between the contributions of the short and long
range correlators is a crucial point for the determination of the
behavior of the elementary amplitudes. Further investigation
along this line can bring new important insights on the
connection between instanton correlators and the physical
quantities which characterize the high-energy hadronic
scattering.
\section{Perspectives and Outlook}
In this section we outline some perspectives in the area of high-energy
elastic hadron scattering from both experimental and theoretical point of views.
Certainly, some ideas may be biased by our own knowledge and our
own personal view.
\subsection{Experiments}
From the experimental point of view the perspectives
are very optimistic due to
the new generation of experiments with both accelerators and
cosmic-ray observatories. Let us quickly summarize some projects
in development.
The upgrade of the Fermilab Tevatron machine, together with upgrades and new
devices in the CDF (Collider Detector Facility) and D0 detectors, are going to
provide improved investigations on $\bar{p}p$ collisions at $\sqrt s \sim 2$
TeV. Although the main purpose of the experiment concerns hard diffraction, it
will be possible to investigate elastic scattering in both high and low $q^2$
regions, the slope parameter, total cross section and single diffraction.
Of topical importance for soft physics, the new
determination of the total cross section shall possibly bring a solution for
the puzzle represented by the discrepant results around $2$ TeV (E710/E811 and
CDF).
At the Relativistic Heavy Ion Collider (RHIC) in the Brookhaven National
Laboratory, $pp$ collisions are presently being investigated at energies never
reached before: $\sqrt s : 50\ -\ 500$ GeV. The experiment ``Total and
Differential Cross Sections and Polarization Effects in $pp$ Elastic
Scattering at RHIC'' (pp2pp) plans to investigate both elastic scattering and
diffraction dissociation (single and double), in addition to spin effects.
This will provide the first opportunity for direct comparison
between $pp$ and $\bar{p}p$ scattering at the highest collider energies.
Although in a bit longer term, at the CERN Large Hadron Collider (LHC), the
TOTEM experiment (Total Cross Section, Elastic Scattering and Diffraction
Dissociation at the LHC) is specifically planned to study soft diffractive
physics in $pp$ collisions at $\sqrt s \sim 16$ TeV. In
particular, diffraction dissociation, total cross
section and elastic scattering at large values of the momentum transfer will be
investigated, up to
$q^2 = 10\ -\ 15$ GeV$^2$. That will certainly allow discrimination and
selection of various models and approaches, giving fundamental information at
large momentum transfer; for example, showing the existence or not of
structures. Moreover, this experiment will probably provide a
final answer on the possible differences between $pp$ and $\bar{p}p$ total
cross sections and the correct power $\gamma$ in the $\ln^{\gamma} s$
dependence of $\sigma_{tot}(s)$.
The most energetic event detected in cosmic-ray experiments had $E_{lab} = 3
\times 10^{20}$ eV, corresponding to an energy of $50$ Joules!
Goals of the Auger project are the measurement of arrival direction
and the
energy and mass composition of cosmic rays above $E_{lab} = 10^{19}$ eV.
For $pp$ collisions this means $\sqrt s $ above $140$ TeV, nearly
$10$ times the LHC energy.
In addition to the astrophysical importance of the experiment, the measurement
of the longitudinal development of showers will provide severe tests on
hadronic interaction models. As a consequence, among others, the puzzles
concerning the extraction of $pp$ cross section from $p-air$ production cross
section may receive better insights, allowing more precise determinations and
at energies possibly never to be reached by accelerator machines.
\subsection{Theory}
Elastic hadron scattering (and soft diffractive processes in general)
is a long distance phenomena and therefore we expect and look for a theoretical
treatment via non-perturbative QCD. Despite all the difficulties mentioned along
this manuscript, we understand that two approaches deserve special attention.
One of them concerns the approach by Nachtmann and the Stochastic Vacuum Model
(Sec. VI.A) \cite{otto1,rev}.
Although under restrictive kinematic conditions (momentum transfer of the
order or below 1 GeV$^2$, and asymptotic energies, $s \rightarrow \infty $) the
formalism has provided interesting results in the investigation of
the physical quantities that characterize the elastic $pp$
and $\bar{p}p$ scattering \cite{dfk,berger}, in special the works
by Ferreira and Pereira, connecting experimental observables and QCD parameters
\cite{fp}.
Attempts to implement the energy dependence in pure QCD grounds
may be an important task for the near future.
The other approach is associated with evidences for finite gluon
propagator and running coupling in the infrared region and
that is the case in some classes of solutions of the
nonperturbative Schwinger-Dyson equations. In particular,
in the solution proposed by Cornwall \cite{cornwall} the gluon
acquires a dynamical mass leading to a freezing of the coupling
constant in the infrared region. As referred before, Natale and
collaborators \cite{natale} have discussed several phenomenological tests,
reaching
interesting results which have permitted the development of the
formalism and the selection of adequate basic inputs.
That opens a new way to investigate long distance phenomena
with a finite calculational approach.
We also understand that the connections between soft and semi-hard processes,
typical of QCD-based models in the eikonal context, may bring
new insights for the development of adequate calculational schemes in
the nonperturbative treatment of high-energy elastic collisions.
\section{Summary and Final Remarks}
Despite its simplicity,
elastic hadron scattering constitutes a topical problem in
high-energy Physics. Although the bulk of experimental data
can be efficiently described in different phenomenological
contexts,
we are still facing the lack of a treatment and of
a reasonable understanding of these processes based exclusively
on QCD.
Our main strategy in investigating elastic scattering has been to look for
descriptions based
on the high-energy principles and theorems from Axiomatic
Quantum Field Theory and, simultaneously, attempting to extract
``empirical" information from all the experimental
data available. Tests of discrepant data and their influence
on the extracted information play a central role.
In that way we hope to get
feedbacks for theoretical development in nonperturbative
and semi-hard QCD.
We can summarize our main recent results as follows.
In the context of the \textit{Analytic Approach},
we have investigated the effects of discrepant experimental information
on the total cross sections in both accelerator and cosmic-ray
energy regions. By means of analytical fits, we have obtained extrema bounds
for the soft Pomeron intercept, namely $\alpha^{upper}_{\tt I\!P}(0) = 1.109$
and $\alpha^{lower}_{\tt I\!P}(0) = 1.081$.
We have also obtained novel constrained bounds for the intercept from spectroscopy
data (fitted Regge trajectories from Chew-Fautschi plots) and extended
the analysis to several reactions.
That information on the Pomeron bounds may be important for
phenomenological developments and projects for new experiments.
We have also shown that the presence of the Odderon in the real
part of the elastic hadronic amplitude is not forbidden by the
bulk of experimental data on $\sigma_{tot}$ and $\rho$. In particular,
the fit with the Kang-Nicolescu parametrization has
indicated a crossing in $\rho(s)$, with $\rho_{pp}$ becoming
greater than $\rho_{\bar{p}p}$ above $\sqrt s$ = 70 GeV. That
parametrization predicts $\rho_{pp}(\sqrt s = 200 \mathrm{GeV})
= 0.134 \pm 0.005$ (RHIC regions). Detailed investigation
on the applicability of DDR have shown that, once the subtraction
constant is used as a free fit parameter, the DDR is
equivalent to the IDR with finite lower limit ($s_0 = 2m^2$).
That result was obtained for the class of entire functions in the
logarithm of the energy (typical of analytic models).
In the context of \textit{Model Independent Analyses}, we have investigated the
correlations
between the experimental data on total cross section and the slope parameter.
The parametrization introduced is based on the empirical behavior of these
quantities and extrapolations to cosmic-ray energies may be useful
in the determination of proton-proton total cross sections
from proton-air cross sections.
By means of a novel model independent fit procedure to the differential cross
section data,
we have found statistical evidence for eikonal zero in the momentum
transfer space and that the position of the zeros decreases as the energy
increases.
The zero position shows agreement with the result recently obtained for
the electromagnetic form factor and inferred from elastic electromagnetic
$e^{-}p$ scattering (polarization transfer experiments). Since our analysis
concerns only hadronic interactions
the results may bring new insights in the investigation of electromagnetic
and hadronic form factors. We are also treating analytic fits with
free parameters depending on the energy so as to develop a model
independent predictive approach.
In the context of \textit{Eikonal Models} we have obtained connections between
the elastic and inelastic channel by means of the geometrical picture,
and correlating $pp$ and $\bar{p}p$ scattering with contact interactions
simulated by $e^{-}e^{+}$ distributions and multiplicities. With the class
of QCD-based or QCD-inspired models we have developed novel gluonic
contributions by means of two approaches. One is based in solutions
of the Schwinger-Dyson equations characterized by the dynamical gluon mass.
The other one is intended to take into account the $Q^2$- scale in updated gluonic
distribution functions. Certainly, the two approaches are not independent, and
we presently investigate their simultaneous implementation.
In the context of the \textit{Stochastic Vacuum Model}, we have obtained
novel results for the elementary scattering amplitudes, making use of
correlators determined either from lattice QCD or from constrained instantons.
In both cases the elementary amplitudes present no zeros (change of
sign in the momentum transfer space). In the context of eikonal models,
in which the eikonal is expressed in terms of the elementary amplitudes
and form factors, Eq. (38), this result corroborates the interpretation of
the zero in the hadronic form factor, and therefore, its dependence on the energy.
\begin{acknowledgments}
I am thankfull to Y. Hama and F.S. Navarra for the invitation
to present this review. For fruitful comments and discussions along these
Workshops I am particularly grateful to E. Ferreira, Y. Hama and T. Kodama.
I am deeply thankful to those who have developed research programs
under my supervision and are co-authors of all the works
reviewed here: R.F. \'Avila,
P.C. Beggio, S.D. Campos, P.A.S. Carvalho, E.G.S. Luna, A.F. Martini,
J. Montanha, and D.S. Thober.
I am also thankful to A. Di Giacomo, P. Valin, A. Dorokhov, A.A. Natale
and R.C. Rigitano for valuable discussions.
This work was supported by FAPESP (Contract N. 00/04422-7).
\end{acknowledgments}
|
{
"timestamp": "2005-02-11T16:05:49",
"yymm": "0411",
"arxiv_id": "hep-ph/0411400",
"language": "en",
"url": "https://arxiv.org/abs/hep-ph/0411400"
}
|
\section{Introduction}
Recently, there has been a considerable interest in cosmological aspects of string
theory (see~\cite{Quevedo} for a recent review).
In~\cite{Kachru, BKQ, Becker, Raise, Vijay}, several methods
of producing de Sitter (dS) vacua in string compactifications were presented.
In~\cite{Kachru, BKQ, Raise}, it was suggested that various
corrections to the supergravity potential energy can raise a supersymmetric
anti de Sitter (AdS) vacuum to a metastable dS vacuum. In~\cite{Becker}, based
on the earlier work~\cite{Curio},
a dS vacuum was created by balancing various exponential superpotentials
and in~\cite{Vijay}, it was argued that a dS vacuum can be created
by taking into account higher order corrections to the moduli
Kahler potential.
In addition, in~\cite{6people, Moh}, it was studied how
effects
of gravity and quantum particle production could trap moduli at enhanced symmetry points.
Furthermore, a substantial progress has been achieved towards inflation
in string
theory~\cite{Dtye, Alexander, Burg, Shiu, Choud, HHK, Raul, Dasgupta, Juan, Kallosh1, Pilo, Burg1, Oliver, Ii, Kallosh2, Kallosh3, 1, 2, 3, 4, 5}.
In these models, inflation was studied within the context of $D$-branes.
Under certain conditions the $D$-brane modulus can be treated as an inflaton.
However, all these models usually have two common problem.
Inflation is often realized in compactification or brane world
scenarios which do not correspond to realistic four-dimensional physics.
The other problem is that, in addition to the inflaton, there are,
usually, other moduli whose stabilization could be a problem
and whose presence can violate the slow roll conditions.
In this paper, we would like to explore the possibility of creating
an inflationary potential within the framework
of strongly coupled heterotic string theory, or hetorotic M-theory~\cite{HW1, HW2, Witten96}.
Such compactifications have a lot of attractive phenomenological features
(see~\cite{Faraggi} for a review on phenomenological aspects of M-theory).
Various GUT- and Standard Model-like theories were obtained
from heterotic compactifications on Calabi-Yau threefold~\cite{DLOW, Standard, Rene, Volker}.
For example, in~\cite{Volker}, vector bundles on Calabi-Yau manifolds
with $Z_3 \times Z_3$ homotopy group were constructed. A compactification
on such a manifold can lead to the Standard Model with suppressed
nucleon decay. The actual particle spectrum in such theories
was studied in~\cite{Yang1, Yang2, Yang3}.
One more
attractive feature of such compactifications is that it is possible
to stabilize moduli in a phenomenologically acceptable range~\cite{BO, Raise}.
The set of moduli considered in~\cite{BO, Raise} was
very general. Nevertheless, it was not complete.
In~\cite{BO, Raise}, it was assumed that the Calabi-Yau threefold had
enough isolated genus zero curves to stabilize all $h^{1,1}$ moduli.
It was also assumed that the five-branes in the bulk wrapped only
isolated genus zero curves. Even though such compactifications can certainly
exist, a generic compactification with $h^{1,1}$ greater than one
contains various, not necessarily isolated genus zero, cycles as well five-branes
wrapped on them. In this case, it is quite possible that
not all $h^{1,1}$ moduli can be stabilized by methods presented in~\cite{BO, Raise}.
The moduli of a five-brane wrapped on a non-isolated genus zero curve or a higher
genus curve cannot be stabilized by methods of~\cite{BO, Raise} either.
In this paper, we add these new moduli. We show that this new
additional $h^{1,1}$ moduli can be stabilized
in a supersymmetric AdS minimum
by the slight modification
of ideas of~\cite{BO, Raise}.
The five-brane moduli cannot be stabilized this way.
Surprisingly, we find that they can be stabilized
in a non-supersymmetric AdS minimum.
Of course, what we really mean by this is that the system of moduli
containing the moduli of this new five-brane admits a
non-supersymmetric AdS vacuum.
However, the potential energy has one more minimum
when the five-brane coincides with the visible brane.
A heterotic M-theory vacuum can contain several five-branes
wrapped on non-isolated genus zero or higher genus
curves. Those which are located relatively far away from
the visible brane will be stabilized.
On the other hand, those which are located close enough to
the visible brane will roll towards it and, eventually, collide with it.
We show that these five-branes can be stabilized as well
by balancing the supergravity potential energy against the Fayet-Iliopoulos
terms~\cite{DSW} induced by an anomalous $U(1)$ gauge group in the hidden sector.
This shows that the most general set of heterotic M-theory moduli can be
stabilized. Furthermore, the cosmological constant can be positive
and fine tuned to be very small.
By balancing the supergravity potential energy against
Fayet-Iliopoulos terms, it is also possible
to create a positive potential satisfying the slow roll conditions
and treat the five-brane translational modulus as an inflaton. Iflation takes place when the five-brane
approaches the visible brane.
However, this potential
has one negative feature. It has a vanishing first derivative when
the five-brane coincides with the visible brane.
This means that it takes infinite time for the branes to collide.
This also means that the primordial fluctuations
will become infinite.
On the other hand, at very short distances, one cannot
trust the low-energy field theory because new states
are expected to become massless. At the present time,
physics at short distances in heterotic M-theory is not known.
Nevertheless, we present an argument how these new
state can terminate inflation before the fluctuations became too big.
Once the five-brane hits the visible brane, it gets dissolved
into it and turns into new moduli of the vector bundle,
so called transition moduli, studied in~\cite{BDO1, BOR}.
This process is called small instanton transition~\cite{Wittensmall, Seiberg, KS, OPP}.
Thus, the post-inflationary phase does not have the inflaton but has
extra moduli of the vector bundle. These moduli are easier to stabilize~\cite{BO}.
Therefore, the new system of moduli can be stabilized, whereas during
inflation this was not the case. In addition,
we argue that, after a small
instanton transition, generically, the cosmological constant changes.
It is possible to decrease the cosmological constant and, by fine tuning,
make it consistent with observations. Let us point out that, though, besides the inflaton,
there are various other moduli during inflation,
they are all taken into account. The potential energy has a minimum in all the other
directions. Therefore, dynamically, one expects that all these moduli will
roll very fast in their minimum leaving the five-brane to roll slowly
towards the visible brane.
This paper is organized as follows. In Section 2, we discuss the
system of moduli in compactifications with $h^{1,1}$ greater than
one. The reason is twofold. First, we would like
to obtain more general results on moduli stabilization. In particular,
we would like to stabilize the $h^{1,1}$ moduli that
do not have a non-perturbative superpotentials and, hence, cannot
be stabilized by methods of~\cite{BO, Raise}.
Second, before we begin to study potentials for the five-brane,
whose stability properties are more complicated,
it is important to understand how the remaining moduli of the system
are stabilized. The system of (complex) moduli considered in this section
includes the complex structure moduli, the volume modulus, two $h^{1,1}$ moduli and the moduli of the
five-brane wrapped on an isolated genus zero cycle.
One of the two $h^{1,1}$ moduli is assumed to be associated with an isolated
genus zero curve and, hence, has a non-perturbative superpotential.
The other one is associated with a non-isolated genus
zero curve or a higher genus curve. The non-perturbative
superpotential does not depend on this modulus.
By the slight modification of ideas of~\cite{BO, Raise}, we show that
this system can be stabilized in a supersymmetric AdS vacuum.
The crucial moment is that, if $h^{1,1}$ is greater that one, it is possible to
choose one of the contributions to the tension of the hidden brane
to be positive without having the gauge coupling constant
stronger in the visible sector.
The system of moduli can be supplemented by vector bundle moduli~\cite{BOR}.
They can be stabilized as well~\cite{BO}.
For simplicity, we will ignore them.
In Section 3, we add one more five-brane to the system.
This five-brane is wrapped on a non-isolated genus zero or higher genus
curve.
Approximately, we can treat the rest of the moduli
fixed and consider an effective potential for the remaining five-brane modulus.
This potential is very difficult to analyze analytically.
A graphical analysis shows that, generically,
it has a non-supersymmteric AdS minimum.
Nevertheless, if a five-brane was originally located close
to the visible brane, it will roll towards it.
In the rest of the paper, we concentrate only on
dynamics of rolling five-branes.
We modify this effective potential with Fayet-Iliopoulos terms and show that addition of a
Fayet-Iliopoulos term in the hidden sector can stabilize a rolling five-brane.
We also show that the cosmological constant in such a vacuum can be
positive and small. The results from Sections 2 and 3 provide
stabilization of heterotic moduli in the most general set-up
in a vacuum with a positive cosmological constant.
They also indicate that it is conceivable to obtain
two distinct dS vacua. One of them is the lift of the non-supersymmetric AdS minimum.
The other one is created by addition of a Fayet-Iliopoulos
term in the hidden sector.
In Section 4, we argue that, by balancing the supergravity
potential energy and Fayet-Iliopoulos terms, it is
also possible to construct a potential with inflationary
properties. One of the slow roll parameters turns out to be naturally
much less than one. The other one can be (not necessarily fine)
tuned to be much less than one.
We also discuss the amount of inflation and primordial fluctuations.
In the last subsection of Section 4, we discuss how the system
can escape from inflation at very short distances.
We give a qualitative argument how the appearance of new light
states in the field theory can provide such an escape.
In Section 5, we discuss the post-inflationary phase.
After inflation, the five-brane hits the visible brane and
disappears through a small instanton transition.
The new system of moduli does not contain the five-brane but has extra vector
bundle moduli. Unlike the five-brane modulus, these moduli can be
stabilized. We show that the new system of moduli can be
stabilized in a vacuum with a positive cosmological constant which can be
fine tuned to be very small.
\section{Supersymmetric AdS Vacua in Models with $h^{1,1}>1$}
\subsection{The System of Moduli}
In this paper, we work in the context of strongly coupled heterotic string
theory~\cite{HW1, HW2} compactified on a Calabi-Yau threefold~\cite{Witten96, LOW4}.
To one of the orbifold fixed planes we will refer as to the visible brane
(or the visible sector). To the other one we will refer as to the hidden brane
(or the hidden sector). Such compactifications also allow five-branes
wrapped on holomorphic cycles in the Calabi-Yau manifold and parallel
to the orbifold fixed planes.
Moduli stabilization in this theory
was performed in a relatively general setting in~\cite{BO, Raise}.
Nevertheless, it was assumed in~\cite{BO, Raise} that the Calabi-Yau manifold has enough
isolated genus zero curves to stabilize all the $h^{1,1}$ and five-brane moduli.
In this paper, we would like to consider a more complicated
set-up when the Calabi-Yau threefold has two-cycles, one represented
by isolated genus zero curve and the other one by curves of a different type.
They could be either non-isolated genus zero curves or curves
of a higher genus. In both of these two cases, no non-perturbative
superpotential for the corresponding $h^{1,1}$ modulus can be
generated by string or, more precisely, open membrane
instantons\footnote{The statement that strings on non-isolated genus zero
curves do not contribute to the non-perturbative superpotential was conjectured
by Witten~\cite{Private}. The author is very grateful to Edward Witten
for discussions on this issue.}~\cite{Private, DSWW2}.
This also means that one cannot generate a non-perturbative
superpotential for moduli of a five-brane wrapped on such a cycle.
However, compactifications on a Calabi-Yau manifold with $h^{1,1}$ represented only by isolated
genus zero cycles are, of course, very restrictive.
A generic compactification scenario involves
a Calabi-Yau manifold with non-isolated genus zero or higher genus cycles
and five-branes wrapped on such cycles. As an example, consider a Calabi-Yau manifold
elliptically fibered over the Hirzebruch surface ${\mathbb F}_r, r=0, 1, \dots$.
The Hirzebruch surface ${\mathbb F}_r$ is a ${\mathbb P}^1$ bundle
over ${\mathbb P}^1$. We denote the class of the base of this bundle by ${\cal S}$ and the
class of the fiber by ${\cal E}$. These Calabi-Yau threefolds are simply connected and
admit a (generically unique) global holomorphic section which we denote by $\sigma$.
For such a manifold, generically, we have
\begin{equation}
h^{1,1}=3
\label{1.1}
\end{equation}
and the basis of curves can be chosen to be
\begin{equation}
\sigma \cdot \pi^*{\cal S}, \quad \sigma \cdot \pi^*{\cal E}, \quad F.
\label{1.2}
\end{equation}
Here $\pi$ is the projection map from the threefold onto the base ${\mathbb F}_r$
and $F$ is the class of the elliptic fiber.
The curves $\sigma \cdot \pi^*{\cal S}$ and $\sigma \cdot \pi^* {\cal E}$ have genus zero. The curve $F$
has genus one. The curve
$\sigma \cdot \pi^*{\cal S}$
has a self-intersection $-r$ and, thus, is an isolated genus zero curve for $r>0$.
It is a non-isolated genus zero curve for $r=0$. The curve $\sigma \cdot \pi^*{\cal S}$
has a self-intersection zero for any $r$ and, thus, is non-isolated genus zero curve.
Therefore, it is important to stabilize the $h^{1,1}$ moduli corresponding
to non-isolated genus zero or higher genus curves.
It is also important to understand whether or not it is possible
to stabilize five-branes wrapped on such cycles.
In this paper, for simplicity, we consider the case $h^{1,1}=2$.
We will assume that there is one isolated genus zero curve and one
curve of a different type. The generalization to the case involving
many curves isolated curves of various types is conceptually straightforward
but technically more difficult.
At this point, we would like to make a remark.
It may happen that
the pullback of more than one
harmonic form $\omega_{I}$ onto a given isolated curve is non-zero.
As a result, the non-perturbative superpotential
associated with this isolated curve may depend on the
linear combination of more than one $h^{1,1}$ modulus. In particular, it may depend
on all $h^{1,1}$ moduli.
In this case, all $h^{1,1}$ moduli can be stabilized by
methods presented in~\cite{BO}. However, one might expect
that, generically, there can be $h^{1,1}$ moduli of two sorts,
those that appear in the non-perturbative superpotential and those
which do not. In this paper, we simply assume that our compactification has
one modulus of each sort.
In this section, we will not consider five-branes wrapped
on non-isolated genus zero or higher genus curves. We will add
such a five-brane in the next section.
The system of moduli that we would like to consider in this section includes
the following complex moduli
\begin{equation}
S, T^1, T^2, {\bf Y}, Z_{\alpha}.
\label{1.3}
\end{equation}
The modulus $S$ is related to the volume of the Calabi-Yau manifold
\begin{equation}
S=V+i\sigma,
\label{1.4}
\end{equation}
where $\sigma$ is the axion. The moduli $T^1$ and $T^2$ are
the $h^{1,1}$ moduli. They are defined as follows~\cite{LOW4, LOSW5}
\begin{equation}
T^I=R b^I+ip^I, \quad I=1, 2.
\label{1.5}
\end{equation}
where $R$ is the size of the eleventh dimension,
$b^I$ are the Kahler moduli of the Calabi-Yau threefold
and $p^I$'s come from the components of the M-theory three-form
$C$ along the interval and the Calabi-Yau manifold.
The moduli $b^I$ are not all independent. They satisfy the constraint
\begin{equation}
\sum_{I,J,K=1}^{2}d_{IJK}b^I b^J b^K =6,
\label{1.6}
\end{equation}
where coefficients $d_{IJK}$ are the Calabi-Yau intersection numbers
\begin{equation}
d_{IJK}=\frac{1}{V}\int_{CY} \omega_I \wedge \omega_J \wedge \omega_K.
\label{1.7}
\end{equation}
The constraint~\eqref{1.6} reduces the number of independent $b$-moduli by one.
We will take $T^1$ to correspond to the area of the isolated genus zero curve and
$T^2$ to the area of the remaining curve.
${\bf Y}$ is the modulus of the five-brane wrapped on the isolated genus zero curve.
In this case, there is only one five-brane modulus~\cite{Five}, whose
real is the position of the five-brane in the bulk
\begin{equation}
{\bf Y}=y+i(a+\frac{p_1}{Rb_1}),
\label{1.8}
\end{equation}
where $a$ is the axion arising from dualizing the three-form field strength
propagating on the five-brane world-volume.
At last, by $Z_{\alpha}$ we denote the complex structure moduli. The
actual number of them is not relevant for us.
A generic heterotic compactification contains also instanton moduli~\cite{BOR}.
Their stabilization was considered in~\cite{BO}. In this section, for simplicity,
we will ignore them. They can be added and treated as in~\cite{BO}.
However, we will come back to them in the last section.
The moduli $V, T^1, T^2$ and $y$ are assumed to be dimensionless
normalized with respect to the following reference scales
\begin{equation}
v_{CY}^{-1/6} \approx 10^{16} GeV, \quad (\pi \rho)^{-1} \approx 10^{14}-10^{15} GeV.
\label{1.9}
\end{equation}
In order to obtain the four-dimensional coupling constants in the correct phenomenological range
\cite{Witten96, Banks},
the corresponding moduli should be stabilized at (or be slowly rolling near) the values
\begin{equation}
V \sim 1 \quad R \sim 1.
\label{1.10}
\end{equation}
The Kahler potential for this system is as follows \cite{Candelas, LOW4, DerS}
\begin{equation}
\frac{K}{M^{2}_{Pl}} = K_Z + K_{S,T^1, T^2, {\bf Y}},
\label{1.11}
\end{equation}
where
\begin{equation}
K_Z= -\ln(-i \int \Omega \wedge \bar \Omega),
\label{1.12}
\end{equation}
and
\begin{equation}
K_{S,T,{\bf Y}} =-\ln(S+\bar S) -
\ln (d_{IJK}(T^I+\bar T^I)(T^J+\bar T^J)(T^K+\bar T^K)) +
2 \tau_5 \frac{({\bf Y}+\bar{\bf Y})^2}{(S+\bar S)(T^1+\bar T^1)}.
\label{1.13}
\end{equation}
Here $M_{Pl}$ is the four-dimensional Planck scale and $\tau_5$ is given by
\begin{equation}
\tau_5 =\frac{T_5 v_5 (\pi \rho)^2}{M^{2}_{Pl}},
\label{1.14}
\end{equation}
where $v_5$ is the area of the cycle on which the five-brane is wrapped and $T_5$ is
\begin{equation}
T_5 =(2 \pi)^{1/3} (\frac{1}{2 \kappa_{11}^2})^{2/3},
\label{1.15}
\end{equation}
with $\kappa_{11}$ being the eleven-dimensional gravitational coupling constant. It is
related to the four-dimensional Planck
mass as
\begin{equation}
\kappa_{11}^2=\frac{\pi \rho v_{CY}}{M^2_{Pl}}.
\label{1.16}
\end{equation}
Evaluating $\tau_5$ by using \eqref{1.16} and \eqref{1.9} gives
\begin{equation}
\tau_5 \approx \frac{v_5}{v_{CY}^{1/3}}.
\label{1.17}
\end{equation}
Generically this coefficient is of order one.
The superpotential for this system consists of three different contributions
\begin{equation}
W=W_{f}-W_{g} -W_{np}.
\label{1.18}
\end{equation}
$W_f$ is the flux-induced superpotential \cite{GVW, Berndt, Constantin}
\begin{equation}
W_f =
\frac{M^2_{Pl}}{v_{CY}\pi \rho} \int dx^{11} \int_{CY} G \wedge \Omega,
\label{1.19}
\end{equation}
where $G$ is the M-theory four-form flux.
The order of magnitude of $W_f$ was estimated in \cite{BO} and was found to be, generically, of order $10^{-8}M_{Pl}^3$.
In fact, this is flexible. The superpotential $W_f$ may receive certain higher order corrections from Chern-Simons
invariants. In~\cite{GKLM} it was argued that these Chern-Simons invariants can reduce the order of magnitude of
$W_f$.
By $W_g$ we denote the superpotential induced by a gaugino condensate in the hidden sector
\cite{DRSW, Horava, LOW, Nonstandard}.
A non-vanishing
gaugino condensate has important phenomenological consequences.
Among other things, it is responsible for
supersymmetry breaking in the hidden sector. When that symmetry
breaking is transported to the observable brane, it leads to
soft supersymmetry breaking terms
for the
gravitino, gaugino and matter fields \cite{KL, BIM, NOY, LT}.
See~\cite{Nil} for a good review on gaugino condensation in string theory.
This superpotential has the following structure
\begin{equation}
W_{g} = h M^{3}_{Pl} exp(-\epsilon S +\epsilon \alpha_1^{(2)} T^1 +
\epsilon \alpha_2^{(2)} T^2
- \epsilon \beta \frac{{\bf Y}^2}{T^1}).
\label{1.20}
\end{equation}
The order of magnitude of $h$ is approximately $10^{-6}$ \cite{LOW}. The coefficient $\epsilon$ is related to the
coefficient $b_0$ of the one-loop beta-function and is given by
\begin{equation}
\epsilon = \frac{6 \pi}{b_0 \alpha_{GUT}}.
\label{1.21}
\end{equation}
For example, for the $E_8$ gauge group $\epsilon \approx 5$.
The coefficients $\alpha_I^{(2)}$ represents the
tension (up to the minus sign) of the hidden brane
measured with respect to the Kahler form $\omega_I$
\begin{equation}
\alpha_I^{(2)} =
\frac{\pi \rho}{16 \pi v_{CY}}
(\frac{\kappa_{11}}{4 \pi})^{2/3}
\int_{CY} \omega_I \wedge (tr F^{(2)} \wedge F^{(2)} - \frac{1}{2} tr {\cal R}
\wedge {\cal R}),
\label{1.22}
\end{equation}
where $F^{(2)}$ is the curvature of the gauge bundle on the hidden brane.
Similarly, the coefficient $\beta$ is the tension of the five-brane. It is given by \cite{Visible}
\begin{equation}
\beta=\frac{2 \pi^2 \rho}{v_{CY}^{2/3}} (\frac{\kappa_{11}}{4 \pi})^{2/3}
\int_{CY} \omega_{1} \wedge {\cal W},
\label{1.23}
\end{equation}
where ${\cal W}$ is the four-form Poincare dual to the holomorphic curve on which
the five-brane is wrapped.
Generically both $\alpha_I^{(2)}$ and $\beta$ are of order one.
In fact, from eqs.~\eqref{1.14}, \eqref{1.15} and and \eqref{1.23} it follows that
\begin{equation}
\beta \approx \tau_5.
\label{1.24}
\end{equation}
Let us note the following fact which will be crucial for stabilization of $T^2$.
If $h^{1,1}=1$, apparently, it is important to have $\alpha^{(2)}$ positive
(and, correspondingly, the tension negative) in the hidden sector.
This, in particular, happens when the bundle on the hidden brane is trivial.
The reason is that the quantity
\begin{equation}
Re(S-\sum_I \alpha_I^{(2)}T^I +\beta \frac{{\bf Y}^2}{T^1})
\label{1.25}
\end{equation}
represents the inverse square of the gauge coupling constant
in the hidden sector $\frac{1}{g^2_{hidden}}$. Furthermore,
the quantity
\begin{equation}
Re(S-\sum_I \alpha_I^{(1)}T^I +T^1(1-\beta \frac{{\bf Y}^2}{T^1}))
\label{1.26}
\end{equation}
represents the inverse square of the gauge coupling constant
in the visible sector $\frac{1}{g^2_{visible}}$.
Here $\alpha^{(1)}$ is the tension (up to the minus sign) of the visible brane
\begin{equation}
\alpha^{(1)} =
\frac{\pi \rho}{16 \pi v_{CY}}
(\frac{\kappa_{11}}{4 \pi})^{2/3}
\int_{CY} \omega \wedge (tr F^{(1)} \wedge F^{(1)} - \frac{1}{2} tr {\cal R}
\wedge {\cal R}),
\label{1.27}
\end{equation}
where $F^{(1)}$ is the curvature of the gauge bundle on the visible brane.
If, for example, $h^{1,1}=1$ and there are no five-branes,
we have
\begin{equation}
\frac{1}{g^2_{hidden}}=Re(S-\alpha^{(2)}T)
\label{1.28}
\end{equation}
and
\begin{equation}
\frac{1}{g^2_{visible}}= Re(S-\alpha^{(1)}T)
\label{1.29}
\end{equation}
The anomaly cancellation condition in the absence of five-branes,
\begin{equation}
c_2(V_{visible})+c_2(V_{hidden})=c_2(TX),
\label{1.30}
\end{equation}
sets
\begin{equation}
\alpha^{(2)}=-\alpha^{(1)}.
\label{1.31}
\end{equation}
Now it is clear that if $\alpha^{(2)} <0$, the gauge coupling
constant in the hidden sector is weaker that the gauge coupling
constant in the visible sector and the whole assumption about the gaugino
condensation in the hidden sector breaks down.
It is unlikely that this statement changes when five-branes are included.
However, when $h^{(1,1)}$ is greater than zero, there is nothing
wrong with having some $\alpha_I^{(2)}$'s negative.
It is still possible to keep the gauge coupling constant
stronger in the hidden sector. We will assume that
\begin{equation}
\alpha_1^{(2)} >0
\label{1.32}
\end{equation}
and
\begin{equation}
\alpha_2^{(2)} <0.
\label{1.33}
\end{equation}
It is important to note that
the quantity given by eq.~\eqref{1.25} must be positive.
This means that the superpotential~\eqref{1.20} cannot be
trusted for large values of the interval size $R$.
One should expect that higher order corrections to the combination~\eqref{1.25}
will make the gauge coupling constant $\frac{1}{g^2_{hidden}}$
well defined for large values of $R$. Partial support for this comes
from~\cite{Curio1, Curio2}.
The last contribution to the superpotential that we have to discuss is the non-perturbative superpotential
$W_{np}$
\cite{DSWW1, DSWW2, Wittensuper, BBS, Witten00, Lima1, Lima2, Moore, BDO2, BDO3}.
Such a superpotential is induced by open membranes wrapped on an isolated genus
zero curve. Therefore, it depends on the $h^{1,1}$ modulus $T^1$ and on the
five-brane modulus ${\bf Y}$. However, it does not depend on the $h^{1,1}$ modulus $T^2$.
The non-perturbative superpotential has three parts
\begin{equation}
W_{np}=W_{vh}+W_{v5} +W_{5h}.
\label{1.34}
\end{equation}
$W_{vh}$ is induced by a membrane stretched between the visible and the hidden branes.
It behaves as
\begin{equation}
W_{vh} \sim e^{-\tau T^1}
\label{1.35}
\end{equation}
$W_{v5}$ is induced by a membrane stretched between the visible brane and the five-brane. It behaves as
\begin{equation}
W_{v5} \sim e^{-\tau {\bf Y}}.
\label{1.36}
\end{equation}
At last, $W_{5h}$ is induced by a membrane stretched between the five-brane and the hidden brane. It behaves as
\begin{equation}
W_{5h} \sim e^{-\tau (T^1-{\bf Y})}.
\label{1.37}
\end{equation}
The coefficient $\tau$ is given by~\cite{Lima1, Lima2}
\begin{equation}
\tau =\frac{1}{2} (\pi \rho) v_i (\frac{\pi}{2 \kappa_{11}})^{1/3},
\label{1.38}
\end{equation}
where $v_i$ is the reference area of the isolated curve.
Generically, $\tau$ is much bigger than one.
As in~\cite{BO, Raise},
we will assume that the five-brane is close to the hidden sector.
It was argued in~\cite{BO, Raise} that only in this case it is possible
to stabilize the size of the interval in a phenomenologically acceptable range.
Therefore, the contributions $W_{vh}$ and $W_{v5}$ decay very fast and
we have
\begin{equation}
W_{np} = W_{5h} = M^{3}_{Pl} a e^{-\tau (T^1-{\bf Y})}.
\label{1.39}
\end{equation}
For concreteness we assume that the coefficient $a \sim 1$.
\subsection{Supersymmetric AdS vacua}
In this subsection, we will argue that this system of moduli has an AdS
minimum. The consideration is, somewhat, similar to~\cite{BO, Raise} and we will be
relatively brief. Let us first discuss the imaginary parts of the moduli.
A consideration analogous to~\cite{BO} shows that the imaginary parts of $T^1$ and ${\bf Y}$ are stabilized at
values
\begin{equation}
ImT^1 \sim \frac{1}{\tau} \approx 0, \quad Im{\bf Y} \approx 0.
\label{1.40}
\end{equation}
The imaginary part of the linear combination
$S-\alpha_2^{(2)}T^2$ is stabilized in such a way that the superpotentials
$W_f$ and $W_g$ are out of phase. Similarly, $W_f$ and $W_{np}$ are also out phase.
We already took this into account in eq.~\eqref{1.18}
by putting the minus sign in appropriate places.
Unfortunately, the superpotential of the form~\eqref{1.18}, \eqref{1.19}, \eqref{1.20}
and~\eqref{1.39} does not allow us to stabilize the remaining linear combination
of $S$ and $T^2$. It can be shown to be a flat direction. This problem
cannot be resolved even by considering a multiple gaugino condensation
in the hidden sector. Nevertheless, it is easy to realize that
this problematic linear combination
can be stabilized by taking into account the higher order $T$-corrections
to the gauge coupling in the hidden sector. We will make a more detailed comment
on it later in this subsection. Therefore, stabilization of the remaining
imaginary part does not represent a conceptual problem.
In the rest of the paper, we will concentrate only on the real parts
of the moduli ignoring their imaginary parts.
Now let us consider the real parts of the moduli and show that
the system under study indeed has an AdS minimum satisfying
\begin{equation}
D_{all{\ }fields}W=0,
\label{1.41}
\end{equation}
where $D$ is the Kahler covariant derivative.
We will not distinguish between the superpotentials and their absolute values.
First, we consider equations
\begin{equation}
D_{Z_{\alpha}}W=0.
\label{1.42}
\end{equation}
Assuming that
\begin{equation}
W_f >>W_g, W_{np}
\label{1.43}
\end{equation}
in the interesting regime, eq.~\eqref{1.43}
can be written as
\begin{equation}
\partial_{Z_{\alpha}}W_{f} + \frac{\partial K_{Z_{\alpha}}}{\partial Z_{\alpha}} W_{f} =0.
\label{1.44}
\end{equation}
In \cite{BO}, it was shown that ineq.~\eqref{1.43} is indeed satisfied. In eq.~\eqref{1.44},
all quantities depend on the complex structure moduli only. We will assume that this equation
fixes all the complex structure moduli. Partial evidence that equations of the type \eqref{1.44}
fix all the complex structure moduli comes, for example, from~\cite{Schulz1}.
The next equation to consider is
\begin{equation}
D_{S}W=0.
\label{1.45}
\end{equation}
By using eqs.~\eqref{1.13}, \eqref{1.20} and~\eqref{1.43}, it can be written as
\begin{equation}
\epsilon W_g =F_1W_f,
\label{1.46}
\end{equation}
where
\begin{equation}
F_1=\frac{1}{2V}(1+\tau_5 \frac{y^2}{(Rb^1)^2}).
\label{1.47}
\end{equation}
By using eqs.~\eqref{1.13}, \eqref{1.20}, \eqref{1.39}, \eqref{1.43} and~\eqref{1.46},
eq.
\begin{equation}
D_{T^1}W=0
\label{1.48}
\end{equation}
can be rewritten as
\begin{equation}
\tau W_{np}=((\alpha_1^{(2)}+\beta\frac{y^2}{(Rb^1)^2})F_1+F_2)W_f,
\label{1.49}
\end{equation}
where
\begin{equation}
F_2=\frac{3\sum_{IJ}d_{1IJ}b^Ib^J}{R\sum_{IJK}d_{IJK}b^Ib^Jb^K}
+\frac{\tau_5 y^2}{V(Rb^1)^2}.
\label{1.50}
\end{equation}
Now let us consider eq.
\begin{equation}
D_{T^2}W=0.
\label{1.51}
\end{equation}
Note that the non-perturbative superpotential~\eqref{1.39} does not depend on $T^2$,
thus, $T^2$ cannot be stabilized by the same mechanism as $T^1$. By using
eqs.~\eqref{1.13}, \eqref{1.20}, \eqref{1.43}, we obtain
\begin{equation}
\epsilon W_g =F_3 W_f,
\label{1.52}
\end{equation}
where
\begin{equation}
F_3=-\frac{3\sum_{IJ}d_{2IJ}b^Ib^J}{\alpha_2^{(2)}R\sum_{IJK}d_{IJK}b^Ib^Jb^K}.
\label{1.53}
\end{equation}
Eqs.~\eqref{1.46} and~\eqref{1.52} are consistent only if
$F_1$ and $F_3$ are equal to each other. In particular, they must have
the same sign. This is possible only if $\alpha_2^{(2)}$ is negative.
As we argued before, this does not lead to any contradictions.
Note, that $F_1$ and $F_3$ are both real. This is the reason why
only one linear combination of the imaginary parts of $S$ and $T^2$ moduli can be
stabilized. On the other hand, if higher order $T$-corrections to the
quantity~\eqref{1.25} are present, $F_3$ is really complex and, hence,
the imaginary parts of both $S$ and $T^2$ moduli can be stabilized.
The last equation to consider is
\begin{equation}
D_{{\bf Y}}W=0.
\label{1.54}
\end{equation}
By using eqs.~\eqref{1.13}, \eqref{1.20}, \eqref{1.39}, \eqref{1.47} and~\eqref{1.50},
we obtain
\begin{equation}
-(\alpha_1^{(2)}+\beta\frac{y^2}{(Rb^1)^2})F_1+F_3+
2\beta\frac{y}{Rb^1}F_1 +2 \tau_5\frac{y}{VRb^1}=0.
\label{1.55}
\end{equation}
Eqs.~\eqref{1.46}, \eqref{1.49}, \eqref{1.52} and \eqref{1.55} are
the four equations with four independent variables $V, R, y$ and one of two $b^I$'s.
Equations of this type were analyzed in detail in~\cite{BO, Raise}
in the case of only one $h^{1,1}$ modulus.
It was shown that they admit a solution
with the following properties
\begin{itemize}
\item
V is of order one.
\item
R is of order one.
\item
The gauge coupling constant $g^2_{hidden}$ does not become imaginary.
\item
The five-brane is close to the hidden brane ($R-y \sim 0.1$).
\end{itemize}
In this paper, we will not perform a detailed analysis.
Let us just point out that in eqs.~\eqref{1.46} and~\eqref{1.49},
the run-away moduli are stabilized by fluxes.
Eqs.~\eqref{1.46} and~\eqref{1.52} lead to eq.
\begin{equation}
F_1=F_3,
\label{1.56}
\end{equation}
which is well defined if $\alpha_2^{(2)}$ is negative.
Eq.~\eqref{1.55} is a purely algebraic equation.
It is possible to show that it admits a numeric solution with the right properties
as in~\cite{BO, Raise}. We will not give a numeric
result in this paper. See~\cite{BO, Raise} for a detailed
analysis of similar equations.
In this subsection, we have provided stabilization of moduli listed in~\eqref{1.3}.
This list includes the modulus $T^2$, corresponding to the area of a non-isolated
genus zero curve or a curve of a higher genus. Stabilization
of such a modulus differs from stabilization of the modulus $T^1$, corresponding
to the area of an isolated genus zero curve. The crucial point
in stabilization of $T^2$ is that, in the case when $h^{1,1}>1$, it possible
to choose the coefficient $\alpha_2^{(2)}$ to be negative.
It is not possible to do in the case when $h^{1,1}=1$, because
it would follow that the gauge coupling coupling in the hidden sector
became weaker than in the visible sector. This would not be consistent
with the assumption about gaugino condensation in the hidden sector.
The AdS vacuum constructed in this section can be raised to a metastable
dS vacuum along the lines of~\cite{Raise}.
This can be achieved by either adding Fayet-Iliopoulos terms to the supergravity
potential energy or by working within the context of $E_8 \times \bar E_8$
theory.
\section{Addition of a Five-Brane and dS Vacua}
\subsection{Effective Potential for a Five-Brane Modulus and Non-Su\-per\-sym\-met\-ric AdS Vacua}
Now we would like to see what happens if we add a five-brane wrapped on a non-isolated
genus zero curve or on a higher genus curve to the system of moduli considered above.
We will denote the complex five-brane modulus by ${\bf X}$
and its real part by $x$. The Kahler potential~\eqref{1.13} receives the contribution
\begin{equation}
\Delta K=2 \tau_5^{'} \frac{({\bf X}+\bar{\bf X})^2}{(S+\bar S)(T^2+\bar T^2)}.
\label{2.1}
\end{equation}
The gaugino condensate superpotential gets modified and becomes
\begin{equation}
W_g \to W_ge^{-\epsilon \beta^{'} \frac{{\bf X}^2}{T^2}}.
\label{2.2}
\end{equation}
The coefficients $\tau_5^{'}$ and $\beta^{'}$ are given by expressions similar
to eqs.~\eqref{1.14} and~\eqref{1.23}.
Unfortunately, if a five-brane wraps a non-isolated cycle, one should expect
other five-brane moduli in addition to ${\bf X}$. Such moduli have
never been considered in the literature in detail. Nevertheless,
one should expect that
the gaugino condensate superpotential~\eqref{2.2} depends on them.
This might provide their stabilization.
Thus, we will assume that these
additional moduli are fixed and not consider them in this paper.
In principle, one can avoid this issue by taking a five-brane
wrapping an isolated higher genus curve.
Let us first see if we can stabilize ${\bf X}$ in an AdS vacuum.
By using eqs.~\eqref{2.1} and~\eqref{2.2},
the
equation
\begin{equation}
D_{{\bf X}}W=0
\label{2.3}
\end{equation}
can be written as
\begin{equation}
\epsilon\beta^{'} \frac{{\bf X}}{T^2}W_g +\tau_5^{'} \frac{x}{VRb^2}W_f=0.
\label{2.4}
\end{equation}
It is easy to realize that the only solution for ${\bf X}$ is
\begin{equation}
{\bf X}=0.
\label{2.5}
\end{equation}
The point $x=0$ corresponds to the five-brane coinciding with the visible brane.
Such a vacuum is unstable in the sense that the five-brane
will disappear through a small instanton transition~\cite{Seiberg, KS, OPP, BDO1}
and turn into new vector bundle moduli.
As an approximation, we will assume that the presence of this extra five-brane will not
modify much the vacuum constructed in the previous section.
As a result, we can talk about the effective potential $U(x)$
describing dynamics of the five-brane.
In fact, it is possible to show that the vacuum value of the moduli $S, T^I, {\bf Y}$
receive corrections of order $x^2$. Therefore, for very small values of $x$, $x << 1$, their vacuum
values will not shift much. This suggest that the effective potential $U(x)$ is
a decent approximation. Of course, in order to describe the system exactly,
one has to solve all the equations for moduli including the equations
for the imaginary parts. This is not possible to do analytically.
However, it is natural to argue that the qualitative behavior of this system
will be captured assuming that there is the effective potential $U(x)$ with
the rest of the moduli fixed along the lines of the discussion in the previous
section.
Thus, we consider dynamics of one field ${\bf X}$ with the Kahler potential
\begin{equation}
\frac{K({\bf X})}{M^2_{Pl}}=K_0+\frac{1}{4}K_1 ({\bf X}+\bar {\bf X})^2
\label{2.6}
\end{equation}
and the superpotential
\begin{equation}
W({\bf X})=W_0-W_1 e^{-\epsilon \gamma {\bf X}^2},
\label{2.7}
\end{equation}
where $K_0$ is a constant independent of ${\bf X}$, $K_1$ is given by
\begin{equation}
K_1=\frac{2 \tau^{'}_5}{VRb^2},
\label{2.8}
\end{equation}
$W_0$ is a constant of order fluxes,
\begin{equation}
W_0 \sim W_f,
\label{2.9}
\end{equation}
$W_1$ is approximately given by (see eq.~\eqref{1.46})
\begin{equation}
W_1 =\frac{F_1}{\epsilon} W_f =\frac{F_1}{\epsilon}W_0
\label{2.10}
\end{equation}
and the coefficient $\gamma$ is given by
\begin{equation}
\gamma=\frac{\beta^{'}}{T^2}.
\label{2.11}
\end{equation}
Without loss of generality, we can set $ImT^2=0$. Then
\begin{equation}
\gamma=\frac{\beta^{'}}{Rb^2}.
\label{2.12}
\end{equation}
The effective potential for the ${\bf X}$ modulus is given by
\begin{equation}
U({\bf X})=e^{\frac{K({\bf X})}{M^2_{Pl}}}
(G^{-1}_{{\bf X}\bar {\bf X}} D_{{\bf X}}W({\bf X})D_{\bar {\bf X}}\bar W(\bar {\bf X})
-3W({\bf X})\bar W(\bar {\bf X})),
\label{2.13}
\end{equation}
where the Kahler covariant derivative is defined as usual
\begin{equation}
D_{{\bf X}}W({\bf X})= \partial_{{\bf X}} W({\bf X})+\frac{1}{M^2_{Pl}}
\partial_{{\bf X}} K({\bf X})W({\bf X}).
\label{2.14}
\end{equation}
As was argued before, the imaginary part of ${\bf X}$ can be stabilized
by this potential. Therefore, the potential $U({\bf X})$ can be treated
as an effective potential for one real field $x$.
We will denote it $U(x)$. From eqs.~\eqref{2.6} and~\eqref{2.7},
we obtain
\begin{equation}
U(x)=U_0 e^{K_1x^2}(-3+2K_1x^2(1+ \lambda e^{-\epsilon \gamma x^2})^2),
\label{2.15}
\end{equation}
where $U_0$ is a constant of order $\frac{W_f^2}{M^2_{Pl}}$,
$K_1$ and $\gamma$ are given by eqs.~\eqref{2.8} and~\eqref{2.12} respectively
and $\lambda$ is given by
\begin{equation}
\lambda =\frac{2 \gamma F_1}{K_1}.
\label{2.16}
\end{equation}
Eq.~\eqref{2.15} gives an effective potential $U(x)$.
Unfortunately, it is very difficult to analyze
this potential analytically. A graphical
analysis shows that, generically,
this potential has a non-supersymmetric AdS vacuum for a non-zero value of $x$.
The form of $U(x)$ for various choices of parameters is shown on
Figures~\ref{f1} and~\ref{f3}.
\begin{figure}
\epsfxsize=4in \epsffile{Fig1.eps}
\begin{picture}(30,30)
\put(-260,180){$\frac{U(x)}{U_0}$}
\put(5, 50){$x$}
\end{picture}
\caption{The graph of $\frac{U(x)}{U_0}$ for $K_1=3, \gamma=3, \lambda =1, \epsilon =10$.
There exists a non-supersymmetric AdS minimum. \label{f1}}
\vspace{3cm}
\epsfxsize=4in \epsffile{Fig3.eps}
\begin{picture}(30,30)
\put(-255,185){$\frac{U(x)}{U_0}$}
\put(5, 15){$x$}
\end{picture}
\caption{The graph of $\frac{U(x)}{U_0}$ for $K_1=5, \gamma=2.5, \lambda =2, \epsilon =10$.
There exists a non-supersymmetric AdS minimum. \label{f3}}
\end{figure}
It is possible to adjust parameters in such a way that
the vacuum becomes dS. However, in this case, the parameter $\lambda$
has to be taken to be sufficiently greater than one, whereas eqs.~\eqref{2.16}, \eqref{2.8} and~\eqref{2.12}
require that $\lambda$ be of order one. Therefore, for reasonable values of the parameters,
the minimum is always AdS. It is possible to adjust parameters so that $x$
is less than the size of the interval, which, as discussed
in the previous section, can be stabilized at a value of order one.
This AdS vacuum can be raised to a metastable
dS vacuum by methods discussed in~\cite{Raise}.
This demonstrates that the most general system of heterotic M-theory
moduli can be stabilized in a dS vacuum.
In the rest of the paper, we will be interested
in dynamics of a five-brane in the regime $x<<1$.
Heterotic M-theory vacua can contain
several five-branes wrapped on non-isolated genus zero or higher
genus curves. We have just argued that those five-branes
which are located sufficiently far away from the visible brane
can be stabilized. Now we would like to understand the fate
of the five-branes which are close to the visible sector.
Such five-brane will roll towards $x=0$.
The potential $U(x)$ in this regime does not lead to any interesting physics.
It does not provide stabilization of $x$. It is also hard to imagine how to use it
in any cosmological framework. On the one hand, it is negative and, hence, cannot
be used for inflation. On the other hand, it does not satisfy conditions necessary
for Ekpyrotic cosmology~\cite{Justin1, Justin2, Justin3}.
To make use of this potential, we will modify it by Fayet-Iliopoulos terms.
Depending on relations among various coefficients,
Fayet-Iliopoulos terms can lead to either stabilization
of $x$ or a potential with certain inflationary properties.
\subsection{Fayet-Iliopoulos Terms and dS Vacua}
In both weakly and strongly coupled heterotic string models, there can be anomalous
$U(1)$ gauge groups. They can arise in both the visible and the hidden sectors.
The anomaly is canceled by a four-dimensional version of the Green-Schwarz
mechanism. This anomalous $U(1)$ gives rise to the Fayet-Iliopoulos term~\cite{DSW},
which, in turn, gives rise to the
moduli effective potential
of the form
\begin{equation}
U_D =M_{Pl}^4 g^2 \frac{b}{V^2},
\label{2.17}
\end{equation}
where $b$ is a constant and $g$ is the gauge coupling constant. In the context of the strongly
coupled heterotic string theory, the coefficient $b$ was estimated in~\cite{Raise}
and was found to be, generically, of order
\begin{equation}
b \sim 10^{-18}.
\label{2.18}
\end{equation}
The potential $U_D$ depends on in what sector there appears an anomalous $U(1)$.
The reason is that the coupling constants in the visible and the hidden sectors are different.
They are given by~\cite{Nonstandard}
\begin{equation}
g_{visible}^2 =\frac{g^2_0}{Re(S+\alpha_1^{(1)}T^1+ \alpha_2^{(1)}T^2+
\beta (T^1-\frac{{\bf Y}^2}{T^1})+ \beta^{'} (T^2-\frac{{\bf X}^2}{T^2}) )}
\label{2.19}
\end{equation}
and
\begin{equation}
g_{hidden}^2 =\frac{g^2_0}{Re(S-\alpha_1^{(2)}T^1-\alpha_2^{(2)}T^2 +
\beta\frac{{\bf Y}^2}{T^1}+\beta^{'}\frac{{\bf X}^2}{T^2})},
\label{2.20}
\end{equation}
where $g_0$ is a moduli independent constant of order $\alpha_{GUT}$.
In~\cite{BKQ, Raise}, Fayet-Iliopoulos potentials $U_D$ were
used to raise AdS vacua to dS vacua. In this paper, we will be
interested in the $x$ dependence of $U_D$. If the anomalous $U(1)$
appears in the hidden sector, the potential $U_D(x)$ takes the form
\begin{equation}
U^{visible}_D(x)=\frac{B_1}{B_2-\gamma x^2},
\label{2.21}
\end{equation}
whereas,
if the anomalous $U(1)$ appears in the hidden sector, the potential $U_D$
is
\begin{equation}
U^{hidden}_D(x)=\frac{C_1}{C_2+\gamma x^2},
\label{2.22}
\end{equation}
where, $B1, B2, C1$ and $C_2$ can be read off from
eqs.~\eqref{2.17}, \eqref{2.19} and~\eqref{2.20} and $\gamma$ is given by
eq.~\eqref{2.12}.
We would like to modify our potential $U(x)$ by $U_{D}(x)$.
In this section, we take $U_D(x)$
to be $U_D^{hidden}(x)$. We will now show that the potential energy
\begin{equation}
\tilde{U}(x)=U(x)+U_{D}^{hidden}(x)
\label{2.23}
\end{equation}
can provide stabilization of $x$ in the regime $x<<1$ in a dS vacuum.
We should point out that, if we modify $U(x)$ by some
other moduli dependent correction, it is not
very obvious that this correction will not destabilize
other, additional to $x$, moduli. However,
in~\cite{Raise}, it was shown that if the order
of magnitude of $U_D$ is the same as (or less than) the order
of magnitude of $U$, it is possible
to find a solution to eqs. $d(U+U_D)=0$ fixing all the moduli
considered in the previous section. Therefore, it is still a
decent approximation to consider the effective potential
$\tilde{U}(x)$ assuming that all the remaining moduli are fixed.
Now note the following simple facts.
Since $x=0$ is the minimum of the function $U$, for small $x$ we have
\begin{equation}
\frac{\partial U(x)}{\partial x} >0
\label{2.24}
\end{equation}
On the other hand, from eq.~\eqref{2.22}, it follows that
\begin{equation}
\frac{\partial U_D^{hidden}(x)}{\partial x} <0.
\label{2.25}
\end{equation}
This means that it should be possible to find a solution to the equation
\begin{equation}
\frac{\partial \tilde{U}(x)}{\partial x} =0
\label{2.26}
\end{equation}
under mild assumptions.
For $x<<1$, the potential $U(x)$ is governed by the quadratic function
\begin{equation}
U(x)=-3U_0+a_2 U_0 x^2,
\label{2.26.1}
\end{equation}
where $a_2$ is given by
\begin{equation}
a_2=K_1(2(1+\lambda)^2-3).
\label{2.28}
\end{equation}
Using eqs.~\eqref{2.8}, \eqref{2.12}, \eqref{2.16} and~\eqref{1.47},
one can
show that $a_2$ is greater than zero for any choice of the parameters.
It is straightforward to solve eq.~\eqref{2.26} in this regime.
The approximate solution is
\begin{equation}
x_{min} \approx
\frac{1}{\sqrt{\gamma}}\left(\sqrt{\frac{\gamma C_1}{a_2U_0}} -C_2\right)^{1/2},
\label{2.27}
\end{equation}
It is possible to adjust the parameters so that $x_{min}$ is real and much less than one.
It is also straightforward to show that, if the solution for $x_{min}$ exists, it is always a minimum.
The simplest way to do it is prove that, if the solution~\eqref{2.27} exists, then
$x=0$ is always a maximum. Since $x_{min}<<1$, the value of the cosmological constant
is approximately given by
\begin{equation}
\Lambda \approx -3U_0 +\frac{C_1}{C_2}.
\label{2.29}
\end{equation}
It is obvious that $\Lambda$ can be of both signs. By fine-tuning it is possible
to set
\begin{equation}
\Lambda \sim 10^{-120}M_{Pl}^4
\label{2.30}
\end{equation}
which is consistent with observations. The form of the potential $\tilde{U}(x)$ in the
regime $x<<1$
is shown on Figure~\ref{f2}.
\begin{figure}
\epsfxsize=4in \epsffile{Fig2.eps}
\begin{picture}(30,30)
\put(-250,165){$\frac{\tilde{U}(x)}{U_0}$}
\put(5, 60){$x$}
\end{picture}
\caption{The graph of $\frac{\tilde{U}(x)}{U_0}$ in the regime $x<<1$
for $a_2=2.95, \gamma=1, \frac{C_1}{U_0} =3.01, C_2 =1$.
There exists a dS minimum. \label{f2}}
\end{figure}
In Sections 2 and 3, we showed that the most general system of heterotic M-theory moduli can be
stabilized in a dS vacuum. In addition to moduli considered in~\cite{Raise},
we also provided stabilization for extra $h^{1,1}$ moduli and an extra five-brane
associated with a non-isolated genus zero curve or with a higher genus curve.
In the presence of such a five-brane,
the system of moduli
can be stabilized by fluxes and non-perturbative effects
in a non-supersymmetric AdS vacuum which then can be lifted to a dS vacuum
as in~\cite{Raise}. This five-brane can also
be stabilized by balancing
the supergravity potential energy against a Fayet-Iliopoulos term induced by an anomalous
$U(1)$ gauge group in the hidden sector.
Thus, the potential energy $\tilde{U}(x)$ might admit two dS vacua.
One of the them is the lift of the non-supersymmetric AdS vacuum.
The other one can additionally arise for $x <<1$, though
it did not existed in the absence of the Fayet-Iliopoulos term.
\section{The Five-Brane Modulus as an Inflaton}
\subsection{Constructing an Inflationary Potential}
We begin this section with modifying the potential energy $U(x)$ by
the Fayet-Iliopoulos term $U_{D}^{visible}(x)$ given by~\eqref{2.21}.
The first derivative of $U_{D}^{visible}(x)$ is positive, hence,
the potential
\begin{equation}
\tilde{U}(x)=U(x)+U_{D}^{visible}(x)
\label{3.1}
\end{equation}
does not have a non-trivial minimum for $x <<1$ and $x$ rolls towards $x=0$.
We will assume that the potential $\tilde{U}(x)$ is positive.
The potential~\eqref{3.1} has the following form
\begin{equation}
\tilde{U}(x)=U_0 e^{K_1x^2}(-3+2K_1x^2(1+\lambda e^{-\epsilon \gamma x^2})^2) +\frac{B_1}{B_2-\gamma x^2}.
\label{3.2}
\end{equation}
Let us recall that the coefficients $\epsilon, K_1, \gamma$ and $\lambda$ are given by
eqs.~\eqref{1.21}, \eqref{2.8}, \eqref{2.12} and~\eqref{2.16},
$U_0$ is a constant of order $\frac{W_f^2}{M_{Pl}^2}$ and
$B_1$ and $B_2$ can be read off from eqs.~\eqref{2.17} and~\eqref{2.19}.
We assume that this potential is positive, that is,
\begin{equation}
\frac{B_1}{B_2} >3U_0.
\label{3.3}
\end{equation}
Our goal is to examine whether this potential can satisfy
the slow roll conditions required by inflation.
As in the previous section, we are interested in the regime
$x <<1$. Then we can expand $\tilde{U}(x)$ in powers of $x$. For
our purposes, it is enough to keep only two leading terms.
We obtain
\begin{equation}
\tilde{U}(x)\approx A_0+A_2x^2,
\label{3.4}
\end{equation}
where
\begin{equation}
A_0 =-3U_0 +\frac{B_1}{B_2}
\label{3.5}
\end{equation}
and
\begin{equation}
A_2=K_1(2(1+\lambda)^2-3)U_0 +\frac{\gamma B_1}{B_2^2}=a_2+\frac{\gamma B_1}{B_2^2}.
\label{3.6}
\end{equation}
In order to study the standard slow roll parameters $\epsilon (x)$ and $\eta (x)$,
we have to canonically normalize the kinetic energy.
From the Kahler potential~\eqref{2.6}, it follows
that we have to redefine $x$ as
\begin{equation}
x \to \sqrt{\frac{2}{K_1}}\frac{x}{M^2_{Pl}}.
\label{3.7}
\end{equation}
This new $x$ is canonically normalized and has dimension one.
The potential energy now looks as follows
\begin{equation}
\tilde{U}(x) = A_0 +\frac{2 A_2}{K_1 M^2_{Pl}} x^2.
\label{3.8}
\end{equation}
To have inflation as $x$ rolls towards $x=0$, the two parameters
\begin{equation}
\epsilon (x) =\frac{M^2_{Pl}}{2} \left( \frac{\tilde{U}^{'}(x)}{\tilde{U}(x)}\right)^2
\label{3.9}
\end{equation}
and
\begin{equation}
\eta (x) =M^2_{Pl} \frac{\tilde{U}^{''}(x)}{\tilde{U}(x)}
\label{3.10}
\end{equation}
have to be much less than one.
From eq.~\eqref{3.8} we obtain
\begin{equation}
\epsilon (x) =\frac{2 A_2^2}{A_0^2 K_1^2 M_{Pl}^2}x^2.
\label{3.11}
\end{equation}
Clearly, for $x <<M_{Pl}$, $\epsilon(x)$ is naturally much less than one.
For the parameter $\eta(x)$ we have
\begin{equation}
\eta (x) =\frac{4A_2}{K_1 A_0}.
\label{3.12}
\end{equation}
Therefore, we need to impose
\begin{equation}
\frac{4}{K_1}A_2 < A_0.
\label{3.13}
\end{equation}
Using eqs.~\eqref{3.5} and~\eqref{3.6}, this condition can be rewritten as
\begin{equation}
\frac{4(2(1+\lambda)^2-3)U_0 +\frac{4 \gamma B_1}{K_1 B_2^2}}{-3U_0 +\frac{B_1}{B_2}} <<1.
\label{3.14}
\end{equation}
The only way this can be fulfilled is when
\begin{equation}
\frac{B_1}{B_2} >> U_0
\label{3.15}
\end{equation}
and
\begin{equation}
\frac{4 \gamma}{K_1 B_2} << 1.
\label{3.16}
\end{equation}
Condition~\eqref{3.15} is a relatively mild constraint.
Using eqs.~\eqref{2.8}, \eqref{2.11}, \eqref{2.17} and~\eqref{2.19}, condition~\eqref{3.16}
can be rewritten as
\begin{equation}
\frac{2V}{Re (S+\alpha_1^{(1)}T^1+ \alpha_2^{(1)}T^2+
\beta (T^1-\frac{{\bf Y}^2}{T^1})+ \beta^{'} T^2)} <<1.
\label{3.17}
\end{equation}
Unfortunately, it does not seem to be possible to fulfill
this condition, at least in the context of low-energy field theory.
Inequality~\eqref{3.17} requires that some of the tensions
$\alpha_I^{(1)}, \beta$ or $\beta^{'}$ be much greater than one. In this case,
one cannot trust, even approximately, expressions~\eqref{2.19} and~\eqref{2.20}
for the coupling constants because they can be substantially modified
by higher order corrections~\cite{Curio1, Curio2}.
On the other hand, eq.~\eqref{3.17} may make perfect
sense in the context of M theory.
However, we would like to stay within the context of low-energy
field theory. All we have to do to make the parameter $\eta(x)$ small is to decrease
the parameter $A_2$ in eq.~\eqref{3.4}. This, in fact, can easily be done. We just have to
replace a Fayet-Iliopoulos term in the visible sector by a Fayet-Iliopoulos term in the
hidden sector. Equivalently, we could just add the Fayet-Iliopoulos term in the
hidden sector to eq.~\eqref{3.1}. In both cases, addition of such a Fayet-Iliopoulos term
increases $A_0$ and decreases $A_2$.
It is possible to (not necessarily fine) tune
the parameter $\eta$ to be much less than one.
Let us consider it in slightly more detail.
Assuming, for simplicity, that only a hidden sector Fayet-Iliopoulos term
is present and using~\eqref{2.22},
we have
\begin{equation}
\eta (x) =
\frac{4(2(1+\lambda)^2-3)U_0 -\frac{4 \gamma C_1}{K_1C_2^2}}
{-3U_0+\frac{C_1}{C_2}} <<1.
\label{3.17.1}
\end{equation}
We can rewrite~\eqref{3.17.1} as
\begin{equation}
\eta (x) =
\frac{4(2(1+\lambda)^2-3-\frac{3\gamma}{K_1C_2})U_0 -\frac{4 \gamma }{K_1C_2}A_0}{A_0} <<1.
\label{3.17.2}
\end{equation}
Since the quantities $U_0$ and $A_0$ are, generically, of the same order of
magnitude~\cite{Raise}, ineq.~\eqref{3.17.2} is a relatively mild constraint.
In the next subsection, we will show that the parameter $\eta$ does
not have to be fine tuned to be very small.
As discussed in the previous section,
addition of a Fayet-Iliopoulos term in the
hidden sector can stabilize $x$.
This happens if the numerator in~\eqref{3.17.2} becomes negative.
In this case, the point $x=0$ becomes a maximum and
the potential $\tilde{U}(x)$ acquires a minimum
at a non-zero value of $x$.
This was studied in the previous section.
In this section, we assume that
the effect of such an addition is to make the potential flat,
rather than to produce a non-trivial minimum.
In this subsection, we showed that the five-brane effective potential,
with various Fayet-Iliopoulos terms included,
can satisfy the slow roll conditions
\begin{equation}
\epsilon (x) <<1, \quad \eta (x) << 1
\label{3.18}
\end{equation}
necessary for inflation.
Let us recall that the system of fields contains various other moduli,
in addition to $x$. The potential energy, by construction,
has a minimum in these directions. Therefore, dynamically, one
expects that they will roll fast in the minimum, leaving
the modulus $x$ to roll slowly. Since ineqs.~\eqref{3.18}
are satisfied for $x <<1$, the five-brane modulus $x$ can be
viewed as an inflaton.
\subsection{The Amount of Inflation and Primordial Fluctuations}
In this subsection, we will consider the amount of inflation and
primordial fluctuations. The amount of inflation is defined by
\begin{equation}
N =\ln \frac{a_f}{a_i},
\label{3.20}
\end{equation}
where $a_i$ and $a_f$ are the initial and final values of the expansion
parameter. The evolution of $a$ and $x$ can be found from
the Friedmann equation
\begin{equation}
H^2 =\frac{1}{3 M^2_{Pl}}(\frac{1}{2} \dot x^2 +\tilde{U}(x))
\label{3.21}
\end{equation}
and the $x$-equation of motion
\begin{equation}
\stackrel{\cdot \cdot}{x} +3H \dot x +\tilde{U}^{'}(x)=0,
\label{3.22}
\end{equation}
where
\begin{equation}
H=\frac{\dot a}{a}
\label{3.23}
\end{equation}
is the Hubble constant.
Since during the period of inflation the kinetic energy is much less than the potential energy,
it follows from eq.~\eqref{3.24} that
\begin{equation}
H \approx \frac{1}{M_{Pl}} \sqrt{\frac{A_0}{3}}.
\label{3.24}
\end{equation}
This gives
\begin{equation}
a(t) \approx a_i e^{\sqrt{\frac{A_0}{3 M^2_{Pl}}} t}.
\label{3.25}
\end{equation}
Similarly, integrating eq.~\eqref{3.22} we find that
\begin{equation}
x(t) \approx x_i e^{-\frac{4 A_2}{K_1 M_{Pl}\sqrt{3 A_0}}t}.
\label{3.27}
\end{equation}
From eqs.~\eqref{3.25} and~\eqref{3.27} we obtain
\begin{equation}
N = \ln \frac{a_f}{a_i} = \frac{A_0 K_1}{4A_2}
\ln \frac{x_i}{x_f} =\frac{1}{\eta} \ln \frac{x_i}{x_f},
\label{3.28}
\end{equation}
where eq.~\eqref{3.12} has been used.
By $x_i$ and $x_f$ we denoted the initial and final positions
of the five-brane during inflation. Taking, as an an example,
\begin{equation}
\eta \sim 0.1, \quad \frac{x_i}{x_f} \sim 10^4.
\label{3.29}
\end{equation}
we get
\begin{equation}
N \sim 80
\label{3.30}
\end{equation}
which is consistent with observations.
Primordial fluctuations are determined by the following quantity
\begin{equation}
\delta^2_H =\frac{4}{25}
\left(\frac{H}{\dot x} \right)^2
\left(\frac{H}{2 \pi} \right) =
\frac{1}{150 \pi^2} \frac{\tilde{U}(x)}{M^4_{Pl}\epsilon(x)}
\approx \frac{1}{150 \pi^2} \frac{A_0}{M^4_{Pl}\epsilon(x)},
\label{3.31}
\end{equation}
where, in our case, $\epsilon (x)$ is given by eq.~\eqref{3.11}.
Note that as $x$ goes to zero, $\epsilon (x)$ goes to zero and $\delta^2_H$
goes to infinity. Therefore, it is important to terminate
inflation before the fluctuations became too big. This will be
discussed in the next subsection. Taking the order of magnitude of
$A_0$ set by the fluxes (see eq.~\eqref{1.19} and discussion below it),
\begin{equation}
A_0 \sim M_{Pl}^4 10^{-18},
\label{3.32}
\end{equation}
$\epsilon (x)$ to be
\begin{equation}
\epsilon (x) \sim 10^{-12},
\label{3.33}
\end{equation}
corresponding, for example, to
\begin{equation}
\eta \sim 0.1, \quad \frac{x_f}{M_{Pl}} \sim 10^{-5},
\label{3.34}
\end{equation}
we obtain
\begin{equation}
\delta^2_{H} \sim 10^{-10},
\label{3.35}
\end{equation}
which is consistent with measurements of the cmb anisotropy.
Thus, this model of inflation gives appropriate values for the
amount of inflation and primordial fluctuations. However, these
results really make sense only if it is possible to escape from inflation
before the fluctuations became too big.
\subsection{Escape from Inflation}
At very small values of $x$, we cannot really trust the potential $\tilde{U}(x)$
because one should expect extra light states to become light as we approach the
singularity $x=0$. At the present time, the new physics at distances much less than the
eleven-dimensional Planck scale is not known. It may happen that these
new states are string-like, rather than particles~\cite{Hanany}.
In this subsection, we would like to give a qualitative argument
how such new states can terminate inflation. Let us emphasize that
we cannot prove that this is the actual mechanism. We just
would like to point out that the appearance of new physics
at short distances can help to terminate inflation.
We will assume that the new states are particles and, in the
absence of fluxes and non-perturbative effects, the moduli space
of heterotic M-theory is describable by the superpotential
\begin{equation}
{\mathbb W}={\mathbb W}(\Phi, {\bf X}),
\label{3.36}
\end{equation}
where the fields $\Phi$ come from a membrane stretching between
the visible brane and the five-brane. These fields are expected to be
charged under $E_8$.
The moduli space, that is the space of solutions of eq.
\begin{equation}
d {\mathbb W}=0,
\label{3.37}
\end{equation}
must consist of two branches. The first branch, the five-brane branch, is characterized by a non-zero expectation value
of the five-brane translational modulus $x$. In this branch, the five-brane multiplet is massless, while
the fields $\Phi$ are massive and integrated out form the low-energy field theory.
The mass of the fields $\Phi$ is proportional to $x$.
The second branch, the instanton branch, is characterized by the vanishing expectation value of $x$
and coincides with the moduli space of transition moduli~\cite{BDO1} of an instanton on our
Calabi-Yau threefold. This five-brane-instanton transition is called small instanton
transition~\cite{Seiberg, KS, OPP, BDO1}. The interpretation of the transition is the following.
As the five-brane hits the visible brane, it changes the vector bundle on the Calabi-Yau
manifold. The second Chern class of the new vector bundle changes by the amount
of the curve on which the five-brane was wrapped. This new bundle has more moduli.
The new moduli are precisely the transition moduli parameterizing the instanton
branch of the superpotential ${\mathbb W}$. In the instanton branch, only those
components of $\Phi$, which correspond to the transition moduli, take non-zero expectation
values and are massless, the remaining ones become massive and get integrated out.
The five-brane is also massive and integrated out in the instanton branch.
The origin represents a singularity, where all the multiplets become massless.
From the bundle viewpoint, the singularity at the origin corresponds to a vector bundle
becoming singular and turning into a torsion free sheaf~\cite{OPP}.
From the five-brane view-point, the singularity at the origin
corresponds to a five-brane coinciding with the visible brane.
An analogous, but simpler, transition takes place in string theory in
the $Dp-D(p+4)$ system~\cite{Douglas}.
The $Dp-D(p+4)$ system is describable by a supersymmetric field theory with eight
supercharges. The moduli space of this system consists of two branches, the Coulomb
branch and the Higgs branch. The Coulomb branch describes positions of the
$Dp$-brane away from the $D(p+4)$-brane. The Higgs branch describes how
the $Dp$-brane can get dissolved into the $D(p+4)$-brane. Geometrically, the Higgs branch
is isomorphic to the one-instanton moduli space on a $4$-manifold which is just the ADHM moduli space.
In the heterotic M-theory case, the analog of the Coulomb branch is the moduli space
of five-branes. The analog of the Higgs branch is the space of transition moduli.
Thus, transition moduli can be understood as a Calabi-Yau threefold generalization
of the ADHM one-instanton moduli space.
Now let us see how this picture changes in the presence of fluxes
and non-perturbative effects. For $x$ very small, we have to include
$\Phi$ into the Lagrangian. The Kahler potential receives an extra contribution
\begin{equation}
K_{\Phi}\sim tr_{E_8} (\Phi \bar \Phi).
\label{3.39}
\end{equation}
Then it is not hard to show that the potential energy
will contain, among others, a term
\begin{equation}
-\frac{U_0}{M^2_{Pl}} tr_{E_8} (\Phi \bar \Phi).
\label{3.40}
\end{equation}
This is a consequence of the following very simple statement.
If we take the theory with the constant superpotential and quadratic
Kahler potential, the potential energy has a maximum
at zero and for small fields is dominated by a negative
quadratic term.
Therefore, the mass of the fields $\Phi$ schematically behaves as
\begin{equation}
|\frac{\partial^2{\mathbb W}(\Phi, {\bf X})}{\partial \Phi^2}\mid_{\Phi =0}|^2
- \frac{U_0}{M^2_{Pl}}.
\label{3.41}
\end{equation}
This means that as the five-brane gets very close to the visible sector,
the fields $\Phi$ become tachyonic and begin to roll downhill.
Since no slow roll conditions on $\Phi$ are satisfied, this terminates
inflation.
Eventually, the five-brane hits the wall and disappears (gets massive and integrated out
from the low-energy field theory) through a small instanton transition.
In addition, those components
of $\Phi$, which do not correspond to the transition instanton moduli, get
a mass, according to the superpotential~\eqref{3.36}, and get integrated out.
Now their mass is set by the vacuum expectation values of the transition moduli.
The new system of moduli after the small instanton transition
involves the moduli discussed in Section 2 plus the new transition moduli.
The physics of them will be considered in the next section.
This escape from inflation is, somewhat, analogous to one in
$D3$-$D7$ inflation studied in~\cite{Dasgupta}.
In~\cite{Dasgupta}, the negative mass for the charged hypermultiplets
(analogs of our fields $\Phi$) was created by Fayet-Iliopoulos terms.
The same Fayet-Iliopoulos terms were responsible for stabilization
at a non-zero value.
\section{The Post-inflationary Phase}
After inflation, the five-brane disappears through a small
instanton transition and turns into new vector bundle moduli, which we
denote by $\phi_i$. The new system of moduli contains now the following fields
\begin{equation}
S, T^1, T^2, {\bf Y}, Z_{\alpha}, \phi_i.
\label{3.42}
\end{equation}
This system of moduli can be stabilized. The moduli
$S, T^1, T^2, {\bf Y}$ and $Z_{\alpha}$ can be stabilized by the same
mechanism as in Section 2. In this section, we will
concentrate on the moduli $\phi_i$. Without loss of generality, we
can assume that there is only one such modulus $\phi$.
Vector bundle moduli have a non-perturbative superpotential.
It appears as a factor in eqs.~\eqref{1.35}, \eqref{1.36} and~\eqref{1.37}.
Since after the small instanton transition only the bundle on the visible brane has
changed, only $W_{v5}$ and $W_{vh}$ will depend on $\phi$ after the transition.
In fact,
\begin{equation}
W_{v5} >>W_{vh},
\label{3.43}
\end{equation}
since the coefficient $\tau$ in eqs.~\eqref{1.35} and~\eqref{1.36}
is much greater than one. Therefore, the potential for $\phi$ is
\begin{equation}
W(\phi)={\cal P}(\phi)e^{-\tau {\bf Y}},
\label{3.44}
\end{equation}
where we have denoted the factor depending on $\phi$ by ${\cal P}(\phi)$.
Let us make a remark. One can worry that, after the transition, the superpotential~\eqref{3.44}
will not depend on $\phi$. Indeed, the superpotential is induced by a string wrapping
isolated genus zero curves. On the other hand, the five-brane that turned into the modulus $\phi$
was wrapped on a non-isolated genus zero curve or a higher genus curve.
It seems possible that the bundle over the isolated curves did not really change
and the bundle moduli contribution to the superpotential will remain unchanged.
However, this logic is not quite correct because the curves might intersect.
If the curve on which the five-brane was wrapped intersects at least one isolated
genus zero curve, the non-perturbative superpotential will
change and it will depend on $\phi$.
Even though the five-brane modulus $x$ could not be stabilized, the new vector
bundle modulus $\phi$ can.
The equation
\begin{equation}
D_{\phi}W=0
\label{3.45}
\end{equation}
has a solution. The analysis of this equation
will be analogous to the one in~\cite{BO}.
Let us recall that the superpotential $W$ is a sum
of various contributions
\begin{equation}
W=W_f+W_{g}+W_{np},
\label{3.46}
\end{equation}
where $W_{np}$ is now the sum of $W_{5h}$ and $W(\phi)$.
Then eq.~\eqref{3.45} can be written as
\begin{equation}
\partial_{\phi}{\cal P}(\phi) e^{-\tau {\bf Y}}=
\partial_{\phi}K(\phi) W_f,
\label{3.47}
\end{equation}
where we have assumed that
\begin{equation}
W_f >> W_{\phi}.
\label{3.48}
\end{equation}
In~\cite{BO}, it was shown that ineq.~\eqref{3.48} is indeed satisfied.
Vector bundle moduli superpotentials were studied in detail in~\cite{BDO2, BDO3}.
They were found to be high degree polynomials.
Therefore, we will take
\begin{equation}
{\cal P}(\phi)=\phi^d.
\label{3.49}
\end{equation}
The Kahler potential $K(\phi)$ represents a more complicated problem.
It was evaluated explicitly only for bundles which can be written
as the pullback of a bundle on a four-manifold~\cite{Gray}.
A generic bundle on a threefold is, obviously, not of this form.
In this paper, we will not need the actual form of the Kahler
potential. Using eq.~\eqref{3.49} and ignoring the imaginary part of ${\bf Y}$,
eq.~\eqref{3.47}
can be written as
\begin{equation}
\phi^d e^{-\tau y}=\partial_{\phi}K(\phi) W_f.
\label{3.50}
\end{equation}
Clearly, such an equation has a solution for a generic function $K(\phi)$.
To present a numeric solution, we need to know the order of magnitude
of $K$. It was estimated in~\cite{BO} and found to be $\sim 10^{-5}$
in Planck units.
Then, if $d$ is sufficiently large, we can approximately write eq.~\eqref{3.50} as follows
\begin{equation}
\phi^d \sim 10^{-5} e^{\tau y} W_f.
\label{3.51}
\end{equation}
Taking, as an example,
\begin{equation}
W_f \sim 10^{-10}, \quad \tau \sim 200, \quad y \sim 0.8, \quad d \sim 40,
\label{3.52}
\end{equation}
we obtain
\begin{equation}
\phi \sim 20.
\label{3.53}
\end{equation}
The relatively large value of $\phi$ means that the gauge connection
is spread out over the Calabi-Yau manifold, rather than
sharply peaked over some curve. As long as we stay away
from singularities in the moduli space of vector bundles,
any value of $\phi$ is acceptable. To be slightly more precise,
eq.~\eqref{3.53} is a solution for the absolute value of $\phi$.
The phase of $\phi$ can also be found form eq.~\eqref{3.47}~\cite{BO}.
Thus, we have shown that the new system of moduli has much simple
stability properties. That is, it is possible to fix
all moduli after inflation. Now we are going to argue that
the cosmological constant in the new vacuum can be positive and
fine tuned to coincide with observations. Recall that the cosmological constant
during inflation is given by three contributions
\begin{equation}
\Lambda_{inflation}=-\Lambda_{SUGRA} +\Lambda_{D}^{hidden}+
\Lambda_{D}^{visible}.
\label{3.54}
\end{equation}
The first term in eq.~\eqref{3.54} is the contribution
from the supergravity potential energy. It is negative and approximately
given by $-\frac{W_f^2}{M_{Pl}^2}$.
The second term comes from the Fayet-Iliopoulos term
in the hidden sector. The last term is the contribution from
the Fayet-Iliopoulos term
in the visible sector. They second and the third terms are positive.
In the previous section, we argued that the existence of Fayet-Iliopoulos terms
in the visible sector is not relevant for inflation.
Nevertheless, we will assume that they are present.
The cosmological constant $\Lambda_{inflation}$ has to be positive and large.
After the small instanton transition, some of the contributions to the cosmological constant
might change, because they depend on the properties of the vector bundle.
The contribution $\Lambda_{SUGRA}$ might change because, as was argued in~\cite{GKLM},
the flux-induced superpotential may receive higher order corrections from Chern-Simons
invariants which depend on the choice of the bundle.
The contribution $\Lambda_{D}^{visible}$ will change because
it depends on the gauge connection. Moreover, since after a small
instanton transition there is a possibility of changing the number
of families of quarks and leptons~\cite{KS, OPP}, the corresponding
$U(1)$ gauge group might not be anomalous anymore.
In this case $\Lambda_{D}^{visible}$ will be zero.
However, $\Lambda_{D}^{hidden}$ will not change because the bundle
on the hidden brane remains unchanged.
All these arguments show that, in principle, the cosmological constant changes after inflation.
Since it receives both negative and positive contributions,
it is possible, by fine tuning, to make it consistent with observations.
\section{Conclusion}
In this paper, we considered dynamics of the five-brane wrapped on
a non-isolated genus zero or higher genus curve.
Non-perturbative superpotentials do not depend on moduli
of such five-branes.
We showed that fluxes and non-perturbative effects can stabilize such a five-brane
in a non-supersymmetric AdS vacuum.
We also
showed
that addition of Fayet-Iliopoulos terms not only
can raise this vacuum to a dS vacuum but also
can create one more dS vacuum.
We also stabilized $h^{1,1}$ moduli
which do not have non-perturbative superpotentials.
The cosmological constant of the vacuum can be positive and
fine tuned to be consistent with observations.
This provides a generalization of results of~\cite{BO, Raise}
and shows that the most complete system of heterotic
string moduli can be stabilized in a vacuum with a positive
cosmological constant. In addition, we
showed that, by modifying the supergravity
potential energy with Fayet-Iliopoulos terms,
one can create an inflationary potential
and treat the five-brane translational modulus
as an inflaton. However, the potential cannot be trusted at very small
distances because one should expect extra light states to appear.
We give a qualitative argument how such new states
can terminate inflation. The idea is that, in the presence
of fluxes and non-perturbative effects, these extra states can become
tachyonic when the five-brane comes very close to the visible brane.
Eventually, the five-brane hits the visible brane.
The system undergoes a small instanton transition which changes the vector
bundle on the visible brane.
The five-brane disappears from the low-energy spectrum
while the vector bundle moduli are created. They have simpler
stability properties and, as a result, the new system
of moduli can be stabilized. We also argue that the
cosmological constant changes after the transition.
The cosmological constant after the transition can be
fine tuned to be consistent with observations.
\section{Acknowledgements}
The author is indebted to Juan Maldacena
for lots of interesting and helpful discussions.
The work is supported by NSF grant PHY-0070928.
|
{
"timestamp": "2005-01-07T01:44:04",
"yymm": "0411",
"arxiv_id": "hep-th/0411062",
"language": "en",
"url": "https://arxiv.org/abs/hep-th/0411062"
}
|
\section*{\S1.~Jet spaces and prolongations}
\subsection*{1.1.~Choice of notations for the jet space variables}
Let $\K = \R$ or $\C$. Let $n\geq 1$ and $m\geq 1$ be two positive
integers and consider two sets of variables $x= (x^1, \dots, x^n) \in
\K^n$ and $y = (y^1, \dots, y^m)$. In the classical theory of Lie
symmetries of partial differential equations, one considers certain
differential systems whose (local) solutions should be mappings of the
form $y = y(x)$. We refer to~\cite{ ol1986} and to~\cite{ bk1989} for
an exposition of the fundamentals of the theory. Accordingly, the
variables $x$ are usually called {\sl independent}, whereas the
variables $y$ are called {\sl dependent}. Not to enter in subtle
regularity considerations (as in~\cite{ m2005}), we shall assume
$\mathcal{ C}^\infty$-smoothness of all functions throughout this
paper.
Let $\kappa \geq 1$ be a positive integer. For us, in a very concrete
way (without fiber bundles), the {\sl $\kappa$-th jet space}
$\mathcal{ J}_{ n, m}^\kappa$ consists of the space $\K^{n+m+ m \frac{
(n+m)!}{n! \ m!}}$ equipped with the affine coordinates
\def5.22}\begin{equation{1.2}\begin{equation}
\left(
x^i, y^j, y_{i_1}^j, y_{i_1,i_2}^j,
\dots\dots,
y_{i_1,i_2,\dots,i_\kappa}^j
\right),
\end{equation}
having the symmetries
\def5.22}\begin{equation{1.3}\begin{equation}
y_{i_1,i_2,\dots,i_\lambda}^j
=
y_{i_{\sigma(1)},i_{\sigma(2)},\dots,i_{\sigma(\lambda)}}^j,
\end{equation}
for every $\lambda$ with $1\leq \lambda \leq \kappa$ and for every
permutation $\sigma$ of the set $\{1, \dots, \lambda \}$. The variable
$y_{i_1, i_2, \dots, i_\lambda}^j$ is an independent coordinate
corresponding to the $\lambda$-th partial derivative $\frac{
\partial^\lambda y^j}{ \partial x^{ i_1} \partial x^{ i_2} \cdots
\partial x^{ i_\lambda }}$, which explains the symmetries~\thetag{
1.3}.
In the classical Lie theory (\cite{ ol1979},
\cite{ ol1986}, \cite{ bk1989}), all the
geometric objects: point transformations, vector fields, {\it etc.},
are local, defined in a neighborhood of some point lying in some
affine space $\K^N$. However, in this paper, the original geometric
motivations are rapidly forgotten in order to focus on combinatorial
considerations. Thus, to simplify the presentation, we shall not
introduce any special notation to speak of certain local open subsets
of $\K^{n+ m}$, or of the jet space $\mathcal{ J}_{ n, m}^\kappa =
\K^{n+m+ m \frac{ (n+m)! }{ n! \ m! }}$, {\it etc.}: we will always
work in global affine spaces $\K^N$.
\subsection*{ 1.4.~Prolongation $\varphi^{ (\kappa)}$
of a local diffeomorphism $\varphi$ to the $\kappa$-th jet space} In
this paragraph, we recall how the prolongation of a diffeomorphism to
the $\kappa$-th jet space is defined (\cite{ ol1979},
\cite{ ol1986}, \cite{ bk1989}).
Let $x_*\in \K^n$ be a central fixed point and let $\varphi : \K^{n+m}
\to \K^{n+m}$ be a diffeomorphism whose Jacobian matrix is close to
the identity matrix at least in a small neighborhood of $x_*$. Let
\def5.22}\begin{equation{1.5}\begin{equation}
J_{x_*}^\kappa
:=
\left(
x_*^i,y_{*i_1}^j,
y_{*i_1,i_2}^j,
\dots\dots,
y_{*i_1,i_2,\dots,i\kappa}^j
\right)
\in
\left.
\mathcal{J}_{n,m}^\kappa
\right\vert_{x_*}
\end{equation}
be an arbitrary $\kappa$-jet based at $x_*$. The goal is to defined
its transformation $\varphi^{(\kappa)} ( J_{x_*}^\kappa)$ by
$\varphi$.
To this aim, choose an arbitrary mapping $\K^n\ni x \mapsto g(x) \in
\K^m$ defined at least in a neighborhood of $x_*$ and representing
this $\kappa$-jet, {\it i.e.} satisfying
\def5.22}\begin{equation{1.6}\begin{equation}
y_{*i_1,\dots,i_\lambda}^j
=
\frac{\partial^\lambda g^j}{
\partial x^{i_1}\cdots \partial x^{i_\lambda}}
(x_*),
\end{equation}
for every $\lambda\in \N$ with $0 \leq \lambda \leq \kappa$, for all
indices $i_1, \dots, i_\lambda$ with $1 \leq i_1, \dots, i_\lambda
\leq n$ and for every $j\in \N$ with $1\leq j\leq m$. In accordance with
the splitting $(x, y)\in \K^n\times \K^m$ of coordinates, split the
components of the diffeomorphism $\varphi$ as $\varphi = (\phi, \psi)
\in \K^n\times \K^m$. Write $\left( \overline{x}, \overline{y}
\right)$ the coordinates in the target space, so that the
diffeomorphism $\varphi$ is:
\def5.22}\begin{equation{1.7}\begin{equation}
\K^{n+m}
\ni
(x,y)
\longmapsto
\left(
\overline{x}, \overline{y}
\right)
=
(\phi(x, y), \psi (x, y))
\in
\K^{n+m}.
\end{equation}
Restrict the variables $(x, y)$ to belong to the graph of $g$, namely
put $y:= g(x)$ above, which yields
\def5.22}\begin{equation{1.8}\begin{equation}
\left\{
\aligned
\overline{x}
&
=
\phi(x, g(x)), \\
\overline{y}
&
=
\psi(x,g(x)).
\endaligned\right.
\end{equation}
As the differential of $\varphi$ at $x_*$ is close to the identity,
the first family of $n$ scalar equations may be solved with respect to
$x$, by means of the implicit function theorem. Denote $x
= \overline{ \chi}(\overline{ x})$ the resulting mapping, satisfying
by definition
\def5.22}\begin{equation{1.9}\begin{equation}
\overline{x}
\equiv
\phi\left(
\overline{ \chi}(\overline{ x}),
g(\overline{ \chi}(\overline{ x}))
\right).
\end{equation}
Replace $x$ by $\overline{ \chi}(\overline{ x})$ in the second family
of $m$ scalar equations~\thetag{1.8} above, which yields:
\def5.22}\begin{equation{1.10}\begin{equation}
\overline{y}
=
\psi\left(
\overline{ \chi}(\overline{ x}),
g(\overline{ \chi}(\overline{ x}))
\right).
\end{equation}
Denote simply by $\overline{ y} = \overline{ g} ( \overline{ x})$ this
last relation, where $\overline{ g} ( \cdot) := \psi \left( \overline{
\chi} (\cdot), g ( \overline{ \chi} (\cdot)) \right)$.
In summary, the graph $y=g(x)$ has been transformed to the graph
$\overline{y} = \overline{ g} (\overline{ x})$ by the diffeomorphism
$\varphi$ whose first order approximation is close to the identity.
Define then the {\sl transformed jet $\varphi^{(\kappa)} \left(
J_{x_*}^\kappa \right)$} to be the $\kappa$-th jet of $\overline{ g}$ at
the point $\overline{ x}_* := \phi ( x_*)$, namely:
\def5.22}\begin{equation{1.11}\begin{equation}
\varphi^{(\kappa)}
\left(
J_{x_*}^\kappa
\right)
:=
\left(
\frac{\partial^\lambda \overline{ g}^j}{
\partial \overline{ x}^{i_1}
\cdots
\partial \overline{ x}^{i_\lambda}}
(\overline{ x}_*)
\right)_{
1\leq i_1, \dots, i_\lambda \leq n, \
0 \leq \lambda \leq \kappa
}^{
1\leq j\leq m}
\left.
\in\mathcal{J}_{n,m}^\kappa
\right\vert_{\overline{x}_*}.
\end{equation}
It may be shown that this jet does not depend on the choice of a local
graph $y = g(x)$ representing the $\kappa$-jet $J_{x_*}^\kappa$ at
$x_*$. Furthermore, if $\pi_\kappa := \mathcal{ J}_{n,m}^\kappa \to
\K^m$ denotes the canonical projection onto the first factor, the
following diagram commutes:
$$
\diagram \mathcal{J}_{n,m}^\kappa \rto^{\varphi^{(\kappa)}}
\dto_{\pi_\kappa}
& \mathcal{J}_{n,m}^\kappa
\dto^{\pi_\kappa} \\
\K^{n+m} \rto^{\varphi} &
\K^{n+m}
\enddiagram.
$$
\subsection*{1.12.~Inductive formulas for the
$\kappa$-th prolongation $\varphi^{(\kappa)}$} To present them, we
change our notations. Instead of $(\overline{ x}, \overline{ y})$, as
coordinates in the target space $\K^n\times \K^m$, we shall use
capital letters:
\def5.22}\begin{equation{1.13}\begin{equation}
\left(
X^1,\dots,X^n,Y^1,\dots,Y^m
\right).
\end{equation}
In the source space $\K^{n+m}$ equipped with the coordinates $(x, y)$,
we use the jet coordinates~\thetag{ 1.2} on the associated
$\kappa$-th jet space. In the target space $\K^{n+m}$ equipped with
the coordinates $(X, Y)$, we use the coordinates
\def5.22}\begin{equation{1.14}\begin{equation}
\left(
X^i, Y^j, Y_{X^{i_1}}^j,
Y_{X^{i_1,i_2}}^j,
\dots\dots,
Y_{X^{i_1,i_2,\dots,i_\kappa}}^j
\right)
\end{equation}
on the associated $\kappa$-th jet space. In these notations, the
diffeomorphism $\varphi$ whose first order approximation is close to
the identity mapping in a neighborhood of $x_*$
may be written under the form:
\def5.22}\begin{equation{1.15}\begin{equation}
\varphi :
\
(x^{i'},y^{j'})
\mapsto
\left(
X^i,Y^j
\right)
=
\left(
X^i(x^{i'},y^{j'}), \ Y^j(x^{i'},y^{j'})
\right),
\end{equation}
for some $\mathcal{ C}^\infty$-smooth functions $X^i(x^{i'},y^{j'})$,
$i = 1, \dots, n$, and $Y^j(x^{i'},y^{j'})$, $j = 1, \dots, m$. The
first prolongation $\varphi^{(1)}$ of $\varphi$ may be written
under the form:
\def5.22}\begin{equation{1.16}\begin{equation}
\varphi^{(1)}
: \
\left(
x^{i'},y^{j'}, y_{i_1'}^{j'}
\right)
\longmapsto
\left(
X^i(x^{i'},y^{j'}), \
Y^j(x^{i'},y^{j'}), \
Y_{X^{i_1}}^j
\left(
x^{i'}, y^{j'}, y_{i_1'}^{j'}
\right)
\right),
\end{equation}
for some functions $Y_{X^{i_1}}^j \left( x^{i'}, y^{j'}, y_{i_1'}^{j'}
\right)$ which depend on the pure first jet variables
$y_{i_1'}^{j'}$. The way how these functions depend on the first order
partial derivatives functions $X_{x^{i'}}^i$, $X_{y^{j'}}^i$,
$Y_{x^{i'}}^j$, $Y_{y^{j'}}^j$ and on the pure first jet variables
$y_{i_1'}^{j'}$ is provided (in principle) by the following compact
formulas (\cite{ bk1989}):
\def5.22}\begin{equation{1.17}\begin{equation}
\left(
\begin{array}{c}
Y_{X^1}^j \\
\vdots \\
Y_{X^n}^j \\
\end{array}
\right)
=
\left(
\begin{array}{ccc}
D_1^1 X^1 & \cdots & D_1^1 X^n \\
\vdots & \cdots & \vdots \\
D_n^1 X^1 & \cdots & D_n^1 X^n \\
\end{array}
\right)^{-1}
\left(
\begin{array}{c}
D_1^1 Y^j \\
\vdots \\
D_n^1 Y^j \\
\end{array}
\right),
\end{equation}
where, for $i' = 1, \dots, n$, the $D_{ i'}^1$ denote the {\sl $i'$-th
first order total differentiation operators}:
\def5.22}\begin{equation{1.18}\begin{equation}
D_{i'}^1
:=
\frac{\partial}{\partial x^{i'}}
+
\sum_{j'=1}^m\,y_{i'}^{j'}\,\frac{\partial}{\partial y^{j'}}.
\end{equation}
Striclty speaking, these formulas~\thetag{ 1.17} are not explicit,
because an inverse matrix is involved and because the terms $D_{i'}^1
X^i$, $D_{i'}^1 Y^j$ are not developed. However, it would be
elementary to write down the corresponding totally explicit complete
formulas for the functions $Y_{X^{i_1}}^j = Y_{X^{i_1}}^j \left(
x^{i'}, y^{j'}, y_{i_1'}^{j'} \right)$.
Next, the second prolongation $\varphi^{ (2)}$ is of the form
\def5.22}\begin{equation{1.19}\begin{equation}
\varphi^{(2)}
: \
\left(
x^{i'},y^{j'}, y_{i_1'}^{j'},
y_{i_1',i_2'}^{j'}
\right)
\longmapsto
\left(
\varphi^{(1)}
\left(
x^{i'},y^{j'}, y_{i_1'}^{j'}
\right),
\
Y_{X^{i_1}X^{i_2}}^j
\left(
x^{i'}, y^{j'}, y_{i_1'}^{j'}, y_{i_1',i_2'}^{j'}
\right)
\right),
\end{equation}
for some functions $Y_{X^{i_1} X^{i_2}}^j \left( x^{ i'}, y^{ j'}, y_{
i_1' }^{j'}, y_{i_1', i_2'}^{j'} \right)$ which depend on the pure
first and second jet variables. For $i = 1, \dots, n$, the
expressions of $Y_{X^{i_1}X^i}^j$ are given by the following compact
formulas (again \cite{ bk1989}):
\def5.22}\begin{equation{1.20}\begin{equation}
\left(
\begin{array}{c}
Y_{X^{i_1}X^1}^j \\
\vdots \\
Y_{X^{i_1}X^n}^j \\
\end{array}
\right)
=
\left(
\begin{array}{ccc}
D_1^1 X^1 & \cdots & D_1^1 X^n \\
\vdots & \cdots & \vdots \\
D_n^1 X^1 & \cdots & D_n^1 X^n \\
\end{array}
\right)^{-1}
\left(
\begin{array}{c}
D_1^2 Y_{X^{i_1}}^j \\
\vdots \\
D_n^2 Y_{X^{i_1}}^j \\
\end{array}
\right),
\end{equation}
where, for $i' = 1, \dots, n$, the $D_{i'}^2$ denote the {\sl
$i'$-th second order
total differentiation operators}:
\def5.22}\begin{equation{1.21}\begin{equation}
D_{i'}^2
:=
\frac{\partial}{\partial x^{i'}}
+
\sum_{j'=1}^m\,y_{i'}^{j'}\,\frac{\partial}{\partial y^{j'}}
+
\sum_{j'=1}^m\,\sum_{i_1'=1}^n\,y_{i',i_1'}^{j'}\,
\frac{\partial}{\partial y_{i_1'}^{j'}}.
\end{equation}
Again, these formulas~\thetag{ 1.20} are not explicit in the sense
that an inverse matrix is involved and the terms $D_{i'}^1 X^i$,
$D_{i'}^2 Y_{ X^{ i_1}}^j$ are not developed. It would already be a
nontrivial computational task to develope these expressions and to
find out nice satisfying combinatorial formulas.
In order to present the general inductive non-explicit formulas for
the computation of the $\kappa$-th prolongation $\varphi^{(\kappa)}$,
we need some more notation. Let $\lambda \in \N$ be an arbitrary
integer. For $i' = 1, \dots, n$, let $D_{i'}^\lambda$ denotes the {\sl
$i'$-th $\lambda$-th order total differentiation operators}, defined
precisely by:
\def5.22}\begin{equation{1.22}\begin{equation}
\left\{
\aligned
D_{i'}^\lambda
&
:=
\frac{\partial}{\partial x^{i'}}
+
\sum_{j'=1}^m\,y_{i'}^{j'}\,\frac{\partial}{\partial y^{j'}}
+
\sum_{j'=1}^m\,\sum_{i_1'=1}^n\,y_{i',i_1'}^{j'}\,
\frac{\partial}{\partial y_{i_1'}^{j'}}
+
\sum_{j'=1}^m\,\sum_{i_1',i_2'=1}^n\,y_{i',i_1',i_2'}^{j'}\,
\frac{\partial}{\partial y_{i_1',i_2'}^{j'}}
+ \\
& \
\ \ \ \ \
+
\cdots
+
\sum_{j'=1}^m\,\sum_{i_1',i_2',\dots,i_{\lambda-1}'=1}^n\,
y_{i',i_1',i_2',\dots,i_{\lambda-1}'}^{j'}\,
\frac{\partial}{\partial y_{i_1',i_2',\dots,i_{\lambda-1}'}^{j'}}.
\endaligned\right.
\end{equation}
Then, for $i = 1, \dots, n$, the expressions of $Y_{X^{i_1}\cdots
X^{i_{\lambda-1}} X^i }^j$ are given by the following compact formulas
(again \cite{ bk1989}):
\def5.22}\begin{equation{1.23}\begin{equation}
\left(
\begin{array}{c}
Y_{X^{i_1}\cdots X^{i_{\lambda-1}}X^1}^j \\
\vdots \\
Y_{X^{i_1}\cdots X^{i_{\lambda-1}}X^n}^j \\
\end{array}
\right)
=
\left(
\begin{array}{ccc}
D_1^1 X^1 & \cdots & D_1^1 X^n \\
\vdots & \cdots & \vdots \\
D_n^1 X^1 & \cdots & D_n^1 X^n \\
\end{array}
\right)^{-1}
\left(
\begin{array}{c}
D_1^\lambda Y_{X^{i_1}\cdots X^{i_{\lambda-1}}}^j \\
\vdots \\
D_n^\lambda Y_{X^{i_1}\cdots X^{i_{\lambda-1}}}^j \\
\end{array}
\right).
\end{equation}
Again, these inductive formulas are incomplete and unsatisfactory.
\def1.32}\begin{problem{1.24}\begin{problem}
Find totally explicit complete formulas
for the $\kappa$-th prolongation $\varphi^{(\kappa)}$.
\end{problem}
Except in the cases $\kappa = 1, 2$, we have not been able to solve
this problem. The case $\kappa = 1$ is elementary. Complete formulas
in the particular cases $\kappa = 2$, $n=1$, $m\geq 1$ and $n\geq 1$,
$m=1$ are implicitely provided in~\cite{ m2004a} and in~\cite{
m2004b}, where one observes the appearance of some modifications of
the Jacobian determinant of the diffeomorphism $\varphi$, inserted in
a clearly understandable combinatorics. In fact, there is a nice
dictionary between the formulas for $\varphi^{ (2)}$ and the formulas
for the second prolongation $\mathcal{ L}^{ (2)}$ of a vector field
$\mathcal{ L}$ which were written in equation~\thetag{ 43} of~\cite{
gm2003} ({\it see} also equations~\thetag{ 2.6}, \thetag{ 3.20},
\thetag{ 4.6} and~\thetag{ 5.3} in the next paragraphs). In the
passage from $\varphi^{ (2)}$ to $\mathcal{ L}^{ (2)}$, a sort of
formal first order linearization may be observed and the reverse
passage may be easily guessed. However, for $\kappa \geq 3$, the
formulas for $\varphi^{ (\kappa)}$ explode faster than the formulas
for the $\kappa$-th prolongation $\mathcal{ L}^{ (\kappa )}$ of a
vector field $\mathcal{ L}$. Also, the dictionary between $\varphi^{
(\kappa)}$ and $\mathcal{ L}^{ ( \kappa )}$ disappears. In fact, to
elaborate an appropriate dictionary, we believe that one should
introduce before a sort of formal $(\kappa-1)$ order linearizations of
$\varphi^{ ( \kappa)}$, finer than the first order linearization
$\mathcal{ L}^{ (\kappa)}$. To be optimistic, we believe that the
final answer to Problem~1.24
is accessible.
The present article is devoted to present totally explicit complete
formulas for the $\kappa$-th prolongation $\mathcal{ L}^{ (\kappa )}$
of a vector field $\mathcal{ L}$ to $\mathcal{ J}_{n,m}^\kappa$, for
$n\geq 1$ arbitrary, for $m\geq 1$ arbitrary and for $\kappa \geq 1$
arbitrary.
\subsection*{ 1.25.~Prolongation of a vector field to the
$\kappa$-th jet space} Consider a vector field
\def5.22}\begin{equation{1.26}\begin{equation}
\mathcal{ L}
=
\sum_{i=1}^n\mathcal{X}^i\,\frac{\partial}{\partial x^i}
+
\sum_{j=1}^m\,\mathcal{Y}^j\,\frac{\partial}{\partial y^j},
\end{equation}
defined in $\K^{ n+m}$. Its flow:
\def5.22}\begin{equation{1.27}\begin{equation}
\varphi_t ( x, y)
:=
\exp
\left(
t \mathcal{ L}
\right)
(x, y)
\end{equation}
constitutes a one-parameter of diffeomorphisms of $\K^{ n+m} $ close
to the identity. The lift $(\varphi_t )^{ ( \kappa )}$ to the
$\kappa$-th jet space constitutes a one-parameter family of
diffeomorphisms of $\mathcal{ J}_{ n,m }^\kappa$. By definition, the
{\sl $\kappa$-th prolongation $\mathcal{ L }^{( \kappa)}$ of
$\mathcal{ L }$ to the jet space $\mathcal{ J}_{n, m }^\kappa$} is the
infinitesimal generator of $(\varphi_t )^{ (\kappa)}$, namely:
\def5.22}\begin{equation{1.28}\begin{equation}
\mathcal{ L}^{(\kappa)}
:=
\left.
\frac{d}{dt}
\right\vert_{t=0}
\left[
(\varphi_t)^{(\kappa)}
\right].
\end{equation}
\subsection*{1.29.~Inductive formulas for the
$\kappa$-th prolongation $\mathcal{ L}^{(\kappa)}$} As a vector field
defined in $\K^{n+m+ m \frac{ (n+m)! }{ n! \ m!}}$, the $\kappa$-th
prolongation $\mathcal{ L}^{ (\kappa) }$ may be written under the
general form:
\def5.22}\begin{equation{1.30}\begin{equation}
\left\{
\aligned
\mathcal{L}^{(\kappa)}
&
=
\sum_{i=1}^n\mathcal{X}^i\,\frac{\partial}{\partial x^i}
+
\sum_{j=1}^m\,\mathcal{Y}^j\,\frac{\partial}{\partial y^j}
+ \\
& \
\ \ \ \ \
+
\sum_{j=1}^m\,\sum_{i_1=1}^n\,{\bf Y}_{i_1}^j\,
\frac{\partial}{\partial y_{i_1}^j}
+
\sum_{j=1}^m\,\sum_{i_1,i_2=1}^n\,{\bf Y}_{i_1,i_2}^j\,
\frac{\partial}{\partial y_{i_1,i_2}^j}
+
\cdots
+ \\
& \
\ \ \ \ \
+
\sum_{j=1}^m\,\sum_{i_1,\dots,i_\kappa=1}^n\,
{\bf Y}_{i_1,\dots,i_\kappa}^j\,
\frac{\partial}{\partial y_{i_1,\dots,i_\kappa}^j}.
\endaligned\right.
\end{equation}
Here, the coefficients ${\bf Y}_{i_1}^j$, ${\bf Y}_{i_1, i_2}^j$,
$\dots$, ${\bf Y}_{i_1, i_2, \dots, i_\kappa}^j$ are uniquely
determined in terms of partial derivatives of the coefficients
$\mathcal{ X}^i$ and $\mathcal{ Y}^j$ of the original vector field
$\mathcal{ L}$, together with the pure jet variables $\left( y_{ i_1
}^j,\dots, y_{i_1, \dots, i_\kappa}^j \right)$, by means of the
following {\sl fundamental inductive formulas} (\cite{ ol1979},
\cite{ ol1986}, \cite{
bk1989}):
\def5.22}\begin{equation{1.31}\begin{equation}
\left\{
\aligned
{\bf Y}_{i_1}^j
&
:=
D_{i_1}^1
\left(
\mathcal{ Y}^j
\right)
-
\sum_{k=1}^n\,D_{i_1}^1
\left(
\mathcal{X}^k
\right)
\, y_k^j,
\\
{\bf Y}_{i_1,i_2}^j
&
:=
D_{i_2}^2
\left(
{\bf Y}_{i_1}^j
\right)
-
\sum_{k=1}^n\,D_{i_2}^1
\left(
\mathcal{X}^k
\right)
\, y_{i_1,k}^j,
\\
\cdots \cdots \cdots
&
\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots
\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots
\\
{\bf Y}_{i_1,i_2,\dots,i_\kappa}^j
&
:=
D_{i_\kappa}^\kappa
\left(
{\bf Y}_{i_1,i_2,\dots,i_{\kappa-1}}^j
\right)
-
\sum_{k=1}^n\,D_{i_\kappa}^1
\left(
\mathcal{X}^k
\right)
\, y_{i_1,i_2,\dots,i_{\kappa-1},k}^j,
\endaligned\right.
\end{equation}
where, for every $\lambda \in \N$ with $0 \leq \lambda \leq \kappa$,
and for every $i\in \N$ with
$1 \leq i \leq n$, the $i$-th $\lambda$-th order total
differentiation operator $D_i^\lambda$ was defined in~\thetag{
1.22}
above.
\def1.32}\begin{problem{1.32}\begin{problem}
Applying these inductive formulas, find totally explicit complete
formulas for the $\kappa$-th prolongation $\mathcal{L}^{(\kappa)}$.
\end{problem}
The present article is devoted to provide all the desired formulas.
\subsection*{ 1.33.~Inductive methodology}
We have the intention of presenting our results in a purely inductive
style, based on several thorough visual comparisons between massive formulas
which will be written and commented in four different cases:
\smallskip
\begin{itemize}
\item[{\bf (i)}]
$n=1$ and $m=1$; $\kappa\geq 1$ arbitrary;
\item[{\bf (ii)}]
$n\geq 1$ and $m=1$; $\kappa\geq 1$ arbitrary;
\item[{\bf (iii)}]
$n=1$ and $m\geq 1$; $\kappa\geq 1$ arbitrary;
\item[{\bf (iv)}]
general case: $n\geq 1$ and $m\geq 1$; $\kappa\geq 1$ arbitrary.
\end{itemize}
\smallskip
Accordingly, we shall particularize and slightly lighten our notations
in each of the three (preliminary) cases (i)
[Section~2], (ii) [Section~3] and (iii) [Section~4].
\section*{\S2.~One independent variable and one dependent variable}
\subsection*{2.1.~Simplified adapted notations}
Assume $n=1$ and $m=1$, let $\kappa\in \N$ with $\kappa \geq 1$ and
simply denote the jet variables by:
\def5.22}\begin{equation{2.2}\begin{equation}
\left(
x,y,y_1,y_2,\dots,y_\kappa
\right) \in \mathcal{J}_{1,1}^\kappa.
\end{equation}
The $\kappa$-th prolongation of a vector field will be denoted by:
\def5.22}\begin{equation{2.3}\begin{equation}
\mathcal{L}
=
\mathcal{X}\,\frac{\partial}{\partial x}
+
\mathcal{Y}\,\frac{\partial}{\partial y}
+
{\bf Y}_1\,\frac{\partial}{\partial y_1}
+
{\bf Y}_2\,\frac{\partial}{\partial y_2}
+
\cdots
+
{\bf Y}_\kappa\,\frac{\partial}{\partial y_\kappa}.
\end{equation}
The coefficients
${\bf Y}_1$, ${\bf Y}_2$, $\dots$, ${\bf Y}_\kappa$
are computed by means of the inductive formulas:
\def5.22}\begin{equation{2.4}\begin{equation}
\left\{
\aligned
{\bf Y}_1
&
:=
D^1(\mathcal{Y})
-
D^1(\mathcal{X})\,y_1, \\
{\bf Y}_2
&
:=
D^2({\bf Y}_1)
-
D^1(\mathcal{X})\,y_2, \\
\cdots
&
\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots
\\
{\bf Y}_\kappa
&
:=
D^\kappa({\bf Y}_{\kappa-1})
-
D^1(\mathcal{X})\,y_\kappa, \\
\endaligned\right.
\end{equation}
where, for $1 \leq \lambda \leq \kappa$:
\def5.22}\begin{equation{2.5}\begin{equation}
D^\lambda
:=
\frac{\partial}{\partial x}
+
y_1\,\frac{\partial}{\partial y}
+
y_2\,\frac{\partial}{\partial y_1}
+
\cdots
+
y_\lambda\,\frac{\partial}{\partial y_{\lambda-1}}.
\end{equation}
By direct elementary computations, for $\kappa = 1$ and for $\kappa =
2$, we obtain the following two very classical formulas :
\def5.22}\begin{equation{2.6}\begin{equation}
\left\{
\aligned
{\bf Y}_1
&
=
\mathcal{Y}_x
+
\left[
\mathcal{Y}_y
-
\mathcal{X}_x
\right]
y_1
+
\left[
-\mathcal{Y}_y
\right]
(y_1)^2,
\\
{\bf Y}_2
&
=
\mathcal{Y}_{x^2}
+
\left[
2\,\mathcal{Y}_{xy}
-
\mathcal{X}_{x^2}
\right]
\,y_1
+
\left[
\mathcal{Y}_{y^2}
-
2\,\mathcal{X}_{xy}
\right](y_1)^2
+
\left[
-
\mathcal{X}_{y^2}
\right]
(y_1)^3
+ \\
& \
\ \ \ \ \
+
\left[
\mathcal{Y}_y
-
2\,\mathcal{X}_x
\right]\,y_2
+
\left[
-
3\,\mathcal{X}_y
\right]\,
y_1\,y_2.
\endaligned\right.
\end{equation}
Our main objective is to {\it devise the general combinatorics}.
Thus, to
attain this aim, we have to achieve patiently formal computations of
the next coefficients ${\bf Y}_3$, ${\bf Y}_4$ and ${\bf Y}_5$. We
systematically use parentheses $\left[ \cdot \right]$ to single out
every coefficient of the polynomials ${\bf Y}_3$, ${\bf Y}_4$
and ${\bf
Y}_5$ in the pure jet variables $y_1, y_2, y_3, y_4$ and
$y_5$, putting every sign inside these parentheses. We always put
the monomials in the pure jet variables $y_1, y_2, y_3, y_4$ and
$y_5$ after the parentheses. For completeness, let us provide the
intermediate computation of the third coefficient ${\bf Y}_3$.
In detail:
$$
\small
\aligned
{\bf Y}_3
&
=
D^3
\left(
{\bf Y}_2
\right)
-
D^1
\left(
\mathcal{ X}
\right)
y_3
\\
&
=
\left(
\frac{\partial}{\partial x}
+
y_1\,\frac{\partial}{\partial y}
+
y_2\,\frac{\partial}{\partial y_1}
+
y_3\,\frac{\partial}{\partial y_2}
\right)
\left(
\mathcal{Y}_{x^2}
+
\left[
2\,\mathcal{Y}_{xy}
-
\mathcal{X}_{x^2}
\right]
y_1
+
\right.
\\
& \
\ \ \ \ \
\left.
+
\left[
\mathcal{ Y}_{y^2}
-
2\,\mathcal{X}_{xy}
\right]
(y_1)^2
+
\left[
-
\mathcal{X}_{y^2}
\right]
(y_1)^3
+
\left[
\mathcal{ Y}_y
-
2\,\mathcal{X}_x
\right]
y_2
+
\left[
-
3\,\mathcal{X}_y
\right]
y_1\,y_2
\right)
\\
\endaligned
$$
\def5.22}\begin{equation{2.7}\begin{equation}
\aligned
&
=
\underline{
\mathcal{Y}_{x^3}
}_{ \fbox{\tiny 1}}
+
\underline{
\left[
2\,\mathcal{Y}_{x^2y}
-
\mathcal{X}_{x^3}
\right]
y_1
}_{ \fbox{\tiny 2}}
+
\underline{
\left[
\mathcal{Y}_{xy^2}
-
2\,\mathcal{X}_{x^2y}
\right]
(y_1)^2
}_{ \fbox{\tiny 3}}
+
\underline{
\left[
-
\mathcal{X}_{xy^2}
\right]
(y_1)^3
}_{ \fbox{\tiny 4}}
+ \\
& \
\ \ \ \ \
+
\underline{
\left[
\mathcal{Y}_{xy}
-
2\,\mathcal{X}_{x^2}
\right]
y_2
}_{ \fbox{\tiny 6}}
+
\underline{
\left[
-
3\,\mathcal{X}_{xy}
\right]
y_1y_2
}_{ \fbox{\tiny 7}}
+
\underline{
\left[
\mathcal{Y}_{x^2y}
\right]
y_1
}_{ \fbox{\tiny 2}}
+ \\
& \
\ \ \ \ \
+
\underline{
\left[
2\,\mathcal{Y}_{xy^2}
-
\mathcal{X}_{x^2y}
\right]
(y_1)^2
}_{ \fbox{\tiny 3}}
+
\underline{
\left[
\mathcal{Y}_{y^3}
-
2\,\mathcal{X}_{xy^2}
\right]
(y_1)^3
}_{ \fbox{\tiny 4}}
+
\underline{
\left[
-
\mathcal{X}_{y^3}
\right]
(y_1)^4
}_{ \fbox{\tiny 5}}
+
\endaligned
\end{equation}
$$
\aligned
& \
\ \ \ \ \
+
\underline{
\left[
\mathcal{Y}_{y^2}
-
2\,\mathcal{X}_{xy}
\right]
y_1y_2
}_{ \fbox{\tiny 7}}
+
\underline{
\left[
-
3\,\mathcal{X}_{y^2}
\right]
(y_1)^2y_2
}_{ \fbox{\tiny 8}}
+
\underline{
\left[
2\,\mathcal{Y}_{xy}
-
\mathcal{X}_{x^2}
\right]
y_2
}_{ \fbox{\tiny 6}}
+ \\
& \
\ \ \ \ \
+
\underline{
\left[
\mathcal{Y}_{y^2}
-
2\,\mathcal{X}_{xy}
\right]
2\,y_1y_2
}_{ \fbox{\tiny 7}}
+
\underline{
\left[
-
\mathcal{X}_{y^2}
\right]
3(y_1)^2y_2
}_{ \fbox{\tiny 8}}
+
\underline{
\left[
-
3\,\mathcal{X}_y
\right]
(y_2)^2
}_{ \fbox{\tiny 9}}
+ \\
& \
\ \ \ \ \
+
\underline{
\left[
\mathcal{Y}_y
-
2\,\mathcal{X}_x
\right]
y_3
}_{ \fbox{\tiny 10}}
+
\underline{
\left[
-
3\,\mathcal{X}_y
\right]
y_1y_3
}_{ \fbox{\tiny 11}}
- \\
& \
\ \ \ \ \
-
\underline{
\left[
\mathcal{X}_x
\right]
y_3
}_{ \fbox{\tiny 10}}
-
\underline{
\left[
\mathcal{X}_y
\right]
y_1y_3
}_{ \fbox{\tiny 11}} \ .
\endaligned
$$
We have underlined all the terms with
a number appended. Each number
refers to the order of appearance
of the terms in the final simplified
expression of ${\bf Y}_3$, also written in~\cite{ bk1989}
with different notations:
\def5.22}\begin{equation{2.8}\begin{equation}
\left\{
\aligned
{\bf Y}_3
&
=
\mathcal{Y}_{x^3}
+
\left[
3\,\mathcal{Y}_{x^2y}
-
\mathcal{X}_{x^3}
\right]
y_1
+
\left[
3\,\mathcal{Y}_{xy^2}
-
3\,\mathcal{X}_{x^2y}
\right]
(y_1)^2
+ \\
& \
\ \ \ \ \
+
\left[
\mathcal{Y}_{y^3}
-
3\,\mathcal{X}_{xy^2}
\right]
(y_1)^3
+
\left[
-
\mathcal{X}_{y^3}
\right]
(y_1)^4
+
\left[
3\,\mathcal{Y}_{xy}
-
3\,\mathcal{X}_{x^2}
\right]
y_2
+ \\
& \
\ \ \ \ \
+
\left[
3\,\mathcal{Y}_{y^2}
-
9\,\mathcal{X}_{xy}
\right]
y_1y_2
+
\left[
-
6\,\mathcal{X}_{y^2}
\right]
(y_1)^2y_2
+
\left[
-
3\,\mathcal{X}_y
\right]
(y_2)^2
+ \\
& \
\ \ \ \ \
+
\left[
\mathcal{Y}_y
-
3\,\mathcal{X}_x
\right]
y_3
+
\left[
-
4\,\mathcal{X}_y
\right]
y_1y_3.
\endaligned\right.
\end{equation}
After similar manual computations, the intermediate details of which
we will not copy in this Latex file, we get the desired expressions of
${\bf Y}_4$ and of ${\bf Y}_5$.
Firstly:
\def5.22}\begin{equation{2.9}\begin{equation}
\small
\left\{
\aligned
{\bf Y}_4
&
=
\mathcal{Y}_{x^4}
+
\left[
4\,\mathcal{Y}_{x^3y}
-
\mathcal{X}_{x^4}
\right]
y_1
+
\left[
6\,\mathcal{Y}_{x^2y^2}
-
4\,\mathcal{X}_{x^3y}
\right]
(y_1)^2
+ \\
& \
\ \ \ \ \
+
\left[
4\,\mathcal{Y}_{xy^3}
-
6\,\mathcal{X}_{x^2y^2}
\right]
(y_1)^3
+
\left[
\mathcal{Y}_{y^4}
-
4\,\mathcal{X}_{xy^3}
\right]
(y_1)^4
+
\left[
-
\mathcal{X}_{y^4}
\right]
(y_1)^5
+ \\
& \
\ \ \ \ \
+
\left[
6\,\mathcal{Y}_{x^2y}
-
4\,\mathcal{X}_{x^3}
\right]
y_2
+
\left[
12\,\mathcal{Y}_{xy^2}
-
18\,\mathcal{X}_{x^2y}
\right]
y_1y_2
+ \\
& \
\ \ \ \ \
+
\left[
6\,\mathcal{Y}_{y^3}
-
24\,\mathcal{X}_{xy^2}
\right]
(y_1)^2y_2
+
\left[
-
10\,\mathcal{X}_{y^3}
\right]
(y_1)^3y_2
+ \\
& \
\ \ \ \ \
+
\left[
3\,\mathcal{Y}_{y^2}
-
12\,\mathcal{X}_{xy}
\right]
(y_2)^2
+
\left[
-
15\,\mathcal{X}_{y^2}
\right]
y_1(y_2)^2
+ \\
& \
\ \ \ \ \
+
\left[
4\,\mathcal{Y}_{xy}
-
6\,\mathcal{X}_{x^2}
\right]
y_3
+
\left[
4\,\mathcal{Y}_{y^2}
-
16\,\mathcal{X}_{xy}
\right]
y_1y_3
+
\left[
-
10\,\mathcal{X}_{y^2}
\right]
(y_1)^2y_3
+ \\
& \
\ \ \ \ \
+
\left[
-
10\,\mathcal{X}_y
\right]
y_2y_3
+
\left[
\mathcal{Y}_y
-
4\,\mathcal{X}_x
\right]
y_4
+
\left[
-
5\,\mathcal{X}_y
\right]
y_1y_4.
\endaligned\right.
\end{equation}
Secondly:
\def5.22}\begin{equation{2.10}\begin{equation}
\small
\left\{
\aligned
{\bf Y}_5
&
=
\mathcal{Y}_{x^5}
+
\left[
5\,\mathcal{Y}_{x^4y}
-
\mathcal{X}_{x^5}
\right]
y_1
+
\left[
10\,\mathcal{Y}_{x^3y^2}
-
5\,\mathcal{X}_{x^4y}
\right]
(y_1)^2
+ \\
& \
\ \ \ \ \
+
\left[
10\,\mathcal{Y}_{x^2y^3}
-
10\,\mathcal{X}_{x^3y^2}
\right]
(y_1)^3
+
\left[
5\,\mathcal{Y}_{xy^4}
-
10\,\mathcal{X}_{x^2y^3}
\right]
(y_1)^4
+ \\
& \
\ \ \ \ \
+
\left[
\mathcal{Y}_{y^5}
-
5\,\mathcal{X}_{xy^4}
\right]
(y_1)^5
+
\left[
-
\mathcal{X}_{y^5}
\right]
(y_1)^6
+
\left[
10\,\mathcal{Y}_{x^3y}
-
5\,\mathcal{X}_{x^4}
\right]
y_2
+ \\
& \
\ \ \ \ \
+
\left[
30\,\mathcal{Y}_{x^2y^2}
-
30\,\mathcal{X}_{x^3y}
\right]
y_1y_2
+
\left[
30\,\mathcal{Y}_{xy^3}
-
60\,\mathcal{X}_{x^2y^2}
\right]
(y_1)^2y_2
+ \\
& \
\ \ \ \ \
+
\left[
10\,\mathcal{Y}_{y^4}
-
50\,\mathcal{X}_{xy^3}
\right]
(y_1)^3y_2
+
\left[
-
15\,\mathcal{X}_{y^4}
\right]
(y_1)^4y_2
+ \\
& \
\ \ \ \ \
+
\left[
15\,\mathcal{Y}_{xy^2}
-
30\,\mathcal{X}_{x^2y}
\right]
(y_2)^2
+
\left[
15\,\mathcal{Y}_{y^3}
-
75\,\mathcal{X}_{xy^2}
\right]
y_1(y_2)^2
+ \\
& \
\ \ \ \ \
+
\left[
-
45\,\mathcal{X}_{y^3}
\right]
(y_1)^2(y_2)^2
+
\left[
-
15\,\mathcal{X}_{y^2}
\right]
(y_2)^3
+ \\
& \
\ \ \ \ \
+
\left[
10\,\mathcal{Y}_{x^2y}
-
10\,\mathcal{X}_{x^3}
\right]
y_3
+
\left[
20\,\mathcal{Y}_{xy^2}
-
40\,\mathcal{X}_{x^2y}
\right]
y_1y_3
+ \\
& \
\ \ \ \ \
+
\left[
10\,\mathcal{Y}_{y^3}
-
50\,\mathcal{X}_{xy^2}
\right]
(y_1)^2y_3
+
\left[
-
20\,\mathcal{X}_{y^3}
\right]
(y_1)^3y_3
+ \\
& \
\ \ \ \ \
+
\left[
10\,\mathcal{Y}_{y^2}
-
50\,\mathcal{X}_{xy}
\right]
y_2y_3
+
\left[
-
60\,\mathcal{X}_{y^2}
\right]
y_1y_2y_3
+
\left[
-
10\,\mathcal{X}_y
\right]
(y_3)^2
+ \\
& \
\ \ \ \ \
+
\left[
5\,\mathcal{Y}_{xy}
-
10\,\mathcal{X}_{x^2}
\right]
y_4
+
\left[
5\,\mathcal{Y}_{y^2}
-
25\,\mathcal{X}_{xy}
\right]
y_1y_4
+
\left[
-
15\,\mathcal{X}_{y^2}
\right]
(y_1)^2y_4
+ \\
& \
\ \ \ \ \
+
\left[
-
15\,\mathcal{X}_y
\right]
y_2y_4
+
\left[
\mathcal{Y}_y
-
5\,\mathcal{X}_y
\right]
y_5
+
\left[
-
6\,\mathcal{X}_y
\right]
y_1y_5.
\endaligned\right.
\end{equation}
\subsection*{2.11.~Formal inspection, formal intuition
and formal induction} Now, we have to comment these formulas. We have
written in length the five polynomials ${\bf Y}_1$, ${\bf Y}_2$, ${\bf
Y}_3$, ${\bf Y}_4$ and ${\bf Y}_5$ in the pure jet variables $y_1,
y_2, y_3, y_4$ and $y_5$. Except the first ``constant'' term
$\mathcal{ Y}_{x^\kappa}$, all the monomials in the expression of
${\bf Y}_\kappa$ are of the general form
\def5.22}\begin{equation{2.12}\begin{equation}
\left(
y_{\lambda_1}
\right)^{\mu_1}
\left(
y_{\lambda_2}
\right)^{\mu_2}
\cdots
\left(
y_{\lambda_d}
\right)^{\mu_d},
\end{equation}
for some positive integer $d\geq 1$, for some collection of
strictly increasing jets indices:
\def5.22}\begin{equation{2.13}\begin{equation}
1 \leq \lambda_1 < \lambda_2 < \cdots < \lambda_d \leq \kappa,
\end{equation}
and for some positive integers $\mu_1, \dots, \mu_d \geq 1$. This and
the next combinatorial facts may be confirmed by reading the formulas
giving ${\bf Y}_1$, ${\bf Y}_2$, ${\bf Y}_3$, ${\bf Y}_4$ and ${\bf
Y}_5$. It follows that the integer $d$ satisfies the inequality
$d\leq \kappa+1$. To include the first ``constant'' term $\mathcal{
Y}_{x^\kappa}$, we shall make the convention that putting $d=0$ in the
monomial~\thetag{ 2.12} yields the constant term $1$.
Furthermore, by inspecting the formulas giving ${\bf Y}_1$, ${\bf
Y}_2$, ${\bf Y}_3$, ${\bf Y}_4$ and ${\bf Y}_5$, we see that the
following inequality should be satisfied:
\def5.22}\begin{equation{2.14}\begin{equation}
\mu_1\lambda_1
+
\mu_2\lambda_2
+
\cdots
+
\mu_d\lambda_d
\leq
\kappa+1.
\end{equation}
For instance, in the expression of ${\bf Y}_4$, the two monomials
$(y_1)^3 y_2$ and $y_1 (y_2)^2$ do appear, but the two monomials
$(y_1)^4 y_2$ and $(y_1)^2 (y_2)^2$ cannot appear. All coefficients
of the pure jet monomials are of the general form:
\def5.22}\begin{equation{2.15}\begin{equation}
\left[
A\,
\mathcal{Y}_{x^\alpha y^{\beta+1}}
-
B\,
\mathcal{X}_{x^{\alpha+1}y^\beta}
\right],
\end{equation}
for some nonnegative integers $A, B, \alpha, \beta \in \N$. Sometimes
$A$ is zero, but $B$ is zero only for the (constant, with respect to
pure jet variables) term $\mathcal{ Y}_{x^\kappa}$. Importantly,
$\mathcal{ X}$ is differentiated once more with respect to $x$ and
$\mathcal{ Y}$ is differentiated once more with respect to $y$. Again,
this may be confirmed by reading all the terms in the formulas for
${\bf Y}_1$, ${\bf Y}_2$, ${\bf Y}_3$, ${\bf Y}_4$ and ${\bf Y}_5$.
In addition, we claim that there is a link between the couple
$(\alpha, \beta)$ and the collection $\{ \mu_1, \lambda_1, \dots,
\mu_d, \lambda_d \}$. To discover it, let us write some of the
monomials appearing in the expressions of ${\bf Y}_4$ (first column)
and of ${\bf Y}_5$ (second column), for instance:
\def5.22}\begin{equation{2.16}\begin{equation}
\left\{
\aligned
&
\left[6\,
\mathcal{Y}_{x^2y^2}
-
4\,\mathcal{X}_{x^3y}
\right]
(y_1)^2,
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
&
\left[
5\,\mathcal{Y}_{xy^4}
-
10\,\mathcal{X}_{x^2y^3}
\right]
(y_1)^4,
&
\\
&
\left[
12\,\mathcal{Y}_{xy^2}
-
18\,\mathcal{X}_{x^2y}
\right]
y_1y_2,
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
&
\left[
30\,\mathcal{Y}_{xy^3}
-
60\,\mathcal{X}_{x^2y^2}
\right]
(y_1)^2y_2,
&
\\
&
\left[
-
10\,\mathcal{X}_{y^3}
\right]
(y_1)^3y_2,
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
&
\left[
-
15\,\mathcal{X}_{y^4}
\right]
(y_1)^4y_2,
&
\\
&
\left[
4\,\mathcal{Y}_{y^2}
-
16\,\mathcal{X}_{xy}
\right]
y_1y_3,
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
&
\left[
10\,\mathcal{Y}_{y^2}
-
50\,\mathcal{X}_{xy}
\right]
y_2y_3,
&
\\
&
\left[
-
10\,\mathcal{X}_{y^2}
\right]
(y_1)^2 y_3,
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
&
\left[
-
60\,\mathcal{X}_{y^2}
\right]
y_1y_2y_3.
& \\
\endaligned\right.
\end{equation}
After some reflection, we discover the hidden intuitive rule: the
partial derivatives of $\mathcal{ Y}$ and of $\mathcal{ X}$ associated
with the monomial $(y_{\lambda_1})^{\mu_1} \cdots
(y_{\lambda_d})^{\mu_d}$ are, respectively:
\def5.22}\begin{equation{2.17}\begin{equation}
\left\{
\aligned
&
\mathcal{ Y}_{
x^{\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d}
\,
y^{\mu_1+\cdots+\mu_d}},
\\
&
\mathcal{ X}_{
x^{\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d+1}
\,
y^{\mu_1+\cdots+\mu_d-1}}.
\\
\endaligned\right.
\end{equation}
This may be checked on each of the $10$ examples~\thetag{ 2.16}
above.
Now that we have explored and discovered the combinatorics of the pure
jet monomials, of the partial derivatives and of the complete sum
giving ${\bf Y}_\kappa$, we may express that it is of the following
general form:
\def5.22}\begin{equation{2.18}\begin{equation}
\left\{
\aligned
{\bf Y}_\kappa
&
=
\mathcal{ Y}_{x^\kappa}
+
\sum_{d=1}^{\kappa+1}
\ \
\sum_{1\leq\lambda_1<\cdots<\lambda_d\leq\kappa}
\ \
\sum_{\mu_1\geq 1,\dots,\mu_d\geq 1}
\
\sum_{
\mu_1\lambda_1
+
\cdots
+
\mu_d\lambda_d\leq \kappa+1}
\\
& \
\ \ \ \ \
\left[
A_\kappa^{
(\mu_1, \lambda_1), \dots, (\mu_d, \lambda_d) }
\cdot
\mathcal{Y}_{
x^{\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d}
\,
y^{\mu_1+\cdots+\mu_d}
}
-
\right. \\
& \
\ \ \ \ \ \ \ \ \ \
\left.
-
B_\kappa^{
(\mu_1, \lambda_1), \dots, (\mu_d, \lambda_d) }
\cdot
\mathcal{X}_{
x^{\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d+1}
\,
y^{\mu_1+\cdots+\mu_d-1}
}
\right]
\cdot
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \
\cdot
(y_{\lambda_1})^{\mu_1}
\cdots
(y_{\lambda_d})^{\mu_d}.
\endaligned\right.
\end{equation}
Here, we separate the first term $\mathcal{ Y}_{x^\kappa}$ from the
general sum; it is the constant term in ${\bf Y}_\kappa$, which is a
polynomial with respect to the jet variables $y_\lambda$. In this
general formula, the only remaining unknowns are the nonnegative
integer coefficients $A_\kappa^{ (\mu_1, \lambda_1), \dots, (\mu_d,
\lambda_d) } \in \N$ and $B_\kappa^{ (\mu_1, \lambda_1), \dots,
(\mu_d, \lambda_d) } \in \N$. In Section~3 below, we shall explain how
we have discovered their exact value.
At present, even if we are unable to devise their explicit
expression, we may observe that the value of the special integer
coefficients $A^{(\mu_1, 1)}_{ \mu_1}$ and $B^{( \mu_1, 1)}_{ \mu_1}$
which are attached to the monomials ${\rm ct.}$, $y_1$, $(y_1)^2$,
$(y_1)^3$, $(y_1)^4$ and $(y_1)^5$ are simple. Indeed, by
inspecting the first terms in the expressions of ${\bf Y}_1$, ${\bf
Y}_2$, ${\bf Y}_3$, ${\bf Y}_4$ and ${\bf Y}_5$, we of course
recognize the binomial coefficients. In general:
\def3.50}\begin{lemma{2.19}\begin{lemma} For $\kappa \geq 1$,
\def5.22}\begin{equation{2.20}\begin{equation}
\left\{
\aligned
{\bf Y}_\kappa
&
=
\mathcal{Y}_{x^\kappa}
+
\sum_{\lambda=1}^\kappa
\left[
\binom{\kappa}{\lambda}
\,\mathcal{Y}_{x^{\kappa-\lambda}y^\lambda}
-
\binom{\kappa}{\lambda-1}
\,\mathcal{X}_{x^{\kappa-\lambda+1}y^{\lambda-1}}
\right]
(y_1)^\lambda
+ \\
& \
\ \ \ \ \
+
\left[
-
\mathcal{X}_{y^\kappa}
\right]
(y_1)^\kappa
+
{\sf remainder},
\endaligned\right.
\end{equation}
where the term {\sf remainder} collects all remaining monomials in
the pure jet variables.
\end{lemma}
In addition, let us remind what we have observed and used in a
previous co-signed work.
\def3.50}\begin{lemma{2.21}\begin{lemma}
\text{\rm (\cite{ gm2003}, p.~536)} For $\kappa \geq 4$, nine among
the monomials of ${\bf Y}_\kappa$ are of the following general
form{\rm :}
\def5.22}\begin{equation{2.22}\begin{equation}
\left\{
\aligned
{\bf Y}_\kappa
&
=
\mathcal{Y}_{x^\kappa}
+
\left[
C_\kappa^1 \, \mathcal{Y}_{x^{\kappa-1}y}
-
\mathcal{X}_{x^\kappa}
\right]
y_1
+
\left[
C_\kappa^2\,
\mathcal{Y}_{x^{\kappa-2}y}
-
C_\kappa^1 \,\mathcal{X}_{x^{\kappa-1}}
\right]
y_2
+ \\
& \
\ \ \ \ \
+
\left[
C_\kappa^2\,
\mathcal{Y}_{x^2y}
-
C_\kappa^3\,
\mathcal{X}_{x^3}
\right]
y_{\kappa-2}
+
\left[
C_\kappa^1 \,\mathcal{Y}_{xy}
-
C_\kappa^2\,
\mathcal{X}_{x^2}
\right]
y_{\kappa-1}
+ \\
& \
\ \ \ \ \
+
\left[
C_\kappa^1 \, \mathcal{Y}_{y^2}
-
\kappa^2 \, \mathcal{ X}_{xy}
\right]
y_1 y_{\kappa-1}
+
\left[
-C_\kappa^2 \, \mathcal{ X}_y
\right]
y_2y_{\kappa-1}
+ \\
& \
\ \ \ \ \
+
\left[
\mathcal{Y}_y
-
C_\kappa^1\,\mathcal{X}_x
\right]
+
\left[
-C_{\kappa+1}^1 \, \mathcal{ X}_y
\right]
y_1 y_\kappa
+
{\sf remainder},
\endaligned\right.
\end{equation}
where the term {\sf remainder} denotes all the remaining monomials,
and where $C_\kappa^\lambda := \frac{ \kappa!}{(\kappa - \lambda)! \
\lambda !}$ is a notation for the binomial coefficient which occupies
less space in Latex ``equation mode'' than the classical notation
\def5.22}\begin{equation{2.23}\begin{equation}
\binom{\kappa}{\lambda}.
\end{equation}
\end{lemma}
Now, we state directly the final theorem, without further inductive or
intuitive information.
\def5.12}\begin{theorem{2.24}\begin{theorem}
For $\kappa \geq 1$, we have{\rm :}
\def5.22}\begin{equation{2.25}\begin{equation}
\boxed{
\aligned
{\bf Y}_\kappa
&
=
\mathcal{ Y}_{x^\kappa}
+
\sum_{d=1}^{\kappa+1}
\ \
\sum_{1\leq\lambda_1<\cdots<\lambda_d\leq\kappa}
\ \
\sum_{\mu_1\geq 1,\dots,\mu_d\geq 1}
\
\sum_{
\mu_1\lambda_1
+
\cdots
+
\mu_d\lambda_d\leq \kappa+1}
\\
& \
\ \
\left[
\frac{\kappa\cdots(\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d+1)}
{(\lambda_1!)^{\mu_1}\,\mu_1!
\cdots
(\lambda_d!)^{\mu_d}\,\mu_d!
}
\cdot
\mathcal{Y}_{
x^{\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d}
\,
y^{\mu_1+\cdots+\mu_d}
}
-
\right. \\
& \
\ \ \ \ \ \ \ \ \ \
\left.
-
\frac{\kappa\cdots(\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d+2)
(\mu_1\lambda_1+\cdots+\mu_d\lambda_d)}
{(\lambda_1!)^{\mu_1}\,\mu_1!
\cdots
(\lambda_d!)^{\mu_d}\,\mu_d!
}
\cdot
\right.
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\cdot
\mathcal{X}_{
x^{\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d+1}
\,
y^{\mu_1+\cdots+\mu_d-1}
}
\Big]
(y_{\lambda_1})^{\mu_1}
\cdots
(y_{\lambda_d})^{\mu_d}.
\endaligned
}
\end{equation}
\end{theorem}
Once the correct theorem is formulated, its proof follows by
accessible induction arguments which will not be developed here. It is
better to continue through and to examine thorougly the case of
several variables, since it will help us considerably to explain how
we discovered the exact values of the integer coefficients $A_\kappa^{
(\mu_1, \lambda_1), \dots, (\mu_d, \lambda_d) }$ and $B_\kappa^{
(\mu_1, \lambda_1), \dots, (\mu_d, \lambda_d) }$.
\subsection*{2.26.~Verification and application}
Before proceeding further, let us rapidly verify that the above
general formula~\thetag{ 2.25} is correct by inspecting two instances
extracted from ${\bf Y}_5$.
Firstly, the coefficient of $(y_1)^3 y_3$ in ${\bf Y}_5$ is obtained
by putting $\kappa = 5$, $d = 2$, $\lambda_1 = 1$, $\mu_1 = 3$,
$\lambda_2 = 3$ and $\mu_2 = 1$ in the general formula~\thetag{ 2.25},
which yields:
\def5.22}\begin{equation{2.27}\begin{equation}
\left[
0
-
\frac{5\cdot 4\cdot 3\cdot 2\cdot 1\cdot 6}{
(1!)^3\ 3!\ (3!)^1\ 1!}
\,\mathcal{X}_{y^3}
\right] =
\left[
-
20\,\mathcal{X}_{y^3}
\right].
\end{equation}
This value is the same as in the original formula~\thetag{ 2.10}:
confirmation.
Secondly, the coefficient of $y_1 (y_2)^2$ in ${\bf Y}_5$ is obtained
by $\kappa = 5$, $d = 2$, $\lambda_1 = 1$, $\mu_1 = 1$, $\lambda_2 =
2$ and $\mu_2 = 2$ in the general formula~\thetag{ 2.25}, which yields:
\def5.22}\begin{equation{2.28}\begin{equation}
\left[
\frac{5\cdot 4\cdot 3\cdot 2\cdot 1}{
(1!)^1\ 1!\ (2!)^2\ 2!}
\,\mathcal{Y}_{y^3}
-
\frac{5\cdot 4\cdot 3\cdot 2\cdot 5}{
(1!)^1\ 1!\ (2!)^2\ 2!}
\,\mathcal{X}_{xy^2}
\right]
=
\left[
15\,\mathcal{Y}_{y^3}
-
75\,\mathcal{X}_{xy^2}
\right].
\end{equation}
This value is the same as in the original formula~\thetag{ 2.10};
again: confirmation.
Finally, applying our general formula~\thetag{ 2.25}, we deduce the value of
${\bf Y}_6$ {\it without having to use ${\bf Y}_5$ and the induction
formulas~\thetag{ 2.4}}, which shortens substantially the
computations. For the pleasure, we obtain:
\def5.22}\begin{equation{2.29}\begin{equation}
\small
\left\{
\aligned
{\bf Y}_6
&
=
\mathcal{Y}_{x^6}
+
\left[
6\,\mathcal{Y}_{x^5y}
-
\mathcal{X}_{x^6}
\right]
y_1
+
\left[
15\,\mathcal{Y}_{x^4y^2}
-
6\,\mathcal{X}_{x^5y}
\right]
(y_1)^2
+ \\
& \
\ \ \ \ \
+
\left[
20\,\mathcal{Y}_{x^3y^3}
-
15\,\mathcal{X}_{x^4y^2}
\right]
(y_1)^3
+
\left[
15\,\mathcal{Y}_{x^2y^4}
-
20\,\mathcal{X}_{x^3y^3}
\right]
(y_1)^4
+ \\
& \
\ \ \ \ \
+
\left[
6\,\mathcal{Y}_{xy^5}
-
15\,\mathcal{X}_{x^2y^4}
\right]
(y_1)^5
+
\left[
\mathcal{Y}_{y^6}
-
6\,\mathcal{X}_{xy^5}
\right]
(y_1)^6
+
\left[
-
\mathcal{X}_{y^6}
\right]
(y_1)^7
+ \\
& \
\ \ \ \ \
+
\left[
15\,\mathcal{Y}_{x^4y}
-
6\,\mathcal{X}_{x^5}
\right]
y_2
+
\left[
60\,\mathcal{Y}_{x^3y^2}
-
45\,\mathcal{X}_{x^4y}
\right]
y_1y_2
+ \\
& \
\ \ \ \ \
+
\left[
90\,\mathcal{Y}_{x^2y^3}
-
120\,\mathcal{X}_{x^3y^2}
\right]
(y_1)^2y_2
+
\left[
60\,\mathcal{Y}_{xy^4}
-
150\,\mathcal{X}_{x^2y^3}
\right]
(y_1)^3y_2
+ \\
& \
\ \ \ \ \
+
\left[
15\,\mathcal{Y}_{y^5}
-
90\,\mathcal{X}_{xy^4}
\right]
(y_1)^4y_2
+
\left[
-
21\,\mathcal{X}_{y^5}
\right]
(y_1)^5y_2
+ \\
& \
\ \ \ \ \
+
\left[
45\,\mathcal{Y}_{x^2y^2}
-
60\,\mathcal{X}_{x^3y}
\right]
(y_2)^2
+
\left[
90\,\mathcal{Y}_{xy^3}
-
225\,\mathcal{X}_{x^2y^2}
\right]
y_1(y_2)^2
+ \\
& \
\ \ \ \ \
+
\left[
45\,\mathcal{Y}_{y^4}
-
270\,\mathcal{X}_{xy^3}
\right]
(y_1)^2(y_2)^2
+
\left[
-
210\,\mathcal{X}_{y^4}
\right]
(y_1)^3(y_2)^2
+ \\
& \
\ \ \ \ \
+
\left[
15\,\mathcal{Y}_{y^3}
-
90\,\mathcal{X}_{xy^2}
\right]
(y_2)^3
+
\left[
-
105\,\mathcal{X}_{y^3}
\right]
y_1(y_2)^3
+ \\
& \
\ \ \ \ \
+
\left[
20\,\mathcal{Y}_{x^3y}
-
15\,\mathcal{X}_{x^4}
\right]
y_3
+
\left[
60\,\mathcal{Y}_{x^2y^2}
-
80\,\mathcal{X}_{x^3y}
\right]
y_1y_3
+ \\
& \
\ \ \ \ \
+
\left[
60\,\mathcal{Y}_{xy^3}
-
150\,\mathcal{X}_{x^2y^2}
\right]
(y_1)^2y_3
+
\left[
20\,\mathcal{Y}_{y^4}
-
120\,\mathcal{X}_{xy^3}
\right]
(y_1)^3y_3
+ \\
& \
\ \ \ \ \
+
\left[
-
35\,\mathcal{X}_{y^4}
\right]
(y_1)^4y_3
+
\left[
60\,\mathcal{Y}_{xy^2}
-
150\,\mathcal{X}_{x^2y}
\right]
y_2y_3
+ \\
& \
\ \ \ \ \
+
\left[
60\,\mathcal{Y}_{y^3}
-
360\,\mathcal{X}_{xy^2}
\right]
y_1y_2y_3
+
\left[
-
210\,\mathcal{X}_{y^3}
\right]
(y_1)^2y_2y_3
+ \\
& \
\ \ \ \ \
+
\left[
-
105\,\mathcal{X}_{y^2}
\right]
(y_2)^2y_3
+
\left[
10\,\mathcal{Y}_{y^2}
-
60\,\mathcal{X}_{xy}
\right]
(y_3)^2
+ \\
& \
\ \ \ \ \
+
\left[
-
70\,\mathcal{X}_{y^2}
\right]
y_1(y_3)^2
+
\left[
15\,\mathcal{Y}_{x^2y}
-
20\,\mathcal{X}_{x^3}
\right]
y_4
+ \\
& \
\ \ \ \ \
+
\left[
30\,\mathcal{Y}_{xy^2}
-
75\,\mathcal{X}_{x^2y}
\right]
y_1y_4
+
\left[
15\,\mathcal{Y}_{y^3}
-
90\,\mathcal{X}_{xy^2}
\right]
(y_1)^2y_4
+ \\
& \
\ \ \ \ \
+
\left[
-
35\,\mathcal{X}_{y^3}
\right]
(y_1)^3y_4
+
\left[
15\,\mathcal{Y}_{y^2}
-
90\,\mathcal{X}_{xy}
\right]
y_2y_4
+ \\
& \
\ \ \ \ \
+
\left[
-
105\,\mathcal{X}_{y^2}
\right]
y_1y_2y_4
+
\left[
-
35\,\mathcal{X}_y
\right]
y_3y_4
+
\left[
6\,\mathcal{Y}_{xy}
-
15\,\mathcal{X}_{x^2}
\right]
y_5
+ \\
& \
\ \ \ \ \
+
\left[
6\,\mathcal{Y}_{y^2}
-
36\,\mathcal{X}_{xy}
\right]
y_1y_5+
\left[
-
21\,\mathcal{X}_{y^2}
\right]
(y_1)^2y_5
+
\left[
-
21\,\mathcal{X}_y
\right]
y_2y_5
+ \\
& \
\ \ \ \ \
+
\left[
\mathcal{Y}_y
-
6\,\mathcal{X}_y
\right]
y_6
+
\left[
-
7\,\mathcal{X}_y
\right]
y_1y_6.
\endaligned\right.
\end{equation}
\subsection*{ 2.30.~Deduction of the classical Fa\`a
di Bruno formula} Let $x,y \in \K$ and let $g = g(x)$, $f = f ( y)$ be
two $\mathcal{ C }^\infty$-smooth functions $\K \to \K$. Consider the
composition $h := f \circ g$, namely $h(x) = f (g (x))$. For $\lambda
\in \N$ with $\lambda \geq 1$, simply denote by $g_\lambda$ the
$\lambda$-th derivative $\frac{ d^\lambda g}{d x^\lambda}$ and
similarly for $h_\lambda$. Also, abbreviate $f_\lambda := \frac{
d^\lambda f}{ d y^\lambda}$.
By the classical formula for the derivative of a composite function,
we have $h_1 = f_1 \, g_1$. Further computations provide the following
list of subsequent derivatives of $h$:
\def5.22}\begin{equation{2.31}\begin{equation}
\left\{
\aligned
h_1
&
=
f_1 \, g_1,
\\
h_2
&
=
f_2 \, (g_1)^2
+
f_1 \, g_2,
\\
h_3
&
=
f_3\,(g_1)^3+3\,f_2\,g_1\,g_2
+
f_1\,g_3,
\\
h_4
&
=
f_4\,(g_1)^4
+
6\,f_3\,(g_1)^2\,g_2
+
3\,f_2\,(g_2)^2
+
4\,f_2\,g_1\,g_3
+
f_1\,g_4,
\\
h_5
&
=
f_5\,(g_1)^5\,
+
10\,f_4\,(g_1)^3\,g_2
+
15\,f_3\,(g_1)^2\,g_3
+
10\,f_3\,g_1\,(g_2)^2
+ \\
& \
\ \ \ \ \
+
10\,f_2\,g_2\,g_3
+
5\,f_2\,g_1\,g_4
+
f_1\,g_5,
\\
h_6
&
=
f_6\,(g_1)^6\,
+
15\,f_5\,(g_1)^4\,g_2
+
45\,f_4\,(g_1)^2\,(g_2)^2
+
15\,f_3\,(g_2)^3
+ \\
& \
\ \ \ \ \
+
20\,f_4\,(g_1)^3\,g_3
+
60\,f_3\,g_1\,g_2\,g_3
+
10\,f_2\,(g_3)^2
+
15\,f_3\,(g_1)^2\,g_4
+ \\
& \
\ \ \ \ \
+
15\,f_2\,g_2\,g_4
+
6\,f_2\,g_1\,g_5
+
f_1\,g_6.
\endaligned\right.
\end{equation}
\def5.12}\begin{theorem{2.32}\begin{theorem}
For every integer $\kappa \geq 1$, the $\kappa$-th derivative of the
composite function $h = f\circ g$ may be expressed as an explicit
polynomial in the partial derivatives of $f$ and of $g$ having integer
coefficients{\rm :}
\def5.22}\begin{equation{2.33}\begin{equation}
\boxed{
\aligned
\frac{ d^\kappa h}{dx^\kappa}
&
=
\sum_{d=1}^\kappa
\
\sum_{1\leq\lambda_1<\cdots<\lambda_d\leq\kappa}
\
\sum_{\mu_1\geq 1,\dots,\mu_d\geq 1}
\
\sum_{\mu_1\lambda_1+\cdots+\mu_d\lambda_d=\kappa}
\\
& \
\ \ \ \ \
\frac{\kappa !}{(\lambda_1!)^{\mu_1}\ \mu_1!
\cdots
(\lambda_d!)^{\mu_d}\ \mu_d!}
\
\frac{d^{\mu_1+\cdots+\mu_d} f}{
dy^{\mu_1+\cdots+\mu_d}}
\
\left(
\frac{d^{\lambda_1}g}{dx^{\lambda_1}}
\right)^{\mu_1}
\cdots\cdots
\left(
\frac{d^{\lambda_d}g}{dx^{\lambda_d}}
\right)^{\mu_d}.
\endaligned
}
\end{equation}
\end{theorem}
This is the classical {\it Fa\`a di Bruno formula}. Interestingly, we
observe that this formula is included as subpart of the general
formula for ${\bf Y}_\kappa$, after a suitable translation. Indeed,
in the formulas for ${\bf Y}_1$, ${\bf Y}_2$, ${\bf Y}_3$, ${\bf
Y}_4$, ${\bf Y}_5$, ${\bf Y}_6$ and in the general sum for ${\bf
Y}_\kappa$, pick only the terms for which $\mu_1\lambda_1 + \cdots +
\mu_d \lambda_d = \kappa$ and drop $\mathcal{ X}$, which yields:
\def5.22}\begin{equation{2.34}\begin{equation}
\aligned
& \
\sum_{d=1}^\kappa
\
\sum_{1\leq\lambda_1<\cdots<\lambda_d\leq \kappa}
\
\sum_{\mu_1\geq 1,\dots,\mu_d\geq 1}
\
\sum_{\mu_1\lambda_1+\cdots+\mu_d\lambda_d=\kappa}
\\
& \
\left[
\frac{\kappa!}{
\mu_1!(\lambda_1!)^{\mu_1}
\cdots
\mu_d!(\lambda_d!)^{\mu_d}
}
\
\mathcal{Y}_{y^{\mu_1+\cdots+\mu_d}}
\right]
\left(
y_{\lambda_1}
\right)^{\mu_1}
\cdots
\left(
y_{\lambda_d}\right)^{\mu_d}.
\endaligned
\end{equation}
The similarity between the two formulas~\thetag{ 2.33} and~\thetag{
2.34} is now clearly visible.
The Fa\`a di Bruno formula may be established by means of
substitutions of power series (\cite{ f1969}, p.~222), by means of the
umbral calculus (\cite{ cs1996}), or by means of some induction
formulas, which we write for completeness. Define the differential
operators
\def5.22}\begin{equation{2.35}\begin{equation}
\small
\aligned
F_2
&
:=
g_2\,\frac{\partial}{\partial g_1}
+
g_1\left(
f_2\,\frac{\partial}{\partial f_1}
\right),
\\
F_3
&
:=
g_2\,\frac{\partial}{\partial g_1}
+
g_3\,\frac{\partial}{\partial g_2}
+
g_1\left(
f_2\,\frac{\partial}{\partial f_1}
+
f_3\,\frac{\partial}{\partial f_2}
\right),
\\
\cdots
&
\cdots\cdots
\cdots\cdots
\cdots\cdots
\cdots\cdots
\cdots\cdots
\cdots\cdots
\cdots\cdots
\\
F_\lambda
&
:=
g_2\,\frac{\partial}{\partial g_1}
+
g_3\,\frac{\partial}{\partial g_2}
+
\cdots
+
g_\lambda\,\frac{\partial}{\partial g_{\lambda-1}}
+
g_1\left(
f_2\,\frac{\partial}{\partial f_1}
+
f_3\,\frac{\partial}{\partial f_2}
+
\cdots
+
f_\lambda\,\frac{\partial}{\partial f_{\lambda-1}}
\right).
\endaligned
\end{equation}
Then we have
\def5.22}\begin{equation{2.36}\begin{equation}
\aligned
h_2
&
=
F^2(h_1),
\\
h_3
&
=
F^3(h_2),
\\
\cdots
&
\cdots
\cdots
\cdots
\cdots
\\
h_\lambda
&
=
F^\lambda(h_{\lambda-1}).
\endaligned
\end{equation}
\section*{\S3.~Several independent variables and
one dependent variable}
\subsection*{3.1.~Simplified adapted notations}
As announced after the statement of Theorem~2.24, it is only after we
have treated the case of several independent variables that we will
understand perfectly the general formula~\thetag{ 2.25}, valid in the
case of one independent variable and one dependent variable. We will
discover massive formal computations, exciting our computational
intuition.
Thus, assume $n\geq 1$ and $m=1$, let $\kappa\in \N$ with $\kappa \geq
1$ and simply denote (instead of~\thetag{ 1.2})
the jet variables by:
\def5.22}\begin{equation{3.2}\begin{equation}
\left(
x^i,y,y_{i_1},y_{i_1,i_2},\dots,y_{i_1,i_2,\dots,i_\kappa}
\right).
\end{equation}
Also, instead of~\thetag{ 1.30}, denote
the $\kappa$-th prolongation of a vector field by:
\def5.22}\begin{equation{3.3}\begin{equation}
\left\{
\aligned
\mathcal{L}^{(\kappa)}
&
=
\sum_{i=1}^n\,\mathcal{X}^i\,\frac{\partial}{\partial x^i}
+
\mathcal{Y}\,\frac{\partial}{\partial y}
+
\sum_{i_1=1}^n\,{\bf Y}_{i_1}\,\frac{\partial}{\partial y_{i_1}}
+
\sum_{i_1,i_2=1}^n\,{\bf Y}_{i_1,i_2}\,
\frac{\partial}{\partial y_{i_1,i_2}}
+ \\
& \
\ \ \ \ \
+
\cdots
+
\sum_{i_1,i_2,\dots,i_\kappa=1}^n\,
{\bf Y}_{i_1,i_2,\dots,i_\kappa}\,
\frac{\partial}{\partial y_{i_1,i_2,\dots,i_\kappa}}.
\endaligned\right.
\end{equation}
The induction formulas are
\def5.22}\begin{equation{3.4}\begin{equation}
\left\{
\aligned
{\bf Y}_{i_1}
&
:=
D_{i_1}^1
\left(
\mathcal{ Y}
\right)
-
\sum_{k=1}^n\,D_{i_1}^1
\left(
\mathcal{X}^k
\right)
\, y_k,
\\
{\bf Y}_{i_1,i_2}
&
:=
D_{i_2}^2
\left(
{\bf Y}_{i_1}
\right)
-
\sum_{k=1}^n\,D_{i_2}^1
\left(
\mathcal{X}^k
\right)
\, y_{i_1,k},
\\
\cdots \cdots \cdots
&
\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots
\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots
\\
{\bf Y}_{i_1,i_2,\dots,i_\kappa}
&
:=
D_{i_\kappa}^\kappa
\left(
{\bf Y}_{i_1,i_2,\dots,i_{\kappa-1}}
\right)
-
\sum_{k=1}^n\,D_{i_\kappa}^1
\left(
\mathcal{X}^k
\right)
\, y_{i_1,i_2,\dots,i_{\kappa-1},k},
\endaligned\right.
\end{equation}
where the total differentiation operators $D_{ i' }^\lambda$ are
defined as in~\thetag{ 1.22}, dropping the sums $\sum_{j ' = 1}^m$ and
the indices $j'$.
\subsection*{3.5.~Two instructing explicit computations}
To begin with, let us compute ${\bf Y}_{i_1}$. With $D_{i_1}^1 =
\frac{ \partial }{\partial x^{i_1}} + y_{i_1}\, \frac{\partial
}{\partial y}$, we have:
\def5.22}\begin{equation{3.6}\begin{equation}
\aligned
{\bf Y}_{i_1}
&
=
D_{i_1}
\left(
\mathcal{Y}
\right)
-
\sum_{k_1=1}^n\,D_{i_1}^1
\left(
\mathcal{X}^{k_1}
\right)
y_{k_1}
\\
&
=
\mathcal{Y}_{x^{i_1}}
+
\mathcal{Y}_y\,y_{i_1}
-
\sum_{k_1=1}^n\,\mathcal{X}_{x^{i_1}}^{k_1}\,y_{k_1}
-
\sum_{k_1=1}^n\,\mathcal{X}_y^{k_1}\,y_{i_1}\,y_{k_1}.
\endaligned
\end{equation}
Searching for formal harmony and for coherence with the formula
$(2.6)_1$, we must include the term $\mathcal{ Y}_y\, y_{i_1}$ inside
the sum $\sum_{ k_1 =1 }^n\, \left[ \cdot \right] y_{ k_1}$. Using
the Kronecker symbol, we may write:
\def5.22}\begin{equation{3.7}\begin{equation}
\mathcal{Y}_y\,y_{i_1}
\equiv
\sum_{k_1=1}^n\,
\left[
\delta_{i_1}^{k_1}\,\mathcal{Y}_y
\right]
y_{k_1}.
\end{equation}
Also, we may rewrite the last term of~\thetag{ 3.6} with a double sum:
\def5.22}\begin{equation{3.8}\begin{equation}
-
\sum_{k_1=1}^n\,
\mathcal{X}_y^{k_1}\,y_{i_1}\,y_{k_1}
\equiv
\sum_{k_1,k_2=1}^n\,
\left[
-
\delta_{i_1}^{k_1}\,\mathcal{X}_y^{k_2}
\right]
y_{k_1}y_{k_2}.
\end{equation}
From now on and up to equation~\thetag{ 3.39}, we shall abbreviate any
sum $\sum_{k=1}^n$ from $1$ to $n$ as $\sum_k$. Putting everything
together, we get the final desired perfect expression of ${\bf
Y}_{i_1}$:
\def5.22}\begin{equation{3.9}\begin{equation}
{\bf Y}_{i_1}
=
\mathcal{Y}_{x^{i_1}}
+
\sum_{k_1}\,
\left[
\delta_{i_1}^{k_1}\,\mathcal{Y}_y
-
\mathcal{X}_{x^{i_1}}^{k_1}
\right]
y_{k_1}
+
\sum_{k_1,k_2}\,
\left[
-
\delta_{i_1}^{k_1}\,\mathcal{X}_y^{k_2}
\right]
y_{k_1}y_{k_2}.
\end{equation}
This completes the first explicit computation.
The second one is about ${\bf Y}_{ i_1, i_2}$. It becomes more
delicate, because several algebraic transformations must be achieved
until the final satisfying formula is obtained. Our goal is to
present each step very carefully, explaining every tiny
detail. Without such a care, it would be impossible to claim that some
of our subsequent computations, for which we will not provide the
intermediate steps, may be redone and verified. Consequently, we will
expose our rules of formal computation thoroughly.
Replacing the value of ${\bf Y}_1$ just obtained in the induction
formula $(3.4)_2$ and developing, we may conduct the very
first steps of the computation:
$$
\small
\aligned
{\bf Y}_{i_1,i_2}
&
=
D_{i_2}^2
\left(
{\bf Y}_{i_1}
\right)
-
\sum_{k_1}\,D_{i_2}^1
\left(
\mathcal{X}^{k_1}
\right)
y_{i_1,k_1}
\\
&
=
\left(
\frac{\partial}{\partial x^{i_2}}
+
y_{i_2}\,\frac{\partial}{\partial y}
+
\sum_{k_1}\,y_{i_2,k_1}\,
\frac{\partial}{\partial y_{k_1}}
\right)
\left(
\mathcal{Y}_{x^{i_1}}
+
\sum_{k_1}\,
\left[
\delta_{i_1}^{k_1}\,\mathcal{Y}_y
-
\mathcal{X}_{x^{i_1}}^{k_1}
\right]
y_{k_1}
+
\right.
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \
\left.
+
\sum_{k_1,k_2}\,
\left[
-
\delta_{i_1}^{k_1}\,\mathcal{X}_y^{k_2}
\right]
y_{k_1}y_{k_2}
\right)
-
\sum_{k_1}\,
\left[
\mathcal{X}_{x^{i_2}}^{k_1}
+
y_{i_2}\,\mathcal{X}_y^{k_1}
\right]
y_{i_1,k_1}
\endaligned
$$
\def5.22}\begin{equation{3.10}\begin{equation}
\small
\aligned
&
=
\left(
\frac{\partial}{\partial x^{i_2}}
\right)
\left(
\mathcal{Y}_{x^{i_1}}
+
\sum_{k_1}\,
\left[
\delta_{i_1}^{k_1}\,\mathcal{Y}_y
-
\mathcal{X}_{x^{i_1}}^{k_1}
\right]
y_{k_1}
+
\sum_{k_1,k_2}\,
\left[
-
\delta_{i_1}^{k_1}\,\mathcal{X}_y^{k_2}
\right]
y_{k_1}y_{k_2}
\right)
+ \\
& \
\ \ \ \ \
+
\left(
y_{i_2}\,\frac{\partial}{\partial y}
\right)
\left(
\mathcal{Y}_{x^{i_1}}
+
\sum_{k_1}\,
\left[
\delta_{i_1}^{k_1}\,\mathcal{Y}_y
-
\mathcal{X}_{x^{i_1}}^{k_1}
\right]
y_{k_1}
+
\sum_{k_1,k_2}\,
\left[
-
\delta_{i_1}^{k_1}\,\mathcal{X}_y^{k_2}
\right]
y_{k_1}y_{k_2}
\right)
+ \\
& \
\ \ \ \ \
+
\left(
\sum_{k_1}\,y_{i_2,k_1}\,
\frac{\partial}{\partial y_{k_1}}
\right)
\left(
\mathcal{Y}_{x^{i_1}}
+
\sum_{k_1}\,
\left[
\delta_{i_1}^{k_1}\,\mathcal{Y}_y
-
\mathcal{X}_{x^{i_1}}^{k_1}
\right]
y_{k_1}
+
\sum_{k_1,k_2}\,
\left[
-
\delta_{i_1}^{k_1}\,\mathcal{X}_y^{k_2}
\right]
y_{k_1}y_{k_2}
\right)
+ \\
& \
\ \ \ \ \
+
\sum_{k_1}\,
\left[
-
\mathcal{X}_{x^{i_2}}^{k_1}
\right]
y_{k_1,i_1}
+
\sum_{k_1}\,
\left[
-
\mathcal{X}_y^{k_1}
\right]
y_{i_2}y_{i_1,k_1}
\endaligned
\end{equation}
$$
\small
\aligned
&
=
\mathcal{Y}_{x^{i_1}x^{i_2}}
+
\sum_{k_1}\,
\left[
\delta_{i_1}^{k_1}\,\mathcal{Y}_{x^{i_2}y}
-
\mathcal{X}_{x^{i_1}x^{i_2}}^{k_1}
\right]
y_{k_1}
+
\sum_{k_1,k_2}\,
\left[
-
\delta_{i_1}^{k_1}\,\mathcal{X}_{x^{i_2}y}^{k_2}
\right]
y_{k_1}y_{k_2}
+ \\
& \
\ \ \ \ \
+
\mathcal{Y}_{x^{i_1}y}\,y_{i_2}
+
\sum_{k_1}\,
\left[
\delta_{i_1}^{k_1}\,\mathcal{Y}_{yy}
-
\mathcal{X}_{x^{i_1}y}^{k_1}
\right]
y_{k_1}y_{i_2}
+
\sum_{k_1,k_2}\,
\left[
-
\delta_{i_1}^{k_1}\,\mathcal{X}_{yy}^{k_2}
\right]
y_{k_1}y_{k_2}y_{i_2}
+ \\
& \
\ \ \ \ \
+
\sum_{k_1}\,
\left[
\delta_{i_1}^{k_1}\,\mathcal{Y}_y
-
\mathcal{X}_{x^{i_1}}^{k_1}
\right]
y_{i_2,k_1}
+
\sum_{k_1,k_2}\,
\left[
-
\delta_{i_1}^{k_1}\,\mathcal{X}_y^{k_2}
\right]
y_{k_2}y_{i_2,k_1}
+
\sum_{k_1,k_2}\,
\left[
-
\delta_{i_1}^{k_1}\,\mathcal{X}_y^{k_2}
\right]
y_{k_1}y_{i_2,k_2}
+ \\
& \
\ \ \ \ \
+
\sum_{k_1}\,
\left[
-
\mathcal{X}_{x^{i_2}}^{k_1}
\right]
y_{k_1,i_1}
+
\sum_{k_1}
\left[
-
\mathcal{X}_y^{k_1}
\right]
y_{i_2}y_{i_1,k_1}.
\endaligned
$$
Some explanations are needed about the computation of the last two
terms of line 11, {\it i.e.} about the passage from line 7 of~\thetag{
3.10} just above to line 11. We have to compute:
\def5.22}\begin{equation{3.11}\begin{equation}
\footnotesize
\left(
\sum_{k_1}\,y_{i_2,k_1}\,
\frac{\partial}{\partial y_{k_1}}
\right)
\left(
\sum_{k_1,k_2}\,
\left[
-
\delta_{i_1}^{k_1}\,
\mathcal{X}_y^{k_2}
\right]
y_{k_1}y_{k_2}
\right).
\end{equation}
This term is of the form
\def5.22}\begin{equation{3.12}\begin{equation}
\footnotesize
\left(
\sum_{k_1}\,A_{k_1}\,
\frac{\partial}{\partial y_{k_1}}
\right)
\left(
\sum_{k_1,k_2}\,
\left[
B_{k_1,k_2}
\right]
y_{k_1}y_{k_2}
\right),
\end{equation}
where the terms $B_{k_1,k_2}$ are independent of the pure first jet
variables $y_{x^k}$.
By the rule of Leibniz for the differentiation
of a product, we may write
\def5.22}\begin{equation{3.13}\begin{equation}
\footnotesize
\aligned
&
\left(
\sum_{k_1}\,A_{k_1}\,
\frac{\partial}{\partial y_{k_1}}
\right)
\left(
\sum_{k_1,k_2}\,
\left[
B_{k_1,k_2}
\right]
y_{k_1}y_{k_2}
\right)
=
\\
&
=
\sum_{k_1,k_2}\,
\left[
B_{k_1,k_2}
\right]
y_{k_2}
\left(
\sum_{k_1'}\,A_{k_1'}\,
\frac{\partial}{\partial y_{k_1'}}
(y_{k_1})
\right)
+
\sum_{k_1,k_2}\,
\left[
B_{k_1,k_2}
\right]
y_{k_1}
\left(
\sum_{k_2'}\,A_{k_2'}\,
\frac{\partial}{\partial y_{k_2'}}
(y_{k_2})
\right)
\\
&
=
\sum_{k_1,k_2}\,
\left[
B_{k_1,k_2}
\right]
y_{k_2}\,A_{k_1}
+
\sum_{k_1,k_2}\,
\left[
B_{k_1,k_2}
\right]
y_{k_1}\,A_{k_2}.
\endaligned
\end{equation}
This is how we have written line 11 of~\thetag{ 3.10}.
Next, the first term $\mathcal{ Y}_{ x^{i_1}y} \, y_{ i_2}$ in line 10
of~\thetag{ 3.10} is not in a suitable shape. For reasons of harmony
and coherence, we must insert it inside a sum of the form
$\sum_{k_1}\, \left[ \cdot \right] y_{k_1}$. Hence, using the
Kronecker symbol, we transform:
\def5.22}\begin{equation{3.14}\begin{equation}
\mathcal{ Y}_{x^{i_1}y} \, y_{i_2}
\equiv
\sum_{k_1}\,
\left[
\delta_{i_2}^{k_1}\,
\mathcal{Y}_{x^{i_1}y}
\right]
y_{k_1}.
\end{equation}
Also, we must ``summify'' the seven other terms, remaining in lines 10,
11 and 12 of~\thetag{ 3.10}. Sometimes, we use the symmetry $y_{i_2,
k_1} \equiv y_{ k_1, i_2}$ without mention. Similarly, we get:
$$
\aligned
\sum_{k_1}\,
\left[
\delta_{i_1}^{k_1}\,\mathcal{Y}_{yy}
-
\mathcal{X}_{x^{i_1y}}^{k_1}
\right]
y_{k_1}y_{i_2}
&
\equiv
\sum_{k_1,k_2}\,
\left[
\delta_{i_1}^{k_1}\,\delta_{i_2}^{k_2}\,
\mathcal{Y}_{yy}
-
\delta_{i_2}^{k_2}\,\mathcal{X}_{x^{i_1}y}^{k_1}
\right]
y_{k_1}y_{k_2}, \\
\sum_{k_1,k_2}\,
\left[
-
\delta_{i_1}^{k_1}\,
\mathcal{X}_{yy}^{k_2}
\right]
y_{k_1}y_{k_2}y_{i_2}
&
\equiv
\sum_{k_1,k_2,k_3}\,
\left[
-
\delta_{i_1}^{k_1}\,
\delta_{i_2}^{k_3}
\mathcal{X}_{yy}^{k_2}
\right]
y_{k_1}y_{k_2}y_{k_3}, \\
\sum_{k_1}
\left[
\delta_{i_1}^{k_1}\,
\mathcal{Y}_y
-
\mathcal{X}_{x^{i_1}}^{k_1}
\right]
y_{k_1,i_2}
&
\equiv
\sum_{k_1,k_2}
\left[
\delta_{i_1}^{k_1}\,
\delta_{i_2}^{k_2}\,
\mathcal{Y}_y
-
\delta_{i_2}^{k_2}\,
\mathcal{X}_{x^{i_1}}^{k_1}
\right]
y_{k_1,k_2},
\endaligned
$$
\def5.22}\begin{equation{3.15}\begin{equation}
\aligned
\sum_{k_1,k_2}\,
\left[
-
\delta_{i_1}^{k_1}\,\mathcal{X}_y^{k_2}
\right]
y_{k_2}y_{k_1,i_2}
&
=
\sum_{k_1,k_2}\,
\left[
-
\delta_{i_1}^{k_2}\,\mathcal{X}_y^{k_1}
\right]
y_{k_1}y_{k_2,i_2}
\\
&
\equiv
\sum_{k_1,k_2,k_3}\,
\left[
-
\delta_{i_1}^{k_2}\,\delta_{i_2}^{k_3}\,
\mathcal{X}_y^{k_1}
\right]
y_{k_1}y_{k_2,k_3},
\endaligned
\end{equation}
$$
\aligned
\sum_{k_1,k_2}\,
\left[
-
\delta_{i_1}^{k_1}\,\mathcal{X}_y^{k_2}
\right]
y_{k_1}y_{k_2,i_2}
&
\equiv
\sum_{k_1,k_2,k_3}\,
\left[
-
\delta_{i_1}^{k_1}\,
\delta_{i_2}^{k_3}\,
\mathcal{X}_y^{k_2}
\right]
y_{k_1}y_{k_2,k_3},
\\
\sum_{k_1}\,
\left[
-
\mathcal{X}_{x^{i_2}}^{k_1}
\right]
y_{k_1,i_1}
&
\equiv
\sum_{k_1,k_2}\,
\left[
-
\delta_{i_1}^{k_2}
\mathcal{X}_{x^{i_2}}^{k_1}
\right]
y_{k_1,k_2},
\\
\sum_{k_1}\,
\left[
-
\mathcal{X}_y^{k_1}
\right]
y_{i_2}y_{k_1,i_1}
&
=
\sum_{k_2}\,
\left[
-
\mathcal{X}_y^{k_2}
\right]
y_{i_2}y_{k_2,i_1}
\\
&
\equiv
\sum_{k_1,k_2,k_3}\,
\left[
-
\delta_{i_2}^{k_1}\,
\delta_{i_1}^{k_3}\,
\mathcal{X}_y^{k_2}
\right]
y_{k_1}y_{k_2,k_3}.
\endaligned
$$
In the sequel, for products of
Kronecker symbols, it will be convenient
to adopt the following self-evident contracted
notation:
\def5.22}\begin{equation{3.16}\begin{equation}
\delta_{i_1}^{k_1}\,
\delta_{i_2}^{k_2}
\equiv
\delta_{i_1,\,i_2}^{k_1,k_2};
\ \ \ \ \ \
{\rm generally:}
\ \ \
\delta_{i_1}^{k_1}\,
\delta_{i_2}^{k_2}
\cdots
\delta_{i_\lambda}^{k_\lambda}
\equiv
\delta_{i_1,\,i_2,\,\cdots\,,i_\lambda}^{
k_1,k_2,\cdots,k_\lambda}.
\end{equation}
Re-inserting plainly these eight summified terms~\thetag{ 3.14},
\thetag{ 3.15} in the last expression~\thetag{ 3.10} of ${\bf Y}_{i_1,
i_2}$ (lines 10, 11 and 12), we get:
\def5.22}\begin{equation{3.17}\begin{equation}
\small
\aligned
{\bf Y}_{i_1,i_2}
&
=
\underline{
\mathcal{Y}_{x^{i_1}x^{i_2}}
}_{ \fbox{\tiny 1}}
+
\underline{
\sum_{k_1}\,
\left[
\delta_{i_1}^{k_1}\,
\mathcal{Y}_{x^{i_2}y}
-
\mathcal{X}_{x^{i_1}x^{i_2}}^{k_1}
\right]
y_{k_1}
}_{ \fbox{\tiny 2}}
+
\underline{
\sum_{k_1,k_2}\,
\left[
-
\delta_{i_1}^{k_1}\,
\mathcal{X}_{x^{i_2}y}^{k_2}
\right]
y_{k_1}y_{k_2}
}_{ \fbox{\tiny 3}}
+ \\
& \
\ \ \ \ \
+
\underline{
\sum_{k_1}\,
\left[
\delta_{i_2}^{k_1}\,
\mathcal{Y}_{x^{i_1}y}
\right]
y_{k_1}
}_{ \fbox{\tiny 2}}
+
\underline{
\sum_{k_1,k_2}\,
\left[
\delta_{i_1,\,i_2}^{k_1,k_2}\,
\mathcal{Y}_{yy}
-
\delta_{i_2}^{k_2}\,
\mathcal{X}_{x^{i_1}y}^{k_1}
\right]
y_{k_1}y_{k_2}
}_{ \fbox{\tiny 3}}
+ \\
& \
\ \ \ \ \
+
\underline{
\sum_{k_1,k_2,k_3}\,
\left[
-
\delta_{i_1,\,i_2}^{k_1,k_3}\,
\mathcal{X}_{yy}^{k_2}
\right]
y_{k_1}y_{k_2}y_{k_3}
}_{ \fbox{\tiny 4}}
+
\underline{
\sum_{k_1,k_2}\,
\left[
\delta_{i_1,\,i_2}^{k_1,k_2}\,
\mathcal{Y}_y
-
\delta_{i_2}^{k_2}\,
\mathcal{X}_{x^{i_1}}^{k_1}
\right]
y_{k_1,k_2}
}_{ \fbox{\tiny 5}}
+ \\
& \
\ \ \ \ \
+
\underline{
\sum_{k_1,k_2,k_3}\,
\left[
-
\delta_{i_1,\,i_2}^{k_2,k_3}\,
\mathcal{X}_y^{k_1}
\right]
y_{k_1}y_{k_2,k_3}
}_{ \fbox{\tiny 6}}
+
\underline{
\sum_{k_1,k_2,k_3}\,
\left[
-
\delta_{i_1,\,i_2}^{k_1,k_3}\,
\mathcal{X}_y^{k_2}
\right]
y_{k_1}y_{k_2,k_3}
}_{ \fbox{\tiny 6}}
+ \\
& \
\ \ \ \ \
+
\underline{
\sum_{k_1,k_2}\,
\left[
-
\delta_{i_1}^{k_2}\,
\mathcal{X}_{x^{i_2}}^{k_1}
\right]
y_{k_1,k_2}
}_{ \fbox{\tiny 5}}
+
\underline{
\sum_{k_1,k_2,k_3}\,
\left[
-
\delta_{i_2,\,i_1}^{k_1,k_3}\,
\mathcal{X}_y^{k_2}
\right]
y_{k_1}y_{k_2,k_3}
}_{ \fbox{\tiny 6}}.
\endaligned
\end{equation}
Next, we gather the underlined terms, ordering them according to their
number. This yields 6 collections of
sums of monomials in the
pure jet variables:
\def5.22}\begin{equation{3.18}\begin{equation}
\small
\aligned
{\bf Y}_{i_1,i_2}
&
=
\mathcal{Y}_{x^{i_1}x^{i_2}}
+
\sum_{k_1}\,
\left[
\delta_{i_1}^{k_1}\,\mathcal{Y}_{x^{i_2}y}
+
\delta_{i_2}^{k_1}\,\mathcal{Y}_{x^{i_1}y}
-
\mathcal{X}_{x^{i_1}x^{i_2}}^{k_1}
\right]
y_{k_1}
+ \\
& \
\ \ \ \ \
+
\sum_{k_1,k_2}\,
\left[
\delta_{i_1,\,i_2}^{k_1,k_2}\,
\mathcal{Y}_{yy}
-
\delta_{i_1}^{k_1}\,
\mathcal{X}_{x^{i_2}y}^{k_2}
-
\delta_{i_2}^{k_2}\,
\mathcal{X}_{x^{i_1}y}^{k_1}
\right]
y_{k_1}y_{k_2}
+ \\
& \
\ \ \ \ \
+
\sum_{k_1,k_2,k_3}\,
\left[
-
\delta_{i_1,\,i_2}^{k_1,k_3}\,
\mathcal{X}_{yy}^{k_2}
\right]
y_{k_1}y_{k_2}y_{k_3}
+ \\
& \
\ \ \ \ \
+
\sum_{k_1,k_2}\,
\left[
\delta_{i_1,\,i_2}^{k_1,k_2}\,
\mathcal{Y}_y
-
\delta_{i_2}^{k_2}\,\mathcal{X}_{x^{i_1}}^{k_1}
-
\delta_{i_1}^{k_2}\,\mathcal{X}_{x^{i_2}}^{k_1}
\right]
y_{k_1,k_2}
+ \\
& \
\ \ \ \ \
+
\sum_{k_1,k_2,k_3}\,
\left[
-
\delta_{i_1,\,i_2}^{k_2,k_3}\,
\mathcal{X}_y^{k_1}
-
\delta_{i_1,\,i_2}^{k_1,k_3}\,
\mathcal{X}_y^{k_2}
-
\delta_{i_2,\,i_1}^{k_1,k_3}\,
\mathcal{X}_y^{k_2}
\right]
y_{k_1}y_{k_2,k_3}.
\endaligned
\end{equation}
To attain the real perfect harmony, this last expression has still to
be worked out a little bit.
\def3.50}\begin{lemma{3.19}\begin{lemma}
The final expression of ${\bf Y}_{i_1,i_2}$ is
as follows{\rm :}
\def5.22}\begin{equation{3.20}\begin{equation}
\left\{
\aligned
{\bf Y}_{i_1,i_2}
&
=
\mathcal{Y}_{x^{i_1}x^{i_2}}
+
\sum_{k_1}\,
\left[
\delta_{i_1}^{k_1}\,\mathcal{Y}_{x^{i_2}y}
+
\delta_{i_2}^{k_1}\,\mathcal{Y}_{x^{i_1}y}
-
\mathcal{X}_{x^{i_1}x^{i_2}}^{k_1}
\right]
y_{k_1}
+ \\
& \
\ \ \ \ \
+
\sum_{k_1,k_2}\,
\left[
\delta_{i_1,\,i_2}^{k_1,k_2}\,
\mathcal{Y}_{yy}
-
\delta_{i_1}^{k_1}\,
\mathcal{X}_{x^{i_2}y}^{k_2}
-
\delta_{i_2}^{k_1}\,
\mathcal{X}_{x^{i_1}y}^{k_2}
\right]
y_{k_1}y_{k_2}
+ \\
& \
\ \ \ \ \
+
\sum_{k_1,k_2,k_3}\,
\left[
-
\delta_{i_1,\,i_2}^{k_1,k_2}\,
\mathcal{X}_{yy}^{k_3}
\right]
y_{k_1}y_{k_2}y_{k_3}
+ \\
& \
\ \ \ \ \
+
\sum_{k_1,k_2}\,
\left[
\delta_{i_1,\,i_2}^{k_1,k_2}\,
\mathcal{Y}_y
-
\delta_{i_1}^{k_1}\,\mathcal{X}_{x^{i_2}}^{k_2}
-
\delta_{i_2}^{k_1}\,\mathcal{X}_{x^{i_1}}^{k_2}
\right]
y_{k_1,k_2}
+ \\
& \
\ \ \ \ \
+
\sum_{k_1,k_2,k_3}\,
\left[
-
\delta_{i_1,\,i_2}^{k_1,k_2}\,
\mathcal{X}_y^{k_3}
-
\delta_{i_1,\,i_2}^{k_3,k_1}\,
\mathcal{X}_y^{k_2}
-
\delta_{i_1,\,i_2}^{k_2,k_3}\,
\mathcal{X}_y^{k_1}
\right]
y_{k_1}y_{k_2,k_3}.
\endaligned\right.
\end{equation}
\end{lemma}
\proof
As promised, we explain every tiny detail.
The first lines of~\thetag{ 3.18} and of~\thetag{ 3.20} are exactly
the same. For the transformations of terms in the second, in the third
and in the fourth lines, we use the following device. Let $\Upsilon_{
k_1, k_2}$ be an indexed quantity which is symmetric: $\Upsilon_{ k_1,
k_2} =\Upsilon_{ k_2, k_1}$. Let $A_{k_1, k_2}$ be an arbitrary
indexed quantity. Then obviously:
\def5.22}\begin{equation{3.21}\begin{equation}
\sum_{k_1,k_2}\,
A_{k_1, k_2}\,
\Upsilon_{k_1, k_2}
=
\sum_{k_1,k_2}\,
A_{k_2,k_1}\,
\Upsilon_{k_1,k_2}.
\end{equation}
Similar relations hold with a quantity $\Upsilon_{i_1, i_2, \dots,
i_\lambda}$ which is symmetric with respect to its $\lambda$ indices.
Consequently, in the second, in the third and in the fourth lines
of~\thetag{ 3.18}, we may permute freely certain indices in some of
the terms inside the brackets. This yields
the passage from lines 2, 3 and 4 of~\thetag{ 3.18}
to lines 2, 3 and 4 of~\thetag{ 3.20}.
It remains to explain how we pass from the fifth (last) line
of~\thetag{ 3.18} to the fifth (last) line of~\thetag{ 3.20}. The
bracket in the fifth line of~\thetag{ 3.18} contains three terms:
$\left[ -T_1 -T_2 -T_3 \right]$. The term $T_3$ involves the product
$\delta_{i_2,\,i_1}^{k_1,k_3}$, which we rewrite as
$\delta_{i_1,\,i_2}^{k_3,k_1}$, in order that $i_1$ appears before
$i_2$. Then, we rewrite the three terms in the new order $\left[ -T_2
-T_3 -T_1 \right]$, which yields:
\def5.22}\begin{equation{3.22}\begin{equation}
\sum_{k_1,k_2,k_3}\,
\left[
-
\delta_{i_1,\,i_2}^{k_1,k_3}\,
\mathcal{X}_y^{k_2}
-
\delta_{i_1,\,i_2}^{k_3,k_1}\,
\mathcal{X}_y^{k_2}
-
\delta_{i_1,\,i_2}^{k_2,k_3}\,
\mathcal{X}_y^{k_1}
\right]
y_{k_1}y_{k_2,k_3}.
\end{equation}
It remains to observe that we can permute $k_2$ and $k_3$ in the first
term $-T_2$, which yields the last
line of~\thetag{ 3.20}. The detailed proof is complete.
\endproof
\subsection*{3.23.~Final perfect expression of ${\bf Y}_{i_1, i_2,
i_3}$} Thanks to similar (longer) computations, we have obtained an
expression of ${\bf Y}_{i_1, i_2, i_3}$ which we consider to be in
final harmonious shape. Without copying the intermediate steps, let
us write down the result. The comments which
are necessary to read it and to interpret it
start just below.
$$
\small
\aligned
{\bf Y}_{i_1,i_2,i_3}
&
=
\mathcal{Y}_{x^{i_1}x^{i_2}x^{i_3}}
+
\sum_{k_1}\,
\left[
\delta_{i_1}^{k_1}\,\mathcal{Y}_{x^{i_2}x^{i_3}y}
+
\delta_{i_2}^{k_1}\,\mathcal{Y}_{x^{i_1}x^{i_3}y}
+
\delta_{i_3}^{k_1}\,\mathcal{Y}_{x^{i_1}x^{i_2}y}
-
\mathcal{X}_{x^{i_1}x^{i_2}x^{i_3}}^{k_1}
\right]
y_{k_1}
+ \\
& \
\ \ \ \ \
+
\sum_{k_1,k_2}\,
\left[
\delta_{i_1,\,i_2}^{k_1,k_2}\,\mathcal{Y}_{x^{i_3}y^2}
+
\delta_{i_3,\,i_1}^{k_1,k_2}\,\mathcal{Y}_{x^{i_2}y^2}
+
\delta_{i_2,\,i_3}^{k_1,k_2}\,\mathcal{Y}_{x^{i_1}y^2}
-
\right. \\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \
\left.
-
\delta_{i_1}^{k_1}\,\mathcal{X}_{x^{i_2}x^{i_3}y}^{k_2}
-
\delta_{i_2}^{k_1}\,\mathcal{X}_{x^{i_1}x^{i_3}y}^{k_2}
-
\delta_{i_3}^{k_1}\,\mathcal{X}_{x^{i_1}x^{i_2}y}^{k_2}
\right]
y_{k_1}y_{k_2}
+ \\
& \
\ \ \ \ \
+
\sum_{k_1,k_2,k_3}\,
\left[
\delta_{i_1,\,i_2,\,i_3}^{k_1,k_2,k_3}\,\mathcal{Y}_{y^3}
-
\delta_{i_1,\,i_2}^{k_1,k_2}\,\mathcal{X}_{x^{i_3}y^2}^{k_3}
-
\delta_{i_1,\,i_3}^{k_1,k_2}\,\mathcal{X}_{x^{i_2}y^2}^{k_3}
-
\delta_{i_2,\,i_3}^{k_1,k_2}\,\mathcal{X}_{x^{i_1}y^2}^{k_3}
\right]
y_{k_1}y_{k_2}y_{k_3}
+ \\
& \
\ \ \ \ \
+
\sum_{k_1,k_2,k_3,k_4}\,
\left[
-\delta_{i_1,\,i_2,\,i_3}^{k_1,k_2,k_3}\,\mathcal{X}_{y^3}^{k_4}
\right]
y_{k_1}y_{k_2}y_{k_3}y_{k_4}
+ \\
\endaligned
$$
\def5.22}\begin{equation{3.24}\begin{equation}
\small
\aligned
& \
\ \ \ \ \
+
\sum_{k_1,k_2}\,
\left[
\delta_{i_1,\,i_2}^{k_1,k_2}\,\mathcal{Y}_{x^{i_3}y}
+
\delta_{i_3,\,i_1}^{k_1,k_2}\,\mathcal{Y}_{x^{i_2}y}
+
\delta_{i_2,\,i_3}^{k_1,k_2}\,\mathcal{Y}_{x^{i_1}y}
-
\right.
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.
-
\delta_{i_1}^{k_1}\,\mathcal{X}_{x^{i_2}x^{i_3}}^{k_2}
-
\delta_{i_2}^{k_1}\,\mathcal{X}_{x^{i_1}x^{i_3}}^{k_2}
-
\delta_{i_3}^{k_1}\,\mathcal{X}_{x^{i_1}x^{i_2}}^{k_2}
\right]
y_{k_1,k_2}
+ \\
& \
\ \ \ \ \
+
\sum_{k_1,k_2,k_3}\,
\left[
\delta_{i_1,\,i_2,\,i_3}^{k_1,k_2,k_3}\,\mathcal{Y}_{y^2}
+
\delta_{i_1,\,i_2,\,i_3}^{k_3,k_1,k_2}\,\mathcal{Y}_{y^2}
+
\delta_{i_1,\,i_2,\,i_3}^{k_2,k_3,k_1}\,\mathcal{Y}_{y^2}
-
\right.
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.
-
\delta_{i_1,\,i_2}^{k_1,k_2}\,\mathcal{X}_{x^{i_3}y}^{k_3}
-
\delta_{i_1,\,i_2}^{k_3,k_1}\,\mathcal{X}_{x^{i_3}y}^{k_2}
-
\delta_{i_1,\,i_2}^{k_2,k_3}\,\mathcal{X}_{x^{i_3}y}^{k_1}
-
\right.
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.
-
\delta_{i_1,\,i_3}^{k_1,k_2}\,\mathcal{X}_{x^{i_2}y}^{k_3}
-
\delta_{i_1,\,i_3}^{k_3,k_1}\,\mathcal{X}_{x^{i_2}y}^{k_2}
-
\delta_{i_1,\,i_3}^{k_2,k_3}\,\mathcal{X}_{x^{i_2}y}^{k_1}
-
\right.
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.
-
\delta_{i_2,\,i_3}^{k_1,k_2}\,\mathcal{X}_{x^{i_1}y}^{k_3}
-
\delta_{i_2,\,i_3}^{k_3,k_1}\,\mathcal{X}_{x^{i_1}y}^{k_2}
-
\delta_{i_2,\,i_3}^{k_2,k_3}\,\mathcal{X}_{x^{i_1}y}^{k_1}
\right]
y_{k_1}y_{k_2,k_3}
+
\\
\endaligned
\end{equation}
$$
\small
\aligned
& \
\ \ \ \ \
+
\sum_{k_1,k_2,k_3,k_4}\,
\left[
-
\delta_{i_1,\,i_2,\,i_3}^{k_1,k_2,k_3}\,\mathcal{X}_{y^2}^{k_4}
-
\delta_{i_1,\,i_2,\,i_3}^{k_2,k_3,k_1}\,\mathcal{X}_{y^2}^{k_4}
-
\delta_{i_1,\,i_2,\,i_3}^{k_3,k_2,k_1}\,\mathcal{X}_{y^2}^{k_4}
-
\right.
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.
-
\delta_{i_1,\,i_2,\,i_3}^{k_3,k_4,k_1}\,\mathcal{X}_{y^2}^{k_2}
-
\delta_{i_1,\,i_2,\,i_3}^{k_3,k_1,k_4}\,\mathcal{X}_{y^2}^{k_2}
-
\delta_{i_1,\,i_2,\,i_3}^{k_1,k_3,k_4}\,\mathcal{X}_{y^2}^{k_2}
\right]
y_{k_1}y_{k_2}y_{k_3,k_4}
+ \\
& \
\ \ \ \ \
+
\sum_{k_1,k_2,k_3,k_4}\,
\left[
-
\delta_{i_1,\,i_2,\,i_3}^{k_1,k_2,k_3}\,\mathcal{X}_y^{k_4}
-
\delta_{i_1,\,i_2,\,i_3}^{k_2,k_3,k_1}\,\mathcal{X}_y^{k_4}
-
\delta_{i_1,\,i_2,\,i_3}^{k_3,k_1,k_2}\,\mathcal{X}_y^{k_4}
\right]
y_{k_1,k_2}y_{k_3,k_4}
+ \\
\endaligned
$$
$$
\small
\aligned
& \
\ \ \ \ \
+
\sum_{k_1,k_2,k_3}\,
\left[
\delta_{i_1,\,i_2,\,i_3}^{k_1,k_2,k_3}\,\mathcal{Y}_y
-
\delta_{i_1,\,i_2}^{k_1,k_2}\,\mathcal{X}_{x^{i_3}}^{k_3}
-
\delta_{i_1,\,i_3}^{k_1,k_2}\,\mathcal{X}_{x^{i_2}}^{k_3}
-
\delta_{i_2,\,i_3}^{k_1,k_2}\,\mathcal{X}_{x^{i_1}}^{k_3}
\right]
y_{k_1,k_2,k_3}
+ \\
& \
\ \ \ \ \
+
\sum_{k_1,k_2,k_3,k_4}\,
\left[
-
\delta_{i_1,\,i_2,\,i_3}^{k_1,k_2,k_3}\,\mathcal{X}_y^{k_4}
-
\delta_{i_1,\,i_2,\,i_3}^{k_4,k_1,k_2}\,\mathcal{X}_y^{k_3}
-
\delta_{i_1,\,i_2,\,i_3}^{k_3,k_4,k_1}\,\mathcal{X}_y^{k_2}
-
\delta_{i_1,\,i_2,\,i_3}^{k_2,k_3,k_4}\,\mathcal{X}_y^{k_1}
\right]
y_{k_1}y_{k_2,k_3,k_4}.
\endaligned
$$
\subsection*{3.25.~Comments, analysis and induction}
First of all, by comparing this expression of ${\bf Y}_{i_1, i_2,
i_3}$ with the expression~\thetag{ 2.8} of ${\bf Y}_3$, we easily
guess a part of the (inductional) dictionary beween the cases $n=1$
and the case $n \geq 1$. For instance, the three monomials $[\cdot ]
(y_1)^3$, $[\cdot]\, y_1 y_2$ and $[\cdot]\, (y_1 )^2\, y_2$ in ${\bf
Y }_3$ are replaced in ${\bf Y}_{ i_1, i_2, i_3}$ by the following
three sums:
\def5.22}\begin{equation{3.26}\begin{equation}
\small
\sum_{k_1,k_2,k_3}\,
\left[
\cdot
\right]
y_{k_1}y_{k_2}y_{k_3},
\ \ \ \ \ \ \ \ \ \
\sum_{k_1,k_2,k_3}\,
\left[
\cdot
\right]
y_{k_1}y_{k_2,k_3},
\ \ \ \ \
{\rm and}
\ \ \ \ \
\sum_{k_1,k_2,k_3,k_4}\,
\left[
\cdot
\right]
y_{k_1}y_{k_2}y_{k_3,k_4}.
\end{equation}
Similar formal correspondences may be observed for all the monomials
of ${\bf Y}_1$, ${\bf Y}_{i_1}$, of ${\bf Y}_2$, ${\bf Y}_{i_1,i_2}$
and of ${\bf Y}_3$, ${\bf Y}_{i_1,i_2,i_3}$. Generally and
inductively speaking, the monomial
\def5.22}\begin{equation{3.27}\begin{equation}
\left[
\cdot
\right]
\left(
y_{\lambda_1}
\right)^{\mu_1}
\cdots
\left(
y_{\lambda_d}
\right)^{\mu_d}
\end{equation}
appearing in the expression~\thetag{ 2.25} of ${\bf Y }_\kappa$ should
be replaced by a certain multiple sum generalizing~\thetag{ 3.26}.
However, it is necessary to think, to pause and to search for an
appropriate formalism before writing down the desired multiple sum.
The jet variable $y_{ \lambda_1}$ should be replaced by a jet variable
corresponding to a $\lambda_1$-th partial derivative, say $y_{ k_1,
\dots, k_{ \lambda_1}}$, where $k_1, \dots,k_{ \lambda_1}= 1, \dots,
n$. For the moment, to simplify the discussion, we leave out the
presence of a sum of the form $\sum_{ k_1, \dots, k_{ \lambda_1}}$.
The $\mu_1$-th power $\left( y_{ \lambda_1 } \right)^{ \mu_1}$ should
be replaced {\it not}\, by $\left( y_{k_1, \dots, k_{ \lambda_1}}
\right)^{\mu_1}$, but by a product of $\mu_1$ different jet variables
$y_{k_1, \dots, k_{ \lambda_1}}$ of length $\lambda$, {\it with all
indices $k_\alpha = 1, \dots, n$ being distinct}. This rule may be
confirmed by inspecting the expressions of ${\bf Y}_{i_1}$, of ${\bf
Y}_{ i_1, i_2}$ and of ${\bf Y}_{i_1, i_2, i_3}$. So $y_{ k_1, \dots,
k_{ \lambda_1}}$ should be developed as a product of the form
\def5.22}\begin{equation{3.28}\begin{equation}
y_{k_1,\dots,k_{\lambda_1}}\,
y_{k_{\lambda_1+1},\dots,k_{2\lambda_1}}
\cdots \
y_{k_{(\mu_1-1)\lambda_1+1},\dots,k_{\mu_1\lambda_1}},
\end{equation}
where
\def5.22}\begin{equation{3.29}\begin{equation}
k_1,\dots,k_{\lambda_1},\dots,k_{\mu_1\lambda_1}
=
1,\dots,n.
\end{equation}
Consider now the product $\left( y_{\lambda_1 } \right)^{ \mu_1}\left(
y_{\lambda_2 } \right)^{ \mu_2}$. How should it develope in the case
of several independent variables? For instance, in the expression of
${\bf Y}_{i_1,i_2,i_3}$, we have developed the product $(y_1)^2\,y_2$
as $y_{k_1} y_{k_2} y_{k_3,k_4}$. Thus, a reasonable proposal of
formalism would be that the product $\left( y_{\lambda_1 } \right)^{
\mu_1}\left( y_{\lambda_2 } \right)^{ \mu_2}$ should be developed as a
product of the form
\def5.22}\begin{equation{3.30}\begin{equation}
\aligned
&
y_{k_1,\dots,k_{\lambda_1}}\,
y_{k_{\lambda_1+1},\dots,k_{2\lambda_1}}
\cdots \
y_{k_{(\mu_1-1)\lambda_1+1},\dots,k_{\mu_1\lambda_1}} \\
&
y_{k_{\mu_1\lambda_1+1},\dots,k_{\mu_1\lambda_1+\lambda_2}}
\cdots \
y_{k_{\mu_1\lambda_1+(\mu_2-1)\lambda_2+1},\dots,
k_{\mu_1\lambda_1+\mu_2\lambda_2}},
\endaligned
\end{equation}
where
\def5.22}\begin{equation{3.31}\begin{equation}
k_1,\dots, k_{\lambda_1},\dots,k_{\mu_1\lambda_1},\dots,
k_{\mu_1\lambda_1+\mu_2\lambda_2}
=
1,\dots,n.
\end{equation}
However, when trying to write down the development of the general
monomial $\left( y_{\lambda_1 } \right)^{ \mu_1}\left( y_{\lambda_2 }
\right)^{ \mu_2} \cdots \left( y_{\lambda_d } \right)^{ \mu_d}$, we
would obtain the complicated product
\def5.22}\begin{equation{3.32}\begin{equation}
\small
\aligned
&
y_{k_1,\dots,k_{\lambda_1}}\,
y_{k_{\lambda_1+1},\dots,k_{2\lambda_1}}
\cdots \
y_{k_{(\mu_1-1)\lambda_1+1},\dots,k_{\mu_1\lambda_1}}
\\
&
y_{k_{\mu_1\lambda_1+1},\dots,k_{\mu_1\lambda_1+\lambda_2}}
\dots
y_{k_{\mu_1\lambda_1+(\mu_2-1)\lambda_2+1},\dots,
k_{\mu_1\lambda_1+\mu_2\lambda_2}}
\\
&
\dots\dots\dots\dots\dots\dots
\dots\dots\dots\dots\dots\dots
\dots\dots\dots\dots\dots\dots
\\
&
y_{k_{\mu_1\lambda_1+\cdots+\mu_{d-1}\lambda_{d-1}+1},
\dots,k_{\mu_1\lambda_1+\cdots+\mu_{d-1}\lambda_{d-1}+\lambda_d}}
\cdots
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\cdots \
y_{k_{\mu_1\lambda_1+\cdots+\mu_{d-1}\lambda_{d-1}+
(\mu_d-1)\lambda_d+1},\dots,
k_{\mu_1\lambda_1+\cdots+\mu_d\lambda_d}}.
\endaligned
\end{equation}
Essentially, this product is still readable. However, in it, some of
the integers $k_\alpha$ have a too long index $\alpha$, often
involving a sum. Such a length of $\alpha$ would be very inconvenient
in writing down and in reading the general Kronecker symbols $\delta_{
i_1, \dots \dots, i_\lambda }^{ k_{ \alpha_1}, \dots, k_{
\alpha_\lambda}}$ which should appear in the final expression of ${\bf
Y}_{ i_1, \dots, i_\kappa}$. One should read in advance Theorem~3.73
below to observe the presence of such multiple Kronecker symbols.
{\sf Consequently, for $\alpha = 1, \dots, \mu_1 \lambda_1, \dots,
\mu_1 \lambda_1 + \cdots + \mu_d \lambda_d$, we have to denote the
indices $k_\alpha$ differently}.
\def3.33}\begin{notationalconvention{3.33}\begin{notationalconvention}
We denote $d$ collection of $\mu_d$ groups of $\lambda_d$ {\rm (a
priori distinct)} integers $k_\alpha = 1, \dots, n$ by
\def5.22}\begin{equation{3.34}\begin{equation}
\aligned
&
\underbrace{
\underbrace{
k_{1:1:1},\dots,k_{1:1:\lambda_1}}_{\lambda_1},
\dots,
\underbrace{
k_{1:\mu_1:1},\dots,k_{1:\mu_1:\lambda_1}}_{\lambda_1}}_{
\mu_1},
\\
&
\underbrace{
\underbrace{
k_{2:1:1},\dots,k_{2:1:\lambda_2}}_{\lambda_2},
\dots,
\underbrace{
k_{2:\mu_2:1},\dots,k_{2:\mu_2:\lambda_2}}_{\lambda_2}}_{
\mu_2},
\\
&
\text{\bf \ \
\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots
\dots\dots\dots\dots\dots
}
\\
&
\underbrace{
\underbrace{
k_{d:1:1},\dots,k_{d:1:\lambda_d}}_{\lambda_d},
\dots,
\underbrace{
k_{d:\mu_d:1},\dots,k_{d:\mu_d:\lambda_d}}_{\lambda_d}}_{
\mu_d}.
\endaligned
\end{equation}
Correspondingly, we identify the set
\def5.22}\begin{equation{3.35}\begin{equation}
\small
\left\{
1,\dots,\lambda_1,\dots,\mu_1\lambda_1,
\dots\dots,
\mu_1\lambda_1+\mu_2\lambda_2,
\dots\dots,
\mu_1\lambda_1+\mu_2\lambda_2
+
\cdots
+
\mu_d\lambda_d
\right\}
\end{equation}
of all integers $\alpha$ from $1$ to $\mu_1 \lambda_1 + \mu_2
\lambda_2 + \cdots + \mu_d \lambda_d$ with the following specific set
\def5.22}\begin{equation{3.36}\begin{equation}
\small
\{
\underbrace{
\underbrace{
\underbrace{
\underbrace{
1\!\!:\!\!1\!\!:\!\!1,
\dots,
1\!\!:\!\!1\!\!:\!\!\lambda_1}_{\lambda_1},
\dots,
1\!\!:\!\!\mu_1\!\!:\!\!\lambda_1}_{\mu_1\lambda_1},
\dots,
2\!:\!\mu_2\!:\!\lambda_2}_{\mu_1\lambda_1+\mu_2\lambda_2},
\dots,
d\!:\!\mu_d\!:\!\lambda_d}_{\mu_1\lambda_1+\mu_2\lambda_2
+\cdots+\mu_d\lambda_d}
\},
\end{equation}
written in a lexicographic way which emphasizes clearly the
subdivision in $d$ collections of $\mu_d$ groups of $\lambda_d$
integers.
\end{notationalconvention}
With this notation at hand, we see that the development, in
several independent variables, of the general monomial $\left(
y_{\lambda_1 } \right)^{ \mu_1}
\cdots \left( y_{ \lambda_d } \right)^{ \mu_d }$, may be written as
follows:
\def5.22}\begin{equation{3.37}\begin{equation}
\aligned
y_{k_{1:1:1},\dots,k_{1:1:\lambda_1}}
\cdots \
y_{k_{1:\mu_1:1},\dots,k_{1:\mu_1:\lambda_1}}
\cdots \
y_{k_{d:1:1},\dots,k_{d:1:\lambda_d}}
\cdots\cdots\,
y_{k_{d:\mu_d:1},\dots,k_{d:\mu_d:\lambda_d}}.
\endaligned
\end{equation}
Formally speaking, this expression is better than~\thetag{ 3.32}. Using
product symbols, we may even write it under the
slightly more compact form
\def5.22}\begin{equation{3.38}\begin{equation}
\prod_{1\leq\nu_1\leq\mu_1}\,
y_{k_{1:\nu_1:1},\dots,k_{1:\nu_1:\lambda_1}}
\cdots \
\prod_{1\leq\nu_d\leq\mu_d}\,
y_{k_{d:\nu_d:1},\dots,k_{d:\nu_d:\lambda_d}}.
\end{equation}
Now that we have translated the monomial, we may add all the summation
symbols: the general expression of ${\bf Y}_\kappa$ (which generalizes
our three previous examples~\thetag{ 3.26}) will be of the form:
\def5.22}\begin{equation{3.39}\begin{equation}
\aligned
{\bf Y}_\kappa
&
=
\mathcal{Y}_{x^{i_1}\cdots x^{i_\kappa}}
+
\sum_{d=1}^{\kappa+1}
\ \
\sum_{1\leq\lambda_1<\cdots<\lambda_d\leq\kappa}
\ \
\sum_{\mu_1\geq 1,\dots,\mu_d\geq 1}
\
\sum_{
\mu_1\lambda_1
+
\cdots
+
\mu_d\lambda_d\leq \kappa+1}
\\
&
\sum_{k_{1:1:1},\dots,k_{1:1:\lambda_1}=1}^n
\cdots \
\sum_{k_{1:\mu_1:1},\dots,k_{1:\mu_1:\lambda_1}=1}^n
\cdots\cdots \
\sum_{k_{d:1:1},\dots,k_{d:1:\lambda_d}=1}^n
\cdots \
\sum_{k_{d:\mu_d:1},\dots,k_{d:\mu_d:\lambda_d}=1}^n
\\
&
\text{\bf[?]}
\prod_{1\leq\nu_1\leq\mu_1}\,
y_{k_{1:\nu_1:1},\dots,k_{1:\nu_1:\lambda_1}}
\ \cdots \
\prod_{1\leq\nu_d\leq\mu_d}\,
y_{k_{d:\nu_d:1},\dots,k_{d:\nu_d:\lambda_d}}.
\endaligned
\end{equation}
From now on,
up to the end of the article, to be very precise, we will restitute
the bounds $\sum_{ k = 1 }^n$ of all the previously abbreviated sums
$\sum_k$. This is justified by the fact that, since we shall deal in
Section~5 below simultaneously with several independent variables
$(x^1, \dots, x^n)$ and with several dependent variables $(y^1, \dots,
y^m)$, we shall encounter sums $\sum_{ l = 1 }^m$, not to be confused
with sums $\sum_{ k = 1 }^n$.
\subsection*{3.40.~Combinatorics of the Kronecker symbols}
Our next task is to determine what appears inside the brackets {\bf
[?]} of the above equation. We will treat this rather delicate
question very progressively. Inductively, we have to guess how we may
pass from the bracketed term of~\thetag{ 2.25}, namely from
\def5.22}\begin{equation{3.41}\begin{equation}
\aligned
&
\left[
\frac{\kappa\cdots(\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d+1)}
{(\lambda_1!)^{\mu_1}\,\mu_1!
\cdots
(\lambda_d!)^{\mu_d}\,\mu_d!
}
\cdot
\mathcal{Y}_{
x^{\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d}
\,
y^{\mu_1+\cdots+\mu_d}
}
-
\right. \\
& \
\ \ \ \ \ \ \ \ \ \
\left.
-
\frac{\kappa\cdots(
\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d+2)
(\mu_1\lambda_1+\cdots+\mu_d\lambda_d)}
{(\lambda_1!)^{\mu_1}\,\mu_1!
\cdots
(\lambda_d!)^{\mu_d}\,\mu_d!
}
\cdot
\right.
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\cdot
\mathcal{X}_{
x^{\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d+1}
\,
y^{\mu_1+\cdots+\mu_d-1}
}
\Big],
\endaligned
\end{equation}
to the corresponding (still unknown) bracketed term
{\bf [?]}.
First of all, we examine the following term, extracted from the
complete expression of ${\bf Y}_{ i_1, i_2, i_3}$ (first line
of~\thetag{ 3.24}):
\def5.22}\begin{equation{3.42}\begin{equation}
\sum_{k_1=1}^n\,
\left[
\delta_{i_1}^{k_1}\,\mathcal{Y}_{x^{i_2}x^{i_3}y}
+
\delta_{i_2}^{k_1}\,\mathcal{Y}_{x^{i_1}x^{i_3}y}
+
\delta_{i_3}^{k_1}\,\mathcal{Y}_{x^{i_1}x^{i_2}y}
-
\mathcal{ X}_{x^{i_1}x^{i_2}x^{i_3}}^{k_1}
\right]
y_{k_1}.
\end{equation}
Here, the coefficient $\left[ 3\, \mathcal{ Y}_{ x^2 y} - \mathcal{
X}_{ x^3 } \right]$ of the monomial $y_1$ in ${\bf Y}_3$ is replaced
by the above bracketed terms.
Let us precisely analyze the combinatorics. Here, $\mathcal{ X}_{x^3}$ is
replaced by $\mathcal{ X}_{x^{ i_1 }x^{ i_2}x^{i_3}}^{k_1}$, where the
lower indices $i_1, i_2, i_3$ come from ${\bf Y}_{i_1, i_2, i_3}$ and
where the upper index $k_1$ is the summation index. Also, the integer
$3$ in $3\, \mathcal{ Y}_{x^2 y}$ is replaced by a sum of exactly
three terms, each involving a single Kronecker symbol $\delta_i^k$, in
which the lower index is always an index $i= i_1, i_2, i_3$ and in
which the upper index is always equal to the summation index $k_1$.
By the way, more generally, we immediately observe that
all the successive positive integers
\def5.22}\begin{equation{3.43}\begin{equation}
1, 3, 1, 3, 3, 1, 3, 1, 3, 3, 3, 9, 6, 3, 1, 3, 4
\end{equation}
appearing in the formula~\thetag{ 2.8} for ${\bf Y}_3$ are replaced,
in the formula~\thetag{ 3.24} for ${\bf Y}_{i_1, i_2, i_3}$, by sums
of exactly the same number of terms involving Kronecker
symbols. This observation will be a precious guide. Finally, in the
symbol $\delta_i^{k_1}$, if $i$ is chosen among the set $\{ i_1, i_2,
i_3\}$, for instance if $i = i_1$, it follows that the development of
$\mathcal{ Y}_{x^2y}$ necessarily involves the remaining indices, for
instance $\mathcal{ Y}_{x^{i_2}x^{i_3}y}$. Since there are three
choices for $i = i_1, i_2, i_3$, we recover the number $3$.
Next, comparing $\left[ \mathcal{ Y}_{yy}
-2\,\mathcal{ X}_{ xy} \right] (y_1)^2$ with
the term
\def5.22}\begin{equation{3.44}\begin{equation}
\sum_{k_1,k_2=1}^n\,
\left[
\delta_{i_1,\,i_2}^{k_1,k_2}\,\mathcal{Y}_{yy}
-
\delta_{i_1}^{k_1}\,\mathcal{X}_{x^{i_2}y}^{k_1}
-
\delta_{i_2}^{k_1}\,\mathcal{X}_{x^{i_1}y}^{k_1}
\right]
y_{k_1}y_{k_2},
\end{equation}
extracted from the complete expression of ${\bf Y}_{i_1,i_2}$ (second
line of~\thetag{ 3.18}), we learn and we guess that the number of
Kronecker symbols before $\mathcal{ Y}_{x^\gamma y^\delta}$ must be
equal to the number of indices $k_\alpha$ minus $\gamma$. This rule
is confirmed by examining the term (second and third line of~\thetag{
3.24})
\def5.22}\begin{equation{3.45}\begin{equation}
\aligned
\sum_{k_1,k_2}
&
\left[
\delta_{i_1,\,i_2}^{k_1,k_2}\,\mathcal{Y}_{x^{i_3}y^2}
+
\delta_{i_3,\,i_1}^{k_1,k_2}\,\mathcal{Y}_{x^{i_2}y^2}
+
\delta_{i_2,\,i_3}^{k_1,k_2}\,\mathcal{Y}_{x^{i_1}y^2}
-
\right.
\\
& \
\ \ \ \ \
\left.
-
\delta_{i_1}^{k_1}\,\mathcal{X}_{x^{i_2}x^{i_3}y}^{k_2}
-
\delta_{i_2}^{k_1}\,\mathcal{X}_{x^{i_1}x^{i_3}y}^{k_2}
-
\delta_{i_3}^{k_1}\,\mathcal{X}_{x^{i_1}x^{i_2}y}^{k_2}
\right]
y_{k_1}y_{k_2},
\endaligned
\end{equation}
developing $\left[ 3\,\mathcal{ Y}_{xy^2} - 3\, \mathcal{
X}_{x^2y} \right] (y_1)^2$.
Also, we may examine the following term
\def5.22}\begin{equation{3.46}\begin{equation}
\aligned
\sum_{k_1,k_2=1}^n\,
&
\left[
\delta_{i_1,\,i_2}^{k_1,k_2}\,\mathcal{Y}_{x^{i_3}x^{i_4}y^2}
+
\delta_{i_1,\,i_3}^{k_1,k_2}\,\mathcal{Y}_{x^{i_2}x^{i_4}y^2}
+
\delta_{i_1,\,i_4}^{k_1,k_2}\,\mathcal{Y}_{x^{i_2}x^{i_3}y^2}
+
\right.
\\
& \
\ \ \ \ \
\left.
+
\delta_{i_2,\,i_3}^{k_1,k_2}\,\mathcal{Y}_{x^{i_1}x^{i_4}y^2}
+
\delta_{i_2,\,i_4}^{k_1,k_2}\,\mathcal{Y}_{x^{i_1}x^{i_3}y^2}
+
\delta_{i_3,\,i_4}^{k_1,k_2}\,\mathcal{Y}_{x^{i_1}x^{i_2}y^2}
-
\right.
\\
& \
\ \ \ \ \
\left.
-
\delta_{i_1}^{k_1}\,\mathcal{X}_{x^{i_2}x^{i_3}x^{i_4}y}^{k_1}
-
\delta_{i_2}^{k_1}\,\mathcal{X}_{x^{i_1}x^{i_2}x^{i_3}y}^{k_1}
-
\delta_{i_3}^{k_1}\,\mathcal{X}_{x^{i_1}x^{i_2}x^{i_4}y}^{k_1}
-
\right.
\\
& \
\ \ \ \ \
\left.
-
\delta_{i_4}^{k_1}\,\mathcal{X}_{x^{i_1}x^{i_2}x^{i_3}y}^{k_1}
\right]
y_{k_1}y_{k_2},
\endaligned
\end{equation}
extracted from ${\bf Y}_{ i_1, i_2, i_3, i_4}$ and developing
$\left[ 6\, \mathcal{ Y}_{ x^2 y^2} - 4\, \mathcal{ X}_{ x^3 y}
\right] (y_1)^2$. We would like to mention that we have not written
the complete expression of ${\bf Y}_{ i_1, i_2, i_3, i_4}$, because it
would cover two and a half printed pages.
By inspecting the way how the indices are permuted in the multiple
Kronecker symbols of the first two lines of this expression~\thetag{
3.46}, we observe that the six terms correspond exactly to the six
possible choices of two complementary ordered couples of integers in
the set $\{ 1, 2, 3, 4\}$, namely
\def5.22}\begin{equation{3.47}\begin{equation}
\aligned
&
\{1,2\}\cup\{3,4\},
\ \ \ \ \ \ \ \
\{1,3\}\cup\{2,4\},
\ \ \ \ \ \ \ \
\{1,4\}\cup\{2,3\},
\\
&
\{2,3\}\cup\{1,4\},
\ \ \ \ \ \ \ \
\{2,4\}\cup\{1,3\},
\ \ \ \ \ \ \ \
\{3,4\}\cup\{1,2\}.
\endaligned
\end{equation}
At this point, we start to devise the general combinatorics. Before
proceeding further, we need some notation.
\subsection*{ 3.48.~Permutation groups}
For every $p \in \N$ with $p \geq 1$, we denote by $\mathfrak{ S}_p$ the
full permutation group of the set $\{ 1, 2, \dots, p-1, p\}$. Its
cardinal equals $p!$. The letters $\sigma$ and
$\tau$ will be used to denote an
element of $\mathfrak{ S}_p$. If $p \geq 2$, and if $q \in \N$
satisfies $1\leq q \leq p-1$, we denote by $\mathfrak{ S}_p^q$ the
subset of permutations $\sigma \in \mathfrak{ S}_p$ satisfying
the two collections of inequalities
\def5.22}\begin{equation{3.49}\begin{equation}
\sigma(1)<\sigma(2)<\cdots<\sigma(q)
\ \ \ \ \ \ \ \ \
{\rm and}
\ \ \ \ \ \ \ \ \
\sigma(q+1)<\sigma(q+2)<\cdots<\sigma(p).
\end{equation}
The cardinal of $\mathfrak{ S}_p^q$
equals $C_p^q = \frac{ p!}{ q! \ (p-q) !}$.
\def3.50}\begin{lemma{3.50}\begin{lemma}
For $\kappa \geq 1$, the development of~\thetag{ 2.20} to several
independent variables $(x^1, \dots, x^n)$ is{\rm :}
\def5.22}\begin{equation{3.51}\begin{equation}
\small
\aligned
&
{\bf Y}_{i_1,i_2,\dots,i_\kappa}
=
\mathcal{Y}_{x^{i_1}x^{i_2}\cdots x^{i_\kappa}}
+
\sum_{k_1=1}^n\,
\left[
\sum_{\tau\in\mathfrak{S}_\kappa^1}\,
\delta_{i_{\tau(1)}}^{k_1}\,\mathcal{Y}_{x^{i_{\tau(2)}}
\cdots x^{i_{\tau(\kappa)}}y}
-
\mathcal{X}_{x^{i_1}x^{i_2}\cdots x^{i_{\kappa}}}^{k_1}
\right]
y_{k_1}
+ \\
& \
\ \ \ \ \
+
\sum_{k_1,k_2=1}^n\,
\left[
\sum_{\tau\in\mathfrak{S}_\kappa^2}\,
\delta_{i_{\tau(1)},i_{\tau(2)}}^{k_1,\ \ \ k_2}\,
\mathcal{Y}_{x^{i_{\tau(3)}}\cdots x^{i_{\tau(\kappa)}}y^2}
-
\sum_{\tau\in\mathfrak{S}_\kappa^1}\,
\delta_{i_{\tau(1)}}^{k_1}\,
\mathcal{X}_{x^{i_{\tau(2)}}\cdots x^{i_{\tau(\kappa)}}y}^{k_2}
\right]
y_{k_1}y_{k_2}
+
\\
& \
\ \ \ \ \
+
\sum_{k_1, k_2, k_3=1}^n
\left[
\sum_{\tau\in\mathfrak{S}_\kappa^3}\,
\delta_{i_{\tau(1)},i_{\tau(2)},i_{\tau(3)}}^{
k_1,\ \ \ k_2,\ \ \ k_3}\,
\mathcal{Y}_{x^{i_{\tau(4)}}\cdots x^{i_{\tau(\kappa)}}y^3}
-
\right.
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \
\left.
-
\sum_{\tau\in\mathfrak{S}_\kappa^2}\,
\delta_{i_{\tau(1),i_{\tau(2)}}}^{k_1,\ \ \ k_2}
\mathcal{X}_{x^{i_{\tau(3)}}\cdots x^{i_{\tau(\kappa)}}y^2}^{k_3}
\right]
y_{k_1}y_{k_2}y_{k_3}
+ \\
& \
\ \ \ \ \
+
\cdots\cdots
+
\\
& \
\ \ \ \ \
+
\sum_{k_1,\dots,k_\kappa=1}^n
\left[
\delta_{i_1,\dots,\ i_\kappa}^{k_1,\dots,k_\kappa}\,
\mathcal{Y}_{y^\kappa}
-
\sum_{\tau\in\mathfrak{S}_\kappa^{\kappa-1}}\,
\delta_{i_{\tau(1)},\dots,i_{\tau(\kappa-1)}}^{
k_1,\dots\dots,k_{\kappa-1}}\,
\mathcal{X}_{x^{i_{\tau(\kappa)}}y^{\kappa-1}}^{k_\kappa}
\right]
y_{k_1}\cdots y_{k_\kappa}
+ \\
& \
\ \ \ \ \
+
\sum_{k_1,\dots,k_\kappa,k_{\kappa+1}=1}^n
\left[
-
\delta_{i_1,\dots,\ i_\kappa}^{k_1,\dots,k_\kappa}\,
\mathcal{X}_{y^\kappa}^{k_{\kappa+1}}
\right]
y_{k_1}\cdots y_{k_\kappa} y_{k_{\kappa+1}}
+
{\sf remainder}.
\endaligned
\end{equation}
Here, the term {\sf remainder} collects all
remaining monomials in the pure jet variables
$y_{ k_1, \dots, k_\lambda }$.
\end{lemma}
\subsection*{3.52.~Continuation}
Thus, we have devised how the part of ${\bf Y}_{i_1,\dots, i_\kappa}$
which involves only the jet variables $y_{k_\alpha}$ must be written.
To proceed further, we shall examine the following term, extracted
from ${\bf Y}_{i_1,i_2,i_3}$ (lines 12 and 13 of~\thetag{ 3.24})
\def5.22}\begin{equation{3.53}\begin{equation}
\aligned
& \
\ \ \ \ \
\sum_{k_1,k_2,k_3,k_4}\,
\left[
-
\delta_{i_1,\,i_2,\,i_3}^{k_1,k_2,k_3}\,\mathcal{X}_{y^2}^{k_4}
-
\delta_{i_1,\,i_2,\,i_3}^{k_2,k_3,k_1}\,\mathcal{X}_{y^2}^{k_4}
-
\delta_{i_1,\,i_2,\,i_3}^{k_3,k_2,k_1}\,\mathcal{X}_{y^2}^{k_4}
-
\right.
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.
-
\delta_{i_1,\,i_2,\,i_3}^{k_3,k_4,k_1}\,\mathcal{X}_{y^2}^{k_2}
-
\delta_{i_1,\,i_2,\,i_3}^{k_3,k_1,k_4}\,\mathcal{X}_{y^2}^{k_2}
-
\delta_{i_1,\,i_2,\,i_3}^{k_1,k_3,k_4}\,\mathcal{X}_{y^2}^{k_2}
\right]
y_{k_1}y_{k_2}y_{k_3,k_4},
\endaligned
\end{equation}
which developes the term $\left[ - 6\,\mathcal{ X}_{ y^2} \right]
(y_1)^2 y_2$ of ${\bf Y}_3$ (third line of~\thetag{ 2.8}). During the
computation which led us to the final expression~\thetag{ 3.24}, we
organized the formula in order that, in the six Kronecker symbols, the
lower indices $i_1,i_2,i_3$ are all written in the same order. But
then, {\it what is the rule for the appearance of the four upper
indices $k_1, k_2, k_3, k_4$}?
In April 2001, we discovered the rule by inspecting both~\thetag{
3.53} and the following complicated term, extracted from the complete
expression of ${\bf Y}_{i_1,i_2,i_3,i_4}$ written in one of our
manuscripts:
\def5.22}\begin{equation{3.54}\begin{equation}
\aligned
\sum_{k_1,k_2,k_3}\,
&
\left[
\delta_{i_1, \ i_2, \ i_3}^{k_1,k_2,k_3}\,\mathcal{Y}_{x^{i_4}y^2}
+
\delta_{i_1, \ i_2, \ i_3}^{k_2,k_1,k_3}\,\mathcal{Y}_{x^{i_4}y^2}
+
\delta_{i_1, \ i_2, \ i_3}^{k_2,k_3,k_1}\,\mathcal{Y}_{x^{i_4}y^2}
+
\right.
\\
& \
\ \
+
\delta_{i_1, \ i_2, \ i_4}^{k_1,k_2,k_3}\,\mathcal{Y}_{x^{i_3}y^2}
+
\delta_{i_1, \ i_2, \ i_4}^{k_2,k_1,k_3}\,\mathcal{Y}_{x^{i_3}y^2}
+
\delta_{i_1, \ i_2, \ i_4}^{k_2,k_3,k_1}\,\mathcal{Y}_{x^{i_3}y^2}
+ \\
& \
\ \
+
\delta_{i_1, \ i_3, \ i_4}^{k_1,k_2,k_3}\,\mathcal{Y}_{x^{i_2}y^2}
+
\delta_{i_1, \ i_3, \ i_4}^{k_2,k_1,k_3}\,\mathcal{Y}_{x^{i_2}y^2}
+
\delta_{i_1, \ i_3, \ i_4}^{k_2,k_3,k_1}\,\mathcal{Y}_{x^{i_2}y^2}
+ \\
& \
\ \
+
\delta_{i_2, \ i_3, \ i_4}^{k_1,k_2,k_3}\,\mathcal{Y}_{x^{i_1}y^2}
+
\delta_{i_2, \ i_3, \ i_4}^{k_2,k_1,k_3}\,\mathcal{Y}_{x^{i_1}y^2}
+
\delta_{i_2, \ i_3, \ i_4}^{k_2,k_3,k_1}\,\mathcal{Y}_{x^{i_1}y^2}
- \\
& \
\ \
-
\delta_{i_1,\ i_2}^{k_1,k_2}\,\mathcal{X}_{x^{i_3}x^{i_4}y}^{k_3}
-
\delta_{i_1,\ i_2}^{k_2,k_1}\,\mathcal{X}_{x^{i_3}x^{i_4}y}^{k_3}
-
\delta_{i_1,\ i_2}^{k_2,k_3}\,\mathcal{X}_{x^{i_3}x^{i_4}y}^{k_1}
- \\
& \
\ \
-
\delta_{i_1,\ i_3}^{k_1,k_2}\,\mathcal{X}_{x^{i_2}x^{i_4}y}^{k_3}
-
\delta_{i_1,\ i_3}^{k_2,k_1}\,\mathcal{X}_{x^{i_2}x^{i_4}y}^{k_3}
-
\delta_{i_1,\ i_3}^{k_2,k_3}\,\mathcal{X}_{x^{i_2}x^{i_4}y}^{k_1}
- \\
& \
\ \
-
\delta_{i_1,\ i_4}^{k_1,k_2}\,\mathcal{X}_{x^{i_2}x^{i_3}y}^{k_3}
-
\delta_{i_1,\ i_4}^{k_2,k_1}\,\mathcal{X}_{x^{i_2}x^{i_3}y}^{k_3}
-
\delta_{i_1,\ i_4}^{k_2,k_3}\,\mathcal{X}_{x^{i_2}x^{i_3}y}^{k_1}
- \\
& \
\ \
-
\delta_{i_2,\ i_3}^{k_1,k_2}\,\mathcal{X}_{x^{i_1}x^{i_4}y}^{k_3}
-
\delta_{i_2,\ i_3}^{k_2,k_1}\,\mathcal{X}_{x^{i_1}x^{i_4}y}^{k_3}
-
\delta_{i_2,\ i_3}^{k_2,k_3}\,\mathcal{X}_{x^{i_1}x^{i_4}y}^{k_1}
- \\
& \
\ \
-
\delta_{i_2,\ i_4}^{k_1,k_2}\,\mathcal{X}_{x^{i_1}x^{i_3}y}^{k_3}
-
\delta_{i_2,\ i_4}^{k_2,k_1}\,\mathcal{X}_{x^{i_1}x^{i_3}y}^{k_3}
-
\delta_{i_2,\ i_4}^{k_2,k_3}\,\mathcal{X}_{x^{i_1}x^{i_3}y}^{k_1}
- \\
& \
\ \
\left.
-
\delta_{i_3,\ i_4}^{k_1,k_2}\,\mathcal{X}_{x^{i_1}x^{i_2}y}^{k_3}
-
\delta_{i_3,\ i_4}^{k_2,k_1}\,\mathcal{X}_{x^{i_1}x^{i_2}y}^{k_3}
-
\delta_{i_3,\ i_4}^{k_2,k_3}\,\mathcal{X}_{x^{i_1}x^{i_2}y}^{k_1}
\right]
y_{k_1}y_{k_2,k_3}.
\endaligned
\end{equation}
This sum developes the term $\left[ 12\, \mathcal{ Y}_{xy^2} -
18\,\mathcal{ X}_{ x^2 y} \right]y_1y_2$ of
${\bf Y}_3$ (third line of~\thetag{
2.9}). Let us explain what are the formal rules.
In the bracketed terms of~\thetag{ 3.53}, there are no permutation of
the indices $i_1,i_2,i_3$, but there is a certain unknown subset of
all the permutations of the four indices $k_1,k_2,k_3,k_4$. In the
bracketed terms of~\thetag{ 3.54}, two combinatorics are present:
\smallskip
\begin{itemize}
\item[$\bullet$]
there are some permutations of the indices $i_1,i_2,i_3,i_4$ and
\item[$\bullet$]
there are some permutations of the indices $k_1,k_2,k_3$.
\end{itemize}
\smallskip
Here, the permutations of the indices $i_1,i_2,i_3,i_4$ are easily
guessed, since they are the same as the permutations which were
introduced in \S3.48 above. Indeed, in the first four
lines of~\thetag{ 3.54}, we
see the four decompositions
\def5.22}\begin{equation{3.55}\begin{equation}
\{i_1,i_2,i_3\}\cup\{i_4\},
\ \ \ \ \ \ \
\{i_1,i_2,i_4\}\cup\{i_3\},
\ \ \ \ \ \ \
\{i_1,i_3,i_4\}\cup\{i_2\},
\ \ \ \ \ \ \
\{i_2,i_3,i_4\}\cup\{i_1\},
\end{equation}
of the set $\{i_1,i_2,i_3,i_4\}$, and in the
last six lines of~\thetag{ 3.54}, we see the
six decompositions
\def5.22}\begin{equation{3.56}\begin{equation}
\aligned
&
\{i_1,i_2\}\cup\{i_3,i_4\},
\ \ \ \ \ \ \
\{i_1,i_3\}\cup\{i_2,i_4\},
\ \ \ \ \ \ \
\{i_1,i_4\}\cup\{i_2,i_3\},
\\
&
\{i_2,i_3\}\cup\{i_1,i_4\},
\ \ \ \ \ \ \
\{i_2,i_4\}\cup\{i_1,i_3\},
\ \ \ \ \ \ \
\{i_3,i_4\}\cup\{i_1,i_2\},
\endaligned
\end{equation}
so that~\thetag{ 3.54} may be written under the form
\def5.22}\begin{equation{3.57}\begin{equation}
\small
\aligned
\sum_{k_1,k_2,k_3}\,
\left[
\sum_{\tau\in\mathfrak{S}_4^3}\,
\sum_{\sigma\in\text{\bf ?}}\,
\delta_{i_{\tau(1)},i_{\tau(2)},i_{\tau(3)}}^{
k_{\tau(1)},k_{\tau(2)},k_{\tau(3)}}\,
\mathcal{Y}_{x^{i_{\tau(4)}}y^2}
-
\sum_{\tau\in\mathfrak{S}_4^2}\,
\sum_{\sigma\in\text{\bf ?}}\,
\delta_{i_{\tau(1)},i_{\tau(2)}}^{
k_{\tau(1)},k_{\tau(2)}}\,
\mathcal{X}_{x^{i_{\tau(3)}}x^{i_{\tau(4)}}y}^{k_{\tau(3)}}
\right]
y_{k_1}y_{k_2,k_3},
\endaligned
\end{equation}
where in the two above sums $\sum_{ \sigma \in \text{\bf ?}}$, the
letter $\sigma$ denotes a permutation of the set $\{1,2,3\}$ and where
the sign {\bf ?} refers to two (still unknown) subset of the full
permutation group $\mathfrak{ S}_3$. {\it The only remaining question
is to determine how the indices $k_\alpha$ are permuted in~\thetag{
3.53} and in~\thetag{ 3.54}}.
The answer may be guessed by looking at the permutations of the set
$\{k_1,k_2,k_3,k_4\}$ which stabilize the monomial
$y_{k_1}y_{k_2}y_{k_3,k_4}$ in~\thetag{ 3.53}: we clearly have the
following four symmetry relations between monomials:
\def5.22}\begin{equation{3.58}\begin{equation}
y_{k_1}y_{k_2}y_{k_3,k_4}
\equiv
y_{k_2}y_{k_1}y_{k_3,k_4}
\equiv
y_{k_1}y_{k_2}y_{k_4,k_3}
\equiv
y_{k_2}y_{k_1}y_{k_4,k_3},
\end{equation}
and nothing more.
Then the number $6$ of bracketed terms in~\thetag{ 3.53}
is exactly equal to the cardinal $24 = 4!$ of the full permutation
group of the set $\{k_1,k_2,k_3,k_4\}$ divided by the number $4$ of
these symmetry relations. The set of permutations $\sigma$ of
$\{1,2,3,4\}$ satisfying these symmetry relations
\def5.22}\begin{equation{3.59}\begin{equation}
y_{k_{\sigma(1)}}y_{k_{\sigma(2)}}
y_{k_{\sigma(3)},k_{\sigma(4)}}
\equiv
y_{k_1}y_{k_2}y_{k_3,k_4}
\end{equation}
consitutes a subgroup of $\mathfrak{ S}_4$ which we will denote by
$\mathfrak{ H}_4^{(2,1),(1,2)}$. Furthermore, the coset
\def5.22}\begin{equation{3.60}\begin{equation}
\mathfrak{ F}_4^{(2,1),(1,2)}
:=
\mathfrak{ S}_4 / \mathfrak{ H}_4^{(2,1),(1,2)}
\end{equation}
possesses the six representatives
\def5.22}\begin{equation{3.61}\begin{equation}
\small
\aligned
\left(
\begin{array}{cccc}
1 & 2 & 3 & 4 \\
1 & 2 & 3 & 4 \\
\end{array}
\right),
\ \ \ \ \ \ \ \ \ \
\left(
\begin{array}{cccc}
1 & 2 & 3 & 4 \\
2 & 3 & 1 & 4 \\
\end{array}
\right),
\ \ \ \ \ \ \ \ \ \
\left(
\begin{array}{cccc}
1 & 2 & 3 & 4 \\
3 & 2 & 1 & 4 \\
\end{array}
\right),
\\
\left(
\begin{array}{cccc}
1 & 2 & 3 & 4 \\
3 & 4 & 1 & 2 \\
\end{array}
\right),
\ \ \ \ \ \ \ \ \ \
\left(
\begin{array}{cccc}
1 & 2 & 3 & 4 \\
3 & 1 & 4 & 2 \\
\end{array}
\right),
\ \ \ \ \ \ \ \ \ \
\left(
\begin{array}{cccc}
1 & 2 & 3 & 4 \\
1 & 3 & 4 & 2 \\
\end{array}
\right), \\
\endaligned
\end{equation}
which exactly appear as the permutations of the upper indices of our
example~\thetag{ 3.53}. Of course, the question arises whether the
choice of such six representatives in the quotient $\mathfrak{ S}_4 /
\mathfrak{ H}_4^{ (2,1), (1,2)}$ is legitimate.
Fortunately, we observe that after conjugation by any permutation
$\sigma \in \mathfrak{ H }_4^{ (2,1), (1,2)}$, we do not perturb any of
the six terms of~\thetag{ 3.53}, for instance the third term
of~\thetag{ 3.53} is not perturbed, as shown by the following
computation
\def5.22}\begin{equation{3.62}\begin{equation}
\small
\aligned
&
\sum_{k_1,k_2,k_3,k_4}
\left[
-
\delta_{i_1, \ \ \ \ i_2, \ \ \ \ i_3}^{
k_{\sigma(3)}, k_{\sigma(2)}, k_{\sigma(1)}}\,
\mathcal{X}_{y^2}^{k_{\sigma(4)}}
\right]
y_{k_1}y_{k_2}y_{k_3,k_4}
= \\
& \
\ \ \ \ \
=
\sum_{k_1,k_2,k_3,k_4}
\left[
-
\delta_{i_1, \ i_2, \ i_3}^{
k_3, k_2, k_1}\,
\mathcal{X}_{y^2}^{k_{\sigma(4)}}
\right]
y_{k_{\sigma^{-1}(1)}}
y_{k_{\sigma^{-1}(2)}}
y_{k_{\sigma^{-1}(3)},k_{\sigma^{-1}(4)}}
\\
& \
\ \ \ \ \
=
\sum_{k_1,k_2,k_3,k_4}
\left[
-
\delta_{i_1, \ i_2, \ i_3}^{
k_3, k_2, k_1}\,
\mathcal{X}_{y^2}^{k_{\sigma(4)}}
\right]
y_{k_1}y_{k_2}y_{k_3,k_4}
\endaligned
\end{equation}
thanks to the symmetry~\thetag{ 3.59}. Thus, as expected, the choice of
$6$ arbitrary representatives $\sigma \in \mathfrak{ F}_4^{(2,1), (1,
2)}$ in the bracketed terms of~\thetag{ 3.53} is free.
In conclusion, we have shown that~\thetag{ 3.53} may
be written under the form:
\def5.22}\begin{equation{3.63}\begin{equation}
\aligned
\sum_{k_1,k_2,k_3,k_4}\,
\left[
-
\sum_{\sigma\in\mathfrak{F}_4^{(2,1),(1,2)}}\,
\delta_{i_1,\ \ \ \ i_2,\ \ \ \ i_3}^{
k_{\sigma(1)},k_{\sigma(2)},k_{\sigma(3)}}\,
\mathcal{X}_{y^2}^{k_{\sigma(4)}}
\right]
y_{k_1}y_{k_2}y_{k_3,k_4},
\endaligned
\end{equation}
This rule is confirmed by inspecting~\thetag{ 3.54} (as
well as all the other terms of ${\bf Y}_{i_1, i_2, i_3}$ and
of ${\bf Y}_{ i_1, i_2, i_3, i_4}$). Indeed, the
permutations $\sigma$ of the set $\{k_1, k_2, k_3\}$ which stabilize the
monomial $y_{k_1} y_{k_2, k_3}$ consist just of the identity
permutation and the transposition of $k_2$ and $k_3$. The coset
$\mathfrak{ S }_3 / \mathfrak{ H }_3^{ (1,1), (1,2)}$ has the three
representatives
\def5.22}\begin{equation{3.64}\begin{equation}
\small
\aligned
\left(
\begin{array}{ccc}
1 & 2 & 3 \\
1 & 2 & 3 \\
\end{array}
\right),
\ \ \ \ \ \ \ \ \ \
\left(
\begin{array}{ccc}
1 & 2 & 3 \\
2 & 1 & 3 \\
\end{array}
\right),
\ \ \ \ \ \ \ \ \ \
\left(
\begin{array}{cccc}
1 & 2 & 3 \\
2 & 3 & 1 \\
\end{array}
\right),
\\
\endaligned
\end{equation}
which appear in the upper index position of each of the ten lines
of~\thetag{ 3.54}. It follows that~\thetag{ 3.54}
may be written under the form
\def5.22}\begin{equation{3.65}\begin{equation}
\aligned
\sum_{k_1,k_2,k_3}\,
&
\left[
\sum_{\tau\in\mathfrak{S}_4^3}\,
\sum_{\sigma\in\mathfrak{F}_3^{(1,1),(1,2)}}\,
\delta_{i_{\tau(1)},i_{\tau(2)},i_{\tau(3)}}^{
k_{\sigma(1)},k_{\sigma(2)},k_{\sigma(3)}}\,
\mathcal{Y}_{x^{i_{\tau(4)}}y^2}
-
\right.
\\
& \
\left.
-
\sum_{\sigma\in\mathfrak{S}_4^2}\,
\sum_{\tau\in\mathfrak{F}_3^{(1,1),(1,2)}}\,
\delta_{i_{\tau(1)},i_{\tau(2)}}^{
k_{\sigma(1)},k_{\sigma(2)}}\,
\mathcal{X}_{x^{i_{\tau(3)}}
x^{i_{\tau(4)}}y}^{k_{\sigma(3)}}
\right]
y_{k_1}y_{k_2,k_3}.
\endaligned
\end{equation}
\subsection*{ 3.66.~General complete expression of ${\bf Y}_{i_1, \dots,
i_\kappa}$} As in the incomplete expression~\thetag{ 3.39} of ${\bf
Y}_{i_1, \dots, i_\kappa}$, consider integers $1\leq \lambda_1 <
\cdots < \lambda_d \leq \kappa$ and $\mu_1\geq 1, \dots, \mu_d\geq 1$
satisfying $\mu_1 \lambda_1 + \cdots + \mu_d \lambda_d \leq \kappa +
1$. By $\mathfrak{ H}_{ \mu_1 \lambda_1 + \cdots + \mathfrak{ H}_{
\mu_d \lambda_d}}$, we denote the subgroup of permutations $\tau \in
\mathfrak{ S}_{\mu_1 \lambda_1 + \cdots + \mathfrak{ H}_{ \mu_d
\lambda_d}}$ that leave unchanged the
general monomial~\thetag{ 3.38}, namely
that satisfy
\def5.22}\begin{equation{3.67}\begin{equation}
\aligned
&
\prod_{1\leq\nu_1\leq\mu_1}\,
y_{k_{\sigma(1:\nu_1:1)},\dots,k_{\sigma(1:\nu_1:\lambda_1)}}
\cdots \
\prod_{1\leq\nu_d\leq\mu_d}\,
y_{k_{\sigma(d:\nu_d:1)},\dots,k_{\sigma(d:\nu_d:\lambda_d)}}
= \\
&
=
\prod_{1\leq\nu_1\leq\mu_1}\,
y_{k_{1:\nu_1:1},\dots,k_{1:\nu_1:\lambda_1}}
\cdots \
\prod_{1\leq\nu_d\leq\mu_d}\,
y_{k_{d:\nu_d:1},\dots,k_{d:\nu_d:\lambda_d}}.
\endaligned
\end{equation}
The structure of this group may be described as follows. For every $e
= 1, \dots, d$, an arbitrary permutation $\sigma$ of the set
\def5.22}\begin{equation{3.68}\begin{equation}
\{
\underbrace{
\underbrace{
e\!:1\!:\!1, \dots, e\!:1\!:\!\lambda_e}_{\lambda_e},
\underbrace{
e\!:2\!:\!1, \dots, e\!:2\!:\!\lambda_e}_{\lambda_e},
\cdots,
\underbrace{
e\!:\mu_e\!:\!1, \dots, e\!:\mu_e\!:\!\lambda_e}_{\lambda_e}}_{
\mu_e}
\}
\end{equation}
which leaves unchanged the monomial
\def5.22}\begin{equation{3.69}\begin{equation}
\prod_{1\leq\nu_e\leq\mu_e}\,
y_{k_{\sigma(e:\nu_e:1)},\dots,k_{\sigma(e:\nu_e:\lambda_e)}}=
\prod_{1\leq\nu_e\leq\mu_e}\,
y_{k_{e:\nu_e:1},\dots,k_{e:\nu_e:\lambda_e}}.
\end{equation}
uniquely decomposes as the composition of
\smallskip
\begin{itemize}
\item[$\bullet$]
$\mu_e$ arbitrary permutations of the $\mu_e$ groups
of $\lambda_e$ integers
$\{e\!:\!\nu_e\!:\!1,
\dots,e\!:\!\nu_e\!:\!\lambda_e\}$,
of total cardinal $(\lambda_e!)^{\mu_e}$;
\item[$\bullet$]
an arbitrary permutation
between these $\mu_e$ groups, of total
cardinal $\mu_e!$.
\end{itemize}
\smallskip
Consequently
\def5.22}\begin{equation{3.70}\begin{equation}
{\rm Card}
\left(
\mathfrak{H}_{\mu_1\lambda_1+\cdots+\mu_d\lambda_d}^{
(\mu_1,\lambda_1),\dots,(\mu_d,\lambda_d)}
\right)
=
\mu_1!(\lambda_1!)^{\mu_1}\cdots
\mu_d!(\lambda_d!)^{\mu_d},
\end{equation}
Finally, define the coset
\def5.22}\begin{equation{3.71}\begin{equation}
\mathfrak{F}_{\mu_1\lambda_1+\cdots+\mu_d\lambda_d}^{
(\mu_1,\lambda_1),\dots,(\mu_d,\lambda_d)}
:=
\mathfrak{S}_{\mu_1\lambda_1+\cdots+\mu_d\lambda_d}
/
\mathfrak{H}_{\mu_1\lambda_1+\cdots+\mu_d\lambda_d}^{
(\mu_1,\lambda_1),\dots,(\mu_d,\lambda_d)}
\end{equation}
with
\def5.22}\begin{equation{3.72}\begin{equation}
\aligned
{\rm Card}
\left(
\mathfrak{F}_{\mu_1\lambda_1+\cdots+\mu_d\lambda_d}^{
(\mu_1,\lambda_1),\dots,(\mu_d,\lambda_d)}
\right)
&
=
\frac{
{\rm Card}
\left(
\mathfrak{S}_{\mu_1\lambda_1+\cdots+\mu_d\lambda_d}
\right)}{
{\rm Card}
\left(
\mathfrak{H}_{\mu_1\lambda_1+\cdots+\mu_d\lambda_d}^{
(\mu_1,\lambda_1),\dots,(\mu_d,\lambda_d)}
\right)}
\\
&
=
\frac{(\mu_1\lambda_1+\cdots+\mu_d\lambda_d)!}{
\mu_1!(\lambda_1!)^{\mu_1}\cdots
\mu_d!(\lambda_d!)^{\mu_d}}.
\endaligned
\end{equation}
In conclusion, by means of this formalism, we may write down the
complete generalization of Theorem~2.24 to several independent
variables.
\def5.12}\begin{theorem{3.73}\begin{theorem}
For every $\kappa \geq 1$ and for every choice of $\kappa$ indices
$i_1,\dots, i_\kappa$ in the set $\{ 1, 2, \dots, n\}$, the general
expression of ${\bf Y}_{i_1, \dots, i_\kappa}$ is as follows{\rm :}
\def5.22}\begin{equation{3.74}\begin{equation}
\small
\boxed{
\aligned
{\bf Y}_{i_1, \dots, i_\kappa}
&
=
\mathcal{Y}_{x^{i_1}\cdots x^{i_\kappa}}
+
\sum_{d=1}^{\kappa+1}
\ \
\sum_{1\leq\lambda_1<\cdots<\lambda_d\leq\kappa}
\ \
\sum_{\mu_1\geq 1,\dots,\mu_d\geq 1}
\
\sum_{
\mu_1\lambda_1
+
\cdots
+
\mu_d\lambda_d\leq \kappa+1}
\\
&
\sum_{k_{1:1:1},\dots,k_{1:1:\lambda_1}=1}^n
\cdots \
\sum_{k_{1:\mu_1:1},\dots,k_{1:\mu_1:\lambda_1}=1}^n
\cdots\cdots \
\sum_{k_{d:1:1},\dots,k_{d:1:\lambda_d}=1}^n
\cdots \
\sum_{k_{d:\mu_d:1},\dots,k_{d:\mu_d:\lambda_d}=1}^n
\\
&
\left[
\aligned
&
\sum_{\sigma\in\mathfrak{F}_{\mu_1\lambda_1+\cdots+\mu_d\lambda_d}^{
(\mu_1,\lambda_1),\dots,(\mu_d,\lambda_d)}}
\
\sum_{\tau\in\mathfrak{S}_\kappa^{
\mu_1\lambda_1+\cdots+\mu_d\lambda_d}}
\\
& \
\ \ \ \ \
\delta_{i_{\tau(1)},\dots,i_{\tau(\mu_1\lambda_1)},\dots,
i_{\tau(\mu_1\lambda_1+\cdots+\mu_d\lambda_d)}}^{
k_{\sigma(1:1:1)},\dots,k_{\sigma(1:\mu_1:\lambda_1)},
\dots,k_{\sigma(d:\mu_d:\lambda_d)}}\,
\frac{\partial^{\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d+
\mu_1+\cdots+\mu_d}
\mathcal{Y}}{
\partial x^{i_{\tau(\mu_1\lambda_1+\cdots+\mu_d\lambda_d+1)}}\cdots
\partial x^{i_{\tau(\kappa)}}
\left(\partial y\right)^{\mu_1+\cdots+\mu_d}}
- \\
&
-
\sum_{\sigma\in\mathfrak{F}_{\mu_1\lambda_1+\cdots+\mu_d\lambda_d}^{
(\mu_1,\lambda_1),\dots,(\mu_d,\lambda_d)}}
\
\sum_{\tau\in\mathfrak{S}_\kappa^{
\mu_1\lambda_1+\cdots+\mu_d\lambda_d-1}}
\\
& \
\ \ \ \ \
\delta_{i_{\tau(1)},\dots,i_{\tau(\mu_1\lambda_1)},\dots,
i_{\tau(\mu_1\lambda_1+\cdots+\mu_d\lambda_d-1)}}^{
k_{\sigma(1:1:1)},\dots,k_{\sigma(1:\mu_1:\lambda_1)},
\dots,k_{\sigma(d:\mu_d:\lambda_d-1)}}\,
\frac{\partial^{\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d
+\mu_1+\cdots+\mu_d}\mathcal{X}^{k_{\sigma(d:\mu_d:\lambda_d)}}}{
\partial x^{i_{\tau(\mu_1\lambda_1+\cdots+\mu_d\lambda_d)}}\cdots
\partial x^{i_{\tau(\kappa)}}
\left(\partial y\right)^{\mu_1+\cdots+\mu_d-1}}
\endaligned
\right]
\cdot
\\
&
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\cdot
\prod_{1\leq\nu_1\leq\mu_1}\,
y_{k_{1:\nu_1:1},\dots,k_{1:\nu_1:\lambda_1}}
\ \cdots \
\prod_{1\leq\nu_d\leq\mu_d}\,
y_{k_{d:\nu_d:1},\dots,k_{d:\nu_d:\lambda_d}}.
\endaligned
}
\end{equation}
\end{theorem}
\subsection*{ 3.75.~Deduction of a multivariate Fa\`a
di Bruno formula} Let $n \in \N$ with $n\geq 1$, let $x = (x^1,\dots,
x^n) \in \K^n$, let $g = g( x^1, \dots, x^n)$ be a $\mathcal{
C}^\infty$-smooth function from $\K^n$ to $\K$, let $y \in \K$ and let
$f = f(y)$ be a $\mathcal{ C}^\infty$ function from $\K$ to $\K$. The
goal is to obtain an explicit formula for the partial derivatives of
the composition $h := f\circ g$, namely $h(x^1, \dots, x^n) := f (
g(x^1,\dots, x^n))$. For $\lambda \in \N$ with $\lambda \geq 1$ and
for arbitrary indices $i_1, \dots, i_\lambda = 1, \dots, n$, we
shall abbreviate the partial derivative $\frac{ \partial^\lambda g}{
\partial x^{i_1} \cdots \partial x^{i_\lambda}}$ by $g_{i_1,\dots,
i_\lambda}$ and similarly for $h_{i_1, \dots, i_\lambda}$. The
derivative $\frac{ d^\lambda f}{ d y^\lambda}$ will be
abbreviated by $f_\lambda$.
Appying the chain rule, we may compute:
\def5.22}\begin{equation{3.76}\begin{equation}
\small
\aligned
h_{i_1}
&
=
f_1
\left[
g_{i_1}
\right],
\\
h_{i_1,i_2}
&
=
f_2
\left[
g_{i_1}\,g_{i_2}
\right]
+
f_1
\left[
g_{i_1,i_2}
\right],
\\
h_{i_1,i_2,i_3}
&
=
f_3
\left[
g_{i_1}\,g_{i_2}\,g_{i_3}
\right]
+
f_2
\left[
g_{i_1}\,g_{i_2,i_3}
+
g_{i_2}\,g_{i_1,i_3}
+
g_{i_3}\,g_{i_1,i_2}
\right]
+
f_1
\left[
g_{i_1,i_2,i_3}
\right],
\\
h_{i_1,i_2,i_3,i_4}
&
=
f_4
\left[
g_{i_1}\,g_{i_2}\,g_{i_3}\,g_{i_4}
\right]
+
f_3
\left[
g_{i_2}\,g_{i_3}\,g_{i_1,i_4}
+
g_{i_3}\,g_{i_1}\,g_{i_2,i_4}
+
g_{i_1}\,g_{i_2}\,g_{i_3,i_4}
+
\right.
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \
\left.
+
g_{i_1}\,g_{i_4}\,g_{i_2,i_3}
+
g_{i_2}\,g_{i_4}\,g_{i_1,i_3}
+
g_{i_3}\,g_{i_4}\,g_{i_1,i_2}
\right]
+ \\
& \
\ \ \ \ \
+
f_2
\left[
g_{i_1,i_2}\,g_{i_3,i_4}
+
g_{i_1,i_3}\,g_{i_2,i_4}
+
g_{i_1,i_4}\,g_{i_2,i_3}
\right]
+ \\
& \
\ \ \ \ \
+
f_2
\left[
g_{i_1}\,g_{i_2,i_3,i_4}
+
g_{i_2}\,g_{i_1,i_3,i_4}
+
g_{i_3}\,g_{i_1,i_2,i_4}
+
g_{i_4}\,g_{i_1,i_2,i_3}
\right]
+ \\
& \
\ \ \ \ \
+
f_1
\left[
g_{i_1,i_2,i_3,i_4}
\right].
\endaligned
\end{equation}
Introducing the derivations
\def5.22}\begin{equation{3.77}\begin{equation}
\small
\aligned
F_i^2
&
:=
\sum_{k_1=1}^n\,g_{k_1,i}\,
\frac{\partial}{\partial g_{k_1}}
+
g_i\left(
f_2\,\frac{\partial}{\partial f_1}
\right), \\
F_i^3
&
:=
\sum_{k_1=1}^n\,g_{k_1,i}\,
\frac{\partial}{\partial g_{k_1}}
+
\sum_{k_1,k_2=1}^n\,g_{k_1,k_2,i}\,
\frac{\partial}{\partial g_{k_1,k_2}}
+
g_i\left(
f_2\,\frac{\partial}{\partial f_1}
+
f_3\,\frac{\partial}{\partial f_2}
\right), \\
\text{\bf
\dots\dots}
& \
\ \ \ \ \
\text{\bf
\dots\dots\dots\dots\dots\dots\dots\dots\dots
\dots\dots\dots\dots\dots\dots\dots\dots\dots
\dots\dots\dots\dots\dots\dots\dots
} \\
\endaligned
\end{equation}
$$
\small
\aligned
F_i^\lambda
&
:=
\sum_{k_1=1}^n\,g_{k_1,i}\,
\frac{\partial}{\partial g_{k_1}}
+
\sum_{k_1,k_2=1}^n\,g_{k_1,k_2,i}\,
\frac{\partial}{\partial g_{k_1,k_2}}
+
\cdots
+ \\
& \
\ \ \ \ \ \ \ \ \ \
+
\sum_{k_1,\dots,k_{\lambda-1}=1}^n\,g_{k_1,\dots,k_{\lambda-1},i}\,
\frac{\partial}{\partial g_{k_1,\dots,k_{\lambda-1}}}
+ \\
& \
\ \ \ \ \ \ \ \ \ \
+
g_i\left(
f_2\,\frac{\partial}{\partial f_1}
+
f_3\,\frac{\partial}{\partial f_2}
+
\cdots
+
f_\lambda\,\frac{\partial}{\partial f_{\lambda-1}}
\right), \\
\endaligned
$$
we observe that the following
induction relations hold:
\def5.22}\begin{equation{3.78}\begin{equation}
\aligned
h_{i_1,i_2}
&
=
F_{i_2}^2
\left(
h_{i_1}
\right),
\\
h_{i_1,i_2,i_3}
&
=
F_{i_3}^3
\left(
h_{i_1,i_2}
\right),
\\
\text{\bf
\dots\dots
}
& \
\ \ \ \ \
\text{\bf
\dots\dots\dots\dots\dots
}
\\
h_{i_1,i_2,\dots,i_\lambda}
&
=
F_{i_\lambda}^\lambda
\left(
h_{i_1,i_2,\dots,i_{\lambda-1}}
\right).
\endaligned
\end{equation}
To obtain the explicit version of the Fa\`a di Bruno in the case of
several variables $(x^1, \dots, x^n)$ and one variable $y$, it
suffices to extract from the expression of ${\bf Y }_{ i_1,\dots,
i_\kappa}$ provided by Theorem~3.73 only the terms corresponding to
$\mu_1 \lambda_1 + \cdots + \mu_d\lambda_d = \kappa$, dropping all the
$\mathcal{ X}$ terms. After some simplifications and after a
translation by means of an elementary dictionary, we obtain a
statement.
\def5.12}\begin{theorem{3.79}\begin{theorem}
For every integer $\kappa \geq 1$ and for every choice of indices
$i_1, \dots, i_\kappa$ in the set $\{ 1, 2, \dots, n\}$, the
$\kappa$-th partial derivative of the composite function $h = h( x^1,
\dots, x^n) = f( g(x^1, \dots, x^n))$ with respect to the variables
$x^{i_1}, \dots, x^{i_\kappa}$ may be expressed as an explicit
polynomial depending on the derivatives of $f$, on the partial
derivatives of $g$ and having integer coefficients{\rm :}
\def5.22}\begin{equation{3.80}\begin{equation}
\boxed{
\aligned
\frac{\partial^\kappa h}{\partial x^{i_1}\cdots
\partial x^{i_\kappa}}
&
=
\sum_{d=1}^\kappa
\
\sum_{1\leq \lambda_1 < \cdots < \lambda_d \leq \kappa}
\
\sum_{\mu_1\geq 1,\dots,\mu_d\geq 1}
\
\sum_{\mu_1\lambda_1+\cdots+\mu_d\lambda_d=\kappa}
\
\frac{d^{\mu_1+\cdots+\mu_d} f}{
dy^{\mu_1+\cdots+\mu_d}}
\\
&
\left[
\aligned
&
\sum_{\sigma\in\mathfrak{F}_\kappa^{
(\mu_1,\lambda_1),\dots,(\mu_d,\lambda_d)}}
\
\prod_{1\leq\nu_1\leq\mu_1}
\
\frac{\partial^{\lambda_1} g}{\partial
x^{i_{\sigma(1:\nu_1:1)}}\cdots
\partial x^{i_{\sigma(1:\nu_1:\lambda_1)}}}
\
\text{\bf \dots}
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\text{\bf \dots}
\prod_{1\leq\nu_d\leq\mu_d}
\
\frac{\partial^{\lambda_d} g}{\partial
x^{i_{\sigma(d:\nu_d:1)}}\cdots
\partial x^{i_{\sigma(d:\nu_ds:\lambda_d)}}}
\endaligned
\right].
\endaligned
}
\end{equation}
\end{theorem}
In this formula, the coset $\mathfrak{ F }_\kappa^{
(\mu_1, \lambda_1 ),\dots, ( \mu_d, \lambda_d)}$
was defined in equation~\thetag{ 3.71}; we have made the identification{\rm :}
\def5.22}\begin{equation{3.81}\begin{equation}
\{1,\dots,\kappa\}
\equiv
\{
1\!:\!1\!:\!1,
\dots,
1\!:\!\mu_1\!:\!\lambda_1,
\text{\rm \dots\dots},
d\!:\!1\!:\!1,
\dots,
d\!:\!\mu_d\!:\!\lambda_d
\};
\end{equation}
and also, for the sake of clarity, we have restituted the complete
(not abbreviated) notation for the (partial) derivatives of $f$ and of
$g$.
\section*{\S4.~Several independent variables and
one dependent variable}
\subsection*{4.1.~Simplified adapted notations}
Assume $n = 1$ and $m \geq 1$, let $\kappa\in \N$ with $\kappa \geq
1$ and simply denote the jet variables by
(instead of~\thetag{ 1.2}):
\def5.22}\begin{equation{4.2}\begin{equation}
\left(
x,y^j,y_1^j,y_2^j,\dots,y_\kappa^j
\right)
\in
\mathcal{J}_{1,m}^\kappa.
\end{equation}
Instead of~\thetag{ 1.30}, denote the $\kappa$-th prolongation
of a vector field by:
\def5.22}\begin{equation{4.3}\begin{equation}
\left\{
\aligned
\mathcal{L}^{(\kappa)}
&
=
\mathcal{X}\,\frac{\partial}{\partial x}
+
\sum_{j=1}^m\,\mathcal{Y}^j\,\frac{\partial}{\partial y^j}
+
\sum_{j=1}^m\,{\bf Y}_1^j\,\frac{\partial}{\partial y_1^j}
+
\sum_{j=1}^m\,{\bf Y}_2^j\,\frac{\partial}{\partial y_2^j}
+ \\
& \
\ \ \ \ \
+
\cdots
+
\sum_{j=1}^m\,{\bf Y}_\kappa^j\,\frac{\partial}{\partial y_\kappa^j}.
\endaligned\right.
\end{equation}
The induction formulas are:
\def5.22}\begin{equation{4.4}\begin{equation}
\left\{
\aligned
{\bf Y}_1^j
&
:=
D^1
\left(
\mathcal{Y}^j
\right)
-
D^1
\left(
\mathcal{ X}
\right)
y_1^j,
\\
{\bf Y}_2^j
&
:=
D^2
\left(
{\bf Y}_1^j
\right)
-
D^1
\left(
\mathcal{ X}
\right)
y_2^j,
\\
\cdots\cdots
&
\cdots\cdots
\\
{\bf Y}_\lambda^j
&
:=
D^\lambda
\left(
{\bf Y}_{\lambda-1}^j
\right)
-
D^1
\left(
\mathcal{ X}
\right)
y_\lambda^j,
\endaligned\right.
\end{equation}
where the total
differentiation operators $D^\lambda$
are denoted by (instead of~\thetag{ 1.22}):
\def5.22}\begin{equation{4.5}\begin{equation}
\aligned
D^\lambda
:=
\frac{\partial}{\partial x}
+
\sum_{l=1}^m\,y_1^l\,
\frac{\partial}{\partial y^l}
+
\sum_{l=1}^m\,y_2^l\,
\frac{\partial}{\partial y_1^l}
+
\cdots
+
\sum_{l=1}^m\,y_\lambda^l\,
\frac{\partial}{\partial y_{\lambda-1}^l}.
\endaligned
\end{equation}
Applying the definitions in the first two lines of~\thetag{ 4.4}, we
compute, we simplify and we organize the results in a harmonious way,
using in an essential way the Kronecker symbol. Here, the
computations are more elementary than the computations of ${\bf
Y}_{i_1}$ and of ${\bf Y}_{i_1, i_2}$ achieved thoroughly
in the previous
Section~3, so that we do not provide a Latex track of the
details. Firstly and secondly:
\def5.22}\begin{equation{4.6}\begin{equation}
\small
\left\{
\aligned
{\bf Y}_1^j
&
=
\mathcal{ Y}_x^j
+
\sum_{l_1=1}^m
\left[
\mathcal{Y}_{y^{l_1}}^j
-
\delta_{l_1}^j\,
\mathcal{X}_x
\right]
y_1^{l_1}
+
\sum_{l_1,l_2=1}^m
\left[
-
\delta_{l_1}^j\,
\mathcal{X}_{y^{l_2}}
\right]
y_1^{l_1}y_1^{l_2}, \\
{\bf Y}_2^j
&
=
\mathcal{ Y}_{x^2}^j
+
\sum_{l_1=1}^m
\left[
2\,\mathcal{Y}_{xy^{l_1}}^j
-
\delta_{l_1}^j\,
\mathcal{X}_{x^2}
\right]
y_1^{l_1}
+
\sum_{l_1,l_2=1}^m
\left[
\mathcal{Y}_{y^{l_1}y^{l_2}}^j
-
\delta_{l_1}^j\,2\,
\mathcal{X}_{xy^{l_2}}
\right]
y_1^{l_1}y_1^{l_2}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2,l_3}
\left[
-
\delta_{l_1}^j\,
\mathcal{X}_{y^{l_2}y^{l_3}}
\right]
y_1^{l_1}y_1^{l_2}y_1^{l_3}
+
\sum_{l_1}
\left[
\mathcal{Y}_{y^{l_1}}^j
-
\delta_{l_1}^j\,2\,
\mathcal{X}_x
\right]
y_2^{l_1}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2=1}^m
\left[
-
\delta_{l_1}^j\,
\mathcal{X}_{y^{l_2}}
-
\delta_{l_2}^j\,2\,
\mathcal{X}_{y^{l_1}}
\right]
y_1^{l_1}y_2^{l_2}.
\endaligned\right.
\end{equation}
Thirdly:
\def5.22}\begin{equation{4.7}\begin{equation}
\small
\aligned
{\bf Y}_3^j
&
=
\mathcal{ Y}_{x^3}^j
+
\sum_{l_1=1}^m
\left[
3\,\mathcal{Y}_{x^2y^{l_1}}^j
-
\delta_{l_1}^j\,
\mathcal{X}_{x^3}
\right]
y_1^{l_1}
+
\sum_{l_1,l_2=1}^m
\left[
3\,\mathcal{Y}_{xy^{l_1}y^{l_2}}^j
-
\delta_{l_1}^j\,
3\,
\mathcal{X}_{x^2y^{l_2}}
\right]
y_1^{l_1}y_1^{l_2}+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2,l_3}
\left[
\mathcal{Y}_{y^{l_1}y^{l_2}y^{l_3}}^j
-
\delta_{l_1}^j\,
3\,\mathcal{X}_{xy^{l_2}y^{l_3}}
\right]
y_1^{l_1}y_1^{l_2}y_1^{l_3}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2,l_3,l_4}
\left[
-
\delta_{l_1}^j\,
\mathcal{X}_{y^{l_2}y^{l_3}y^{l_4}}
\right]
y_1^{l_1}y_1^{l_2}y_1^{l_3}y_1^{l_4}
+
\sum_{l_1=1}^m
\left[
3\,\mathcal{Y}_{xy^{l_1}}^j
-
\delta_{l_1}^j\,
3\,\mathcal{X}_{x^2}
\right]
y_2^{l_1}
\endaligned
\end{equation}
$$
\small
\aligned
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2=1}^m
\left[
3\,\mathcal{Y}_{y^{l_1}y^{l_2}}^j
-
\delta_{l_1}^j\,
3\,\mathcal{X}_{xy^{l_2}}
-
\delta_{l_2}^j\,
6\,\mathcal{X}_{xy^{l_1}}
\right]
y_1^{l_1}y_2^{l_2}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2,l_3=1}^m
\left[
-
\delta_{l_1}^j\,
3\,\mathcal{X}_{y^{l_2}y^{l_3}}
-
\delta_{l_3}^j\,
3\,\mathcal{X}_{y^{l_1}y^{l_2}}
\right]
y_1^{l_1}y_1^{l_2}y_2^{l_3}
+
\sum_{l_1,l_2=1}^m
\left[
-
\delta_{l_3}^j\,
3\,\mathcal{X}_{y^{l_2}}
\right]
y_2^{l_1}y_2^{l_2}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1=1}^m
\left[
\mathcal{Y}_{y^{l_1}}^j
-
\delta_{l_1}^j\,
3\,\mathcal{X}_x
\right]
y_3^{l_1}
+
\sum_{l_1,l_2=1}^m
\left[
-
\delta_{l_1}^j\,
\mathcal{X}_{y^{l_2}}
-
\delta_{l_2}^j\,
3\,\mathcal{X}_{y^{l_1}}
\right]
y_1^{l_1}y_3^{l_2}.
\endaligned
$$
Fourthly:
$$
\small
\aligned
{\bf Y}_4^j
&
=
\mathcal{ Y}_{x^4}^j
+
\sum_{l_1=1}^m
\left[
4\,\mathcal{Y}_{x^3y^{l_1}}^j
-
\delta_{l_1}^j
\mathcal{X}_{x^4}\,
\right]
y_1^{l_1}
+
\sum_{l_1,l_2=1}^m
\left[
6\,\mathcal{Y}_{x^2y^{l_1}y^{l_2}}^j
-
\delta_{l_1}^j\,
4\,
\mathcal{X}_{x^3y^{l_2}}
\right]
y_1^{l_1}y_1^{l_2}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2,l_3=1}^m
\left[
4\,\mathcal{Y}_{xy^{l_1}y^{l_2}y^{l_3}}^j
-
\delta_{l_1}^j\,
6\,\mathcal{X}_{x^2y^{l_2}y^{l_3}}
\right]
y_1^{l_1}y_1^{l_2}y_1^{l_3}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2,l_3,l_4=1}^m
\left[
\mathcal{Y}_{xy^{l_1}y^{l_2}y^{l_3}y^{l_4}}^j
-
\delta_{l_1}^j\,
4\,\mathcal{X}_{xy^{l_2}y^{l_3}y^{l_4}}
\right]
y_1^{l_1}y_1^{l_2}y_1^{l_3}y_1^{l_4}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2,l_3,l_4,l_5=1}^m
\left[
-
\delta_{l_1}^j\,
\mathcal{X}_{y^{l_2}y^{l_3}y^{l_4}y^{l_5}}
\right]
y_1^{l_1}y_1^{l_2}y_1^{l_3}y_1^{l_4}y_1^{l_5}
+
\sum_{l_1=1}^m
\left[
6\,\mathcal{Y}_{x^2y^{l_1}}^j
-
\delta_{l_1}^j\,
4\,\mathcal{X}_{x^3}
\right]
y_2^{l_1}
+
\endaligned
$$
\def5.22}\begin{equation{4.8}\begin{equation}
\small
\aligned
& \
\ \ \ \ \
+
\sum_{l_1,l_2=1}^m
\left[
12\,\mathcal{Y}_{xy^{l_1}y^{l_2}}^j
-
\delta_{l_1}^j\,
6\,\mathcal{X}_{x^2y^{l_2}}
-
\delta_{l_2}^j\,
12\,\mathcal{X}_{x^2y^{l_1}}
\right]
y_1^{l_1}y_2^{l_2}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2,l_3=1}^m
\left[
6\,\mathcal{Y}_{y^{l_1}y^{l_2}y^{l_3}}^j
-
\delta_{l_1}^j\,
12\,\mathcal{X}_{xy^{l_2}y^{l_3}}
-
\delta_{l_3}^j\,
12\,\mathcal{X}_{xy^{l_1}y^{l_2}}
\right]
y_1^{l_1}y_1^{l_2}y_2^{l_3}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2,l_3,l_4=1}^m
\left[
-
\delta_{l_1}^j\,
6\,\mathcal{X}_{y^{l_2}y^{l_3}y^{l_4}}
-
\delta_{l_4}^j\,
4\,\mathcal{X}_{y^{l_1}y^{l_2}y^{l_3}}
\right]
y_1^{l_1}y_1^{l_2}y_1^{l_3}y_2^{l_4}
+ \\
\endaligned
\end{equation}
$$
\small
\aligned
& \
\ \ \ \ \
+
\sum_{l_1,l_2=1}^m
\left[
3\,\mathcal{Y}_{y^{l_1}y^{l_2}}^j
-
\delta_{l_1}^j\,
12\,\mathcal{X}_{xy^{l_2}}
\right]
y_2^{l_1}y_2^{l_2}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2,l_3=1}^m
\left[
-
\delta_{l_1}^j\,
3\,\mathcal{X}_{y^{l_2}y^{l_3}}
-
\delta_{l_2}^j\,
12\,\mathcal{X}_{y^{l_1}y^{l_3}}
\right]
y_1^{l_1}y_2^{l_2}y_2^{l_3}
+
\sum_{l_1=1}^m
\left[
4\,\mathcal{Y}_{xy^{l_1}}^j
-
\delta_{l_1}^j\,
6\,\mathcal{X}_{x^2}
\right]
y_3^{l_1}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2=1}^m
\left[
4\,\mathcal{Y}_{y^{l_1}y^{l_2}}^j
-
\delta_{l_1}^j\,
4\,\mathcal{X}_{xy^{l_2}}
-
\delta_{l_2}^j\,
12\,\mathcal{X}_{xy^{l_1}}
\right]
y_1^{l_1}y_3^{l_2}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2,l_3=1}^m
\left[
-
\delta_{l_1}^j\,
4\,\mathcal{X}_{y^{l_2}y^{l_3}}
-
\delta_{l_3}^j\,
6\,\mathcal{X}_{y^{l_1}y^{l_2}}
\right]
y_1^{l_1}y_1^{l_2}y_3^{l_3}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2=1}^m
\left[
-
\delta_{l_1}^j\,
4\,\mathcal{X}_{y^{l_2}}
-
\delta_{l_2}^j\,
6\,\mathcal{X}_{y^{l_1}}
\right]
y_2^{l_1}y_3^{l_2}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1=1}^m
\left[
\mathcal{Y}_{y^{l_1}}^j
-
\delta_{l_1}^j\,
4\,\mathcal{X}_x
\right]
y_4^{l_1}
+
\sum_{l_1,l_2=1}^m
\left[
-
\delta_{l_1}^j\,
\mathcal{X}_{y^{l_2}}
-
\delta_{l_2}^j\,
4\,\mathcal{X}_{y^{l_1}}
\right]
y_1^{l_1}y_4^{l_2}.
\endaligned
$$
\subsection*{4.9.~Inductive elaboration of the
general formula} Now we compare the formula~\thetag{ 2.9} for ${\bf
Y}_4$ with the above formula~\thetag{ 4.8} for ${\bf Y}_4^j$. The goal
is to find the rules of transformation and of development by
inspecting several instances, in order to devise how to transform and
to develope the formula~\thetag{ 2.25} to several dependent variables.
First of all, we have to develope the general monomial $(y_{ \lambda_1
})^{ \mu_1} \cdots (y_{ \lambda_d })^{ \mu_d}$. In every monomial
present in the expressions of ${\bf Y}_1^j$, of ${\bf Y}_2^j$, of
${\bf Y}_3^j$ and of ${\bf Y}_4^j$ above, we see that the number
$\alpha$ of indices $l_\beta$ appearing in all the sums $\sum_{l_1,
\dots, l_\alpha = 1}^m$ is exactly equal to $\mu_1 + \dots+ \mu_d$. To
denote these $\mu_1 + \cdots + \mu_d$ indices $l_\beta$, we shall use
the notation:
\def5.22}\begin{equation{4.10}\begin{equation}
\underbrace{
\underbrace{
l_{1:1},\dots,l_{1:\mu_1}}_{\mu_1},
\dots,
\underbrace{
l_{d:1},\dots,l_{d:\mu_d}}_{\mu_d}}_{
\mu_1+\cdots+\mu_d},
\end{equation}
inspired by Convention~3.33. With such a choice of notation, we may
avoid long subscripts in the indices $l_\beta$, like $l_{\mu_1+\cdots+
\mu_d}$. It follows that the development of the general monomial $(y_{
\lambda_1 })^{ \mu_1} \cdots (y_{ \lambda_d })^{ \mu_d}$ to several
dependent variables yields $m^{\mu_1+ \cdots + \mu_d}$ possible
choices:
\def5.22}\begin{equation{4.11}\begin{equation}
\prod_{1\leq\nu_1\leq\mu_1}
y_{\lambda_1}^{l_{1:\nu_1}}
\ \cdots\cdots
\prod_{1\leq\nu_d\leq\mu_d}
y_{\lambda_d}^{l_{d:\nu_d}},
\end{equation}
where the indices $l_{1 :1}, \dots, l_{1: \mu_1 }, \dots, l_{ d:1},
\dots, l_{d: \mu_d }$ take their values in the set $\{ 1, 2, \dots, m
\}$. Consequently, the general expression of ${\bf Y }_\kappa^j$ must
be of the form:
\def5.22}\begin{equation{4.12}\begin{equation}
\small
\aligned
{\bf Y}_\kappa^j
&
=
\mathcal{Y}_{x^\kappa}^j
+
\sum_{d=1}^{\kappa+1}
\
\sum_{1\leq\lambda_1<\cdots<\lambda_d\leq\kappa}
\
\sum_{\mu_1\geq 1,\dots,\mu_d\geq 1}
\
\sum_{\mu_1\lambda_1+\cdots+\mu_d\lambda_d\leq\kappa+1}
\
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\sum_{l_{1:1}=1}^m
\cdots
\sum_{l_{1:\mu_1}=1}^m
\cdots\cdots
\sum_{l_{d:1}=1}^m
\cdots
\sum_{l_{d:\mu_d}=1}^m
\
\text{\bf [?]}
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\prod_{1\leq\nu_1\leq\mu_1}
y_{\lambda_1}^{l_{1:\nu_1}}
\ \cdots\cdots
\prod_{1\leq\nu_d\leq\mu_d}
y_{\lambda_d}^{l_{d:\nu_d}},
\endaligned
\end{equation}
where the term in brackets {\bf [?]} is still unknown. To determine
it, let us examine a few instances.
According to~\thetag{ 4.8} (fourth line), the term $\left[ 6\,
\mathcal{ Y}_{ x^2y} - 4\, \mathcal{ X}_{ x^3} \right] y_2$ of ${\bf
Y}_4$ developes as $\sum_{ l_1 =1}^m \, \left[ 6\, \mathcal{ Y}_{ x^2
y^{l_1}}^j - \delta_{ l_1}^j \, 4 \, \mathcal{ X}_{ x^3} \right] y_2^{
l_1}$ in ${\bf Y}_4^j$. Here,
$6\, \mathcal{ Y}_{ x^2y}$ just becomes
$6\, \mathcal{ Y}_{ x^2y^{l_1}}^j$. Thus, we suspect that the term
$\frac{\kappa\cdots(\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d+1)}
{(\lambda_1!)^{\mu_1}\,\mu_1! \cdots (\lambda_d! )^{\mu_d}\,\mu_d! }
\cdot \mathcal{ Y}_{ x^{ \kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d}
\, y^{\mu_1+\cdots+\mu_d} } $ of the second line of~\thetag{ 2.25}
should simply be developed as
\def5.22}\begin{equation{4.13}\begin{equation}
\small
\aligned
&
\frac{\kappa(\kappa-1)\cdots(\kappa-\mu_1\lambda_1-\cdots
-\mu_d\lambda_d+1)}{
(\lambda_1!)^{\mu_1}\ \mu_1!\cdots
(\lambda_d!)^{\mu_d}\ \mu_d!}
\cdot
\\
& \
\ \ \ \ \ \ \ \
\cdot
\frac{\partial^{\kappa-\mu_1\lambda_1-\cdots
-\mu_d\lambda_d+\mu_1+\cdots+\mu_d}\mathcal{Y}^j}{
(\partial x)^{\kappa-\mu_1\lambda_1-\cdots
-\mu_d\lambda_d}
\partial y^{l_{1:1}}
\cdots
\partial y^{l_{1:\mu_1}}
\cdots
\partial y^{l_{d:1}}
\cdots
\partial y^{l_{d:\mu_d}}
}.
\endaligned
\end{equation}
This rule is confirmed by inspecting all the other monomials of ${\bf
Y}_1^j$, of ${\bf Y}_2^j$, of ${\bf Y}_3^j$ and of ${\bf Y}_4^j$.
It remains to determine how we must develope the term in $\mathcal{
X}$ appearing in the last two lines of~\thetag{ 2.25}. To begin with,
let us rewrite in advance this term in the slightly different
shape, emphasizing a factorization:
\def5.22}\begin{equation{4.14}\begin{equation}
\small
\aligned
\frac{\kappa\cdots(
\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d+2)}
{(\lambda_1!)^{\mu_1}\,\mu_1!
\cdots
(\lambda_d!)^{\mu_d}\,\mu_d!
}
\left[
(\mu_1\lambda_1+\cdots+\mu_d\lambda_d)
\mathcal{X}_{
x^{\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d+1}
\,
y^{\mu_1+\cdots+\mu_d-1}
}
\right].
\endaligned
\end{equation}
Then we examine four instances extracted from the complete expression
of ${\bf Y}_4^j$:
\def5.22}\begin{equation{4.15}\begin{equation}
\small
\left\{
\aligned
&
\sum_{l_1,l_2,l_3=1}^m
\left[
4\,\mathcal{Y}_{xy^{l_1}y^{l_2}y^{l_3}}^j
-
\delta_{l_1}^j\,
6\,\mathcal{X}_{x^2y^{l_2}y^{l_3}}
\right]
y_1^{l_1}y_1^{l_2}y_1^{l_3},
\\
&
\sum_{l_1,l_2=1}^m
\left[
12\,\mathcal{Y}_{xy^{l_1}y^{l_2}}^j
-
\delta_{l_1}^j\,
6\,\mathcal{X}_{x^2y^{l_2}}
-
\delta_{l_2}^j\,
12\,\mathcal{X}_{x^2y^{l_1}}
\right]
y_1^{l_1}y_2^{l_2},
\\
&
\sum_{l_1,l_2,l_3,l_4=1}^m
\left[
-
\delta_{l_1}^j\,
6\,\mathcal{X}_{y^{l_2}y^{l_3}y^{l_4}}
-
\delta_{l_4}^j\,
4\,\mathcal{X}_{y^{l_1}y^{l_2}y^{l_3}}
\right]
y_1^{l_1}y_1^{l_2}y_1^{l_3}y_2^{l_4},
\\
&
\sum_{l_1,l_2,l_3=1}^m
\left[
-
\delta_{l_1}^j\,
4\,\mathcal{X}_{y^{l_2}y^{l_3}}
-
\delta_{l_3}^j\,
6\,\mathcal{X}_{y^{l_1}y^{l_2}}
\right]
y_1^{l_1}y_1^{l_2}y_3^{l_3},
\endaligned\right.
\end{equation}
and we compare them to the corresponding terms of ${\bf Y}_4$:
\def5.22}\begin{equation{4.16}\begin{equation}
\small
\left\{
\aligned
&
\left[
4\,\mathcal{Y}_{xy^3}
-
6\,\mathcal{X}_{x^2y^2}
\right]
(y_1)^3,
\\
&
\left[
12\,\mathcal{Y}_{xy^2}
-
18\,\mathcal{X}_{x^2y}
\right]
y_1y_2,
\\
&
\left[
-
10\,\mathcal{X}_{y^3}
\right]
(y_1)^3y_2,
\\
&
\left[
-
10\,\mathcal{X}_{y^2}
\right]
(y_1)^2y_3.
\endaligned\right.
\end{equation}
In the development from~\thetag{ 4.16} to~\thetag{ 4.15}, we see that
the four integers just before $\mathcal{ X}$, namely $6 = 6$, $18 = 6
+ 12$, $10 = 6 + 4$ and $10 = 4 + 6$, are split in a certain
manner. Also, a single Kronecker symbol $\delta_{l_\alpha}^j$ is added
as a factor. {\it What are the rules}?
In the second splitting $18 = 6 + 12$, we see that the relative weight
of $6$ and of $12$ is the same as the relative weight of $1$ and $2$
in the splitting $3 = 1 + 2$ issued from the lower indices of the
corresponding monomial $y_1^{l_1} y_2^{l_2}$. Similarly, in the third
splitting $10 = 6 + 4$, the relative weight of $6$ and of $4$ is the
same as the relative weight of $1+1+1$ and of $2$ issued from the
lower indices of the corresponding monomial $y_1^{l_1} y_1^{l_2}
y_1^{l_3} y_2^{l_4}$. This rule may be confirmed by inspecting all
the other monomials of ${\bf Y}_2$, ${\bf Y}_2^j$, of ${\bf Y}_3$,
${\bf Y}_3^j$ and of ${\bf Y}_4$, ${\bf Y}_4^j$. For a general $\kappa
\geq 1$, the splitting of integers just amounts to decompose the sum
appearing inside the brackets of~\thetag{ 4.14} as $\mu_1\lambda_1,
\mu_2\lambda_2, \dots, \mu_d\lambda_d$. In fact, when we
wrote~\thetag{ 4.14}, we emphasized in advance the decomposable
factor $(\mu_1 \lambda_1 + \cdots + \mu_d \lambda_d)$.
Next, we have to determine what is the subscript $\alpha$ in the
Kronecker symbol $\delta_{l_\alpha}^j$. We claim that in the four
instances~\thetag{ 4.15}, the subscript $\alpha$ is intrinsically
related to weight splitting. Indeed, recall that in the second line
of~\thetag{ 4.15}, the number $6$ of the splitting $18 = 6 + 12$ is
related to the number $1$ in the splitting $3 = 1 + 2$ of the lower
indices of the monomial $y_1^{l_1} y_2^{l_2}$. It follows that the
index $l_\alpha$ {\it must be}\, the index $l_1$ of the monomial
$y_1^{l_1}$. Similarly, also in the second line of~\thetag{ 4.15}, the
number $12$ of the splitting $18 = 6 + 12$ being related to the number
$2$ in the splitting $3 = 1 + 2$ of the lower indices of the monomial
$y_1^{l_1} y_2^{l_2}$, it follows that the index $l_\alpha$ attached
to the second $\mathcal{ X}$ term must be the index $l_2$ of the
monomial $y_2^{l_2}$.
This rule is still ambiguous. Indeed, let us examine the third line
of~\thetag{ 4.15}. We have the splitting $10 = 6 + 4$, homologous to
the splitting of relative weights $5 = (1+1+1) + 2$ in the monomial
$y_1^{ l_1} y_1^{ l_2} y_1^{ l_3} y_2^{ l_4}$. Of course, it is clear
that we must choose the index $l_4$ for the Kronecker symbol
associated to the second $\mathcal{ X}$ term $-4\, \mathcal{ X}_{
y^3}$, thus obtaining $-\delta_{ l_4}^j \, 4 \, \mathcal{ X}_{ y^{
l_1} y^{ l_2} y^{ l_3}}$. However, since the monomial $y_1^{ l_1}
y_1^{ l_2} y_1^{ l_3}$ has three indices $l_1$, $l_2$ and $l_3$, there
arises a question: {\it what index $l_\alpha$ must we choose for the
Kronecker symbol $\delta_{ l_\alpha}^j$ attached to the first
$\mathcal{ X}$ term $6\,\mathcal{ X}_{y^3}${\rm :} the index $l_1$,
the index $l_2$ or the index $l_3$}?
The answer is simple: {\it any of the three indices $l_1$, $l_2$ or
$l_3$ works}. Indeed, since the monomial $y_1^{ l_1} y_1^{ l_2}
y_1^{ l_3}$ is symmetric with respect to all permutations of the set
of three indices $\{ l_1, l_2, l_3\}$, we have
\def5.22}\begin{equation{4.17}\begin{equation}
\small
\aligned
\sum_{ l_1, l_2, l_3, l_4 = 1}^m\,
\left[
-
\delta_{l_1}^j\,6\,\mathcal{X}_{y^{l_2}y^{l_3}y^{l_4}}
\right]
y_1^{l_1}y_1^{l_2}y_1^{l_3}y_2^{l_4}
&
=
\sum_{ l_1, l_2, l_3, l_4 = 1}^m\,
\left[
-
\delta_{l_2}^j\,6\,\mathcal{X}_{y^{l_1}y^{l_3}y^{l_4}}
\right]
y_1^{l_1}y_1^{l_2}y_1^{l_3}y_2^{l_4}
=
\\
&
=
\sum_{ l_1, l_2, l_3, l_4 = 1}^m\,
\left[
-
\delta_{l_3}^j\,6\,\mathcal{X}_{y^{l_1}y^{l_2}y^{l_4}}
\right]
y_1^{l_1}y_1^{l_2}y_1^{l_3}y_2^{l_4}.
\endaligned
\end{equation}
In fact, we have systematically used such symmetries during the
intermediate computations (not exposed here) which we achieved
manually to obtain the final expressions of ${\bf Y}_1^j$, of ${\bf
Y}_2^j$, of ${\bf Y}_3^j$ and of ${\bf Y}_4^j$. To fix ideas, we have
always choosen the first index. Here, the first index is $l_1$; in the
first sum of line~9 of~\thetag{ 4.8}, the first index $l_\alpha$ for
the second weight $12$ is $l_2$.
This rule may be confirmed by inspecting all the monomials of ${\bf
Y}_2^j$, of ${\bf Y}_3^j$, of ${\bf Y}_4^j$ (and also of ${\bf
Y}_5^j$, which we have computed in a
manuscript, but not copied in this Latex file).
From these considerations, we deduce that for the general formula, the
weight decomposition is simply $\mu_1\lambda_1, \dots, \mu_d\lambda_d$
and that the Kronecker symbol $\delta_\alpha^j$ is intrinsically
associated to the weights. In conclusion,
building on inductive reasonings, we have formulated the following
statement.
\def5.12}\begin{theorem{4.18}\begin{theorem}
For one independent variable $x$, for several dependent variables
$(y^1, \dots, y^m)$ and for $\kappa \geq 1$, the general expression of
the coefficient ${\bf Y }_\kappa^j$ of the prolongation~\thetag{ 4.3}
of a vector field is{\rm :}
\def5.22}\begin{equation{4.19}\begin{equation}
\boxed{
\small
\aligned
{\bf Y}_\kappa^j
&
=
\mathcal{Y}_{x^\kappa}^j
+
\sum_{d=1}^{\kappa+1}
\
\sum_{1\leq\lambda_1<\cdots<\lambda_d\leq\kappa}
\
\sum_{\mu_1\geq 1,\dots,\mu_d\geq 1}
\
\sum_{\mu_1\lambda_1+\cdots+\mu_d\lambda_d\leq\kappa+1}
\\
& \
\ \ \ \ \
\sum_{l_{1:1}=1}^m
\cdots
\sum_{l_{1:\mu_1}=1}^m
\cdots\cdots
\sum_{l_{d:1}=1}^m
\cdots
\sum_{l_{d:\mu_d}=1}^m
\
\frac{\kappa(\kappa-1)\cdots
(\kappa-\mu_1\lambda_1+\cdots+\mu_d\lambda_d+2)}{
(\lambda_1!)^{\mu_1}\ \mu_1!\cdots(\lambda_d!)^{\mu_d}\ \mu_d!
}
\\
& \
\left[
\aligned
&
(\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d+1)
\frac{\partial^{\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d+
\mu_1+\cdots+\mu_d}\mathcal{Y}^j}{
(\partial x)^{\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d}
\partial y^{l_{1:1}}
\cdots
\partial y^{l_{1:\mu_1}}
\cdots
\partial y^{l_{d:1}}
\cdots
\partial y^{l_{d:\mu_d}}}
- \\
& \
-
\delta_{l_{1:1}}^j\,\mu_1\lambda_1\,
\frac{\partial^{\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d+
\mu_1+\cdots+\mu_d}\mathcal{X}}{
(\partial x)^{\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d+1}
\widehat{\partial y^{l_{1:1}}}
\cdots
\partial y^{l_{1:\mu_1}}
\cdots
\partial y^{l_{d:1}}
\cdots
\partial y^{l_{d:\mu_d}}}
-
\\
&
\ \ \ \ \ \ \ \ \
-
\cdots
-
\\
& \
-
\delta_{l_{d:1}}^j\,\mu_d\lambda_d\,
\frac{\partial^{\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d+
\mu_1+\cdots+\mu_d}\mathcal{X}}{
(\partial x)^{\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d+1}
\partial y^{l_{1:1}}
\cdots
\partial y^{l_{1:\mu_1}}
\cdots
\widehat{\partial y^{l_{d:1}}}
\cdots
\partial y^{l_{d:\mu_d}}}
\endaligned
\right]
\cdot
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\cdot
\prod_{1\leq\nu_1\leq\mu_1}
y_{\lambda_1}^{l_{1:\nu_1}}
\ \cdots\cdots
\prod_{1\leq\nu_d\leq\mu_d}
y_{\lambda_d}^{l_{d:\nu_d}}.
\endaligned
}
\end{equation}
Here, the notation $\widehat{ \partial y^l}$ means
that the partial derivative is dropped.
\end{theorem}
\subsection*{ 4.20.~Deduction of a multivariate Fa\`a
di Bruno formula} Let $m \in \N$ with $m\geq 1$, let $y = (y^1,\dots,
y^m) \in \K^m$, let $f = f( y^1, \dots, y^m)$ be a $\mathcal{
C}^\infty$-smooth function from $\K^m$ to $\K$, let $x \in \K$ and let
$g^1 = g^1(x), \dots, g^m = g^m( x)$ be $\mathcal{ C}^\infty$
functions from $\K$ to $\K$. The goal is to obtain an explicit formula
for the derivatives, with respect to $x$, of the composition $h :=
f\circ g$, namely $h(x) := f \left( g^1(x), \dots, g^m(x)
\right)$. For $\lambda \in \N$ with $\lambda \geq 1$, and for $j= 1,
\dots, m$, we shall abbreviate the derivative $\frac{ d^\lambda g^j}{
dx^\lambda}$ by $g_\lambda^j$ and similarly for $h_\lambda$. The
partial derivatives $\frac{ \partial^\lambda f}{ \partial
y^{l_1}\cdots \partial y^{l_\lambda}}$ will be abbreviated by $f_{l_1,
\dots, l_\lambda }$.
Appying the chain rule, we may compute:
\def5.22}\begin{equation{4.21}\begin{equation}
\small
\aligned
h_1
&
=
\sum_{l_1=1}^m\,f_{l_1}\,g_1^{l_1},
\\
h_2
&
=
\sum_{l_1,l_2=1}^m\,f_{l_1,l_2}\,g_1^{l_1}\,g_1^{l_2}
+
\sum_{l_1=1}^m\,f_{l_1}\,g_2^{l_1},
\\
h_3
&
=
\sum_{l_1,l_2,l_3=1}^m\,f_{l_1,l_2,l_3}\,
g_1^{l_1}\,g_1^{l_2}\,g_1^{l_3}
+
3\,\sum_{l_1,l_2=1}^m\,f_{l_1,l_2}\,
g_1^{l_1}\,g_2^{l_2}
+
\sum_{l_1=1}^m\,f_{l_1}\,g_3^{l_1},
\\
h_4
&
=
\sum_{l_1,l_2,l_3,l_4=1}^m\,f_{l_1,l_2,l_3,l_4}\,
g_1^{l_1}\,g_1^{l_2}\,g_1^{l_3}\,g_1^{l_4}
+
6\,\sum_{l_1,l_2,l_3=1}^m\,f_{l_1,l_2,l_3}\,
g_1^{l_1}\,g_1^{l_2}\,g_2^{l_3}
+ \\
& \
\ \ \ \ \
+
3\,\sum_{l_1,l_2=1}^m\,f_{l_1,l_2}\,
g_2^{l_1}\,g_2^{l_2}
+
4\,\sum_{l_1,l_2=1}^m\,f_{l_1,l_2}\,
g_1^{l_1}\,g_3^{l_2}
+
\sum_{l_1=1}^m\,
f_{l_1}\,g_4^{l_1},
\\
h_5
&
=
\sum_{l_1,l_2,l_3,l_4,l_5=1}^m\,f_{l_1,l_2,l_3,l_4,l_5}\,
g_1^{l_1}\,g_1^{l_2}\,g_1^{l_3}\,g_1^{l_4}\,g_1^{l_5}
+
10\,\sum_{l_1,l_2,l_3,l_4=1}^m\,f_{l_1,l_2,l_3,l_4}\,
g_1^{l_1}\,g_1^{l_2}\,g_1^{l_3}\,g_2^{l_4}
+ \\
& \
\ \ \ \ \
+
15\,\sum_{l_1,l_2,l_3=1}^m\,f_{l_1,l_2,l_3}\,
g_1^{l_1}\,g_2^{l_2}\,g_2^{l_3}
+
10\,\sum_{l_1,l_2,l_3=1}^m\,f_{l_1,l_2,l_3}\,
g_1^{l_1}\,g_1^{l_2}\,g_3^{l_3}
+ \\
& \
\ \ \ \ \
+
10\,\sum_{l_1,l_2=1}^m\,f_{l_1,l_2}\,
g_2^{l_1}\,g_3^{l_2}
+
5\,\sum_{l_1,l_2=1}^m\,f_{l_1,l_2}\,
g_1^{l_1}\,g_4^{l_2}
+
\sum_{l_1=1}^m\,
f_{l_1}\,g_5^{l_1}.
\endaligned
\end{equation}
Introducing the derivations
\def5.22}\begin{equation{4.22}\begin{equation}
\small
\aligned
F^2
&
:=
\sum_{l_1=1}^m\,g_2^{l_1}\,
\frac{\partial}{\partial g_1^{l_1}}
+
\sum_{l_1=1}^m\,g_1^{l_1}
\left(
\sum_{l_2=1}^m\,
f_{l_1,l_2}\,\frac{\partial}{\partial f_{l_2}}
\right), \\
F^3
&
:=
\sum_{l_1=1}^m\,g_2^{l_1}\,
\frac{\partial}{\partial g_1^{l_1}}
+
\sum_{l_1=1}^m\,g_3^{l_1}\,
\frac{\partial}{\partial g_2^{l_1}}
+
\sum_{l_1=1}^m\,g_1^{l_1}
\left(
\sum_{l_2=1}^m\,
f_{l_1,l_2}\,\frac{\partial}{\partial f_{l_2}}
+
\sum_{l_2,l_3=1}^m\,
f_{l_1,l_2,l_3}\,\frac{\partial}{\partial f_{l_2,l_3}}
\right),
\\
& \
\ \ \ \ \
\text{\bf
\dots\dots\dots\dots\dots\dots\dots\dots\dots
\dots\dots\dots\dots\dots\dots\dots\dots\dots
\dots\dots\dots\dots\dots\dots\dots\dots\dots
} \\
F^\lambda
&
:=
\sum_{l_1=1}^m\,g_2^{l_1}\,
\frac{\partial}{\partial g_1^{l_1}}
+
\sum_{l_1=1}^m\,g_3^{l_1}\,
\frac{\partial}{\partial g_2^{l_1}}
+
\cdots
+
\sum_{l_1=1}^m\,g_\lambda^{l_1}\,
\frac{\partial}{\partial g_{\lambda-1}^{l_1}}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1=1}^m\,g_1^{l_1}
\left(
\sum_{l_2=1}^m\,
f_{l_1,l_2}\,\frac{\partial}{\partial f_{l_2}}
+
\sum_{l_2,l_3=1}^m\,
f_{l_1,l_2,l_3}\,\frac{\partial}{\partial f_{l_2,l_3}}
+
\cdots
+
\sum_{l_2,\dots,l_\lambda=1}^m\,
f_{l_1,l_2,\dots,l_\lambda}\,
\frac{\partial}{\partial f_{l_2,\dots,l_\lambda}}
\right),
\endaligned
\end{equation}
we observe that the following
induction relations hold:
\def5.22}\begin{equation{4.23}\begin{equation}
\aligned
h_2
&
=
F^2
\left(
h_1
\right),
\\
h_3
&
=
F^3
\left(
h_2
\right),
\\
\text{\bf
\dots\dots
}
& \
\ \ \ \ \
\text{\bf
\dots\dots\dots
}
\\
h_\lambda
&
=
F^\lambda
\left(
h_{\lambda-1}
\right).
\endaligned
\end{equation}
To obtain the explicit version of the Fa\`a di Bruno in the case of
one variable $x$ and several variables $(y^1, \dots, y^m)$, it
suffices to extract from the expression of ${\bf Y}_\kappa^j$ provided
by Theorem~4.18 only the terms corresponding to $\mu_1 \lambda_1 +
\cdots + \mu_d\lambda_d = \kappa$, dropping all the $\mathcal{ X}$
terms. After some simplifications and after a translation by means of
an elementary dictionary, we may formulate a statement.
\def5.12}\begin{theorem{4.24}\begin{theorem}
For every integer $\kappa \geq 1$, the $\kappa$-th partial derivative
of the composite function $h = h( x) = f \left(
g^1(x), \dots, g^m(x) \right)$ with
respect to $x$ may be expressed as an explicit polynomial depending on
the partial derivatives of $f$, on the derivatives of $g$ and having
integer coefficients{\rm:}
\def5.22}\begin{equation{4.25}\begin{equation}
\boxed{
\aligned
\frac{d^\kappa h}{dx^\kappa}
&
=
\sum_{d=1}^\kappa
\
\sum_{1\leq \lambda_1 < \cdots < \lambda_d \leq \kappa}
\
\sum_{\mu_1\geq 1,\dots,\mu_d\geq 1}
\
\sum_{\mu_1\lambda_1+\cdots+\mu_d\lambda_d=\kappa}
\
\frac{\kappa!}{
(\lambda_1!)^{\mu_1}\ \mu_1!
\cdots
(\lambda_d!)^{\mu_d}\ \mu_d!
}
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\sum_{l_{1:1},\dots,l_{1:\mu_1}=1}^m
\ \cdots \
\sum_{l_{d:1},\dots,l_{d:\mu_d}=1}^m
\\
\\
& \
\ \ \ \ \
\frac{\partial^{\mu_1+\cdots+\mu_d}f}{
\partial y^{l_{1:1}}
\cdots
\partial y^{l_{1:\mu_1}}
\cdots
\partial y^{l_{d:1}}
\cdots
\partial y^{l_{d:\mu_d}}
}
\
\prod_{1\leq\nu_1\leq\mu_1}
\frac{d^{\lambda_1} g^{l_{1:\nu_1}}}{d x^{\lambda_1}}
\ \cdots
\prod_{1\leq\nu_d\leq\mu_d}
\frac{d^{\lambda_d} g^{l_{d:\nu_d}}}{d x^{\lambda_d}}.
\endaligned
}
\end{equation}
\end{theorem}
\section*{\S5.~Several independent variables and
several dependent variables}
\subsection*{5.1.~Expression of ${\bf Y}_{i_1}^j$,
of ${\bf Y}_{i_1,i_2}^j$ and of ${\bf Y}_{i_1,i_2,i_3}^j$} Applying
the induction~\thetag{1.31} and working out the obtained formulas
until they take a perfect shape, we obtain firstly:
\def5.22}\begin{equation{5.2}\begin{equation}
\small
{\bf Y}_{i_1}^j
=
\mathcal{Y}_{x^{i_1}}^j
+
\sum_{l_1=1}^m\ \sum_{k_1=1}^n
\left[
\delta_{i_1}^{k_1}\,\mathcal{Y}_{y^{l_1}}^j
-
\delta_{l_1}^j\,
\mathcal{X}_{x^{i_1}}^{k_1}
\right]
y_{k_1}^{l_1}
+
\sum_{l_1,l_2=1}^m\ \sum_{k_1,k_2=1}^n
\left[
-
\delta_{l_2}^j\,
\delta_{i_1}^{k_1}\,\mathcal{X}_{y^{l_1}}^{k_2}
\right]
y_{k_1}^{l_1}y_{k_2}^{l_2}.
\end{equation}
Secondly:
\def5.22}\begin{equation{5.3}\begin{equation}
\small
\aligned{\bf Y}_{i_1,i_2}^j
&
=
\mathcal{Y}_{x^{i_1}x^{i_2}}^j
+
\sum_{l_1=1}^m\ \sum_{k_1=1}^n
\left[
\delta_{i_1}^{k_1}\,\mathcal{Y}_{x^{i_2}y^{l_1}}^j
+
\delta_{i_2}^{k_1}\,\mathcal{Y}_{x^{i_1}y^{l_1}}^j
-
\delta_{l_1}^j\,
\mathcal{X}_{x^{i_1}x^{i_2}}^{k_1}
\right]
y_{k_1}^{l_1}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2=1}^m\ \sum_{k_1,k_2=1}^n
\left[
\delta_{i_1, \ i_2}^{k_1,k_2}\,
\mathcal{Y}_{y^{l_1}y^{l_2}}^j
-
\delta_{l_2}^j\,\delta_{i_1}^{k_1}\,
\mathcal{X}_{x^{i_2}y^{l_1}}^{k_2}
-
\delta_{l_2}^j\,\delta_{i_2}^{k_1}\,
\mathcal{X}_{x^{i_1}y^{l_1}}^{k_2}
\right]
y_{k_1}^{l_1}y_{k_2}^{l_2}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2,l_3=1}^m\ \sum_{k_1,k_2,k_3=1}^n
\left[
-
\delta_{l_3}^j\,\delta_{i_1,\ i_2}^{k_1,k_2}\,
\mathcal{X}_{y^{l_1}y^{l_2}}^{k_3}
\right]
y_{k_1}^{l_1}y_{k_2}^{l_2}y_{k_3}^{l_3}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1=1}^m\ \sum_{k_1,k_2=1}^n
\left[
\delta_{i_1,\ i_2}^{k_1,k_2}\,
\mathcal{Y}_{y^{l_1}}^j
-
\delta_{l_1}^j\,\delta_{i_1}^{k_1}\,
\mathcal{X}_{x^{i_2}}^{k_2}
-
\delta_{l_1}^j\,\delta_{i_2}^{k_1}\,
\mathcal{X}_{x^{i_1}}^{k_2}
\right]
y_{k_1,k_2}^{l_1}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2=1}^m\ \sum_{k_1,k_2,k_3=1}^n
\left[
-
\delta_{l_1}^j\,\delta_{i_1,\ i_2}^{k_2,k_3}\,
\mathcal{X}_{y^{l_2}}^{k_1}
-
\delta_{l_2}^j\,\delta_{i_1,\ i_2}^{k_3,k_1}\,
\mathcal{X}_{y^{l_1}}^{k_2}
-
\delta_{l_2}^j\,\delta_{i_1,\ i_2}^{k_1,k_2}\,
\mathcal{X}_{y^{l_1}}^{k_3}
\right]
y_{k_1}^{l_1}y_{k_2}^{l_2}y_{k_3}^{l_3}.
\endaligned
\end{equation}
Thirdly:
$$
\small
\aligned
{\bf Y}_{i_1,i_2,i_3}^j
&
=
\mathcal{Y}_{x^{i_1}x^{i_2}x^{i_3}}^j
+
\sum_{l_1=1}^m\ \sum_{k_1=1}^n
\left[
\delta_{i_1}^{k_1}\,\mathcal{Y}_{x^{i_2}x^{i_3}y^{l_1}}^j
+
\delta_{i_2}^{k_1}\,\mathcal{Y}_{x^{i_1}x^{i_3}y^{l_1}}^j
+
\delta_{i_3}^{k_1}\,\mathcal{Y}_{x^{i_1}x^{i_2}y^{l_1}}^j
-
\delta_{l_1}^j\,
\mathcal{X}_{x^{i_1}x^{i_2}x^{i_3}}^{k_1}
\right]
y_{k_1}^{l_1}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2=1}^m\ \sum_{k_1,k_2=1}^n
\left[
\delta_{i_1, \ i_2}^{k_1,k_2}\,
\mathcal{Y}_{x^{i_3}y^{l_1}y^{l_2}}^j
+
\delta_{i_3, \ i_1}^{k_1,k_2}\,
\mathcal{Y}_{x^{i_2}y^{l_1}y^{l_2}}^j
+
\delta_{i_2, \ i_3}^{k_1,k_2}\,
\mathcal{Y}_{x^{i_1}y^{l_1}y^{l_2}}^j
-
\right.
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.
-
\delta_{l_2}^j\,\delta_{i_1}^{k_1}\,
\mathcal{X}_{x^{i_2}x^{i_3}y^{l_1}}^{k_2}
-
\delta_{l_2}^j\,\delta_{i_2}^{k_1}\,
\mathcal{X}_{x^{i_1}x^{i_3}y^{l_1}}^{k_2}
-
\delta_{l_2}^j\,\delta_{i_3}^{k_1}\,
\mathcal{X}_{x^{i_1}x^{i_2}y^{l_1}}^{k_2}
\right]
y_{k_1}^{l_1}y_{k_2}^{l_2}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2,l_3=1}^m\ \sum_{k_1,k_2,k_3=1}^n
\left[
\delta_{i_1, \ i_2, \ i_3}^{k_1,k_2,k_3}\,
\mathcal{Y}_{y^{l_1}y^{l_2}y^{l_3}}^j
-
\delta_{l_3}^j\,\delta_{i_1,\ i_2}^{k_1,k_2}\,
\mathcal{X}_{x^{i_3}y^{l_1}y^{l_2}}^{k_3}
-
\right.
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \
\left.
-
\delta_{l_3}^j\,\delta_{i_1,\ i_3}^{k_1,k_2}\,
\mathcal{X}_{x^{i_2}y^{l_1}y^{l_2}}^{k_3}
-
\delta_{l_3}^j\,\delta_{i_2,\ i_3}^{k_1,k_2}\,
\mathcal{X}_{x^{i_1}y^{l_1}y^{l_2}}^{k_3}
\right]
y_{k_1}^{l_1}y_{k_2}^{l_2}y_{k_3}^{l_3}
+
\endaligned
$$
$$
\small
\aligned
& \
\ \ \ \ \
+
\sum_{l_1,l_2,l_3,l_4=1}^m\ \sum_{k_1,k_2,k_3,k_4=1}^n
\left[
-
\delta_{l_4}^j\,\delta_{i_1,\ i_2,\ i_3}^{k_1,k_2,k_3}\,
\mathcal{X}_{y^{l_1}y^{l_2}y^{l_3}}^{k_4}
\right]
y_{k_1}^{l_1}y_{k_2}^{l_2}y_{k_3}^{l_3}y_{k_4}^{l_4}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1=1}^m\ \sum_{k_1,k_2=1}^n
\left[
\delta_{i_1,\ i_2}^{k_1,k_2}\,
\mathcal{Y}_{x^{i_3}y^{l_1}}^j
+
\delta_{i_3,\ i_1}^{k_1,k_2}\,
\mathcal{Y}_{x^{i_2}y^{l_1}}^j
+
\delta_{i_2,\ i_3}^{k_1,k_2}\,
\mathcal{Y}_{x^{i_1}y^{l_1}}^j
-
\right. \\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.
-
\delta_{l_1}^j\,\delta_{i_1}^{k_1}\,
\mathcal{X}_{x^{i_2}x^{i_3}}^{k_2}
-
\delta_{l_1}^j\,\delta_{i_2}^{k_1}\,
\mathcal{X}_{x^{i_1}x^{i_3}}^{k_2}
-
\delta_{l_1}^j\,\delta_{i_3}^{k_1}\,
\mathcal{X}_{x^{i_1}x^{i_2}}^{k_2}
\right]
y_{k_1,k_2}^{l_1}
+
\endaligned
$$
\def5.22}\begin{equation{5.4}\begin{equation}
\small
\aligned
& \
\ \ \ \ \
+
\sum_{l_1,l_2=1}^m\ \sum_{k_1,k_2,k_3=1}^n
\left[
\delta_{i_1,\ i_2,\ i_3}^{k_1,k_2,k_3}\,
\mathcal{Y}_{y^{l_1}y^{l_2}}^j
+
\delta_{i_1,\ i_2,\ i_3}^{k_3,k_1,k_2}\,
\mathcal{Y}_{y^{l_1}y^{l_2}}^j
+
\delta_{i_1,\ i_2,\ i_3}^{k_2,k_3,k_1}\,
\mathcal{Y}_{y^{l_1}y^{l_2}}^j
-
\right.
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \
\left.
-
\delta_{l_1}^j\,\delta_{i_1,\ i_2}^{k_2,k_3}\,
\mathcal{X}_{x^{i_3}y^{l_2}}^{k_1}
-
\delta_{l_1}^j\,\delta_{i_1,\ i_3}^{k_2,k_3}\,
\mathcal{X}_{x^{i_2}y^{l_2}}^{k_1}
-
\delta_{l_1}^j\,\delta_{i_2,\ i_3}^{k_2,k_3}\,
\mathcal{X}_{x^{i_1}y^{l_2}}^{k_1}
-
\right.
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \
\left.
-
\delta_{l_2}^j\,\delta_{i_1,\ i_2}^{k_3,k_1}\,
\mathcal{X}_{x^{i_3}y^{l_1}}^{k_2}
-
\delta_{l_2}^j\,\delta_{i_1,\ i_3}^{k_3,k_1}\,
\mathcal{X}_{x^{i_2}y^{l_1}}^{k_2}
-
\delta_{l_2}^j\,\delta_{i_2,\ i_3}^{k_3,k_1}\,
\mathcal{X}_{x^{i_1}y^{l_1}}^{k_2}
-
\right.
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \
\left.
-
\delta_{l_2}^j\,\delta_{i_1,\ i_2}^{k_1,k_2}\,
\mathcal{X}_{x^{i_3}y^{l_1}}^{k_3}
-
\delta_{l_2}^j\,\delta_{i_1,\ i_3}^{k_1,k_2}\,
\mathcal{X}_{x^{i_2}y^{l_1}}^{k_3}
-
\delta_{l_2}^j\,\delta_{i_2,\ i_3}^{k_1,k_2}\,
\mathcal{X}_{x^{i_1}y^{l_1}}^{k_3}
\right]
y_{k_1}^{l_1}y_{k_2,k_3}^{l_2}
+ \\
\endaligned
\end{equation}
$$
\small
\aligned
& \
\ \ \ \ \
+
\sum_{l_1,l_2,l_3=1}^m\ \sum_{k_1,k_2,k_3,k_4=1}^n
\left[
-
\delta_{l_3}^j\,\delta_{i_1,\ i_2,\ i_3}^{k_1,k_2,k_3}\,
\mathcal{X}_{y^{l_1}y^{l_2}}^{k_4}
-
\delta_{l_3}^j\,\delta_{i_1,\ i_2,\ i_3}^{k_2,k_3,k_1}\,
\mathcal{X}_{y^{l_1}y^{l_2}}^{k_4}
-
\delta_{l_3}^j\,\delta_{i_1,\ i_2,\ i_3}^{k_3,k_2,k_1}\,
\mathcal{X}_{y^{l_1}y^{l_2}}^{k_4}
-
\right.
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \
\left.
-
\delta_{l_2}^j\,\delta_{i_1,\ i_2,\ i_3}^{k_3,k_4,k_1}\,
\mathcal{X}_{y^{l_1}y^{l_3}}^{k_2}
-
\delta_{l_2}^j\,\delta_{i_1,\ i_2,\ i_3}^{k_3,k_1,k_4}\,
\mathcal{X}_{y^{l_1}y^{l_3}}^{k_2}
-
\right.
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.
-
\delta_{l_2}^j\,\delta_{i_1,\ i_2,\ i_3}^{k_1,k_3,k_4}\,
\mathcal{X}_{y^{l_1}y^{l_3}}^{k_2}
\right]
y_{k_1}^{l_1}y_{k_2}^{l_2}y_{k_3,k_4}^{l_3}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2=1}^m\ \sum_{k_1,k_2,k_3,k_4=1}^n
\left[
-
\delta_{l_2}^j\,\delta_{i_1,\ i_2,\ i_3}^{k_1,k_2,k_3}\,
\mathcal{X}_{y^{l_1}}^{k_3}
-
\delta_{l_2}^j\,\delta_{i_1,\ i_2,\ i_3}^{k_2,k_4,k_1}\,
\mathcal{X}_{y^{l_1}}^{k_3}
-
\delta_{l_2}^j\,\delta_{i_1,\ i_2,\ i_3}^{k_4,k_1,k_2}\,
\mathcal{X}_{y^{l_1}}^{k_3}
\right]
y_{k_1,k_2}^{l_1}y_{k_3,k_4}^{l_2}
+ \\
\endaligned
$$
$$
\small
\aligned
& \
\ \ \ \ \
+
\sum_{l_1=1}^m\ \sum_{k_1,k_2,k_3=1}^n
\left[
\delta_{i_1,\ i_2, \i_3}^{k_1,k_2,k_3}\,
\mathcal{Y}_{y^{l_1}}^j
-
\delta_{l_1}^j\,\delta_{i_1,\ i_2}^{k_1,k_2}\,
\mathcal{X}_{x^{i_3}}^{k_3}
-
\delta_{l_1}^j\,\delta_{i_1,\ i_3}^{k_1,k_2}\,
\mathcal{X}_{x^{i_2}}^{k_3}
-
\delta_{l_1}^j\,\delta_{i_2,\ i_3}^{k_1,k_2}\,
\mathcal{X}_{x^{i_1}}^{k_3}
\right]
y_{k_1,k_2,k_3}^{l_1}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2=1}^m\ \sum_{k_1,k_2,k_3,k_4=1}^n
\left[
-
\delta_{l_2}^j\,\delta_{i_1,\ i_2,\ i_3}^{k_1,k_2,k_3}\,
\mathcal{X}_{y^{l_1}}^{k_4}
-
\delta_{l_2}^j\,\delta_{i_1,\ i_2,\ i_3}^{k_4,k_1,k_2}\,
\mathcal{X}_{y^{l_1}}^{k_3}
-
\delta_{l_2}^j\,\delta_{i_1,\ i_2,\ i_3}^{k_3,k_4,k_1}\,
\mathcal{X}_{y^{l_1}}^{k_2}
-
\right.
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \
\left.
-
\delta_{l_1}^j\,\delta_{i_1,\ i_2,\ i_3}^{k_2,k_3,k_4}\,
\mathcal{X}_{y^{l_2}}^{k_1}
\right]
y_{k_1}^{l_1}y_{k_2,k_3,k_4}^{l_2}.
\endaligned
$$
\subsection*{5.5.~Final synthesis}
To obtain the general formula for ${\bf Y}_{ i_1, \dots, i_\kappa}^j$,
we have to achieve the synthesis between the two formulas~\thetag{
3.74} and~\thetag{ 4.19}. We start with~\thetag{ 3.74} and we modify
it until we reach the final formula for ${\bf Y}_{ i_1, \dots,
i_\kappa }^j$.
We have to add the $\mu_1+\cdots+\mu_d$
sums $\sum_{ l_{ 1:1} =1 }^m \cdots \sum_{
l_{ 1: \mu_1 } =1 }^m \cdots \cdots \sum_{ l_{ d:1} =1}^m \cdots
\sum_{l_{ d: \mu_d }= 1}^m$, together with various indices
$l_\alpha$. About these indices, the only point which is not obvious
may be analyzed as follows.
A permutation $\sigma \in \mathfrak{ F}_{
\mu_1\lambda_1 + \cdots + \mu_d \lambda_d }^{ (\mu_1, \lambda_1),
\dots, (\mu_d, \lambda_d)}$ yields the list:
\def5.22}\begin{equation{5.6}\begin{equation}
\aligned
& \
\sigma(1\!:\!1\!:\!1),\dots,\sigma(1\!:\!1\!:\!\lambda_1),
\dots
\sigma(1\!:\!\mu_1\!:\!1),\dots,\sigma(1\!:\!\mu_1\!:\!\lambda_1),
\dots
\\
& \
\ \ \ \ \ \ \ \ \
\dots,
\sigma(d\!:\!1\!:\!1),\dots,\sigma(1\!:\!1\!:\!\lambda_d),
\dots
\sigma(d\!:\!\mu_d\!:\!1),\dots,\sigma(d\!:\!\mu_d\!:\!\lambda_d),
\endaligned
\end{equation}
In the end of the
sixth line of~\thetag{ 3.74}, the last term $\sigma ( d\! : \!
\mu_d \! : \! \lambda_d )$ of the above list appears as the subscript
of the upper index $k_{ \sigma (d: \mu_d: \lambda_d )}$ of the term
$\mathcal{ X }^{ k_{ \sigma( d:\mu_d: \lambda_d) }}$. According to the
formal rules of Theorem~4.19, we have to multiply the partial
derivative of $\mathcal{ X }^{ k_{ \sigma( d:\mu_d: \lambda_d ) }}$ by
a certain Kronecker symbol $\delta_{ l_\alpha}^j$. The question is:
{\it what is the subscript $\alpha$ and how
to denote it}?
As explained before the statement of Theorem~4.19, the subscript
$\alpha$ is obtained as follows. The term $\sigma ( d\! : \! \mu_d \!
: \! \lambda_d )$ is of the form $(e\! : \! \nu_e \! : \! \gamma_e
)$, for some $e$ with $1\leq e \leq d$, for some $\nu_e$ with $1\leq
\nu_e \leq \mu_e$ and for some $\gamma_e$ with $1\leq \gamma_e \leq
\lambda_e$. The single pure jet variable
\def5.22}\begin{equation{5.7}\begin{equation}
\aligned
y_{k_{e:\nu_e:1},\dots,
k_{e:\nu_e:\gamma_e},\dots,
k_{e:\nu_e:\lambda_e}}^{l_{e:\nu_e}}
\endaligned
\end{equation}
appears inside the total monomial
\def5.22}\begin{equation{5.8}\begin{equation}
\aligned
\prod_{1\leq\nu_1\leq\mu_1}\,
y_{k_{1:\nu_1:1},\dots,k_{1:\nu_1:\lambda_1}}^{l_{1:\nu_1}}
\ \cdots \
\prod_{1\leq\nu_d\leq\mu_d}\,
y_{k_{d:\nu_d:1},\dots,k_{d:\nu_d:\lambda_d}}^{l_{d:\nu_d}},
\endaligned
\end{equation}
placed at the end of the formula for ${\bf Y }_{ i_1, \dots,
i_\kappa}^j$ ({\it see} in advance formula~\thetag{ 5.13} below; this
total monomial generalizes to several dependent variables the total
monomial appearing in the last line of~\thetag{ 3.74}). According to
the rule explained before the statement of Theorem~4.18, the index
$l_\alpha$ must be equal to $l_{e : \nu_e }$, since
$l_{e : \nu_e }$ is attached to the monomial~\thetag{ 5.7}.
Coming back to the term
$\sigma ( d\! : \! \mu_d \! : \! \lambda_d )$, we shall denote this
index by
\def5.22}\begin{equation{5.9}\begin{equation}
\aligned
l_{e:\nu_e}
=:
l_{\pi(e:\nu_e:\gamma_e)}
=:
l_{\pi\sigma(d:\mu_d:\lambda_d)},
\endaligned
\end{equation}
where the symbol $\pi$ denotes the projection
from the set
\def5.22}\begin{equation{5.10}\begin{equation}
\aligned
\{
1\!:\!1\!:\!1,\dots,1\!:\!\mu_1\!:\!\lambda_1,
\dots\dots,
d\!:\!1\!:\!1,\dots,d\!:\!\mu_d\!:\!\lambda_d
\}
\endaligned
\end{equation}
to the set
\def5.22}\begin{equation{5.11}\begin{equation}
\aligned
\{
1\!:\!1,\dots,1\!:\!\mu_1,
\dots,
d\!:\!1,\dots,d\!:\!\mu_d
\}
\endaligned
\end{equation}
simply defined by $\pi(e \! : \! \nu_e \! : \! \gamma_e) := (e \! : \!
\nu_e)$.
In conclusion, by means of this formalism, we may write down the
complete generalization of Theorems~2.24, 3.73 and~4.18 to several
independent variables and several dependent variables
\def5.12}\begin{theorem{5.12}\begin{theorem}
For $j = 1, \dots, m$, for every $\kappa \geq 1$ and for every choice
of $\kappa$ indices $i_1,\dots, i_\kappa$ in the set $\{ 1, 2, \dots,
n\}$, the general expression of ${\bf Y}_{i_1, \dots, i_\kappa }^j$ is
as follows{\rm :}
\def5.22}\begin{equation{5.13}\begin{equation}
\small
\boxed{
\aligned
{\bf Y}_{i_1, \dots, i_\kappa}^j
&
=
\mathcal{Y}_{x^{i_1}\cdots x^{i_\kappa}}^j
+
\sum_{d=1}^{\kappa+1}
\ \
\sum_{1\leq\lambda_1<\cdots<\lambda_d\leq\kappa}
\ \
\sum_{\mu_1\geq 1,\dots,\mu_d\geq 1}
\
\sum_{
\mu_1\lambda_1
+
\cdots
+
\mu_d\lambda_d\leq \kappa+1}
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \
\sum_{l_{1:1}=1}^m
\cdots
\sum_{l_{1:\mu_1}=1}^m
\cdots\cdots
\sum_{l_{d:1}=1}^m
\cdots
\sum_{l_{d:\mu_d}=1}^m
\\
&
\sum_{k_{1:1:1},\dots,k_{1:1:\lambda_1}=1}^n
\cdots \
\sum_{k_{1:\mu_1:1},\dots,k_{1:\mu_1:\lambda_1}=1}^n
\cdots\cdots \
\sum_{k_{d:1:1},\dots,k_{d:1:\lambda_d}=1}^n
\cdots \
\sum_{k_{d:\mu_d:1},\dots,k_{d:\mu_d:\lambda_d}=1}^n
\\
&
\left[
\aligned
&
\sum_{\sigma\in\mathfrak{F}_{\mu_1\lambda_1+\cdots+\mu_d\lambda_d}^{
(\mu_1,\lambda_1),\dots,(\mu_d,\lambda_d)}}
\
\sum_{\tau\in\mathfrak{S}_\kappa^{
\mu_1\lambda_1+\cdots+\mu_d\lambda_d}}\,
\delta_{i_{\tau(1)},\dots,i_{\tau(\mu_1\lambda_1)},\dots,
i_{\tau(\mu_1\lambda_1+\cdots+\mu_d\lambda_d)}}^{
k_{\sigma(1:1:1)},\dots,k_{\sigma(1:\mu_1:\lambda_1)},
\dots,k_{\sigma(d:\mu_d:\lambda_d)}}
\cdot
\\
& \
\ \ \ \ \
\cdot
\frac{\partial^{\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d+
\mu_1+\cdots+\mu_d}
\mathcal{Y}^j}{
\partial x^{i_{\tau(\mu_1\lambda_1+\cdots+\mu_d\lambda_d+1)}}\cdots
\partial x^{i_{\tau(\kappa)}}
\partial y^{l_{1:1}}\cdots\partial y^{l_{d:\mu_d}}}\
- \\
&
-
\sum_{\sigma\in\mathfrak{F}_{\mu_1\lambda_1+\cdots+\mu_d\lambda_d}^{
(\mu_1,\lambda_1),\dots,(\mu_d,\lambda_d)}}
\
\sum_{\tau\in\mathfrak{S}_\kappa^{
\mu_1\lambda_1+\cdots+\mu_d\lambda_d-1}}\,
\delta_{i_{\tau(1)},\dots,i_{\tau(\mu_1\lambda_1)},\dots,
i_{\tau(\mu_1\lambda_1+\cdots+\mu_d\lambda_d-1)}}^{
k_{\sigma(1:1:1)},\dots,k_{\sigma(1:\mu_1:\lambda_1)},
\dots,k_{\sigma(d:\mu_d:\lambda_d-1)}}
\cdot
\\
& \
\ \ \ \ \
\cdot
\delta_{l_{\pi\sigma(d:\mu_d:\lambda_d)}}^j
\cdot
\frac{\partial^{\kappa-\mu_1\lambda_1-\cdots-\mu_d\lambda_d
+\mu_1+\cdots+\mu_d}\mathcal{X}^{k_{\sigma(d:\mu_d:\lambda_d)}}}{
\partial x^{i_{\tau(\mu_1\lambda_1+\cdots+\mu_d\lambda_d)}}\cdots
\partial x^{i_{\tau(\kappa)}}
\partial y^{l_{1:1}}\cdots
\widehat{\partial y^{l_{\pi\sigma(d:\mu_d:\lambda_d)}}}
\cdots\partial y^{l_{d:\mu_d}}}
\endaligned
\right]
\cdot
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\cdot
\prod_{1\leq\nu_1\leq\mu_1}\,
y_{k_{1:\nu_1:1},\dots,k_{1:\nu_1:\lambda_1}}^{l_{1:\nu_1}}
\ \cdots \
\prod_{1\leq\nu_d\leq\mu_d}\,
y_{k_{d:\nu_d:1},\dots,k_{d:\nu_d:\lambda_d}}^{l_{d:\nu_d}}.
\endaligned
}
\end{equation}
\end{theorem}
In this formula, the coset $\mathfrak{ F }_{ \mu_1 \lambda_1 + \cdots
+ \mu_d \lambda_d }^{ ( \mu_1, \lambda_1 ),\dots, ( \mu_d,
\lambda_d)}$ was defined in equation~\thetag{ 3.71};
as in Theorem~3.73, we have made the
identification{\rm :}
\def5.22}\begin{equation{5.14}\begin{equation}
\{1,\dots,\kappa\}
\equiv
\{
1\!:\!1\!:\!1,
\dots,
1\!:\!\mu_1\!:\!\lambda_1,
\text{\rm \dots\dots},
d\!:\!1\!:\!1,
\dots,
d\!:\!\mu_d\!:\!\lambda_d
\}.
\end{equation}
\subsection*{ 5.15.~Deduction of the
most general multivariate Fa\`a di Bruno formula} Let $n \in \N$ with
$n\geq 1$, let $x = (x^1,\dots, x^n) \in \K^n$, let $m\in \N$ with
$m\geq 1$, let $g^j = g^j ( x^1, \dots, x^n)$, $j=1, \dots, m$, be
$\mathcal{ C}^\infty$-smooth functions from $\K^n$ to $\K^m$, let $y =
(y^1, \dots, y^m) \in \K^m$ and let $f = f(y^1, \dots, y^m)$ be a
$\mathcal{ C}^\infty$ function from $\K^m$ to $\K$. The goal is to
obtain an explicit formula for the partial derivatives of the
composition $h := f\circ g$, namely
\def5.22}\begin{equation{5.16}\begin{equation}
h(x^1, \dots, x^n) := f
\left(
g^1(x^1,\dots, x^n), \dots, g^m(x^1,\dots, x^n)
\right).
\end{equation}
For $j= 1,\dots, m$, for $\lambda \in \N$ with $\lambda \geq 1$ and
for arbitrary indices $i_1, \dots, i_\lambda = 1, \dots, n$, we shall
abbreviate the partial derivative $\frac{ \partial^\lambda g^j}{
\partial x^{i_1} \cdots \partial x^{i_\lambda}}$ by $g_{i_1,\dots,
i_\lambda }^j$ and similarly for $h_{i_1, \dots, i_\lambda}$. For
arbitrary indices $l_1, \dots, l_\lambda = 1, \dots, m$, the partial
derivative $\frac{ \partial^\lambda f}{ \partial y^{l_1} \cdots
\partial y^{l_\lambda }}$ will be abbreviated by $f_{ l_1, \dots,
l_\lambda }$.
Appying the chain rule, we may compute:
\def5.22}\begin{equation{5.17}\begin{equation}
\small
\aligned
h_{i_1}
&
=
\sum_{l_1=1}^m\,f_{l_1}
\left[
g_{i_1}^{l_1}
\right],
\\
h_{i_1,i_2}
&
=
\sum_{l_1,l_2=1}^m\,f_{l_1,l_2}
\left[
g_{i_1}^{l_1}\,g_{i_2}^{l_2}
\right]
+
\sum_{l_1=1}^m\,f_{l_1}
\left[
g_{i_1,i_2}^{l_1}
\right],
\\
h_{i_1,i_2,i_3}
&
=
\sum_{l_1,l_2,l_3=1}^m\,f_{l_1,l_2,l_3}
\left[
g_{i_1}^{l_1}\,g_{i_2}^{l_2}\,g_{i_3}^{l_3}
\right]
+
\sum_{l_1,l_2=1}^m\,f_{l_1,l_2}
\left[
g_{i_1}^{l_1}\,g_{i_2,i_3}^{l_2}
+
g_{i_2}^{l_1}\,g_{i_1,i_3}^{l_2}
+
g_{i_3}^{l_1}\,g_{i_1,i_2}^{l_2}
\right]
+ \\
& \
\ \ \ \ \
+
\sum_{l_1=1}^m\,f_{l_1}
\left[
g_{i_1,i_2,i_3}^{l_1}
\right],
\endaligned
\end{equation}
$$
\small
\aligned
h_{i_1,i_2,i_3,i_4}
&
=
\sum_{l_1,l_2,l_3,l_4=1}^m\,f_{l_1,l_2,l_3,l_4}
\left[
g_{i_1}^{l_1}\,g_{i_2}^{l_2}\,g_{i_3}^{l_3}\,g_{i_4}^{l_4}
\right]
+ \\
& \
\ \ \ \ \
\sum_{l_1,l_2,l_3=1}^m\,f_{l_1,l_2,l_3}
\left[
g_{i_2}^{l_1}\,g_{i_3}^{l_2}\,g_{i_1,i_4}^{l_3}
+
g_{i_3}^{l_1}\,g_{i_1}^{l_2}\,g_{i_2,i_4}^{l_3}
+
g_{i_1}^{l_1}\,g_{i_2}^{l_2}\,g_{i_3,i_4}^{l_3}
+
\right.
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.
+
g_{i_1}^{l_1}\,g_{i_4}^{l_2}\,g_{i_2,i_3}^{l_3}
+
g_{i_2}^{l_1}\,g_{i_4}^{l_2}\,g_{i_3,i_1}^{l_3}
+
g_{i_3}^{l_1}\,g_{i_4}^{l_2}\,g_{i_1,i_2}^{l_3}
\right]
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2=1}^m\,f_{l_1,l_2}
\left[
g_{i_1,i_2}^{l_1}\,g_{i_3,i_4}^{l_2}
+
g_{i_1,i_3}^{l_1}\,g_{i_2,i_4}^{l_2}
+
g_{i_1,i_4}^{l_1}\,g_{i_2,i_3}^{l_2}
\right]
+ \\
& \
\ \ \ \ \
+
\sum_{l_1,l_2=1}^m\,f_{l_1,l_2}
\left[
g_{i_1}^{l_1}\,g_{i_2,i_3,i_4}^{l_2}
+
g_{i_2}^{l_1}\,g_{i_1,i_3,i_4}^{l_2}
+
g_{i_3}^{l_1}\,g_{i_1,i_2,i_4}^{l_2}
+
g_{i_4}^{l_1}\,g_{i_1,i_2,i_3}^{l_2}
\right]
+ \\
& \
\ \ \ \ \
+
\sum_{l_1=1}^m\,f_{l_1}
\left[
g_{i_1,i_2,i_3,i_4}^{l_1}
\right].
\endaligned
$$
Introducing the derivations
\def5.22}\begin{equation{5.18}\begin{equation}
\small
\aligned
F_i^2
&
:=
\sum_{k_1=1}^n\,\sum_{l_1=1}^m\,g_{k_1,i}^{l_1}\,
\frac{\partial}{\partial g_{k_1}^{l_1}}
+
\sum_{l_1=1}^m\,g_i^{l_1}\left(
\sum_{l_2=1}^m\,f_{l_1,l_2}\,\frac{\partial}{\partial f_{l_2}}
\right), \\
F_i^3
&
:=
\sum_{k_1=1}^n\,\sum_{l_1=1}^m\,g_{k_1,i}^{l_1}\,
\frac{\partial}{\partial g_{k_1}^{l_1}}
+
\sum_{k_1,k_2=1}^n\,\sum_{l_1=1}^m\,g_{k_1,k_2,i}^{l_1}\,
\frac{\partial}{\partial g_{k_1,k_2}^{l_1}}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1=1}^m\,g_i^{l_1}
\left(
\sum_{l_2=1}^m\,f_{l_1,l_2}\,\frac{\partial}{\partial f_{l_2}}
+
\sum_{l_2,l_3=1}^m\,f_{l_1,l_2,l_3}\,
\frac{\partial}{\partial f_{l_2,l_3}}
\right),
\\
\text{\bf
\dots\dots}
& \
\ \ \ \ \
\text{\bf
\dots\dots\dots\dots\dots\dots\dots\dots\dots
\dots\dots\dots\dots\dots\dots\dots\dots\dots
\dots
} \\
F_i^\lambda
&
:=
\sum_{k_1=1}^n\,\sum_{l_1=1}^m\,g_{k_1,i}^{l_1}\,
\frac{\partial}{\partial g_{k_1}^{l_1}}
+
\sum_{k_1,k_2=1}^n\,\sum_{l_1=1}^m\,g_{k_1,k_2,i}^{l_1}\,
\frac{\partial}{\partial g_{k_1,k_2}^{l_1}}
+
\cdots
+ \\
& \
\ \ \ \ \
+
\sum_{k_1,k_2,\dots,k_{\lambda-1}=1}^n\,\sum_{l_1=1}^m\,
g_{k_1,k_2,\dots,k_{\lambda-1},i}
\
\frac{\partial}{\partial g_{k_1,\dots,k_{\lambda-1}}^{l_1}}
+ \\
& \
\ \ \ \ \
+
\sum_{l_1=1}^m\,g_i^{l_1}
\left(
\sum_{l_2=1}^m\,f_{l_1,l_2}\,\frac{\partial}{\partial f_{l_2}}
+
\sum_{l_2,l_3=1}^m\,f_{l_1,l_2,l_3}\,
\frac{\partial}{\partial f_{l_2,l_3}}
+
\right.
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \
\left.
+
\cdots
+
\sum_{l_2,l_3,\dots,l_\lambda}\,
f_{l_1,l_2,l_3,\dots,l_\lambda}\,
\frac{\partial}{\partial f_{l_2,l_3,\dots,l_\lambda}}
\right),
\endaligned
\end{equation}
we observe that the following
induction relations hold:
\def5.22}\begin{equation{5.19}\begin{equation}
\aligned
h_{i_1,i_2}
&
=
F_{i_2}^2
\left(
h_{i_1}
\right),
\\
h_{i_1,i_2,i_3}
&
=
F_{i_3}^3
\left(
h_{i_1,i_2}
\right),
\\
\text{\bf
\dots\dots
}
& \
\ \ \ \ \
\text{\bf
\dots\dots\dots\dots\dots
}
\\
h_{i_1,i_2,\dots,i_\lambda}
&
=
F_{i_\lambda}^\lambda
\left(
h_{i_1,i_2,\dots,i_{\lambda-1}}
\right).
\endaligned
\end{equation}
To obtain the explicit version of the Fa\`a di Bruno in the case of
several variables $(x^1, \dots, x^n)$ and several variables $(y^1,
\dots, y^m)$, it suffices to extract from the expression of ${\bf Y
}_{ i_1,\dots, i_\kappa }^j$ provided by Theorem~5.12 only the terms
corresponding to $\mu_1 \lambda_1 + \cdots + \mu_d\lambda_d = \kappa$,
dropping all the $\mathcal{ X}$ terms. After some simplifications and
after a translation by means of an elementary dictionary, we obtain
the fourth and the most general multivariate Fa\`a di Bruno formula.
\def5.12}\begin{theorem{5.20}\begin{theorem}
For every integer $\kappa \geq 1$ and for every choice of indices
$i_1, \dots, i_\kappa$ in the set $\{ 1, 2, \dots, n\}$, the
$\kappa$-th partial derivative of the composite function
\def5.22}\begin{equation{5.21}\begin{equation}
h = h( x^1,
\dots, x^n) =
f \left( g^1 (x^1, \dots, x^n),\dots, g^m (x^1, \dots,
x^n) \right)
\end{equation}
with respect to the variables $x^{i_1}, \dots,
x^{i_\kappa}$ may be expressed as an explicit polynomial depending on
the partial derivatives of $f$, on the partial derivatives of the
$g^j$ and having integer coefficients{\rm :}
\def5.22}\begin{equation{5.22}\begin{equation}
\boxed{
\aligned
\frac{\partial^\kappa h}{\partial x^{i_1}\cdots
\partial x^{i_\kappa}}
&
=
\sum_{d=1}^\kappa
\
\sum_{1\leq \lambda_1 < \cdots < \lambda_d \leq \kappa}
\
\sum_{\mu_1\geq 1,\dots,\mu_d\geq 1}
\
\sum_{\mu_1\lambda_1+\cdots+\mu_d\lambda_d=\kappa}
\\
& \
\ \ \ \ \
\sum_{l_{1:1},\dots,l_{1:\mu_1}=1}^m
\cdots
\sum_{l_{d:1},\dots,l_{d:\mu_d}=1}^m
\
\frac{\partial^{\mu_1+\cdots+\mu_d}f}{
\partial y^{l_{1:1}}\cdots
\partial y^{l_{1:\mu_1}}\cdots
\partial y^{l_{d:1}}\cdots
\partial y^{l_{d:\mu_d}}
}
\\
&
\ \ \ \ \ \ \ \ \ \
\left[
\aligned
&
\sum_{\sigma\in\mathfrak{F}_\kappa^{
(\mu_1,\lambda_1),\dots,(\mu_d,\lambda_d)}}
\
\prod_{1\leq\nu_1\leq\mu_1}
\
\frac{\partial^{\lambda_1} g^{l_{1:\nu_1}}}{\partial
x^{i_{\sigma(1:\nu_1:1)}}\cdots
\partial x^{i_{\sigma(1:\nu_1:\lambda_1)}}}
\
\text{\bf \dots}
\\
& \
\ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \
\text{\bf \dots}
\prod_{1\leq\nu_d\leq\mu_d}
\
\frac{\partial^{\lambda_d} g^{l_{d:\nu_d}}}{\partial
x^{i_{\sigma(d:\nu_d:1)}}\cdots
\partial x^{i_{\sigma(d:\nu_d:\lambda_d)}}}
\endaligned
\right].
\endaligned
}
\end{equation}
\end{theorem}
|
{
"timestamp": "2004-11-30T11:25:31",
"yymm": "0411",
"arxiv_id": "math/0411650",
"language": "en",
"url": "https://arxiv.org/abs/math/0411650"
}
|
\section{Infrared spectroscopy and the 1.083-\boldmath $\mu$m He\,{\sc i} P Cygni profile}
The WC7+O4-5 binary WR\,140 (HD 193793) is the archetypal episodic dust-forming
Wolf-Rayet Colliding Wind Binary. It is particularly luminous in X-rays and was the
first WR system to show non-thermal radio emission and episodic dust formation.
Variations in its X-ray, radio and infrared properties were linked to its
binary motion by Williams et al. (1990), and it has since been the subject of
theoretical and observational studies of colliding-wind phenomena. As part of
the campaign planned for the 2001 periastron passage, we observed a series of
near-infrared spectra having resolutions $R \simeq 1000$ using the United Kingdom
Infrared Telescope (UKIRT), Hawaii, and the Mt Abu Infrared Telescope, India
(Varricatt, Williams \& Ashok 2004). The 1.083-$\mu$m He\,{\sc i} line was
also observed at $R = 4700$ to look for colliding-wind effects (cf. Stevens
\& Howarth 1999). Previous observations of WR\,140 (e.g. Eenens \& Williams 1994)
had shown flat-topped profiles for this line, but they were taken far from
periastron.
Our observation at $\phi = 0.96$, however, showed the appearance of a strong,
blue-shifted sub-peak on the emission profile (Fig.\,1), which we interpret
as being formed in a shell of compressed WC7 stellar wind material flowing along
the wind-collision region (WCR). The shape of the WCR can be approximated (e.g.
Usov 1992., L\"uhrs 1996) by a hollow cone symmetric about the axis joining the
stars and having its apex towards the WC7 star, which has the higher
mass-loss rate.
\begin{figure}[!ht]
\plottwo{WilliamsFig1a.eps}{WilliamsFig1b.eps}
\caption{Left: profiles of the 1.083-$\mu$m He\,{\sc i} line in 2000 October
($\phi = 0.96$), 2001 March ($\phi = 0.01$) and 2003 ($\phi = 0.29$), which
is typical of most of the orbit).
Right: constraints on the opening angle ($\theta$) of the wind-collision
region as a function of orbital inclination set by the varying strength
of the absorption component of the He\,{\sc i} line.}
\end{figure}
At $\phi = 0.96$, the system is near conjunction with the O star in front of
the WC7 star, so we expect the compressed WC7 material to be flowing towards
us. At the same time, we are viewing the two stars mostly through the wind
of the O star, accounting for our observation (Fig.\,1) that the absorption
component of the He\,{\sc i} line is weaker than at phases when the stars
are observed through the He-rich WC7 stellar wind. At this phase, the angle
$\psi$ between our sightline and the axis joining the two stars is smaller
than the opening angle ($\theta$) of the wind-collision `cone'.
The value of $\psi$ varies round the orbit and can be calculated for the
phases of our observations for a range of values of the inclination
so we can use successive measurements of the strength of the absorption
component to explore permitted values of $\theta$ and inclination.
At the same time, we model the variation of the radial velocity (RV) of the subpeak, including
its width, following L\"uhrs (1997) but with the difference that, instead of
treating the velocity of the compressed wind as a free parameter to be fit,
we calculate it from those of the WC7 and O star winds and $\theta$ following
Cant\'o, Raga \& Wilkins (1996).
This models the movement of the sub-peak to the red end of the profile by
the time of our first post-periastron observation ($\phi = 0.01$, Fig.\,1).
At this time, the absorption component was strong because we observed both
stars through the WC7 stellar wind. Two years later, by $\phi = 0.29$, the
emission sub-peak had vanished as the stars had moved further apart, but
the absorption component was still strong. This puts a useful upper limit
on $\theta$ (Fig.\,1). Modelling the RVs of the sub-peak indicates an
orbital inclination $i \simeq 65\deg\pm10\deg$, implying (Fig.\,1)
$\theta \simeq 53\deg\pm10\deg$.
\section{Infrared imaging of the dust emission}
We imaged WR\,140 in the infrared with four different instruments:
PHARO +AO on the Hale telescope, INGRID+AO (NAOMI) on the William Herschel
Telescope (WHT), and UIST and Michelle, both on UKIRT. The UKIRT observations
were made at longer wavelengths, where the contrast between dust and stellar
emission was greater, especially as the dust cooled, but the resolution is
lower. Single stars were observed to determine the image psfs and the images
of WR\,140 were restored using maximum entropy methods.
A test of the validity of our procedures is provided by two images of
WR\,140 (Fig.\,2) observed at about the same time with two different
instruments and reduced with different software packages. The basic
structures are similar, with a `bar' of dust emission to the south
and another feature to the east. Similar structures are evident in all the
images, e.g. a 3.99-$\mu$m image observed with UIST in 2003 (Fig.\,3).
\begin{figure}[!ht]
\plottwo{WilliamsFig2a.ps}{WilliamsFig2b.ps}
\caption{Two 2-$\mu$m images of WR\,140 observed in 2002 July with different
instruments: NAOMI/INGRID on the WHT (left) and AO/PHARO system on the Hale (right).
Both images have NE at top left and the scale is arcsec from the central star (`S').
Dust emission knots (`C', `D' and `E') correspond to those identified in the
2001 images of Monnier et al. (2002).}
\end{figure}
We can use the contemporaneous 3.6-$\mu$m and 3.99-$\mu$m UIST observations to
measure the infrared colours of the emission features relative to that of the star,
confirming that the the features are 0.4 mag. redder in [3.6]--[3.99] than the star,
consistent with their being heated dust.
We identified several knots of dust emission with those identified in the
aperture-masking images observed in 2001 by Monnier, Tuthill \& Danchi (2002)
from the similarity of their position angles (P.A.) relative to the star,
and use these and other knots to track the expansion of the dust cloud.
The observations are summarized in Table 1, together with the P.A.s and
radial distances ($r$) of the emission knot, `C'. These distances, together
with those from the images of Monnier et al. and measured from the image
given by Tuthill et al. (2003), and similar data for knot `E' to the east,
are plotted against date in Fig.\,3, and show remarkably linear expansion.
\begin{table}
\caption{Log of imaging observations of WR\,140 with position angle and
distance of dust-emission knot `C' from the star}
\smallskip
\begin{center}
{\small
\begin{tabular}{lcccccc}
\tableline
\noalign{\smallskip}
Instrument & Date & Phase & Scale &$\lambda$obs & P.A. & r \\
& & & (mas) & ($\mu$m) & ($\deg$) & (mas)\\
\noalign{\smallskip}
\tableline
\noalign{\smallskip}
Hale: PHARO+AO & 2001.68 & 0.073 & 25 & 2.2 & & \\
Hale: PHARO+AO & 2002.31 & 0.153 & 25 & 2.2 & 216 & 278 \\
WHT: INGRID+AO & 2002.51 & 0.178 & 26.3 & 2.27 & 219 & 341 \\
Hale: PHARO+AO & 2002.56 & 0.184 & 25 & 2.2 & 217 & 352 \\
UKIRT: UIST & 2002.89 & 0.226 & 16.5 & 3.6 & 217 & 498 \\
UKIRT: UIST & 2002.89 & 0.226 & 16.5 & 3.99 & 216 & 451 \\
UKIRT: UIST & 2003.42 & 0.293 & 16.5 & 3.6 & 210 & 630 \\
UKIRT: UIST & 2003.42 & 0.293 & 16.5 & 3.99 & 217 & 601 \\
UKIRT: Michelle & 2004.25 & 0.397 & 210 & 10.52 & 215 & 874 \\
UKIRT: UIST & 2004.49 & 0.426 & 16.5 & 3.99 & 214 & 948 \\
UKIRT: UIST & 2004.49 & 0.426 & 16.5 & 4.68 & 218 & 928 \\
\noalign{\smallskip}
\tableline
\end{tabular}
}
\end{center}
\end{table}
\begin{figure}[!ht]
\plottwo{WilliamsFig3a.ps}{WilliamsFig3b.eps}
\caption{Left: A 3.99-$\mu$m image of WR\,140 observed on 2003 June 4 with UIST
on UKIRT showing homologous expansion of the dust emission features observed in
earlier images. The resolution is lower than in Fig.\,2 owing to the longer
wavelength. Right: proper motions of knots `C' and `E' of dust emission measured
from our images and the 2001 images of Monnier et al. and Tuthill et al. (2003).}
\end{figure}
\begin{table}[!ht]
\caption{Proper motions, projected velocities and ejection dates of dust-emission
`knots'.}
\smallskip
\begin{center}
{\small
\begin{tabular}{lllll}
\noalign{\smallskip}
\tableline
\noalign{\smallskip}
Knot & P. M. & Proj. vel. & Date started & Phase started \\
& (mas/y) & (km/s) & & \\
\noalign{\smallskip}
\tableline
\noalign{\smallskip}
C & 294$\pm$7 & 2575$\pm$63 & 2001.28$\pm$0.07 & 0.023$\pm$0.009 \\
C0 & 240$\pm$17 & 2108$\pm$151 & 2001.34$\pm$0.21 & 0.030$\pm$0.027 \\
D & 313$\pm$18 & 2745$\pm$158 & 2001.22$\pm$0.11 & 0.015$\pm$0.014 \\
E & 304$\pm$2 & 2664$\pm$14 & 2001.03$\pm$0.01 & 0.991$\pm$0.002 \\
\noalign{\smallskip}
\tableline
\end{tabular}
}
\end{center}
\end{table}
The proper motions of selected knots from linear fits to the distances of
selected knots are given in Table 2, together with projected velocities
adopting the revised distance of 1.85 kpc (Dougherty et al., this meeting).
These velocities are comparable to that (2470 km/s) of the compressed WC7
wind material responsible for the He\,{\sc i} sub-peaks and presumably the
gas in which the dust formed. The observation of relatively high projected
velocities for three of the four well observed knots suggests either that
they are clumps moving in the plane of the sky or, more probably, that they
are limb-brightened edges of a hollow dust cloud and not physical clumps.
To model the dust-emission images, we assume that the dust moves radially
along the projection of the WCR and therefore need to know the changing
configuration of the WCR during dust formation. From the NIR light curves
(Williams 1990), we infer that dust formation occurs for only $2-3\%$ of
the period on either side of periastron. The high eccentricity of the
orbit (Marchenko et al. 2003), however, means that the axis of the WCR
moves through more than $180\deg$ in this short time and, allowing for
$\theta \simeq 53\deg$, that the dust is spread around three-quarters of
the obital plane in a small fraction of the period. This will not give a
dust `spiral', nor a `jet', but a `splash'. The orientation of this
`splash' on the sky using the orientation of the orbit from values of
$\Omega$ and $i$ derived by Dougherty et al. has the dust starting to
the east and running clockwise round the star to finish to the south,
making the southern `bar' the most recently formed dust. The absence of
significant dust emission to the NW, the projected axis at the time
of periastron passage, suggests that dust formation is quenched at the
very closest separation.
\acknowledgments{We gratefully acknowledge the Service Observing Programmes
of the Isaac Newton Group (WHT) and the Joint Astronomy Centre (UKIRT).}
|
{
"timestamp": "2004-11-16T12:25:23",
"yymm": "0411",
"arxiv_id": "astro-ph/0411440",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411440"
}
|
\section{Introduction}
We consider here labelled graphs, i.e., graphs with labelled
vertices, undirected edges and without self-loops or multiple edges as well
as labelled \textit{multigraphs} which are labelled graphs with self-loops
and/or multiple edges. A $(n,q)$ graph (resp. multigraph) is one having $n$
vertices and $q$ edges.
On one hand, classical papers \cite{ER59, ER60, FKP89, JKLP93} provide
algorithms and analysis of algorithms that deal with random graphs or
multigraphs generation, estimating relevant characteristics of their
evolution. Starting with an initially empty graph of $n$ vertices, we enrich
it by successively adding edges. As random graph evolves, it displays a
phase transition similar to the typical phenomena observed with percolation
process. On the other hand, various authors such as \aut{Wright}
\cite{Wr77, Wr80} or \aut{Bender, Canfield} and \aut{McKay}
\cite{BCM90, BCM92} studied exact enumeration or
asymptotic properties of labelled
\textit{connected} graphs.
A lot of research is devoted to graphs not containing
a prefixed set of subgraphs as copies and various approaches
exist for these problems. Most of them, following \aut{Erd\"{o}s}
and R\'enyi's seminal papers \cite{ER59, ER60}, are
probabilistic; moment methods, tail inequalities, or
probabilistic inequalities are then essential as well explained
in \cite{Bollobas}. These approaches take
advantage over enumerative ones by allowing treatments
under the edges independence assumption \cite{Bollobas}.
The situation changes radically if we consider connected components,
and results relative to connectedness are few. Related
works include \cite{Wr77, Wr78, Wr80, BCM90, BCM92, BCM97, FKP89, JKLP93}
Let $H$ be a fixed connected graph;
by a \textit{copy of $H$}, we mean any subgraph,
not necessarily induced, isomorphic to $H$.
Let $\mathcal{F}$ be a family of graphs none of which contains a copy
of $H$. In this case, we say that the family $\mathcal{F}$
is \textit{$H$-free}. Otherwise, a graph
containing a copy of $H$ is called a \textit{supergraph}
of $H$. The highly non-trivial task of enumerating
\textit{triangle-free} or \textit{quadrilateral-free}
components goes back to the book of
\aut{Harary} and \aut{Palmer} \cite{HP73}.
Mostly forbidden configurations are triangle, quadrilateral, ...,
$C_{p}$, $K_{p}$, $K_{p,q}$ or
any combination of them
(see \cite[Chapter IV]{Bollobas}, \cite[Chapter III]{JLR00}).
$C_{p}$ shall always denote the
cycle on $p$ vertices, $K_{p}$ the complete graph with $p$ vertices and $%
K_{p,q}$ the complete bipartite graph with $p$ vertices on the first side
and $q$ vertices on the second side. For example, we can work with the
family of graphs which do not contain a copy of
triangle ($C_{3}$) or of $K_{3,3}$, i.e.,
$\{C_{3},K_{3,3}\}$-free graphs. Following the authors of \cite{FKP89},
we refer as \textit{bicyclic} graphs all connected
graphs with $n$ vertices and $(n+1)$ edges
and in general $(q+1)$-\textit{cyclic} graphs are
connected $(n,n+q)$ graphs. If we define the \textit{excess} of a graph as
the difference between its number of edges and its number of vertices,
\textit{$(q+1)$-cyclic} graphs are referred also as
\textit{$q$-excess} connected graphs.
In general, we refer as \textit{multicyclic} a
connected graph which is not acyclic. The same nomenclature holds for
multigraphs. More generally, denote by $\xi
=\{H_{1},H_{2},H_{3},...\}$ a set of connected multicyclic graphs (resp.
multigraphs); a ${\xi}$\textit{-free} graph is then one which
does not contain any copy of $H_i$ for all $H_i \in \xi$ as subgraph.
Throughout this paper, unless explicitly mentioned, $\xi$ denotes
a \textit{finite} set of forbidden configurations.
Our aim in this paper is
\begin{itemize}
\item[1.] to study randomly generated graphs
with $n$ vertex and approximately $\frac{n}{2}$ edges
focusing our attention
on the appearance or not of the forbidden configurations,
\item[2.] to compute the asymptotic number of $\xi$-free connected
graphs when $\xi$ is finite.
\end{itemize}
The results obtained here show that some characteristics of random
generation as well as asymptotic enumeration of labelled
graphs or multigraphs, can be read within the forms of the
exponential generating functions (EGF for short) of the sparse components.
In fact, denote by $\gr{W}_{k}$ ($k \geq -1$) the EGFs
of $(k+1)$-cyclic (connected) graphs.
In a series of important papers, \cite{Wr77, Wr78, Wr80},
E.~M. Wright proved that $\gr{W}_{k}(z)$ $(k \geq 1)$, where $z$
is the variable marking the number of vertices in the graph,
can be expressed as finite sums of power of
$1/(1-T(z))$ where $T(z) = \sum_{n \geq 1} n^{n-1} \frac{z^n}{n!}$
is the EGF for rooted labelled trees \cite{Cay89, Moo67}.
Starting with a functional equation satisfied
by our $(k+1)$-cyclic
$\xi$-free graphs; we will show that their EGF, denoted $\gr{W}_{k,\xi}$,
have the same global forms as
those of $(k+1)$-cyclic graphs, i.e.,
$\gr{W}_{k}$. These forms will allow us to study random
graphs without forbidden configurations and also to
enumerate asymptotically connected components of these objects under
some restrictions. Similar results related to multigraphs will be
treated and carried along this paper, in parallel. Since our results
concern graphs and multigraphs,
we will be frequently assuming throughout this paper that the
term \textit{component} is the general term for connected graph
as well as for connected multigraph.
\subsection{Asymptotic number of $\xi$-free $(n,n+o(n^{1/3}))$ components}
In the first part of this paper, we will compute
the asymptotic number of triangle-free connected $(n,n+k)$-graphs, whenever
$k=o(n^{1/3})$.
To do this, we will rely heavily on the results in \cite{Wr80} to prove that
the power series $\gr{W}_{k,C_3}$ satisfy the same
inequalities as for $\gr{W}_k$ which we shall call here
``\textit{Wright's inequalities}''. Next,
we will investigate the asymptotic
behavior of the coefficient of $z^n$ in $\frac{1}{(1-T(z))^{k(n)}}$
(where $T$ is the EGF for Cayley's rooted labelled trees)
by means of saddle point method. The combination of these computations
will permit us to show
\textit{almost all}
connected $(n,n+o(n^{1/3}))$ graphs,
i.e., connected graphs with $n$ vertices and $n+o(n^{1/3})$ edges are
triangle-free. These asymptotic results are related
to the interesting problems posed by \aut{Harary} and \aut{Palmer}
in their reference book (see \cite[Sect. 10.4, 10.5 and 10.6]{HP73}).
The purpose of this part is also to introduce methods by which the asymptotic
number of connected $\xi$-free $(n,n+k)$ graphs
can be computed systematically, whenever $k=o(n^{1/3})$.
\subsection{Forbidden subgraphs in random $(n,\frac{n}{2})$ components}
The two models of graph evolution, explicitly introduced
in \cite{FKP89}, are considered in the second part of this note, in
order to generate randomly graphs and multigraphs. We will study the
structure of evolving graphs and multigraphs when edges are added one at
time and at random, mainly looking at the presence or absence of certain
configurations. In \cite[Theorem 5]{JKLP93}, the authors proved that
the probability that a random graph or multigraph with $n$ vertices and $%
\frac{n}{2}+O(n^{1/3})$ edges has $r_{1}$ bicyclic components, $r_{2}$
tricyclic components, ..., $r_{q}$ $(q+1)$-cyclic components
and no components of higher-cyclic order is
\begin{equation}
\left( \frac{4}{3}\right) ^{r} %
\sqrt{\frac{2}{3}}\frac{b_{1}^{r_{1}}}{r_{1}!}%
\frac{b_{2}^{r_{2}}}{r_{2}!} \cdots %
\frac{b_{q}^{r_{q}}}{r_{q}!}\frac{r!}{%
(2r)!}+O(n^{-\frac{1}{3}}) \label{JKLP93Theorem5}
\end{equation}
where $r=r_{1}+2r_{2}+ \cdots + qr_{q}$ and
the $b_i$ are \textit{Wright's constants} also found
by \aut{Louchard} and \aut{Tak\'acs}
($b_{1}=\frac{5}{24}, \, b_{2}=\frac{5}{16}$, \, ...), and
are involved in an important series of papers
\cite{Lo84a,Lo84b,Vo87,Ta91a,Ta91b,JKLP93,Sp97,FPV98}.
Given a finite collection
$\xi =\{H_{1},H_{2},H_{3},...,H_{q}\}$ of
multicyclic connected components, with slight
modifications of the results in
\cite{JKLP93}, we show that for a random graph or multigraph with $n$
vertices and
$m(n)=\frac{n}{2}(1+\mu n^{-\frac{1}{3}})$ edges,
$|\mu |\leq n^{1/12}$
(in this paper, we will often choose $\mu=O(n^{- \frac{1}{3} } ) $ so
$m(n)= \frac{n}{2} +O(n^{\frac{1}{3}})$), the
probability of finding only acyclic and unicyclic components without
copy of $H_{i}$, $\forall H_{i}\in \xi $, is
asymptotically the same value as for ``general''
random graphs \textit{times}
${\exp{\left( -\sum_{k\in \Theta }\frac{1}{2p}\right)} }$ where $%
\Theta $ is the subset (possibly empty) of the lengths of all
polygons in $\xi$:
$\Theta=\{p,H_{i}\in \xi $ and $H_{i}$ is a $p$-\textit{gon}$\}$.
For example, if $%
\xi =\{C_{3},C_{4}\}$, $\Theta=\{3,4\}$ and
the probability that a random graph or a multigraph
with $n$ vertices and $\frac{n}{2}+O(n^{1/3})$ edges has only trees and
unicyclic components without \textit{triangles}
or \textit{quadrilaterals} as
induced subgraphs is
\begin{equation}
\sqrt{\frac{2}{3}}e^{-\frac{1}{6}-\frac{1}{8}} \sim 0.6099\cdots \, .
\label{SIMPLEWITHOUTC3C4}
\end{equation}
Recall that an \textit{elementary contraction} of
a graph $G$ is obtained by
identifying two adjacent points $x$ and $y$, that is, by the removal of $x$
and $y$ and the addition of a new point $z$
adjacent to those points to which
$x$ or $y$ were adjacent. Then a graph $G_{1}$ is
\textit{contractible }to a
graph $G_{2} $ if $G_{2}$ can be obtained from $G_{1}$ by a sequence of
elementary contractions. We show that
a sufficient condition to change the coefficient $b_{i}$,
for any $i>0$, of (\ref{JKLP93Theorem5}) in
this probability is to force $\xi$ to
contains the entire family of graphs \textit{%
contractible} to certain graphs $H_{1}, H_{2}, \cdots$ (in
this case $\xi $ is \textit{infinite}). We then give the corresponding
probability.
The ideas of sections 4, 5 and 6 may be summarized by the figure
\ref{FIG:ABSTRACT456}. \\
\begin{figure}
\begin{center} \epsfig{file= abstract456.eps,scale=0.50}
\end{center}
\caption[sections 4, 5, 6.]
{Summarizing sections 4, 5, 6 and the methods therein.}
\label{FIG:ABSTRACT456}
\end{figure}
\subsection{An outline of the paper}
The rest of this paper is organized as follows. In
section 2, we recall some useful definitions and notations
of the stuff we will encounter
along this document. In section 3, we will work with the
example of bicyclic graphs. The enumeration of these
graphs was discovered, as far as we know, independently by \aut{Bagaev}
\cite{Bag73} and by \aut{Wright} \cite{Wr77}. The purpose of this example
is two-fold. First, it brings a simple new combinatorial point of view to
the relationship between the generating functions of some \textit{integer
partitions}, on one hand, and \textit{graphs}, on
the other hand. Next, this example gives us ideas, regarding the
\textit{simplest complex components}, i.e., simplest non-acyclic
components, of what will happen if we force our
graphs to contain some specific configurations
(especially the form of the generating functions). In section 4,
we start giving the functional equation satisfied by our
$\xi$-free connected graphs involving also the first
components containing copies of
forbidden configurations. This equation is difficult to solve but
leads to the general forms of the EGFs of all $(k+1)$-cyclic $\xi$-free
components. In fact, general combinatorial techniques are
presented and used to enumerate the first low-order cyclic triangle-free
components.
Section 5 presents methods to estimate asymptotically the number of
connected components built
with $n$ vertices and $n+k$ edges as $n \rightarrow \infty$ and $k
\rightarrow \infty$ but $k=o(n^{1/3})$. The obtained results show that
\textit{almost all} $(n,n+o(n^{1/3}))$
connected components are triangle-free and the methods used
show that this fact can be generalized to any finite set $\xi$
of forbidden subgraphs.
We then turn on the computation of the
probability of random graphs/multigraphs without forbidden
configurations in section 6.
Along this paper, \textit{triangle-free} graphs
will be treated as significant example but many results stand for any
\textit{finite} set $\xi$ of forbidden multicyclic graphs or multigraphs.
\section{Notations}
Definitions and tools are given in this section. Because they are mostly
well known, they are quickly sketched. Powerful tools in all
combinatorial approaches, \textit{generating functions} will be used for our
concern. If $F(z)$ is a power series, we write $\left[ z^{n}\right] F(z)$
for the coefficient of $z^{n}$ in $F(z)$.
We say that $F(z)$ is the \textit{exponential generating function}
(EGF for brief) for a collection
$\mathcal{F}$ of \textit{labelled} objects
if $n!\left[ z^{n}\right] F(z)$ is
the number of ways to attach objects in $\mathcal{F}$ that
have $n$ elements (see for instance \cite{FZvC94} or \cite{Wi90}).
The generating functions for labelled unrooted and labelled rooted
trees are nice examples of EGFs. The
mathematical theory of labelled trees, as first discussed by \aut{Cayley}
in 1889 \cite{Cay89} was concerned in their enumeration aspect. This study
initiated the enumeration of labelled graphs. In fact, a labelled tree is a
connected graph with $n$ vertices labelled from $1$ to $n$ and $n-1$ edges.
It is well known that the number of such structures upon $n$ points is $%
n^{n-2}$. Let $T$ be the EGF for labelled rooted trees.
A tree consists of a root to which is attached a set of
rooted subtrees, thus
\begin{equation}
T(z)=z\left( \sum_{n\geq 0}\frac{T(z)^{n}}{n!}\right) = %
\sum_{n\geq 1}n^{n-1}%
\frac{z^{n}}{n!} \, .
\label{EGF-ROOTED-TREES}
\end{equation}
In (\ref{EGF-ROOTED-TREES}), the exponent of the variable $z$ reflects
the number of nodes. One can use
\textit{bivariate exponential generating function} to count
labelled rooted trees. Throughout this paper, the variable $z$
is the variable recording the number of nodes and
$w$ is the variable for the number of edges.
For e.g., a tree with $n$ vertices is a connected graph with
$n-1$ edges and we have
\begin{equation}
T(w,z)=z\, \exp{\left(w \,T(w,z)\right)} = %
\sum_{n>0}(wn)^{n-1}\frac{z^{n}}{n!} \, .
\label{BGF-ROOTED-TREES}
\end{equation}
This bivariate EGF satisfies
\begin{equation}
T(w,z)=\frac{T(wz)}{w} \, .
\label{BGF-EGF-ROOTED-TREES}
\end{equation}
We will denote by $W_{k}$, resp. $\gr{W}_{k}$, the EGF for labelled
multicyclic connected multigraphs, resp. graphs, with $k$ edges more than
vertices. For $k\geq 1$, these EGFs have been computed in \cite{Wr77}
and in \cite{JKLP93}.
A connected graph is of excess $k$ which is always greater than or
equal to $-1$. Let $\gr{W}_{-1}$ be the EGF of unrooted labelled trees.
One can obtain at generating function level the relation
\begin{equation}
\gr{W}_{-1}(z) = \int_{0}^{z} T(x) \frac{dx}{x}\, ,
\label{ROOTED-UNROOTED}
\end{equation}
which reflects the fact that any
node of an unrooted tree can be taken as the root. The integration
of (\ref{ROOTED-UNROOTED}) leads to the classical relation
\begin{equation}
\gr{W}_{-1}(z) = T(z) - \frac{T(z)^2}{2} \, .
\label{UNROOTED}
\end{equation}
It is convenient to work with bivariate EGFs and
the bivariate EGFs that enumerate the family
$\gr{\mathcal{W}}_{k}$ of labelled $k$-excess graphs, for all
$k \geq -1$, can be expressed
using the corresponding univariate EGFs as follows
\begin{equation}
\gr{W}_{k}(w,z) = w^k \gr{W}_{k}(wz) \, .
\label{UNIVARIATE-BIVARIATE}
\end{equation}
The factor $w^k$ in the right side of (\ref{UNIVARIATE-BIVARIATE})
reflects the excess of the component, that is
its number of edges minus its number of vertices.
The same remark holds between the univariate and bivariate EGFs,
$\mg{W}_{k}$, of $k$-excess multigraphs.
Without ambiguity, one can also associate a given configuration of labelled
graph or multigraph with its EGF. For instance, a triangle
can be labelled in only one way and we have the following
informal relation
\begin{equation}
C_{3} \rightarrow C_{3}(w,z)=\frac{1}{3!}w^{3}z^{3} \, .
\label{TRIANGLE-TRIANGLE-EGF}
\end{equation}
For any given multicyclic component $H$,
denote by $W_{k,H}$ (resp. $\gr{W}_{k,H}$) the EGF of
multicyclic $H$-free multigraphs (resp. graphs)
with $k$ edges more than vertices.
In these notations, the second index refers to the forbidden
configuration(s). Recall that a \textit{smooth}
graph or multigraph is one with all
vertices of degree $\geq 2$ (see \cite{Wr78}). Throughout the rest of this
paper, the ``\textit{widehat}'' notation will be used for EGF of graphs and
``\textit{underline}'' notation corresponds to the
\textit{smoothness} of the species. E.g.,
$\underline{\gr{W}_k}$, resp. $\underline{W_k}$,
are EGF for connected $(n,n+k)$ smooth graphs, resp. smooth
multigraphs.
\begin{rem} \label{REM_MULTIGRAPH}
We follow the authors of \cite{JKLP93} and the widehat notation will
be used for graphs generating functions. Although, our main concern is graphs,
one can extend the results presented in this paper to multigraphs. In fact,
in the giant paper \cite{JKLP93}, the uniform model of random graphs which
allows self-loops and multiple edges is treated and shown to be easier to
analyze than the classical model of random graphs due to Erd\"{o}s and R\'{e}nyi
\cite{ER60} since the multigraphs EGFs have better expressions.
\end{rem}
We need additional definitions corresponding to the first appearance
of the forbidden configurations in some random evolving graphs/multigraphs.
For sake of simplicity, we suppose temporarily that $\xi=\{C_3\}$.
Consider the random graph process which starts
with $n$ initially disconnected
nodes. When enriching it by successively adding edges,
one at time and at random, the first time a new copy of
triangle is created with the last added edge in some
connected component, there are two possibilities:
\begin{itemize}
\item[1.] the last edge creates \textit{exactly}
one and only one triangle,
\item[2.] there are many occurrences of triangles but sharing
the last added edge which deletion will
suppress all copies of triangle in
the considered component. We shall call this sort of configuration
``\textit{juxtaposition}'' of triangles.
\end{itemize}
The same nomenclature holds when considering a set $\xi$ of forbidden
configurations. For example if $\xi=\{C_3,C_4\}$, a ``\textit{house}''
can appear in some component. More formally, we have the following
reformulation related to these kinds of construction:
\begin{defn} \label{DEF_JUXTA}
Given a subset $\{H_{i_1}, \, H_{i_2}, \, \cdots, \, H_{i_q}\}$
of $\xi$, we define the \textit{juxtaposition} of
$H_{i_1}, \, H_{i_2}, \, \cdots, \, H_{i_q}$
as a subgraph containing at least one
copy of each $H_{i_j}$ but such that there exists an edge
which deletion will suppress all the occurrences of
$H_{i_1}, \, H_{i_2}, \, \cdots, \, H_{i_q}$. When there exists
$s$ shared edges such that the deletion of any of them will suppress
all the occurrences of $H_{i_1}, \, H_{i_2}, \, \cdots, \, H_{i_q}$,
we define this specific configuration as a $s$-juxtaposition.
\end{defn}
\begin{exmp}
We have the figure \ref{HOUSE}
depicting a $1$-juxtaposition of $C_3$ and $C_4$,
representing a ``house''. In figure \ref{2K4}, we have
a $1$-juxtaposition and a $3$-juxtaposition
of two $K_4$.
\begin{figure}[h]
\begin{minipage}[t]{6.0cm}
\begin{center} \epsfig{file=1234juxta.eps,scale=0.5}
\end{center}
\caption[$1$-juxtaposition of $C_3$ and $C_4$ ($\xi=\{C_3, \, C_4\}$)]
{The ``house'': $1$-juxtaposition of $C_3$
and $C_4$ ($\xi=\{C_3, \, C_4\}$).}
\label{HOUSE}
\end{minipage}
\hfill
\begin{minipage}[t]{7.0cm}
\begin{center} \epsfig{file=123juxta.eps,scale=0.5}
\end{center}
\caption[$1$-juxtaposition and $3$-juxtaposition of $K_4$]
{$1$-juxtaposition and $3$-juxtaposition of $K_4$ ($\xi=\{K_4\}$).}
\label{2K4}
\end{minipage}
\end{figure}
\end{exmp}
\begin{defn} \label{DEF:JKXI}
For any $H \in \xi$, denote by $\gr{S}_{k,H}$ the
EGF of $(k+1)$-cyclic graphs with exactly one copy of $H$
(copies of other graphs of $\xi$ are not allowed).
Define by $\gr{S}_{k,\xi} = \sum_{H \in \xi} \gr{S}_{k,H}$,
the EGF of $(k+1)$-cyclic graphs with one occurrence of a member of $\xi$.
For any subset $\xi^{'} \subseteq \xi$, denote by $\gr{J}_{k,\xi^{'}}^{(p)}$
the EGF of $p$-juxtaposition of $\xi^{'}$. We let
$\gr{J}_{k,\xi} = \sum_{\xi^{'} \subseteq \xi} \sum_{p} %
p \, \gr{J}_{k,\xi^{'}}^{(p)}$.
Respectively, $S_{k,\xi}$ and $J_{k,\xi}$ are the
EGFs for multigraphs with the same characteristics.
\end{defn}
Furthermore, denote by $\vartheta_w$, resp. $\vartheta_z$,
the differential operator $w \frac{\partial}{\partial w}$, resp. $z
\frac{\partial}{\partial z}$. The operator $\vartheta_w$
corresponds to marking an edge of a graph (or a multigraph).
Similarly, $\vartheta_z$ corresponds to marking a vertex .
For the use of pointing and marking, we refer to
\cite{GJ83} and for general techniques
concerning graphical enumerations we refer to \cite{HP73}.
The following observation will take its importance as we will
see later:
\begin{rem} \label{JKXI}
$\gr{J}_{k,\xi}$ is the EGF of $(k+1)$-cyclic graphs
with a shared edge of the juxtaposition marked.
\end{rem}
\begin{rem} \label{PRECEDENCE} Throughout this paper, we will
frequently use the following notation when comparing the
coefficients of two generating functions.
If $A$ and $B$ are two formal power series such that
for all $n \geq 0$ we have $\coeff{z^n} A(z) \leq \coeff{z^n}B(z)$ then
we denote this relation $A \preceq B$ (or $A(z) \preceq B(z)$).
\end{rem}
\section{The link between the EGF of bicyclic graphs and integer partitions}
\label{SEC:LINK}
At least in 1967, there were $10$ different proofs for the EGF for trees
according to the paper of \aut{Moon} \cite{Moo67} and $16$ proofs
related in \cite{K73}. Then,
\aut{R\'{e}nyi}
\cite{Ren59} found the formula to enumerate unicyclic graphs which
can be expressed in terms of the generating function of rooted
labelled trees, namely
\begin{equation}
\gr{W}_{0}(z)=\frac{1}{2}\ln{\frac{1}{1-T(z)}}%
-\frac{T(z)}{2}-\frac{T(z)^{2}}{4} \, .
\label{UNICYCLIC-GRAPH}
\end{equation}
We refer here to the symbolic methods developed in \cite{FS96}
for modern computation of formulae like (\ref{UNICYCLIC-GRAPH}).
The formula for unicyclic multigraphs is very similar
and there are terms due to self-loops and multiple edges
\begin{equation}
W_0(z)=\frac{1}{2}\ln{\frac{1}{1-T(z)}} \, .
\label{UNICYCLIC-MULTIGRAPH}
\end{equation}
It may be noted that in some connected graphs, as well as
multigraphs the number of edges exceeding the number of vertices
can be seen as useful enumerating parameter.
The term \textit{bicyclic} graphs, appeared
first in the seminal paper of \aut{Flajolet} \textit{et al.}
\cite{FKP89} followed
few years later by the huge one of \aut{Janson} \textit{et al.}
\cite{JKLP93} and
was concerned with all connected graphs with $(n+1)$
edges and $n$ vertices.
The authors of these documents choose then the word \textit{bicyclic}
for connected component
which is constructed by adding a random
edge to a unicyclic component.
\aut{Bagaev} \cite{Bag73} first found a method to count
such graphs. His method of \textit{shrinking-and-expanding} graphs is
well explained in \cite{BV98}. \aut{Wright} \cite{Wr77} found a recurrent
formula well adapted for formal calculation to compute the number of all
connected graphs of excess $k$ (for all $k \geq 1$). Our aim
in this section is to show that the
problem of the enumeration of \textit{bicyclic graphs} can also be solved
with techniques involving integer partitions. We present here
a simple treatment very close to the Wright's method
as a warm-up for the forthcoming results in the next sections.
Given a fixed set of $n$ vertices, there exist two types
of graphs which are connected and have $(n+1)$ edges as
described in the figure \ref{examples_bicyclic}. \\
\begin{figure}[h]
\begin{minipage}[t]{7.0cm}
\begin{center} \epsfig{file=bicyclic.eps,scale=0.5}
\end{center}
\caption[Examples of bicyclic components.]
{Examples of bicyclic components.}
\label{examples_bicyclic}
\end{minipage}
\begin{minipage}[t]{7.0cm}
\begin{center} \epsfig{file=smooth.eps,scale=0.5}
\end{center}
\caption[Smooth bicyclic components.]
{Smooth bicyclic components.}
\label{smooth_bicyclic}
\end{minipage}
\end{figure}
\noindent
\aut{Wright} \cite{Wr77} showed with his \textit{reduction} method that
the EGF of all multicyclic graphs, namely bicyclic graphs, can be expressed
in terms of the EGF of labelled rooted trees. In order to
count the number of
ways to label a graph, we can repeatedly \textit{prune }it by suppressing
recursively any vertex of degree $1$. We then remove as many vertices as
edges. As these structures present many symmetries, our experiences suggest
us so far that we ought to look at our previously described object without
symmetry and without the possible rooted subtrees.
There are
\[
{n \choose p}{ {n-p} \choose q}%
\frac{(p-1)!}{2}p\frac{(q-1)!}{2}q(n-p-q)!=\frac{n!}{4}
\]
manners to label the graph represented by the figure
\ref{smooth_bicyclic} (a)
whenever $p\neq q$. In the graph of figure \ref{smooth_bicyclic} (b),
if $r \neq s$, $s \neq t$, $t \neq r$, there are $\frac{n!}{2}$
ways to label the graph. Note that these results are
independent from the size of the subcycles. One can obtain all smooth
bicyclic graphs after considering possible symmetry criterions. In figure
\ref{smooth_bicyclic} (a), if the subcycles have the same length, $p=q$, a
factor $\half$ must be considered and we have $n!/8$ ways to label the
graph. Similarly, the graph of figure \ref{smooth_bicyclic} (b)
can have the 3 arcs with the same number of vertices.
In this case, a factor $1/6$ is introduced. If only two arcs have the same
number of vertices, we need a symmetrical factor $1/2$. Thus, the
enumeration of smooth bicyclic graphs can be viewed as specific problem of
integer partitioning into 2 or 3 parts following the dictates of the basic
graphs in figure \ref{basic_bicyclic}. \\
\begin{figure}[h]
\begin{center} \epsfig{file=basic3.eps,scale=0.40}
\end{center}
\caption[The different basic smooth bicyclic graphs.]
{The different basic smooth bicyclic graphs.}
\label{basic_bicyclic}
\end{figure}
\noindent
With the same notations as in \cite{Co70}, denote by $P_{i}(z)$,
respectively $Q_{i}(z)$, the generating functions of the number of
partitions of an integer in $i$ parts, respectively in $i$ different parts.
Let $\underline{\gr{W}_{1} }(z)$ be the univariate EGF for smooth bicyclic graphs, then we
have $\underline{\gr{W}_{1}}(z)=f\big(P_2(z),P_3(z),Q_2(z),Q_3(z)\big)$, i.e.,
\begin{equation}
\begin{array}{cc}
\underline{\gr{W}_{1}}(z)= & %
\underbrace{\frac{1}{2}z^{2}(Q_{3}(z)+Q_{2}(z))}_{%
\mbox{figures \ref{basic_bicyclic} (a), \ref{basic_bicyclic} (b)}}
+ \underbrace{\frac{1}{12}\frac{z^{5}}{1-z^{3}}}_{%
\mbox{\ref{basic_bicyclic} (c)}} \\
& +\underbrace{\frac{1}{4}\left( \frac{z^{4}}{1-z^{2}}+\frac{z^{5}}{%
(1-z)(1-z^{2})}-\frac{z^{5}}{(1-z^{3})}\right) }_{%
\mbox{\ref{basic_bicyclic} (d), \ref{basic_bicyclic} (e)}} \\
& +\underbrace{\frac{1}{4}\frac{z^{6}}{(1-z)^{2}(1-z^{2})}}_{%
\mbox{\ref{basic_bicyclic} (f)}}+%
\underbrace{\frac{1}{8}\frac{z^{5}}{(1-z)(1-z^{2})}}_{%
\mbox{\ref{basic_bicyclic} (g)}} \, .
\end{array}
\label{DECOMPOSITION-PARTITION}
\end{equation}
In formula (\ref{DECOMPOSITION-PARTITION}) or equivalently $%
\underline{\gr{W}_{1}}(z)=\frac{z^{4}}{24}\frac{(6-z)}{(1-z)^{3}}$, the
denominator $\frac{1}{(1-z)^{3}}$ denotes the fact that there is at most $3$
arcs or $3$ \textit{degrees of liberty} of integer partitions of the
vertices in a bicyclic graph. The same remark holds for the denominators $%
\frac{1}{(1-T(z))^{3k}}$ in Wright's formulae \cite{Wr77} for all
$(k+1)$-cyclic connected labelled graphs.
To get the whole EGF for bicyclic graphs, we have to substitute
$z$ by $T(z)$ in $\sth{\gr{W}_1}(z)$ in order to replace all
(shrinked) vertices of the smooth graphs by labelled rooted trees.
The form of these EGF
takes its importance when studying
the asymptotic behavior of random graphs or multigraphs with a
given excess. In fact, the known expansion of the Cayley's
function, $T$, at its singularity $z=\frac{1}{e}$ is
(see \cite{KP89, FO90, FS+})
\begin{equation}
T(z) = 1 - \sqrt{2}\delta + \frac{2}{3}{\delta}^2 %
- \frac{11}{36}\sqrt{2}{\delta}^3 + \cdots \, , \, \, (\delta=\sqrt{1-ez})\, .
\label{SINGULARITY-CAYLEY}
\end{equation}
As the EGFs of multicyclic components can be expressed
in terms of $T$, the key point of
their characteristics corresponds directly to
the analytical properties of \textit{tree
polynomial} $t_{n}(y)$ defined as follow
\begin{equation}
\frac{1}{(1-T(z))^y}=\sum_{n\geq 0}t_{n}(y)\frac{z^{n}}{n!} \, .
\label{TREE-POLYNOMIAL}
\end{equation}
($t_{n}(y)$ is a polynomial of degree $n$ in $y$.) \aut{Knuth}
and \aut{Pittel} \cite{KP89} studied their properties. For
\textit{fixed} $y$ as $n\rightarrow \infty $, we have (see
\cite[lemma 2]{KP89})
\begin{equation}
t_{n}(y)=\frac{\sqrt{2\pi }n^{(n-1/2+y/2)}}{2^{y/2}\Gamma (y/2)
+O(n^{n-1+y/2}) \, .
\label{TREE-POLYNOMIAL-ASYMPT}
\end{equation}
This equation tells us that in the EGF, $\gr{W}_{1}$ of bicyclic graphs,
expressed here as a sum of powers of $1/(1-T(z))$
\begin{eqnarray}
\gr{W}_{1}(z) & = & \frac{T(z)^{4}}{24}\frac{(6-T(z))}{(1-T(z))^{3}} \cr
& = & \frac{5}{%
24}\frac{1}{(1-T(z))^{3}}-\frac{19}{24}\frac{1}{(1-T(z))^{2}} + %
\cdots \, ,
\label{ANOTHER-DVPT-BICYCLIC}
\end{eqnarray}
only the coefficient $\frac{5}{24}$ of $t_{n}(3)$ is
asymptotically significant.
\section{Functional equation for $\xi$-free graphs/multigraphs
and the forms of their EGFs}
\subsection{Differential recurrence for $\xi$-free components}
\label{SUBSUB:FUNCTIONAL}
EGFs of triangle-free unicyclic components can be easily obtained
when avoiding cycle of length $3$ in the general formulae for
unicyclic graphs (\ref{UNICYCLIC-GRAPH}), resp. multigraphs
(\ref{UNICYCLIC-MULTIGRAPH}). Denote respectively by
$W_{0,C_{3}}$ and $\gr{W}_{0,C_{3}}$ the EGFs for unicyclic multigraphs and
graphs without triangle ($C_{3}$), we have
\begin{equation}
W_{0,C_{3}}(z)=\frac{1}{2}\ln{\frac{1}{1-T(z)}}-\frac{T(z)^{3}}{6} \, ,
\label{ACYCLIC-MC3FREE}
\end{equation}
\begin{equation}
\gr{W}_{0,C_{3}}(z)=\frac{1}{2}\ln{\frac{1}{1-T(z)}}-\frac{T(z)}{2}-\frac{%
T(z)^{2}}{4}-\frac{T(z)^{3}}{6} \, .
\label{ACYCLIC-GC3FREE}
\end{equation}
Enumerating components of higher
cyclic order without triangle is much more difficult. However, we
have the following lemma:
\begin{lem} \label{LEMMA0}
For all $i \geq -1$, denote by $\gr{W}_{i,C_3}$ the EGF for
triangle-free $(i+1)$-cyclic graphs. Let $\gr{S}_{i,C_3}$ and
$\gr{J}_{i,C_3}$ be the EGFs described as in definition
\ref{DEF:JKXI}.
Then, the bivariate EGFs $\gr{W}_{k+1,C_3}$,
$\gr{S}_{k+1,C_3}$, $\gr{J}_{k+1,C_3}$ and
$\gr{W}_{p,C_3}$ for
$-1 \leq p \leq k$ are related by the differential recurrence:
\begin{eqnarray}
\vartheta_w \gr{W}_{k+1,C_3} \, \, \, &+& \, \, 3 \gr{S}_{k+1,C_3}
\, \, + \, \, \gr{J}_{k+1,C_3} \, \, = \, \, %
\, \, w \Big( \frac{{\vartheta_z}^2 - \vartheta_z}{2} - \vartheta_w \Big) \gr{W}_{k,C_3} \cr
& + & \, \, w \left(%
\sum_{ -1\leq p \leq q \leq k+1, \, p+q=k} %
\frac{1}{1+\delta_{p,q}} (\vartheta_z \gr{W}_{p,C_3})%
(\vartheta_z \gr{W}_{q,C_3}) \right) \,
\label{FUNCTIONAL_TRIANGLE_GRAPH}
\end{eqnarray}
where $\delta_{p,q}=1$ iff $p=q$, otherwise $\delta_{p,q}=0$.
Similarly, we have for multigraphs (with the same parameters):
\begin{eqnarray}
\vartheta_w {W_{k+1,C_3}} \, \, \, &+& \, \, 3 {S_{k+1,C_3}}
\, \, + \, \, {J_{k+1,C_3}} \, \, = \, \, %
\, \, w \Big(\frac{{\vartheta_z}^2}{2} {W_{k,C_3}}\Big) \cr
& + & \, \, w \left(\sum_{-1\leq p \leq q \leq k+1, \, p+q=k} %
\frac{1}{1+\delta_{p,q}} (\vartheta_z {W_{p,C_3}})%
(\vartheta_z {W_{q,C_3}}) \right) \, .
\label{FUNCTIONAL_TRIANGLE_MULTIGRAPH}
\end{eqnarray}
\end{lem}
\noindent \textbf{Proof.} There are two ways to obtain a
$(k+2)$-cyclic component from components of lower cyclic order,
which are in the right part of (\ref{FUNCTIONAL_TRIANGLE_GRAPH})
and are assumed to be triangle-free. For multigraphs,
we have to employ the combinatorial
operation $\frac{{\vartheta_z}^2}{2}$.
First of all, consider a triangle-free
$(k+1)$-cyclic component. To add a new edge to this component,
we have to choose two vertices,
different and already not adjacent for graphs,
and not necessarily different for multigraphs. For graphs,
the combinatorial operator used to choose two different vertices
is $ \frac{{\vartheta_z}^2 - \vartheta_z}{2} $. Then, we have to avoid the adjacent
vertices by means of the operator $- \vartheta_w$ (see \cite[Section 10]{JKLP93}
or \cite{GJ83} for the use of marking and pointing).
If the new $(k+2)$-cyclic component contains a triangle, the triangle
can only occur in the following cases:
\begin{itemize}
\item[1.] The new edge creates exactly a triangle. In this case,
the last added edge is necessarily one of
the $3$ edges of the new triangle.
\item[2.] The last edge creates many triangles but necessarily
juxtaposed as defined above (definition \ref{DEF_JUXTA}),
and in this latter case, the last edge
is necessarily the one which is shared
between all the occurrences of triangle.
\end{itemize}
Thus, the left side of (\ref{FUNCTIONAL_TRIANGLE_GRAPH}), resp. of
(\ref{FUNCTIONAL_TRIANGLE_MULTIGRAPH}), distinguishes the last added edge in
the new $(k+2)$-cyclic component.
Next, a $(k+2)$-cyclic triangle-free component can be
built when creating an edge between a $(p+1)$-cyclic
and a $(q+1)$-cyclic triangle-free components such that
$p+q=k$ and $ -1 \leq p \leq q \leq k+1$ (note that
the case $p=-1$ and $q=k+1$ corresponds to the case
where a tree is attached to a $(k+1)$-cyclic
triangle-free component).
This construction is done by choosing one vertex
belonging to the $(p+1)$-cyclic component and another vertex from
the $(q+1)$-cyclic component. A symmetry factor, $\frac{1}{2!}$, occurs
when $p=q$.
The right side of (\ref{FUNCTIONAL_TRIANGLE_GRAPH}) simply reflects
the constructions used to build a $(k+2)$-cyclic connected graph
In (\ref{FUNCTIONAL_TRIANGLE_MULTIGRAPH}), the term
$\frac{{\vartheta_z}^2}{2}{W_{k,C_3}}$ represents all $(k+1)$-cyclic
multigraphs with an ordered pair $\langle x,y \rangle$ of
marked vertices
(see also \cite[Sect. 4, Eq. (4.2) and following]{JKLP93}). \hfill \qed
When considering a finite set $\xi$ of forbidden configurations,
we have the following generalization of lemma \ref{LEMMA0}:
\begin{lem} \label{LEMMA_GENERAL0}
Suppose that $\xi = \{H_1, \, \cdots, H_p\}$, $|\xi| < \infty$.
Let $\gr{W}_{k+1,\xi}$, $\gr{S}_{k+1,H_i}$, $\gr{J}_{k+1,\xi}$
and $\gr{W}_{k+1,\xi}$ be the EGFs defined as in above (definition
\ref{DEF:JKXI}). Let $\rho_s$
be the finite set of all $s$-juxtapositions of member(s) of $\xi$ and
denote by $e(H_i)$ the number of edges of $H_i$.
Then, we have for graphs
\begin{eqnarray}
\vartheta_w \gr{W}_{k+1,\xi} \, \, \, & + & \, \, %
\sum_{H_i \in \xi}e(H_i) \gr{S}_{k+1,H_i}
\, \, + \, \,\gr{J}_{k+1,\xi} \, \, = \, \, %
\, \, w \Big( \frac{{\vartheta_z}^2 - \vartheta_z}{2} - \vartheta_w \Big) \gr{W}_{k,\xi} \cr
& + & \, \, w \left(\sum_{-1\leq p \leq q \leq k+1, \, p+q=k} %
\frac{1}{1+\delta_{p,q}} (\vartheta_z \gr{W}_{p,\xi})%
(\vartheta_z \gr{W}_{q,\xi}) \right) \, .
\label{FUNCTIONAL_XI_GRAPH}
\end{eqnarray}
For the EGFs of connected $\xi$-free multigraphs, we have
\begin{eqnarray}
\vartheta_w {W_{k+1,\xi}} \, \, \, & + & \, \, %
\sum_{H_i \in \xi} e(H_i) {S_{k+1,H_i}}
\, \, + \, \, J_{k+1, \xi} \, \, = \, \, %
\, \, w \Big(\frac{{\vartheta_z}^2}{2} {W_{k,\xi}}\Big) \, \, + \cr
& & \, \, w \left(\sum_{-1\leq p \leq q \leq k+1, \, p+q=k} %
\frac{1}{1+\delta_{p,q}} (\vartheta_z {W_{p,\xi}})%
(\vartheta_z {W_{q,\xi}}) \right) \, .
\label{FUNCTIONAL_XI_MULTIGRAPH}
\end{eqnarray}
\end{lem}
(\ref{FUNCTIONAL_XI_GRAPH}) and (\ref{FUNCTIONAL_XI_MULTIGRAPH})
are simply generalization of (\ref{FUNCTIONAL_TRIANGLE_GRAPH}) and
(\ref{FUNCTIONAL_TRIANGLE_MULTIGRAPH}).
\subsection{Bicyclic components without triangle}
\label{SUBSUB:BICYCLIC-TRIANGLE-FREE}
EGFs for respectively bicyclic graphs with one triangle
and with exactly one juxtaposition of triangles can be
obtained using the method developed in section \ref{SEC:LINK},
with the help of figures \ref{FIG:SMOOTH_S1} and \ref{FIG:SMOOTH_J1}. \\
\noindent
\begin{rem} \label{rem:WRIGHT}
Since Wright's reduction method\footnote{the second method
in \cite{Wr77}, see also the proof of lemma
\ref{LEMMA_INDECOMPOSABLE} in \S \ref{SUBSUB:GENERAL-FORM}}
suggests us to work with labelled smooth components,
figures such as \ref{FIG:SMOOTH_S1} and \ref{FIG:SMOOTH_J1}
represent the situation after smoothing. Also for any family
$\mathcal{F}_k$ of $(k+1)$-cyclic components with EGF
$F_k(z)$, the EGF of smooth species of $\mathcal{F}_k$ is
simply obtained by means of substitutions of all occurrences
of $T(z)$ in $F_k(z)$ by $z$. Conversely, if $\sth{F_k}(z)$
is the EGF of smooth species of $\mathcal{F}_k$, then
$F_k(z) = \sth{F_k}(T(z))$ gives the EGF associated to the
whole family $\mathcal{F}_k$.
\end{rem}
\begin{rem} \label{rem:TTz}
Since all EGFs we deal with can be expressed in terms of
$T(z)$ in the univariate case, and of $w$ and $T(wz)$ in the
bivariate case, we assume that $T \equiv T(z)$ to express
univariate EGFs. In the case of bivariate
EGFs, we let $T \equiv T(wz)$. These notations should
not induce ambiguity to the reader who can read the
meaning within the context.
\end{rem}
The following figures can be used to compute the EGFs
$\gr{S}_{1,C_3}$ and $\gr{J}_{1,C_3}$ \\
\begin{figure}[h]
\noindent
\begin{minipage}[t]{6.5cm}
\epsfig{file=smooths1.eps,scale=0.35}
\caption[Smooth bicyclic graphs with one occurrence of triangle.]
{Smooth bicyclic graphs with one occurrence of triangle.}
\label{FIG:SMOOTH_S1}
\end{minipage}
\hfill
\begin{minipage}[t]{6.5cm}
\tiny
\epsfig{file=smoothj1.eps,scale=0.45}
\caption[Smooth bicyclic graph with a $1$-juxtaposition of 2 triangles.]
{Smooth bicyclic graph with a $1$-juxtaposition of 2 triangles.}
\label{FIG:SMOOTH_J1}
\end{minipage}
\normalsize
\end{figure}
\noindent
Using similar techniques as for (\ref{DECOMPOSITION-PARTITION})
with the help of the previous figures,
we have for $\sth{\gr{S}_{1,C_3}}$ and $\sth{\gr{J}_{1,C_3}}$
\begin{equation}
\begin{array}{cc}
\sth{\gr{S}_{1,C_3}} (z) = & \underbrace{\frac{1}{2}z^{5}\frac{1}{1-z}}_{%
\mbox{figure \ref{FIG:SMOOTH_S1} (b)}} + %
\underbrace{\frac{z^{6}}{4}\frac{1}{(1-z)^{2}}}_{%
\mbox{figure \ref{FIG:SMOOTH_S1} (a)}}
\end{array}
\end{equation}
and
\begin{equation}
\sth{\gr{J}_{1,C_3}} (z) = \frac{z^{4}}{4} \, .
\end{equation}
Again, to obtain the whole EGFs we have to substitute $z$ by $T \equiv T(z)$,
replacing all shrinked vertices of the smooth graphs by labelled rooted trees.
\begin{equation}
\gr{S}_{1,C_3}(z) = \frac{T^5}{4}\frac{(2-T)}{(1-T)^2} \, , %
\, \, \gr{J}_{1,C_3}(z) = \frac{T^4}{4} \, .
\label{EQ:S_1_C_3}
\end{equation}
Thus, using (\ref{EQ:S_1_C_3}) and (\ref{FUNCTIONAL_TRIANGLE_GRAPH})
we have
\begin{equation}
\gr{W}_{1,C_{3}}(z)=\frac{T^{5}}{24}\frac{(2+6T-3T^{2})}
(1-T)^{3}} \, .
\label{BICYCLIC-MC3FREE}
\end{equation}
We know from (\ref{TREE-POLYNOMIAL-ASYMPT}) that
the decomposition of formula such as
(\ref{BICYCLIC-MC3FREE}) into
sums of powers of $\frac{1}{1-T}$,
are useful in order to study the asymptotic behavior of the number of
such objects.
We have
\begin{equation}
\begin{array}{ccc}
\gr{W}_{1,C_{3}}(z) & = & \sum_{n\geq 0}\Big(\frac{5}{24}t_{n}(3)-\frac{%
25}{24}t_{n}(2)+\frac{47}{24}t_{n}(1) %
-\frac{35}{24}-\frac{5}{24}t_{n}(-1) \\
& & +\frac{25}{24}t_{n}(-2)%
-\frac{5}{8}t_{n}(-3)+\frac{1}{8}t_{n}(-4)\Big) %
\frac{z^{n}}{n!} \, .
\end{array}
\label{BICYCLIC-GC3FREE-POLYNOMIAL}
\end{equation}
In order to enumerate the first multicyclic $\xi$-free components
for general $\xi$, we introduce some more techniques in the next
paragraphs.
\subsection{General techniques for first multicyclic components %
and instantiations}
\label{SUBSUB:GRAPH-SURGERY}
In this paragraph, we give methods that can be applied to
enumerate first low-order cyclic components, i.e., with excess
$1$ and $2$ for a forbidden $p$-gon and in general for
an excess up to $l+1$ and $l+2$ for all forbidden components of excess $l$.
For e.g., the EGF of $C_3$-free tricyclic graphs are given
as instantiation of these methods and follows
the formula (\ref{BICYCLIC-MC3FREE})
given above. Also, we will see
later that these techniques are useful
to obtain the forms of the EGFs $\gr{W}_{k,\xi}$ and
$\mg{W}_{k,\xi}$ by induction (see \S \ref{SUBSUB:GENERAL-FORM}).
We consider here only connected graphs with exactly one
occurrence of $H$ since if $H^{^{\prime }}$ represents
any juxtaposition of $H$, we can work directly in the same manner
with a single occurrence of $H^{^{\prime }}$.
First of all, we have to prune recursively all vertices of degree $1$.
The obtained graphs are smooth. We can subdivide these graphs
containing an occurrence of $H$ in 3 types:
types (a) and (b) are such as those represented by
figure \ref{FIG:SMOOTH_S1} and type (c) is as in the figure
\ref{FIG:3} below where $H$ represents a triangle.
\noindent
\begin{figure}[h]
\begin{center}
\begin{minipage}[t]{14cm}
\small
\begin{center}
\epsfig{file=cutcutchroma.eps,scale = 0.35}
\end{center}
\caption[ ]
{ }
\label{FIG:3}
\end{minipage}
\end{center}
\end{figure}
\noindent
The first two types (a) and (b) of figure \ref{FIG:SMOOTH_S1}
can be described as follows:
\begin{itemize}
\item[$\bullet$] (a) represents the concatenation of two components $H$
and $F$ (respectively non $H$-free and $H$-free) by a common vertex
or more generally by a path between the two components.
In the figure, $H$ is simply a triangle. Note that a cutpoint
(a vertex whose removal increases the number of connected components)
belongs to the triangle after the recursive deletions of vertices
of degree $1$. This is referred here as a
\textit{serial composition} of components.
\item[$\bullet$] (b) is the concatenation of the same components but
by a common edge. This construction is referred as a
\textit{parallel composition} of components.
\item[$\bullet$] Figure \ref{FIG:3} (c) represents components which are not in
figure \ref{FIG:SMOOTH_S1} (a) nor in figure \ref{FIG:SMOOTH_S1} (b).
\end{itemize}
\subsubsection{The serial composition or concatenation by a vertex}
\label{SUBSUB:SERIAL}
Since a graph with one cutpoint belonging to a forbidden
configuration may be considered to
be rooted at this cutpoint, the number of connected
graphs with one cutpoint can be expressed in terms of the EGFs of the
different subgraphs rooted at the same cutpoint
(cf. \cite{HP73} or \cite{Selkow}). This construction
may be interpreted combinatorially as follows.
\begin{lem} \label{LEMMA7} Let $\mathcal{F}$ be a family of
connected $H$-free graph. Denote by $\sth{F}$ the EGF
of the graphs obtained when
smoothing a graph of $\mathcal{F}$. Let $A_1$ be the
EGF of connected graphs containing
possibly many copies of $H$ and obtained
as the concatenation of graphs of $\mathcal{F}$ and of
$H$ by a vertex belonging to $H$. Then, $A_1$ satisfies
\begin{equation}
A_1 \preceq {\left[\frac{1}{z} \, \, \left(z\frac{\partial }{\partial z}%
\sth{F}(z)\right) \, \left( z\frac{\partial }%
{\partial z}H(z)\right)\right]}_{| z=T(z)}
\label{PRE-ONECUTPOINT}
\end{equation}
and let $A_2$ be the EGF of all connected graphs obtained
when allowing a path starting at a vertex
belonging to $H$ and joining
any graph of $\mathcal{F}$. $A_2$ satisfies
\begin{equation}
A_2 \preceq {\left[\frac{1}{z} \, \left(\frac{1}{1-z}\right)%
\, \left(z\frac{\partial }{\partial z}%
\sth{F}(z)\right) \, \left( z\frac{\partial }%
{\partial z}H(z)\right) \right]}_{| z=T(z)} \, .
\label{ONECUTPOINT}
\end{equation}
In (\ref{PRE-ONECUTPOINT}) and (\ref{ONECUTPOINT}), equalities
hold when $H$ is two-connected.
\end{lem}
\noindent \textbf{Proof.} Recall that for two EGFs $A$ and $B$,
$A \preceq B$ means that $\forall n, \,
\coeff{z^n} A(z) \leq \coeff{z^n}B(z)$ (cf. remark \ref{PRECEDENCE}).
First, let us consider
the case where $H$ is two-connected. In this case, the concatenation
of $H$ with a graph of $\mathcal{F}$, by a vertex of $H$, leads to a
graph with a single copy of $H$ in the resulting graph.
Thus, the fact that there is
\textit{exactly} one occurrence of copy of $H$ in the concatenation
insures the \textit{uniqueness of the decomposition} into
two graphs such that one belongs to $\mathcal{F}$ and the other
is (necessarily) $H$. The lemma is a combination of the
approach presented in \cite{Selkow} and Wright's reduction method
\cite{Wr77}. We have to introduce a
factor $\frac{1}{z}$ to relabel the common
cutpoint considered here as shared between the smooth components.
$\vartheta_z \sth{F}(z) = z\frac{\partial}{\partial z}\sth{F}(z)$
and $\vartheta_z H(z) = z\frac{\partial}{\partial z}H(z)$ are used
to distinguish the vertex to be shared between
pruned components of $\mathcal{F}$ and of $H$.
In (\ref{ONECUTPOINT}) to
represent a possible \textit{path}, we insert the term
$\frac{1}{1-z}$ i.e., a sequence of
vertices of degree 2 except the two extremal nodes, between
the two sides. When substituting $z$ by $T(z)$,
we reverse the \textit{vertexectomy} process
starting with a smooth graph and sprout rooted trees
from each node. Hence, in the case where $H$ is two-connected,
we have the equalities in (\ref{PRE-ONECUTPOINT}) and (\ref{ONECUTPOINT}).
The situation changes a bit for more general configurations.
Typically, we can have concatenations of $H$ and graphs of
$\mathcal{F}$ which can lead to a new graph with two (or more)
occurrences of $H$. This is the case depicted by figure
\ref{FIG:COUNTER-EXAMPLE}
\begin{figure}
\begin{center} \epsfig{file= counterex.eps,scale=0.40}
\end{center}
\caption[Serial composition with symmetric factor $\frac{1}{2}$]
{Serial composition with symmetric factor $\frac{1}{2}$.}
\label{FIG:COUNTER-EXAMPLE}
\end{figure}
where $H$ is made with a triangle and a square attached by a vertex
and the graph of $\mathcal{F}$ is simply a triangle. In this special case,
we just have to introduce a symmetry factor $\frac{1}{2!}$ and then the
upper bound of (\ref{PRE-ONECUTPOINT}) is valid. In fact, the upper bound
enumerates graphs where the concatenation
such as the one obtained in figure \ref{FIG:COUNTER-EXAMPLE} are
counted twice or more.
\hfill \qed
\subsubsection{The parallel composition or concatenation by an edge}
\label{SUBSUB:PARALLEL}
Graphs of the type represented by the figure \ref{FIG:SMOOTH_S1} (b)
can be enumerated in a very close way.
\begin{lem} \label{LEMMA8} Let $\mathcal{F}$
and $\sth{F}$ be defined as in lemma \ref{LEMMA7} above.
Let $B$ be the EGF associated to the graphs containing
copies of $H$ and obtained as the
concatenation of two graphs of
$\mathcal{F}$ and of $H$ sharing a
common edge. $B$ satisfies
\begin{equation}
B \preceq {\left[ \frac{2}{wz^{2}} \, \, %
\left(w \frac{\partial }{\partial w}\sth{F}(w,z)\right) \, \, %
\left(w \frac{\partial }{\partial w} %
H(w,z)\right)\right]}_{| wz=T(wz)} \, .
\label{ONECUTEDGE}
\end{equation}
\end{lem}
\noindent \textbf{Proof.} The formula (\ref{ONECUTEDGE})
differs slightly from the one in (\ref{ONECUTPOINT}). The
factor $\frac{2}{wz^{2}}$ comes from
the fact that we have here, as in the figure \ref{FIG:SMOOTH_S1} (b),
a common edge which is defined by his two common vertices
and can be seen as a root-edge. A graph such as those represented by the
figure \ref{FIG:SMOOTH_S1} (b) can be considered as pendant
to this edge. Also, we have the equality whenever
$H$ is two-connected. Otherwise symmetries can arise but the
upper bound of (\ref{ONECUTEDGE}) remains valid for the same reasons
as for (\ref{PRE-ONECUTPOINT}) and (\ref{ONECUTPOINT}). \hfill \qed
Unfortunately, equation likes (\ref{FUNCTIONAL_XI_GRAPH})
of lemma \ref{LEMMA_GENERAL0} are much
easier to propose than to really solve. However, we can derive
the EGF of the first multicyclic $H$-free components
by applying the techniques presented above.
\subsubsection{The example of triangle-free graphs}
The EGFs of unicyclic and bicyclic graphs without triangles
are given by formulae (\ref{ACYCLIC-GC3FREE})
and (\ref{BICYCLIC-MC3FREE}).
For graphs having $2$ excesses, the removal of all edges and
vertices by the Wright's reduction method leads to the set of graphs
represented by figure \ref{FIG:solo2C3} for graphs
containing $1$ triangle and
figure \ref{FIG:JUXTA2C3} for graphs with a juxtaposition of triangles.
\noindent
\begin{figure}[h]
\begin{center}
\begin{minipage}[t]{12cm}
\small
\begin{center}
\epsfig{file= soloC3-2.eps,scale= 0.5}
\end{center}
\caption{Basic tricyclic graphs with exactly one triangle. The
subgraph in grey represent bicyclic triangle-free components.}
\label{FIG:solo2C3}
\end{minipage}
\end{center}
\end{figure}
\noindent
As before, given a family $\mathcal{F}$ of graphs, we denote by
$\sth{F}$ the EGF of \textit{smooth} elements of $\mathcal{F}$,
i.e., graphs without endvertices (vertices of degree $1$).
The bivariate EGF of bicyclic triangle-free smooth graphs,
$\sth{\gr{W}_{1,C_3}}$ is obtained from (\ref{BICYCLIC-MC3FREE}), namely
\begin{equation}
\sth{\gr{W}_{1,C_3}}(w,z)=w\frac{w^{5}z^{5}}{24} %
\frac{(2+6wz-3w^{2}z^{2})}{(1-wz)^{3}}
\label{eqn:FreeC3-1-smooth}
\end{equation}
Note that $\vartheta_w C_3(w,z)= \vartheta_z C_3(w,z) = \frac{w^3z^3}{2}$.
Thus, the application of the lemmas \ref{LEMMA7} and
\ref{LEMMA8} to the smooth graphs depicted by
figures \ref{FIG:solo2C3} (a) and
\ref{FIG:solo2C3} (b) gives
\begin{equation}
{\frac{w^{3}z^{2}}{2(1-wz)} \, \, \vartheta_z %
\big(\sth{\gr{W}_{1,C_3}}(w,z)\big) %
+ w^{2}z \, \, \vartheta_w\big(\sth{\gr{W}_{1,C_{3}}}(w,z)\big) } \, .
\label{eqn:SoloC3-2-lemma78}
\end{equation}
Similarly, we have for smooth graphs represented by the figure
\ref{FIG:solo2C3} (d)
\begin{equation}
{\frac{1}{z(1-wz)} \Big(\frac{2}{wz^2} \, %
\big(\vartheta_w \sth{\gr{W}_{0,C_3}}(w,z) \big) \, \big(\frac{w^3z^3}{2}%
\Big) \Big) %
\Big( \vartheta_z \sth{\gr{W}_{0,C_3}}(w,z) \Big)}
\label{eqn:solo2C3-d}
\end{equation}
and for figure \ref{FIG:solo2C3} (e), we find
\begin{equation}
\frac{2}{wz^2} %
\left( \frac{2}{wz^2} \Big( \frac{w^3z^3}{2} \Big) %
\Big( \vartheta_w \sth{\gr{W}_{0,C_3}}(w,z) \Big)^2 \right) \, .
\label{eqn:solo2C3-e}
\end{equation}
A simple way to enumerate the smooth graphs represented by
the figure \ref{FIG:solo2C3} (c) is to consider
that the three paths between the triangle and the vertex
$v$ are symmetric. Taking into account the fact
that only one of these three paths can be reduced
to a simple edge (to avoid another triangle), we have
the following EGF associated to these smooth graphs
\begin{equation}
\frac{z^7}{3!(1-z)^3} + \frac{z^6}{2!(1-z)^2} \, .
\label{eqn:solo2C3-c}
\end{equation}
In total, the bivariate EGF for all graphs such that smooth species are
depicted by the figures \ref{FIG:solo2C3} (c), \ref{FIG:solo2C3} (d) and
\ref{FIG:solo2C3} (e) is given by
\begin{equation}
w^2 \frac{T^6}{6}\frac{(3-2T)}{(1-T)^3} + \frac{w^2T^7}{%
2(1-T)^2} + \frac{w^2T^8}{4(1-T)^3} \, .
\label{eqn:SoloC3-2-others}
\end{equation}
Summing (\ref{eqn:SoloC3-2-lemma78}) and (\ref{eqn:SoloC3-2-others}%
), one can deduce the bivariate EGF for tricyclic graphs
containing exactly a triangle
\begin{equation}
\gr{S}_{2,C_3}(w,z) = \frac{w^2T^6}{48} %
\frac{(48+18T-140T^2+119T^3-30T^4)}{(1-T)^5} \, .
\label{eqn:SoloC3-2}
\end{equation}
We turn now to the enumeration of tricyclic graphs
with one occurrence of juxtaposition of triangles.
The figure \ref{FIG:JUXTA2C3} represents the
2-excess smooth graphs with juxtapositions of
triangles.
\begin{figure}[h]
\begin{center}
\begin{minipage}[t]{12cm}
\small
\begin{center}
\epsfig{file= juxtaC3-2.eps,scale=0.5}
\end{center}
\caption{Basic tricyclic graphs with juxtapositions of triangles.}
\label{FIG:JUXTA2C3}
\end{minipage}
\end{center}
\end{figure}
\noindent
We observe that figures \ref{FIG:JUXTA2C3} (b) and \ref{FIG:JUXTA2C3} (c)
can be handled with the techniques of lemma \ref{LEMMA8} using the EGF
$\sth{\gr{W}_{0,C_3}}$ and $\frac{w^5z^4}{2!2!}$ (which is the EGF of the
smooth juxtaposition of $2$ triangles).
Similarly, we can use lemma \ref{LEMMA7} for the figures
\ref{FIG:JUXTA2C3} (d) and \ref{FIG:JUXTA2C3} (e).
The EGF associated to the smooth graph of figure \ref{FIG:JUXTA2C3} (a)
is simply $\frac{w^7z^5}{2!3!}$, and the one for
smooth graphs depicted by the figure \ref{FIG:JUXTA2C3} (f)
is $\frac{w^7z^5}{4(1-wz)}$. In fact, graphs such as
the one drawn in figure \ref{FIG:JUXTA2C3} (f) can be obtained
by replacing an edge of the complete graph $K_4$ with
a path of length at least $2$.
The EGF that corresponds to the figure \ref{FIG:JUXTA2C3} is
then
\begin{eqnarray}
& & \sth{\gr{J}_{2,C_3}}(w,z) = \frac{w}{z(1-wz)} %
\vartheta_z\big(\frac{w^5z^4}{4}\big) \vartheta_z\big(\sth{\gr{W}_{0,C_3}}(w,z)\big) \cr
& & + \frac{2}{wz^2} \vartheta_w\big(\frac{w^5z^4}{4}\big) %
\vartheta_w\big(\sth{\gr{W}_{0,C_3}}(w,z)\big) %
+ {w^2 \frac{(wz)^5}{2!3!}} + {w^2\frac{(wz)^5}{4(1-wz)}} \, .
\end{eqnarray}
Thus, the bivariate EGF of tricyclic graphs containing
exactly a juxtaposition of triangles is
\begin{equation}
\gr{J}_{2,C_3}(w,z) = \frac{w^2T^5}{6}%
\frac{(2+5T-4T^2)}{(1-T)^2} \, .
\label{eqn:JuxtaC3-2}
\end{equation}
The bivariate EGF of tricyclic triangle-free graphs is then obtained using
(\ref{eqn:SoloC3-2}), (\ref{eqn:JuxtaC3-2})
and (\ref{FUNCTIONAL_TRIANGLE_GRAPH}), namely,
\begin{equation}
\gr{W}_{2,C_3}(w,z)=w^2\frac{T^6}{48} %
\frac{(7+36T-18T^2-40T^3+40T^4-10T^5)}{(1-T)^6} \, .
\label{eqn:FreeC3-2}
\end{equation}
\subsection{General forms of the EGFs of $\xi$-free components}
\label{SUBSUB:GENERAL-FORM}
Although lemmas \ref{LEMMA0} and \ref{LEMMA_GENERAL0} do not allow us
to solve completely the problems of
enumerating $\xi$-free connected graphs with
a given number of vertices and edges, the combination of
these lemmas with subtle combinatorial
constructions provides alternative solutions to get the general forms
of the EGFs $\gr{W}_{k,\xi}$ and ${W_{k,\xi}}$. Recall the following
theorem due to \aut{Wright}
\begin{thm}[Wright 1977] \label{TH_WRIGHT_FORM}
For $k \geq 1$, the EGFs, $\gr{W_{k}}$,
of $(k+1)$-cyclic graphs can be expressed as a finite sum
of powers of $\frac{1}{1-T(z)}$ with rational coefficients and
we have
\begin{equation}
\gr{W}_{k}(z) = \frac{b_k}{(1-T(z))^{3k}} - \frac{c_k}{(1-T(z))^{3k-1}} %
+ \sum_{ 2 \leq s \leq 3k-2} \frac{\omega_{k,s}}{(1-T(z))^{s}} \, .
\label{EQ:TH_WRIGHT_FORM}
\end{equation}
The $(b_k)_{k \geq 1}$ are called
the Wright's constants of first order (also called
Wright-Louchard-Tak\'acs constants, see for e.g. \cite{Sp97}).
$b_1=\frac{5}{24}$ and for $k \geq 1$, $b_k$ is defined recursively by
\begin{equation}
2 (k+1) b_{k+1} = 3k(k+1)b_k+ 3 \sum_{t=1}^{k-1} t(k-t)b_t b_{k-t} \, .
\label{EQ:B_K}
\end{equation}
The $(c_k)_{k \geq 1}$ are the Wright's constants of
second order and are defined recursively, using (\ref{EQ:B_K}), by
$c_1 = \frac{19}{24}$ and for $k \geq 1$
\begin{eqnarray}
2 (3k+2) c_{k+1} &=& 8 (k+1) b_{k+1} + 3k b_k + (3k+2)(3k-1) c_k \cr
& + & 6\sum_{t=1}^{k-1} t (3k - 3t -1) b_t c_{k-t} \, .
\label{EQ:C_K}
\end{eqnarray}
\end{thm}
The proof of theorem \ref{TH_WRIGHT_FORM} is an interesting
combinatorial exercise involving essentially the pointing
operators $\vartheta_w$ and $\vartheta_z$ (see \cite{Wr77, JKLP93}).
Note that formulae (\ref{EQ:TH_WRIGHT_FORM}), (\ref{EQ:B_K}) and
(\ref{EQ:C_K}) are obtained with \aut{Wright}'s fundamental
differential recurrence (well explained in \cite[section 6]{JKLP93})
and which is written here with the notations of this paper
\begin{eqnarray}
\vartheta_w \gr{W}_{k+1} &=& \, \, %
\, \, w \Big( \frac{{\vartheta_z}^2 - \vartheta_z}{2} - \vartheta_w \Big) \gr{W}_{k} \cr
& + & \, \, w \left(%
\sum_{ -1\leq p \leq q \leq k+1, \, p+q=k} %
\frac{1}{1+\delta_{p,q}} (\vartheta_z \gr{W}_{p})%
(\vartheta_z \gr{W}_{q}) \right) \, .
\label{FUNCTIONAL_WRIGHT}
\end{eqnarray}
For our connected $(k+1)$-cyclic triangle-free graphs, we have the
following existence theorem on the forms of their EGFs:
\begin{thm} \label{THM_GENERAL_FORM}
There exists rational $\omega_{k,i}^{(C_3)}$ such that for all
$k \geq 2$, the univariate EGF, $\gr{W}_{k,C_3}$, associated
to $(k+1)$-cyclic triangle-free graphs,
is of the form:
\begin{equation}
\gr{W}_{k,C_3}(z) = \frac{b_k}{( 1-T)^{3k}} - %
\frac{ \cpt{k} } %
{ (1-T )^{3k-1}} + %
\sum_{i \leq 3k-2} %
\frac{ {\omega}_{k,i}^{(C_3)} }{(1-T)^{i}}
\label{PWK}
\end{equation}
where $T\equiv T(z)$, the summation is finite and
the coefficients $\cpt{k}$ are defined, for all $k \geq 1$, by
\begin{equation}
\begin{array}\{{rl}.
& \cpt{1}=\frac{25}{24} \, , \\
& \cpt{k+1} = c_{k+1} + \frac{3}{2}k b_k \, .
\end{array}
\label{CPRIME_0}
\end{equation}
\end{thm}
Before proving theorem \ref{THM_GENERAL_FORM},
the connected components with one occurrence of triangle
are subdivided into $3$ kinds of constructions, according
to the degrees of the vertices of the unique triangle (after smoothing).
Let us define these classifications. A smooth graph containing
a triangle is of three kinds:
\begin{itemize}
\item[-] exactly one vertex of the triangle is of degree $ \geq 3$,
\item[-] exactly two vertices of the triangle are of degree $\geq 3$,
\item[-] the $3$ vertices of the triangle are all of degree $\geq 3$.
\end{itemize}
\noindent
\begin{figure}[h]
\noindent
\begin{minipage}[t]{4.0cm}
\epsfig{file=solo1.eps,scale=0.25}
\caption[Type I]
{One vertex of the triangle
is of degree $\geq 3$.}
\label{FIG:SOLO1}
\end{minipage}
\hfill
\begin{minipage}[t]{4.0cm}
\tiny
\epsfig{file=solo2.eps,scale=0.25}
\caption[Type II]
{Two vertices of the triangle
are of degree $\geq 3$.}
\label{FIG:SOLO2}
\end{minipage}
\hfill
\begin{minipage}[t]{4.0cm}
\tiny
\epsfig{file=solo3.eps,scale=0.25}
\caption[Type III]
{Other smooth components.}
\label{FIG:SOLO3}
\end{minipage}
\normalsize
\end{figure}
\noindent
Graphs whose situations after smoothing
are depicted by figures \ref{FIG:SOLO1} and \ref{FIG:SOLO2}
can be handled by the techniques of lemmas \ref{LEMMA7}
and \ref{LEMMA8}, and will be considered more
precisely later. Note that in the figures, the right
parts (in grey) of the constructions
correspond to multicyclic structures without triangle.
The lemma \ref{LEMMA_INDECOMPOSABLE}
gives the form of the EGF of the connected component with
exactly one occurrence of
triangle depicted by the figure \ref{FIG:SOLO3}.
\begin{lem} \label{LEMMA_INDECOMPOSABLE}
The EGF of $(k+1)$-cyclic graphs containing one occurrence of
triangle with all of its vertices of degree at least $3$ has the
following form
\begin{equation}
\sum_{s \leq 3k-3} \frac{\epsilon_{k,s}}{(1-T(z))^{s}} \,
\end{equation}
where the summation is finite and the coefficients $\epsilon_{k,s}$
are rational numbers.
\end{lem}
\noindent \textbf{Proof.} Our idea is to apply
Wright's reduction method on our specific configuration.
Since this method is known but is not that
familiar, we repeat here the main steps. Suppose that
we have a connected graph with $k$ edges more than vertices containing one triangle
and suppose that the recursive suppressions of vertices of degree $1$ lead
to a graph of the type depicted by figure \ref{FIG:SOLO3}.
That is, the obtained smooth graph has $t$ vertices of
degree at least $2$ and $t+k$ edges (here, $t$ is less that or
equal to the number of vertices of the original graph).
This way, we get a smooth graph with $r$ vertices of degree
at least $3$, $r \leq 2k$. These vertices of degree $\geq 3$
are called \textit{special vertices} and let us \textit{color}
the edges of the triangle in order to distinguish them.
The paths between these points, except the colored edges of the
triangle, are of four kinds and we apply the following special
operations on them (see \cite[Sect.~ 6]{Wr77}):
\begin{itemize}
\item[1.] An $\alpha$-path begins and ends with the same special point
and so must have at least two interior points. We elide all its interior
points except two of them.
\item[2.] A $\beta$-path joins two different special vertices
and we elide all its interior points.
\item[3.] If two different special vertices are joined by more than
one special path, at most one of these paths is reduced to a single edge
which we call a $\delta-path$.
\item[4.] The remaining paths, or all the paths if there is
no $\delta$-path, are called $\gamma$-paths and for each
$\gamma$-path, we elide all its interior points except one of them.
\end{itemize}
The obtained graph is called \textit{Wright's basic graph}.
Denote respectively by $a$, $b$, $c$ and $d$
the number of $\alpha$-, $\beta$-, $\gamma$-
and $\delta$- paths. Since each elision has removed exactly
one edge and one vertex, the number of vertices of the
basic graph is exactly $r+2a+c$. Taking into account,
the colored edges of the triangle and the operations made upon
the special paths, the number of edges in the basic graph is
$r+2a+c+k = 3a+b+2c+d+3$. Thus, we have $a+b+c+d+3 = r+k \leq 3k$.
We find
\begin{equation}
a+b+c \leq 3k -3 \, .
\label{ELIDEELIDE}
\end{equation}
To obtain any of the original graphs without vertices of degree $1$,
we distribute the previously $t-r-2a-c$ elided nodes
on the $\alpha$-, $\beta$- and $\gamma$- paths. (\ref{ELIDEELIDE}) gives
us ideas on the number of ways to redistribute these points:
suppose that $f(n)$ is the number of labelings of the $(n,n+k)$-graphs
which can produce the considered basic graph. Let $F(z)$ be their EGF:
\begin{equation}
F(z) = \sum_n f(n) \frac{z^n}{n!} \, .
\end{equation}
To obtain each of the original $(t,t+k)$ graphs without endvertices,
the distribution of the $(t-r-2a-c)$ nodes on the
$(a+b+c)$ $\alpha$-, $\beta$- and $\gamma$-paths can be done
in $y$ ways where $y$ is the number of
partitions of $(t-r-2a-c)$ into $(a+b+c)$ parts.
Relabel the obtained graph and replace the $t$ vertices with
$t$ rooted and labelled trees.
All the graphs are
enumerated but they are not all different. In fact, they
are enumerated $g$ times where $g$ is the order of
the automorphisms of the current Wright's basic graph.
Thus, we have
\begin{equation}
g F(z) = \sum_t y T(z)^t = \frac{T(z)^{r+2a+c}}{(1-T(z))^{a+b+c}} \, .
\end{equation}
Summing over all the finitely many possible basic graphs,
we obtain the lemma.
\hfill \qed
\noindent \textbf{Proof of theorem \ref{THM_GENERAL_FORM}.} Denote by
\PW{k}, \PS{k} and \PJ{k} the following properties:
\begin{itemize}
\item[$\bullet$] \PW{k}~: $\gr{W}_{k,C_3}$ is of the form
given by the equation (\ref{PWK}).
\item[$\bullet$] \PS{k}~: \\
If $k=1$,
\begin{equation}
\gr{S}_{1,C_3}(z) = %
\frac{1}{4\, (1-T)^{2}}-\frac{1}{(1-T)}-\frac{1}{4}{T}^{4}+%
\frac{1}{4}{T}^{2}+\frac{1}{2}T +\frac{3}{4} \,
\end{equation}
and for all $k \geq 2$, $\gr{S}_{k,C3}$ is of the form
\begin{equation}
\gr{S}_{k,C_3}(z) = \frac{3 (k-1) b_{k-1}}{2 \, \big( 1-T(z) \big)^{3k-1}} %
+ \sum_{i \leq 3k-2} %
\frac{\sigma_{k,i}^{(C_3)}}{\big( 1-T(z) \big)^{i}} \, .
\label{PSK}
\end{equation}
\item[$\bullet$] \PJ{k}~: \\
If $k=1$
\begin{equation}
\gr{J}_{1,C_3}(z) = \frac{T^4}{4} \,
\end{equation}
and if $k=2$, we have
\begin{equation}
\gr{J}_{2,C_3}(z) = \frac{1}{2 (1-T)^2 }- %
\frac{2}{(1-T)} + \frac{3}{2} + T + \frac{T^2}{2} %
- \frac{T^4}{2} - \frac{2 T^5}{3}\, .
\label{DECOMPOSITION_J2}
\end{equation}
For all $k \geq 3$, $\gr{J}_{k,C_3}$ is of the form
\begin{equation}
\gr{J}_{k,C_3}(z) = \frac{3 (k-2) b_{k-2}}{\big( 1-T(z) \big)^{3k-4}} %
+ \sum_{i \leq 3k-5} %
\frac{\upsilon_{k,i}^{(C_3)}}{\big( 1-T(z) \big)^{i}} \, .
\label{PJK}
\end{equation}
\end{itemize}
where the coefficients $(\omega_{k,i}^{(C_3)})$, $(\sigma_{k,i}^{(C_3)})$
and $(\upsilon_{k,i}^{(C_3)})$ are rational numbers and the summations
in (\ref{PWK}), (\ref{PSK}) and (\ref{PJK}) are \textbf{finite}.
We will show by \textit{induction} on $k$, that for all $k \geq 1$,
the properties \PW{k}, \PS{k} and \PJ{k} described above
are simultaneously verified. To do this,
we have
\PW{1}, \PS{1}, \PJ{1} and \PJ{2}
and we have to check that if
\PW{i}, \PS{i} and \PJ{i} are true for all $i$ such that
$ 1 \leq i \leq k-1$ then
\PW{k}, \PS{k} and \PJ{k} are also satisfied.
Note that due to the presence of the factor $(k-1)$ in
(\ref{PSK}), resp. $(k-2)$ in (\ref{PJK}),
we have to give $\gr{S}_{1,C_3}$, $\gr{J}_{1,C_3}$
and $\gr{J}_{2,C_3}$.
Rewriting (\ref{eqn:FreeC3-2}) and (\ref{eqn:SoloC3-2})
as sums of powers of $\frac{1}{1-T}$, we have
\begin{eqnarray}
& & \gr{W}_{2,C_3}(z) = \frac{5}{16 (1-T)^6} %
- \frac{5}{3 (1-T)^5} + \frac{167}{48 (1-T)^4 }%
- \frac{91}{24 (1-T)^3} \cr
& & + \frac{55}{16 (1-T)^2 } %
- \frac{35}{8 (1-T) } + \frac{125}{48} %
+ \frac{17 T}{12} + \frac{11 T^2}{24} - \frac{5 T^3}{24} %
- \frac{5 T^4}{12} - \frac{5 T^5}{24} \, , \cr
& & \gr{S}_{2,C_3}(z) = \frac{5}{16 (1-T)^5 } %
- \frac{65}{48 (1-T)^4} + \frac{7}{3 (1-T)^3} %
- \frac{73}{24 (1-T)^2} \cr
& & + \frac{61}{12 (1-T)} %
- \frac{10}{3} - \frac{103 T}{48} - \frac{53 T^2}{48} %
- \frac{5 T^3}{48} %
+ \frac{31 T^4}{48} + \frac{5 T^5}{8} \, .
\end{eqnarray}
Thus, $\gr{S}_{2,C_3}(z)$, $\gr{J}_{2,C_3}(z)$,
and $\gr{W}_{2,C_{3}}(z)$ can be formulated as finite sums
of power of $\frac{1}{(1-T)}$ and properties \PW{2}, \PS{2} and \PJ{2}
are satisfied. Note that we let $b_0 = \frac{1}{2}$, due to the fact that
$\vartheta_z \, \gr{W}_{0,C_3}(z) = \frac{1}{2} \, \frac{T^4}{(1-T)}$.
Now, suppose that \PW{i}, \PS{i} and \PJ{i} are true for
$i \in \coeff{1,\, k-1}$. If we want to compute directly
$\gr{W}_{k,C_3}$, the differential recurrence relation
(\ref{FUNCTIONAL_TRIANGLE_GRAPH}) of
lemma \ref{LEMMA0} is not useful except if we know
the EGFs $\gr{S}_{k,C_3}$ and $\gr{J}_{k,C_3}$. However, assuming
that \PW{i}, \PS{i} and \PJ{i} are true for
$i \in \coeff{2,\, k-1}$, we can compute the forms of $\gr{S}_{k,C_3}$ and
$\gr{J}_{k,C_3}$ using combinatorial decompositions of these
graphs. In the rest of this proof, our attention will be
focused on the terms involving $\frac{1}{(1-T(z))^{3k}}$
and $\frac{1}{(1-T(z))^{3k-1}}$ for
$\gr{W}_{k,C_3}$ and $\frac{1}{(1-T(z))^{3k-1}}$
for $\gr{S}_{k,C_3}$.
Under the hypothesis of the induction, let us compute
the forms of $\gr{S}_{k,C_3}$ and $\gr{J}_{k,C_3}$.
More specifically, the components represented by figures \ref{FIG:SOLO1} and
\ref{FIG:SOLO2} can be decomposed and the forms of their EGFs can be
computed using the EGF of the triangle
(eq. (\ref{TRIANGLE-TRIANGLE-EGF})), the operator $\vartheta_z$ (to distinguish
the common point) and the form of the EGF $\gr{W}_{k-1,C_3}$ which is
assumed by the induction hypothesis.
Recall that $\sth{\gr{W}_{k-1,C_3}}$ denotes the EGF of
\textit{$k$-cyclic smooth graphs} without triangle obtained
when deleting recursively all vertices of degree $1$.
Using lemma \ref{LEMMA7}, we obtain the univariate
EGF of all the graphs such that the
situation after smoothing is depicted by figure \ref{FIG:SOLO1}, namely
\begin{equation}
{\left[ \frac{1}{z} \, \frac{1}{1-z} \, %
\vartheta_z (\frac{z^3}{3!}) \vartheta_z \sth{\gr{W}_{k-1,C_3}}(z) \right]}_{| z=T(z)}
\end{equation}
Similarly, the smooth graph represented by figure \ref{FIG:SOLO2} can be
enumerated using the operator $\vartheta_w$. We obtain the following
bivariate EGF
\begin{equation}
{\left[ \frac{2}{wz^2} \vartheta_w \big( \frac{w^3 z^3}{3!} \big) %
\vartheta_w \big( \sth{\gr{W}_{k-1,C_3}}(w,z)\big) \right]}_{| wz = T(wz)} \,
\end{equation}
Using the form of the EGF of $(k+1)$-cyclic components given
by lemma \ref{LEMMA_INDECOMPOSABLE}, we find the form of the bivariate
EGF of smooth graphs of $\sth{\gr{S}_{k,C_3}}$,
\begin{eqnarray}
& & \sth{\gr{S}_{k,C_3}}(w,z) = \left( \frac {w} {z(1-wz)} %
\vartheta_z \big( \frac {w^3 z^3} {3!} \big) %
\vartheta_z \big( \sth{\gr{W}_{k-1,C_3}} (w,z) \big) \right) \cr
& & + \left( \frac{2}{wz^2} \vartheta_w \big( \frac{w^3 z^3}{3!} \big) %
\vartheta_w \big( \sth{\gr{W}_{k-1,C_3}}(w,z)\big) \right) %
+ w^k \sum_{i \leq 3k-2} %
\frac{ s_{k,i}^{(C_3)} }{\big( 1-wz \big)^i} \, .
\label{SMOOTH_S}
\end{eqnarray}
Remark that the constants $s_{k,i}^{(C_3)}$ are not those described by
eq. (\ref{PSK}) because we have to take into account the terms
from $\frac{2}{wz^2} \vartheta_w \big( \frac{w^3 z^3}{3!} \big)
\vartheta_w \big( \sth{\gr{W}_{k-1,C_3}}(w,z)\big)$.
Thus, we find
\begin{eqnarray}
& & \sth{\gr{S}_{k,C_3}}(w,z) \, \, = \frac{w^3z^2}{2(1-wz)} \times %
w^{k-1} \, \vartheta_z \left( \frac{b_{k-1}}{(1-wz)^{3k-3}}
+ \sum_{i \leq 3k-4}%
\frac{s_{k-1,i}^{(C_3)}}{(1-wz)^i} \right) \cr
& & + w^2 z \vartheta_w \left(\frac{w^{k-1}b_{k-1}}{(1-wz)^{3k-3}} %
+ \sum_{i \leq 3k-4
\frac{w^{k-1} s_{k-1,i}^{(C_3)}}{(1-wz)^i} \right) %
+ w^k \sum_{i \leq 3k-2} %
\frac{ s_{k,i}^{(C_3)} }{\big( 1-wz \big)^i} \, .
\end{eqnarray}
A bit of calculus leads to the EGF of $(k+1)$-cyclic
components with exactly one triangle
\begin{equation}
\gr{S}_{k,C_3}(w,z) = w^k \left( %
\frac{3 (k-1) b_{k-1}}{2 \big( 1-T \big)^{3k-1}} %
+ \sum_{i \leq 3k-2} %
\frac{\sigma_{k,i}^{(C_3)}}{\big( 1-T \big)^{i}} \right) \, .
\label{FORME_S_C3}
\end{equation}
and \PS{k} is verified. Similarly,
the same principles can be used to compute
the form of $\gr{J}_{k,C_3}$ when replacing the single occurrence of
triangle by a single occurrence of juxtaposition of triangles which can
be considered in its turn as a single subgraph. For this purpose,
we have to replace the EGF $\frac{w^3z^3}{3!}$ of the triangle
by EGFs of juxtapositions of triangles, viz.
$\frac{w^5z^4}{2!2!}$ (EGF of the smooth graph
depicted by figure \ref{FIG:SMOOTH_J1}),
$\frac{w^7z^5}{2!3!}$, $\cdots$,
$\frac{w^{2i+1}z^{i+2}}{2!i!}$, $\cdots$ . We find
\begin{eqnarray}
& & \sth{\gr{J}_{k,C_3}} (w,z) \, \,= \frac {w} {z(1-wz)}%
\vartheta_z\big( \frac {w^5 z^4} {4} \big) \, %
\vartheta_z \big( \sth{\gr{W}_{k-2,C_3}} (w,z) \big) \cr
& & \, + \, \frac {w} {z(1-wz)} %
\vartheta_z\big( \frac {w^7 z^5} {12} \big) %
\vartheta_z \big( \sth{\gr{W}_{k-3,C_3}} (w,z) \big) \cr
& & + \frac{2}{wz^2} \vartheta_w \big( \frac{w^5 z^4}{4} \big) %
\vartheta_w \big( \sth{\gr{W}_{k-2,C_3}}(w,z)\big) %
+ w^k \sum_{i \leq 3k-3} %
\frac{ \iota_{k,i}^{(C_3)} }{\big( 1-wz) \big)^i} \, .
\end{eqnarray}
Hence, we have the form of $3\gr{S}_{k,C_3}+\gr{J}_{k,C_3}$
which starts with $\frac{9(k-1)b_{k-1}}{2 (1-T)^{3k-1}}$.
We need some useful notations, mainly related to those of Wright
\cite{Wr77,Wr80}.
Denote by $\mathbb{X}$ the following EGF
\begin{equation}
\mathbb{X} \equiv 1-T \, .
\label{eq:DEFTHETA}
\end{equation}
Let $\Lambdat{1} = 0$ and for all $k \geq 2$, let
$\Lambdat{k}$ be the following formal power series
\begin{equation}
\Lambdat{k}: \, \Lambdat{k}(z) = \sum_{t=1}^{k-1} %
\Big(\vartheta_z \gr{W}_{t,C_3}(z) \Big) %
\Big(\vartheta_z \gr{W}_{k-t,C_3}(z)\Big) \, .
\label{EQ:LAMBDA_K}
\end{equation}
Let $F$ be an EGF. For all $k \geq 1$,
we denote by $\Delta$ and $\Omegat{k}$ the following operators
\begin{equation}
\Delta_{k+1}: \, \, \Delta_{k+1} \, \big(F \big) %
= 2 \Big( k+1 - T \frac{\partial}{\partial T}\Big) \, \big(F\big) \,
\label{OP:DELTA_K}
\end{equation}
and
\begin{equation}
\Omegat{k}: \, \, \Omegat{k} \, \big(F\big) =%
\Big( \big( \vartheta_z^2 - 3 \vartheta_z - 2k\big) +%
2 \big( \vartheta_z \gr{W}_{0,C_3}(z) \big) \vartheta_z \Big) \, \big(F \big) \, .
\label{OP:OMEGAT_K}
\end{equation}
Using these notations, we remark that the functional equation
(\ref{FUNCTIONAL_TRIANGLE_GRAPH}) of lemma \ref{LEMMA0}
can be reformulated as follows
\begin{eqnarray}
& & \Delta_{k+1} \gr{W}_{k+1,C_3} %
+ 6 \gr{S}_{k+1,C_3} + 2 \gr{J}_{k+1,C_3} = \cr
& & \, \, \, \, %
\Omegat{k} \gr{W}_{k,C_3} + \Lambdat{k} \, , \, \, (k \geq 1) \, .
\label{AVEC_OP_C3}
\end{eqnarray}
Then, we remark that
\begin{equation}
\Delta_{k} {\mathbb{X}}^{-t} = \Delta_{k} \frac{1}{(1-T)^{t}} = %
2 \mathbb{X}^{-t} (t \mathbb{X}^{-1} + k-t) \, .
\label{EQ:DELTA}
\end{equation}
We also have
\begin{equation}
\vartheta_z \gr{W}_{0,C_3}(z) = \frac{T^4}{2(1-T)^2} %
= \frac{\mathbb{X}^{-2} }{2} - 2\mathbb{X}^{-1} + 3 - 2 \mathbb{X} + \frac{{\mathbb{X}}^2}{2} \, .
\end{equation}
\begin{eqnarray}
& & (\vartheta_z^2 - \vartheta_z -2(k-1)) \mathbb{X}^{-t} + %
2(\vartheta_z \gr{W}_{0,C_3})(\vartheta_z \mathbb{X}^{-t}) = \cr
& & \, \, \, %
t(t+3) \mathbb{X}^{-t-4} - t(2t+8) \mathbb{X}^{-t-3} + \cdots \, .
\end{eqnarray}
Using these formulae, the induction hypothesis, the form
of the generating function $6\gr{S}_{k,C_3} + 2\gr{J}_{k,C_3}$ and the
formula (\ref{FUNCTIONAL_TRIANGLE_GRAPH}) of lemma \ref{LEMMA0},
when looking after the coefficients of
$\mathbb{X}^{-3k+1}$ and $\mathbb{X}^{-3k}$, we find
\[
\gr{W}_{k,C_3} = b_k \mathbb{X}^{-3k} - \cpt{k} \mathbb{X}^{-3k+1} + \cdots
\]
where the sequences $(b_k)$ and $(\cpt{k})$ satisfy
exactly the recurrences given by (\ref{EQ:B_K}) and
\begin{equation}
\begin{array}\{{rl}.
& \cpt{1}=\frac{25}{24} \, , \\
& \, \, %
2(3k+2)\cpt{k+1} = 8(k+1)b_{k+1} \\
& \, \, + \, \, 6 kb_k + (3k-1)(3k+2)\cpt{k} \\
& \, \, + \, \, 6 \sum_{t=1}^{k-1}t(3k-3t-1)b_t\cpt{k-t} \, .
\end{array}
\label{PRE_CPRIME_0}
\end{equation}
Now, we can show (\ref{CPRIME_0}) by induction. We have
$\cpt{1} = \frac{25}{24}$, $b_1 = \frac{5}{24}$ and
$c_2 = \frac{65}{48}$ and we can check
$\cpt{2} = \frac{5}{3} = c_2 + \frac{3}{2} b_1$.
Suppose that for $i$ from $1$ to $k-1$,
$\cpt{i}$ verifies
\[
\cpt{i+1} = c_{i+1} + \frac{3}{2} i b_i \, .
\]
Using (\ref{PRE_CPRIME_0}) and the induction hypothesis, we have
for $i=k$ (we have to be careful with $\cpt{1}=c_{1} + \frac{1}{4}$)
\begin{eqnarray}
2(3k+2)\cpt{k+1} = & & 8(k+1)b_{k+1} + 6kb_k \cr
& & + (3k-1)(3k+2)c_{k} + \frac{3}{2} (3k-1)(3k+2)(k-1)b_{k-1} \cr
& & + 12(k-1)b_{k-1}c_1 + 3(k-1)b_{k-1} \cr
& & + 6 \sum_{t=1}^{k-2} t(3k-3t-1) b_t c_{k-t} \cr
& & + 9 \sum_{t=1}^{k-2} t(3k-3t-1)(k-t-1) b_t b_{k-t-1} \, .
\end{eqnarray}
And as already remarked by Wright,
\cite[eq. (3.5)]{Wr80}, for any given sequence $(\alpha_k)$ we have
\begin{equation}
\sum_{t=1 }^{k-1 } t \alpha_t \alpha_{k-t} = %
\frac{ k}{2} \sum_{t=1 }^{k-1 } \alpha_t \alpha_{k-t} \, .
\label{REMARK_ALPHA}
\end{equation}
Rearranging, we find using the definition of $c_{k+1}$ given by
(\ref{EQ:C_K}) and (\ref{REMARK_ALPHA})
\begin{eqnarray}
2(3k+2)\cpt{k+1} & & = 2(3k+2)c_{k+1} + 3kb_k \cr
& & + (3 + \frac{3}{2}(3k-1)(3k+2) )(k-1)b_{k-1} \cr
& & + \frac{9}{2}(3k+1) \sum_{t=1}^{k-2} t b_t (k-t-1) b_{k-t-1} \, .
\end{eqnarray}
Since $3 \sum_{t=1}^{k-2} tb_t (k-t-1)b_{k-t-1} =
2k b_k - 3(k-1)kb_{k-1}$, we obtain
\begin{eqnarray}
2(3k+2)\cpt{k+1} & & = 2(3k+2)c_{k+1} + 3kb_k \cr
& & + (3 + \frac{3}{2}(3k-1)(3k+2) )(k-1)b_{k-1} \cr
& & + 3(3k+1)kb_k - \frac{9}{2}(k-1)k(3k+1) b_{k-1} \, .
\end{eqnarray}
Finally, we find $2(3k+2)\cpt{k+1} = 2(3k+2)c_{k+1} + 3(3k+2)kb_k$. \hfill \qed
As a consequence, if we want to work with a
forbidden subgraph $H$ which is not unicyclic (e.g. $K_4$),
the decomposition of $\gr{W}_{k,H}$ into sums of
negative powers of $\mathbb{X}$ (i.e. tree polynomials)
starts
\[
\gr{W}_{k,H} = b_k \mathbb{X}^{-3k} - c_{k} \mathbb{X}^{-3k+1} + \cdots \, .
\]
The same remark holds for any finite collection
of forbidden subgraphs which are not unicyclic.
In the next theorem, we will generalize the case $\xi = \{ C_3 \}$.
\begin{thm} \label{THM_GENERAL_FORM_MANY}
Let $\xi = \{H_1,H_2,\cdots, H_p\}$ a finite collection of multicyclic
components. Suppose that $\xi$ contains $r$, $r > 0$,
distinct polygons (unicyclic smooth graphs).
Denote by $\gr{W}_{k,\xi}$ the EGF of $(k+1)$-cyclic
$\xi$-free labelled graphs. For all $k \geq 2$, $\gr{W}_{k,\xi}$ can
be expressed as a finite sum of powers of $\frac{1}{1-T}$
and has the following form:
For $k=1$, we have
\begin{equation}
\gr{W}_{1,\xi}(z) = \frac{5}{24 \big( 1-T(z)\big)^3}
- \frac{\big(19/24 + r/4\big)}{\big(1-T(z)\big)^2}
+ \sum_{i \leq 1} %
\frac{{\psi_{i,1}}^{(\xi)}}{\big(1-T(z)\big)^{i} } \,
\end{equation}
and for $k > 1$
\begin{equation}
\gr{W}_{k,\xi}(z) = \frac{b_k}{\big(1-T(z)\big)^{3k} } %
- \frac{ \cpxi{k} }{ \big(1-T(z) \big)^{3k-1} } %
+ \sum_{i \leq 3k-2} %
\frac{{\psi_{i,k}}^{(\xi)}}{\big(1-T(z)\big)^{i}} \,
\label{FORME_GENERALE}
\end{equation}
where $b_k$ is Wright's coefficient
of first order given by (\ref{EQ:B_K}) and
$\cpxi{k}$ is given recursively by $\cpxi{1} = \frac{19+6r}{24}$
and for $k \geq 1$
\begin{equation}
\cpxi{k+1} = c_{k+1} + \frac{3}{2} r k b_{k} \, .
\label{EQ:CPXI}
\end{equation}
\end{thm}
\noindent \textbf{Proof.} The proof of this theorem is very close
to that of theorem \ref{THM_GENERAL_FORM}.
Suppose that $\xi$ contains $r$ polygons $(r > 0)$. Furthermore,
suppose that $C_q$ is the greatest polygon of $\xi$. That is
\[
\gr{W}_{0,\xi} = \frac{1}{2} \ln{\frac{1}{1-T}} - \frac{T}{2} %
- \frac{T^2}{4} - \sum_{i} \frac{T^i}{2i}
\]
where in the summation $i$ describes all lengths (less than
or equal to $q$) of the forbidden polygons.
Then, since
\[
\frac{T^q}{(1-T)^2 } = \mathbb{X}^{-2} - (q+1) \mathbb{X}^{-1} +%
\sum_{j=1}^{q} T^{q-j} \, ,
\]
we have
\[
2 \vartheta_z \gr{W}_{0,\xi}(z) =%
\frac{T^{q+1} }{(1-T)^2} + \sum_{j} \frac{T^j}{1-T} \,
\]
where the summation is over all lengths of the $q-r-2$
authorized (distinct) polygons. So,
\begin{equation}
2 \vartheta_z \gr{W}_{0,\xi}(z) = \mathbb{X}^{-2} - (r+3)\mathbb{X}^{-1} %
+ \mbox{Polynomial}_{\xi}(T) \,
\end{equation}
and $2 (\vartheta_z \gr{W}_{0,\xi}(z))(\vartheta_z \mathbb{X}^{-t})$ starts with
\begin{equation}
t \mathbb{X}^{-t-4} - (r+4) t \mathbb{X}^{-t-3} + \cdots \, .
\end{equation}
Defining the operator $\Omegaxi{k}$ as
\begin{equation}
\Omegaxi{k}: \, \, \Omegaxi{k} =%
\Big( \big( \vartheta_z^2 - 3 \vartheta_z - 2k\big) +%
2 \big( \vartheta_z \gr{W}_{0,\xi}(z) \big) \vartheta_z \Big) \,
\label{OP:OMEGAXI_K}
\end{equation}
and $\Lambdaxi{k}$ as the formal power seriers
\begin{equation}
\Lambdaxi{k}: \, \Lambdaxi{k}(z) = \sum_{t=1}^{k-1} %
\Big(\vartheta_z \gr{W}_{t,\xi}(z) \Big) %
\Big(\vartheta_z \gr{W}_{k-t,\xi}(z)\Big) \, ,
\label{EQ:LAMBDAXI_K}
\end{equation}
we can generalize (\ref{AVEC_OP_C3})
\begin{eqnarray}
\Delta_{k+1} \gr{W}_{k+1,\xi} + 2 \sum_{\mathcal{H} \in \xi} %
e(\mathcal{H}) \gr{S}_{k+1,\mathcal{H}} %
+ 2 \gr{J}_{k+1,\xi} = \cr
\Omegaxi{k} \gr{W}_{k,\xi} + \Lambdaxi{k} \, , \, \, (k \geq 1) \, .
\label{AVEC_OP}
\end{eqnarray}
Then, we find
\begin{equation}
\Omegaxi{k}\mathbb{X}^{-t} = t(t+3) \mathbb{X}^{-t-4} - %
t(2t+r+7) \mathbb{X}^{-t-3} + \cdots
\end{equation}
As for theorem \ref{THM_GENERAL_FORM}, we find that
$\cpxi{k+1}$ satisfies $\cpxi{1}=c_1 + \frac{r}{4}$ and for
$k \geq 1$
\begin{eqnarray}
& & 2 (3k +2) \cpxi{k+1} = 8(k+1) b_{k+1} + 3k(r+1)b_{k} + \cr
& & \, \,\,\,\,\,\, %
(3k-1)(3k+2) \cpxi{k} + 6\sum_{t=1}^{k-1} t(3k-3t-1) b_t \cpxi{k-t} \, .
\label{EQ:PRE_CPXI}
\end{eqnarray}
We can now argue as for the proof of theorem \ref{THM_GENERAL_FORM}
to verify that the sequence $(\cpxi{k})$ satisfies
(\ref{EQ:CPXI}).
\hfill \qed
In the next section, we will determine the asymptotic number of
triangle-free labelled components
when the number of exceeding edges satisfies $k=o(n^{1/3})$.
\section{Asymptotic number of sparsely connected labelled triangle-free
components}
The methods we give are based on the fundamental work of
Wright in \cite{Wr80} with some ingredients from analytic combinatorics.
First of all, we will study the behavior of
\[
t_n(a\, n+\beta)= n! \, \coeff{z^n} \frac{1}{(1-T(z))^{a \, n+\beta}} \,
\]
where $a \equiv a(n)$ tends to $0$ as $n \rightarrow \infty$
and $\beta$ is fixed. Then, we will show that if
$\beta_1 < \beta_2$,
$a \equiv a(n) \rightarrow 0$ as $n \rightarrow \infty$
but $\frac{a \, n}{{\ln{n}}^3} \rightarrow \infty$,
then $\frac{t_n(a \, n+ \beta_1)}{t_n(a \, n+\beta_2)} \rightarrow 0$.
Next, we will give a general framework analogous to that
of Wright in \cite{Wr80}. More precisely, let $(b_k)$
and $(\cpt{k})$ be the coefficients given by (\ref{EQ:B_K})
and (\ref{CPRIME_0}).
We will show that
the coefficients of the EGFs $\gr{W}_{k,C_3}$
satisfy the following inequalities
\begin{eqnarray}
& & n! \coeff{z^n} \gr{W}_{k,C_3}(z) \leq n! \coeff{z^n} %
\frac{b_k}{\big(1-T(z)\big)^{3k}} \, \, \, \, \, \, \, and \cr
& & n! \coeff{z^n} \left( \frac{b_k}{\big(1-T(z)\big)^{3k}} - %
\frac{\cpt{k}}{\big(1-T(z)\big)^{3k-1}} \right) %
\leq n! \coeff{z^n} \gr{W}_{k,C_3}(z) \,
\label{COEFF_INEGALITES}
\end{eqnarray}
which we shall call \textit{Wright's inequalities} for triangle-free graphs.
Thus, the inequalities in (\ref{COEFF_INEGALITES}) and the fact that
$\frac{t_n(a \, n - 1)}{t_n(a \, n)} \rightarrow 0$
imply that almost all connected components
with $n$ vertices and $n+o(n^{1/3})$ edges
are $\xi$-free whenever $k=o(n^{1/3})$. Equivalently, we will show
that the number $c_{C_3}(n,n+k)$ of triangle-free $(k+1)$-cyclic
graphs is asymptotically the same as the number $c(n,n+k)$ of
$(k+1)$-cyclic general graphs computed by Wright in \cite{Wr80}
(see \cite{BCM90} for the extension of Wright's asymptotic results).
\subsection{Saddle point method for tree polynomials}
\label{SEC:COL}
In \cite{KP89}, Knuth and Pittel studied combinatorially
and analytically the polynomial $t_n(y)$ defined as follows
\begin{equation}
t_n(y) = n! \coeff{z^n} \frac{1}{\big( 1-T(z)\big)^y}
\label{EQ:TREE_POLYNOMIAL}
\end{equation}
which they call \textit{tree polynomial}.
In fact, the authors of \cite{KP89} observed that
the analysis of these polynomials can also be used
to study random graphs.
The lemma below is an application of the saddle point method
\cite{Bruijn,FS+} to study the asymptotic behavior of the
coefficients $n! \coeff{z^n} \big(1-T(z) \big)^{-m(n)}$ as $m, \, n$
tend to infinity but $m=o(n)$.
\begin{lem} \label{TREE_POLYNOMIAL_INFINITY}
Let $a \equiv a(n)$ such that $a \rightarrow 0$
but $\frac{a \, n}{{\ln{n}}^3} \rightarrow \infty$, and
$\beta$ a fixed number. Then, the tree polynomial $t_n(a\,n+\beta)$
defined in (\ref{EQ:TREE_POLYNOMIAL}) satisfies
\begin{equation}
t_n(a \, n+\beta) = \frac{n!}{2 \sqrt{\pi n}} %
\frac{\exp{ (n u_0) } (1-u_0)^{(1-\beta)}}{{u_0}^n (1-u_0)^{a\,n}} %
\left(1+O\big(\sqrt{a}\big) + O\big(\frac{1}{\sqrt{a\,n}}\big)\right)
\label{EQ:TREE_POLYNOMIAL_INFINITY}
\end{equation}
where $u_0 = 1 + \frac{a}{2} - \sqrt{a(1+\frac{a}{4})}$.
\end{lem}
\noindent \textbf{Proof.} Cauchy's integral formula gives
\begin{eqnarray}
t_n(a\,n+\beta) & = & n! \coeff{z^n} %
\frac{1}{\big(1-T(z) \big)^{a\,n+\beta}} \cr
& = & \frac{n!}{2 \pi i} %
\oint \frac{1}{\big(1-T(z) \big)^{a\,n+\beta}} \frac{dz}{z^{n+1}} \,
\label{COL_UN}
\end{eqnarray}
where we integrate around a small circle enclosing the origin
and whose radius is smaller than $1/e$ (since $1/e$ is
the radius of convergence of the formal power series
$T(z) = \sum_{n \geq 1} n^{(n-1)} \frac{z^n}{n!}$).
We make the substitution $u=T(z)$ and
get $dz = e^{-u}(1-u)du$. Thus,
\begin{equation}
t_n(a\,n+\beta) = \frac{n!}{2 \pi i} \oint \frac{e^{nu}\, du}%
{(1-u)^{a\,n+\beta-1} \, u^{n+1} } \, .
\label{COL_DEUX}
\end{equation}
The power $\big(\exp{(u)}/(1-u)^a \big)^n$ suggests us to use the
saddle point method. We will describe briefly this
method for our case and refer to
de Bruijn \cite[Chap. 5]{Bruijn},
Flajolet and Sedgewick \cite{FS+} or Bender \cite{Be74}
for more details on general asymptotic methods.
We set $h(u) = u - \ln(u) - a \ln(1-u)$.
Starting with (\ref{COL_DEUX}), we now have
\begin{equation}
t_n(a\,n+\beta) = \frac{n!}{2 \pi i} %
\oint (1-u)^{1-\beta} \exp(n h(u))\frac{du}{u} \, .
\label{COL_TROIS}
\end{equation}
Let $F(r,\theta)$ be the integrand of
\begin{eqnarray}
& & \frac{1}{2 \pi r^n} \int_{-\pi}^{\pi} (1- re^{i \theta})^{1-\beta} %
\exp(n h(r e^{i \theta}) ) d\theta \cr
& = & \frac{1}{2 \pi r^n} \int_{-\pi}^{\pi} F(r,\theta) d\theta \, .
\end{eqnarray}
The saddle point method consists to remark that
$F(r,\theta)$ turns very quickly as $n \rightarrow \infty$
such that the essential of the integral is captured
by only few values of $\theta$, say
$\theta \in \coeff{-\theta_0, \, \theta_0}$ (with $\theta_0 \rightarrow 0$).
Then, we have to choose the radius $r$
in order to concentrate the main contribution
of the integral, viz. for $\theta \in \coeff{-\theta_0, \, \theta_0}$,
$|F(r,\theta)|$ represents the essential of the integral.
In other words, we have to find a vicinity
of $\theta=0$ where $| F(r,\theta) |$ takes its maximum.
Hence, we investigate the roots of $h^{'}(u) = 0$ and we find
two saddle points, at $u_0 = 1 + \frac{a}{2} - \sqrt{a(1+\frac{a}{4})}$
and $u_1 = 1 + \frac{a}{2} + \sqrt{a(1+\frac{a}{4})}$. We notice that
$h^{''}(u) = \frac{1-2u + (1+a)u^2}{u^2(1-u)^2}$,
$h^{''}(u_0) = 2 + 3 \sqrt{a} + O(a)$ and
$h^{''}(u_1) = 2 - 3 \sqrt{a} + O(a)$.
The main point of the application of the saddle
point method here is that $h^{'}(u_0)=0$ and
$h^{''}(u_0) > 0$, hence $nh(u_0 \exp{(i\theta)})$
is approximately $nh(u_0) - n {u_0}^2 h^{''}(u_0) \frac{\theta^2}{2}$
in the vicinity of $\theta = 0$.
If we integrate (\ref{COL_TROIS}) around a circle
passing vertically through $u=u_0$, we obtain:
\begin{equation}
t_n(a\,n+\beta) = \frac{n!}{2 \pi i}
\int_{-\pi}^{\pi} %
(1-u_0 e^{i \theta})^{1-\beta} \exp( n h(u_0 e^{i\theta}) ) d \theta \,
\label{COL_QUATRE}
\end{equation}
where
\begin{equation}
h(u_0 e^{i\theta}) = u_0 \cos \theta + i u_0 \sin \theta %
- \ln u_0 - i \theta - a \ln (1-u_0 e^{i \theta}) \, \, .
\end{equation}
Denote by $\EuFrak{Re}(z)$ the real part of $z$, we have
\begin{eqnarray}
f(\theta) & = & \EuFrak{Re}( h(u_0 e^{i \theta})) \cr
& = & u_0 \cos \theta - \ln u_0 - %
a \ln ( |1-u_0e^{i\theta}|) \cr
& = & u_0 \cos \theta - \ln u_0 -
a \ln u_0 - \frac{a}{2} %
\ln \big( 1+ \frac{1}{u_0^2} - \frac{2}{u_0} \cos \theta \big) \, .
\end{eqnarray}
It comes
\begin{equation}
f^{'}(\theta) = %
\frac{d}{d \theta} \EuFrak{Re}(h(u_0 e^{i\theta})) = %
- u_0 \sin \theta - %
\frac{\frac{a}{2}\big( \frac{2}{u_0} \sin \theta \big)^2}%
{\big(1 + \frac{1}{u_0^2} - \frac{2}{u_0} \cos \theta \big)}
\end{equation}
and $f^{'}(\theta) = 0$ if $\theta = 0$. Also, $f(\theta)$ is
a symmetric function of $\theta$ and in
$\left[ -\pi, -\theta_0 \right] \cup \left[ \theta_0, \pi \right]$, for
a given $\theta_0$, $0 < \theta_0 < \pi$, it takes it maximum value for
$\theta = \theta_0$.
Since $|\exp( h(u))| = \exp( \EuFrak{Re}(h(u)) )$,
when splitting
the integral in (\ref{COL_QUATRE}) into three parts, viz.
``$\int_{-\pi}^{-\theta_0} + \int_{-\theta_0}^{\theta_0}
+ \int_{\theta_0}^{\pi}$'', we know that it suffices to integrate
from $-\theta_0$ to $\theta_0$, for a convenient value of
$\theta_0$, because
the others can be bounded by the magnitude of the integrand
at $\theta_0$. In fact, we have
\begin{eqnarray}
h(u_0 e^{i\theta}) & & = h(u_0)+ \frac{{u_0}^2 %
(e^{i\theta}-1)^2}{2!}h^{''}(u_0) %
+\frac{{u_0}^3 (e^{i\theta}-1)^3}{3!}h^{(3)}(u_0) \cr
& & +\frac{{u_0}^4 (e^{i\theta}-1)^4}{4!}h^{(4)}(u_0) %
+ \sum_{p \geq 5} %
\frac{{u_0}^p (e^{i\theta}-1)^p}{p!}h^{(p)}(u_0) %
\cr
& & = h(u_0) + \sum_{p\geq 2} \alpha_p (e^{i\theta} -1)^p %
\, ,
\label{ETOILE4}
\end{eqnarray}
where $\alpha_p = \frac{{u_0}^p}{p!} h^{(p)}(u_0)$. We compute
$ h^{(p)}(u_0) = (-1)^p (p-1)! \Big( \frac{1}{{u_0}^{p}} -
\frac{a}{{(1-u_0)}^{p}}\Big)$, for $p \geq 2$. Then, on first hand we obtain
\begin{eqnarray}
\alpha_p & & = \frac{ (-1)^p}{p} \Big(1 - \frac{a {u_0}^p}%
{(1-u_0)^p} \Big) \cr
& & = \frac{ (-1)^p}{p} + %
\frac{ (-1)^{p+1}}{p} \, %
\frac{a (1+\frac{a}{2} - \sqrt{a(1+\frac{a}{4})})^p}{{a}^{\frac{p}{2}} %
(\sqrt{1+\frac{a}{4}} - \frac{\sqrt{a}}{2} )^p } \cr
& & = \frac{(-1)^p}{p} +
\frac{(-1)^{p+1}}{p} \, \frac{2^p}{{a}^{\frac{p}{2}-1}} \, %
\frac{(1+\frac{a}{2} %
- \sqrt{a(1+\frac{a}{4})})^p}{(\sqrt{1+\frac{a}{4}} - \frac{\sqrt{a}}{2} )^p } \, .
\label{ALPHA_P}
\end{eqnarray}
Hence,
\begin{equation}
| \alpha_p | \leq O\Big( \frac{2^p}{a^{\frac{p}{2}-1}}\Big) \,, %
\, \, \, (a \rightarrow 0) \,.
\end{equation}
On the other hand,
\begin{equation}
| e^{i\theta} - 1 | = \sqrt{ 2(1- \cos \theta)} < \theta %
\,, \,\, (\theta > 0)\, .
\end{equation}
Thus, the summation in (\ref{ETOILE4}) can be bounded for values of
$\theta$ and $a$ such that $\theta \rightarrow 0$,
$a \rightarrow 0$ but $\frac{\theta}{\sqrt{a}} \rightarrow 0$ and we have
\begin{eqnarray}
| \sum_{p \geq 4} \alpha_p (e^{i\theta} - 1)^p | %
& & \leq \sum_{p \geq 4} | \alpha_p \theta^p | \cr
& & \leq \sum_{p \geq 4} O\Big( \frac{2^p \theta^p}{a^{\frac{p}{2}-1}}\Big) %
= O\Big( \frac{\theta^4}{a} \Big) \, .
\end{eqnarray}
It follows that for $\theta \rightarrow 0$,
$a \rightarrow 0$ and $\frac{\theta}{\sqrt{a}} \rightarrow 0$
\begin{eqnarray}
h(u_0e^{i\theta}) & & = h(u_0) %
- \frac{1}{2} \, \frac{u_0}{(1-u_0)^2} (1+a -2u_0+{u_0}^2) \theta^2 \cr
& & + i \frac{u_0}{6(1-u_0)^3} \, (1+a+(a-3)u_0+3{u_0}^2 - {u_0}^3) \theta^3 %
+ O\Big(\frac{\theta^4}{a}\Big)\, ,
\end{eqnarray}
where the term in the big-oh takes into account the terms from
$(e^{i\theta}-1)^2$ and $(e^{i\theta}-1)^3$ of (\ref{ETOILE4})
which we can neglect since $(e^{i\theta} -1)^2 = - \theta^2 - i \theta^3
+ O(\theta^4)$ and $(e^{i\theta}-1)^3 = -i\theta^3 + \frac{3}{2}\theta^4 +
iO(\theta^5)$.
Therefore, if $a \rightarrow 0$ but
$\frac{an}{{(\ln{n})}^2} \rightarrow \infty$, if we let
$\theta_0 = \frac{\ln n}{\sqrt{n \rho}}$ with
$\rho = \frac{u_0(1+a-2u_0+{u_0}^2)}{(1-u_0)^2} = 2 - \sqrt{a} + O(a)$,
we can remark (as already said) that it suffices to
integrate (\ref{COL_QUATRE}) from $- \theta_0$ to
$\theta_0$, using the magnitude of the integrand at $\theta_0$
to bound the resulting error. Hence,
\begin{eqnarray}
& & |(1-u_0 e^{i\theta_0})^{(1-\beta)} %
\left(\exp{( n h(u_0 e^{i \theta_0} ))} - n u_0 + n \ln u_0 %
+ a \ln(1-u_0) \right) | = \cr
& & |1-u_0 e^{i\theta_0} |^{(1-\beta)} %
\exp \Big( - \frac{n}{2} \rho \, {\theta_0}^2 +
+ O\big( n \frac{{\theta_0}^4}{a} \big) \Big) =
O\Big( e^{-\frac{(\ln{n})^2}{2} } \Big) \, .
\end{eqnarray}
To estimate $t_n(a\,n+\beta)$, it proves convenient to compute
\begin{equation}
J_n = \int_{-\theta_0}^{\theta_0} (1-u_0 e^{i \theta})^{(1-\beta)} %
\exp{(nh(u_0 e^{i\theta}))} d\theta \, .
\end{equation}
If we make the substitution $\theta = \frac{t}{\sqrt{n \rho}}$,
we have (recall that $\theta_0 = \frac{\ln n}{\sqrt{n \rho}} $)
\begin{equation}
J_n = \frac{1}{\sqrt{n \rho}} \int_{ - \ln{n}}^{\ln{n}} %
\Big( 1 - u_0 e^{\frac{it}{\sqrt{n\rho}}}\Big)^{(1-\beta)} %
\exp\Big(nh(u_0 e^{\frac{it}{\sqrt{n\rho}}}) \Big) dt \, .
\end{equation}
Since $(1-u_0 e^{\frac{it}{\sqrt{n\rho}}})^{(1-\beta)} =
(1-u_0)^{(1-\beta)}(1+O(t/\sqrt{na})) $, $J_n$ becomes
\[
J_n = \frac{1}{\sqrt{n \rho}} \lambda_n
\]
where $\lambda_n = \int_{-\ln{n} }^{ \ln{n} }
(1-u_0)^{(1-\beta)} \exp\Big(nh(u_0) -\frac{t^2}{2} +
i f_3 \frac{t^3}{\sqrt{na}} +
O\big(\frac{t^4}{na}\big) \Big) \Big(1+O\big(\frac{t}{\sqrt{na}}\big) \Big) \,
dt$ and
$f_3 = - \frac{\sqrt{a}(1+a+(a-3)u_0+3u_0^2-u_0^3)}%
{\sqrt{u_0}(1+a-2u_0+u_0^2)^{\frac{3}{2}}}= %
- \frac{\sqrt{2}}{12} -\frac{5}{48}\sqrt{a} + O(a)$.
We obtain
\begin{eqnarray}
J_n & = & \frac{(1-u_0)^{(1-\beta)}}{\sqrt{n\rho}} e^{(nh(u_0))} \, \, \times \cr
& & \left[ \, \int_{-\ln{n}}^{\ln{n} } e^{- \frac{t^2}{2}}%
\cos{\big( f_3\frac{t^3}{\sqrt{na}}\big)}%
\Big( 1+ O\big(\frac{t}{\sqrt{na}} \big) + %
O\big( \frac{t^4}{na}\big)\Big)\, dt \right] \cr
&=& \frac{(1-u_0)^{(1-\beta)}}{\sqrt{n\rho}} e^{(nh(u_0))} \, \, \times \cr
& & \left[ \, \int_{-\infty}^{\infty} e^{- \frac{t^2}{2}}%
\Big( 1+ O\big(\frac{t}{\sqrt{na}} \big) + %
O\big( \frac{t^6}{na} \big) \Big)\, dt %
\, \, + \, \, O\big( e^{- \frac{ (\ln{n})^2}{2} } \big) \right]\cr
&=& \frac{\sqrt{2\pi} (1-u_0)^{(1-\beta)} e^{(nh(u_0))} } {\sqrt{n\rho}} %
\Big( 1+ O\big(\frac{1}{\sqrt{na}}\big)\Big) \cr
&=& \sqrt{\frac{\pi}{n}}(1-u_0)^{(1-\beta)} e^{(nh(u_0))} %
\Big( 1+ O\big(\sqrt{a}\big) + O\big( \frac{1}{\sqrt{na}} \big) \Big) \, .
\label{COL_THE_END}
\end{eqnarray}
We used $\cos{(x)} = 1+O(x^2)$ and $\exp{(O(x))} = 1+O(x)$ when $x=O(1)$.
Since $t_n(a\,n+\beta) = \frac{n!}{2 \pi} J_n$, the proof of
lemma \ref{TREE_POLYNOMIAL_INFINITY} is now complete.
\hfill \qed
\subsection{Wright's inequalities}
\label{INEGALITES_WRIGHT}
In order to adapt the techniques of Wright to
our $\xi$-free components, we need to
bound the \textit{perturbative terms}, i.e., the EGFs containing
the first apparitions of the forbidden configurations
$\gr{S}_{k,\xi}$ and $\gr{J}_{k,\xi}$.
\subsubsection{Upper bounds of $\gr{S}_{k,\xi}$ and $\gr{J}_{k,\xi}$}
To take control on these EGFs, let us recall briefly
the \textit{shrinking-and-expanding} Bagaev's method \cite{BV98}:
In order to enumerate
graphs of a given type, an induced subgraph with special properties should
be chosen and shrunk to a marked vertex. Separately, we
have to calculate:
\begin{itemize}
\item the number of the obtained graphs, rooted at a fixed vertex of degree $d$,
\item the number of the shrunk subgraphs,
\item the number of ways to reconstruct the initial graphs.
\end{itemize}
We note that this technique generalizes the
methods of lemmas \ref{LEMMA7} and \ref{LEMMA8}.
\begin{figure}
\begin{center} \epsfig{file= bagaev.eps,scale=0.50}
\end{center}
\caption[Illustration of Bagaev's method.]
{Illustration of Bagaev's method.}
\label{FIG:BAGAEV}
\end{figure}
As an illustration of this method, consider the graph
depicted by figure \ref{FIG:BAGAEV} where $H$ is represented
by the juxtaposition of triangles. The number of ways to label
this graph can be computed easily using Bagaev's techniques. In fact,
we have
\[
{7 \choose 3} \times \underbrace{2 \times 1}_{reconstruction} %
\times 3 \times 6 = \frac{7!}{4} \,
\]
manners to label the graph of figure \ref{FIG:BAGAEV} (3 manners to label the
path with $3$ vertices and $6$ manners to label the juxtaposition of triangles).
This method is very useful to bound graph typified by the one in
figure \ref{FIG:BAGAEV} (where our interest is focused on the juxtaposition
of triangles). The difficulties arise mainly from the number of
possible reconstructions. In the current example, we have to rely
the vertices $1$ and $3$ to $2$ vertices belonging to
$\{a, \, b, \, c, \, d\}$. Thus, the number of reconstructions is at most $4^2$
(including graphs different from the one in figure \ref{FIG:BAGAEV}).
Consider now $\gr{S}_{k,\xi}$ with the special case
$\xi = \{C_3\}$.
\begin{lem}
\label{LEMMA_BOUND_S3}
For all $k \geq 1$ and $\forall \varepsilon > 0$
\begin{equation}
\gr{S}_{k+1,C_3} %
\preceq \Big(\frac{3}{2}+ \varepsilon\Big) \frac{kb_k}{{\mathbb{X}}^{3k+2}} \, .
\label{EQ:LEMMA_BOUND_S3}
\end{equation}
\end{lem}
\noindent \textbf{Proof.}
The bound of (\ref{EQ:LEMMA_BOUND_S3}) is inspired by the forms of
the EGF $\gr{S}_{k+1,C_3}$. We will prove (\ref{EQ:LEMMA_BOUND_S3})
by induction. We can verify that $\gr{S}_{2,C_3} \preceq \frac{5}{12\mathbb{X}^5}$,
using (\ref{eqn:SoloC3-2}). Suppose that
$\gr{S}_{i,C_3} \preceq \frac{2(i-1)b_{i-1}}{\mathbb{X}^{3i-1}}$, for
$i \in \left[2,\, k-1\right]$ and let us prove that
$\gr{S}_{k,C_3} \preceq \frac{2(k-1)b_{k-1}}{{\mathbb{X}}^{3k-1}}$.
Split the set
of $(k+1)$-cyclic graphs with exactly one occurrence of triangle
into three subsets as follows~:
\begin{itemize}
\item[1-] the first subset $\Sigma_1$
contains all graphs whose situations
after smoothing are characterized by the fact that
exactly one vertex of the triangle is of degree $ \geq 3$,
\item[2-] similarly, the second subset $\Sigma_2$
is built with all graphs whose situations
after smoothing are characterized by the fact that
exactly two vertices of the triangle are of degree $ \geq 3$,
\item[3-] $\Sigma_3$ contains all other graphs of $\mathcal{\gr{S}}_{k,C_3}$
not in $\Sigma_1 \cup \Sigma_2$.
\end{itemize}
We can bound the number of the graphs of the subsets $\Sigma_1$ and
$\Sigma_2$, using lemmas \ref{LEMMA7}, \ref{LEMMA8},
$\sth{\gr{W}_{k-1,C_3}} \preceq \frac{b_{k-1}}{ {(1-z)}^{3k-3} }$
(since Wright showed $\gr{W}_{k-1} \preceq \frac{b_{k-1}}{{\mathbb{X}}^{3k-3}}$
\cite{Wr80})
and the fact that
$\vartheta_z (\frac{1}{ {\mathbb{X}}^t}) \preceq \frac{t}{{\mathbb{X}}^{t+2}}$ for $t\geq 0$. In fact,
\begin{eqnarray}
\Sigma_1(z) + \Sigma_2(z) & & = {\left[ \frac{1}{z(1-z)} \left(\vartheta_z %
\frac{b_{k-1}}{(1-z)^{3k-3}} \right) \, \left( \vartheta_z %
\frac{z^3}{3!}\right) \right]}_{| z=T} + %
\cr
& & %
{\left[ \frac{2}{wz^{2}} \, %
\left( \vartheta_w \frac{w^{k-1} b_{k-1}}{(1-wz)^{3k-3}}\right)%
\, %
\left( \vartheta_w %
\frac{w^3 z^3}{3!}\right)\right]}_{| wz=T} \cr
& & \, \, \, %
\preceq \frac{3}{2} \, \frac{(k-1)b_{k-1}}{ {\mathbb{X}}^{3k-3}} \big( {\mathbb{X}}^{-2} -
{\mathbb{X}}^{-1} + \frac{5}{3} - \mathbb{X} \big) \cr
& & \preceq \frac{3}{2} \, \frac{(k-1)b_{k-1}}{ {\mathbb{X}}^{3k-1}} \, .
\label{EQ:SIGMA1SIGMA2}
\end{eqnarray}
For graphs of $\Sigma_3$, we have two subcases. Denote by
$\Sigma_{3}^{'}$, resp. $\Sigma_{3}^{''}$, the graphs
of $\Sigma_3$ such that the deletion
of the $3$ vertices and the $3$ edges of the triangle will
leave a connected graph, resp. disconnected graphs.
The figures \ref{FIG:solo2C3} (c) and
\ref{FIG:solo2C3} (e) illustrate these 2 classifications.
In the first case, i.e. $\Sigma_{3}^{'}$, we will not use
the induction hypothesis. In fact, to build a graph of
$\Sigma_{3}^{'}$, we have to rely $d$ vertices ($d \geq 3$) of
a graph of $\mathcal{\gr{W}}_{k-d,C_3}$ to the triangle.
Thus, the number of manners to construct a graph of $\Sigma_{3}^{'}$
of order $n$ this way is at most
\begin{eqnarray}
& & 3^d {n \choose 3} {{n-3} \choose d} (n-3)! %
\coeff{z^{n-3}}\gr{W}_{k-d,C_3}(z) \cr
& & \leq \frac{3^d}{6} {{n-3} \choose d} n! %
\coeff{z^{n-3}}\gr{W}_{k-d,C_3}(z) \cr
& & \leq \frac{3^d}{6 \, d!} n^d \, n! %
\coeff{z^{n}}\gr{W}_{k-d,C_3}(z) \cr
& & \leq \frac{3^d}{6 \,d!} n! %
\coeff{z^{n}} \vartheta_z^d \, \gr{W}_{k-d,C_3}(z) \, , (3 \leq d \leq k+1) \, .
\end{eqnarray}
In terms of generating function, we then have (summing over $d$)
\begin{eqnarray}
\Sigma_{3}^{'}(z) & & \preceq \sum_{d \geq 3} \frac{3^d}{6 \, d!} %
\vartheta_z^d \, \gr{W}_{k-d,C_3}(z) \, .
\label{SUM_OVER_D}
\end{eqnarray}
First, let us treat the cases $d=k+1$ and $d=k$.
We have
\[
\vartheta_z^{(k+1)} \, \gr{W}_{-1} = \vartheta_z^{k} \, T = \vartheta_z^{k-1}\, \frac{T}{\mathbb{X}} \,
\]
and
\[
\vartheta_z^{k} \, \gr{W}_{0,C_3} = \vartheta_z^{k-1}\, \Big( \frac{T^4}{2 \mathbb{X}^2 }\Big) \, .
\]
Since $\frac{T}{X} \preceq \frac{1}{\mathbb{X}^2}$, we have
\[
\frac{3^k}{6 k!} \Big(1+\frac{3}{k+1} \Big)%
\Big( \vartheta_z^{(k+1)} \, \gr{W}_{-1} + \vartheta_z^{k} \, \gr{W}_{0,C_3}\Big) \preceq %
\frac{3^k}{k!} \vartheta_z^{(k-1)} \Big( \frac{1}{\mathbb{X}^2}\Big) \, .
\]
Similarly
\[
\vartheta_z^{(k-1)} \frac{1}{\mathbb{X}^2} \preceq \vartheta_z^{(k-2)} \frac{2}{\mathbb{X}^4} %
\cdots %
\preceq \frac{2 \times 4 \times \cdots \times 2(k-1)}{\mathbb{X}^{2k}} %
= \frac{2^k(k-1)}{\mathbb{X}^{2k}} \,
\]
and we obtain for $d=k+1$ and $d=k$ in (\ref{SUM_OVER_D})
\begin{equation}
\frac{3^{k+1}}{6(k+1)!} \vartheta_z^{(k+1)} \, \gr{W}_{-1} +
\frac{3^{k}}{6k!} \vartheta_z^{k} \, \gr{W}_{0,C_3} \preceq %
\frac{6^k}{6 k!} \frac{(k-1)!}{\mathbb{X}^{2k}} \, .
\end{equation}
Next, we have
\[
\frac{b_{k+1}}{b_k} \geq \frac{3}{2}k
\]
since $b_k = (\frac{3}{2})^k (k-1)! d_k$ and $(d_k)$ is an
increasing sequence (cf. \cite[eq. (1.4)]{Wr80}). Thus,
\[
b_k \geq \frac{3}{2}(k-1) b_{k-1} \geq %
{(\frac{3}{2})}^2 (k-1)(k-2) b_{k-2} %
\geq \cdots \geq {(\frac{3}{2})}^{k-1} (k-1)! \, \, b_1
\]
and
\begin{equation}
(k-1)! \leq 6 (k-1) b_{k-1} \, .
\label{FRAC_VS_B}
\end{equation}
Finally,
\begin{equation}
\frac{3^{k+1}}{6(k+1)!} \vartheta_z^{(k+1)} \, \gr{W}_{-1} +
\frac{3^{k}}{6k!} \vartheta_z^{k} \, \gr{W}_{0,C_3} \preceq %
\frac{6^k}{6 k!} \frac{(k-1)b_{k-1}}{\mathbb{X}^{2k}} \, .
\label{ETOILE-ETOILE}
\end{equation}
Summing (\ref{SUM_OVER_D}) over $d$ for $d \in \left[3, \, k-2\right]$,
we obtain
\begin{eqnarray}
& & \sum_{d=3}^{k-2} \frac{3^d}{6 \, d!} %
\vartheta_z^d \, \gr{W}_{k-d,C_3}(z)
\preceq \sum_{d=3}^{k-2} \frac{3^d}{6 \, d!} %
\vartheta_z^d \, \frac{b_{k-d}}{\mathbb{X}^{3k-3d}} \cr
& & \preceq \sum_{d=3}^{k-2} \frac{3^d}{6 \, d!} %
\frac{(3k -3d)(3k-3d+2) \cdots (3k-3d+2(d-1)) b_{k-d}}{\mathbb{X}^{3k-d}} \cr
& & \preceq \sum_{d=3}^{k-2} \frac{3^d}{6 \, d!} %
\frac{3^d (k-d)(k-d+\frac{2}{3})(k-d+\frac{4}{3}) \cdots %
(k - \frac{1}{3}d -\frac{2}{3}) b_{k-d}}{\mathbb{X}^{3k-d}} \cr
& & \preceq \sum_{d=3}^{k-2} \frac{3^d}{6 \, d!} \, \, %
3^d \frac{ (k-d)(k-d+1)(k-d+2) \cdots (k-1) b_{k-d}}{\mathbb{X}^{3k-3}}
\label{EQ:131}
\end{eqnarray}
So using (\ref{FRAC_VS_B}) and (\ref{ETOILE-ETOILE}), we get
after a bit of algebra
\begin{equation}
\Sigma_{3}^{'} \preceq %
\sum_{d=3}^{k+1} \frac{6^{d-1}}{d!} \, \, \frac{(k-1)b_{k-1}}{\mathbb{X}^{3k-3}} %
\preceq 379 \, \frac{(k-1)b_{k-1}}{\mathbb{X}^{3k-3}} \, .
\end{equation}
We can apply the same techniques as above for graphs of $\Sigma_{3}^{''}$.
However, we need here the help of the induction hypothesis where we will choose
$\varepsilon=\frac{1}{2}$ for sake of simplicity.
\begin{figure}
\begin{center} \epsfig{file= sigma31.eps,scale=0.50}
\end{center}
\caption[A representative graph of $\Sigma_{3}^{''}$ and its %
reconstruction.]
{A representative graph of $\Sigma_{3}^{''}$ and its %
reconstruction.}
\label{FIG:SIGMA31}
\end{figure}
In fact, a graph from $\Sigma_{3}^{''}$ can be seen
as the composition of two graphs: the first from $\mathcal{\gr{S}}_{e_1,C_3}$
and the second from $\mathcal{\gr{W}}_{e_2,C_3}$ (e.g. the graph
in the dashed box of figure \ref{FIG:SIGMA31}).
Furthermore, suppose that the first graph is of order
$p$, the second $n-p$ and that we have to rely $d$ vertices of the
second to the triangle (e.g. in the figure \ref{FIG:SIGMA31},
$d=3$, $p=5$ and $n=8$).
The number of manners to label such composition is less than or equal to
\begin{eqnarray}
& & 3^d {n \choose p}{ {n-p} \choose d} %
p! \coeff{z^p} \gr{S}_{e_1,C_3} \, %
(n-p)! \coeff{z^{n-p}} \gr{W}_{e_2,C_3} \cr
& & \leq 3^d { {n-p} \choose d} n! \coeff{z^n} %
\gr{S}_{e_1,C_3} \, \times \, \gr{W}_{e_2,C_3} \cr
& & \leq \frac{3^d}{d!} \, n! \coeff{z^n} \vartheta_z^d \, %
(\gr{S}_{e_1,C_3} \, \times \, \gr{W}_{e_2,C_3}) \, .
\end{eqnarray}
We have $d+e_1+e_2=k$ and using the induction hypothesis on
$\gr{S}_{e_1,C_3}$ with the fact that
$\gr{W}_{e_2,C_3} \preceq \gr{W}_{e_2} \preceq \frac{b_{e_2}}{\mathbb{X}^{3e_2}}$,
we obtain
\begin{eqnarray}
\Sigma_{3}^{''} %
& & \preceq \sum_{d+e_1+e_2=k} %
\frac{3^d}{d!} \vartheta_z^d (\gr{S}_{e_1,C_3} \, \times \, \gr{W}_{e_2,C_3}) \cr
& & \preceq \sum_{d+e_1+e_2=k} %
\frac{3^d}{d!} \vartheta_z^d %
\frac{2(e_1-1)b_{e_1-1}b_{e_2}}{\mathbb{X}^{3e_1+3e_2-1}} \cr
& & \preceq 2 \, \sum_{d+e_1+e_2=k} %
\frac{3^d}{d!} (3k-3d-1)\cdots (3k-d-3)%
\frac{(e_1-1)b_{e_1-1}b_{e_2}}{\mathbb{X}^{3k-d-1}} \cr
& & \preceq 2\, \sum_{d+e_1+e_2=k} %
\frac{3^d}{d!} 3^d (k-d-\frac{1}{3})%
\cdots (k-\frac{d}{3}-1)%
\frac{(e_1-1)b_{e_1-1}b_{e_2}}{\mathbb{X}^{3k-d-1}} \cr
& & \preceq 2\, \sum_{d+e_1+e_2=k} %
\frac{3^d}{d!} 3^d (k-d)(k-d+1) %
\cdots (k-1) \frac{b_{k-d}}{\mathbb{X}^{3k-d-1}} \, ,
\end{eqnarray}
because we have
\begin{equation}
(e_1-1)b_{e_1-1}b_{e_2} \leq b_{k-d}
\end{equation}
since
\begin{eqnarray}
(e_1-1)b_{e_1-1} \, b_{e_2} & & = %
\Big( \frac{3}{2}\Big)^{e_1+e_2-1}%
(e_1-1)!\, (e_2-1)! \, d_{e_1-1} \, d_{e_2} \cr
& & \leq %
\Big( \frac{3}{2}\Big)^{k-d-1} %
(e_1-1)! \, e_2! \, d_{e_1-1} \, d_{e_2} \cr
& & \leq %
\Big( \frac{3}{2}\Big)^{k-d-1} %
(e_1+e_2)! \, d_{e_1+e_2-1} \cr
& & \leq \Big( \frac{3}{2}\Big)^{k-d} %
(e_1+e_2)! \, d_{k-d} \cr
& & = b_{k-d} \, .
\end{eqnarray}
(We used $ (k+1)!d_{k+1} = (k+1)!d_k + %
\sum_{h=1}^{k-1} h! (k-h)! d_h d_{k-h}$ \cite[eq. (1.4)]{Wr80}.)
Hence,
\begin{equation}
\Sigma_{3}^{''} \preceq 2 \sum_{d \geq 1} \frac{6^d}{d!} \, %
\frac{kb_k}{\mathbb{X}^{3k-2}} %
\preceq 805 \frac{(k-1)b_{k-1}}{\mathbb{X}^{3k-2}}\, .
\end{equation}
We have $ \coeff{z^n} \frac{1}{\mathbb{X}^{3k-3}} %
\Big(\frac{\varepsilon}{{\mathbb{X}}^2}-\frac{805}{\mathbb{X}} - 379 \Big) \geq 0$,
$\forall n \geq 1$ since $\forall n > 0, \, %
\coeff{z^n}(\varepsilon-805 \mathbb{X} - 379 \mathbb{X}^2) \geq 0$
and
$\coeff{z^n}\gr{S}_{k,C_3} = 0$ for $0 \leq n \leq 2$. (In fact,
$\forall a, \, b, \,c > 0$, we have
$0 \preceq a-b\mathbb{X}-c\mathbb{X}^2 = (a-b-c)+bT+2c(T-\frac{T^2}{2}) \, .)$
Finally, we obtain
$\gr{S}_{k,C_3} \preceq %
\Big(\frac{3}{2}+ \varepsilon\Big) \frac{(k-1)b_{k-1}}{{\mathbb{X}}^{3k-1}}$. \hfill \qed
\noindent
By similar methods, one can prove
\begin{lem}
\label{LEMMA_BOUND_J3}
For $\varepsilon > 0$ and $k\geq 2$,
\begin{equation}
\gr{J}_{k+1,C_3} \preceq \Big(6+\varepsilon\Big) %
\frac{(k-1)b_{k-1}}{\mathbb{X}^{3k-1}} \, .
\label{EQ:LEMMA_BOUND_J3}
\end{equation}
\end{lem}
Before proving lemma \ref{LEMMA_BOUND_J3}, we notice that
working with juxtaposition of $t$ triangles as subgraph is much
easier.
\begin{defn}
Denote by
${\gr{J}}_{k,C_3}^{(t)}$ the EGF that counts $k$-excess graphs with
a juxtaposition of \textit{exactly} $t$ triangles sharing a common edge.
\end{defn}
(For instance, the graph of figure \ref{FIG:BAGAEV} belongs to
the family ${\gr{\mathcal{J}}_{2,C_3}}^{(2)}$.)
\begin{lem}
\label{WITH_T_TRIANGLES}
$\forall \varepsilon > 0$, $k>t>1$, $k \geq 3$, we have
\begin{equation}
{\gr{J}_{k,C_3}}^{(t)} \preceq (3+\varepsilon) %
\, \, \frac{(t+2)}{2!t!} \, \, \, %
\frac{(k-t)b_{k-t}}{\mathbb{X}^{3k-3t+2}} \, .
\label{EQ:WITH_T_TRIANGLES}
\end{equation}
\end{lem}
\noindent \textbf{Proof (sketch).} Smooth members of
${\gr{\mathcal{J}}_{t-1,C_3}}^{(t)}$ are counted by
\begin{equation}
\sth{\gr{\mathcal{J}}_{t-1,C_3}}^{(t)}(w,z) = %
\frac{w^{2t+1} z^{t+2}}{2! t!} \, .
\end{equation}
Thus, the reader can remark that the bound in
(\ref{EQ:WITH_T_TRIANGLES}) is suggested by serial concatenation of
graphs of ${\gr{\mathcal{J}}_{t-1,C_3}}^{(t)}$ and of
${\gr{\mathcal{W}}}_{k-t,C_3}$. At this stage, (\ref{EQ:WITH_T_TRIANGLES})
can be proved as it was be done for the bound of $\gr{S}_{k,C_3}$ in
lemma (\ref{LEMMA_BOUND_S3}). The main change is that the ``unique occurrence
of triangle'' has been replaced by a ``unique occurrence of
juxtaposition of $t$ triangles'' with EGF $\frac{w^{2t+1} z^{t+2}}{2! t!}$.
\hfill \qed
\noindent \textbf{Proof of lemma \ref{LEMMA_BOUND_J3}.}
It suffices to sum over all possible values of $t$. We have
\begin{eqnarray}
\gr{J}_{k,C_3} & \preceq & \frac{(3+\frac{\varepsilon}{2})}{2 \mathbb{X}^{3k-4}} %
\sum_{t=2}^{k} \frac{ (t+2)(k-t)b_{k-t}}{t!} \quad %
(\mbox{we use } \frac{1}{\mathbb{X}^{3k-3t+2}} \preceq \frac{1}{\mathbb{X}^{3k-4}} ) \, , %
\cr
& \preceq & \frac{(3+\frac{\varepsilon}{2}) (k-2)b_{k-2}}{\mathbb{X}^{3k-4}} %
\sum_{t=2}^{k} \frac{t}{t!} \preceq %
\frac{(6+\varepsilon)(k-2)b_{k-2}}{\mathbb{X}^{3k-4}} \, .
\end{eqnarray}
\hfill \qed
Lemmas \ref{LEMMA_BOUND_S3} and \ref{LEMMA_BOUND_J3} suggest
themselves for generalization for any finite set $\xi$.
Although, we do not intend to present such generalization here,
we are convinced that this can be done practically in the same ways
as we did for (\ref{EQ:LEMMA_BOUND_S3}) and (\ref{EQ:LEMMA_BOUND_J3}).
\subsubsection{Bounds of $\gr{W}_{k,\xi}$}
In this paragraph, we present results that are strongly related
to those of Wright. In fact, the Wright's seminal paper
contains general techniques that are well suited for our triangle-free
graphs. In paragraph \S \ref{SUBSUB:GENERAL-FORM},
we obtained the general forms of the EGFs $\gr{W}_{k,\xi}$
(see theorem \ref{THM_GENERAL_FORM_MANY}).
Recall that $(b_k)$ and $(c_k)$ are given respectively by
(\ref{EQ:B_K}) and (\ref{EQ:C_K}). The lemmas
\ref{LEMMA_0} -- \ref{LEMMA_F} stated below
will serve us to show by induction
the inequalities (\ref{COEFF_INEGALITES}). Before,
let us specify some useful notations.
\noindent \textbf{Notations.}
\noindent For all $k \geq 1$, define by ${\EuScript{L}}_k$
and ${\EuScript{R}}_k$ the generating functions given by
(recall that $\mathbb{X} = 1-T$)
\begin{equation}
{\EuScript{L}}_k(z) = \gr{W}_{k,C_3}(z) - \frac{b_k}{{\mathbb{X}}^{3k}} %
+ \frac{\cpt{k}}{{\mathbb{X}}^{3k-1}} \,
\label{GAUCHE}
\end{equation}
and
\begin{equation}
{\EuScript{R}}_k(z) = \frac{b_k}{{\mathbb{X}}^{3k}} - W_{k,C_3}(z) \, .
\label{DROITE}
\end{equation}
\noindent Recall that we just have to prove
that $\mathbf{{\EuScript{L}}_k \succeq 0}$
for all ${k \geq 1}$ since $\mathbf{{\EuScript{R}}_k \succeq 0}$ was proved by
Wright \cite{Wr80}.
First of all, the following lemma gives bounds of $\cpxi{k}$ by means of $b_k$:
\begin{lem}
\label{LEMMA_0}
For all $k \geq 1$, we have $kb_k \leq \cpxi{k} \leq \frac{19+6r}{5} kb_k$,
where $r$ is the number of polygons of $\xi$.
\end{lem}
\noindent \textbf{Proof.}
We let $\cpxi{k} = k b_k (1+\betaxi{k})$.
Hence, $\betaxi{1}= \frac{14+6r}{5}$
(where $r$ is the number of the forbidden polygons of distinct lengths).
After a bit of algebra, we find
\begin{eqnarray}
2(3k+2)(k+1) b_{k+1}& & (1+ \betaxi{k+1}) = 8(k+1)b_{k+1} +3k(r+1)b_k \cr
& & + (3k-1)(3k+2)kb_k (1+ \betaxi{k}) \cr
& & + 6 \sum_{t=1 }^{k-1 } t(k-t)(3k-3t-1) b_t b_{k-t} (1 + \betaxi{k-t} )\, .
\end{eqnarray}
Let $\mathcal{B}_k$ and $\Ckxi{k}$ be
the rational numbers defined with the help of $(b_k)$ and
$(\cpxi{k})$ by
\begin{equation}
\mathcal{B}_k = \sum_{t=1}^{k-1} t (k-t) b_t b_{k-t} \, .
\label{BB_K}
\end{equation}
\begin{equation}
\Ckxi{k} = \sum_{t=1}^{k-1} t (3k - 3t -1) b_t \cpxi{k-t} \, .
\label{CC_K}
\end{equation}
Using (\ref{REMARK_ALPHA}), we find
\begin{eqnarray}
6 \sum_{t=1}^{k-1} & & t (k-t) (3k - 3t -1) b_t b_{k-t} %
= 3(3k-2) \mathcal{B}_k \cr
& & = 2(k+1)(3k-2) b_{k+1} - 3k(k+1)(3k-2) b_k \, .
\end{eqnarray}
Thus,
\begin{eqnarray}
& & 2 (3k+2)(k+1)b_{k+1} \betaxi{k+1} = %
(3r+7)kb_k + k(3k+2)(3k-1)b_k \betaxi{k} \cr
& & + 6 \sum_{t=1}^{k-1} %
t (k-t) (3k - 3t -1) b_t b_{k-t} \betaxi{k-t}
\label{eq:**}
\end{eqnarray}
and we have $\betaxi{k} > 0 $ for all $\xi$ and $k > 0$.
We let $\Rhoxi{k} = \mbox{max}_{1 \leq t \leq k} %
\betaxi{t} \geq \frac{14+6r}{5}$.
Then, (\ref{eq:**}), (\ref{REMARK_ALPHA}) and (\ref{EQ:B_K}) give
\begin{eqnarray}
2 (3k+2)(k+1)b_{k+1} \betaxi{k+1} %
& \leq & (3r+7)kb_k + \cr
& & \Big( k(3k+2)(3k-1)b_k \cr
& & + 6 \sum_{t=1}^{k-1} t (k-t) (3k - 3t -1) b_t b_{k-t} \Big) %
\Rhoxi{k} \cr
& \leq & (3r+7)kb_k + \cr
& &\Big( k(3k+2)(3k-1)b_k + 3(3k-2) \mathcal{B}_k\Big) \Rhoxi{k} \cr
& \leq & (3r+7)kb_k \cr
& +& (2(3k-2)(k+1)b_{k+1} + 4kb_k) \Rhoxi{k} \, .
\end{eqnarray}
Now, if we suppose that $\betaxi{k+1} > \Rhoxi{k}$, we will have
\begin{eqnarray}
8(k+1)b_{k+1}\betaxi{k+1} & \leq & 4k b_k \betaxi{k+1} + (3r+7)kb_k \cr
12k(k+1)b_{k}\betaxi{k+1} + 12 \mathcal{B}_k \betaxi{k+1} %
& \leq & 4k b_k \betaxi{k+1} + (3r+7)kb_k \,
\end{eqnarray}
so that
\begin{equation}
12(k+1)b_{k}\betaxi{k+1} \leq 4 b_k \betaxi{k+1} + (3r+7)b_k \,
\end{equation}
and
\begin{equation}
\betaxi{k+1} \leq \frac{3r+7}{4(3k+2)}
\end{equation}
which is in contradiction with the fact that
$\betaxi{k+1} > \Rhoxi{k} \geq \frac{14+6r}{5}$
(this will lead us to $3r+7 > 18kr + 42k$).
So, $(\betaxi{k})$ is a nonincreasing
sequence and $\Rhoxi{k} = \frac{14+6r}{5}$ for all $k > 0$.
\hfill \qed
Next, we have the lemmas
\ref{LEMMA_A}, \ref{LEMMA_B}, \ref{LEMMA_C}, \ref{LEMMA_D}, \ref{LEMMA_E}
stated below, corresponding to the lemmas 6, 7, 8, 9 and 10 of
\cite{Wr80} but adapted for our $\xi$-free graphs. Lemmas 3
and 4 of \cite{Wr80} are contained in lemma \ref{LEMMA_F}.
\begin{lem}
\label{LEMMA_A}
If $\Big( \gr{W}_{t,\xi} - %
\frac{b_t}{{\mathbb{X}}^{3t}} %
+ \frac{ \cpxi{t} }{{\mathbb{X}}^{3t-1}} \Big) %
\succeq 0$, for $t$ such that $1 \leq t \leq k-1$ then
\begin{equation}
\Lambdaxi{k} \succeq \frac{T^2}{{\mathbb{X}}^{3k+4}} %
\, \Big(9 \mathcal{B}_k - 6 \Ckxi{k} \, {\mathbb{X}} \Big) \,
\label{LEMMA6_WRIGHT}
\end{equation}
where $\Ckxi{k}$ is given by (\ref{CC_K}) and $\Lambdaxi{k}$
is given by (\ref{EQ:LAMBDAXI_K}).
\end{lem}
\noindent \textbf{Proof.} If $x_1, \, \cdots, \, x_6$ are
positive real numbers and $x_1 \geq x_2 -x_3$, $x_4 \geq x_5 - x_6$ then
\begin{equation}
x_1 x_4 \geq x_2 x_5 - x_2 x_6 - x_5 x_3 \, .
\label{42}
\end{equation}
In fact, if $x_2 < x_3$ and/or $x_5 < x_6$, the right side of
the above inequality is negative. Otherwise,
if $x_2 \geq x_3$ and $x_5 \geq x_6$, we have:
\[
x_1 x_4 \geq (x_2 - x_3)(x_5 - x_6) \geq x_2 x_5 - x_2 x_6 - x_5 x_3 \, .
\]
Assume now that $1 \leq t \leq k-1$. We have
$\gr{W}_{t,\xi} \succeq 0$, $b_t/ {\mathbb{X}}^{3t} \succeq 0$,
$(c_t + 3/2r (t-1)b_{t-1})/{\mathbb{X}}^{3t-1} \succeq 0$ and
${\EuScript{L}}_t \succeq 0$ for $1 \leq t \leq k-1$.
Consequently, the coefficients of $\vartheta_z \gr{W}_{t,\xi}(z)$ are
positive for the same value of $t$.
Setting
\begin{eqnarray}
x_1 & =& s! \coeff{z^s} \vartheta_z \gr{W}_{t,\xi}(z) \, , \cr
x_2 &=& b_t s! \coeff{z^s} \vartheta_z \frac{1}{{\mathbb{X}}^{3t}} \, , \cr
x_3 &=& \cpxi{t} s! \coeff{z^s} %
\vartheta_z \frac{1}{{\mathbb{X}}^{3t-1}} \, , \cr
x_4 &=& (n-s)! \coeff{z^{n-s}} \vartheta_z \gr{W}_{k-t,\xi}(z) \, , \cr
x_5 &=& b_{k-t} (n-s)! \coeff{z^{n-s}} \vartheta_z \frac{1}{{\mathbb{X}}^{3k-3t}} %
\mbox{ and } \cr
x_6 &=& \cpxi{k-t} %
(n-s)! \coeff{z^{n-s}} \vartheta_z \frac{1}{{\mathbb{X}}^{3k-3t-1}} \,
\end{eqnarray}
where $s \in \coeff{0, \, n}$, after substituting the values of $x_i$,
$i \in \coeff{1, \, 6}$ in (\ref{42}) and summing over $s$
and $t$, $t \in \coeff{1, \, k-1}$, we obtain (\ref{LEMMA6_WRIGHT}).
\hfill \qed
\noindent Similarly, we have
\begin{lem}
\label{LEMMA_B}
If $\Big(\frac{b_t}{{\mathbb{X}}^{3t}} - \gr{W}_{t,\xi} \Big) \succeq 0$
for $1 \leq t \leq k-1$ then
\begin{equation}
\Lambdaxi{k} \preceq 9 \mathcal{B}_k \frac{T^2}{{\mathbb{X}}^{3k+4}} \, .
\label{LEMMA7_WRIGHT}
\end{equation}
\end{lem}
In the following lemmas, we work again with the
special case $\xi=\{C_3\}$ for sake of clarity.
\begin{lem}
\label{LEMMA_C}
Define by $\Yt{k}$ and $\Zt{k}$ the formal power series
\begin{eqnarray}
\Yt{k}(z) & & = \Delta_{k+1} \frac{b_{k+1}}{{\mathbb{X}}^{3k+3}} %
- \Omegat{k} \frac{b_k}{{\mathbb{X}}^{3k}} %
- 9 \mathcal{B}_k \frac{T^2}{{\mathbb{X}}^{3k+4}}
\label{LEMMA8Y_WRIGHT}
\end{eqnarray}
\begin{eqnarray}
\Zt{k}(z) & & = \Delta_{k+1} \frac{\cpt{k+1}}{{\mathbb{X}}^{3k+2}} %
- \Omegat{k} \frac{\cpt{k}}{{\mathbb{X}}^{3k-1}} %
- 6\Ckt{k} \frac{T^2}{{\mathbb{X}}^{3k+3}} \, .
\label{LEMMA8Z_WRIGHT}
\end{eqnarray}
For all $k \geq 1$, we have $\Zt{k} \succeq \Yt{k} + 6 \gr{S}_{k+1,C_3} +
2 \gr{J}_{k+1,C_3} \succeq 0$.
\end{lem}
\noindent \textbf{Proof.}
First, we remark that
\begin{eqnarray}
\Omegat{k}(\mathbb{X}^{-t}) & & = t(t+3) \mathbb{X}^{-t-4} - t(2t+8) \mathbb{X}^{-t-3} \cr
& & + t(t+8) \mathbb{X}^{-t-2} - 7t \mathbb{X}^{-t-1} + (5t-2k) \mathbb{X}^{-t} - t \mathbb{X}^{-t+1} \, .
\end{eqnarray}
Thus, using this (\ref{EQ:DELTA}) and (\ref{BB_K}),
we have
\begin{eqnarray}
\Yt{k}(z) & & = \Big(6kb_k + 8(k+1) b_{k+1}\Big) \mathbb{X}^{-3k-3} \cr
& & - \Big(15kb_k + 6(k+1)b_{k+1}\Big) \mathbb{X}^{-3k-2} \cr
& & + 21kb_k \mathbb{X}^{-3k-1} -13kb_k \mathbb{X}^{-3k} + 3kb_k \mathbb{X}^{-3k+1} \, .
\label{\Yt{k}}
\end{eqnarray}
Similarly, we find
\begin{eqnarray}
\Zt{k}(z) & & = \Big(6kb_k + 8(k+1) b_{k+1}\Big) \mathbb{X}^{-3k-3} \cr
& & +\Big(2(4k+3) \cpt{k+1} + 2(3k-1)\cpt{k}%
- 16(k+1)b_{k+1} - 12kb_k\Big) %
\mathbb{X}^{-3k-2} \cr
& & + \Big(8(k+1) b_{k+1} + 6kb_k %
- 2(3k+2)\cpt{k+1} - 5(3k-1) \cpt{k}\Big) %
\mathbb{X}^{-3k-1} \cr
& & + 7(3k-1) \cpt{k} - (13k-5) \cpt{k} \mathbb{X} + (3k-1) \cpt{k} \mathbb{X}^2 \, .
\label{Zt_k}
\end{eqnarray}
Rearranging (\ref{\Yt{k}}), we obtain
\begin{eqnarray}
\Yt{k}(z) & & = 3kb_k \mathbb{X}^{-3k-2} \big( 2\mathbb{X}^{-1} -5 \big) \cr
& & + 2(k+1)b_{k+1} \mathbb{X}^{-3k-3} \big( 4 \mathbb{X}^{-1} -3 \big) \cr
& & + kb_k \mathbb{X}^{-3k} \big( 21\mathbb{X}^{-1} -13 \big) + 3kb_k \mathbb{X}^{-3k+1}
\end{eqnarray}
and so $\Yt{k} \succeq 0$.
By (\ref{EQ:LEMMA_BOUND_S3}) and (\ref{EQ:LEMMA_BOUND_J3}),
we have
$\gr{S}_{k+1,C_3} + \gr{J}_{k+1,C_3} \preceq \frac{2 k b_k}{\mathbb{X}^{3k+2}}$
Hence,
\begin{equation}
\Zt{k} - \Yt{k} - 6 \gr{S}_{k+1,C_3} - 2 \gr{J}_{k+1,C_3}
\succeq \Zt{k} - \Yt{k} - 12 kb_k \mathbb{X}^{-3k-2}
\end{equation}
and
\begin{eqnarray}
& & \Zt{k} - \Yt{k} - 6 \gr{S}_{k+1,C_3} - 2 \gr{J}_{k+1,C_3} \, \, \, \, %
\succeq \cr
& & \Big( 2(4k+3) \cpt{k+1} + 2(3k-1) \cpt{k} %
- 9 kb_k -10(k+1)b_{k+1} \Big) \mathbb{X}^{-3k-2} \cr
& & + \Big( 8(k+1)b_{k+1} -15kb_k %
- 2(3k+2) \cpt{k+1} - 5(3k-1) \cpt{k} \Big) \mathbb{X}^{-3k-1} \cr
& & + \Big(7(3k-1)\cpt{k} + 13kb_k \Big) \mathbb{X}^{-3k} %
-\Big((13k-5) \cpt{k} +3kb_k \Big) \mathbb{X}^{-3k+1} \cr
& & + (3k-1) \cpt{k} \mathbb{X}^{-3k+2} \, .
\end{eqnarray}
Rewriting, we have
\begin{eqnarray}
& & \Zt{k} - \Yt{k} - 6 \gr{S}_{k+1,C_3} - 2 \gr{J}_{k+1,C_3} \, \, \, \, %
\succeq \cr
& & \Big(2(4k+3)\cpt{k+1} + 2(3k-1)\cpt{k} %
- 9kb_k -10(k+1) b_{k+1} \Big) (\mathbb{X}^{-1}-2)^2 \cr
& & + \Big(2(13k+10)\cpt{k+1} + 3(3k-1)\cpt{k} %
- 51kb_k -32(k+1)b_{k+1} \Big) (\mathbb{X}^{-1}-2) \cr
& & + \Big( (44k+28)\cpt{k+1} + 9(3k-1)\cpt{k} %
- 69kb_k -40(k+1)b_{k+1} \Big) \cr
& & + \Big( (13k-5)\cpt{k} + 3kb_k \Big) (T - 1) \cr
& & + (3k-1)\cpt{k} \mathbb{X}^{-2}
\end{eqnarray}
and by lemma \ref{LEMMA0}, (\ref{CPRIME_0}) and
(\ref{EQ:B_K}) after some calculations we find
$\Zt{k} - \Yt{k} - 6 \gr{S}_{k+1,C_3} - 2 \gr{J}_{k+1,C_3} \succeq 0$
\hfill \qed
\begin{lem}
\label{LEMMA_D}
For all $t \in \coeff{1,\, k-1}$, if
\begin{equation}
\left( \gr{W}_{t,C_3} - \frac{b_t}{{\mathbb{X}}^{3t}} %
+ \frac{ \cpt{t} }{{\mathbb{X}}^{3t-1}} \right) \succeq 0
\end{equation}
then
\begin{equation}
\Delta_{k+1} \left[ \gr{W}_{k+1,C_3} - \frac{b_{k+1}}{{\mathbb{X}}^{3k+3}} %
+ \frac{ \cpt{k+1} }{{\mathbb{X}}^{3k+2}} \right] \succeq %
\Omegat{k} %
\left[ \gr{W}_{k,C_3} - \frac{b_{k}}{{\mathbb{X}}^{3k}} %
+ \frac{ \cpt{k} }{{\mathbb{X}}^{3k-1}} \right] \, .
\label{LEMMA9b_WRIGHT}
\end{equation}
\end{lem}
\noindent \textbf{Proof.}
Using lemmas \ref{LEMMA_A}, \ref{LEMMA_C} and
(\ref{AVEC_OP}), we infer that
\begin{eqnarray}
& & \Delta_{k+1}{\EuScript{L}}_{k+1} - \Omegat{k} {\EuScript{L}}_{k} =
- 6 \gr{S}_{k+1,C_3} - 2 \gr{J}_{k+1,C_3} + %
\Lambdat{k} - Y_k + Z_k \cr
& - & {\mathbb{X}}^{-3k-2} T^2/{\mathbb{X}}^2 %
(9\mathcal{B}_k - 6\Ckt{k}/{\mathbb{X}}) %
\succeq (Z_k - Y_k) \succeq 0 \, .
\end{eqnarray}
\hfill \qed
\begin{lem}
\label{LEMMA_E}
Let $n_0 = n_0(k) = \frac{3}{2} + \sqrt{(2k + \frac{9}{4})}$.
If $k \geq 2$ and $ 0 \leq n \leq n_0(k)$ then the coefficients
of $\EuScript{L}_k$.
\end{lem}
\noindent \textbf{Proof.} If $n < n_0(k)$ then ${n \choose 2} < n+k$ and
\textit{a fortiori} there is no $(k+1)$-cyclic connected graphs.
Let $k \geq 2$ and $n < n_0(k)$. Since $n! \coeff{z^n} \gr{W}_{k,\xi}(z)=0$,
we have to prove only that
\begin{equation}
n! \coeff{z^n} \Big( \frac{\cpt{k}}{{\mathbb{X}}^{3k-1}} %
- \frac{b_k}{{\mathbb{X}}^{3k}} \Big) \geq 0 \, ,
\label{APROVER}
\end{equation}
because $n! \coeff{z^n} \frac{b_k}{{\mathbb{X}}^{3k}} \geq 0$. As
$\cpt{k} \geq c_k$, it suffices to show that
\[
n! \coeff{z^n} \Big( \frac{c_k}{{\mathbb{X}}^{3k-1}} %
- \frac{b_k}{{\mathbb{X}}^{3k}} \Big) \geq 0 \, .
\]
Let
\begin{equation}
M(z) = 1+\sum_{n\geq 1} n^n \frac{z^n}{n!} \, ,
\label{M&Ms}
\end{equation}
i.e., $M = \frac{1}{\mathbb{X}} = \frac{1}{1-T}$.
Note that
if $n < j$ and $t < n_0$ then $t < 3k-1$ and
lemma \ref{LEMMA_A} tells us that $(3k-t)c_k \geq 3kb_k$ and
${{3k-1} \choose {t}}c_k \geq {{3k} \choose {t}}b_k$. Thus,
\begin{eqnarray}
& & n! \coeff{z^n} \left[ \frac{c_k}{{\mathbb{X}}^{3k-1}} %
- \frac{b_k}{{\mathbb{X}}^{3k}} \right] = \cr
& & \,\,\,\,\,\,\,\,\,\,\,\, n! \coeff{z^n} \left[ c_k(1+M(z))^{3k-1} %
- b_k(1+M(z))^{3k} \right] = \cr
& & \,\,\,\,\,\,\,\,\,\,\,\, \sum_{t=0}^{n_0} n! \coeff{z^n}
\left[{{3k-1} \choose {t}}c_k - {{3k} \choose {t}}b_k \right] M(z)^t \, .
\end{eqnarray}
\hfill \qed
We are now ready to prove (\ref{COEFF_INEGALITES}).
\begin{lem}
\label{LEMMA_F}
For all $k \geq 1$, the formal power series
${\EuScript{L}}_k$ satisfies
\[
{\EuScript{L}}_k(z) = \gr{W}_{k,C_3}(z) - \frac{b_k}{{\mathbb{X}}^{3k}} + %
\frac{\cpt{k}}{{\mathbb{X}}^{3k-1}} \succeq 0\, .
\]
\end{lem}
\noindent \textbf{Proof.}
First, $\EuScript{L}_1 \succeq 0$ by (\ref{BICYCLIC-MC3FREE}).
Suppose that $\EuScript{L}_i \succeq 0$
for all $i \in \coeff{1,k-1}$ and we have to show
that $\EuScript{L}_k \succeq 0$.
Hence, we can use lemma \ref{LEMMA_D}.
By definition,
\[
\Omegat{k-1} \EuScript{L}_{k-1}(z) = %
\big( \vartheta_z^2 - 3 \vartheta_z - 2(k-1) \big) \EuScript{L}_{k-1}(z) +%
2 \big( \vartheta_z W_{0,C_3}(z) \big) %
\big( \vartheta_z \, \EuScript{L}_{k-1}(z) \big) \, .
\]
If $n > n_0(k)$ then $n^2 -3n -2k > 0$ and we have
\[
\begin{array}{ccc}
\coeff{z^n} \Omegat{k-1} \EuScript{L}_{k-1}(z) & = & %
(n^2 -3n -2k+2) \coeff{z^n} \EuScript{L}_{k-1}(z) \cr
& + & 2 \coeff{z^n} %
\big( \vartheta_z W_{0,C_3}(z) \big) %
\big( \vartheta_z \, \EuScript{L}_{k-1}(z) \big) \geq 0
\end{array}
\]
Lemma \ref{LEMMA_D} tells us that ${\Delta_{k} \EuScript{L}_{k}} \geq
0$. Taking into account the definition of $\Delta$ given by
(\ref{OP:DELTA_K}), we obtain for $n \geq n_0(k-1)$~:
\begin{equation}
2 (n+k) \coeff{z^n} \EuScript{L}_{k} %
\geq 2\coeff{z^n}{T \vartheta_z\EuScript{L}_k}= %
2 \sum_{s=1}^{n-1} {n \choose s} s(n-s)^{n-s-1} %
\coeff{z^s} \EuScript{L}_k(z) \, .
\label{433333}
\end{equation}
And lemma \ref{LEMMA_E} leads to $\coeff{z^n} \EuScript{L}_{k}(z) \geq 0$,
if $n < n_0(k)$. Since $n_0(k-1) < n_0(k)$ we can infer by induction
on $n$ using (\ref{433333}) that $\EuScript{L}_k \succeq 0$. \hfill \qed
\subsection{Asymptotic results}
\label{SEC:N_UN_TIERS}
Denote by $c(n,n+k)$ the number of connected graphs having
$n$ vertices and $n+k$ edges.
Our aim of this paragraph is to
establish that the number $c_\xi(n,n+k)$ of $\xi$-free connected graphs
with $n$ vertices and $n+k$ edges is asymptotically
the same as $c(n,n+k)$ whenever $k=o(n^{1/3})$.
Combining lemmas \ref{TREE_POLYNOMIAL_INFINITY},
\ref{LEMMA_A} and \ref{LEMMA_F}, we obtain
the following important results:
\begin{thm} \label{THEOREM_ASYMPT1}
Almost all graphs having $n$ vertices and
$n+k$ edges are triangle-free when $n, \, k \rightarrow \infty$
but $k=o(n^{1/3})$.
\end{thm}
\noindent \textbf{Proof.} On one hand, lemma \ref{TREE_POLYNOMIAL_INFINITY}
shows that if $a \equiv a(n) \rightarrow 0$ as $n \rightarrow \infty$,
and if $b_1$ and $b_2$ are two fixed numbers such that $b_1 < b_2$,
then we have $t_n(an+\beta_1) \ll t_n(an+\beta_2)$ since
in (\ref{EQ:TREE_POLYNOMIAL_INFINITY}) we obtain a factor
$(1-u_0)^{(1-\beta)} = %
(\sqrt{a(1+\frac{a}{4})} - \frac{a}{2})^{(1-\beta)} = %
a^{\frac{1-\beta}{2}} + O(a)$.
On the other hand, we have
\[
kb_k \leq \cpt{k} \leq \frac{25}{5}kb_k
\]
and
\[
\frac{b_k}{{\mathbb{X}}^{3k}} - \frac{\cpt{k}}{{\mathbb{X}}^{3k-1}} \preceq %
\widehat{W}_{k,C_3} \preceq \frac{b_k}{{\mathbb{X}}^{3k}}
\, \, (k \geq 1) \, .
\]
Since $\cpt{k} = c_k + O((k-1)b_{k-1})= O(kb_k)$,
we have to find the values of $k$ for which
\[ k b_k t_n(3k-1) \ll b_k t_n(3k) \, . \]
We will use formula (\ref{EQ:TREE_POLYNOMIAL_INFINITY})
of lemma \ref{TREE_POLYNOMIAL_INFINITY}
to estimate $t_n(a\, n+\beta_1)$ and $t_n(a\, n+\beta_2)$,
with $an=3k, \, \beta_1=-1$, resp. $\beta_2= 0$.
It proves convenient to compute
$\frac{kt_n(a\, n+\beta_1)}{t_n(a\, n+\beta_2)}$ and we have
\begin{eqnarray}
\frac{kt_n(a\, n+\beta_1)}{t_n(a\, n+\beta_2)} %
& = & \frac{k t_n(3k-1)}{t_n(3k)} \cr
& = & k(1-u_0) = k(\sqrt{a}+O(a)) \cr
& = & \frac{n}{3}(a^{\frac{3}{2}} + O(a^2)) \, .
\end{eqnarray}
Consequently, if $k=o(n^{1/3})$ the number
$c_\xi(n,n+k)$ is asymptotically the same
as $c(n,n+k)$.
\hfill \qed
Also, we have
\begin{thm}[Wright 1980] \label{WRIGHT_THM}
As $n, \, k \rightarrow \infty$ but $k=o(n^{1/3})$, we have
\begin{eqnarray}
c(n,n+k) & = & d_k\, (3 %
\pi)^{1/2}(e/12k)^{k/2}n^{n+1/2(3k-1)} \cr
& & \, \, \, \, \, \, \, \, %
\times \Big(1+O(k^{-1})+O(k^{3/2}/n^{1/2})\Big) \,
\label{EQ:ASYMPT_N13}
\end{eqnarray}
where $d_k = \frac{1}{2 \pi} + O(1/k)$.
\end{thm}
Note that the value
$d=\frac{1}{2 \pi}=\lim_{k\rightarrow \infty} \, d_k$
was independently found by Voblyi \cite{Vo87} and by Meertens \cite{BCM90}.
\noindent
As a corollary of theorems \ref{THEOREM_ASYMPT1} and \ref{WRIGHT_THM},
we obtain
\begin{cor} \label{THEOREM_ASYMPT2}
If $n, \, k \rightarrow \infty$ but $k=o(n^{1/3})$
the asymptotic number of \\
$(n,n+k)$ triangle-free connected graphs
is given by
\begin{equation}
d_k\, (3 %
\pi)^{1/2}(e/12k)^{k/2}n^{n+1/2(3k-1)} %
\Big(1+O(k^{-1})+O(k^{3/2}/n^{1/2})\Big) \, .
\end{equation}
\end{cor}
\section{Random graphs and forbidden subgraphs}
\label{SEC:RANDOM-GRAPHS}
As shown in \cite{FKP89,JKLP93},
the machinery of generating functions permits to study the limit
distribution of random graphs and multigraphs with great precision.
In this section, we will show that probabilistic results on random \
$\xi$-free
graphs and multigraphs can be obtained when looking at the form of their
generating functions, mainly looking at the so-called \textit{leading
coefficients} of their decompositions into tree polynomials, i.e.,
using the results of the previous sections
and some analytical facts contained in \cite{JKLP93}.
We consider here two models of random graphs, namely the
\textit{permutation model} and the
\textit{multigraph process}. The idea is to
start with $n$ totally disconnected vertices and to add
successive edges one at time and at random \cite{ER59,ER60}.
In the first model, also called \textit{graph process}, we consider
all $N={n \choose 2}$ possible edges $x \rbar y$ with $x <y$
which are introduced in random order, allowing all $N!$
permutations with the same probability.
In the second model, also called \textit{uniform model},
ordered pairs $\langle x,y \rangle$ are generated
repeatedly ($1 \leq x, y \leq n$) and the edge $x \rbar y$ is added
to the multigraph. Thus, this process can generate
self-loops and multiple edges. Remark that we follow
Janson \textit{et al.} and for purposes of analysis,
we assign a \textit{compensation factor} to a multigraph $M$, viz.
a multigraph $M$ on $n$ labelled vertices can be defined by a symmetric
$n\times n$ matrix of nonnegative integers
$m_{xy}$, where $m_{xy}=m_{yx}$ is the number of undirected edges
$x\rbar y$ in~$G$. The {\it compensation factor\/} associated to $M$
is given by
\begin{equation}
\kappa(M)=
1\left/\prod_{x=1}^n\left(2^{m_{xx}}\prod_{y=x}^nm_{xy}!\right)\right.
\label{COMPENSATION_FACTOR}
\end{equation}
Thus, if $m=\sum_{x=1}^n\sum_{y=x}^nm_{xy}$ is the total number of edges,
the number of sequences $\langle x_1,y_1\rangle\langle x_2,y_2\rangle
\,\ldots\,\langle x_m,y_m\rangle$ that lead to~$M$ is then exactly
\begin{equation}
2^m\,m!\,\kappa(M) \,.
\label{COMPENSATION_FACTOR2}
\end{equation}
(We refer to \cite[Sect. 1]{JKLP93} for more details about $\kappa$.)
At generating function level, it follows that
after adding $m$ edges, the uniform model
on $n$ vertices will produce a
multigraph in a family $\mathcal{F}$ with
probability
\begin{equation}
{2^m\,m!\,n!\over n^{2m}}\,\,[w^mz^n]\,\,F(w,z)\,.
\label{PROBA-MULTIGRAPH}
\end{equation}
Similarly, if ${\mathcal{F}}$ is a family of
graphs with labelled vertices, the probability that $m$ steps
of the permutation model will produce
a graph in $\mathcal{F}$ is
\begin{equation}
{n!\over{N\choose m}}\,[w^mz^n]\,F(w,z)\,,\qquad N={n\choose 2}\,.
\label{PROBA-GRAPH}
\end{equation}
In \cite[Theorem 5]{JKLP93}, the authors
proved that only leading coefficients of $t_{n}(3k)$ are
relevant to compute
the probability that randomly generated graphs or multigraphs
will produce $r_1$ bicyclic components, $r_2$
tricyclic components, $\cdots$
We have the following results about $\xi$-free components and
random graphs:
\begin{thm} \label{THEOREM-RG-XI-FREE}
The probability that a random graph or
multigraph with $n$ vertices and $\frac{n}{2}$ edges has
only acyclic, unicyclic, bicyclic components all triangle-free is
\begin{equation}
\sqrt{\frac{2}{3}}\cosh \left( \sqrt{\frac{5}{18}}\right) e^{-\frac{1}{6}%
}+O(n^{-1/3})\approx 0.789... \, .
\label{PROPOSITION1}
\end{equation}
More generally, let
$\Theta = \{ p \in \mathbb{N}, p \geq 3 \mbox{ and } C_p \in \xi \}$.
The probability that a random graph or multigraph with $n$
vertices and $\frac{n}{2}$ edges has only acyclic, unicyclic,
bicyclic components all $C_{p}$-free, $p\in \Theta$, is
\begin{equation}
\sqrt{\frac{2}{3}}\cosh \left( \sqrt{\frac{5}{18}}\right) %
e^{-\sum_{p\in \Theta }\frac{1}{2p}}+O(n^{-1/3}) \, .
\label{PROPOSITION11}
\end{equation}
\end{thm}
\noindent \textbf{Proof.} This is a corollary of \cite[eq (11.7)]{JKLP93}
using the formulae (\ref{ACYCLIC-MC3FREE}), (\ref{ACYCLIC-GC3FREE})
and (\ref{BICYCLIC-GC3FREE-POLYNOMIAL}).
Incidentally, random graphs and multigraphs have the same asymptotic
behavior as shown by the proof of \cite[Theorem 4]{JKLP93}. As
multigraphs graphs without cycles of length $1$
and $2$, the forbidden cycles of length $1$ and $2$ bring a
factor $e^{-3/4}$ which is cancelled by a factor $e^{+3/4}$ because of the
ratio between weighting functions that convert the EGF of graphs and
multigraphs into probabilities. Indeed,
formulae (\ref{PROBA-MULTIGRAPH}) and (\ref{PROBA-GRAPH}) are
asymptotically related by the formula
\begin{equation}
{ {n \choose 2} \choose m} =\left( \frac{n^{2m}}{2^{m}m!}\right) %
\exp{\big(-\frac{m}{n}-\frac{%
m^{2}}{n^{2}}+O(\frac{m}{n^{2}})+O(\frac{m^{3}}{n^{4}})\big)},\, m \leq {n
\choose 2} \, .
\label{RAPPORT-GRAPH/MULTI}
\end{equation}
The situation changes radically when cycles of length greater to or less
than $3$ are forbidden. Equations (\ref{ACYCLIC-MC3FREE}), (\ref
{ACYCLIC-GC3FREE}) and the ``significant coefficient'' $\frac{5}{24}$ of $%
t_{n}(3)$
in (\ref{BICYCLIC-GC3FREE-POLYNOMIAL}) and the demonstration of \cite[Lemma 3]
{JKLP93} show us that the term $-\frac{T(z)^p}{2p}$, introduced in (\ref
{ACYCLIC-MC3FREE}) and (\ref{ACYCLIC-GC3FREE}) for each forbidden $p$-gon,
simply changes the result by a factor of $e^{-1/2p}+O(n^{-1/3})$.\hfill \qed
The example of forbidden $p$-gon suggests itself for a
generalization.
\begin{thm}
\label{THEOREM4} \textit{Let} $\xi
=\{H_{1},H_{2},H_{3},...H_{q}\} $ \textit{be a finite collection of
multicyclic connected graphs or multigraphs}. \textit{Then the probability
that a random graph with }$n$ \textit{vertices and} $\frac{1}{2}n+O(n^{%
\frac{1}{3}})$ \textit{edges} \textit{has} $r_{1}$ \textit{bicyclic
components}, $r_{2}$ \textit{tricyclic components},$\cdots $, $(k+1)$-%
\textit{cyclic components}, \textit{all components }$%
\{H_{1},H_{2},H_{3},...H_{q}\}$\textit{-free}
\textit{and no components of
higher cyclic order is}
\begin{equation}
\big(\frac{4}{3}\big)^{r}\exp{\big(-\sum_{p\in \Theta }\frac{1}{2p}\big)}\,%
\sqrt{\frac{2}{3}}\,\frac{b_{1}^{r_{1}}}{r_{1}!}%
\,\frac{b_{2}^{r_{2}}}{r_{2}!%
}\,\,\cdots \,\,\frac{b_{k}^{r_{k}}}{r_{k}!}\frac{r!}{(2r)!}+O(n^{-1/3})
\label{theorem1}
\end{equation}
\textit{where }$\Theta =\{p\geq 3$ , $ \exists i\in [1,q]$
\textit{such that} $H_{i}$ \textit{is a} $p$\textit{-gon}$\}$.
\end{thm}
Theorem \ref{THEOREM4} raised a natural question. Under what
conditions on the
forbidden configurations of graphs will the coefficients $(b_{i})$
change? The theorem \ref{THEOREM5} below shows
that a sufficient condition to
change a coefficient $b_{i}$ of (\ref{theorem1}) is
that $\xi$ must contain all graphs
\textit{contractible} to a certain $i$-excess graph $H$.
\begin{thm} \label{THEOREM5} Let $H$ be a $k$-excess
multicyclic graph (resp. multigraph) with $k>0$. Suppose
that $c(H)\, n!$ is the number of ways to label $H$
(for example $c(K_{4})=1/24$).
Denote by $\mathcal{A}_{k}^{(H)}$ the set of all $k$-excess
graphs contractible to $H$. Then the probability
that a random graph (resp. multigraph) with $n$ vertices and
$m(n)=\frac{n}{2}+O(n^{1/3})$ edges has $r_{1}$ bicyclic, $r_{2}$
tricyclic, ..., $r_{p}$ $(p+1)$-cyclic components, all
without component isomorphic to any member of the set
$\mathcal{A}_{k}^{(H)}$ and with $r=r_1+2 r_2+\cdots+p r_p$ is
\begin{equation}
\big(\frac{4}{3}\big)^{r}\sqrt{\frac{2}{3}}\,\frac{b_{1}^{r_{1}}}{r_{1}!}\,%
\cdots \,\,\frac{b_{k-1}^{r_{k-1}}}{r_{k-1}!}\,%
\frac{(b_{k}-c(H))^{r_{k}}}{r_{k}!}\,\frac{b_{k+1}^{r_{k+1}}}{r_{k+1}!}%
\,\,\cdots \,\,\frac{b_{p}^{r_{p}}}{r_{p}!}\frac{r!}{(2r)!}+O(n^{-1/3}) \, .
\end{equation}
\end{thm}
\noindent \textbf{Proof.} The EGF associated to $\mathcal{A}_{k}^{(H)}$
is simply
\begin{equation}
A_{k}^{(H)}(w,z)=w^k \, c(H) \, \frac{T(wz)^n}{(1-T(wz))^{3k}} \, .
\label{proof of THEOREM5}
\end{equation}
Thus in (\ref{theorem1}) if we want to avoid all graphs \textit{%
contractible} to $H$, we have to subtract (\ref{proof of THEOREM5}) from
the EGF of connected $k$-excess graphs. $\hfill \qed $
Note that in \cite[lemma 3]{JKLP93}, theorems
\ref{THEOREM-RG-XI-FREE}, \ref{THEOREM4} and \ref{THEOREM5},
the number of edges $m=m(n)$ varies from $\frac{n}{2}$
to $\frac{n}{2}+ O(n^{1/3})$. The discrepancy in the windows is
a consequence of the parameter $\mu$ in \cite[lemma 3]{JKLP93},
where $m(n) = \frac{1}{2}n(1+\mu n^{-1/3})$ and $|\mu| \leq n^{1/12}$.
Hence, when choosing very small $\mu$, such as $\mu=O(n^{-1/3})$, one
can get results like theorems 4-5 in \cite{JKLP93} or
theorems \ref{THEOREM-RG-XI-FREE}, \ref{THEOREM4} and \ref{THEOREM5}
here.
\begin{ack}
The authors wish to thank C. Banderier, P. Flajolet and
G. Schaeffer for helpful discussions and encouragements
relating to this research and also all the anonymous referees for
their efforts reading and improving the quality of this paper.
\end{ack}
\bibliographystyle{alpha}
|
{
"timestamp": "2004-11-25T10:32:26",
"yymm": "0411",
"arxiv_id": "cs/0411093",
"language": "en",
"url": "https://arxiv.org/abs/cs/0411093"
}
|
\section{Introduction}
\label{sec:int}
As was shown in \cite{Se}, the category of locally trivial vector
bundles over an affine regular algebraic variety $X$ is equivalent
to that of finitely generated projective ${\Bbb K}(X)$--modules where
${\Bbb K}(X)$ is the coordinate algebra of $X$. (In what follows the
ground field ${\Bbb K}$ is assumed to be ${\Bbb C}$ or ${\Bbb R}$, the latter case
will be specified each time.) A similar equivalence is valid for
compact smooth varieties (cf. \cite{Ro}). This is the reason why
in the non-commutative geometry finitely generated projective
modules over non-commutative algebras are considered as appropriate
analogs of vector bundles. Thus, the problem of constructing and
classifying such ${\cal A}$-modules for a given non-commutative (NC)
algebra ${\cal A}$ is of great interest. Unfortunately, besides
${\Bbb C}^*$-algebras very few examples of algebras with a significant
family of projective modules are known.
In \cite{GS1} we have suggested a way of constructing projective
${\cal A}$-modules via an NC version of the Cayley-Hamilton (CH)
identities. The existence of these identities (as well as the
Newton identities) is the very remarkable property of the
so-called reflection equation algebras (REA) or their modified
versions (mREA). The algebras of this type can be associated to a
large class of braidings (solutions of the quantum Yang-Baxter
equation) of the Hecke type.
In the present paper we consider some quotients of the mREA --- the
non-commutative (NC) orbits. The terminology is motivated by a close
connection of these quotients with the coordinate algebras of orbits
in $gl(n)^*$.
For each generic NC orbit ${\cal A}$ we establish certain combinatorial
relations among its central elements. The relations can be
interpreted as the higher NC counterparts of the Newton identities.
Recall, that the classical Newton relations connect the elementary
symmetric functions of some commutative variables and the power
sums of the same variables. Besides, we construct a large family of
projective ${\cal A}$-modules (in what follows all modules are assumed to
be finitely generated and one-sided).
Also, we introduce an algebra $Q({\cal A})$ which is an analog of
$K^0(\rm Fl({\Bbb C}^n))$, where $\rm Fl({\Bbb C}^n)$ is a flag variety, and define a
$q$-analog of the Euler characteristic of line bundles on the flag
variety which is is well defined on $Q({\cal A})$.
A particular case of the NC orbits is the set of the so-called
quantum orbits arising from the quantization of a certain Poisson
pencil on semisimple orbits in $gl(n)^*$ (i.e., $GL(n)$ orbits of
semisimple elements of $gl(n)^*$). The Poisson pencil is generated
by the two brackets: the Kirillov bracket and another one, related
to a classical $r$-matrix. One of the main properties of the algebra
${\cal A}$ arising from the quantization of the Poisson pencil is that
the product $\mu: {\cal A}\otimes {\cal A} \rightarrow {\cal A}$ is equivariant
(covariant) with respect to the action of the quantum group $U_q(sl(n))$
\begin{equation}
U\triangleright\mu(a\otimes
b)=\mu(U_1\triangleright a\otimes U_2\triangleright b),\quad a,b
\in {\cal A},\quad U\inU_q(sl(n)), \label{equi}
\end{equation}
where $\Delta(U)=U_1\otimes U_2$. In this case (the $U_q(sl(n))$ case for short)
at the limit $q\rightarrow 1$ we get the $SL(n)$-equivariant (or,
if we pass to the compact form, $SU(n)$-equivariant) algebras which
are also called {\em the fuzzy orbits}.
Let us describe the algebras in question in more detail. We start
from the definition of the reflection equation algebra connected
with a braiding of the Hecke type. The initial data for its
construction is a braiding $R$
$$
R:\quadV^{\ot 2}\toV^{\ot 2},
$$
which is a solution of the quantum Yang-Baxter equation (\ref{YBE})
satisfying additionally the second order equation (\ref{Hecke}). Here
V is a finite dimensional vector space, ${\rm dim} V = n$. Such a braiding
will be called {\em the Hecke symmetry}. A well-known example is
connected with the quantum group $U_q(sl(n))$ when the corresponding Hecke
symmetry is the image of the universal $R$-matrix in the fundamental
vector representation of $U_q(sl(n))$.
In \cite{G} there were constructed other examples of Hecke symmetries
such that the Hilbert-Poincar\'e series of the associated "symmetric"
and "skew-symmetric" subalgebras of the tensor algebra $T(V)$ differ
from the classical ones. Below we shall additionally assume the
Hilbert-Poincar\'e series of the "skew-symmetric" subalgebra to be a
monic polynomial. Such a Hecke symmetry will be called {\em even} and
the degree $p$ of this polynomial will be called {\it the symmetry
rank} of $R$.
Consider now an associative unital algebra generated by $n^2$
indeterminates $l_i^{\,j}$, $1\le i,j\le n$ satisfying the following
relation
\begin{equation}
R\, L_1\,R \,L_1 - L_1\,R\, L_1\,R = \hbar(R\,L_1 - L_1\,R),
\qquad L_1\equiv L\otimes I, \label{defREA}
\end{equation}
where $\hbar$ is a formal parameter, $I$ is an $n\times n$ unit
matrix and $L =\|l_i^{\,j}\|$ is a matrix with entries $l_i^{\,j}$.
If ${\hbar}=0$, we call this algebra the {\it reflection equation
algebra} (REA) and denote it ${\cal L}_q$; if ${\hbar}\not=0$, we call it
the {\it modified reflection equation algebra} (mREA) and denote
it $\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi$.
In the $U_q(sl(n))$ case, being specialized at $q=1$, the mREA coincides
with the enveloping algebras $U(gl(n)_{\hbar})$ (hereafter, given a Lie
algebra $\mbox{$\frak g$}$ with the bracket $[\,\cdot\,,\cdot\,]$, the symbol
$\gg_{\h}$ will denote the Lie algebra with the bracket
${\hbar}[\,\cdot\,,\cdot\,]$). In some sense the mREA is a "braided"
analog of the enveloping algebra $U(gl(n)_{\hbar})$. For a motivation
of this treatment, see section \ref{sec:rea-br}.
Now we list some properties of the mREA. First of all, the
definition implies that the generators $l_i^j$ of any mREA obey the
quadratic-linear commutation relations. Second, the category ${\lhq}\mbox{--}{\rm Mod}$
of {\em equivariant} finite dimensional representations of the mREA
corresponding to an even Hecke symmetry is close to the category
$U(gl(p))$--Mod, where $p$ is the symmetry rank of the Hecke
symmetry $R$ (see section \ref{sec:rea-br}). For example, simple
objects of the category ${\lhq}\mbox{--}{\rm Mod}$ can be labelled by signatures
(partitions) $\lambda=(\lambda_1,\dots,\lambda_p)$, $ \lambda_1\geq\dots \geq
\lambda_p$, similarly to the category of $gl(p)$-modules. Besides, the
Grothendieck rings of these categories are isomorphic \cite{GLS1}.
To explain the term "equivariant" we restrict ourselves to the $U_q(sl(n))$
case. Consider an $U_q(sl(n))$ module $V$. Then, as is well known, the
space ${\rm End\, }(V)$ of the internal endomorphisms of $V$ is endowed with
an $U_q(sl(n))$-action, too. The mREA can as well be equipped with an
$U_q(sl(n))$-action satisfying (\ref{equi}). In this case a representation
$\pi:\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi\to{\rm End\, }(V)$ is called {\it equivariant} if $\pi$ commutes
(as a mapping) with the $U_q(sl(n))$-action.
The next important property of the mREA is the existence of a monic
polynomial ${\cal CH}(x)$ of degree $p$ such that the matrix $L$
entering formula (\ref{defREA}) satisfies the Cayley-Hamilton (CH)
identity
$$
{\cal CH}(L) = \sum_{k=0}^p(-L)^{p-k}\sigma_k(L) \equiv 0,\qquad
\sigma_0(L)={\rm id}_{\cal L}
$$
where the coefficients $\sigma_k(L), \,0\le k\le p$ are linear independent
generators of the center $Z(\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi)$
of the
mREA. This identity was proved in \cite{GPS} for the non-modified
REA (the $U_q(sl(n))$ case was previously considered in \cite{PS}). The
above CH identity for the mREA can be established by a linear change
of generators \cite{GS1}.
Let $\chi:Z(\lhq)\to{\Bbb K}$ be a character of the center. The character
$\chi$ is completely fixed by its values on $\sigma_k$
$$
\chi(\sigma_k)=\sum_{1\leq i_1<\dots<i_k\leq p}\mu_{i_1}\dots
\mu_{i_k}, \quad 1\le k\le p
$$
where the numbers $\mu_i\in{\Bbb K}$ are assumed to be distinct. The
quotient of the algebra $\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi$ modulo the ideal generated by the
elements $z-\chi(z)$, where $z\inZ(\lhq)$, will be called an {\it NC
orbit} and denoted ${\cal L}_{\h,q}^{\chi}$ (although as explained in
remark \ref{DM} such an "NC orbit" in the $U_q(sl(n))$ case
can arise from a quantization of a union of some orbits in
$gl(n)^*$).
Observe that, being switched to the
algebra ${\cal L}_{\h,q}^{\chi}$, the CH identity for $L$ takes the form
$$
\prod_i(L-\mu_i I)=0.
$$
(Thus, the numbers $\mu_i$ are thought of as eigenvalues of the
matrix $L$ with entries from ${\cal L}_{\h,q}^{\chi}$.)
A great importance of the CH identity is occasioned by the fact that
it allows us to construct a family of idempotents from
${\rm Mat}({\cal L}_{\h,q}^{\chi})$ and therefore projective modules. Moreover, the
family of idempotents (and the corresponding projective modules) can
be essentially enlarged. For this purpose, we construct a series of
higher order matrices ${ L}_{(m)}$ and polynomials
${\cal CH}_{(m)},\,\,m=2,3,\dots$ with central coefficients such that the
higher order CH identities ${\cal CH}_{(m)}({ L}_{(m)})=0$ are satisfied. We also
set $L_{(1)}=L$, ${\cal CH}_{(1)}={\cal CH}$.
Upon restricting to the NC orbit ${\cal L}_{\h,q}^{\chi}$, we come to the series of
polynomials ${\cal CH}_{(m)}^{\chi}$ with the numerical
coefficients. Assuming the roots of these polynomials to be distinct
for any $m$ we find $p_m = \deg\,{\cal CH}_{(m)}$ idempotents $e_\mathbf{k}(m)\in
{\rm Mat}({\cal L}_{\h,q}^{\chi})$, each corresponding to a projective
${\cal L}_{\h,q}^{\chi}$-module.
One of the main aims of this paper is to prove the existence of the
higher order CH identities and to compute the coefficients of the
polynomials ${\cal CH}_{(m)}$. This is rigorously done for the mREA
associated with any even Hecke symmetries of rank $p=2$. For
$p\geq 3$ we present an explicit formula for these polynomials as a
conjecture.
Our other aim is to compute the values of the central elements
${\rm Tr}_R{ L}_{(m)}^s$, $s\ge 1$, for a generic NC orbit. Here ${\rm Tr}_R:
{\rm Mat}(\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi)\rightarrow \ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi$ is the trace defined by the initial
Hecke symmetry $R$. It is closely related to the {\it categorical
trace} which is discussed in section \ref{sec:rea-br} (see
\cite{GLS1} for more detail). For example, in the $U_q(sl(n))$ case ${\rm Tr}_R$
is the well known quantum trace which is a weighed sum of the
diagonal entries, but in the general case it can be more
complicated. We express ${\rm Tr}_R { L}_{(m)}^s$ (up to the aforementioned
conjecture) in terms of the eigenvalues of the matrix $L$. For $m=1$
we treat these expressions as a parametric resolution of the Newton
relations. For $m>1$ they are thought of as higher analogs of the
Newton relations.
As a byproduct we compute the value of ${\rm Tr}_R \,e_\mathbf{k}(m)$ on generic
NC orbits. In contrast with the usual orbits when these quantities
are always equal to 1 (since the related projective modules
correspond to line bundles), for the NC orbits they are more
informative. We show that the assignment $M_\mathbf{k}(m) \mapsto {\rm Tr}_R
e_\mathbf{k}(m)$, where the module $M_\mathbf{k}(m)$ corresponds to the idempotent
$e_\mathbf{k}(m)$, can be considered as an analog of the Euler
characteristic of a line bundle over a flag variety. Developing the
analogy with a flag variety, we can introduce a multiplicative
structure on the set of the modules $M_\mathbf{k}(m)$ and construct an
algebra $Q({\cal L}_{\h,q}^{\chi})$ playing the role of $K^0$ of the flag variety.
Completing the Introduction, we consider the $U_q(sl(n))$ case in more
detail. As we have already mentioned, in this case the NC orbits
arise from a quantization of a Poisson pencil on some algebraic
varieties. So, it is natural to consider the problem of quantization
of the vector bundles over these varieties in terms of projective
modules. In this case the projective modules $M_\mathbf{k}(m)$ over the
quantum orbits are nothing but deformations of the corresponding
modules over the coordinate rings of the initial varieties.
The problem of quantization of semisimple orbits in $\mbox{$\frak g$}^*$ was
considered in numerous papers. In \cite{DM1} the full solution of
this problem was given in terms of the mREA and its appropriate
quotients related to the minimal polynomials of orbits in question.
In \cite{DM2} this method of quantization is compared in the
$U(sl(n))$ case with an approach based on the so-called generalized
Verma modules. In particular, the authors produce a formula
describing the eigenvalues $\mu_i$ to be functions of the
corresponding generalized Verma module weight
$\lambda_i$\footnote{In fact, this formula is well-known, cf. for
example, \cite{BR} where the eigenvalues of the Casimir elements
${\rm Tr} L^s$ in finite dimensional representations are computed.}.
We find a q-analog of this formula by using different methods. (Note
that in the $U(sl(n))$ case this formula is obtained by means of the
coproduct, see section 2, whereas in the mREA we did not find any
convenient coproduct.)
The paper is organized as follows. In the next section we consider
the NC orbits arising from the quantization of the Kirillov bracket
on some semisimple orbits in $gl(n)$ (or $su(n)$). We call them {\em
the fuzzy orbits}, the term is motivated by numerous papers devoted
to "the fuzzy physics". In contrast with all those papers, we
present a general scheme of constructing a large family of
projective modules via the CH identities. In subsequent sections we
generalize this scheme to the mREA associated with a large class of
Hecke symmetries and thereby we construct a similar family of
projective modules for their appropriate quotients
("$q$-fuzzy orbits").
Note that the methods of section \ref{sec:fuzzy} are close to those
going back to the pioneering paper \cite{Ko} and having been used in
numerous works devoted to the {\it characteristic identities} (which
are nothing but a specialization of the CH identities to concrete
representations of the algebras in question, cf. \cite{Go}, \cite{O}
and references therein\footnote{Note that there are known $q$-analogs
of such characteristic identities related to the quantum groups (cf.
\cite{GZB}). However, the quantum groups are not deformations of
commutative algebras and are not convenient objects for constructing
projective modules and the related Newton identities. A set of
specific New\-ton-Cay\-ley-Ha\-mil\-ton identities exists in
algebras dual to the quantum groups, cf. \cite{IOP}. However, they
are useless for constructing the projective modules over these
algebras.}).
In section \ref{sec:rea-def}, we introduce the REA and list its
basic properties in detail.
In section \ref{sec:rea-br}, we give reasons allowing us to treat
the mREA as a braided analog of the enveloping algebra.
Section \ref{sec:5} is devoted to the proof of a parametric
resolution of the {\em basic Newton identity}.
In section \ref{sec:6} we give a proof of the relation between
eigenvalues of the matrix $L$ and those of the highest matrices
$L_{(m)}$ in the particular case $p=2$. At the end of the section
we compute the values of the central elements ${\rm Tr}_R { L}_{(m)}^k$. As a
byproduct we get the aforementioned formula featuring the
eigenvalues of the matrix $L$ as a function of a partition $\lambda$.
Finally, in section \ref{sec:q-eiler} we introduce the algebra
$Q({\cal L}_{\h,q}^{\chi})$ and define a $q$-analog of the Euler characteristic.
{\bf Acknowledgement} The authors are grateful to the
Max-Planck-Institut f\"ur Mathematik (Bonn) where the paper was
started and the Institute Mittag-Leffler (Stockholm) where the paper
was completed for the warm hospitality and the stimulating
atmosphere. One of the authors (D.G.) is grateful to A. Mudrov for
valuable discussions.
\section{Fuzzy orbits}
\label{sec:fuzzy}
In this section we are dealing with algebras arising from the
quantization of the Kirillov bracket alone. First, we describe our
initial object --- the coordinate algebra of a generic classical
orbit. Let us fix a diagonal matrix
\begin{equation}
M={\rm diag}(\mu_1,\dots , \mu_n),\quad \mu_i\in {\Bbb C}
\label{elem}
\end{equation}
with simple $\mu_i$. We treat this matrix as an element of
$\mbox{$\frak g$}^* = gl(n)^*$ (identified with $\mbox{$\frak g$}=gl(n)$) and consider its
orbit ${\cal O}_M$ with respect to the $GL(n)$-action
\begin{equation}
GL(n)\ni g:\quad M \mapsto M_g=g^{-1}\, M\, g.
\label{act}
\end{equation}
The orbit ${\cal O}_M$ is a closed affine algebraic variety and its
coordinate algebra ${\Bbb K}({\cal O}_M)$ can be described as follows. Let
${\Bbb K}(\mbox{$\frak g$}^*)$ be the polynomial algebra in $n^2$ commutative
indeterminates $l_i^j,\,1\le i,j \le n$, which are nothing but the
coordinate functions. Then we have ${\Bbb K}({\cal O}_M)={\Bbb K}(\mbox{$\frak g$}^*)/{\cal I}$
where ${\cal I}\subset {\Bbb K}(\mbox{$\frak g$}^*)$ is the ideal generated by the
elements
\begin{equation}
{\rm Tr}\,L^k-\beta_k,\quad\,k=1,2, \dots , n.
\label{lk}
\end{equation}
Here the matrix $L=\|l_i^j\|$ is composed of the indeterminates
$l_i^j$ ($i$ labels the rows and $j$ --- the columns) and
\begin{equation}
\beta_k=\sum_{i=1}^n \mu_i^k,\quad k=1,2, \dots , n.
\label{alp}
\end{equation}
\begin{remark}
\label{rem:nongener}
{\rm
Note that if ${\cal O}_M$ is semisimple but not generic (i.e., the
eigenvalues $\mu_i$ of a given diagonal matrix $M$ are not simple)
the coordinate algebra is the quotient ${\Bbb K}({\cal O}_M)={\Bbb K}(\mbox{$\frak g$}^*)/
{\cal I}'$, where the ideal ${\cal I}'$ is of the following form.
Let
$$
{\cal P}(M) = (M-\mu_1I)\dots (M-\mu_rI)
$$
be the minimal polynomial of the matrix $M$ ($I$ stands for the
$n\times n$ unit matrix). Then $n^2$ entries of the matrix
${\cal P}(L)$ are polynomials in generators $l_i^j$. The ideal
${\cal I}'$ is generated by these polynomials and by elements
(\ref{lk}) with $k=1,\dots ,r-1$. Note, that if we disregard the
latter elements we get a union of all semisimple orbits possessing
the same minimal polynomial.}
\end{remark}
If the initial matrix $M$ is such that ${\rm Tr}\, M=0$, then the
corresponding orbit is embedded into $sl(n)^*$. Since
$$
{\Bbb K}(sl(n)^*)={\Bbb K}(gl(n)^*)/\{{\rm Tr}\, L\},
$$
the coordinate algebra of the corresponding orbit can be realized as
above but with $\beta_1=0$. (Hereafter $\{S\}$ stands for the ideal
generated by a set $S$.)
If all eigenvalues of the matrix $M$ are real, we can consider the
matrix ${\rm i} M$ (here ${\rm i}=\sqrt{-1}$) as an element of $u(n)^*$
(or $su(n)^*$ if ${\rm Tr}\, M=0$). Choosing the generators $x_i^j$ for
$i\le j$ and $y_i^j$ for $i<j$ such that
$$
l_i^j=x_i^j+ {\rm i} y_i^j\quad{\rm for}\;\;i<j, \qquad
l_i^j=x_j^i-{\rm i} y_j^i\quad{\rm for}\;\;i>j, \qquad{\rm and}
\qquad
l_i^i=x_i^i,
$$
we get a compact real variety which is an $SU(n)$-orbit (i.e., in
(\ref{act}) we assume that $g\in SU(n)$). Consequently, we consider
its coordinate algebra as an ${\Bbb R}$-algebra.
Now let us pass to the fuzzy orbits. In the spirit of "NC affine
algebraic geometry" we realize them via some explicit relations on
generators. Consider again the matrix $L=\|l_i^j\|$, but now we let
the generators $l_i^j$ to satisfy the defining relations of the
algebra $U(\gg_{\h})$ with $\mbox{$\frak g$}=gl(n)$:
$$
l_i^j\,l_m^n-l_m^n\,l_i^j-{\hbar}(l_i^n\,\delta_m^j-l_m^j\,\delta_i^n)=0.
$$
Then the matrix $L\in{\rm Mat }_n(U(\gg_{\h}))$ obeys a polynomial relation
\begin{equation}
{\cal CH}(L)=\sum_{k=0}^n(- L)^{n-k}\sigma_k( L) = 0,
\label{CH1}
\end{equation}
such that the coefficients $\sigma_k( L)$ are central and
$\sigma_0( L) = 1$. This fact is well known. An expression of the
coefficients $\sigma_k( L)$ in terms of the generators $l_i^{\,j}$
can be found in \cite{GS1}. Below we present them in a convenient
form for a more general case of NC orbit (see section \ref{sec:5}).
It is also well known that the center $Z(U(\gg_{\h}))$ of the algebra
$U(\gg_{\h})$ is generated by $\sigma_k( L)$ for $1\le k\le n$. Another
family generating the center is $s_k(L)={\rm Tr} L^k$, $ 1\le k \le n$.
Therefore, any character
$$
\chi:\; Z(U(\gg_{\h}))\to {\Bbb K}
$$
is completely determined by its values on the generators of the
center $\chi(\sigma_k( L)) = {\alpha}_k$ or $\chi(s_k(L))=\beta_k$, $k=
1,\dots,n$.
Consider the quotient algebra
\begin{equation}
{\cal L}_{\h}^{\chi} = U(\gg_{\h})/\{z-\chi(z)\;|\; z\in Z(U(\gg_{\h})) \},
\label{alg}
\end{equation}
where $\chi$ is a fixed character. Being switched to ${\cal L}_{\h}^{\chi}$,
relation (\ref{CH1}) takes the form
\begin{equation}
{\cal CH^{\chi}}(L)\stackrel{\mbox{\tiny def}}{=}\sum_{k=0}^n(- L)^{n-k}
\chi(\sigma_k(
L)) = \sum_{k=0}^n(- L)^{n-k}{\alpha}_k=0.
\label{CH2}
\end{equation}
In what follows the superscript $\chi$ means that we have passed
from $U(\mbox{$\frak g$}_{\hbar})$ to ${\cal L}_{\h}^{\chi}$ (and similarly, for other algebras
below).
Also, consider the associated numerical equation
\begin{equation}
\sum_{k=0}^n(- \mu)^{n-k}{\alpha}_k = 0. \label{CH3}
\end{equation}
Denoting the roots of this equation by $\mu_i$ we get
\begin{equation}
\alpha_k = \chi(\sigma_k) =
\sum_{1\le i_1<i_2<\dots <i_k\le n}\mu_{i_1}\mu_{i_2}\dots
\mu_{i_k}.
\label{gen}
\end{equation}
Hereafter (up to the last section) we assume the numbers
$\mu_i\in{\Bbb K}$ to be fixed and the character $\chi$ to be defined
by the set of values (\ref{gen}).
\begin{definition}
\label{def:fuz-orb}
The roots of equation (\ref{CH3}) will be called the eigenvalues of
the matrix $L$ on the orbit ${\cal L}_{\h}^{\chi}$ (that is, when entries of $L$
belong to the algebra ${\cal L}_{\h}^{\chi}$). The algebra ${\cal L}_{\h}^{\chi}$ will be called
the 1-generic fuzzy orbit if the eigenvalues $\mu_i$ are simple.
\end{definition}
In what follows we shall only consider the 1-generic fuzzy orbits
without specifying it each time. Note, that the 1-generic fuzzy
orbits can arise from a quantization of semisimple but not generic
orbits (see remark \ref{DM} below).
For any fuzzy orbit, we introduce $n$ idempotents:
\begin{equation}
e_j=\prod_{i\not=j}{{(L-\mu_iI)}\over{(\mu_j-\mu_i)}},\qquad
j=1,\dots,n. \label{proj1}
\end{equation}
Identity (\ref{CH2}) leads to
$$
e_i\,e_j=\delta_{ij}\,e_i,\qquad \sum_{i=1}^n e_i=I.
$$
The CH identity (\ref{CH1}) and all related objects (idempotents
(\ref{proj1}), corresponding projective ${\cal L}_{\h}^{\chi}$-modules, etc.) will
be called {\em basic}.
Now, we shall describe a regular way of constructing some
{\em higher} analogs of these objects. To this end, consider the
category of finite dimensional representations of the algebra
$U(\mbox{$\frak g$})$. Its simple objects $V_{\lambda}$ are labelled by sequences of
numbers
\begin{equation}
\lambda=(\lambda_1,\dots ,\lambda_{n}),
\label{signature}
\end{equation}
where $\lambda_i-\lambda_{i+1}$ are nonnegative integers. Following \cite{W}
we call these sequences the {\em signatures}. If moreover, $\lambda_{i}$
themselves are nonnegative integers and $\sum \lambda_i=m$ we call the
signature $\lambda$ {\it ordered partition} of the integer $m$.
Since the algebras $U(\gg_{\h})$ and $U(\mbox{$\frak g$})$ are isomorphic to each
other, the objects $V_{\lambda}$ can be equipped with an $U(\gg_{\h})$-action.
Consider a left $U(\gg_{\h})$-module $V_{\lambda}$ and let
$$
\pi_{\lambda}:\; U(\gg_{\h})\to{\rm End\, }(V_{\lambda})
$$
be the corresponding left irreducible representation. All
representations in question are assumed to be {\em equivariant},
i.e., they commute with the $GL(n)$-action, where we suppose
$U(\gg_{\h})$ to be equipped with the adjoint $GL(n)$-action.
Introduce now the map
$$
\pi_\lambda^{(2)}=I\otimes\pi_\lambda :\quad U(\gg_{\h})\otimes U(\mbox{$\frak g$})\to U(\gg_{\h})\otimes
{\rm End\, }(V_{\lambda})
$$
(note, that in the second factor we put ${\hbar}=1$). On fixing a basis
in the space $V_{\lambda}$, we can identify the spaces ${\rm End\, }(V_{\lambda})$ and
${\rm Mat }_{n_\lambda}({\Bbb K})$, where $n_\lambda=\dimV_{\lambda}$. Consequently, the spaces
\begin{equation}
U(\gg_{\h})\otimes {\rm End\, }(V_{\lambda})\quad
{\rm and}\quad U(\gg_{\h})\otimes {\rm Mat }_{n_\lambda}({\Bbb K})={\rm Mat }_{n_\lambda}(U(\gg_{\h}))
\label{ident}
\end{equation}
can be identified. Therefore, the above map $\pi_\lambda^{(2)}$ is of
the form
$$
\pi_\lambda^{(2)}:\quad U(\gg_{\h})\otimes U(\mbox{$\frak g$})\to {\rm Mat }_{n_\lambda}(U(\gg_{\h})).
$$
\begin{remark}
\label{rem:3}
{\rm
Note that the spaces $V_\lambda$ and $V_\mu$ whose signatures are
different by a constant shift
$$
\lambda_i -\mu_i = z,\quad 1\le \forall\, i\le n,\quad
z\in{\Bbb K}
$$
have equal dimensions $n_\lambda = n_\mu$. Making the
transformation
\begin{equation}
\pi_\lambda(e_i^{\,j})\mapsto \pi_\lambda(e_i^{\,j}) -
z\,\delta_i^{\,j}
{\rm id}_{V_\lambda}\quad \forall\,z\in{\Bbb K}
\label{ed-sdv}
\end{equation}
where $e_i^{\,j}$ are the $U(gl(n))$ generators, one can convert the
$U(gl(n))$-representation $\pi_\lambda$ into $\pi_\mu$. In
particular, setting $z=\lambda_n$ we come to the representation
$\pi_{\hat
\lambda}$ where $\hat\lambda$ is the {\it partition} of the form
\begin{equation}
\hat \lambda = (\hat \lambda_1, \dots, \hat \lambda_{n-1},0),
\quad \hat\lambda_i = \lambda_i-\lambda_n.
\label{lambda-hat}
\end{equation}
Note that the objects $V_\lambda $ and $V_{\hat \lambda}$ differ as
$U(gl(n))$-modules but coincide as $U(sl(n))$-ones (recall that
simple $U(sl(n))$-modules are labelled by partitions
(\ref{lambda-hat})). Therefore,
${\rm dim} V_\lambda = {\rm dim} V_{\hat \lambda}$.
Moreover, to the space $V^*_\lambda$ (dual to $V_\lambda$) there
corresponds the space $V_{\lambda^{\!*}}$ in the category of finite
dimensional $U(gl(n))$-modules. Here the signature $\lambda^{\!*}$
is defined as
\begin{equation}
\lambda^{\!*} = (-\lambda_n,\dots,-\lambda_1).
\label{lambda-star}
\end{equation}
Note that in the category of $U(sl(n))$-modules the dual object is
labelled by the partition $\hat{\lambda^{\!*}}$. }
\end{remark}
Now we apply $\pi_\lambda^{(2)}$ to the {\it split Casimir element}
\begin{equation}
{\bf Cas}= l_i^j\otimes l_j^i.
\label{cas-kl}
\end{equation}
Hereafter the summation over the repeated indices is understood. For
${\hbar}= 1$, the image of the split Casimir element under the product
map $U(\mbox{$\frak g$})^{\otimes 2}\to U(\mbox{$\frak g$})$ becomes the usual quadratic Casimir
element $s_2={\rm Tr} L^2\in U(\mbox{$\frak g$})$.
Denote the matrix transposed to $\pi^{(2)}_\lambda({\bf Cas})$ by
$L_{\lambda}$, i.e.
$$
L_{\lambda}^t = \pi^{(2)}_\lambda({\bf Cas}) = l_i^j\otimes \pi_\lambda(l_j^i).
$$
It can be easily seen that if $\lambda_0=(1,0,\dots ,0)$, then
$L_{\lambda_0}=L$ provided that the basis $\{x_i\}\in V$ is fixed in
such a way that $\pi_{\lambda_0}(l_i^j)\triangleright x_k =
\delta_k^j x_i$.
We emphasize, that the element $\pi_{\lambda}^{(2)}({\bf Cas})$ actually
belongs to the algebra ${\rm Mat }^{\rm Inv}_{n_\lambda}(U(\gg_{\h}))$ --- the
$GL(n)$-invariant subalgebra of ${\rm Mat }_{n_\lambda}(U(\gg_{\h}))$. This follows
from the fact that we are working with equivariant representations
$\pi_\lambda: U(\gg_{\h}) \to {\rm End\, }(V_{\lambda}) = {\rm Mat }_{n_\lambda}({\Bbb K})$. Here we assume
that the action of $GL(n)$ on ${\rm Mat }_{n_\lambda}(U(\gg_{\h}))$ is defined on the
base of identification (\ref{ident}). In \cite{K} the algebras
${\rm Mat }^{\rm Inv}_{n_\lambda}(U(\gg_{\h}))$ were called {\it the family algebras}
(classical if ${\hbar}=0$ and quantum if ${\hbar}\not=0$), see also \cite{R}.
\noindent
{\bf Example}\ \ Let $n=2$. Then
$$
{\bf Cas}=a\otimes a+b\otimes c +c \otimes b+ d\otimes d,\quad {\rm where} \quad a=
l_1^1, \,\,
b=l_1^2,\,\, c=l_2^1,\,\,d=l_2^2.
$$
Set $\lambda=(l+1,l),\,\,l\in{\Bbb K}$ and consider the representation
$\pi_\lambda$ of $U(gl(2))$. We have
$$
\pi_\lambda(a)=\left(\matrix{1+l&0\cr
0&l}\right),\;\pi_\lambda(b)=\left(\matrix{0&1\cr 0&0}\right),\;
\pi_\lambda(c)=\left(\matrix{0&0\cr
1&0}\right),\;\pi_\lambda(d)=\left(\matrix{l&0\cr 0&1+l}\right).
$$
The corresponding matrix $L_\lambda$ is
$$
L_\lambda=\left(\matrix{a&b\cr c&d}\right)
+l(a+d)
\left(\matrix{1&0\cr 0&1}\right).
$$
If $l=0$, we have just the matrix $L$.
Below we shall restrict ourselves to the matrices $L_\lambda$
corresponding to the partitions $\lambda=(m)=(m,0,0,\dots ,0)$. These
matrices, as well as related objects, will be denoted ${ L}_{(m)}$,
$\pi_{(m)}$, $V_{(m)}$, etc.
The matrix ${ L}_{(m)}$ as well as $L=L_{(1)}$ satisfies the CH identity.
Now we describe the construction of the corresponding CH polynomial.
In what follows we shall often use the set of all possible
(non-ordered) partitions $\mathbf{k}$ of the integer $m$ with the length not
greater than $n$. By definition, $\mathbf{k}$ is the set of $n$ integers
$k_i$ with the following properties
\begin{equation}
\mathbf{k}=(k_1,\dots ,k_n),\quad k_i\geq 0,\quad |\mathbf{k}|=k_1+\dots +k_n=m.
\label{part}
\end{equation}
For any partition $\mathbf{k}\vdash m$ from the set (\ref{part}), let
\begin{equation}
\mu_\mathbf{k}(m)=\sum_{i=1}^n k_i\mu_i+{\hbar}\sum_{1\le i<j\le n} k_i k_j
\label{high}
\end{equation}
where $\mu_i$, $1\le i\le n$, are elements of the algebraic closure
of the center $Z(U(\mbox{$\frak g$}_{\hbar}))$. They are the solutions of the set of
polynomial relations
\begin{equation}
\sum_{1\le i_1<i_2<\dots <i_k\le
n}\mu_{i_1}\mu_{i_2}\dots \mu_{i_k}=\sigma_k(L),
\quad 1\le k\le n,
\label{elsym}
\end{equation}
where $\sigma_k(L)$ are the coefficients of the basic CH polynomial
(\ref{CH1}).
Let us define the monic polynomial
\begin{equation}
{\cal CH}_{(m)}(x) = \prod_{k\vdash m}(x-\mu_\mathbf{k}(m)).
\label{pol-form}
\end{equation}
Since its coefficients are symmetric functions in $\mu_i$ they can
be expressed via the elementary symmetric functions. By virtue of
(\ref{elsym}) we conclude that the coefficients of ${\cal CH}_{(m)}$
belongs to $Z(U(\mbox{$\frak g$}_{\hbar}))$. Besides,
$\deg {\cal CH}_{(m)}(x) = n_m=\dimV_{(m)}$.
\begin{proposition}
\label{prop:3}
The polynomial ${\cal CH}_{(m)}(x)$ is a CH polynomial for ${ L}_{(m)}$, namely,
we have
$$
{\cal CH}_{(m)}({ L}_{(m)})\equiv 0.
$$
\end{proposition}
When switching to the 1-generic orbit ${\cal L}_{\h}^{\chi}$ (\ref{alg}), we fix the
eigenvalues $\mu_i$ of the matrix $L$ (see (\ref{gen})) and thereby
fix the values of the quantities $\mu_\mathbf{k}(m)$. Similarly to the case
$m=1$, the quantities $\mu_\mathbf{k}(m)$ are called {\it eigenvalues} of
the matrix ${ L}_{(m)}$ {\it on the orbit ${\cal L}_{\h}^{\chi}$}. Besides, for the sake of
uniformity, we put $\mu_\mathbf{k}(1)=\mu_i$ for $|\mathbf{k}|=1$ and
$k_j= \delta_{i,j}$.
\medskip \noindent
{\bf Proof\ \ } In what follows, besides the representations
$\pi_\lambda$, we also need the right representations $\overline \pi_\lambda$. The
representation $\overline \pi_\lambda$ is defined in the space $V_\lambda^{\!*}$ dual
to $V_\lambda$
$$
\langle X\triangleleft\overline \pi_\lambda(a), Y\rangle=\langle X,\pi_\lambda(a)
\triangleright Y \rangle,\quad X\in V_\lambda^{\!*},\; Y\in V_\lambda,\;
a\in U(\gg_{\h}).
$$
Hereafter, $\triangleleft$ (resp., $\triangleright$) stands for the
right (resp., left) action of a given operator on a given vector.
Replacing the operators $\overline \pi_\lambda(a)$ by $-\overline \pi_\lambda(a)^*$ where $^*$
is an involution, we get a left representation $\pi_{\lambda^{\!*}}$ of
the algebra $U(\gg_{\h})$ which is called contragradient to $\pi_\lambda$ and
is labelled by the signature $\lambda^{\!*}$ defined in
(\ref{lambda-star}) (cf. \cite{W}).
The idea of our proof consists in the following. To verify a
relation in the algebra $U(\gg_{\h})$, we consider the image of this
relations in an arbitrary representation $\overline \pi_\lambda$ and prove that
this relation is true in all such representations. Then, since any
element of the enveloping algebra with trivial image in any
representation $\overline \pi_\lambda$ is trivial (cf. \cite{D}), we conclude
that our relation is valid in the algebra itself. Moreover, it
suffices to check the relation in question for ${\hbar}= 1$, since by
rescaling we can pass to an arbitrary ${\hbar}\not=0$.
Let us introduce the following notation:
$$
{ L}_{(\lambda,m)}=\overline \pi_\lambda(L_{(m)}),
$$
where applying a representation $\overline \pi_\lambda$ to a matrix means
applying it to any entry of the matrix in question. Thus,
$\overline \pi_\lambda({ L}_{(m)})$ is a matrix with entries from ${\rm End\, }(V_{\lambda})$ which in
turn is identified with ${\rm Mat }_{n_\lambda}({\Bbb K})$.
So, given a fixed $m$ and an arbitrary $\lambda$, we should prove
the identity ${\cal CH}_{(m)}({ L}_{(\lambda,m)})\equiv 0$ with the polynomial
${\cal CH}_{(m)}(x)$ defined in (\ref{pol-form}).
For a fixed $m$, let us first consider a subset of representations
$\overline \pi_\lambda$ corresponding to signatures
$\lambda=(\lambda_1,\lambda_2,\dots ,\lambda_n)$ such that
\begin{equation}
\lambda_i- \lambda_{i+1}\geq m\quad {\rm for}\quad 1\le i\le n-1.
\label{proper}
\end{equation}
For such signatures, irreducible components in the tensor product
$V_\lambda^*\otV_{(m)}$ are of the form $V_{\lambda^{\!*}+\mathbf{k}}$, where $\mathbf{k}$ runs
over the full set of all possible partitions (\ref{part}). The sum
$\lambda+\mathbf{k}$ means the signature with the components
\begin{equation}
(\lambda+\mathbf{k})_i = \lambda_i+k_i.
\label{sum-lk}
\end{equation}
Since all $k_i\le m$, then, due to restriction (\ref{proper}), we
have $(\lambda^{\!*}+k)_{i}\geq (\lambda^{\!*}+k)_{i+1}$ and the space
$V_{\lambda^{\!*}+\mathbf{k}}$ is well-defined.
Now, assuming $\lambda$ to satisfy (\ref{proper}), we use the split
Casimir element (\ref{cas-kl}) in order to calculate the eigenvalues
of the matrices $L_{(\lambda, 1)}$ and ${ L}_{(\lambda,m)}$. With this purpose we
insert the split Casimir ${\bf Cas}\in U(\mbox{$\frak g$})\otimes U(\mbox{$\frak g$})$ between the
factors of the tensor product $V_\lambda\otimes V_{(m)}$ and consider the image
\begin{equation}
(\overline \pi_\lambda\otimes \pi_{(m)})(V_\lambda^*\otimes{\bf Cas}\otimes V_{(m)})
\stackrel{\mbox{\tiny def}}
{=}
\sum_{1\le i,j\le n} (V_\lambda^* \triangleleft \overline \pi_\lambda(l_i^j))
\otimes (\pi_{(m)}(l_j^i) \trianglerightV_{(m)}).
\label{cas-lr}
\end{equation}
Thus, we have represented ${\bf Cas}$ as a linear operator ${\bf Cas}_{(\lambda,m)}$ in
the space $V_\lambda^*\otimes V_{(m)}$
$$
{\bf Cas}_{(\lambda,m)}:\quad V_\lambda^*\otimes V_{(m)}\to V_\lambda^*\otimes V_{(m)}.
$$
This operator commutes with the action of the group $GL(n)$ and
therefore it is scalar on each irreducible component
$V_{\lambda^{\!*}+\mathbf{k}}\subset V_\lambda^*\otimes V_{(m)} $. Since in an appropriate
basis of $V_\lambda^*\otimes V_{(m)}$ the matrix of the operator ${\bf Cas}_{(\lambda,m)}$
coincides with ${ L}_{(\lambda,m)}$, then the eigenvalues of the matrix
$L_{(\lambda, m)}$ are nothing but the eigenvalues of the operator
${\bf Cas}_{(\lambda,m)}$.
Let $\mu_\mathbf{k}(\lambda,m)$ be the eigenvalue of ${\bf Cas}_{(\lambda,m)}$ on the component
$V_{\lambda^{\!*}+\mathbf{k}}$ (if $m=|\mathbf{k}|=1,\,k_j=\delta_{i,j}$ we shall also
use the notation $\mu_i(\lambda)$ for the eigenvalues of
${\bf Cas}_{(\lambda,1)} = L_{(\lambda,1)}$). One can prove that
$$
\mu_\mathbf{k}(\lambda,m)=-{{1}\over{2}}(s_2(\lambda^{\!*}+\mathbf{k})-s_2(\lambda^{\!*})
-s_2((m))),
$$
where $s_2(\lambda)$ stands for the value of the quadratic Casimir
element on the irreducible module $V_{\lambda}$. This formula is an
immediate consequence of the Leibnitz rule (or in other words, the
coproduct in the algebra $U(\mbox{$\frak g$})$). The negative sign appears due to
the passage from the representation $\overline \pi_\lambda$ to $\pi_{\lambda^{\!*}}$.
The straightforward calculation gives
$$
s_2(\lambda)=\sum_{1\le i\le n} (\lambda_i^2+ \lambda_i(n+1-2i)).
$$
Therefore, for any $\lambda$ satisfying (\ref{proper}), we have
\begin{equation}
\mu_\mathbf{k}(\lambda, m)=-\sum_{i=1}^n k_i(-\lambda_{n-i+1}-i+1)+
\sum_{1\le i<j\le n} k_i k_j.
\label{muk}
\end{equation}
In particular, if $m=1$, we get the eigenvalues of $L_{(\lambda,1)}$
\begin{equation}
\mu_i(\lambda)=\lambda_{n-i+1}+i-1,\quad i=1,\dots ,n.
\label{mu}
\end{equation}
To pass to a generic ${\hbar}$, we should multiply all these eigenvalues
by ${\hbar}$. Finally, we conclude that for all ${\hbar}$ and the signatures
$\lambda$ satisfying (\ref{proper}) the eigenvalues of ${ L}_{(\lambda,m)}$ and
$L_{(\lambda,1)}$ are indeed connected by (\ref{high}):
\begin{equation}
\mu_\mathbf{k}(\lambda, m)=\sum_i k_i \mu_i(\lambda)+ {\hbar}\,\sum_{1\le i<j\le n}
k_i k_j.
\label{mu-lam}
\end{equation}
So, if we apply such a representation $\overline \pi_\lambda$ to the matrix
${\cal CH}_{(m)}({ L}_{(m)})$ we obtain 0. Now we must get rid of restriction
(\ref{proper}).
For this purpose, observe the following. Let $m=1$. Then once
$\lambda_i\geq \lambda_{i+1}+1$ the eigenvalues $\mu_i(\lambda)$ are given by
(\ref{mu}). However, this formula is valid even if the condition
(\ref{proper}) with $m=1$ is not fulfilled. Indeed, formula
(\ref{reduc}) below gives correct values of central elements
${\rm Tr} L^s_{(\lambda, 1)}$ if we assume the eigenvalues $\mu_i(\lambda)$ to be
always given by (\ref{mu}) without any restrictions on $\lambda$. It can
be verified through calculating ${\rm Tr} L^s_{(\lambda, m)}$ by other means.
For instance, it can be done as in \cite{BR}\footnote{\label{foot:2}
In order to compare the formula from \cite{BR} with our result it
suffices to invert the numbering of the eigenvalues $\mu_i$.} or
${\rm Tr} L^s_{(\lambda, 1)}$ can be obtained from the $q$-Newton identities
below. By inverting the Newton identities we can express the central
elements $\sigma_k(L)$ in terms of ${\rm Tr} L^s$. This implies that
formulae (\ref{mu}) are true for any representation $\overline \pi_\lambda$.
Now, we are able to complete the proof. Even if $\lambda$ does not
satisfy (\ref{proper}) for a given $m$, the eigenvalues of the split
Casimir ${\bf Cas}_{(\lambda,m)}$ form a subset of the set $\{\mu_\mathbf{k}(m)\}$ with
$\mu_\mathbf{k}(m)$ defined in (\ref{high}). So, if we apply the
representation $\overline \pi_\lambda$ to the matrix ${\cal CH}(L_{(m)})$ we get 0.
This implies the statement. \hfill\rule{6.5pt}{6.5pt}
\begin{remark} \label{DMO}
{\rm
As we have seen above, the degree of the polynomial ${\cal CH}_{(\lambda,1)}$
is always equal to $n$, even though the number of the eigenvalues of
the split Casimir ${\bf Cas}_{(\lambda,1)}$ can be smaller. This means that
not all eigenvalues $\mu_i(\lambda)$ of the matrix $L_{(\lambda,1)}$ can be
found from the split Casimir but only those appearing in the minimal
polynomial (see remark \ref{DM}). However, formula (\ref{reduc})
below is always true since extra eigenvalues which do not come in
the split Casimir spectrum have vanishing coefficients in the r.h.s.
of (\ref{reduc}).}
\end{remark}
\begin{definition}
For a given natural $m$, a fuzzy orbit will be called $m$-generic if
it is 1-generic and the eigenvalues $\mu_\mathbf{k}(m)$ are simple. A fuzzy
orbit will be called generic if it is $m$-generic for any $m$.
\end{definition}
For any $m$-generic fuzzy orbit, we define
$n_m={\rm dim} V_{(m)}=\left(\matrix{n+m\cr m}\right)$ idempotents in the
usual way:
\begin{equation}
e_\mathbf{k}(m)=\prod_{\mathbf{k}'\not=\mathbf{k}}{\frac{(L_{(m)}-\mu_{\mathbf{k}'}(m)I)}
{(\mu_\mathbf{k}(m)-
\mu_{\mathbf{k}'} (m))}}.
\label{high-idemp}
\end{equation}
The following proposition is an easy corollary of the CH identity
${\cal CH^{\chi}}_{(m)}({ L}_{(m)})=0$ and it is treated as an NC version of the
spectral decomposition.
\begin{proposition} \label{mult}
For any $m$-generic fuzzy orbit ${\cal L}_{\h}^{\chi}$ characterized by eigenvalues
$\mu_i,\,\,1\le i\le n,$ there exist quantities $d_\mathbf{k}(m)$ such that
$$
\TrL_{(m)}^s=\sum_{|\mathbf{k}|=m}\mu_\mathbf{k}(m)^s\, d_\mathbf{k}(m), \quad s=1,2,\dots,
$$
where $\mu_\mathbf{k}(m)$ are given by (\ref{high}). Moreover, we have
$$
d_\mathbf{k}(m)={\rm Tr} e_\mathbf{k}(m).
$$
\end{proposition}
The quantities $d_\mathbf{k}(m)$ will be called {\it the quantum
multiplicities}. Their precise values on fuzzy orbits are given by
the following proposition.
\begin{proposition}
\label{prop:6}
For any $m$-generic orbit ${\cal L}_{\h}^{\chi}$ characterized by eigenvalues
$\mu_i,\,\,1\le i\le n$ we have
\begin{equation}
d_\mathbf{k}(m)=\prod_{1\le i<j\le n}{{\mu_i-\mu_j-(k_i-k_j){\hbar}} \over
{\mu_i-\mu_j}}. \label{razmer}
\end{equation}
Equivalently, the following "higher Newton identities" are valid
\begin{equation}
{\rm Tr} { L}_{(m)}^s=\sum_{|\mathbf{k}|=m}(\sum_{i=1}^n k_i\mu_i+{\hbar}\sum_{1\le i<j\le
n}
k_ik_j)^s \prod_{1\le i<j\le n}{{\mu_i-\mu_j-(k_i-k_j){\hbar}}
\over{\mu_i-\mu_j}}.
\label{princip}
\end{equation}
\end{proposition}
\medskip\noindent
{\bf Proof\ \ } The proof is based on the following observation. It
is not difficult to see that the quantities ${\rm Tr} L_{(m)}^s$ in
(\ref{princip}) are (symmetric) polynomials in $\mu_i$. To define a
polynomial unambiguously, it suffices to fix it at a finite set of
values of its arguments.
As such a set we take the values $\mu_i(\lambda)$ (\ref{mu})
connected with the finite dimensional representations
parametrized by signatures $\lambda$. However, a subtle point here
is that not any representation $\overline \pi_\lambda$ is admissible for our
purpose. In the proof we use the split Casimir element, therefore we
should take a signature $\lambda$ in such a way that the
corresponding eigenvalues $\mu_\mathbf{k}(\lambda,m)$ (\ref{mu-lam}) would
be all present in the spectrum of ${\bf Cas}_{(\lambda,m)}$ and, besides, they would
be all distinct. A simple analysis of the structure of
$\mu_i(\lambda)$ and $\mu_\mathbf{k}(\lambda,m)$ shows that these conditions
can always be met if the differences $\lambda_i-\lambda_{i+1}$ are
sufficiently large. Let us introduce the corresponding notion.
\begin{definition}
\label{def:m-ad}
A representation $\overline \pi_\lambda$ (and the corresponding signature $\lambda$)
will be called $m$-ad\-mis\-sible if condition (\ref{proper}) is
fulfilled and the eigenvalues $\mu_\mathbf{k}(\lambda,m)$ (\ref{mu-lam}) are all
distinct.
\end{definition}
So, in an $m$-admissible representation $\overline \pi_\lambda$ we have
\begin{equation}
\overline \pi_\lambda({\rm Tr} L_{(m)}^s)=\sum_{|\mathbf{k}|=m} \mu_\mathbf{k}(\lambda,m)^s d_\mathbf{k}(\lambda,m)
\,{\rm Id}_{V_\lambda^{\!*}}
\label{trlmk}
\end{equation}
with some multiplicities $d_\mathbf{k}(\lambda,m)$. The identity operator in the
right hand side appears since ${\rm Tr} L_{(m)}^s\in Z(U(\mbox{$\frak g$}_\hbar))$ and
$\overline \pi_\lambda$ is an irreducible representation.
Now, on the one hand, the multiplicities $d_\mathbf{k}(\lambda,m)$ should be
specializations of $d_\mathbf{k}(m)$ (\ref{razmer}) at
$\mu_i = \mu_i(\lambda)$. On the other hand, $d_\mathbf{k}(\lambda,m)$ can be
computed independently by means of the split Casimir element. To
prove the proposition, we have to show that these two ways of
computation give the same results.
So, we proceed in the way analogous to proof of proposition
\ref{prop:3}. We apply the representation $\overline \pi_\lambda$ to the
matrix $L^s_{(m)}$ passing thereby to $L^s_{(\lambda,m)}$. Then, the
transposed matrix $(L^s_{(\lambda,m)})^t$ is nothing but the matrix
of ${\bf Cas}_{(\lambda,m)}^s$, i.e., the $s$-th power of the linear operator
(\ref{cas-lr}) constructed via the split Casimir element (for
detail, see \cite{GLS2}).
Applying the transposition operator to the CH identities ${\cal CH^{\chi}}_{(m)}
(L_{(\lambda,m)})=0$ where $\chi$ is fixed by the values of $\mu_i$
(\ref{mu}), we conclude that the set of eigenvalues of the matrix
$L_{(\lambda,m)}$ coincides with that of the operator ${\bf Cas}_{(\lambda,m)}$.
Since the operator ${\bf Cas}_{(\lambda,m)}$ is scalar on irreducible components
$V_{\lambda^{\!*}+\mathbf{k}}$, we get
\begin{equation}
{\bf Tr}\,{\bf Cas}_{(\lambda,m)}^s=\sum_{|\mathbf{k}|=m} \mu_\mathbf{k}(\lambda,m)^s{\rm dim} V_{\lambda^{\!*}+\mathbf{k}}.
\label{altern}
\end{equation}
Here ${\bf Tr}={\rm Tr}\otimes{\rm Tr}:{\rm End\, }(V_{\lambda})\otimes{\rm End\, }(V_{(m)})\to{\Bbb K}$. Taking the traces
in (\ref{trlmk}) and comparing the result with (\ref{altern}) we
conclude that
\begin{equation}
d_\mathbf{k}(\lambda,m) = \frac{{\rm dim} V_{\lambda^{\!*}+\mathbf{k}}}{{\rm dim} V_{\lambda^{\!*}}}.
\label{dk}
\end{equation}
At last, using the well known Frobenius formula
$$
{\rm dim} V_{\lambda} = \prod_{i<j} {(\lambda_i-\lambda_j-i+j)\over (j-i)}
$$
and taking into account (\ref{mu}), we find that $d_\mathbf{k}(\lambda,m)$ are
indeed given by (\ref{razmer}) with the substitution
$\mu_i = \mu_i(\lambda)$ and ${\hbar}=1$. The passage to a generic ${\hbar}\not=0$
is standard.
So, formulae (\ref{razmer}) and (\ref{princip}) have been proved for
any $m$-admissible representation. But since the family of
$m$-admissible signatures $\lambda$ is a large enough (actually
infinite) set, then by virtue of reasons discussed at the beginning
of the proof we conclude that (\ref{princip}) is true in general.
\hfill\rule{6.5pt}{6.5pt}
\medskip
If $m=1$, then (\ref{princip}) reduces to
\begin{equation}
{\rm Tr} L^s=\sum_{j=1}^n \mu_j^s \prod_{i\not=j}
\frac{\mu_j-\mu_i-{\hbar}}{\mu_j-\mu_i}.
\label{reduc}
\end{equation}
As we have said above, at ${\hbar}=1$ and $\mu_i=\mu_i(\lambda)$ this formula
is equivalent to that from \cite{BR}.
Formula (\ref{reduc}) together with (\ref{elsym}) can be treated as
an $n$-parametric resolution of the Newton relations in the algebra
$U(\gg_{\h})$. Indeed, the right hand side of (\ref{reduc}) is a
symmetric polynomial in $\mu_i$, and therefore it can be expressed
as a polynomial in elementary symmetric functions in $\mu_i$. Upon
replacing these functions by $\sigma_k$, we find explicit relations
between two central families $\{\sigma_k\}$ and $\{s_k={\rm Tr} L^k\}$ in
the algebra $U(\gg_{\h})$. The variables $\mu_i$ can be thought of as the
elements of the algebraic closure of the center of $U(\gg_{\h})$.
In the same sense we consider (\ref{princip}) as the higher order
counterparts of the Newton relations. In section \ref{sec:6} we
present the $q$-analogs of these relations.
\section{Reflection Equation Algebra: definition and basic
properties}
\label{sec:rea-def}
This section is devoted to the detailed description of the Hecke
symmetries $R$ and corresponding reflection equation algebras which
were shortly outlined in the Introduction.
Let $V$ be a vector space over ${\Bbb K}$, ${\rm dim}\, V=n$ and let
$R \in {\rm End\, }(V^{\ot 2})$ be an endomorphism. We call $R$ {\em braiding} if
it satisfies the {\it Yang-Baxter equation}
\begin{equation}
R_{12}R_{23}R_{12} = R_{23}R_{12}R_{23},
\label{YBE}
\end{equation}
where $R_{12}=R\otimes {\rm id}_V$ and $R_{23}={\rm id}_V\otimes R$ are
treated as elements of ${\rm End\, }(V^{\otimes 3})$. On fixing a basis
$\{x_i\}\in V$, $1\le i \le n$, in the space $V$, we can realize the
endomorphism $R$ as a numerical $n^2\times n^2$ matrix $R$ for which
we use the same notation and call it {\em $R$-matrix}:
\begin{equation}
(x_i\otimes x_j)\triangleleft R=R^{\,kl}_{ij}\, x_k\otimes x_l.
\label{R-matr}
\end{equation}
In what follows we shall deal with a special case of {\it Hecke
type} $R$-matrices satisfying the four additional conditions listed
below under the items {\bf C1) -- C4)}.
\medskip
\noindent{\bf C1)}\
First of all, the $R$-matrix should obey the {\it Hecke condition}
\begin{equation}
(R-qI)(R+q^{-1}I) = 0,
\label{Hecke}
\end{equation}
where $q$ is a fixed nonzero number from the ground field $\Bbb K$
with the only constraint
\begin{equation}
q^m\not=1, \quad\forall\,m\in{\Bbb N}.
\label{generic}
\end{equation}
As a consequence, the $q$-analogs of all integers are nonzero
\begin{equation}
m_q\equiv\frac{q^m-q^{-m}}{q-q^{-1}}\not=0,\quad \forall\, m\in
{\Bbb Z}.
\label{q-num}
\end{equation}
In some cases it proves to be convenient to consider $q$ as a formal
parameter and extend $\Bbb K$ to the field of rational functions in
the indeterminate $q$. We shall always bear in mind this extension
when considering the classical limit $q\rightarrow 1$.
\medskip
\noindent{\bf C2)}\
To formulate the further restriction on $R$, we need to recall some
properties of the Hecke algebras of $A_{m-1}$ series and to describe
their relation to the Hecke type $R$-matrices.
Fix a nonzero number $q\in \Bbb K$. The {\it Hecke algebra of
$A_{m-1}$ series} ($m\ge 2$) is an associative algebra $H_m(q)$ over
the field $\Bbb K$ generated by the unit element $1_{H}$ and $m-1$
generators $\sigma_i$ subject to the following relations:
$$
\left.
\begin{array}{l}
\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}\\
\rule{0pt}{5mm}
\sigma_i\sigma_j = \sigma_j\sigma_i\qquad\qquad {\rm if\ }|i-j|\ge
2\\
\rule{0pt}{5mm}
(\sigma_i-q\,1_{H})(\sigma_i+q^{-1}\,1_{H}) = 0\\
\end{array}
\right\}\quad i=1,2,\dots m-1.
$$
If the parameter $q$ satisfies (\ref{generic}), then for any
positive integer $m$ the Hecke algebra $H_m(q)$ is isomorphic to the
group algebra ${\Bbb K}S_m$ of the $m$-th order permutation group
$S_m$. As a consequence, $H_m(q)$ is isomorphic to the following
direct sum of matrix algebras
\begin{equation}
H_m(q)\cong \bigoplus_{\lambda\vdash m}{\rm Mat}_{d_\lambda}(\Bbb
K)
\label{H-iso-Mat}
\end{equation}
where the summation goes over all possible ordered partitions
$\lambda$ of the integer $m$. The parameter $d_\lambda$ is equal to
the number of all standard Young tableaux $\lambda(a)$ which can be
constructed for the given partition $\lambda$.
As is known, the associative $\Bbb K$-algebra ${\rm Mat}_k(\Bbb K)$
possesses the linear basis of $k^2$ generators (matrix units)
$E_{ab}$ $1\le a,b \le k$ with the multiplication law
$$
E_{ab}E_{cd} = \delta_{bc}E_{ad}.
$$
Due to isomorphism (\ref{H-iso-Mat}) in the Hecke algebra $H_m(q)$
one can choose a system of generators
${\cal Y}^\lambda_{ab}(\sigma) \in H_m(q)$, $\lambda\vdash m$,
$1\le a,b \le d_\lambda$, which form a linear basis in $H_m(q)$ and
satisfy the following multiplication rule
\begin{equation}
{\cal Y}^\lambda_{ab}(\sigma){\cal Y}^\mu_{cd}(\sigma) =
\delta^{\lambda\mu}\delta_{bc} {\cal Y}^\lambda_{ad}(\sigma).
\label{Y-Y}
\end{equation}
The diagonal "matrix units" ${\cal Y}^\lambda_{aa}(\sigma)$ are the
primitive idempotents of the Hecke algebra. Evidently, they are in
one-to-one correspondence with the set of standard Young tableaux
$\lambda(a)$ constructed for the partition $\lambda$. Below we shall
use the compact notation
${\cal Y}^\lambda_{aa}\equiv{\cal Y}_{\lambda(a)}$ for the primitive
idempotents. The idempotent, corresponding to the partition $(1^m)$
(the one-column Young diagram) will be called the
$q$-antisymmetrizer and denoted ${\cal A}^{(m)}(\sigma)$.
Given a Hecke type $R$-matrix, we can construct a {\em local
representation} of $H_m(q)$ in $V^{\otimes m}$ by the following
rule
\begin{equation}
\sigma_i\mapsto \rho_R(\sigma_i)= R_{ii+1} = I^{\otimes(i-1)}
\otimes
R\otimes I^{\otimes(m-i-1)}\in {\rm End}(V^{\otimes m}).
\label{loc-rep}
\end{equation}
In the local representation (\ref{loc-rep}) the idempotents
${\cal Y}_{\lambda(a)}(\sigma)$ are realized as some projection
operators in $V^{\otimes m}$. With respect to the action of these
projectors the space $V^{\otimes m}$ splits into the direct sum of
subspaces $V_{\lambda(a)}$:
\begin{equation}
V^{\otimes m} = \bigoplus_{\lambda\vdash m} \bigoplus_{a=1}
^{d_\lambda}
V_{\lambda(a)},\quad V_{\lambda(a)} = Y_{\lambda(a)}(R)
\triangleright
V^{\otimes m}. \label{razl-prostr}
\end{equation}
The projector $Y_{\lambda(a)}(R) = \rho_R({\cal Y}_{\lambda(a)})$ is
given by a polynomial in matrices $R_{ii+1}$. For a detailed
treatment of these questions, explicit formulae for $q$-projectors
and the extensive list of original papers, see \cite{OgP}.
So, we shall assume the Hecke symmetry in question to be even. By
definition this means that there exists an integer $p>0$ such that
the image of the $q$-an\-ti\-sym\-met\-ri\-zer
${\cal A}^{(p+1)}(\sigma)$ in the local $R$-matrix representation
$\rho_R$ identically vanishes while the image of the
$q$-antisymmetrizer ${\cal A}^{(p)}(\sigma)$ is a projector of the
unit rank in the space $V^{\otimes m}$ (for any $m>p$)
\begin{equation}
\exists\,p\in{\Bbb N}:\quad
\left\{
\begin{array}{l}
{\cal A}^{(p+1)}(\sigma)\mapsto A^{(p+1)}(R)\equiv 0,\\
\rule{0pt}{6mm} {\cal A}^{(p)}(\sigma)\mapsto A^{(p)}(R),
\quad {\rm rank}\,A^{(p)}(R) = 1.
\end{array}
\right.
\label{f-rank}
\end{equation}
Such a number $p$ will be called {\it the symmetry rank} of the
matrix $R$. For example, the symmetry rank of the $R$-matrix
connected with the quantum universal enveloping algebra $U_q(sl(n))$
is equal to $n$. Examples of $n^2\times n^2$ $R$-matrices with $p<n$
(for $n\geq 3$) can be found in \cite{G}.
\medskip
\noindent{\bf C3)}
Besides, we assume the $R$-matrix to be {\it skew-invertible}; that
is there exists an $n^2\times n^2$ matrix $\Psi$ with the property
$$
\sum_{a,b}R_{ia}^{\;jb}\Psi_{bk}^{\;as} = \delta_i^{\,s}
\delta_k^{\,j} =
\sum_{a,b} \Psi_{ia}^{\;jb}R_{bk}^{\;as}.
$$
In the compact notations the above formula reads
\begin{equation}
{\rm Tr}_{(2)}R_{12}\Psi_{23} = P_{13} = {\rm Tr}_{(2)}\Psi_{12}R_{23},
\label{closed}
\end{equation}
where the symbol ${\rm Tr}_{(2)}$ means applying the trace in the second
space and $P$ is the permutation matrix. Note, that this condition
does not depend on a choice of basis, therefore we can treat $\Psi$
as an endomorphism as well.
Let now $B$ and $C$ be two endomorphisms of the space V represented
by the following $n\times n$ matrices
\begin{equation}
B = {\rm Tr}_{(2)}\Psi_{21}, \quad C = {\rm Tr}_{(2)}\Psi_{12},
\label{BC}
\end{equation}
where $\Psi$ is defined in (\ref{closed}). If the $R$-matrix has the
symmetry rank $p$, we have
\begin{equation}
B\cdot C = \frac{1}{q^{2p}}\,I
\label{nonsing}
\end{equation}
(cf. \cite{G}). This implies that the matrices $B$ and $C$ are
invertible. Moreover, we have
\begin{equation}
{\rm Tr} B={\rm Tr} C=\frac{p_q}{q^p}.
\label{BC-norm}
\end{equation}
Note that {\bf C3)} is in fact a consequence of {\bf C2)} since any
even Hecke symmetry is skew-invertible automatically.
\medskip
\noindent{\bf C4)}\
As the last requirement on the $R$-matrix we shall assume, that
the space $V^*$ dual to $V$ can be identified with the $(p-1)$-th
wedge $q$-power of $V$
$$
V^* = \wedge^{p-1}_q V.
$$
Explicitly this identification is constructed as follows. Since
$A^{(p)}(R)$ is the unit rank projector then in the basis
$x_{i_1}\otimes\dots \otimes x_{i_p}$ (see (\ref{R-matr})) its
matrix can be represented in the form
$$
{A^{(p)}}_{i_1i_2\dots i_p}^{\;\;j_1j_2\dots j_p}
= u_{i_1i_2\dots i_p}v^{j_1j_2\dots j_p}
$$
with some structure tensors $u$ and $v$. One can show that the
vectors
\begin{equation}
x^i = v^{ia_2\dots a_p}x_{a_2}\otimes \dots \otimes x_{i_p}
\label{dual-basis}
\end{equation}
are linear independent and by definition form the basis of the space
$\wedge^{p-1}_q V\subset V^{\otimes (p-1)}$. We shall not explicitly
describe the property of $R$ which allows us to identify $V^*$ and
$\wedge^{p-1}_q V$. The thorough treatment of this problem is
presented in \cite{GLS1}.
In the particular case $p=2$ which will be studied in detail below
the components of structure tensor $v^{ij}$ form a nondegenerated
matrix \cite{G}. Therefore, as follows from (\ref{dual-basis}), the
space $V^*$ is isomorphic to $V$ itself
\begin{equation}
V^*\cong V\quad {\rm at}\quad p=2.
\label{v-dua-v}
\end{equation}
\medskip
As the next step, we define the reflection equation algebra.
Consider a unital associative ${\Bbb K}$-algebra $\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi$ generated by $n^2$
elements $l_i^{\,j}$, $1\le i,j\le n$ satisfying the following
relations
\begin{equation}
R L_1R L_1 - L_1R L_1R = \hbar(RL_1
- L_1R), \quad L_1\equiv L\otimes I,
\label{RE}
\end{equation}
where $\hbar$ is a numerical parameter and $ L =\|l_i^{\,j}\|$ is a
matrix composed of $l_i^{\,j}$. The algebra $\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi$ is said to be the
{\it modified reflection equation algebra} (mREA). In the particular
case ${\hbar}= 0$ this algebra will be called (non-modified) REA and
denoted ${\cal L}_q$.
\begin{remark}\label{rem-1}
{\rm Similarly to the algebra $U(\gg_{\h})$ the mREA corresponding to
different {\it nonzero} parameters $\hbar$ are isomorphic --- one
can easily pass from one $\hbar\not=0$ to another $\hbar'\not =0$ by
a trivial renormalization of generators.
Moreover, at $q\not=\pm 1$ the generators $l_i^{\,j}$ of the
algebra \ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi{} is connected with the generators $\hat l_i^{\,j}$
of ${\cal L}_q$ via a linear shift by the unit element ${\rm id}_{\cal L}$
\begin{equation}
l_i^{\,j} = \hat l_i^{\,j} + \frac{\hbar}{\zeta} \,\delta_i^{\,j}\,
{\rm id}_{\cal L},\qquad \zeta = q-q^{-1},
\label{shift}
\end{equation}
and therefore these two kinds of REA are isomorphic, too.
Nevertheless, their classical limits are different, since the above
isomorphism is broken as $q\rightarrow 1$.}
\end{remark}
Let us now define an important map ${\rm Tr}_R:{\rm Mat}_n(\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi)
\rightarrow \ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi$
\begin{equation}
{\rm Tr}_R(X)\stackrel{\mbox{\tiny def}}{=} {\rm Tr} (C\cdot X),\quad X\in
{\rm
Mat}_n(\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi).
\label{q-sled}
\end{equation}
For the $U_q(sl(n))$ $R$-matrix, such a map is called the
{\it quantum trace} \cite{FRT} and is often denoted ${\rm Tr}_q$. Note
that in the sequel we apply the trace (\ref{q-sled}) only to the
matrices from the space ${\rm Mat}_n(\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi)$ which are in some sense
invariant.
Similarly to the classical case, a generating set for the center of
the algebra \ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi{} consists of the elements:
\begin{equation}
s_0\equiv {\rm id}_{\cal L},\quad
s_k = q\,{\rm Tr}_RL^k,\quad 1\le k\le p,
\end{equation}
where the factor $q$ is chosen for the future convenience. These
elements become scalar operators on each irreducible module over
the mREA (see section \ref{sec:rea-br}).
One of the basic properties of the REA (modified or not) consists in
the following. In this algebra, similarly to $U(su(n))$ (and to some
other Lie algebras, cf. \cite{Go}), one can find a series of
the Cayley-Hamilton identities. Let us discuss the first of them,
which will be called the {\it basic CH identity}.
As was shown in \cite{GPS}, the matrix $\hat L=\|\hat l_i^{\,j}\|$
of generators of REA satisfies the CH identity of the form
\begin{equation}
\sum_{k=0}^p(-\hat
L)^{p-k}\sigma_k(\hat L) = 0,\quad \sigma_0(\hat L) \equiv
{\rm id}_{\cal L},\quad \hat L^0\equiv I
\label{CH}
\end{equation}
where $\{\sigma_k\}$ is another set of generators of the center of
REA connected with the set $\{s_k\}$ by means of the $q$-Newton
relations (\ref{NI}) and $p$ is the symmetry rank of the
corresponding $R$.
Applying the shift (\ref{shift}) it is possible to get analogous
relations for the matrix $L$ with entries belonging to the
corresponding mREA:
$$
\sum_{k=0}^p(-L)^{p-k}\sigma_k(L) = 0,\qquad \sigma_0(L)
\equiv {\rm id}_{\cal L}.
$$
As above the coefficients $\sigma_k(L)$ are central element of the
algebra $\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi$.
An explicit form of these coefficients was obtained in \cite{GS1}.
However, this form is somewhat cumbersome. Here we only present an
example of such a CH identity for the mREA algebra related to a
Hecke symmetry with the symmetry rank $p=2$:
\begin{equation}
L^2 - \Big(q{\rm Tr}_R L +\frac{\hbar}{q}\Big)\,L+\Big(\frac{q^2}{2_q}
\Big(q({\rm Tr}_RL)^2 - {\rm Tr}_RL^2\Big)+ \hbar\,\frac{q}{2_q}\,
{\rm Tr}_R L \Big)\,I = 0.
\end{equation}
In this formula the coefficients $\sigma_i(L)$ are expressed via the
quantities ${\rm Tr}_R(L^k)$. In what follows we will find some inverse
relations expressing the latter quantities via $\sigma_i(L)$ but in
a parametric form.
By analogy with the case of fuzzy orbit (see definition
\ref{def:fuz-orb} and formula (\ref{gen})) we define a character
$\chi: Z(\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi) \to {\Bbb K}$ of the center of the mREA \ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi{} by
fixing its values on the central elements $\sigma_k$
\begin{equation}
\chi(\sigma_k(L)) = \alpha_k = \sum_{1\le i_1<\dots <
i_k\le p}\mu_{i_1} \dots \mu_{i_k},
\label{q-gen}
\end{equation}
where the numbers $\mu_i$ $1\le i\le p$ are assumed to be all
distinct.
Then we define a {\it 1-generic NC orbit} ${\cal L}_{\h,q}^{\chi}$ as the quotient
of the mREA (\ref{RE}) modulo the two sided ideal ${\cal I}^\chi$
\begin{equation}
{\cal L}_{\h,q}^{\chi} = \ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi/{\cal I}^\chi
\label{q-orb}
\end{equation}
where ${\cal I}^\chi$ is generated by the set of relations
\begin{equation}
\sigma_k(L)-\alpha_k, \quad {\rm for } \quad 1\le k
\le p.
\label{ideal}
\end{equation}
Finally, we will compute the quantities ${\rm Tr}_R L^k$ restricted on
the NC orbit ${\cal L}_{\h,q}^{\chi}$ in terms of $\mu_i$.
Following the pattern of the case considered in section
\ref{sec:fuzzy} we will also compute the quantities
${\rm Tr}_R L^k_{(m)}$ for some higher extensions $L_{(m)}$ of the
matrix $L$. The crucial role in this computing is played by a split
Casimir element which is defined for the mREA in the following way
\begin{equation}
{\bf Cas} = q^{2p}\,l_i^{\,k}\otimes l_k^{\,j}C_j^{\,i}\;\in
\;\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi\otimes \ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi.
\label{quant-Cas}
\end{equation}
As in the classical case considered in section \ref{sec:fuzzy}, this
element allows us to introduce the aforementioned extensions
$L_{(m)}$ of the matrix $L$ and to construct the higher CH
identities for them. Also note, that in the $U_q(sl(n))$ case at the limit
$q=1$ we get just the split Casimir element (\ref{cas-kl})
described above.
\section{mREA as a braided enveloping algebra}
\label{sec:rea-br}
In this section we give a short review of the representation theory
of REA which is necessary for the subsequent sections. Besides, we
adduce some arguments which allow us to consider the mREA as a
"braided" analog of the enveloping algebra.
First of these arguments originates from the consideration of the
mREA for an {\it involutive} $R$-matrix: $R^2=I$. This is a
particular case of the Hecke condition (\ref{Hecke}) corresponding
to $q=1$. The mREA (\ref{RE}) with involutive $R$-matrix turns out
to be the enveloping algebra of some generalized Lie algebra.
The notions of generalized Lie algebra and its enveloping algebra
were introduced in \cite{G} (see also the references therein). Given
an involutive $R$-matrix (treated as an endomorphism of
$V^{\otimes 2}$), the generalized enveloping algebra is defined as
the quotient of the free tensor algebra of the space ${\rm End\, }(V)$ over
the two-sided ideal generated by the relations
$$
X\otimes Y-R_{{\rm End\, }(V)}(X\otimes Y)=\circ X\otimes Y-\circ
R_{{\rm End\, }(V)}(X\otimes Y),\quad X,Y\in {\rm End\, }(V),
$$
where $\circ:{\rm End\, }(V)^{\otimes 2}\to {\rm End\, }(V)$ is the usual product in the
space of endomorphisms and $R_{{\rm End\, }(V)}$ is the extension of the
initial braiding $R$ to the space ${\rm End\, }(V)$ (which is well defined
as $R$ is skew-invertible).
If we realize ${\rm End\, }(V)$ as the space of left endomorphisms, fix a
natural basis $\{h_i^{\,j}\}$ such that $h_i^{\,j}\circ h_k^{\,l} =
\delta_k^j h_i^{\,l}$ and compute $R_{{\rm End\, }(V)}$ in this basis, we
recover the enveloping algebra of the generalized Lie algebra in
terms of generators $h_i^{\,j}$. Note, that $h_i^{\,j}$ can be
identified with $x_i\otimes x^j$ where $\{x^j\}$ is the basis of the
left dual space to $V$, see \cite{G} for detail.
The point is that in ${\rm End\, }(V)$ we can choose another basis
$\{l_i^{\,j}\}$ such that
$l_i^{\,j}\circ l_k^{\,l} = l_i^{\,l}\, B_k^j$, where $B_i^j$ is
the matrix element of the endomorphism $B$ (\ref{BC}) written in the
basis $\{x_i\}$. Being expressed in terms of the new generators
$l_i^{\,j}$, the enveloping algebra in question turns into the mREA
with ${\hbar}=1$. The elements $l_i^{\,j}$ can be identified with
$x_i\otimes{^j}x$ where $\{{^j}x\}$ is the basis of the right dual
space to $V$.) The details are left to the reader.
Let us also mention that in the $U_q(sl(n))$ case (in contrast with quantum
groups corresponding to other simple Lie algebras) the mREA is a
two parameter deformation of the symmetric algebra of $gl(n)$ (a
close treatment of this algebra is given in \cite{IP}). The
corresponding Poisson structure is the aforementioned pencil which,
in fact, is well defined on the whole $gl(n)^*$.
The most important property of the mREA which enable us to treat
this algebra as a braided analog of the enveloping algebra is that
the category of its equivariant finite dimensional representations
is close to that of $U(gl(p))$--Mod where $p$ is the symmetry rank
of the $R$-matrix.
Let us first consider the quotient of the mREA \ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi{} over the ideal
generated by the relation ${\rm Tr}_RL = 0$. We denote this quotient
algebra as ${\cal SL}_{q,{\hbar}}$. This is an analog of the $U(sl(p))$
subalgebra in the $U(gl(p))$. The category of finite dimensional
completely reducible modules over the ${\cal SL}_{q,{\hbar}}$ is the
{\it quasitensor Schur-Weyl category} introduced in \cite{GLS1}.
Its simple objects (irreducible modules) are labelled by the
partitions $\lambda$ whose height (the number of nonzero parts) are
not greater than $p-1$. Besides, the Grothendieck ring of the
Schur-Weyl category is isomorphic to that of the category
$U(sl(p))$--Mod.
The representations of \ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi{} are labelled by the partitions
$\lambda$ whose height is not greater than $p$ and a number
$z\in {\Bbb K}$ which is analog of the shift (\ref{ed-sdv}) of the
$U(gl(p))$ representations (see remark \ref{rem:14} below). In full
analogy with the classical case discussed in remark \ref{rem:3}, the
vector spaces $V_{\lambda,z}$ and $V_{\hat \lambda}$ are isomorphic
as ${\cal SL}_{q,{\hbar}}$ modules, $\hat\lambda$ being constructed from
$\lambda$ in accordance with (\ref{lambda-hat}).
The map ${\rm Tr}_R$ defined in (\ref{q-sled}) is closely related to
the {\em categorical trace}
$\mbox{\bf\sf Tr}_{V_\lambda}:{\rm End\, }(V_\lambda)\to{\Bbb K}$. The
categorical trace is a morphism of the Schur-Weyl category which
plays the same role as the usual trace does in the category
$U(sl(p))$--Mod. In particular, the categorical trace allows one
to define the notion of the {\it $q$-dimension}
\begin{equation}
{\rm dim}_qV_\lambda = \mbox{\bf\sf Tr}_{V_\lambda}
({\rm id}_{V_\lambda}),
\label{def:q-dim}
\end{equation}
which is a multiplicative-additive functional on the Grothendieck
ring of the Schur-Weyl category. Namely we have
\begin{equation}
\begin{array}{l}
{\rm dim}_q(V_\lambda\otimes V_\mu)={\rm dim}_q(V_\lambda)\,{\rm dim}_q(V_\mu),\\
\rule{0pt}{5mm}{\rm dim}_q(V_\lambda\oplus V_\mu) = {\rm dim}_q(V_\lambda)+
{\rm dim}_q(V_\mu)
\end{array}
\label{ma-func}
\end{equation}
and the $q$-dimensions of the isomorphic spaces are equal to each
other (see \cite{GLS1} for detail).
Now let us give a short review of some facts from the representation
theory of the mREA. Their detailed description can be found in
\cite{S}.
The mREA \ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi{} possesses a profound representation theory. We shall
confine ourselves to considering the {\it finite dimensional,
completely reducible, equivariant} modules over $\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi$ (the term
"equivariant" will be explained at the end of the section). Besides,
we can set $\hbar = 1$ by virtue of the isomorphisms mentioned in
Remark~\ref{rem-1}.
First of all, we define the so-called {\it left fundamental module
of B type}. Let $V$ be an $n$-dimensional vector space with a fixed
basis $\{x_i\}$ $1\le i\le n$. Putting ${\hbar}=1$, we consider the
homomorphism $\pi:{\cal L}_{1,q}\rightarrow {\rm End}_l(V)$ defined as follows
\begin{equation}
\pi(l_i^{\,j})\triangleright x_k = x_iB_k^{\,j},
\label{B-rep}
\end{equation}
where the matrix $B$ is introduced in (\ref{BC}). Since $B$ is
invertible (see (\ref{nonsing})) the representation $\pi$ is
irreducible.
The tensor power $V^{\otimes m}$, $m\in {\Bbb N}$, can be endowed
with the structure of a (reducible) mREA module. The corresponding
homomorphism $\rho_m:{\cal L}_{1,q}\rightarrow {\rm End}_l(V^{\otimes m})$
is of the form
\begin{equation}
\rho_m(l_i^{\;j}) = \pi_1(l_i^{\;j}) + {\cal R}_{12}^{-1}
\pi_1(l_i^{\;j}) {\cal R}_{12}^{-1} + \dots +
{\cal R}_{m-1,m}^{-1}\dots {\cal R}_{12}^{-1}\pi_1(l_i^{\;j})
{\cal R}_{12}^{-1}\dots {\cal R}_{m-1,m}^{-1},
\label{m-rep}
\end{equation}
where
$$
\pi_1 = \pi \otimes I^{\otimes (m-1)}, \quad
{\cal R}_{k,k+1} = I^{k-1}\otimes {\cal R} \otimes
I^{m-k-1}\quad 1\le k\le m-1.
$$
Here ${\cal R}$ is an automorphism of $V^{\otimes 2}$ connected with
the matrix $R$ of the Hecke symmetry by the following definition
\begin{equation}
{\cal R}\triangleright(x_i\otimes x_j) = \sum_{r,s}R_{ij}^{\;kl}
x_k\otimes x_l.
\label{R}
\end{equation}
The representation (\ref{m-rep}) is reducible. In accordance with
(\ref{razl-prostr}) it decomposes into the direct sum of mREA
submodules $V_{\lambda(a)}$, where $\lambda$ is an ordered partition
of $m$. The representations $\pi_{\lambda(a)}$ are extracted from
$\rho_m$ by the action of the corresponding projectors
$Y_{\lambda(a)}(R)$.
We write down the explicit form of the representation $\pi_{(m)}$,
corresponding to the partition $(m)$, since it will play an
important role in what follows. The subspace
$V_{(m)}\subset V^{\otimes m}$ is an image of the
{\it $q$-symmetrizer} $S^{(m)}({\cal R})$ whose matrix is
iteratively defined as follows (see \cite{G})
\begin{equation}
S^{(1)}\equiv I,\qquad
S^{(m)}_{12\dots m} = \frac{1}{m_q}\,S^{(m-1)}_{2\dots m}(q^{1-m}
I+(m-1)_q
R_{12})S^{(m-1)}_{2\dots m}.
\label{q-symm-S}
\end{equation}
The following proposition holds true \cite{S}.
\begin{proposition} \label{prop:pim}
Consider an arbitrary tensor power $V^{\otimes m}$ of the left
fundamental module $V$. Its $q$-symmetric subspace $V_{(m)}$ is a
left ${\cal L}_{1,q}$-submodule. On the generators of the mREA the
homomorphism $\pi_{(m)}:{\cal L}_{1,q} \rightarrow {\rm End}_l(V_{(m)})$ reads
as follows:
\begin{equation}
\pi_{(m)}(l_i^{\,j}) = q^{1-m}m_q\,S^{(m)}({\cal R})
\Big[\pi(l_i^{\,j})
\otimes I^{\otimes (m-1)}\Big]S^{(m)}({\cal R}),
\end{equation}
where $\pi$ and ${\cal R}$ are defined in (\ref{B-rep}) and
(\ref{R}) respectively.
\end{proposition}
Given the representation (\ref{B-rep}), we can realize the matrix
$L$ satisfying the commutation relations (\ref{RE}) as an image of
the split Casimir element {\bf Cas} (\ref{quant-Cas}) under the map
$$
{\rm id}\otimes \pi:\; \ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi\otimes{\cal L}_{1,q}\rightarrow \ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi\otimes
{\rm Mat}_n(\Bbb K).
$$
(Note that in the second factor we put ${\hbar}=1$.) Indeed, taking
(\ref{nonsing}) into account one easily gets
\begin{equation}
L^t = ({\rm id}\otimes \pi)({\bf Cas}),\quad L\in
{\rm Mat}_n(\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi).
\label{L-real}
\end{equation}
Hereafter $L^t$ stands for the matrix transposed to $L$.
By analogy with the $U(\mbox{$\frak g$})$ case considered in section
\ref{sec:fuzzy} we also introduce the symmetric matrix $L_{(m)}$ as
the following image of ${\bf Cas}$
\begin{equation}
L_{(m)}^t = ({\rm id}\otimes \pi_{(m)})({\bf Cas}),
\quad L_{(m)}\in {\rm Mat}_{n_m}(\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi)
\label{L-symm}
\end{equation}
where $n_m = {{\rm dim}}\,V_{(m)}$.
In addition to the left ${\cal L}_{1,q}$-modules we also need the right
ones. For a generic $p\geq 2$ such a representation can be defined
in the dual space $V^*$. By using the method of the paper \cite{S}
we can extend this representation to the tensor power
$(V^*)^{\otimes m}$ and decompose it into the direct sum of submodules
associated with the projectors $Y_\lambda$. However, if $p=2$ the
space $V^*$ can be identified with $V$ itself via a categorical
pairing arising from the projector of the space $V^{\otimes 2}$ onto its
skew-symmetric component (see \ref{v-dua-v}). The spaces
$V_{(k)}^*$ and $V_{(k)}$ can also be identified via such a pairing
(see \cite{GLS1} for detail). This pairing allows us to equip the
space $V_{(k)}$ with a structure of the right ${\cal L}_{1,q}$-module. More
precisely, the following proposition takes place.
\begin{proposition}\label{prop:r-m}
Let the symmetry rank of the Hecke symmetry $R$ is $p=2$. Consider
an arbitrary tensor power $V^{\otimes m}$ of the left fundamental
module $V$. Its $q$-symmetric subspace $V_{(m)}$ is a right
${\cal L}_{1,q}$-submodule. On the generators of the mREA the homomorphism
$\overline\pi_{(m)}:{\cal L}_{1,q} \rightarrow {\rm End}_r(V_{(m)})$ reads as
follows:
\begin{equation}
\overline\pi_{(m)}(l_i^{\,j}) = q^{1-m}m_q\,S^{(m)}(R)
\Big[I^{\otimes (m-1)}
\otimes \overline\pi(l_i^{\,j})\Big]S^{(m)}(R),
\end{equation}
where the homomorphism
$\overline\pi:{\cal L}_{1,q} \rightarrow {\rm End}_r(V)$ has the form
\begin{equation}
x_k\triangleleft \overline\pi(l_i^{\,j})=
\frac{2_q}{q^2} {A^{(2)}}_{ki}^{\;sj}x_s,\qquad
A^{(2)}\equiv \frac{1}{2_q}\,(qI - R),
\label{rep:pi-r}
\end{equation}
and the right action of $R$ is defined in (\ref{R-matr}).
\end{proposition}
In what follows we shall also need the representations of the
generators $\hat l_i^{\,j}$ of the REA connected with those of mREA
by shift (\ref{shift}). A simple calculation proves the following
corollary of the proposition \ref{prop:r-m}.
\begin{corollary}
\label{prop:rea-r}
The right representation of the generators $\hat l_i^{\,j}$ of the
REA in the $q$-symmetric component $V_{(m)}\subset V^{\otimes m}$ is
given by the homomorphism
\begin{equation}
\overline \pi_{(m)}(\hat l_i^{\,j}) =
q^{1-m}m_q\,S^{(m)}(R)\Big[I^{\otimes (m-1)} \otimes
\overline\pi(\hat l_i^{\,j})\Big]S^{(m)}(R),
\label{rep:pim-r}
\end{equation}
where
$$
x_k\triangleleft \overline\pi(\hat l_i^{\,j}) =
\Phi_{ki}^{\;sj}x_s ,\quad \Phi \equiv q^{1-m}m_q\,I -
\zeta\, \frac{2_q}{q^2}\,A^{(2)}.
$$
\end{corollary}
\begin{remark}\label{rem:14}
{\rm
Up to now, the representations of mREA were labelled by partitions
$\lambda$, whereas in general the finite dimensional representations
of $U(gl(n))$ are labelled by {\it signatures} $\lambda$
(\ref{signature}). What is an analog of these representations in the
mREA case? To answer the question, observe the following.
As was mentioned in remark \ref{rem:3}, with any finite dimensional
$U(gl(n))$ representation $\pi_\lambda$ labelled by a signature
$\lambda$ we can associate the representation $\pi_{\hat \lambda}$
where the partition $\hat \lambda$ is connected with the signature
by relation (\ref{lambda-hat}). These representations are connected
by the unit ope\-ra\-tor shift (\ref{ed-sdv}) and the
corresponding modules $V_\lambda$ and $V_{\hat \lambda}$ are
isomorphic as $U(sl(n))$-modules and therefore as vector spaces.
For the mREA representations there exists a transformation analogous
to (\ref{ed-sdv}). Namely, a simple calculation shows that if
$\pi_\lambda$ is a representation of mREA in the space $V_\lambda$
where $\lambda$ is a partition then the operators
\begin{equation}
\pi_\lambda^z(l_i^{\,j}) = z\,\pi_\lambda(l_i^{\,j}) +
\delta_i^{\,j}\, \frac{1-z}{\zeta}\,{\rm id}_{V_\lambda},
\quad z\in{\Bbb K}\backslash 0
\label{pi-z}
\end{equation}
also realize an mREA representation in the same space $V_\lambda$.
It can be shown that the representation of the subalgebra
${\cal SL}_{\hbar,q}$ which is a quotient of $\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi$ over the ideal
generated by the relation ${\rm T}_RL = 0$ does not change under the
shift (\ref{pi-z}). This subalgebra is an analog of $U(sl(n))$ in
the classical case. However, in contrast with the classical case
discussed in remark \ref{rem:3}, we cannot put any signature into
correspondence to $\pi_\lambda^z$ since our approach to the
representation theory of the mREA is not based on the technique of
the highest weight vectors.}
\end{remark}
Now we explain the meaning of the statement that the proposed
presentation theory of mREA is equivariant.
To any $R$-matrix we can assign an associative bialgebra $\cal T$
generated by the elements $t_i^{\,j}$ subject to the following
commutation relations
$$
R\, T_1 T_2 = T_1 T_2R, \quad {\rm where}\quad T=
\|t_i^{\,j}\|,\quad T_1 = T\otimes I,\quad T_2 = I\otimes T.
$$
If the $R$-matrix is skew-invertible (see (\ref{closed})), then
the bialgebra structure can be extended to the Hopf algebra
one\footnote{In order to get such a structure it suffices to
formally invert the quantum determinant in the extended algebra
$\cal T$. It can be done by an appropriate localization, cf.
\cite{G}.}. When $R$ is the image of the universal $U_q(sl(n))$
$R$-matrix (in the fundamental vector representation), the Hopf
algebra $\cal T$ is the well known quantization of the algebra of
regular functions on the group $GL(n)$ \cite{FRT}. Since the REA can
always be endowed with the structure of the left adjoint comodule
over $\cal T$
$$
l_i^{\,j}\mapsto \sum_{r,s}t_i^{\,r}S(t_s^{\,j})\otimes l_r^{\,s},
$$
$S(t_i^{\,j})$ being the antipode of $t_i^{\,j}$, then REA is also a
module over the dual Hopf algebra $\cal T^*$ (in the $U_q(sl(n))$
case this dual is the quantum group $U_q(gl(n))$ itself).
All the finite dimensional modules $V_\lambda$ over the REA are also
the modules over $\cal T^*$. The representations $\pi_\lambda$ of
the REA constructed in \cite{S} commute (as the mappings) with the
action of $\cal T^*$. We call them {\it the equivariant
representations} precisely in this sense.
At the end of the section we would like to discuss the problem
whether the category $\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi-\rm Rep$ of finite dimensional equivariant
representations of the algebra $\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi$ is "big enough".
\begin{definition} \label{def:faithful}
Let $A$ be an algebra and $A-\rm Rep$ be the category of its
representations. We say that this category is faithful if for any
nonzero element $a\in A$ there exists an object $V\in A-\rm Rep$ such
that $\pi_V(a)\not=0$ where $\pi_V$ is the representation
corresponding to $V$.
\end{definition}
For an involutive $R$-matrix (that is $R^2 = I$) the category
$\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi-\rm Rep$ of equivariant representations $\pi_\lambda^z$ of the
algebra $\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi$ is faithful. The same is true for any Hecke symmetry
$R$ which is a flat deformation of an involutive $R$-matrix (in
particular, for the $U_q(sl(n))$ $R$-matrix). This can be
established by a direct modification of the proof of the analogous
classical statement about universal enveloping algebras given in
\cite{D}. For an arbitrary Hecke symmetry we shall suppose that the
above property of the representation category takes place as a
plausible conjecture.
\section{The basic q-Newton identities}
\label{sec:5}
In this section we deal with a non-modified REA and generalize
formulae (\ref{reduc}) to this case.
First, we introduce some convenient notations. Let us denote
${\frak D}(t_1,t_2,\dots,t_n)$ the following Vandermonde determinant
$$
{\frak D}(t_1,t_2,\dots,t_n) = \left|
\matrix{1&1&\dots &1\cr
t_1&t_2&\dots&t_n\cr
t_1^2&t_2^2&\dots&t_n^2\cr
\dots &\dots &\dots &\dots \cr
t_1^{n-1}&t_2^{n-1}&\dots&t_n^{n-1}}
\right| = \prod_{i>j}(t_i-t_j),
$$
where $t_i$ are some variables. In general, this variables can be
elements of a commutative algebra (a ring).
Besides this, we consider the elementary symmetric functions in the
variables $t_i$:
\begin{equation}
e_0\equiv 1,\quad
e_k = \sum_{1\le i_1<i_2<\dots <i_k\le n}t_{i_1}t_{i_2}
\dots t_{i_k},\quad 1\le k\le n.
\label{elem-symm}
\end{equation}
With each $e_k$ we associate the series of the following quantities
\begin{equation}
e_k(\hat t_i)\stackrel{\mbox{\tiny def}}{=}{e_k}_{
\rule{0.25pt}{3.5mm}_{\;t_i=0}},\quad
e_k(\hat t_i,\hat t_j)\stackrel{\mbox{\tiny def}}{=}{e_k}_{
\rule{0.25pt}{3.5mm}_{\;t_i=0,t_j=0}},\quad 1\le i,j\le n,
\quad{\rm and\ so\ on}.
\label{hat}
\end{equation}
It is evident, that the quantity $e_k(\hat t_i)$ is an elementary
symmetric function in the set of $(n-1)$ variables $t_j$, $j\not=i$,
etc.
Note some useful properties of the above quantities (their proof is
a simple exercise)
\begin{eqnarray}
&&e_k = e_k(\hat t_i) + t_ie_{k-1}(\hat t_i),\label{pro:1}\\
\rule{0pt}{5mm}
&&e_k(\hat t_i) - e_k(\hat t_j) = (t_j-t_i)
e_{k-1}(\hat t_i, \hat t_j) \label{pro:2}\\
\rule{0pt}{5mm}
&& ke_k = \sum_{i=1}^{n}t_ie_{k-1}(\hat t_i).
\label{pro:3}
\end{eqnarray}
Here we assume $1\le k\le n$ and, besides, $1\le i,j\le n$ in the
first two lines.
The following lemma is easy to verify.
\begin{lemma}
\label{lem:1}
$$
\left|
\matrix{1&1&\dots &1\cr
e_1(\hat t_1)&e_1(\hat t_2)&\dots&e_1(\hat t_n)\cr
e_2(\hat t_1)&e_2(\hat t_2)&\dots&e_2(\hat t_n)\cr
\dots &\dots &\dots &\dots \cr
e_{n-1}(\hat t_1)&e_{n-1}(\hat t_2)&\dots&
e_{n-1}(\hat t_n)}
\right|=\prod_{i<j}(t_i-t_j)
\equiv {\frak D}(t_n,t_{n-1},\dots,t_1).
$$
\end{lemma}
\medskip
\noindent
{\bf Proof\ \ } The lemma is proved by induction in the size of the
determinant. Being based on (\ref{pro:2}), the induction proceeds in
the same way as when calculating the Vandermonde determinant.
\hfill\rule{6.5pt}{6.5pt}
\medskip
Consider now two sets of independent central elements of REA ${\cal L}_q$
$$
\sigma_k(\hat L) = q^k {\rm Tr}_{R(12\dots k)}A^{(k)}\hat L_{\bar 1}\dots
\hat L_{\bar k}, \quad{\rm and}\quad s_k(\hat L) =q\,{\rm Tr}_R(\hat L^k),
\quad 1\le k\le p,
$$
were $\hat L_{\bar k}$ is defined as follows:
$$
\hat L_{\bar 1} = \hat L_1, \quad \hat L_{\bar k} =
R_{k-1}\hat L_{\overline{k-1}}\, R_{k-1}^{-1},\quad k\ge 2.
$$
We also set by definition
$$
\sigma_0(\hat L) = s_0(\hat L) = {\rm id}_{\cal L}.
$$
These two sets of central elements are connected by the Newton
identities \cite{GPS,PS}
\begin{equation}
\begin{array}{rcl}
s_1 &=& \sigma_1 \\
\rule{0pt}{4mm}
-s_2+s_1\sigma_1& =&
2_qq^{-1}\,\sigma_2 \\
\rule{0pt}{4mm}
s_3-s_2\sigma_1+s_1\sigma_2 &=&
3_qq^{-2}\,\sigma_3 \\
\rule{0pt}{4mm}
\dots&\dots &\dots \\
\rule{0pt}{4mm}
(-1)^{p-1}s_p+(-1)^{p-2}s_{p-1}\sigma_1 + \dots +
s_{1}\sigma_{p-1}
&=& p_qq^{1-p}\,\sigma_p \\
\end{array}
\label{NI}
\end{equation}
Using (\ref{NI}) one can in principle express the quantities $s_k$
(and, therefore, ${\rm Tr}_{R}(\hat L^k)$) in terms of $\sigma_i$,
$1\le i\le k$. But the corresponding expressions are very cumbersome
and provide no advantage in working with ${\rm Tr}_{R}(\hat L^k)$.
On the other hand, there exists a useful and handy parametric
resolution of the system of Newton identities.
Namely, we shall assume the central elements $\sigma_k(\hat L)$ to
be represented in the form, analogous to (\ref{elem-symm}) (see also
(\ref{elsym}))
\begin{equation}
\sigma_k(\hat L) = \sum_{1\le i_1<i_2<\dots <i_k\le p}\mu_{i_1}
\mu_{i_2} \dots \mu_{i_k},\quad 1\le k\le p.
\label{q-elem-symm}
\end{equation}
The elements $\mu_i$, $1\le i\le p$, belong to the algebraic closure
of the center of the REA. On passing to an orbit ${\cal L}_q^\chi$,
the quantities $\mu_i$ take numerical values from the ground field.
In this case we assume that all these numbers are distinct pairwise.
So, we have a 1-generic orbit. (Hereafter, all the notions are used
by analogy with those introduced in section \ref{sec:fuzzy}.)
The above mentioned parametric resolution of (\ref{NI}) is given in
the following proposition.
\begin{proposition}
Let the central elements $\sigma_k(\hat L)$ be parametrized by
(\ref{q-elem-symm}). Then
\begin{equation}
q^{-1}s_k(\hat L)\equiv {\rm Tr}_R(\hat L^k) =
q^{-p} \sum_{i=1}^p\mu_i^kd_i
\label{trl}
\end{equation}
where
\begin{equation}
d_i = \prod_{j\not=i}^p{{q\mu_i-q^{-1}\mu_j}\over{\mu_i-\mu_j}}.
\label{DoM}
\end{equation}
\end{proposition}
\noindent
{\bf Proof\ \ } Let us denote
$$
x_i = q^{1-p}d_i =
\prod_{j\not=i}^p{{\mu_i-q^{-2}\mu_j}\over{\mu_i-\mu_j}}.
$$
Then we should prove that $s_k = \sum\mu_i^kx_i$ is a solution of
(\ref{NI}), provided that $\sigma_k$ is given by
(\ref{q-elem-symm}).
First of all, note the following representation for $x_i$:
\begin{equation}
x_i = q^{(p-1)(p-2)}\,
\frac{{\frak D}(q^{-2}\mu_1,q^{-2}\mu_2,\dots,\mu_i,\dots,q^{-2}
\mu_p)}{{\frak D}(\mu_1,\mu_2,\dots,\mu_p)}.
\label{x-vmd}
\end{equation}
It is a direct consequence of the explicit form of $x_i$.
Now we substitute the ansatz $s_k=\sum\mu_i^kx_i$ into the set of
Newton identities and prove that the corresponding system of linear
equation in the variables $x_i$ has a unique solution which
coincides with (\ref{x-vmd}).
Using (\ref{pro:1}) and (\ref{q-elem-symm}), we transform (\ref{NI})
to the following system of linear equations:
\begin{equation}
\sum_{i=1}^p\mu_i\sigma_{k-1}(\hat\mu_i)x_i = k_q q^{1-k}
\,\sigma_k,
\qquad 1\le k\le p,
\label{system}
\end{equation}
where the quantities $\sigma_k(\hat \mu_i)$ have the same meaning as
$e_k(\hat t_i)$ in (\ref{hat}).
The determinant of the system is
$$
\Delta(\mu) =
\left|
\matrix{\mu_1&\mu_2&\dots &\mu_p\cr
\mu_1\sigma_1(\hat \mu_1)&\mu_2\sigma_1(\hat \mu_2)
&\dots&\mu_p\,\sigma_1(\hat \mu_p)\cr
\mu_1\sigma_2(\hat \mu_1)&\mu_2\sigma_2(\hat \mu_2)&
\dots&\mu_p\,\sigma_2(\hat \mu_p)\cr
\dots &\dots &\dots &\dots \cr
\mu_1\sigma_{p-1}(\hat \mu_1)&\mu_2\sigma_{p-1}(\hat \mu_2)&\dots&
\mu_p\,\sigma_{p-1}(\hat \mu_p)}
\right|.
$$
Taking into account Lemma \ref{lem:1}, one can rewrite the above
determinant in an equivalent form:
\begin{equation}
\Delta(\mu) = \left(\prod_{i=1}^p\mu_i\right)
{\frak D}(\mu_p,\mu_{p-1},\dots,\mu_1).
\label{det-sys}
\end{equation}
Since $\Delta(\mu)\not=0$, the system (\ref{system}) has a unique
solution. To find the solution we use the Cramer's formula.
Evidently, it suffices to find the value of $x_1$ say, since the
values of other variables can be obtained from the letter one by
simple permutation of $\mu_i$.
So, we shall find $x_1$. In accordance with the Cramer's rule, it is
equal to the ratio of the following determinants:
$$
x_1 = \frac{1}{\Delta(\mu)}\,
\left|
\matrix{\sigma_1&\mu_2&\dots &\mu_p\cr
2_qq^{-1}\sigma_2&\mu_2\sigma_1(\hat \mu_2)
&\dots&\mu_p\,\sigma_1(\hat \mu_p)\cr
\displaystyle
3_qq^{-2}\sigma_3&\mu_2\sigma_2(\hat \mu_2)&
\dots&\mu_p\,\sigma_2(\hat \mu_p)\cr
\dots &\dots &\dots &\dots \cr
p_qq^{1-p}\sigma_p&\mu_2\sigma_{p-1}(\hat \mu_2)
&\dots&\mu_p\,\sigma_{p-1}(\hat \mu_p)}
\right|\equiv \frac{\Delta_1(\mu)}{\Delta(\mu)}.
$$
Let us now identically transform the numerator of the above
expression --- the determinant $\Delta_1(\mu)$.
First of all, from the first column of $\Delta_1(\mu)$ we subtract
the sum of all other columns. Using (\ref{pro:1}) and (\ref{pro:3}),
one gets for the general element of the first column
\begin{equation}
k_qq^{1-k}\,\sigma_k -
\sum_{i=2}^p\mu_i\sigma_{k-1}(\hat \mu_i) = z_k\,\sigma_k(\hat
\mu_1)+z_{k-1}\,\mu_1\sigma_{k-1}(\hat \mu_1) + q^{2(1-k)}\mu_1\,
\sigma_{k-1}(\hat \mu_1).
\label{1-sum}
\end{equation}
where for the sake of compactness we have introduced a notation
$$
z_n = n_qq^{1-n} - n.
$$
So, we find that each element of the first column of $\Delta_1(\mu)$
is the sum of several terms and therefore one can expand
$\Delta_1(\mu)$ into the sum of determinants
$$
\Delta_1(\mu) = \Delta_1'(\mu) + \Delta_1''(\mu),
$$
where the $k$-th element of the first column of $\Delta_1'(\mu)$ is
equal to $q^{2(1-k)}\mu_1 \sigma_{k-1}(\hat \mu_1)$ while the $k$-th
element of the first column of $\Delta_1''(\mu)$ contains the sum
$\eta_k(\mu)$ of two rest terms in the right hand side of
(\ref{1-sum})
$$
\eta_k(\mu)\equiv z_k\,\sigma_k(\hat \mu_1)+z_{k-1}\,\mu_1
\sigma_{k-1}(\hat \mu_1).
$$
First, consider the determinant
$$
\Delta_1''(\mu) = \prod_{i=2}^p\mu_i
\left|
\matrix{0&1&1& \dots &1\cr
\eta_2 &\sigma_1(\hat \mu_2)&\sigma_1(\hat \mu_3)
&\dots&\sigma_1(\hat \mu_p)\cr
\eta_3 &\sigma_2(\hat \mu_2)&\sigma_2(\hat \mu_3)&
\dots&\sigma_2(\hat \mu_p)\cr
\dots &\dots &\dots &\dots \cr
\eta_p&\sigma_{p-1}(\hat \mu_2)&\sigma_{p-1}(\hat \mu_3)
&\dots&\sigma_{p-1}(\hat \mu_p)}
\right|.
$$
We shall prove that $\Delta_1''(\mu) = 0$.
Subtracting the second column consecutively from the third one,
the fourth one and so on, and taking into account (\ref{pro:2}), one
gets
$$
\Delta_1''(\mu) = \prod_{i=2}^p\mu_i\prod_{j=3}^p(\mu_2-\mu_j)
\left|
\matrix{0&1&0& \dots &0\cr
\eta_2 &\sigma_1(\hat \mu_2)&1&\dots&1\cr
\eta_3 &\sigma_2(\hat \mu_2)&\sigma_1(\hat \mu_2,\hat \mu_3)&
\dots&\sigma_1(\hat \mu_2,\hat \mu_p)\cr
\dots &\dots &\dots &\dots &\dots\cr
\eta_p&\sigma_{p-1}(\hat \mu_2)&\sigma_{p-2}(\hat \mu_2,\hat
\mu_3)
&\dots&\sigma_{p-2}(\hat \mu_2,\hat \mu_p)}
\right|.
$$
Then we repeat this procedure subtracting the third column from each
$j$-th column with $j>3$ and so on. As a result we come to
$$
\Delta_1''(\mu) = N(\mu)
\left|
\matrix{0 & 1 & 0 & 0 & \dots & 0 &0\cr
\eta_2 &\sigma_1(\hat \mu_2)&1& 0 &\dots&0 &0\cr
\eta_3 &\sigma_2(\hat \mu_2)&\sigma_1(\hat \mu_2,\hat \mu_3)& 1&
\dots& 0 & 0\cr
\dots &\dots &\dots &\dots &\dots &\dots &\dots\cr
\eta_{p-1}&\sigma_{p-2}(\hat \mu_2)&\sigma_{p-3}(\hat \mu_2,
\hat \mu_3)&
\sigma_{p-4}(\hat \mu_2,\hat \mu_3,\hat \mu_4)&
\dots&\sigma_{1}(\hat \mu_2,\dots)& 1\cr
\eta_p&\sigma_{p-1}(\hat \mu_2)&\sigma_{p-2}(\hat \mu_2,
\hat \mu_3) &\sigma_{p-3}(\hat \mu_2,\hat \mu_3,\hat \mu_4)
&\dots&\sigma_{2}(\hat \mu_2,\dots)& \mu_1
}
\right|
$$
with
$$
N(\mu) = \prod_{i=2}^p\mu_i\prod_{2\le j<k \le p}(\mu_j-\mu_k).
$$
The result obtained admits the further simplification. We multiply
the last column by $\mu_p$ and subtract it from the $(p-1)$-th
column, then multiply the last column by $\mu_{p-1}\mu_p$ and
subtract it from the $(p-2)$-th column, etc. Then we repeat the
procedure starting from the $(p-1)$-th column of the resulting
determinant and so on. It is not difficult to see that we end up
with the following form of $\Delta_1''(\mu)$
$$
\Delta_1''(\mu) = N(\mu)
\left|
\begin{array}{lcccccc}
0 & 1 & 0 & 0 & \dots & 0 &0\\
z_2\sigma_2(\hat\mu_1) &\mu_1&1& 0 &\dots&0 &0\\
z_3\sigma_3(\hat\mu_1)+z_2\sigma_2(\hat\mu_1) \mu_1
&0 &\mu_1& 1& \dots& 0 & 0\\
z_4\sigma_4(\hat\mu_1)+z_3\sigma_3(\hat\mu_1)\mu_1 &0 &0& \mu_1&
\dots& 0 & 0\\
\dots &\dots &\dots &\dots &\dots &\dots &\dots\\
z_{p-1}\sigma_{p-1}(\hat\mu_1)+z_{p-2}\sigma_{p-2}(\hat\mu_1)\mu_1
&0&0&0&
\dots&\mu_1& 1\\
z_{p-1}\mu_1\sigma_{p-1}(\hat\mu_1) &0&0&0&\dots&0&\mu_1
\end{array}
\right|
$$
where we have restored the explicit form of $\eta_k$ and have taken
into account that $\sigma_p(\hat\mu_1)\equiv 0$.
At last, we multiply the third column by $z_2\sigma_2(\hat\mu_1)$,
the fourth one by $z_3\sigma_3(\hat\mu_1)$, etc., and then subtract
all these columns from the first one. We get all elements of the
first column of $\Delta_1''(\mu)$ to be zero, therefore
$\Delta_1''(\mu) = 0$.
Turn now to the determinant
$$
\Delta_1'(\mu) = \prod_{i=1}^p\mu_i
\left|
\matrix{1 & 1 & 1 & \dots & 1 \cr
q^{-2}\sigma_1(\hat \mu_1) &\sigma_1(\hat \mu_2)&\sigma_1
(\hat \mu_3) &\dots&\sigma_1(\hat \mu_p)\cr
q^{-4}\sigma_2(\hat \mu_1) &\sigma_2(\hat \mu_2)&\sigma_2
(\hat \mu_3)& \dots&\sigma_2(\hat \mu_p)\cr
\dots &\dots &\dots &\dots &\dots \cr
q^{2(1-p)}\sigma_{p-1}(\hat \mu_1)&\sigma_{p-1}(\hat \mu_2)&
\sigma_{p-1}(\hat \mu_3) &\dots&\sigma_{p-1}(\hat \mu_p)}
\right|.
$$
With the same operations which were applied to $\Delta_1''(\mu)$ we
convert the determinant into the form
$$
\Delta_1'(\mu) = \prod_{i=1}^p\mu_i\prod_{2\le j<k\le p}
(\mu_j - \mu_k) \left|
\matrix{1 & 1 & 0 & \dots & 0 \cr
q^{-2}\sigma_1(\hat \mu_1) &\mu_1&1&\dots&0\cr
q^{-4}\sigma_2(\hat \mu_1) &0&\mu_1& \dots&0\cr
\dots &\dots &\dots &\dots &\dots\cr
q^{2(1-p)}\sigma_{p-1}(\hat \mu_1)&0&
0 &\dots&\mu_1}
\right|.
$$
Let us introduce new parameters
$$
\nu_1 = \mu_1,\quad \nu_i = q^{-2}\mu_i,\quad 2\le i\le p.
$$
Since the function $\sigma_k(\hat \mu_1)$ is a homogeneous
polynomial of the $k$-th order in the variables $\mu_i$, $i\ge 2$,
then
$$
q^{-2k}\sigma_k(\hat \mu_1) = \sigma_k(\hat \nu_1)
$$
and therefore
$$
\Delta_1' = \left(q^{2(p-1)}\prod_{i=1}^p\nu_i\right)q^{(p-1)(p-2)}
\left|
\matrix{1 & 1 & 1 & \dots & 1 \cr
\sigma_1(\hat \nu_1) &\sigma_1(\hat \nu_2)&\sigma_1(\hat \nu_3)
&\dots&
\sigma_1(\hat \nu_p)\cr
\sigma_2(\hat \nu_1) &\sigma_2(\hat \nu_2)&\sigma_2(\hat \nu_3)&
\dots&
\sigma_2(\hat \nu_p)\cr
\dots &\dots &\dots &\dots &\dots\cr
\sigma_{p-1}(\hat \nu_1)&\sigma_{p-1}(\hat \nu_2)&
\sigma_{p-1}(\hat \nu_3) &\dots&\sigma_{p-1}(\hat \nu_p)}
\right|.
$$
Applying Lemma \ref{lem:1} and changing the set of parameters
$\{\nu_i\}$ back to the set of $\{\mu_i\}$ we get
$$
\Delta_1(\mu) = \Delta_1'(\mu) = q^{(p-1)(p-2)}
\left(\prod_{i=1}^p\mu_i\right)
{\frak D}(q^{-2}\mu_p,q^{-2}\mu_{p-1},\dots,q^{-2}\mu_{2},\mu_1).
$$
Using the value (\ref{det-sys}) of the determinant $\Delta(\mu)$,
one comes to the final result
$$
x_1 = \frac{\Delta_1(\mu)}{\Delta(\mu)} =
q^{(p-1)(p-2)}\frac{{\frak D}(q^{-2}\mu_p,q^{-2}\mu_{p-1},\dots,\mu_1)}
{{\frak D}(\mu_p,\mu_{p-1},\dots,\mu_1)}
$$
which is obviously equivalent to (\ref{x-vmd}).
\hfill\rule{6.5pt}{6.5pt}
\begin{remark}
\label{DM} {\rm
In \cite{DM1} a way of quantization of semisimple (but not necessary
generic) orbit in $gl(n)^*$ was suggested and in this connection
another form of the basic $q$-Newton identity was given. Let us
briefly describe the quantization procedure from \cite{DM1} and
compare two forms of basic $q$-Newton identities. (The normalization
of the quantum trace in \cite{DM1} differs from that accepted in the
present paper.) Here we restrict ourselves to the $U_q(sl(n))$ case.
Consider the $GL(n)$ orbit ${\cal O}_M$ of an arbitrary semisimple
matrix $M\in gl(n)^*$ characterized by $r\le n$ pairwise distinct
eigenvalues $\mu_i$ with the multiplicities $m_i\geq 1$
\begin{equation}
m_1+m_2+\dots +m_r=n
\label{spec-dat}
\end{equation}
(see Remark 1).
Let $P(x)=\prod_{i=1}^r(x-\mu_i)$ be the degree $r$ minimal
polynomial of this orbit (i.e. each $\mu_i$ is a simple root of
$P(x)$). Then the quotient of the mREA over the ideal generated by
the entries of the matrix $P(L)=\prod_{i=1}^r(L-\mu_iI)$ and by
the elements
\begin{equation}
{\rm Tr}_R L^k-\bar{\beta_k},\quad k=1,\dots ,r-1
\label{nong-orb}
\end{equation}
with appropriate values of $\bar{\beta_k}$ is a flat deformation
(quantization) of the commutative algebra ${\Bbb K}({\cal O}_M)$. This
fact was established in \cite{DM1} where the exact values of
$\bar{\beta_k}$ were expressed in terms of the roots of the minimal
polynomial. However, the same values of these quantities can be
obtained from our parametric resolution of the basic $q$-Newton
identities given in (\ref{trl})--(\ref{DoM}).
For this purpose, we associate to any root $\mu=\mu_i$ of the
minimal polynomial the following {\it string} of $m_i$ quantities
\begin{equation}
\nu_1=\mu,\;\;
\nu_2=q^{-2}\nu_1+q^{-1}{\hbar},\;\;\nu_3=q^{-2}\nu_2+q^{-1}
{\hbar},\;\;\dots,\;\;
\nu_{m_i}=q^{-2} \nu_{m_i-1}+q^{-1}{\hbar}.
\label{string}
\end{equation}
Consider the set of all $\nu_{i_k}$ belonging to strings
(\ref{string}) and define the character $\chi:Z(\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi)\to{\Bbb K}$ on
central elements $\sigma_i$ as in (\ref{q-gen}) but with eigenvalues
running over all $\nu$ from the mentioned set. Let us pass to the
corresponding NC orbit ${\cal L}_{\h,q}^{\chi}$ (\ref{q-orb}). Emphasize that the NC
orbit ${\cal L}_{\h,q}^{\chi}$ thus obtained turns out to be bigger than the result
of quantization of the Poisson pencil (see Introduction) on the
initial $GL(n)$ orbit ${\cal O}_M$. In order to get the genuine
quantum orbit we should quotient ${\cal L}_{\h,q}^{\chi}$ over the ideal generated by
the entries of the matrix $P(L)$ constructed from the minimal
polynomial of the orbit ${\cal O}_M$. So, assuming $|q-1|$ and ${\hbar}$
to be small enough in order to avoid any casual coincidence of the
elements from the above union of the strings, we get $n$
{\it pairwise distinct} eigenvalues $\nu_{i_k}$ of the matrix $L$
with entries considered as elements of ${\cal L}_{\h,q}^{\chi}$. This is well
coordinated with the empirical principle that the quantization
decreases the degeneracy.
Finally, we have got a 1-generic orbit ${\cal L}_{\h,q}^{\chi}$ and the quantities
${\rm Tr}_R L^k$ can be computed via (\ref{trl})-(\ref{DoM}). However,
the multiplicities $d_i$ corresponding to extra eigenvalues (i.e.
those which are not roots of the minimal polynomial) vanish.
So, given a 1-generic NC orbit ${\cal L}_{\h,q}^{\chi}$, we can be sure that it is a
quantization of a classical generic orbit iff the set of eigenvalues
of the matrix $L$ corresponding to this NC orbit contains no string.
If it is not the case, we construct the minimal polynomial $P(x)$
taking the first element of each string as its simple root and
consider a two sided ideal in ${\cal L}_{\h,q}^{\chi}$ generated by the entries of
the matrix $P(L)$. Then, quotienting the given 1-generic NC orbit
over this ideal, we get a quantization of a semisimple but not
generic orbit whose eigenvalues are the first elements of strings
and the multiplicity of each eigenvalue is equal to the length of
the corresponding string.}
\end{remark}
\section{The higher Cayley-Hamilton and Newton identities}
\label{sec:6}
Consider a 1-generic quantum orbit ${\cal L}_{\h,q}^{\chi}$ defined by relations
(\ref{q-gen})--(\ref{ideal}) where $p$ is the symmetry rank of the
Hecke $R$-matrix.
In \cite{GLS2} the following conjecture was formulated.
\begin{conjecture}
On a 1-generic NC orbit ${\cal L}_{\h,q}^{\chi}$ the matrix $L_{(m)}$ (\ref{L-symm})
satisfies the Cayley-Hamil\-ton identity
$$
{\cal CH}_{(m)}^{\chi}(L_{(m)}) = 0,
$$
where the degree of the polynomial ${\cal CH}_{(m)}^{\chi}$ is
\begin{equation}
{\rm deg\,}{\cal CH}_{(m)}^{\chi} = {m+p-1\choose m}
\label{CH-order}
\end{equation}
and its roots $\mu_\mathbf{k}(m)$ are
\begin{equation}
q^{m-1}\mu_\mathbf{k}(m) = \sum_{i=1}^p \frac{(k_i)_q}{q^{m-k_i}} \,
\mu_i + \hbar\,\xi_p(k_1,\dots,k_p),
\label{q-high}
\end{equation}
where $\mathbf{k}$ is a partition (\ref{part}) of the integer $m$
$$
\mathbf{k}=(k_1,\dots ,k_p),\quad k_i\geq 0,\quad |\mathbf{k}|=k_1+\dots +k_p=m
$$
and $\xi_p(k_1,\dots,k_p)$ is the symmetric function in $k_i$ of the
form
$$
\xi_p(k_1,\dots,k_p) = \sum_{s=2}^{p} q^{k_1+k_2+\dots+k_s-m}
(k_s)_q (k_1+k_2+\dots+k_{s-1})_q.
$$
\end{conjecture}
\begin{remark}{\rm
The fact that $\xi_p$ is a symmetric function in $k_i$ can be easily
verified upon expanding all $q$-numbers in accordance with their
definition (\ref{q-num}).
Note that (\ref{q-high}) is a generalization of the analogous
formula (\ref{high}). In the classical limit $q\to 1$ the above
formula transforms into (\ref{high}) for the 1-generic fuzzy orbit
in $U(gl(p))$.}
\end{remark}
The Conjecture is justified by explicit calculations for small
values of $m$ but we still have no general proof of it. Here we
present a proof for the particular case $p=2$.
Since at $q\not=\pm 1$ the mREA $\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi$ is isomorphic to the
nonmodified REA ${\cal L}_q$, we first prove the Conjecture for ${\cal L}_q$ and
then pass to the corresponding result for $\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi$ by means of the
shift of generators.
\begin{proposition}\label{main-prop}
Let the symmetry rank $p$ of the $R$-matrix be equal to $2$. Then
the roots $\hat \omega_s$ $0\le s\le m$ of the CH polynomial for the
symmetric matrix $\hat L_{(m)}$ are given by
\begin{equation}
q^{1-m} \hat \omega_s =q^{s-m}s_q\hat \mu_1+q^{-s}(m-s)_q
\hat \mu_2,\quad s=0,1,\dots,m
\label{omegas}
\end{equation}
where $\hat \mu_i$ are the roots of the basic CH polynomial for the
matrix $\hat L_{(1)} = \|\hat l_i^{\,j}\|$ composed of the
generators of the REA ${\cal L}_q$
$$
(\hat L_{(1)} - \hat \mu_{1}I_e)(\hat L_{(1)} -
\hat \mu_{2}I_e) = 0, \qquad I_e = {\rm id}_{\cal L}\otimes I.
$$
\end{proposition}
\noindent{\bf Proof\ \ }
Let us shortly outline the strategy of the proof. We shall use the
same approach as in proposition \ref{prop:3}. Namely, we prove the
claim in each finite dimensional representation $V_{(k)}$ of the
REA. This means that, for a given fixed $m$, we associate to
$\hat L_{(m)}$ a series of numerical matrices $\hat L_{(k,m)}$ of
the form
\begin{equation}
\hat L_{(k,m)}
= (\bar \pi_{(k)}\otimes \pi_{(m)})({\bf Cas}),\quad k=m,m+1,\dots
\label{def:Lkm}
\end{equation}
and prove the claim for each value of $k\ge m$. We should consider
the values $k\ge m$ since at $k<m$ the matrix $\hat L_{(k,m)}$ can
satisfy the CH identity of an order lower than (\ref{CH-order}).
This is a peculiarity of low dimensional representations connected
with the combinatorics of the Young diagrams.
The above matrix $\hat L_{(k,m)}$ coincides with that of a linear
operator ${\bf Cas}_{(k,m)} \in {\rm End}_r (V_{(k)})\otimes
{\rm End}_l(V_{(m)})$ which is obtained from ${\bf Cas}$
(\ref{quant-Cas}) in the full analogy with the fuzzy case
construction (\ref{cas-lr}). Recall, that at $p=2$ the spaces
$V_{(k)}$ and $V^*_{(k)}$ are isomorphic (see section
\ref{sec:rea-br}). We consider the decomposition of the
tensor product $V_{(k)}\otimes V_{(m)}$ into the direct sum of
subspaces $V_{(\nu_1,\nu_2)}$ and prove that the restriction of
${\bf Cas}_{(k,m)}$ on each subspace $V_{(\nu_1,\nu_2)}$ is a
multiple of the unit operator. The corresponding factors will be the
roots of the CH polynomial for $\hat L_{(k,m)}$ and we should verify
that they are given by (\ref{omegas}).
{\bf Step 1.} Let us fix an arbitrary integer $k\ge m$ and find the
values $\hat \mu_1(k)$ and $\hat \mu_2(k)$ of the roots of the basic
CH polynomial for
$\hat L_{(k,1)} = \|\bar\pi_{(k)}(\hat l_i^{\,j})\|$. For this
purpose we substitute the matrix $\hat L_{(k,1)}$ into identity
(\ref{CH})
$$
\hat L^2 - \sigma_1(\hat L)\hat L +\sigma_2(\hat L)\,I = 0
$$
and calculate the spectrum of the central elements $\sigma_i$. Since
$$
\sigma_1(\hat L) = q{\rm Tr}_R\hat L\quad \sigma_2(\hat L) =
\frac{q^2}{2_q}\,\Bigl(q({\rm Tr}_R\hat L)^2 - {\rm Tr}_R\hat L^2\Bigr),
$$
then basing on the explicit form (\ref{rep:pim-r}) of the right
representation $\bar\pi_{(k)}$ one can show
$$
\sigma_1(\hat L_{(k,1)}) = (1+q^{-2k-2})\,S^{(k)}(R)\equiv
(1+q^{-2k-2})\,{\rm id}_{V_{(k)}}
$$
and
$$
\sigma_2(\hat L_{(k,1)}) = q^{-2k-2}\,S^{(k)}(R)\equiv q^{-2k-2}\,
{\rm id}_{V_{(k)}}.
$$
So, on the subspace $V_{(k)}\subset V^{\otimes k}$ the
Cayley-Hamilton identity for $\hat L_{(k,1)}$ takes the form
$$
(\hat L_{(k,1)} - I)(\hat L_{(k,1)}-q^{-2k-2}I) = 0
$$
that is
\begin{equation}
\hat \mu_1(k) = 1, \quad \hat \mu_2(k) = q^{-2k-2}.
\label{mui}
\end{equation}
{\bf Step 2.} Now we pass to the matrix $\hat L_{(k,m)}$
(\ref{def:Lkm}) treated as that of a linear operator from
${\rm End}_r (V_{(k)})\otimes {\rm End}_l(V_{(m)})$. In the case
$p=2$ the general decomposition (\ref{razl-prostr}) reduces to
$$
V_{(k)}\otimes V_{(m)} = \bigoplus_{s=0}^{m}V_{(k+s,m-s)}.
$$
The subspace $V_{(k+s,m-s)}$ can be represented as an image of the
operator ${\cal P}_s\in {\rm End}(V^{\otimes (k+m)})$
$$
{\cal P}_s = S^{(k)}_{1\dots k}S^{(m)}_{k+1 \dots k+m}
Y_{(k+s,m-s)}S^{(k)}_{1\dots k}S^{(m)}_{k+1 \dots k+m}.
$$
Here the lower indices of the $q$-symmetrizers $S^{(k)}$ and
$S^{(m)}$ indicate explicitly the numbers of factors in the tensor
product $V^{\otimes (k+m)}$ in which these symmetrizers act. The
$q$-projector $Y_{(k+s,m-s)}$ corresponds to the following Young
tableau
\begin{equation}
Y_{(k+s,m-s)}\;\leftrightarrow\;
\begin{array}{|c@{\hspace{-5pt}}c|}\hline
1\;\;\;\dots \;\;\;k \;\;\; \dots\;\;\; k&+\;s\\ \hline
\multicolumn{1}{|c|}{k+s+1\;\dots\; k+m }\\ \cline{1-1}
\end{array}\equiv [1,\,\dots\,, k+s\,|\,k+s+1,\,\dots\,, k+m].
\label{Y-diag}
\end{equation}
Here in the last equality we have introduced a more convenient
notation for an explicit enumeration of a tableau corresponding to
the two row partition $\lambda =(k+s,m-s)$, $0\le s\le m$.
Due to the fact that
$$
S^{(k)}_{1\dots k} Y_{(k+s,m-s)} = Y_{(k+s,m-s)}
S^{(k)}_{1\dots k} = Y_{(k+s,m-s)}
$$
the expression for ${\cal P}_s$ can be simplified to
\begin{equation}
{\cal P}_s = S^{(m)}_{k+1 \dots k+m} Y_{(k+s,m-s)}
S^{(m)}_{k+1\dots k+m}.
\label{Ps}
\end{equation}
The operator ${\cal P}_s$ is a projector up to a normalizing factor
$$
{\cal P}_s^2 = \gamma_s {\cal P}_s,
$$
the exact value of $\gamma_s$ is not important for us.
To prove the proposition it is suffice to show that the matrix
$\hat L_{(k,m)}$ commutes with ${\cal P}_s$ for any $0\le s\le m$
and that the following relation holds
\begin{equation}
{\hat L}_{(k,m)}{\cal P}_s = \hat \omega_s{\cal P}_s =
{\cal P}_s{\hat L}_{(k,m)}.
\label{eigenv}
\end{equation}
{\bf Step 3.} Consider the product ${\hat L}_{(k,m)}{\cal P}_s$ in
detail. On the base of definition (\ref{def:Lkm}) with the explicit
form of representations $\pi_{(m)}$ and $\bar \pi_{(k)}$ given in
propositions \ref{prop:pim} and \ref{prop:rea-r} we find the
following expression for the matrix $\hat L_{(k,m)}$
\begin{equation}
q^{m-1}\hat L_{(k,m)} = \frac{m_q}{q^{2k+2}}\,S^{(k)}_{1\dots k}
S^{(m)}_{k+1\dots k+m}+\frac{\zeta m_q (k+1)_q}{q^{k+1}}\,
S^{(m)}_{k+1\dots k+m} S^{(k+1)}_{1\dots k+1}
S^{(m)}_{k+1\dots k+m}.
\label{Lkm-expl}
\end{equation}
where we have extracted the overall factor $q^{m-1}$ to simplify the
subsequent calculations. In deriving this formula one should use the
relation
$$
S^{(k)}_{1\dots k}A^{(2)}_{kk+1}S^{(k)}_{1\dots k} =
\frac{(k+1)_q}{2_qk_q} \,(S^{(k)}_{1\dots k}
I_{\raisebox{-1.75pt}{\scriptsize $k+1$}}-S^{(k+1)}_{1\dots k+1}).
$$
Now we are to multiply the above expression for $\hat L_{(k,m)}$ on
the matrix ${\cal P}_s$ from the right. The multiplication of the
first summand in (\ref{Lkm-expl}) results in
$q^{-2k-2}m_q{\cal P}_s$ and the main difficulty concentrates in the
second term
\begin{equation}
\frac{\zeta m_q (k+1)_q}{q^{k+1}}\, S^{(m)}_{k+1\dots k+m}
S^{(k+1)}_{1\dots k+1}S^{(m)}_{k+1\dots k+m} Y_{(k+s,m-s)}
S^{(m)}_{k+1\dots k+m} \equiv \frac{\zeta m_q (k+1)_q}{q^{k+1}}
\Omega_s
\label{Om}
\end{equation}
were the left hand side is a definition of the symbol $\Omega_s$. At
$s=m$ this is obviously the multiple of ${\cal P}_m\equiv S^{(k+m)}$
therefore below we shall suppose $0\le s\le m-1$.
To prove that $\Omega_s$ is a multiple of ${\cal P}_s$ one should
somehow get rid of the $q$-symmetrizer $S^{(k+1)}_{1\dots k+1}$ in
its expression. For this purpose we decompose the projector
$S^{(m)}$ standing between $S^{(k+1)}$ and $Y_{(k+s,m-s)}$ into the
product of the factors (\ref{q-symm-S})
$$
S^{(m)}_{k+1\dots k+m} = \frac{(m-1)_q}{m_q}\,
S^{(m-1)}_{k+2\dots k+m} \Bigl(\frac{q^{1-m}}{(m-1)_q}\,I +
R_{k+1}\Bigr)S^{(m-1)}_{k+2\dots k+m}.
$$
The first $S^{(m-1)}$ in the right hand side of the above relation
commutes with $S^{(k+1)}$ and can be absorbed into the most left
$S^{(m)}$ in (\ref{Om})
$$
S^{(m)}_{k+1\dots k+m}S^{(m-1)}_{k+2\dots k+m} =
S^{(m)}_{k+1\dots k+m}.
$$
As for the second $S^{(m-1)}$, we shall continue its decomposition
in the same way until descending to $S^{(m-s)}$ which is absorbed by
the projector $Y_{(k+s,m-s)}$. So, we come to the result
$$
\Omega_s = \frac{(m-s)_q}{m_q} \,S^{(m)}S^{(k+1)}
\Bigl(\frac{q^{1-m}}{(m-1)_q}\,I +R_{k+1}\Bigr)\dots
\Bigl(\frac{q^{s-m}}{(m-s)_q}\,I+R_{k+s}\Bigr)Y_{(k+s,m-s)}
S^{(m)}.
$$
The next step is to draw the factors of the type $(zI+R)$ through
$Y_{(k+s,m-s)}$ and then absorb them into the right
$S^{(m)}_{k+1\dots k+m}$ on the base of the following property of
the $q$-symmetrizer
$$
R_{k+i}\,S^{(m)}_{k+1\dots k+m} = q\, S^{(m)}_{k+1\dots k+m}\qquad
1\le i\le m-1.
$$
The commutation of the linear in $R$-matrix terms with
$Y_{(k+s,m-s)}$ is done on the base of the identity \cite{OgP}
\begin{equation}
\Bigl(\frac{q^{-i}}{i_q}\,I +R_i\Bigr)\;
\begin{array}{|cc|}\hline
1\;\;\;\dots\;\;\; i&\dots\\ \hline
\multicolumn{1}{|c|}{i+1\;\;\dots}
\\ \cline{1-1}
\end{array}
=
\begin{array}{|cc|}\hline
1\;\;\;\dots& i+1\;\;\dots\\ \hline
\multicolumn{1}{|l|}{i\;\;\dots}
\\ \cline{1-1}
\end{array}
\;\Bigl(R_i - \frac{q^{i}}{i_q}\,I \Bigr) \label{link}
\end{equation}
On applying such like formulae $s$ times we get the following result
for $\Omega_s$
\begin{equation}
\Omega_s = \frac{(m-s)_q}{(k+s)_q}\sum_{r=1}^s
\frac{(k-m+2r)_q}{(m-r)_q(m-r+1)_q}\,S^{(m)}Y_{(k+s,m-s)}^{[r]}
S^{(m)}, \quad 0\le s\le m-1.
\label{Om-prom}
\end{equation}
Here the $q$-projector $Y_{(k+s,m-s)}^{[r]}$ corresponds to the
following Young tableau
$$
Y_{(k+s,m-s)}^{[r]}\leftrightarrow [1,\,\dots\,,k+r,k+r+2,\,
\dots\,,k+s+1\,|\,k+r+1,k+s+2,\,\dots\,,k+m].
$$
The unwanted $q$-symmetrizer $S^{(k+1)}$ has been absorbed into
$Y^{[r]}$
$$
S^{(k+1)}_{1\dots k+1}Y^{[r]}_{(k+s,m-s)} = Y^{[r]}_{(k+s,m-s)},
\quad 1\le r\le s.
$$
And at last, we transform all $Y^{[r]}_{(k+s,m-s)}$ in
(\ref{Om-prom}) back to $Y_{(k+s,m-s)}$. This can be done by the
multiple successive application of the following consequence of
(\ref{link})
$$
\begin{array}{|cc|}\hline
1\;\;\;\dots& i+1\;\;\dots\\ \hline
\multicolumn{1}{|l|}{i\;\;\dots}
\\ \cline{1-1}
\end{array} =
\frac{i_q^2}{(i-1)_q(i+1)_q}\, \Bigl(\frac{q^{-i}}{i_q}\,I
+R_i\Bigr)\;
\begin{array}{|cc|}\hline
1\;\;\;\dots\;\;\; i&\dots\\ \hline
\multicolumn{1}{|c|}{i+1\;\;\dots}
\\ \cline{1-1}
\end{array}
\;\Bigl(\frac{q^{-i}}{i_q}\,I + R_i\Bigr).
$$
The brackets with $R$-matrices appearing in this way are absorbed
into $S^{(m)}$ which gives rise to the accumulation of an overall
numerical factor. The final result for $\Omega_s$ reads
$$
\Omega_s = \beta_s \,S^{(m)} Y_{(k+s,m-s)}S^{(m)}, \quad
0 \le s \le m-1,
$$
where
$$
\beta_s = (m-s)_q(k+s+1)_q\sum_{r=1}^s
\frac{(k-m+2r)_q}{(k+r)_q(k+r+1)_q(m-r)_q(m-r+1)_q}.
$$
The sum in this relation can be easily calculated by induction in
$s$ and one gets
$$
\beta_s = \frac{s_q(k+s+1-m)_q}{m_q(k+1)_q}.
$$
Now, gathering together the results from the both terms in the right
hand side of (\ref{Lkm-expl}), we find
$$
q^{m-1}\hat L_{(k,m)}{\cal P}_s = \Bigl(\frac{m_q}{q^{2k+2}}
+\frac{\zeta}{q^{k+1}}\, m_q(k+1)_q\beta_s\Bigr) {\cal P}_s =
\Bigl(\frac{m_q}{q^{2k+2}} +
\frac{\zeta}{q^{k+1}}s_q(k+s+1-m)_q\Bigr) {\cal P}_s.
$$
Moreover, since the expressions for $\hat L_{(k,m)}$
(\ref{Lkm-expl}) and ${\cal P}_s$ (\ref{Ps}) are symmetric with
respect to the $q$-projectors, the same result can be obtained for
${\cal P}_s\hat L_{(k,m)}$ that is ${\cal P}_s$ and $\hat L_{(k,m)}$
commute.
The last step is to show that the coefficient in the right hand
side of the above expression for $\hat L_{(k,m)}{\cal P}_s $ is
equal to $\hat \omega_s$. It is a matter of a short straightforward
calculation to verify the identity
$$
\frac{m_q}{q^{2k+2}} + \frac{\zeta}{q^{k+1}}s_q(k+s+1-m)_q
\equiv \frac{s_q}{q^{m-s}}+\frac{(m-s)_q}{q^{2k+2+s}}.
$$
Taking into account the values of the roots $\hat \mu_i(k)$
(\ref{mui}) we come to the desired result
$$
\frac{s_q}{q^{m-s}}+\frac{(m-s)_q}{q^{2k+2+s}} = q^{s-m}{s_q}\hat
\mu_1(k)+q^{-s}(m-s)_q \hat \mu_2(k) = q^{m-1}\hat \omega_s
$$
and therefore
$$
\hat L_{(k,m)}{\cal P}_s = \hat \omega_s {\cal P}_s.
$$
As was mentioned above, the same result is valid for
${\cal P}_s \hat L_{(k,m)}$. \hfill\rule{6.5pt}{6.5pt}
\smallskip
The roots of the CH polynomial for the symmetric matrix $L_{(m)}$
composed of the generators $l_i^{\,j}$ of the mREA can be found as a
simple corollary of the proposition \ref{main-prop}.
\begin{corollary}
Let the symmetry rank $p$ of the $R$-matrix be equal to $2$. Then
the roots $\omega_s$ $0\le s\le m$ of the CH polynomial for the
symmetric matrix $L_{(m)}$ are given by
\begin{equation}
q^{1-m} \omega_s =q^{s-m}s_q \mu_1+q^{-s}(m-s)_q\mu_2 +
{\hbar} s_q(m-s)_q,\quad s=0,1,\dots,m
\label{omgas}
\end{equation}
where $\mu_i$ are the roots of the basic CH polynomial for the
matrix $L_{(1)} = \|l_i^{\,j}\|$ composed of the generators of the
mREA $\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi$
$$
( L_{(1)} - \mu_{1}I_e)(L_{(1)} - \mu_{2}I_e) = 0,
\qquad I_e = {\rm id}_{\cal L}\otimes I.
$$
\end{corollary}
\noindent {\bf Proof\ \ }
Consider in detail how the shift of generators presented in
(\ref{shift}) affects the CH identity. If the matrix
$\hat L = \|\hat l_i^{\,j}\|$ of the REA generators satisfies the CH
identity
$$
(\hat L - \hat \mu_1 I_e)(\hat L - \hat \mu_2 I_e) = 0,
$$
then the matrix $L$ of the mREA generators satisfies the same
identity but with $\mu_i = \hat\mu_i + \hbar\zeta^{-1}$. This is a
trivial consequence of (\ref{shift}).
As for the higher order CH identity for symmetrical matrix
$L_{(m)}$ (\ref{L-symm}) the modification is as follows. Suppose, we
know the CH identity for $\hat L_{(m)}$
$$
\prod_{s=0}^{m}(\hat L_{(m)}- \hat\omega_s I_e^{(m)}) = 0,\quad
q^{m-1}\hat\omega_s = q^{s-m}s_q\hat \mu_1+q^{-s}(m-s)_q\hat \mu_2,
$$
where $I_e^{(m)} = {\rm id}_{\cal L}\otimes I_{V_{(m)}}$. To pass to
the mREA case we should take into account the connection of the
matrices $\hat L_{(m)}$ and $L_{(m)}$
$$
\hat L_{(m)} = L_{(m)} - \frac{\hbar}{\zeta}\,q^{1-m}
m_q\,I_e^{(m)}.
$$
This relation follows from (\ref{shift}), (\ref{L-symm}) and
the explicit form of the representation $\pi_{(m)}$ given in
proposition \ref{prop:pim}. Besides, one should take into account
the shift from $\hat \mu_i$ to $\mu_i$ described above. Therefore,
we come to the following result
\begin{eqnarray*}
\hat L_{(m)} - \hat \omega_sI_e^{(m)}&=&L_{(m)} - q^{1-m}
\Bigl(\frac{\hbar}{\zeta}\,m_q + q^{s-m}s_q(\mu_1 -
\frac{\hbar}{\zeta}) +q^{-s}(m-s)_q(\mu_2 -
\frac{\hbar} {\zeta}) \Bigr)\,I_e^{(m)}\\
&=& L_{(m)} - q^{1-m}(q^{s-m}s_q\,\mu_1+q^{-s}(m-s)_q\,\mu_2 +
\hbar s_q(m-s)_q)I_e^{(m)} \\
&=& L_{(m)} - \omega_s I_e^{(m)}.
\end{eqnarray*}
This completes the proof.\hfill\rule{6.5pt}{6.5pt}
\smallskip
To sum up, we conclude that on any 1-generic NC orbit ${\cal L}_{\h,q}^{\chi}$
(\ref{ideal}) the extended matrix $L_{(m)}$ satisfies the
$(m+1)$-th order polynomial identity with the roots (\ref{omgas})
($p=2$). Fixing the finite dimensional representation
$\bar\pi_{(k)}$ affects only the particular form of the roots
$\mu_i$ of the basic CH polynomial for the matrix $L$.
In full analogy with the constructions of section \ref{sec:fuzzy},
we can introduce idempotents $e_\mathbf{k}(m)$ on any $m$-generic NC orbit.
Moreover, the following proposition holds true.
\begin{proposition}
\label{prop:21}
For any m-generic NC orbit ${\cal L}_{\h,q}^{\chi}$ defined by
(\ref{q-gen})--(\ref{ideal}) we have
\begin{equation}
{\rm Tr}_R { L}_{(m)}^s=q^{-p}\sum_{|\mathbf{k}|=m}\mu_\mathbf{k}(m)^s\,d_\mathbf{k}(m)
\label{lm-tr}
\end{equation}
with $\mu_\mathbf{k}$ introduced in (\ref{q-high}). If the symmetry rank $p$
of $R$ is equal to 2, then
\begin{equation}
d_\mathbf{k}(m)=\prod_{1\le i< j\le
p}{\frac{q^{k_i-k_j}\mu_i-q^{k_j-k_i}\mu_j-{\hbar}\,(k_i-k_j)_q}
{\mu_i-\mu_j}}.
\label{mulp}
\end{equation}
If the symmetry rank of $R$ $p>2$ then the above expression for
$d_\mathbf{k}(m)$ is valid provided that the category of finite dimensional
equivariant representations of the corresponding mREA \ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi{} is
faithful (see definition \ref{def:faithful}).
\end{proposition}
{\bf Proof\ \ } The proof of (\ref{lm-tr}) is straightforward.
As for the proof of (\ref{mulp}), it is sufficient to establish this
formula for ${\hbar}=0$. Similarly to the proof of proposition
\ref{prop:6} we consider $\overline \pi_\lambda({ L}_{(m)})$, compute the
corresponding multiplicities $d_\mathbf{k}(\lambda,m) = \overline \pi_\lambda(d_\mathbf{k}(m))$
and prove relation (\ref{mulp}) with $\mu_i = \mu_i(\lambda)$.
Then, due to the faithfulness of the representation category, we
conclude that (\ref{mulp}) takes place at the level of the algebra
itself.
So, assuming the representation $\overline \pi_\lambda$ to be $m$-admissible
(see definition \ref{def:m-ad}) we find
\begin{equation}
d_\mathbf{k}(\lambda, m)={{{\rm dim}_q (V_{\lambda}^*\otV_{(m)})_\mathbf{k}}\over{{\rm dim}_q
V_{\lambda}^*}}
\label{raz}
\end{equation}
where $(V_{\lambda}^*\otV_{(m)})_\mathbf{k}$ is the irreducible component with the
label $\mathbf{k}$ (\ref{part}) in the tensor product $V_{\lambda}^*\otV_{(m)}$ and
the $q$-dimension is defined in (\ref{def:q-dim}). This formula is
the $q$-analog of (\ref{dk}) and can be obtained by the same method.
Now let us take into account that the decomposition rules for the
tensor product of the \ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi{} modules $V_\lambda$ are the same as
those for the $U(gl(p))$ modules, $p$ being the symmetry rank of $R$
(see the beginning of section \ref{sec:rea-br}). Bearing in mind the
property {\bf C4)} of the Hecke symmetry $R$ (section
\ref{sec:rea-def}), we conclude that $V^*_\lambda$ is isomorphic to
$V_{\hat{\lambda^{\!*}}}$ as a vector space where
$\hat{\lambda^{\!*}}$ is defined in (\ref{lambda-star}) and
(\ref{lambda-hat}). And secondly,
$(V_{\lambda}^*\otimes V_{(m)})_\mathbf{k}\cong V_{\hat{\lambda^{\!*}}+\mathbf{k}}$ (as
vector spaces). Since we are interested in calculating the
$q$-dimensions which are the same for isomorphic spaces, then in all
formulae we can change the spaces $V^*_\lambda$ and
$(V_{\lambda}^*\otimes V_{(m)})_\mathbf{k}$ for the corresponding isomorphic spaces.
On the other hand, by virtue of (\ref{DoM}) and the isomorphism
mentioned above, we have in the case $m=1$
\begin{equation}
d_j(\lambda,1)=\prod_{i\not=j}{{q\mu_j(\lambda)-q^{-1}q\mu_i(\lambda)}
\over{\mu_j(\lambda) - \mu_i(\lambda)}}= {{{\rm dim}_q (V_{\hat{\lambda^{\!*}}}
\otimes V)_j} \over{{\rm dim}_q V_{\hat{\lambda^{\!*}}}}},\quad j=1,\dots ,p.
\label{d-j}
\end{equation}
The explicit form of the $q$-dimension reads
\begin{equation}
{\rm dim}_q V_{\lambda}=s_\lambda(q^{p-1}, q^{p-3},\dots ,q^{-p+3}, q^{-p+1}) =
\prod_{1\le i<j\le p}{{(\lambda_i-\lambda_j-i+j)_q} \over({j-i})_q}
\label{dva}
\end{equation}
where $s_\lambda$ is the Schur function corresponding to the partition
$\lambda$. The first equality in (\ref{dva}) was established in
\cite{GLS1}. The second one can be proved by a straightforward
calculations. Its equivalent form can be also found in \cite{Ma}.
So, on taking into account (\ref{d-j}) and (\ref{dva}) we come to
the system of $p$ equations
\begin{equation}
\prod_{i\not=j}{{q\mu_j(\lambda)- q^{-1} \mu_i(\lambda)} \over{\mu_j(\lambda)-
\mu_i(\lambda)}} = \prod_{i\not=j}{{(\lambda_{p-i+1}-\lambda_{p-j+1} +
i-j+1)_q} \over {(\lambda_{p-i+1}-\lambda_{p-j+1} +i-j)_q}},\quad
1\le j\le p.
\label{razm}
\end{equation}
Let us find $\mu_i(\lambda)$ from this system. It is easy to see that
there exist a solution of the form
\begin{equation}
\mu_i(\lambda)=\eta(\lambda) q^{-2(\lambda_{p-i+1}+i)},
\label{mu-sol}
\end{equation}
where $\eta(\lambda)$ is an arbitrary nonzero multiplier. Indeed,
given such $\mu_i(\lambda)$ and taking into account definition
(\ref{q-num}) of the $q$-numbers, we get
$$
{\frac{q\mu_j(\lambda)-q^{-1}\mu_i(\lambda)}
{\mu_j(\lambda)-\mu_i(\lambda)}} =
{\frac{(\lambda_{p+1-i}-\lambda_{p+1-j}+i-j+1)_q}
{(\lambda_{p+1-i}-\lambda_{p+1-j}+i-j)_q}}.
$$
At $p=2$ the above solution (\ref{mu-sol}) is unique since the system
(\ref{razm}) is actually linear in $\mu_i(\lambda)$.
\begin{conjecture}
At an arbitrary value of $p$ the solution (\ref{mu-sol})
of the system (\ref{razm}) is unique.
\end{conjecture}
Up to this Conjecture we can extend relation (\ref{mulp}) for an
arbitrary $p\ge 2$ (at $p=2$ it is true rigorously). Indeed, from
(\ref{raz}) it follows that
$$
d_\mathbf{k}(\lambda, m)=\frac{{\rm dim}_qV_{\hat{\lambda^{\!*}}+\mathbf{k}}}
{{\rm dim}_qV_{\hat{\lambda^{\!*}}}}.
$$
On the other hand, taking $\mu_i$ as in (\ref{mu-sol}) we find
$$
{\frac{q^{k_i-k_j}\mu_i(\lambda)-q^{k_j-k_i}\mu_j(\lambda)}
{\mu_i(\lambda)-\mu_j(\lambda)}} =
{\frac{(\lambda_{p+1-j}-\lambda_{p+1-i}+k_i-k_j-i+j)_q}{(
\lambda_{p+1-j}-\lambda_{p+1-i}-i+j)_q}}.
$$
This relation together with above form of $d_\mathbf{k}(\lambda, m)$ and
(\ref{dva}) entails result (\ref{mulp}) for ${\hbar}=0$. A passage
to the general case can be performed by the shift of generators
(\ref{shift}).
To complete the proof, we point out, that since the category of
finite dimensional equivariant representation of \ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi{} is faithful,
then relation (\ref{mulp}), being valid in all representations, must
take place at the level of the algebra itself.\hfill
\rule{6.5pt}{6.5pt}
\medskip
The above relation (\ref{mu-sol}) shows which values of $\mu_i$
(up to a normalizing factor) correspond to a finite dimensional
representation $\pi_\lambda$ of the NC orbit ${\cal L}_q^\chi$. To
fix the factor $\eta(\lambda)$ one should consider the value of
$\pi_\lambda({\rm Tr}_RL)$ for the mREA \ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi{} and use the connection
(\ref{shift}). The following corollary of proposition \ref{prop:21}
holds true.
\begin{corollary}
Let ${\cal L}_{q,1}^\chi$ be the NC orbit defined by
(\ref{q-gen})--(\ref{ideal}). In the finite dimensional
representation $\pi_\lambda$ of the mREA ${\cal L}_{q,1}$
the eigenvalues $\bar \mu_i(\lambda)$ of the matrix $L$ of the mREA
generators on the orbit ${\cal L}_{q,1}^\chi$ are as follows
\begin{equation}
\bar\mu_i(\lambda) = \frac{(\lambda_{p-i+1}+i-1)_q}{
q^{\lambda_{p-i+1}+i-1}}.
\label{q-mu-mrea}
\end{equation}
Here the eigenvalues $\bar\mu_i$ are the roots of the CH polynomial
for $L$
$$
\prod_{i=1}^p(L-\bar\mu_i\,I) \equiv 0.
$$
\end{corollary}
\noindent
{\bf Proof\ \ }
As was found in \cite{S}, the central element ${\rm Tr}_R L$ has the
following form in the representation $\pi_\lambda$
$$
\pi_\lambda({\rm Tr}_R L) = \chi_1{\rm id}_{V_\lambda}, \quad
\chi_1 = q^{-2p}\sum_{r=1}^pq^{2r-1-\lambda_r} (\lambda_r)_q.
$$
On the other hand, by virtue of (\ref{trl}), (\ref{q-elem-symm}) and
(\ref{NI}) the $R$-trace of the matrix $\hat L$ of the generators of
non-modified REA can be presented as follows
$$
\pi_\lambda({\rm Tr}_R\hat L) = \frac{1}{q}\sum_{i=1}^p\mu_i\,
{\rm id}_{V_\lambda}
$$
were $\mu_i$ are given by (\ref{mu-sol}). Taking into account the
above expressions for the traces and the fact that $L$ and $\hat L$
are connected by means of shift (\ref{shift}), we find
$$
q^{-1}\eta(\lambda)\sum_{r=1}^pq^{-2(\lambda_{p-i+1}+i)} =
q^{-2p}\sum_{r=1}^pq^{2r-1-\lambda_r}(\lambda_r)_q -
\frac{p_q}{q^p(q-q^{-1})}.
$$
This entails
$$
\eta(\lambda) = - \frac{q^2}{(q-q^{-1})}.
$$
And at last, using the connection
$\bar\mu_i = \mu_i + (q-q^{-1})^{-1}$ and values (\ref{mu-sol}), we
come to (\ref{q-mu-mrea}) which is the $q$-generalization of the
classical result (\ref{mu}).\hfill \rule{6.5pt}{6.5pt}
\section{q-Euler characteristic and group $Q({\cal L}_{\h,q}^{\chi})$}
\label{sec:q-eiler}
In what follows we shall consider the multiplicities $d_\mathbf{k}(m)$ on
the set of all generic NC orbits as functions in $\mu_i$. As was
pointed out in remark \ref{DM}, there exist orbits on which some of
the basic multiplicities $d_i$ can vanish. But even if all $d_i$ are
nonzero, some of the higher multiplicities $d_\mathbf{k}(m)$ can vanish as
well. We shall restrict ourselves to the set of generic NC orbits
such that $d_\mathbf{k}(m)\not= 0$ for all $m=1,2,\dots $ and all for all
partitions $\mathbf{k}\vdash m$. Such NC orbits will be called {\em strictly
generic}. However, in considering the multiplicities as functions in
$\mu_i$ it does not matter whether we exclude a low-dimensional
subset of values of $\mu_i$ (corresponding to non-strictly generic
orbits).
Let $M_\mathbf{k}=M_\mathbf{k}(m)$ be the left (for the definiteness) projective
module corresponding to the idempotent $e_\mathbf{k}(m)$ defined on a NC
orbit ${\cal L}_{\h,q}^{\chi}$ in full analogy with (\ref{high-idemp}).
For a strictly generic orbit we consider the set of projective
modules $M_\mathbf{k}$ with the assignment
\begin{equation}
M_\mathbf{k}\mapsto {\rm Tr}_R \,e_\mathbf{k}(m).
\label{char}
\end{equation}
In the classical limit $q=1$, ${\hbar}=0$ of the $U_q(sl(n))$ case this
assignment is out of interest (its value is identically equal to 1
since the modules $M_\mathbf{k}$ correspond to line bundles). But for
generic $q$ and ${\hbar}$ it is not so. Let us show that the
characteristic (\ref{char}) is close to the Euler characteristic
of a line bundle over the flag variety $\rm Fl({\Bbb C}^n)$.
It is known that the set of $SU(n)$-equivariant line bundles over
the flag variety is in the one-to-one correspondence with the set of
holomorphic one-dimensional representations of the torus
$T\subset SU(n)$. Therefore, the bundles can be labelled by the
vectors $\mathbf{k}=(k_1,\dots ,k_n), k_i \in {\Bbb Z}$ since any such a
representation is of the form
$T\ni (t_1,\dots ,t_n) \mapsto \prod t_i^{k_i}$. Two vectors
$\mathbf{k}=(k_1,\dots ,k_n)$ and $\mathbf{k}'= (k'_1,\dots ,k'_n)$ give rise to the
same representation and hence to the same line bundle iff
$k_i=k'_i+a$ with an integer $a$ (shortly iff $\mathbf{k}=\mathbf{k}'+a$) since
$\prod t_i=1$. So, applying (if necessary) a shift by an integer we
can assume that the labels $\mathbf{k}$ of these line bundles obey the
condition $k_i\geq 0$. It can be easily shown (cf. \cite{L}) that
the Euler characteristic of the line bundle corresponding to
$\mathbf{k}=(k_1,\dots ,k_n)$ equals
\begin{equation}
\prod_{i<j}{(k_i-k_j+i-j)\over (i-j)}.
\label{Ec}
\end{equation}
Now let us pass to the NC orbits. Recall the notion of the
$q$-index introduced in \cite{GLS2}. It is defined as the following
paring
$$
\langle M_\mathbf{k}, \overline \pi_\lambda\rangle=\mbox{\bf \sf Tr}\,
\overline \pi_\lambda(e_\mathbf{k}(m))
$$
of a projective ${\cal L}_{\h,q}^{\chi}$ module $M_\mathbf{k}$ and a finite dimensional
equivariant representation $\overline \pi_\lambda$. (Note that the character
$\chi$ depends on the representation $\overline \pi_\lambda$.) Here
$\mbox{\bf \sf Tr}$ stands for the categorical trace applied to the
space ${\rm End\, }(V_{\lambda})\otimes {\rm End\, }(V_{(m)})$. (In contrast with the quantities
${\rm Tr}_R L_{(m)}^k$ considered above where ${\rm Tr}_R$ is applied to the
second factor only.)
As follows from constructions of the previous section, for any
$m$-admissible $\lambda$ we have
\begin{equation}
\langle M_\mathbf{k}(m), \overline \pi_\lambda \rangle = \prod_{i<j} {(\lambda_i-\lambda_j+k_i-
k_j+i-j)_q \over (i-j)_q}.
\label{Hir}
\end{equation}
Putting $\lambda_i=0$ in (\ref{Hir}) we get the quantity\footnote{Since
$\lambda=(0,0,\dots ,0)$ is not an $m$-admissible partition we cannot
directly apply the formula (\ref{Hir}) to the corresponding
representation. However, we consider the extension of (\ref{Hir}) to
this point.}
$$
\chi_q(M_\mathbf{k})=\prod_{i<j}{(k_i-k_j+i-j)_q\over (i-j)_q}
$$
which is a $q$-analog of (\ref{Ec}). We call it {\em the $q$-Euler
characteristic}.
The characteristic $\chi_q(M_\mathbf{k})$ does not depend on a concrete form
of the initial braiding $R$. Note that $\chi_q(M_\mathbf{k})$ is introduced
without any holomorphic structure which is usually employed in the
definition of the Euler characteristic of a line bundle since the
operator $\bar \partial$ is essentially involved in the definition
(cf., e.g. \cite{H}).
In this connection we recall that a generic orbit in $su(n)^*$
(treated as a real algebraic variety) can be equipped with
different complex structures. These structures are labelled by
elements of the Weyl group and the Euler characteristic of a given
line bundle essentially depends on the choice of such an element. In
our setting the quantity $\chi_q(M_\mathbf{k})$ depends on the ordering of
the eigenvalues $\mu_i$. These orderings are also labelled by
elements of the Weyl group.
\begin{remark} {\rm Considering a semisimple orbit ${\cal O}$ in
$su(n)^*$ to be a real algebraic variety, we treat its coordinate
ring as an ${\Bbb R}$-algebra. The algebra arising from the quantization
of the Kirillov bracket on such a real variety can be realized as a
quotient of the enveloping algebra $U(su(n)_{\hbar})$ (cf. \cite{GS2}).
It is also treated as an ${\Bbb R}$-algebra. However, for the
corresponding quotients of the mREA it is not possible to get rid of
complex numbers (cf. \cite{DGH} where an example of the quantum
sphere is considered). That is why we treat the above quotient as a
${\Bbb C}$-algebra and consider it as the quantization of the
complexification of the orbit ${\cal O}$. }
\end{remark}
Considering the $q$-Euler characteristic as a function in $\mu_i$
we can see that
$$
\chi_q(M_\mathbf{k})=\chi_q(M_{\mathbf{k}'})\quad \Leftrightarrow \quad \exists
\,a\in{\Bbb Z}:\;\mathbf{k}=\mathbf{k}'+a.
$$
Being motivated by this property, we introduce the following
equivalence relation on the set of our projective modules
$$
M_\mathbf{k}\sim M_{\mathbf{k}'}\quad \Leftrightarrow\quad \exists
\,a\in{\Bbb Z}:\; \mathbf{k}=\mathbf{k}'+a.
$$
The equivalence class of the module $ M_\mathbf{k}$ will be denoted
$[M_\mathbf{k}]$.
\begin{remark}{\rm In NC geometry one usually employs the following
type of equivalence of projective modules (cf. \cite{Ro}). Two
${\cal A}$-modules are called equivalent if the corresponding idempotents
(being extended by 0 if necessary) are similar. This equivalence is
compatible with the usual trace. Namely, for two such idempotents we
have ${\rm Tr}\,\pi(e_1)={\rm Tr}\,\pi(e_2)$ where $\pi$ is any finite
dimensional representation of the algebra ${\cal A}$. However, such an
equivalence is not compatible with the categorical trace ${\rm Tr}_R$.
The matter is that the categorical trace is not invariant with
respect to a similarity transformation $M\to P^{-1}\, M\, P$ since
the matrices $B$ and $C$ (and their higher extensions) entering
definition (\ref{q-sled}) of the categorical trace are not invariant
under this map (if $R$ differs from the usual flip).}
\end{remark}
Now, taking the flag variety as a pattern, we define a product on
the set of the projective modules $M_\mathbf{k}$:
$$
M_\mathbf{k}\cdot M_{\mathbf{k}'}=M_{\mathbf{k}+\mathbf{k}'}.
$$
It should be pointed out that the above relation does not mean any
product of {\it elements} of the modules in question.
It is evident that thus defined product of modules factorizes to the
set of equivalence classes. This set equipped with the product
becomes a group. The role of the unity is played by the class of the
trivial module. This group is an analog of the Picard group of the
flag variety.
In a similar way we can introduce an analog of G-equivariant
K-theory. Usually the algebra $K_G(pt)$ of a point is defined as an
algebra of all virtual representations of the group $G$. Let $[V]$
be the equivalence class of a finite dimensional module $V$ over the
algebra $\ifmmode {\cal L}_{\hbar,q}\else ${\cal L}_{\hbar,q}$\fi$. Consider the $\Bbb{Z}_q$-algebra additively generated by
all these classes where $\Bbb{Z}_q$ is in turn a ${\Bbb Z}$-algebra generated by
all $q$-numbers $m_q$, $m\in {\Bbb Z}$ (\ref{q-num}). As the sum of
equivalence classes we take $[U]+[V] = [U\oplus V]$ and as the
product --- $[U]\cdot [V]=[U\otimes V]$. The quotient of this algebra
over the subalgebra generated by
$$
[V]-{\rm dim}_q V\,[1]
$$
will be denoted $Q(pt)$. Here $[1]$ is the class corresponding to
the trivial object $V={\Bbb K}$. Thus, the algebra $Q(pt)$ is smaller
than $K_G(pt)$ and it is somewhat similar to the group $K(pt)$ in
the non-equivariant K-theory.
Turning to the NC orbit ${\cal L}_{\h,q}^{\chi}$ we treat the class $[V]$ as that of
the free ${\cal L}_{\h,q}^{\chi}$ module ${\cal L}_{\h,q}^{\chi}\otimes V$. The corresponding idempotent
is the identity operator $I_V$ and its trace equals ${\rm dim}_q V$.
Note, that a direct sum of two projective modules is a projective
module as well. The corresponding idempotent is a direct sum of
idempotents related to the summands. However, the modules $M_\mathbf{k}(m)$
and $M_{\mathbf{k}'}(m)$ with the same $m$ admit the usual sum. The
corresponding idempotents are added as equal size matrices. So, we
get
$$
\sum_{|\mathbf{k}|=m} \,e_\mathbf{k}(m)=I_{V_{(m)}}.
$$
Regardless of the addition type we associate to the sum of two or
more modules the sum of their equivalence classes.
Now, we consider the vector space spanned by the classes $[M_\mathbf{k}]$
and $[1]$ with coefficients from $\Bbb{Z}_q$. Its quotient over subspace
generated by
$$
\sum_{|k|=m} [M_\mathbf{k}]-{\rm dim}_qV_{(m)} [1]
$$
will be denoted $Q({\cal L}_{\h,q}^{\chi})$. Moreover, defining $[M_\mathbf{k}]\cdot [1]$
to be $[M_\mathbf{k}]$ we equip $Q({\cal L}_{\h,q}^{\chi})$ with a ${\Bbb Z}_q$-algebra
structure. Thus, we have $[M_\mathbf{k}]\cdot [V]={\rm dim}_q V\,[M_\mathbf{k}]$ (i.e.
we define the class of the tensor product $M_\mathbf{k}\otimes V$ to be
${\rm dim}_q V\,[M_\mathbf{k}]$).
Proofs of the following two propositions are easy and left to the
reader.
\begin{proposition}
The algebra $Q({\cal L}_{\h,q}^{\chi})$ is generated by the unity $[1]$ and $p$
generators $x_i=[M_i],\,\,1\le i\le p$ subject to the following
relations
\begin{equation}
\sum_{1\le i_1< i_2<\dots< i_k\le p} x_{i_1} \cdot \dots \cdot
x_{i_k} = {\rm dim}_q V_{(1^k)}\,[1]= {p_q! \over {k_q!(p-k)_q!}} [1].
\label{Weyl}
\end{equation}
Here as usual $p_q!=1_q\,2_q\dots p_q$.
\end{proposition}
Canceling $[1]$ in this formula we get the relations close to those
arising in quantum co\-ho\-mo\-lo\-gy theory in a completely
different context.
\begin{example}
Let $p=3$ and $x,\, y,\, z$ be the generators of the algebra
$Q({\cal L}_{\h,q}^{\chi})$. Then they satisfy the system
$$
x+y+z=3_q,\quad x\cdot
y+x \cdot z + y \cdot z = 3_q,\quad x\cdot y \cdot z =1.
$$
\end{example}
\begin{proposition}
The $q$-Euler characteristic being extended at $[1]$ by
$\chi_q([1])=1$ gives rise to a linear map
$$
\chi_q:\; Q({\cal L}_{\h,q}^{\chi})\to {\Bbb Z}_q.
$$
\end{proposition}
|
{
"timestamp": "2004-11-25T16:33:00",
"yymm": "0411",
"arxiv_id": "math/0411579",
"language": "en",
"url": "https://arxiv.org/abs/math/0411579"
}
|
\section{Introduction\label{sec:intro}}
The hard dynamics of processes involving hadrons is nowadays remarkably
well described by QCD predictions. An important role has been played
in this achievement by the ability to compute the relevant reactions
to next-to-leading order (NLO) accuracy, a mandatory step in view
of the large value of the coupling constant $\as$. Although fairly
successful in their phenomenological applications, NLO predictions
are affected by uncertainties that can be of the order of a few
tens of percent, typically estimated by varying the renormalization and
factorization scales. This is a consequence of the fact that, in some cases,
NLO corrections are numerically as important as the
leading order (LO) contributions. By increasing the accuracy of
the perturbative predictions, through the computation of the
next-to-next-to-leading order (NNLO) contributions, one would
certainly reduce the size of the uncertainties, and obtain a firmer
estimate of the rates~\cite{Giele:2002hx}.
Such a task, however, implies finding the
solution of a few highly non trivial technical problems. Recently,
two major bottlenecks have been cleared: two-loop functions with up to
four legs have been computed (with zero or one massive leg)~\cite{twoloop},
and so have the three-loop Altarelli-Parisi kernels~\cite{Moch:2004pa},
which opens the way to {\em exact} NNLO PDF fits.
The present situation is thus fairly similar to that of the early 80's.
Back then, the computation of the
two-loop Altarelli-Parisi kernels~\cite{Curci:1980uw},
and the ability to compute all the tree-level and one-loop functions
for specific processes, left open the problem of achieving explicitly
the cancellation of soft and collinear singularities, as predicted
by the Kinoshita-Lee-Nauenberg (KLN) theorem for inclusive,
infrared-safe observables. Early approaches pioneered
the subtraction~\cite{Ellis:1980wv}
and phase-space slicing~\cite{Fabricius:1981sx} techniques,
for computing analytically the divergent
part of the real corrections, in the context of the prediction for a
given observable; a different observable required a novel computation.
Later, it was realized that the KLN cancellation could be
proven in an observable- and process-independent manner, which allows
one to predict any observable (for which the relevant matrix elements
can be computed) to NLO accuracy without actually using the observable
definition in the intermediate steps of the
computation~\cite{Giele:1991vf}--\cite{Nagy:1996bz}.
Apart from being very flexible, these universal methods have the virtue of
clarifying the fundamental structure of the soft and collinear
regimes of QCD.
Observable- and process-specific calculations have certain advantages over
universal formalisms; the possibility of exploiting the observable definition
and the kinematics of a given reaction in the intermediate steps of the
computation generally leads to more compact expressions to integrate.
This is evident if we consider the fact that the first NNLO results for
total rates~\cite{sigmatot,Hamberg:1990np} pre-date the universal NLO
formalisms by some years. More recently, the first calculation of a
rapidity distribution at NNLO has been performed using these techniques,
for single vector boson
hadroproduction~\cite{Anastasiou:2003yy,Anastasiou:2003ds}.
In the last two years a fair amount of work has been carried out in
the context of process-specific calculations. The structure of NNLO
infrared-singular contributions necessary to implement the subtraction
method has been discussed in the case of
$e^+e^-\to 2~{\rm jets}$~\cite{Gehrmann-DeRidder:2004tv},
and the $\CF^3$ contribution to $e^+e^-\to 3~{\rm jets}$
has been computed~\cite{Gehrmann-DeRidder:2004xe}.
Moreover, an observable-independent method~\cite{Anastasiou:2003gr},
based on sector decomposition~\cite{sector}, has been recently proposed.
This approach allows one to handle and cancel infrared singular
contributions appearing in the intermediate steps of an NNLO calculation
in a fully automatic (numerical) way, and thus significantly differs from the
semi-analytical approaches of refs.~\cite{Giele:1991vf}--\cite{Nagy:1996bz}.
This method has been applied to the computations of
$e^+e^-\to 2~{\rm jets}$~\cite{Anastasiou:2004qd} and Higgs
hadroproduction~\cite{Anastasiou:2004xq} cross sections.
One may argue that, since the number of two-loop amplitudes computed so
far is limited, process-specific (but observable-independent) computations
such as those of
refs.~\cite{Gehrmann-DeRidder:2004xe,Anastasiou:2004qd,Anastasiou:2004xq}
are all what is needed for phenomenology for several years to come.
Although this is a legitimate claim, we believe that universal formalisms
are interesting in themselves, and that their achievement should be pursued
in parallel to, and independently of, that of process-specific computations.
Work in this direction is currently being performed by various groups,
and partial results are becoming
available~\cite{Weinzierl:2003fx}--\cite{Somogyi:2005xz}.
We stress that the formulation of universal subtraction formalisms at NNLO
would pave the way to the matching between fixed-order computations and
parton-shower simulations, similarly to what recently done at
NLO~\cite{Frixione:2002ik,Frixione:2003ei}, thus resulting in predictions
with a much broader range of applicability for phenomenological studies.
The purpose of the present paper is to propose a general framework for the
implementation of an observable- and process-independent subtraction method.
A complete formalism would require the construction of {\em all} of the NNLO
counterterms necessary to cancel soft and collinear singularities in the
intermediate steps of the calculation, as well as their integration over the
corresponding phase spaces. Most of the recent work on the subject
concentrated on the former aspect, without performing the
analytical integration of the kernels. In this paper we take a
different attitude: we propose a subtraction formula, and we give a
general prescription for the construction of all of the counterterms.
However, we do not construct most of them explicitly here,
since we limit ourselves to considering only those
relevant to the $\CF\TR$ part of the dijet cross section in $e^+e^-$
collisions. On the other hand, we specifically address the problem of their
integration. We propose general formulae for the phase spaces necessary for
the integration of the subtraction kernels, and we use them to integrate the
counterterms mentioned above. In this way, we directly prove that the
subtraction formula we propose does allow us to cancel the singularities
at least in the simple case of the $\CF\TR$ contribution to the
$e^+e^-\to 2~{\rm jets}$ cross section, and we construct a numerical code with
which we recover the known results for dijet and total rates. Since the
subtraction formula, the counterterms, and the phase-space measures are
introduced in a way which is fully independent of the hard process considered,
this result gives us confidence that the framework we propose here is general
enough to lead to the formulation of a complete subtraction formalism, whose
explicit construction is, however, beyond the scope of the present paper.
The paper is organized as follows: in sect.~\ref{sec:NLO} we review
the strategy adopted by NLO universal subtraction formalisms,
and we formulate it in language suited to its extension to NNLO.
Such an extension is discussed in sect.~\ref{sec:NNLO}, and our
main NNLO subtraction formula is introduced there. This subtraction
procedure is shown to work in a simple case in sect.~\ref{sec:CFTR}.
We present a short discussion in sect.~\ref{sec:comm}, and give
our conclusions in sect.~\ref{sec:concl}. A few useful formulae are
collected in appendices~\ref{sec:simpl} and~\ref{sec:phsp}.
\section{Anatomy of subtraction at NLO\label{sec:NLO}}
Let us denote by $r$ any real (tree-level) matrix element squared
contributing to the NLO correction of a given process, possibly times
a measurement function that defines an infrared-safe observable,
and by $d\psp$ the phase space. For example, when considering two-jet
production in $\epem$ collisions, $r$ is the product of the
$\gamma^*\to q\bq g$ matrix element squared times the $\delta$ functions
that serve to define the jets within a given jet-finding algorithm,
and $d\psp$ is the three-body phase-space for the final-state partons
$q$, $\bq$, and $g$. As is well known, the integral
\begin{equation}
R=\int r d\psp
\label{Rcont}
\end{equation}
is in general impossible to compute analytically. In the context of
the subtraction method, one rewrites eq.~(\ref{Rcont}) as follows:
\begin{equation}
R=\int\left(r d\psp - \tilde{r}d\tilde{\psp}\right)
+\int\tilde{r}d\tilde{\psp}\,.
\label{subtNLO}
\end{equation}
The quantities $\tilde{r}$ and $d\tilde{\psp}$ are completely arbitrary,
except for the fact that $\tilde{r}d\tilde{\psp}$ must fulfill two conditions:
the first integral in eq.~(\ref{subtNLO}) must be finite, and the second
integral must be calculable analytically; when this happens,
$\tilde{r}d\tilde{\psp}$ is called a {\em subtraction counterterm}.
The former condition guarantees that
the divergences of $\int\!\tilde{r}d\tilde{\psp}$ are the same as those
of $\int\! r d\psp$. Thus, by computing $\int\!\tilde{r}d\tilde{\psp}$ one
is able to cancel explicitly, without any numerical inaccuracies, the
divergences of the one-loop contribution, as stated by the KLN
theorem; when using dimensional regularization with
\begin{equation}
d=4-2\vep
\label{ddim}
\end{equation}
the divergences will appear as poles $1/\vep^k$, with $k=1,2$.
As far as the first integral in eq.~(\ref{subtNLO}) is concerned,
its computation is still unfeasible analytically. However, being
finite, one is allowed to remove the regularization, by letting
$\vep\to 0$, and to compute it numerically.
In the context of the subtraction method, the computation of an
observable to NLO accuracy therefore amounts to finding suitable forms
for $\tilde{r}$ and $d\tilde{\psp}$, that fulfill the conditions given
above. Clearly, it is the asymptotic behaviour of $r$ in the soft
and collinear configurations that dictates the form of $\tilde{r}$.
The major difficulty here is that soft and collinear singularities
overlap, and $\tilde{r}$ should not oversubtract them. To show how
to construct $\tilde{r}$ systematically, let us introduce a few
notations. We call {\em singular limits} those configurations of
four-momenta which may lead to divergences of the real matrix elements.
Whether the matrix elements actually diverge in a given limit depends
on the identities of the partons whose momenta are involved in the limit;
for example, no singularities are generated when a quark is soft, or
when a quark and an antiquark of different flavours are collinear.
We call {\em singular (partonic) configurations} the singular limits
associated with given sets of partons, and we denote them as follows:
\begin{itemize}
\item[$\bullet$] Soft: one of the partons has vanishing energy.
\begin{equation}
S_i:\;\;\;\;\;\;p_i^0\to 0.
\label{sNLO}
\end{equation}
\item[$\bullet$] Collinear: two partons have parallel three-momenta.
\begin{equation}
C_{ij}:\;\;\;\;\;\;\vp_i\parallel\vp_j.
\label{cNLO}
\end{equation}
\item[$\bullet$] Soft-collinear: two partons have parallel three-momenta,
and one of them is also soft.
\begin{equation}
SC_{ij}:\;\;\;\;\;\;(p_i^0\to 0,\vp_i\parallel\vp_j).
\label{scNLO}
\end{equation}
\end{itemize}
The singular limits will be denoted by $S$, $C$, and $SC$, i.e. by
simply removing the parton labels in eqs.~(\ref{sNLO})--(\ref{scNLO}).
In practice, in what follows parton labels will be often understood,
and singular limits will collectively indicate the corresponding
partonic configurations.
For any function $f$, which depends on a collection of four-momenta,
we introduce the following operator:
\begin{equation}
\ENLO(f)=f-\sum_i f(S_i)-\sum_{i<j}f(C_{ij})-\sum_{ij}f(SC_{ij}),
\label{ENLOdef}
\end{equation}
where $f(L_\alpha)$, $L_\alpha=S_\alpha,C_\alpha,SC_\alpha$ is the
asymptotic behaviour of $f$ in the singular partonic
configuration $L_\alpha$. Notice that
the role of indices $i$ and $j$ is symmetric in the collinear limit,
but is not symmetric in the soft-collinear one (see eqs.~(\ref{cNLO})
and~(\ref{scNLO})), and this is reflected in the sums appearing in
the third and fourth terms on the r.h.s. of eq.~(\ref{ENLOdef}).
The asymptotic behaviour $f(L_\alpha)$ is always defined
up to non-singular terms; however, what follows is independent
of the definition adopted for such terms. Although all of the singularities
of $f$ are subtracted on the r.h.s. of eq.~(\ref{ENLOdef}), $\ENLO(f)$ is
not finite, owing to the overlap of the divergences. To get rid of this
overlap, we introduce a set of formal rules, that we call the
$\E$-{\em prescription}:
\begin{enumerate}
\item Apply $\ENLO$ to $f$, getting
\mbox{$f+\sum_{\alpha_1} f_{\alpha_1}^{(1)}$}.
\item Apply $\ENLO$ to every $f_{\alpha_1}^{(1)}$ obtained in this way
and substitute the result in the previous expression,
getting \mbox{$f+\sum_{\alpha_1} f_{\alpha_1}^{(1)}
+\sum_{\alpha_2} f_{\alpha_2}^{(2)}$}.
\item Iterate the procedure until
\mbox{$\ENLO(f_{\alpha_n}^{(n)})=f_{\alpha_n}^{(n)}$}, $\forall\,\alpha_n$,
at fixed $n$.
\item Define \mbox{$\tilde{f}=-\sum_i\sum_{\alpha_i} f_{\alpha_i}^{(i)}$}.
\end{enumerate}
In order to show explicitly how the $\E$-prescription works, let us
apply it step by step. After the first iteration, we find eq.~(\ref{ENLOdef}).
With the second iteration, we need to compute
\begin{eqnarray}
\ENLO(f(S_i))&=&f(S_i)-\sum_{j\ne i} f(S_i\oplus S_j)-
\sum_{j<k}f(S_i\oplus C_{jk})-
\sum_{jk}^{j\ne i}f(S_i\oplus SC_{jk}),
\phantom{aaa}
\label{Etone}
\\
\ENLO(f(C_{ij}))&=&f(C_{ij})-\sum_k f(C_{ij}\oplus S_k)-
\sum_{k<l}^{\{i,j\}\ne\{k,l\}}f(C_{ij}\oplus C_{kl})
\nonumber\\*&&
-\sum_{kl}^{\{i,j\}\ne\{k,l\}}f(C_{ij}\oplus SC_{kl}),
\label{Ettwo}
\\
\ENLO(f(SC_{ij}))&=&f(SC_{ij})-\sum_k^{k\ne i} f(SC_{ij}\oplus S_k)-
\sum_{k<l}^{\{i,j\}\ne\{k,l\}}f(SC_{ij}\oplus C_{kl})
\nonumber\\*&&
-\sum_{kl}^{\{i,j\}\ne\{k,l\}}f(SC_{ij}\oplus SC_{kl}),
\label{Etthree}
\end{eqnarray}
where we denoted by $f(L_{\alpha_1}\oplus L_{\alpha_2})$ the asymptotic
behaviour of the function $f(L_{\alpha_1})$ in the singular partonic
configuration $L_{\alpha_2}$. Although in general the operation $\oplus$
is non commutative, we shall soon encounter examples in which
$f(L_{\alpha_1}\oplus L_{\alpha_2})=f(L_{\alpha_2}\oplus L_{\alpha_1})$.
Notice that we included all possible singular parton configurations
in eqs.~(\ref{Etone})--(\ref{Etthree}), except for the redundant ones --
an example of which would be the case $k=i$ in the second term on the
r.h.s. of eq.~(\ref{Etthree}).
We now have to take into account the fact that we are performing a
computation to NLO accuracy. Thus, the definition of an observable
will eventually be encountered (for example, embedded in a measurement
function), which will kill all matrix element singularities associated
with a partonic configuration that cannot contribute to the observable
definition at NLO. For example, the limit in which two partons are soft
is relevant only to beyond-NLO results, and this allows us to set
$f(S_i\oplus S_j)=0$ $\forall \{i,j\}$; analogously
\begin{equation}
\sum_k f(C_{ij}\oplus S_k)=
f(C_{ij}\oplus S_i)\,,
\end{equation}
since the case $k\ne i$ would result into two unresolved partons,
which is again a configuration that cannot contribute to NLO.
In general, it is easy to realize that the operator $\ENLO$ is
equivalent to the identity when it acts on the terms generated in the
second iteration of the $\E$-prescription (i.e., the terms with negative
signs in eqs.~(\ref{Etone})--(\ref{Etthree})), and thus that, according
to the condition in item 3 above, the $\E$-prescription requires at most
two iterations at NLO (it should be stressed that this would not be true
in the case of an infrared-unsafe observable, which would lead to an
infinite number of iterations). It follows that
\begin{equation}
\tilde{f}=\sum_i f(S_i)-\sum_{ij} f(S_i\oplus C_{ij})+
\sum_{i<j} f(C_{ij})-\sum_{ij} f(C_{ij}\oplus S_i)+\sum_{ij} f(SC_{ij}).
\end{equation}
This expression can be further simplified by observing that the soft
and the collinear limits commute. This allows one to write
\begin{equation}
f(S_i\oplus C_{ij})=f(C_{ij}\oplus S_i)=f(SC_{ij}),
\label{commNLO}
\end{equation}
and therefore
\begin{equation}
\tilde{f}=\sum_i f(S_i)+\sum_{i<j}f(C_{ij})-\sum_{ij}f(SC_{ij}).
\label{ftilNLO}
\end{equation}
Let us now identify $f$ with $rd\psp$. Eq.~(\ref{ftilNLO}) implies that
the term added and subtracted in eq.~(\ref{subtNLO}) reads
\begin{equation}
\tilde{r}d\tilde{\psp}=\sum_i r(S_i)d\psp(S_i)+
\sum_{i<j} r(C_{ij})d\psp(C_{ij})-
\sum_{ij} r(SC_{ij})d\psp(SC_{ij}).
\label{rdmuNLO}
\end{equation}
All that is needed for the construction of a subtraction counterterm
at NLO is eq.~(\ref{rdmuNLO}), and the definition of the rules for
the computations of $r(L)$ and $d\psp(L)$. In other words, all subtraction
procedures at NLO are implementation of eq.~(\ref{rdmuNLO}), i.e. of
the $\E$-prescription, within a computation scheme for the asymptotic
behaviours of the matrix elements and the phase spaces. Although not
strictly necessary in principle, it is always convenient to adopt the
same procedure for the computation of $r(L)$ and of $d\psp(L)$; most
conveniently, this is done by first choosing a parametrization for
the phase space, and by eventually using it to obtain $r(L)$.
\begin{figure}[htb]
\begin{center}
\epsfig{figure=softsq2.eps,width=0.40\textwidth}
~~~~~~\epsfig{figure=collsq2.eps,width=0.40\textwidth}
\caption{\label{fig:limits}
Configurations contributing to soft (left panel)
and collinear (right panel) limits of squared amplitudes.
}
\end{center}
\end{figure}
In order to be more specific, we shall consider two explicit constructions
of subtraction counterterms, namely those of
ref.~\cite{Frixione:1995ms} and of
ref.~\cite{Catani:1996vz}. Since the subtractions always need to be
performed at the level of amplitudes squared, the relevant diagrams
(in a physical gauge) are those depicted in fig.~\ref{fig:limits},
for the soft (left panel) and collinear (right panel) limits
respectively. According to the notations introduced before and
the parton labelings that appear in the diagrams, we denote the
corresponding singular partonic configurations by $S_i$ and $C_{ij}$.
As is well known, the leading (singular) behaviours of the real matrix
elements squared will be given by the following factors
(see e.g. ref.~\cite{Bassetto:1984ik})
\begin{eqnarray}
&&S_i\;\longrightarrow\;\frac{p_j\mydot p_k}{p_i\mydot p_j p_i\mydot p_k}\,,
\label{softfc}
\\
&&C_{ij}\;\longrightarrow\;\frac{1}{p_i\mydot p_j}\,,
\label{collfc}
\end{eqnarray}
with the parton $k$ playing no role in the collinear limit.
In order to integrate the subtraction counterterms analytically,
a phase space parametrization must be chosen such that the leading
divergences displayed in eqs.~(\ref{softfc}) and~(\ref{collfc})
have as trivial as possible a dependence upon the integration
variables. Furthermore, the eikonal and collinear factors of
eqs.~(\ref{softfc}) and~(\ref{collfc}) have manifestly overlapping
divergences; thus, a matching treatment of the two, while not
strictly necessary, would allow an easy identification of the
overlapping contributions.
In ref.~\cite{Frixione:1995ms} the phase space is first decomposed
in a manner which is largely arbitrary, but such that in each of
the resulting regions only one soft and one collinear
singularity at most can arise (i.e., the other singularities are
damped by the $\stepf$ functions which are used to achieve
the partition); the two may also occur simultaneously.
Thus, each region is identified by a pair of parton indices -- say,
$i$ and $j$ -- and no singularity other than \mbox{$1/p_i\mydot p_j$}
can occur in that region. This implies that the eikonal factor in
eq.~(\ref{softfc}) will not contribute a divergence to the region
above when $\vp_i\parallel\vp_k$.
\FIGURE[htb]{
\epsfig{figure=top12.eps,width=0.30\textwidth}
\caption{\label{fig:topone}
Splitting topology contributing to NLO computations.
}}
In turn, this makes possible to choose a parametrization of the phase space,
based on {\em exact} factorization formulae (see app.~\ref{sec:phsp}),
in which a pseudo-parton $i\!j$ (thick line in fig.~\ref{fig:topone})
branches into on-shell partons $i$ and $j$ (narrow lines); in other
words, the phase space of eq.~(\ref{Rcont}) is written as follows:
\begin{equation}
d\psp = d\psp_{n-1}(i\!j)d\pspb_2(i,j)\,,
\label{psfactFKS}
\end{equation}
where $d\pspb$ is the two-body phase space, times the measure over
the virtuality of the branching pseudo-parton $i\!j$.
The parton $k$ may or may not serve to define the integration variables,
but is irrelevant in the treatment of the singularities. Clearly, this
parametrization is suggested by a collinear-like configuration,
but thanks to the partition of the phase space it also allows a
straightforward integration over soft singularities. Graphically,
this is equivalent to squaring the parts of the diagrams in
fig.~\ref{fig:limits} which lay to the left of the cut. The contribution
due to the part to the right of the cut in the diagram on the left panel
is clearly recovered once the sum over parton labels is carried out,
since the role of indices $j$ and $k$ is fully symmetric. Once the
exact parametrization of eq.~(\ref{psfactFKS}) is fixed,
ref.~\cite{Frixione:1995ms} proceeds by defining
\begin{equation}
d\psp(L) = d\psp_{n-1}(\widetilde{i\!j})d\pspb_2(i,j;L)\,,
\label{psfactFKSs}
\end{equation}
where $\widetilde{i\!j}$ is an on-shell parton, and
\begin{eqnarray}
d\pspb_2(i,j;S_i)&=&d\pspb_2(i,j)\Big|_{p_i^0\to 0}\,,
\\
d\pspb_2(i,j;C_{ij})&=&d\pspb_2(i,j)\Big|_{\vp_i\parallel\vp_j}\,,
\\
d\pspb_2(i,j;SC_{ij})&=&d\pspb_2(i,j)\Big|_{p_i^0\to 0,\vp_i\parallel\vp_j}\,,
\end{eqnarray}
are obtained by neglecting constant terms in the corresponding limits.
Obviously, $d\psp(L)$ is not an exact representation of the full phase space
any longer, i.e. $d\psp(L)\ne d\psp$; however, this does not introduce
any approximation in the procedure, since the subtraction counterterm is
subtracted and added back in the physical cross section (eq.~(\ref{subtNLO})).
Finally, the asymptotic behaviours $r(L)$ appearing in eq.~(\ref{rdmuNLO})
are directly taken from factorization formulae, eqs.~(\ref{softfc})
and~(\ref{collfc}), without any further manipulation.
In the dipole formalism of ref.~\cite{Catani:1996vz}, the
following identity is exploited
\begin{equation}
\frac{1}{p_i\mydot p_j p_i\mydot p_k}=
\frac{1}{p_i\mydot p_j p_i\mydot (p_j+p_k)}+
\frac{1}{p_i\mydot p_k p_i\mydot (p_j+p_k)}
\label{eikid}
\end{equation}
for the eikonal factor of eq.~(\ref{softfc}).
The two terms on the r.h.s. of eq.~(\ref{eikid}) are symmetric
for $j\leftrightarrow k$, and thus only the first
one actually needs to be considered.
As far as collinear configurations are concerned, this term is singular
only when $\vp_i\parallel\vp_j$, but not when $\vp_i\parallel\vp_k$.
Thus, the identity in eq.~(\ref{eikid}) has the same function as the
partition of the phase space of ref.~\cite{Frixione:1995ms}.
Furthermore, an exact parametrization of the phase space is chosen
\begin{equation}
d\psp = d\psp_{n-1}(\widetilde{i\!j};k)d\pspb_2(i,j;k)
\label{psfactCS}
\end{equation}
which differs from the one of eq.~(\ref{psfactFKS}) in that the parton
$k$ (called the spectator) plays a fundamental role, since it allows
to put on shell the splitting pseudo-parton even if $i$ and $j$ are
not exactly collinear, or $i$ is not soft. Thanks to this property,
in ref.~\cite{Catani:1996vz} we have
\begin{equation}
d\psp(L) = d\psp\,,\;\;\;\;\;\;\;\;
L=S_i,C_{ij},SC_{ij},
\end{equation}
and thus eq.~(\ref{rdmuNLO}) becomes
\begin{eqnarray}
&&\tilde{r}d\tilde{\psp}=D_{ij;k} d\psp\,,
\label{dipsubt}
\\*
&&D_{ij;k}=r(S_{i;k})+r(C_{ij})-r(SC_{ij})\,,
\label{NLOdip}
\end{eqnarray}
where $r(S_{i;k})$ is obtained by taking the soft limit of the real
matrix element, and keeping only the first term on the r.h.s. of
eq.~(\ref{eikid}).
In summary, the common feature of
refs.~\cite{Frixione:1995ms,Catani:1996vz} is the fact
that, by disentangling (with different techniques) the two collinear
singularities that appear in each eikonal factor, they define subtraction
formalisms based on building blocks which all have a collinear-like topology;
we shall denote by $\To$ this topology, depicted in fig.~\ref{fig:topone}
(the parton labels are obviously irrelevant for topological considerations),
a notation which is reminiscent of a branching after which the list of
resolved partons is diminished by one unity.
The explicit expressions for the building blocks, which originate from
the soft, collinear, and soft-collinear limits and include the treatment
of the phase space, depend on the formalism; however, in all
cases they are combined according to eq.~(\ref{rdmuNLO}), i.e. according
to the $\E$-prescription, in order to construct the sought subtraction
counterterm.
\section{Subtraction at NNLO\label{sec:NNLO}}
In this section, we shall introduce a general framework for the
implementation of a subtraction method to NNLO accuracy.
We shall consider the process
\begin{equation}
\epem\,\longrightarrow\,n~{\rm jets}\,.
\label{physproc}
\end{equation}
In this way, all the intricacies are avoided due to initial-state
collinear singularities, which allows us to simplify the notation
considerably. The systematic construction of the subtraction counterterms
that we propose in the following will however be valid also in the
case of processes with initial-state hadrons, since the procedure is
performed at the level of short-distance partonic cross sections.
On the other hand, we do not present here the explicit parametrizations
of the phase spaces for the case of initial-state partons, and we do
not consider the contributions of initial-state collinear counterterms
which are necessary
in order to achieve the complete cancellation of infrared singularities
for processes with QCD partons in the initial state.
\subsection{Generalities\label{sec:NNLOintro}}
At NNLO, the process~(\ref{physproc}) receives contribution from the
following partonic subprocesses
\begin{equation}
\epem\,\longrightarrow\,m~{\rm partons},\;\;\;\;\;\;
m=n,\,n+1,\,n+2.
\label{partproc}
\end{equation}
We write the amplitude corresponding to eq.~(\ref{partproc}) in the
following way
\begin{equation}
\Am=\gs^{m-2}\Tm + \gs^m \LOm + \gs^{m+2} \LTm+\dots,
\label{ampm}
\end{equation}
where $\Tm$, $\LOm$ and $\LTm$ are the tree-level, one-loop and
two-loop contributions to the process~(\ref{partproc}) respectively.
Squaring eq.~(\ref{ampm}) we get:
\begin{eqnarray}
\abs{\An}^2&=&\gs^{2n-4}\abs{\Tn}^2
+\gs^{2n-2}\left(\Tn\LOnStar+\TnStar\LOn\right)
\nonumber \\*&&
+\gs^{2n}\left(\abs{\LOn}^2+\Tn\LTnStar+\TnStar\LTn\right)
+{\cal O}(\gs^{2n+2}),
\label{ampsqn}
\\
\abs{\Anone}^2&=&\gs^{2n-2}\abs{\Tnone}^2
+\gs^{2n}\left(\Tnone\LOnoneStar+\TnoneStar\LOnone\right)
+{\cal O}(\gs^{2n+2}),
\label{ampsqnone}
\\
\abs{\Antwo}^2&=&\gs^{2n}\abs{\Tntwo}^2
+{\cal O}(\gs^{2n+2}).
\label{ampsqntwo}
\end{eqnarray}
In eq.~(\ref{ampsqn}) the number of partons coincides with the number
of jets of the physical process, and this implies that all partons
must be {\em resolved}, i.e. hard and well separated. On the other
hand, in eqs.~(\ref{ampsqnone}) and~(\ref{ampsqntwo}) the number of
partons exceeds that of jets, which means that one and two partons
respectively are unresolved in these contributions. The accuracy
with which the various terms in eqs.~(\ref{ampsqn})--(\ref{ampsqntwo})
enter the cross section can be read from the power of $\gs$.
The Born contribution is proportional to $\gs^{2n-4}$, and appears
solely in eq.~(\ref{ampsqn}). The NLO contributions are proportional to
$\gs^{2n-2}$, and appear in eqs.~(\ref{ampsqn}) and~(\ref{ampsqnone}).
The divergences of the former are entirely due to the loop integration
implicit in $\LOn$, whereas those of the latter are obtained analytically
after applying the subtraction procedure described in sect.~\ref{sec:NLO}
(there, $\gs^{2n-2}\abs{\Tnone}^2$ has been denoted by $r$). We understand
that, in the actual computation of an infrared-safe observable, the
matrix elements in eqs.~(\ref{ampsqn})--(\ref{ampsqntwo}) are multiplied
by the relevant measurement functions.
The NNLO contributions are proportional to $\gs^{2n}$, and we can
classify them according to the number of unresolved partons.
In the {\em double-virtual} contribution,
$\abs{\LOn}^2+(\Tn\LTnStar+\TnStar\LTn)$, all partons are resolved.
The term in round brackets is identical to the NLO virtual contribution,
except for the fact that one-loop results are formally replaced
by two-loop ones; on the other hand, the former term is typologically
new. However, for both the structure of the singularities is explicit
once the loop computations are carried out. We then have the
{\em real-virtual} contribution, $\Tnone\LOnoneStar+\TnoneStar\LOnone$,
in which one parton is unresolved. This is again formally identical
to the NLO virtual contribution, but there is a substantial difference:
in addition to the singularities resulting from loop integration,
there are singularities due to the unresolved parton, which will appear
explicitly only after carrying out the integration over its phase space.
In order to do this analytically, a subtraction procedure will be
necessary, and the methods of sect.~\ref{sec:NLO} may be applied.
When doing so, however, at variance with an NLO computation the analogue
of the first term on the r.h.s. of eq.~(\ref{subtNLO}) will not be finite,
because of the presence of the explicit divergences due to the one-loop
integration. This prevents us from setting $\vep\to 0$ as in
eq.~(\ref{subtNLO}), and thus ultimately from computing
the integral, since this integration can only be done numerically.
It follows that the straightforward application of an NLO-type
subtraction procedure to the real-virtual contribution {\em alone}
would not lead to the analytical cancellation of all the divergences.
We shall show that such a cancellation can be achieved by adding to the
real-virtual contribution a set of suitably-defined terms obtained from
the {\em double-real} contribution, $\abs{\Tntwo}^2$. This contribution
is characterized by the fact that two partons are unresolved and, analogously
to the case of the real contribution to an NLO cross section, all of the
divergences are obtained upon phase space integration, a task which is overly
complicated due to the substantial amount of overlapping among the various
singular limits.
\subsection{The subtracted cross section\label{sec:NNLOrr}}
In this section we shall use the findings of sect.~\ref{sec:NLO} as a template
for the systematic subtraction of the phase-space singularities of the
double-real contribution, which we shall obtain by suitably generalize
the $\E$-prescription. In order to do this, we start from listing all
singular limits that lead to a divergence of the double-real matrix
elements. We observe that the $S_\alpha$, $C_\alpha$, and $SC_\alpha$
configurations described in sect.~\ref{sec:NLO} are also relevant to
the double-real case. In addition, we have the following
configurations~\cite{Campbell:1997hg,Catani:1999ss}:
\begin{itemize}
\item[$\bullet$] Soft-collinear: two partons have parallel three-momenta,
and a third parton has vanishing energy\footnote{Note therefore that,
upon removing the parton labels, the $SC$ symbol denotes both the cases
with one or two unresolved partons, eqs.~(\ref{scNLO}) and~(\ref{IRSCt})
respectively.}.
\begin{equation}
SC_{ijk}:\;\;\;\;\;\;(p_i^0\to 0,\vp_j\parallel\vp_k).
\label{IRSCt}
\end{equation}
\item[$\bullet$] Double soft: two partons have vanishing energy.
\begin{equation}
SS_{ij}:\;\;\;\;\;\;(p_i^0\to 0,p_j^0\to 0).
\end{equation}
\item[$\bullet$] Double collinear: three partons have parallel three-momenta,
or two pairs of two partons have parallel three-momenta\footnote{The
former case is usually denoted as triple collinear. We prefer this
notation since, if $\vec{a}\parallel\vec{b}$ and $\vec{b}\parallel\vec{c}$,
necessarily $\vec{a}\parallel\vec{c}$. Furthermore, the present notation is
more consistent with the strongly-ordered limit case. It gives minimal but
sufficient information.}.
\begin{eqnarray}
&&CC_{ijk}:\;\;\;\;\;\;(\vp_i\parallel\vp_j\parallel\vp_k),
\label{IRCCt}
\\*
&&CC_{ijkl}:\;\;\;\;\;\;(\vp_i\parallel\vp_j,\vp_k\parallel\vp_l).
\label{IRCCopo}
\end{eqnarray}
\item[$\bullet$] Double soft and collinear: two partons have vanishing energy,
and two partons have parallel three-momenta.
\begin{eqnarray}
&&SSC_{ij}:\;\;\;\;\;\;(p_i^0\to 0,p_j^0\to 0,\vp_i\parallel\vp_j),
\\*
&&SSC_{ijk}:\;\;\;\;\;\;(p_i^0\to 0,p_j^0\to 0,\vp_j\parallel\vp_k).
\end{eqnarray}
\item[$\bullet$] Double collinear and soft: as in the case of double
collinear, but one of the collinear partons has also vanishing energy.
\begin{eqnarray}
&&SCC_{ijk}:\;\;\;\;\;\;(p_i^0\to 0,\vp_i\parallel\vp_j\parallel\vp_k),
\\*
&&SCC_{ijkl}:\;\;\;\;\;\;(p_i^0\to 0,\vp_i\parallel\vp_j,\vp_k\parallel\vp_l).
\end{eqnarray}
\item[$\bullet$] Double soft and double collinear: as in the case of double
collinear, but two of the collinear partons have also vanishing energy.
\begin{eqnarray}
&&SSCC_{ijk}:\;\;\;\;\;\;
(p_i^0\to 0,p_j^0\to 0,\vp_i\parallel\vp_j\parallel\vp_k),
\\*
&&SSCC_{ijkl}:\;\;\;\;\;\;
(p_i^0\to 0,p_k^0\to 0,\vp_i\parallel\vp_j,\vp_k\parallel\vp_l).
\label{IRSSCCopo}
\end{eqnarray}
\end{itemize}
The soft and collinear limits in each of eqs.~(\ref{IRSCt})--(\ref{IRSSCCopo})
are understood to be taken simultaneously, following for example
the rules given in ref.~\cite{Catani:1999ss} (see in particular
eqs.~(23) and~(98) there).
\begin{figure}[htb]
\begin{center}
\epsfig{figure=top22.eps,width=0.25\textwidth}
~~~~~~~~~~~~\epsfig{figure=top1p12.eps,width=0.25\textwidth}
\caption{\label{fig:topNNLO}
NNLO-type topologies: $\Tt$ (left panel) and $\Topo$ (right panel).
}
\end{center}
\end{figure}
The necessity of introducing the notion of
topology emerges at NNLO even without considering the soft
limits and the problem of overlapping divergences, as is clear
by inspection of the purely collinear limits, eqs.~(\ref{IRCCt})
and~(\ref{IRCCopo}). These are associated with the
branching processes depicted in fig.~\ref{fig:topNNLO}; we denote
the corresponding topologies by $\Tt$ and $\Topo$ respectively
(again, this notation serves as a reminder of the number of
partons to be removed from the list of resolved partons).
Some of the singular limits in which one or two partons are soft
cannot be straightforwardly associated with either topology. However,
as in the case of NLO computations, this can be done after
some formal manipulations, whose nature (be either a partition
of the phase space, or a partial fractioning, or something else) we do
not need to specify at this stage. Suffice here to say that, after such
manipulations, all singular limits in eqs.~(\ref{IRSCt})--(\ref{IRSSCCopo})
that feature at least one soft parton will in general contribute to
both $\Tt$ and $\Topo$ topologies (we may formally write, for
example, $f(SS)=f(SS^{(\Tt)})+f(SS^{(\Topo)})$). The same kind
of procedure can be applied to the singular limits of
eqs.~(\ref{sNLO})--(\ref{scNLO}), since topology $\To$ can always
be seen as a sub-topology of $\Tt$ or $\Topo$; thus, $S$, $C$, and
$SC$ limits will be manipulated, if need be, so as to be associated
with topologies $\Tt$ and $\Topo$. In the case we shall need to
distinguish between the singular limits in the various topologies,
we shall denote them by $L^{(\toplab)}$, with $\toplab=\To,\Tt,\Topo$.
However, as for parton labels, topology labels may be understood
in what follows.
We now claim that by applying the $\E$-prescription defined in
sect.~\ref{sec:NLO} we can systematically subtract the singularities
of the double-real contribution to the NNLO cross section, provided
that the operator $\ENLO$ is replaced by $\ENNLO$, where
\begin{eqnarray}
\ENNLO(f)&=&\ENLO(f)-\sum_{\toplab=\Tt,\Topo}\sum_\alpha
\Bigg\{f\left(SC_\alpha^{(\toplab)}\right)+f\left(SS_\alpha^{(\toplab)}\right)+
f\left(CC_\alpha^{(\toplab)}\right)
\nonumber\\*&&\phantom{\sum_{\toplab=\Tt,\Topo}\sum_\alpha}
+f\left(SSC_\alpha^{(\toplab)}\right)+f\left(SCC_\alpha^{(\toplab)}\right)+
f\left(SSCC_\alpha^{(\toplab)}\right)\Bigg\},
\label{ENNLOdef}
\end{eqnarray}
and $\alpha$ denotes all indices relevant to the corresponding singular
limits, which can be read in eqs.~(\ref{IRCCt})--(\ref{IRSSCCopo}).
As in the case of NLO computations, the iterative $\E$-prescription
comes to an end thanks to the infrared safety of the observables.
However, at NNLO up to four iterations are necessary in order to
define $\tilde{f}$, which can be easily generated by means of an
algebraic-manipulation code. In such a way, one obtains up to 51
terms for each topology; fortunately, such a massive counterterm
can be greatly simplified. We start by observing that the freedom
in the definition of the asymptotic form of the matrix elements
associated with a given singular limit allows us to exploit commutation
properties as done in eq.~(\ref{commNLO}). Furthermore, the presence of
$\ENLO$ in the definition of $\ENNLO$ implies that some of the terms
obtained with the $\E$-prescription will be formally identical to those
appearing in an NLO subtraction. This suggests us to write
\begin{equation}
\tilde{f}=\tfmo+\tfmt^{(\Tt)}+\tfmt^{(\Topo)}\,,
\label{ftilNNLOt}
\end{equation}
where
\begin{equation}
\tfmo=\sum_i f\left(S_i^{(\To)}\right)+
\sum_{i<j}f\left(C_{ij}^{(\To)}\right)-
\sum_{ij}f\left(SC_{ij}^{(\To)}\right),
\label{tfmodef}
\end{equation}
and
\begin{eqnarray}
\tfmt^{(\toplab)}&=&\sum_\alpha\Bigg\{
f\left(CC_\alpha^{(\toplab)}\right)+f\left(SS_\alpha^{(\toplab)}\right)
-f\left(SC_\alpha^{(\toplab)}\right)
-f\left([C\oplus SS]_\alpha^{(\toplab)}\right)
\nonumber \\*&&\phantom{\sum_\alpha}
-f\left([CC\oplus S]_\alpha^{(\toplab)}\right)
-f\left([CC\oplus SS]_\alpha^{(\toplab)}\right)
-f\left([S\oplus S]_\alpha^{(\toplab)}\right)
-f\left([C\oplus C]_\alpha^{(\toplab)}\right)
\nonumber \\*&&\phantom{\sum_\alpha}
+f\left([C\oplus S\oplus S]_\alpha^{(\toplab)}\right)
+f\left([CC\oplus S\oplus S]_\alpha^{(\toplab)}\right)
+f\left([C\oplus C\oplus S]_\alpha^{(\toplab)}\right)
\nonumber \\*&&\phantom{\sum_\alpha}
+f\left([C\oplus C\oplus SS]_\alpha^{(\toplab)}\right)
-f\left([C\oplus C\oplus S\oplus S]_\alpha^{(\toplab)}\right)\Bigg\},
\label{tfmtdef}
\end{eqnarray}
where we used the fact that (for example) $SSC=SS\oplus C=C\oplus SS$,
and $\alpha$ denotes symbolically the relevant parton indices,
not indicated explicitly in order to simplify the notation. The physical
meaning of eqs.~(\ref{ftilNNLOt})--(\ref{tfmtdef}) is clear: in the
double real contribution to an NNLO cross section, there are terms with
one or two unresolved partons. The former have the same kinematics as
those relevant to a pure NLO subtraction. More interestingly, they also
have the same kinematics as the real-virtual contribution. Although this
fact formally results from the application of the $\E$-prescription,
it can also be understood intuitively: by requiring more stringent
jet-finding conditions (for example, by enlarging the minimum
$\pt$ which defines a tagged jet), the NNLO cross section turns
into an NLO one, which receives contributions only from the real-virtual
term, and from the pieces obtained by applying $\ENLO$ to the double-real
matrix elements. Clearly, if $f$ is associated with an $(n+2)$-body final
state, each term in $\tfmo$ and $\tfmt$ factorize $(n+1)$-body and $n$-body
hard matrix elements and measurement functions respectively.
In order to give more details on the structure of the subtraction that
emerges from the $\E$-prescription, let us denote by
\begin{eqnarray}
vv&=&\gs^{2n}\left(\abs{\LOn}^2+\Tn\LTnStar+\TnStar\LTn\right),
\\
rv&=&\gs^{2n}\left(\Tnone\LOnoneStar+\TnoneStar\LOnone\right),
\\
rr&=&\gs^{2n}\abs{\Tntwo}^2,
\end{eqnarray}
the double-virtual, real-virtual, and double-real matrix elements
squared respectively, possibly times measurement functions that
we understand. The jet cross section is
\begin{eqnarray}
d\sigma&=&d\sigma_{rr}+d\sigma_{rv}+d\sigma_{vv}
\\*
&=&\int rr d\psp_{n+2}+\int rv d\psp_{n+1}+\int vv d\psp_n\,.
\label{jetxsec}
\end{eqnarray}
We start by applying the $\E$-prescription to the double-real contribution
\begin{equation}
d\sigma_{rr}=\int\left(rr d\psp_{n+2}-\trrmt d\tilde{\psp}_{n+2}^{-2}
-\trrmo d\tilde{\psp}_{n+2}^{-1}\right)
+\int\trrmt d\tilde{\psp}_{n+2}^{-2}
+\int\trrmo d\tilde{\psp}_{n+2}^{-1},
\label{rrsubt}
\end{equation}
where we allowed the possibility of adopting two different parametrizations
for the phase spaces attached to $\trrmt$ and to $\trrmo$.
The first integral on the r.h.s. of eq.~(\ref{rrsubt}) is finite, and can
be computed numerically after removing the regularization by letting
$\vep\to 0$. The second and the third integrals will contain all the
divergences of the double-real contribution, to be cancelled by those
of the real-virtual and double-virtual contributions. However, only the
second term can, at this stage, be integrated analytically. In fact,
$\trrmo$ is, according to its definition, obtained by considering
the asymptotic behaviour of the double-real matrix element squared
in the singular limits $S$, $C$, and $SC$ of NLO nature. These limits
will render manifest only part of the singular structure of the matrix
elements, preventing a complete analytical integration of the divergent
terms. For example, if parton $i$ becomes soft, $rr(S_i)$ factors the
eikonal term of eq.~(\ref{softfc}), and the integration over the
variables of parton $i$ can be carried out analytically, as in the
case of NLO. However, at NNLO we must take into account that
another singularity may appear -- say, parton $l$ may also become
soft, and $rr(S_i)$ has too complicated a dependence upon the variables
of parton $l$ to perform the necessary analytical integration. Notice that
this is not true for, say, $rr(SS_{il})$, which is why the second term on the
r.h.s. of eq.~(\ref{rrsubt}) can indeed be integrated. This suggests
combining $\trrmo$ with the real-virtual contribution which, as discussed in
sect.~\ref{sec:NNLOintro}, cannot be integrated analytically too
(although for different reasons). In order to achieve this combination,
we must choose the phase-space measure appropriately; in particular,
we shall use
\begin{equation}
d\tilde{\psp}_{n+2}^{-1}=
d\psp_{n+1}d\psp_2^{-1}\,,
\label{rvphsp}
\end{equation}
where $d\psp_{n+1}$ is the exact $(n+1)$-body massless phase space,
and $d\psp_2^{-1}$ is related to the phase space relevant to the partons
whose contributions to the singular behaviour of the double-real
matrix elements is explicit in $\trrmo$, and includes the measure over
the virtuality of the branching parton; we shall show in
what follows how to construct explicitly the phase spaces
of eq.~(\ref{rvphsp}). Having done that, we define a subtracted
real-virtual contribution as follows
\begin{equation}
rv^{(s)}=rv+\int\trrmo d\psp_2^{-1}\,.
\label{rvsubt}
\end{equation}
In an NLO computation, eq.~(\ref{rvsubt}) would amount to the
full NLO correction, with the first and the second term on the
r.h.s. playing the roles of virtual and real contributions respectively.
This implies that the {\em explicit} poles in $1/\vep$ that appear
in $rv$ because of the loop integration will be exactly cancelled by those
of $\trrmo$ which result from the phase-space integration $d\psp_2^{-1}$.
Thus, although in an NNLO computation $rv^{(s)}$ still contains
phase-space divergences, we can manipulate it in the same manner
as a real contribution to an NLO cross section:
\begin{equation}
\int rv^{(s)}d\psp_{n+1}=
\int\left(rv^{(s)} d\psp_{n+1} - \widetilde{rv}^{(s)}d\tilde{\psp}_{n+1}\right)
+\int\widetilde{rv}^{(s)}d\tilde{\psp}_{n+1}\,.
\label{rvfinal}
\end{equation}
The last term on the r.h.s. of eq.~(\ref{rvfinal}) can now be integrated
analytically, whereas the first is finite and can be integrated numerically.
Therefore, by combining part of the double-real and the real-virtual
contributions we have managed to define a scheme in which the analytical
integration of all the divergent terms is possible. By combining
eqs.~(\ref{jetxsec}), (\ref{rrsubt}), and~(\ref{rvfinal}) we get
\begin{eqnarray}
d\sigma&=&
\int\left(rr d\psp_{n+2}-\trrmt d\tilde{\psp}_{n+2}^{-2}
-\trrmo d\tilde{\psp}_{n+2}^{-1}\right)
\nonumber\\*
&+&\int\left(rv^{(s)} d\psp_{n+1}-
\widetilde{rv}^{(s)}d\tilde{\psp}_{n+1}\right)
\nonumber\\*
&+&\int\trrmt d\tilde{\psp}_{n+2}^{-2}
+\int\widetilde{rv}^{(s)}d\tilde{\psp}_{n+1}
+\int vv d\psp_n\,,
\label{jetxsecsub}
\end{eqnarray}
where the first two terms on the r.h.s. are finite, and can be integrated
numerically after letting $\vep\to 0$; the sum of the remaining terms is
also finite, but they are individually divergent and must be computed
analytically.
In order to show explicitly how our master subtraction formula
eq.~(\ref{jetxsecsub}) works, we consider the unphysical case in
which only singular limits of collinear nature can contribute to
the cross section. In such a situation, the application of the
$\E$-prescription is trivial, and we readily arrive at\footnote{The
parton indices play an obvious role here, and we omit them in order
to simplify the notation.}:
\begin{eqnarray}
&&\trrmt\, d\tilde{\psp}_{n+2}^{-2}=
rr(CC)\,d\tilde{\psp}_{n+2}^{-2}(CC)-
rr(C\oplus C)\,d\tilde{\psp}_{n+2}^{-2}(C\oplus C)\,,
\label{unprrmt}
\\
&&\trrmo\, d\tilde{\psp}_{n+2}^{-1}=rr(C)\,d\tilde{\psp}_{n+2}^{-1}(C)\,.
\label{unprrmo}
\end{eqnarray}
As in the general formula for the subtraction counterterm at NLO,
eq.~(\ref{rdmuNLO}), we leave the possibility open of associating
different phase spaces measures with different terms in $\trrmt$. This
has the advantage that each parametrization can be tailored in order
to simplify as much as possible the analytical integration. The
drawback is that, in general, there could be algebraic simplifications
among the contributions to $\trrmt$ (here, between $rr(CC)$ and
$rr(C\oplus C)$), which can be explicitly carried out only upon
factorizing the phase space. We shall discuss this point further
in the following, in the context of a more physical case.
By construction, $rr(C\oplus C)$ is the strongly-ordered, double-collinear
limit. This may coincide with $rr(CC)$, and it does in particular in
topology $\Topo$, but in general $\trrmt$ is different from
zero\footnote{Here and in what follows, ``zero'' means non-divergent.},
and corresponds to the non-strongly-ordered part of the double-collinear
limit. As such, when a kinematic configuration is generated in which
two, and only two partons are collinear, we have $\trrmt\to 0$; in
other words, the $C$ limit of $\trrmt$ is zero. This is what should
happen: in the collinear limit, $rr(C)$ (which appears in $\trrmo$)
is sufficient to cancel locally the divergences of $rr$, and thus
$\trrmt$ should not diverge in this limit, since otherwise the first
term on the r.h.s. of eq.~(\ref{jetxsecsub}) would not be finite.
Analogously, in the double-collinear limit the local counterterm
for $rr$ is $rr(CC)$; therefore, in order to avoid divergences,
in such a limit the contribution of $rr(C\oplus C)$ (in $\trrmt$)
must cancel that of $rr(C)$ (in $\trrmo$). Notice that, for these
cancellations to happen not only at the level of matrix elements,
but also at the level of cross sections, suitable choices of the phase
spaces must be made. The $C$ limit thus relates $d\tilde{\psp}_{n+2}^{-2}(CC)$
to $d\tilde{\psp}_{n+2}^{-2}(C\oplus C)$, whereas the $CC$ limit relates
$d\tilde{\psp}_{n+2}^{-2}(C\oplus C)$ to $d\tilde{\psp}_{n+2}^{-1}(C)$.
A suitable choice in the former case clearly includes the trivial one,
where the two phase spaces are taken to be identical.
We finally note that the subtraction formula of eq.~(\ref{jetxsecsub}) appears
in a very similar form in ref.~\cite{Weinzierl:2003fx}, where a first
discussion was given on the extension of the subtraction method to NNLO.
Ref.~\cite{Weinzierl:2003fx} constructs the NNLO subtraction formula
building upon the NLO dipole subtraction~\cite{Catani:1996vz}, and provides
the explicit expressions of the NNLO counterterms for the leading-colour
contribution to $e^+e^-\to 2~{\rm jets}$. Ref.~\cite{Weinzierl:2003fx},
however, does not discuss the way in which the subtraction kernels
can be constructed for more general processes. In our approach, the
$\E$-prescription provides a general framework for the construction of the
counterterms $\trrmo$, $\trrmt$ and $\widetilde{rv}^{(s)}$, and explicitly
suggests the subtractions of eq.~(\ref{jetxsecsub}). We find it reassuring
that we arrive at a subtraction structure consistent with that of
ref.~\cite{Weinzierl:2003fx}, given the fact that neither here nor in
ref.~\cite{Weinzierl:2003fx} a formal proof is given that
eq.~(\ref{jetxsecsub}) achieves a complete cancellation of the infrared
singularities. We do obtain such a cancellation explicitly, for the $\CF\TR$
colour factor of the dijet cross section, as we shall show in the next
section. On the other hand, to the best of our knowledge the counterterms
presented in ref.~\cite{Weinzierl:2003fx} have not yet been integrated over
the corresponding phase spaces.
\section{An application: the $\CF\TR$ part of
$\epem\to 2$ jets\label{sec:CFTR}}
In this section, we shall apply the subtraction procedure
discussed in sect.~\ref{sec:NNLO} to a physical case, namely the
contribution proportional to the colour factor $\CF\TR$ of the
dijet cross section in $\epem$ collisions. This is a relatively
simple part of the complete calculation of an observable to NNLO,
but it allows us to discuss the practical implementation of most
of the features of the subtraction procedure we propose in this
paper. The branching kernels that we shall introduce are
universal, i.e. they can be used in any other computations where
they are relevant. We shall also define precisely the phase spaces
needed to integrate the above kernels over the variables of
unresolved partons, i.e. the quantities $d\tilde{\psp}$ used in
the formal manipulations of sect.~\ref{sec:NNLO}. We shall show
that the subtraction procedure leads to the expected KLN
cancellation, and that the numerical integration of the finite
remainder gives a result in excellent agreement with that
obtained in ref.~\cite{Anastasiou:2004qd}.
Although not necessary for the implementation of eq.~(\ref{jetxsecsub}),
in this section we shall use partial fractioning to deal with the
eikonal factors associated with soft singularities, and combine them
with the corresponding collinear factors by using the {\em same}
kinematics in the hard matrix elements that factorize. Since such
kinematics will also enter the measurement functions appearing
in the counterterms, all of the manipulations involving the hard matrix
elements will also apply to the measurement functions; for this reason,
the latter will be left implicit in the notation.
We shall reinstate here the notation commonly used
for QCD amplitudes, which can be written as vectors in a colour space
including the coupling constant. Thus, eq.~(\ref{ampm}) is
now rewritten as
\begin{equation}
\colket{\cm_{a_1a_2\dots a_m}}=\colket{\cmt_{a_1a_2\dots a_m}}+
\colket{\cmo_{a_1a_2\dots a_m}}+\colket{\cmd_{a_1a_2\dots a_m}}+\dots
\end{equation}
where ${a_1a_2\dots a_m}$ are flavour indices, and particles other than
QCD partons are always understood. When not necessary, flavour labels
and colour vector symbols may also be understood.
UV renormalization is performed in the ${\overline {\rm MS}}$ scheme,
just by expressing
the bare coupling $\asu$ in terms of the renormalized coupling $\as(\mu^2)$
at the renormalization scale $\mu$. We use the following expression
\begin{equation}
\label{msbarreno}
\asu\,\mu_0^{2\ep}\,S_{\ep} = \as(\mu^2)\,\mu^{2\ep}
\left[ 1 - \f{\as(\mu^2)}{2\pi} \;\frac{\beta_0}{\ep} +\dots\right]\,,
\end{equation}
where $\beta_0=\f{11}{6} \CA-\f{2}{3}\nf\TR$, and
\begin{equation}
S_\ep=(4\pi)^\ep e^{-\ep\Euler}
\end{equation}
is the typical phase space factor in $d=4-2\ep$ dimensions
($\Euler=0.5772...$ being the Euler number).
\begin{figure}[htb]
\begin{center}
\epsfig{figure=dijetrr.eps,width=0.25\textwidth}
~~~~~~~~~~~~~~~~~~\epsfig{figure=dijetrv2.eps,width=0.25\textwidth}
\caption{\label{fig:diagrm}
Sample diagrams for double-real (left panel) and real-virtual (right panel)
contributions to the $\CF\TR$ part of the dijet cross section.
}
\end{center}
\end{figure}
The diagrams contributing to the $\CF\TR$ part of $\cmt_4$ (i.e., to the
double real) are of the kind of that displayed in the left panel of
fig.~\ref{fig:diagrm}; thus, the singular limits we are interested in are
associated with the branchings $g\to q\bq$ and $q\to q\qp\bqp$ (the
identical-flavour branching $q\to qq\bq$ is also accounted for by considering
$q\to q\qp\bqp$, since its interference contributions are not proportional to
$\CF\TR$). As far as the real-virtual contribution is concerned, a sample
diagram of $\cmo_3$ is depicted in the right panel of fig.~\ref{fig:diagrm};
the singular limits of phase-space origin which, according to the discussion
given in sect.~\ref{sec:NNLOrr}, are obtained upon applying the
$\E$-prescription, are associated with the branching $q\to qg$.
\subsection{The double-real contribution\label{sec:djrr}}
Thanks to the universality properties of soft and collinear emissions,
any matrix element with
singularities due to the branchings $g\to q\bq$ and $q\to q\qp\bqp$ can
be used to define the subtraction terms relevant to the double-real
contribution. Thus, we write the double-real matrix element
squared as
\begin{equation}
rr=\langle\cmt_{\rrtbrc\dots a_{n+2}}
\colket{\cmt_{\rrtbrc\dots a_{n+2}}}\,,
\label{djrr}
\end{equation}
where the labels imply that the four-momenta of $\qp$, $\bqp$
and $q$ are $k_1$, $k_2$ and $k_3$ respectively. When applying the
$\E$-prescription to eq.~(\ref{djrr}) we obtain from
eqs.~(\ref{tfmodef}) and~(\ref{tfmtdef})
\begin{eqnarray}
\trrmt&=&rr(CC)+rr(SS)-rr(CC\oplus SS)
\nonumber \\*
&-&rr(C\oplus C)-rr(C\oplus SS)+rr(C\oplus C\oplus SS)\,,
\label{djtrrmt}
\\
\trrmo&=&rr(C)\,,
\label{djtrrmo}
\end{eqnarray}
where clearly $C\equiv C_{12}$, $CC\equiv CC_{123}$, and so forth.
In order to compute explicitly the quantities that appear in
eqs.~(\ref{djtrrmt}) and~(\ref{djtrrmo}), we shall use the results
of ref.~\cite{Catani:1999ss}
(see also refs.~\cite{Campbell:1997hg,DelDuca:1999ha}).
The asymptotic behaviours obtained through
successive iterations of $\ENNLO$, characterized by the $\oplus$ symbol,
can be freely defined to a certain extent. We give such definitions
within a given parametrization of the phase space, which we shall
introduce in the next subsection.
\subsubsection{Choices of phase spaces\label{sec:djrrps}}
As discussed in sect.~\ref{sec:NNLOrr} (see in particular
eq.~(\ref{rrsubt})), the definition of the double-real subtraction
terms implies the necessity of defining $d\tilde{\psp}_{n+2}^{-2}$
and $d\tilde{\psp}_{n+2}^{-1}$ which are related to some extent.
As a preliminary step, we make here the choice of associating
the same $d\tilde{\psp}_{n+2}^{-2}$ with all of the terms that
appear in eq.~(\ref{djtrrmt}); this is in principle not necessary
(see eq.~(\ref{unprrmt})), but we find it non restrictive in
the computations that follow (more complicated kernels {\em may}
require a different choice). We then write the exact $(n+2)$-body
phase space using eq.~(\ref{ps:phspfact})
\begin{equation}
d\psp_{n+2}=
\frac{ds_{123}}{2\pi}\,
d\psp_{n}(123)\,d\psp_3(1,2,3)\,,
\label{djrrEpst}
\end{equation}
where we use the momenta labels rather than the four-momenta to shorten
the notation (consistently with eq.~(\ref{psfactFKS}), $123$ means
$k_1+k_2+k_3$), and
\begin{equation}
s_{123}=(k_1+k_2+k_3)^2\,.
\end{equation}
As in the case of NLO computations, eq.~(\ref{djrrEpst}) is unsuited for
the analytical integration over the variables of unresolved partons since
$123$, which enters the phase space associated with the non-singular part
of the matrix element, has an off-shellness $s_{123}$, while the reduced
matrix element which corresponds to such non-singular part has all of the
final-state QCD partons with zero mass. We follow the same strategy
as in eq.~(\ref{psfactFKSs}), i.e. that of
ref.~\cite{Frixione:1995ms}: we introduce a
four-momentum $\widetilde{123}$ with invariant mass equal to
zero\footnote{$\widetilde{123}$ can be defined to have the same
three-momentum as $123$ in the $\epem$ rest frame, and zero mass.
However, the precise definition is irrelevant in what follows.}, and define
\begin{equation}
d\tilde{\psp}_{n+2}^{-2}=
d\psp_{n}(\widetilde{123})\,
\frac{ds_{123}}{2\pi}\,d\psp_3(1,2,3)\,.
\label{psrrmt}
\end{equation}
This definition is equivalent to that in eq.~(\ref{psfactFKSs}),
and differs from that in eq.~(\ref{psfactCS}) in that no spectator
is used to keep the branching parton on shell.
For the definition of $d\tilde{\psp}_{n+2}^{-1}$, we use
again eq.~(\ref{ps:phspfact})
\begin{equation}
d\psp_{n+2}=
\frac{ds_{12}}{2\pi}\,
d\psp_{n+1}(12)\,d\psp_2(1,2)\,,
\label{djrrEpso}
\end{equation}
from which the analogue of eq.~(\ref{psrrmt}) can readily follow:
\begin{equation}
d\tilde{\psp}_{n+2}^{-1}=
d\psp_{n+1}(\widetilde{12})\,\frac{ds_{12}}{2\pi}\,d\psp_2(1,2)\,.
\label{psrrmow}
\end{equation}
There is however a subtlety: as discussed in sect.~\ref{sec:NNLOrr},
the choice of $d\tilde{\psp}_{n+2}^{-1}$ must match that of
$d\tilde{\psp}_{n+2}^{-2}$. In order to see how this can happen,
we use eq.~(\ref{ps:threebdso}) to write
\begin{equation}
d\tilde{\psp}_{n+2}^{-2}=
d\psp_{n}(\widetilde{123})\,
\frac{ds_{123}}{2\pi}\,\frac{ds_{12}}{2\pi}\,
d\psp_2(3,12)\,d\psp_2(1,2)\,.
\label{tbtemp}
\end{equation}
The part relevant to the $12\to 1+2$ branching is identical in
eqs.~(\ref{psrrmow}) and~(\ref{tbtemp}), except for the fact that the
invariant mass of the $12$ system, $s_{12}$, enters $d\psp_2(3,12)$, and
not only $d\psp_2(1,2)$, in eq.~(\ref{tbtemp}). If this dependence could
be safely neglected in the limit $s_{12}\to 0$, i.e. in the singular region
in which $d\tilde{\psp}_{n+2}^{-1}$ and $d\tilde{\psp}_{n+2}^{-2}$ must
match, the choice of eq.~(\ref{psrrmow}) would be appropriate. However,
as can be seen from eqs.~(\ref{ps:tdsoinn}) and~(\ref{ps:gramsa}),
$d\psp_2(3,12)$ contains the factor
\begin{equation}
\left(z_3 z_{12} s_{123} - z_3 s_{12}\right)^{-\ep}\,,
\label{psreg}
\end{equation}
which acts as a regulator in the integrations in $d\psp_2(3,12)$;
since these are in general divergent, the regulator affects finite
(and possibly also divergent) terms, and thus cannot be ignored even if the
limit $s_{12}\to 0$ is considered. The regulator of eq.~(\ref{psreg})
is implicit in $d\psp_{n+1}(12)$ in eq.~(\ref{djrrEpso}); the
effect of neglecting it as done by defining $d\tilde{\psp}_{n+2}^{-1}$
in eq.~(\ref{djrrEpso}) therefore leads to neglecting contributions
to the cross section if $d\psp_{n+1}(\widetilde{12})$ is involved
in the integration of divergent terms. This is what happens, since
the subtracted real-virtual contribution is indeed divergent.
This problem may seem to be cured by
inserting the regulator of eq.~(\ref{psreg}) into the r.h.s. of
eq.~(\ref{psrrmow}). However, the definition of $d\tilde{\psp}_{n+2}^{-1}$
should not depend on whether the branching $12\to 1+2$ is followed
by the branching $123\to 3+12$, which is what the presence of
$s_{123}$ in eq.~(\ref{psreg}) implies. The most general form
for the regulator to be inserted in eq.~(\ref{psrrmow}) can be
deduced by writing
\begin{equation}
d\psp_{n+2}=
\frac{ds_{12}}{2\pi}\,\frac{ds_{3\dots(n+2)}}{2\pi}\,
d\psp_2(12,3\dots(n+2))\,d\psp_2(1,2)\,d\psp_n(3,\dots,(n+2))\,.
\label{djrrpstt}
\end{equation}
The first phase space on the r.h.s. features the regulator
\begin{equation}
\left(z_{12} z_{3\dots(n+2)} Q^2 - z_{12} s_{3\dots(n+2)}
- z_{3\dots(n+2)} s_{12}\right)^{-\ep}\equiv
\left(z_{3\dots(n+2)} s_{12}^\Mx\right)^{-\ep}
\left(1-\frac{s_{12}}{s_{12}^\Mx}\right)^{-\ep}\,,
\label{grampsmo}
\end{equation}
which can be obtained from eqs.~(\ref{ps:twobd}) and~(\ref{ps:twobdgram}),
and where
\begin{equation}
s_{12}^\Mx=\left(z_{12} z_{3\dots(n+2)} Q^2 - z_{12} s_{3\dots(n+2)}\right)/
z_{3\dots(n+2)}\,,
\label{s12maxdef}
\end{equation}
with \mbox{$Q^2=(k_1+\dots+k_{n+2})^2$}. Eq.~(\ref{grampsmo}) suggests
to replace eq.~(\ref{psrrmow}) with
\begin{equation}
d\tilde{\psp}_{n+2}^{-1}=
d\psp_{n+1}(\widetilde{12})\,\frac{ds_{12}}{2\pi}\,
\left(1-\frac{s_{12}}{s_{12}^\Mx}\right)^{-\ep}
d\psp_2(1,2)\,,
\label{psrrmo}
\end{equation}
where the quantity $s_{12}^\Mx$ is a constant with respect to the integration
in $s_{12}$.
\subsubsection{Computation of the divergent terms\label{sec:djrrdiv}}
In this section, we integrate $\trrmt$ and $\trrmo$ given in
eqs.~(\ref{djtrrmt}) and~(\ref{djtrrmo}) over the phase spaces
defined in sect.~\ref{sec:djrrps}. We start from the case
of $\trrmt$, and write
\begin{equation}
\trrmt=\colbra{\cmt_{q\dots a_{n+2}}}
\bKth_{\rrtbrc}
\colket{\cmt_{q\dots a_{n+2}}}\,.
\label{djrrmt}
\end{equation}
The kernel $\bKth_{\rrtbrc}$ is a matrix in the colour space, and
collects all singular behaviours associated with the branching
$q\to\rrtbrc$ according to the combination given in eq.~(\ref{djtrrmt}).
For its explicit construction we shall use the factorization
formulae given in ref.~\cite{Catani:1999ss}.
In general, we shall denote the branching kernels as follows
\begin{equation}
\bK^{(n)}_{a_1\dots a_n}\;\;\;\;\;\;{\rm for}\;\;\;\;\;\;
S(a_1,\dots,a_n)\to a_1+\dots+a_n\,,
\label{kernnot}
\end{equation}
where $S(a_1,\dots,a_n)$ is the parton flavour which matches the
combination $a_1+\dots+a_n$. In this paper, only $n=2,3$ will be
considered. Notice that the notation of eq.~(\ref{kernnot}) does
not coincide with the one typically used for two-parton branchings,
where the Altarelli-Parisi kernel is denoted by $P_{a_1S(a_1,a_2)}$.
There is a remarkable simplification that occurs in the computation
of eq.~(\ref{djrrmt}); namely, one can prove that (see app.~\ref{sec:simpl})
\begin{equation}
rr(SS)=rr(C\oplus SS)\,,\;\;\;\;\;\;\;\;
rr(CC\oplus SS)=rr(C\oplus C\oplus SS)\,.
\end{equation}
Thus
\begin{equation}
\trrmt=rr(CC)-rr(C\oplus C)\,.
\label{djtrrmtsim}
\end{equation}
Since in the collinear limits colour correlations do not appear,
eq.~(\ref{djtrrmtsim}) implies that $\bKth_{\rrtbrc}$ has a trivial
structure in colour space. Using the same normalization in the
factorization formulae as in ref.~\cite{Catani:1999ss} (see
eq.~(29) there), we thus rewrite the kernel as follows
\begin{equation}
\bKth_{\rrtbrc}=
\f{(8\pi\asu\mu_0^{2\ep})^2}{s_{123}^2}
\left(K_{\rrtbrc}^{CC}-K_{\rrtbrc}^{\CpC}\right)\,,
\label{Kthsim}
\end{equation}
where the first term can be read from eq.~(57) of ref.~\cite{Catani:1999ss}
\begin{equation}
K_{\rrtbrc}^{CC}=
\f{1}{2} \, \CF\TR \,\f{s_{123}}{s_{12}}
\Bigg[- \f{t_{12,3}^2}{s_{12}s_{123}}
+\f{4z_3+(z_1-z_2)^2}{z_1+z_2}+(1-2\ep)
\left(z_1+z_2-\f{s_{12}}{s_{123}}\right)\Bigg]\,,
\label{K3cc}
\end{equation}
and
\begin{equation}
t_{ij,k} \equiv 2 \;\f{z_i s_{jk}-z_j s_{ik}}{z_i+z_j} +
\f{z_i-z_j}{z_i+z_j} \,s_{ij}\,.
\label{tvar}
\end{equation}
The quantity $K_{\rrtbrc}^{\CpC}$ in eq.~(\ref{Kthsim}) corresponds
to the successive branchings $q\to q_3g$, $g\to\qp_1\bqp_2$. We can
therefore construct it by using the polarized Altarelli-Parisi kernels
$\Ph_{gq}^{\mu\nu}$ and $\Ph_{q\bq}^{\mu\nu}$, as shown in
app.~\ref{sec:simpl}. The result is
\begin{equation}
K^{\CpC}_{\rrtbrc}=\f{1}{2} \,
\CF \TR \,\f{s_{123}}{s_{12}}
\Bigg[- \f{t_{12,3}^{(S)^2}}{s_{12}s_{123}}
+\f{4z_3+(z_1-z_2)^2}{z_1+z_2}+(1-2\ep)\left(z_1+z_2\right)\Bigg]\,,
\label{K3cpc}
\end{equation}
where
\begin{equation}
t_{ij,k}^{(S)}\equiv 2 \;\f{z_i s_{jk}-z_j s_{ik}}{z_i+z_j}\,.
\end{equation}
We note that the result in eq.~(\ref{K3cpc}) corresponds to the naive
$s_{12}\to 0$ limit of $K_{\rrtbrc}^{CC}$ in eq.~(\ref{K3cc}),
since $t_{12,3}\to t_{12,3}^{(S)}\propto\sqrt{s_{12}}$ in such a limit
(see app.~\ref{sec:phsp}).
The kernel $\bKth_{\rrtbrc}$ is therefore manifestly simpler than
its two contributions in eq.~(\ref{Kthsim}); using the results of
eqs.~(\ref{K3cc}) and~(\ref{K3cpc}), we find
\begin{equation}
\bKth_{\rrtbrc}=
\f{(8\pi\asu\mu_0^{2\ep})^2}{s_{123}^2}\,
\f{1}{2}\CF\TR \f{s_{123}}{s_{12}}
\Bigg[- \f{t_{12,3}^2-t_{12,3}^{(S)^2}}{s_{12}s_{123}}
-(1-2\ep)\f{s_{12}}{s_{123}}\Bigg]\,.
\label{Kthfin}
\end{equation}
We can now obtain the analytical expressions of the $1/\ep$ poles in
$\trrmt$. Using eq.~(\ref{psrrmt})
\begin{equation}
\int\trrmt d\tilde{\psp}_{n+2}^{-2}=
\abs{\cmt_{q\dots a_{n+2}}}^2 d\psp_{n}
\int \frac{ds_{123}}{2\pi}\bKth_{\rrtbrc}\,d\psp_3(1,2,3)\,.
\label{djrrmtI}
\end{equation}
We use the parametrization of eq.~(\ref{ps:pstbso}) to perform the
integration in eq.~(\ref{djrrmtI}). The invariants are expressed in
terms of the integration variables according to eqs.~(\ref{ps:sot})
and~(\ref{ps:stt}); however, one must be careful when replacing
these expressions into $t_{12,3}^{(S)}$, since in doing so
finite terms are generated when $s_{12}\to 0$; such terms should not
appear, since they have been explicitly neglected when working
$K^{\CpC}_{\rrtbrc}$ out. To avoid this, we use
\begin{equation}
z_1 s_{23}-z_2 s_{13}\to \sqrt{s_{12}}\xi_0\,,
\;\;\;\;\;\;\;\;
\xi_0=2\sqrt{z_3 \zeta_2(1-\zeta_2)s_{123}(1-z_3)}\, x
\end{equation}
in the computation of $t_{12,3}^{(S)}$. In a more general case,
the replacements
\begin{equation}
s_{13}\to s_{123}(1-\zeta_2)\,,\;\;\;\;\;\;\;\;
s_{23}=s_{123}\, \zeta_2
\end{equation}
should also be made. The result of the integral in
eq.~(\ref{djrrmtI}) is
\begin{equation}
\int \frac{ds_{123}}{2\pi}\bKth_{\rrtbrc}\,d\psp_3(1,2,3)=
\left(\f{\as}{2\pi}\right)^2\CF\TR
\left(\f{s_{123}^{\Mx}}{\mu^2}\right)^{-2\ep}
\f{1}{6\ep}\left(1+\f{31}{6}\ep\right)
+{\cal O(\ep)}\,,
\label{intKth}
\end{equation}
where we denoted by $s_{123}^{\Mx}$ the upper limit of the integration
in $s_{123}$, whose form does not need be specified here; a definite
choice will be made in sect.~\ref{sec:res}.
We now turn to the case of $\trrmo$, and write the analogue of
eq.~(\ref{djrrmt})
\begin{equation}
\trrmo=\colbra{\cmt_{gq_3\dots a_{n+2}}}
\bKtw_{\rrobrc}
\colket{\cmt_{gq_3\dots a_{n+2}}}\,.
\label{djrrmo}
\end{equation}
As in the case of $\bKth_{\rrtbrc}$, the kernel $\bKtw_{\rrobrc}$
is purely collinear (see eq.~(\ref{djtrrmo})), and therefore has a
trivial colour structure. Using again the factorization formulae
with the normalization of ref.~\cite{Catani:1999ss} (see eq.~(7)
there), we have
\begin{equation}
\langle\mu|\bKtw_{\rrobrc}|\nu\rangle
=\f{8\pi\asu\mu_0^{2\ep}}{s_{12}}\Ph_{q\bq}^{\mu\nu}\,,
\label{Ktw}
\end{equation}
where $\Ph_{q\bq}^{\mu\nu}$ is given in eq.~(\ref{Pqq}).
Spin correlations are essential for the kernel
$\bKtw$ to be a local subtraction counterterm,
but they do not contribute to the analytic integration which follows,
where the spin-dependent collinear kernel $\Ph_{q\bq}^{\mu\nu}$
can be replaced by
\begin{equation}
\langle P_{q\bq}\rangle=\TR\Bigg[1-\frac{2z(1-z)}{1-\ep}\Bigg]\, .
\end{equation}
Using eq.~(\ref{psrrmo}) we obtain
\begin{equation}
\int\trrmo d\tilde{\psp}_{n+2}^{-1}=
\abs{\cmt_{gq_3\dots a_{n+2}}}^2 d\psp_{n+1}
\int \frac{ds_{12}}{2\pi}\bKtw_{\rrobrc}
\left(1-\frac{s_{12}}{s_{12}^\Mx}\right)^{-\ep}d\psp_2(1,2)\,.
\label{djrrmoI}
\end{equation}
The integral can be performed straightforwardly, with the result
\begin{eqnarray}
&&\int \frac{ds_{12}}{2\pi}\bKtw_{\rrobrc}
\left(1-\frac{s_{12}}{s_{12}^\Mx}\right)^{-\ep}d\psp_2(1,2)=
\f{\as}{2\pi}\, \f{e^{\ep\Euler}}{\Gamma(1-\ep)}\,
\left(\f{\mu^2}{s_{12}^{\Mx}}\right)^{\ep}\TR\left(-\frac{1}{\ep}\right)
\nonumber \\*&&\phantom{aaaaaaaaaaa}\times
\Bigg[\f{2}{3}+\f{10}{9} \ep+\left(\f{56}{27}-\f{\pi^2}{9}\right)\ep^2
+\left(\f{328}{81}-\f{5}{27}\pi^2-\f{4}{3}\zeta_3\right)\ep^3
\Bigg]
\nonumber \\*&&\phantom{aaaaaaaaaaa}\times
\left(1-\f{\pi^2}{6}\ep^2-2\ep^3\zeta_3\right)
+{\cal O}(\ep^3)\,,
\label{intKtw}
\end{eqnarray}
where all the terms up to ${\cal O}(\ep^2)$ have been kept, since
eq.~(\ref{intKtw}) will be used to define the subtracted real-virtual
contribution as given in eq.~(\ref{rvsubt}), which will generate
further poles up to $1/\ep^2$.
\subsection{The real-virtual contribution\label{sec:djrv}}
In this section, we shall construct the subtracted real-virtual
contribution, defined in eq.~(\ref{rvsubt}). As discussed there,
$rv^{(s)}$ has no explicit $1/\vep$ poles, since those resulting
from the one-loop integrals contributing to the (unsubtracted)
real-virtual contribution $rv$ are cancelled by those of $\trrmo$.
We can show this explicitly for the $\CF\TR$ part of the $\epem$
dijet cross section: the contribution of
\begin{equation}
rv=\langle\cmt_{gq\bq}\colket{\cmo_{gq\bq}}+
\langle\cmo_{gq\bq}\colket{\cmt_{gq\bq}}
\label{djrv}
\end{equation}
to this colour factor is entirely due to the $\TR$ part resulting
from UV renormalization:
\begin{equation}
rv|_{\TR}=\f{\as}{2\pi}\f{2}{3}\f{\TR\nf}{\vep}
\abs{\cmt_{gq\bq}}^2.
\label{djrvTR}
\end{equation}
By using eqs.~(\ref{rvphsp}), (\ref{djrrmoI}), and~(\ref{intKtw}),
with $gq_3\dots a_{n+2}\equiv gq\bq$ and summing over $\nf$ quark
flavours, it is apparent that $rv^{(s)}$ is indeed free of explicit
$1/\vep$ poles:
\begin{eqnarray}
rv^{(s)}d\psp_3&\equiv& rv|_{\TR}d\psp_3+
\sum_{flav}\int\trrmo d\tilde{\psp}_4^{-1}=
\f{\as}{2\pi}\f{\TR\nf}{\vep}\abs{\cmt_{gq\bq}}^2\,d\psp_3
\nonumber\\*&&\times
\Bigg\{\f{2}{3}-\left(\f{\mu^2}{s_{12}^{\Mx}}\right)^{\ep}\,
\f{e^{\ep\Euler}}{\Gamma(1-\ep)}\Bigg[\f{2}{3}+\f{10}{9} \ep+
\left(\f{56}{27}-\f{\pi^2}{9}\right)\ep^2
\nonumber\\*&&\phantom{\times}
+\left(\f{328}{81}-\f{5}{27}\pi^2-\f{4}{3}\zeta_3\right)\ep^3\Bigg]
\left(1-\f{\pi^2}{6}\ep^2-2\ep^3\zeta_3\right)\Bigg\}
+{\cal O}(\ep^3)\phantom{aaaaa}
\label{djrvsub}
\\*&=&
-\f{\as}{2\pi}\f{2\TR\nf}{9}
\left(5+3\log\left(\f{\mu^2}{s_{12}^{\Mx}}\right)\right)
\abs{\cmt_{gq\bq}}^2\,d\psp_3+{\cal O}(\ep)\,.
\label{djrvsubfin}
\end{eqnarray}
We underline the presence of $s_{12}^{\Mx}$ in eqs.~(\ref{djrvsub})
and~(\ref{djrvsubfin}), due to the integration performed in
eqs.~(\ref{djrrmoI}) and~(\ref{intKtw}), and to the choice of phase
space of eq.~(\ref{psrrmo}). Its specific form depends on the
branchings involved in the construction of $\trrmo$, and can
be read from eq.~(\ref{s12maxdef}); we shall write it explicitly
in the following, after choosing a parametrization for the phase
spaces $d\psp_{n+1}$ and $d\tilde{\psp}_{n+1}$.
Implicit divergences remain in eq.~(\ref{djrvsub}), whose analytic
computation requires the definition of the subtraction counterterm
$\widetilde{rv}^{(s)}$ which appears in eq.~(\ref{rvfinal}). It is clear
that these divergences are those of the matrix element $\cmt_{gq\bq}$
which factorizes in eq.~(\ref{djrvsub}) and thus are due to the
branching $q\to qg$. In order to study them with full generality, we consider
\begin{equation}
rv^{(0)}=\langle\cmt_{gq\dots a_{n+1}}
\colket{\cmt_{gq\dots a_{n+1}}}\,,
\label{djrvz}
\end{equation}
from which we construct the subtraction counterterm by applying the
$\E$-prescription:
\begin{equation}
\widetilde{rv}^{(0)}=rv^{(0)}(C)+rv^{(0)}(S)-rv^{(0)}(SC).
\label{djrvzsub}
\end{equation}
Clearly, $\widetilde{rv}^{(s)}$ is obtained by multiplying
$\widetilde{rv}^{(0)}$ by the r.h.s. of eq.~(\ref{djrvsub}),
divided by $\abs{\cmt_{gq\bq}}^2\,d\psp_3$; the same manipulations
can be carried out on the r.h.s. of eq.~(\ref{djrvsubfin}), in
this way obtaining the local subtraction counterterm to be
used in the numerical computation of the integral that appears
in the first term on the r.h.s. of eq.~(\ref{rvfinal}), or in
the second line on the r.h.s of eq.~(\ref{jetxsecsub}).
We stress that eq.~(\ref{djrvzsub}) coincides with eq.~(\ref{ftilNLO}),
since the subtraction of the phase-space singularities of $rv^{(s)}$
is identical to that performed in the context of an NLO computation.
In what follows, similarly to what done in sect.~\ref{sec:djrr},
we shall choose the same parametrization of the phase space for
the three terms on the r.h.s. of eq.~(\ref{djrvzsub}), whose sum
will therefore be essentially identical to one of the dipole kernels
of ref.~\cite{Catani:1996vz}. As before, we shall however not use the
dipole parametrization for the phase space, and our integrated kernel
will thus be different from that of ref.~\cite{Catani:1996vz}.
We obtain
\begin{equation}
\widetilde{rv}^{(0)}=\colbra{\cmt_{q\dots a_{n+1}}}
\bKtw_{qg}
\colket{\cmt_{q\dots a_{n+1}}}\,,
\label{djrvzfac}
\end{equation}
where the kernel has now a non-trivial colour structure
\begin{equation}
\bKtw_{qg}=-\f{8\pi\mu_0^{2\ep}\asu}{s_{qg}}\sum_{k\ne q}
\bVtw_{qg,k}\,\bT_q\mydot\bT_k\,.
\end{equation}
Here
\begin{eqnarray}
\bVtw_{qg,k}&=&\f{2p_q\mydot p_k}{(p_q+p_k)\mydot p_g}+
(1-z)(1-\ep),
\label{Vqg}
\\
z&=&\f{p_q\mydot p_k}{(p_q+p_g)\mydot p_k}\,,
\label{rvzdef}
\end{eqnarray}
and in order to obtain the result in eq.~(\ref{Vqg}) we have decomposed the
eikonal factor that appears in $rv^{(0)}(S)$ as shown
in eq.~(\ref{eikid})\footnote{We note that in this calculation
the kernel needed to construct $\widetilde{rv}^{(s)}$
coincides with the NLO one owing to eq.~(\ref{djrvTR}).
In more general cases, a formula similar to (\ref{djrvzsub})
for $\widetilde{rv}^{(s)}$ still holds,
but $rv^{(s)}(C)$, $rv^{(s)}(S)$ and $rv^{(s)}(SC)$ will have to be computed
using the results of refs.~\cite{bern,Kosower:1999rx,Catani:2000pi}.}.
According to our master subtraction formula, eq.~(\ref{jetxsecsub}), we
have now to integrate $\widetilde{rv}^{(s)}$ over the phase space, and
thus we have to choose a parametrization for $d\tilde{\psp}_{n+1}$.
A possibility is that of using again the form of eq.~(\ref{psrrmo}),
which with the parton labeling of the case at hand would read as follows
\begin{equation}
d\tilde{\psp}_{n+1}=
d\psp_n(\widetilde{qg})\,\frac{ds_{qg}}{2\pi}\,
\left(1-\frac{s_{qg}}{s_{qg}^\Mx}\right)^{-\ep}
d\psp_2(q,g)\,.
\label{psrvtil}
\end{equation}
On the other hand, the present case is simpler than that discussed in
sect.~\ref{sec:djrr}. The integration of $\widetilde{rv}^{(s)}$ over
$d\tilde{\psp}_{n+1}$ will not be followed by another analytical integration;
thus, the regulator that appears explicitly in eq.~(\ref{psrvtil}) is
not actually necessary. Furthermore, its presence would require the
presence of an analogous regulator for $s_{123}$ in eq.~(\ref{psrrmt}).
Thus, in order to simplify as much as possible the analytic computations,
we rather adopt the following form
\begin{equation}
d\tilde{\psp}_{n+1}=
d\psp_n(\widetilde{qg})\,\frac{ds_{qg}}{2\pi}\,
d\psp_2(q,g)\,.
\label{psrvfin}
\end{equation}
Having made this choice, we can deduce the expression of $s_{12}^{\Mx}$
from eq.~(\ref{s12maxdef}):
\begin{equation}
s_{12}^{\Mx}=(1-z)s_{qg}\,.
\label{s12maxch}
\end{equation}
In order to integrate the kernel $\bKtw_{qg}$ over $ds_{qg}d\psp_2(q,g)$,
we use eq.~(\ref{ps:twobd}), choosing the reference four-vector $n$ there
to coincide with $p_k$; in such a way, the variable $z_1$ of
eq.~(\ref{ps:zidef}) coincides with $z$ of eq.~(\ref{rvzdef}).
We also define
\begin{equation}
Q_{qgk}^2=(p_q+p_g+p_k)^2\,,
\label{Qqgkdef}
\end{equation}
which is treated as a constant during the integration. We obtain
\begin{eqnarray}
\int \widetilde{rv}^{(0)}d\tilde{\psp}_{n+1}&=&
-\f{\as}{2\pi}\,\f{e^{\ep\Euler}}{\Gamma(1-\ep)}
\sum_{k\ne q}\left(\f{\mu^2}{Q_{qgk}^2}\right)^{\ep}
I_{qg,k}^{(A)}\nn\\
&\times &\colbra{\cmt_{q\dots a_{n+1}}}\bT_q\mydot\bT_k
\colket{\cmt_{q\dots a_{n+1}}}\,d\psp_n\,.
\label{rv0int}
\end{eqnarray}
In eq.~(\ref{djrvsub}) we also need
\begin{eqnarray}
\int \widetilde{rv}^{(0)}\left(\f{\mu^2}{s_{12}^\Mx}\right)^{\ep}
d\tilde{\psp}_{n+1}&=&
-\f{\as}{2\pi}\,\f{e^{\ep\Euler}}{\Gamma(1-\ep)}
\sum_{k\ne q}\left(\f{\mu^2}{Q_{qgk}^2}\right)^{\ep}
I_{qg,k}^{(B)}\nn\\
&\times &\colbra{\cmt_{q\dots a_{n+1}}}\bT_q\mydot\bT_k
\colket{\cmt_{q\dots a_{n+1}}}\,d\psp_n\,,
\label{rv0intB}
\end{eqnarray}
with $s_{12}^\Mx$ given in eq.~(\ref{s12maxch}), and
\begin{eqnarray}
\f{(4\pi)^\ep}{\Gamma(1-\ep)}
\left(\f{\mu^2}{Q_{qgk}^2}\right)^{\ep}I_{qg,k}^{(A)}&=&
8\pi\mu^{2\ep}\int\frac{ds_{qg}}{s_{qg}}\,d\psp_2(q,g)\,
\bVtw_{qg,k}\,,
\\
\f{(4\pi)^\ep}{\Gamma(1-\ep)}
\left(\f{\mu^2}{Q_{qgk}^2}\right)^{2\ep}I_{qg,k}^{(B)}&=&
8\pi\mu^{2\ep}\int\frac{ds_{qg}}{s_{qg}}\,d\psp_2(q,g)\,
\left(\f{\mu^2}{s_{12}^{\Mx}}\right)^{\ep}\bVtw_{qg,k}\,.
\end{eqnarray}
A somewhat lengthy computation returns the following results
\begin{eqnarray}
I_{qg,k}^{(A)}&\equiv & \int_0^1 dz \int_0^{\ymax}dy
\left[\f{2z(1-y)}{1-z+y z}+(1-z)(1-\ep)\right]
\left(z(1-z)\right)^{-\ep} y^{-1-\ep}
\nn\\*
&=&\f{1}{\ep^2}+\f{3}{2}\f{1}{\ep}+\f{7}{2}-\f{\pi^2}{6}-
\log(\ymax)\left(\log(\ymax)+\f{3}{2}\right)-2\Li_2(1-\ymax)
\nonumber\\*
&+&\ep\Bigg[7-\f{\pi^2}{4}-2\zeta_3+\log(\ymax)
\left(\f{\pi^2}{3}-\f{7}{2}\right)-2\log(\ymax)\Li_2(\ymax)
\nonumber\\*&&
+\log^2 (\ymax)\left(\f{3}{4}-2\log(1-\ymax)+\f{2}{3}\log(\ymax)\right)
\nonumber\\*&&
-2\Li_3(1-\ymax)
+2\Li_3\left(\f{\ymax-1}{\ymax}\right)\Bigg]+{\cal O}(\ep^2)\,,
\label{IAres}
\\
I_{qg,k}^{(B)}&\equiv & \int_0^1 dz \int_0^{\ymax}dy
\left[\f{2z(1-y)}{1-z+y z}+(1-z)(1-\ep)\right]
z^{-\ep}(1-z)^{-2\ep} y^{-1-2\ep}\nn
\\*
&=&\f{1}{4\ep^2}+\f{3}{4\ep}+\f{21}{8}-
\log(\ymax)\left(\f{3}{2}+\log(\ymax)\right)
-2\Li_2(1-\ymax)
\nonumber\\*
&+&\ep\Bigg[\f{127}{16}-\f{\pi^2}{4}-\zeta_3
+\log(\ymax)\left(\f{\pi^2}{3}-\f{21}{4}\right)
-6\log(\ymax)\Li_2(\ymax)
\nonumber\\*&&
+\log^2(\ymax)\left(\f{3}{2}-4\log(1-\ymax)+\f{4}{3}\log(\ymax)\right)
\nonumber\\*&&
-2\Li_3(1-\ymax)+4\Li_3\left(\f{\ymax-1}{\ymax}\right)
+4\Li_3(\ymax)\Bigg]+{\cal O}(\ep^2)\,,
\label{IBres}
\end{eqnarray}
where we have introduced the {\em arbitrary} parameter $\ymax$, which
defines the upper limit of the $s_{qg}$ integration
\begin{equation}
s_{qg}\le\ymax\,Q_{qgk}^2\,,\;\;\;\;\;\;\;\;
0<\ymax\le 1\,.
\label{sqgupp}
\end{equation}
The physical results must not depend on $\ymax$, whose variation amounts
to changing finite contributions to the subtraction counterterms; whether
this condition is fulfilled represents a powerful check on the correctness
of the subtraction procedure. The parameter $\ymax$ is the analogue of the
free parameters $\xi_{cut}$, $\delta_{\sss I}$ and $\delta_{\sss O}$
of ref.~\cite{Frixione:1995ms}.
\subsection{Results\label{sec:res}}
In this section, we shall use the results obtained in sects.~\ref{sec:djrr}
and~\ref{sec:djrv} in order to implement our master subtraction formula
eq.~(\ref{jetxsecsub}). As a preliminary step, we need the $\CF\TR$ part
of the double-virtual contribution
\begin{equation}
vv=\langle\cmt_{q\bq}\colket{\cmd_{q\bq}}+
\langle\cmd_{q\bq}\colket{\cmt_{q\bq}}\,,
\label{djvv}
\end{equation}
which can be straightforwardly obtained from ref.~\cite{Matsuura:1988sm}:
\begin{eqnarray}
vv|_{\CF\TR}&=&\CF\TR\nf
\left(\f{\as}{2\pi}\right)^2 \abs{\cmt_{q\bq}}^2
\nonumber\\*
&\times&\Bigg\{
\left(\f{Q^2}{\mu^2}\right)^{-2\ep}\Bigg[\f{1}{3\ep^3}+\f{14}{9\ep^2}+
\left(\f{353}{54}-\f{11}{18}\pi^2\right)\f{1}{\ep}+\f{7541}{324}
-\f{77}{27}\pi^2-\f{26}{9}\zeta_3\Bigg]
\nonumber\\*&&
+\left(\f{Q^2}{\mu^2}\right)^{-\ep}\Bigg[-\f{4}{3\ep^3}-\f{2}{\ep^2}+
\left(\f{7}{9}\pi^2-\f{16}{3}\right)\f{1}{\ep}-\f{32}{3}+\f{7}{6}\pi^2+
\f{28}{9}\zeta_3\Bigg]
\Bigg\}\,,\phantom{aaaaaa}
\end{eqnarray}
where $Q^2$ is the $\epem$ c.m. energy squared.
The subtraction counterterms in the first and second integrals on the r.h.s.
of eq.~(\ref{jetxsecsub}) are constructed using the kernels $\bKtw$ and
$\bKth$ defined in the previous sections. It must be stressed that
we shall have to subtract $\bKth$ twice, since we have two configurations
contributing to topology $\Tt$: the first is that depicted on the left panel
of fig.~\ref{fig:diagrm}, the second being identical except for the
fact that the gluon is attached to the other quark leg emerging from the
$\gamma\to q\bq$ branching. This also implies that the double-real
contribution to the last line on the r.h.s. of eq.~(\ref{jetxsecsub})
will be obtained upon multiplying by two the result presented in
eq.~(\ref{intKth}) (an overall factor of $\nf$ appears when summing
over flavours). Finally, the largest kinematically-allowed value of
the virtuality of the branching parton in a $1\to 3$ splitting is $Q^2$.
For consistency with what done in eq.~(\ref{sqgupp}), we thus set
\begin{equation}
s_{123}^{\Mx}=\ymax\,Q^2
\label{s123upp}
\end{equation}
as the upper limit of the integration over $s_{123}$ performed
in sect.~\ref{sec:djrrdiv}. The real-virtual contribution
$\int\widetilde{rv}^{(s)}d\tilde{\psp}_{n+1}$ can be obtained from
eqs.~(\ref{djrvsub}), (\ref{rv0int}) and (\ref{rv0intB});
for the dijet cross section,
$q\dots a_{n+1}\equiv q\bq$, and the colour algebra is trivial:
\begin{equation}
\colbra{\cmt_{q\bq}}\bT_q\mydot\bT_{\bq}
\colket{\cmt_{q\bq}}=-\CF\abs{\cmt_{q\bq}}^2\,.
\end{equation}
In the computation of the real-virtual contribution, we also
necessarily have $Q_{qgk}^2\equiv Q^2$ (see eq.~(\ref{Qqgkdef})).
When putting this all together, we obtain what follows for the
analytic-computed part of the subtraction formula of
eq.~(\ref{jetxsecsub}):
\begin{eqnarray}
&&\int\trrmt d\tilde{\psp}_4^{-2}
+\int\widetilde{rv}^{(s)}d\tilde{\psp}_3
+\int vv d\psp_2\,\Bigg|_{\CF\TR}=
\CF\TR\nf
\left(\f{\as}{2\pi}\right)^2 \abs{\cmt_{q\bq}}^2 d\psp_2
\nonumber\\*&&\phantom{aaaa}
\times\Bigg\{\f{19}{9}-\f{29}{27}\pi^2
-\f{2}{3}\log\ymax
+\log\ymax \left(\f{17}{3}+\f{16}{3}\Li_2(\ymax)\right)
\nonumber\\*&&\phantom{aaaa\times}
+\log^2\ymax\left(\f{11}{9}+\f{8}{3}\log(1-\ymax)-\f{8}{9}\log\ymax\right)
\nonumber\\*&&\phantom{aaaa\times}
+\f{40}{9}\Li_2(1-\ymax)
-\f{8}{3}\Li_3\left(\f{\ymax-1}{\ymax}\right)-\f{16}{3}\Li_3(\ymax)
\nonumber\\*&&\phantom{aaaa\times}
+\left(\f{2}{3}-\f{4}{9}\pi^2+\f{4}{3}\log^2\ymax+2\log\ymax
+\f{8}{3}\Li_2(1-\ymax)\right)\log\f{\mu^2}{Q^2}
\Bigg\}\,.\phantom{aaaaaa}
\label{analytic}
\end{eqnarray}
This result is non-divergent, which proves the successful cancellation,
as dictated by the KLN theorem, of the soft and collinear divergences
which arose in the intermediate steps of the computation.
The cancellation of the divergences does not guarantee that the finite
parts resulting from the implementation of the subtraction procedure are
all included correctly. In order to check this, we started by computing
the $\CF\TR$ part of the NNLO contribution to the total hadronic cross
section, and compared it with the well-known analytic result, which can
be read from~\cite{sigmatot}
\begin{eqnarray}
R&=&\sum_q e^2_q\Bigg\{1+\left(\f{\as}{2\pi}\right)\f{3}{2}\CF
+\left(\f{\as}{2\pi}\right)^2
\Bigg[-\f{3}{8}\CF^2+\CF\CA\left(\f{123}{8}-11\zeta_3\right)
\nn\\*&&
\phantom{\sum_q e^2_q\Bigg\{1+aaaaa}
+\CF\TR\nf\left(-\f{11}{2}+4\zeta_3\right)\Bigg]+{\cal O}(\as^3)\Bigg\}\,,
\end{eqnarray}
where
\begin{equation}
R=\f{\sigma(e^+e^-\to {\rm hadrons})}{\sigma(e^+e^-\to \mu^+\mu^-)}\,.
\end{equation}
Although the physical result must not depend on $\ymax$, the three lines
on the r.h.s. of eq.~(\ref{jetxsecsub}) (which we call 4-parton,
3-parton, and analytic contributions respectively) separately do.
We thus compute the rate for different choices of $\ymax$. The
results are presented in table~\ref{tab:rates}.
\begin{table}
\begin{center}
\begin{tabular}{crrrrc}
\hline
$\ymax$ & 4-parton & 3-parton & analytic & total & pull\\
\hline
0.2 & $-$0.692 $\pm$ 0.002 & 1.4241 $\pm$ 0.0004 & $-$1.4236
& $-$0.692 $\pm$ 0.002& 0.11\\
0.4 & $-$0.229 $\pm$ 0.004 & 9.8001 $\pm$ 0.0003 & $-$10.2610
& $-$0.690 $\pm$ 0.004& 0.44\\
0.6 & 0.074 $\pm$ 0.02~$\,$ & 12.3440 $\pm$ 0.0003& $-$13.0753
& $-$0.657 $\pm$ 0.02~$\,$ & 1.74\\
0.8 & 0.235 $\pm$ 0.005 & 13.3759 $\pm$ 0.0003& $-$14.2989
& $-$0.688 $\pm$ 0.005& 0.75\\
1.0 & 0.383 $\pm$ 0.004 & 13.8284 $\pm$ 0.0003& $-$14.9005
& $-$0.689 $\pm$ 0.004& 0.69\\
\hline
\end{tabular}
\caption{\label{tab:rates}
Numerical results for the $\CF\TR\nf$ term of the $R$ ratio.
The three unphysical contributions, called 4-parton, 3-parton,
and analytic, correspond to the three lines on the r.h.s. of
eq.~(\ref{jetxsecsub}) respectively; their sums are reported under
the column ``total''. The pulls are defined in eq.~(\ref{pulldef}).
}
\end{center}
\end{table}
As can be seen there, the dependence on $\ymax$ of the three
contributions to our subtraction formula is fairly large. However,
such a dependence cancels in the sum, within the statistical
accuracy of the numerical computation. We defined
\begin{eqnarray}
{\rm pull}&=&\f{\abs{{\rm total}-{\rm exact}}}{{\rm error}}\,,
\label{pulldef}
\\
{\rm exact}&=&-\f{11}{2}+4\zeta_3=-0.69177\,,
\end{eqnarray}
the error being that due to the numerical computation, reported
under the column ``total'' in table~\ref{tab:rates}. We note that
the 4-parton result changes sign when $\ymax$ is increased,
which implies that the integrand gives both positive and negative
contributions. As is well known, when such contributions are
close in absolute value, as when choosing $\ymax=0.6$, the computation
of the integral is affected by a relatively large error; in fact, the
worst agreement with the exact result is obtained with $\ymax=0.6$.
We finally computed a proper dijet total rate, by reconstructing
the jets using the JADE algorithm with $y_{cut}=0.1$ (for a discussion on
$\epem$ jet algorithms, see e.g. ref.~\cite{Bethke:1991wk}). Using the
same normalization as in ref.~\cite{Anastasiou:2004qd}, we find for
the NNLO contribution proportional to $\nf$
\begin{equation}
1.7998\pm 0.0016\;\;\;\;\;\;\;\;
{\rm with}\;\;\;\;\;\;\;\;
\ymax=0.6\,,
\label{JADEres}
\end{equation}
which is in nice agreement with the result of ref.~\cite{Anastasiou:2004qd}.
We point out that the small error in eq.~(\ref{JADEres}),
resulting from a run with lower statistics with respect to the results
in table \ref{tab:rates}, is obtained with the value
of $\ymax$ that gives the worst convergence performance in the case
of the total rate. We verified that, as for total rates, dijet
cross sections are independent of $\ymax$ within the error
of the numerical computation (for example, with $\ymax=0.4$ we
obtain $1.7992\pm 0.0015$).
\section{Comments\label{sec:comm}}
We have shown in sect.~\ref{sec:CFTR} that our master subtraction
formula, eq.~(\ref{jetxsecsub}), allows us to cancel analytically
the soft and collinear divergences relevant to an NNLO computation.
By using the $\E$-prescription, the subtraction counterterms
$\trrmt$, $\trrmo$, and $\widetilde{rv}^{(s)}$ are constructed
from the basic kernels that account for the matrix element
singularities. The successful implementation of the subtraction
procedure also requires a sensible definition of the phase spaces
used to integrate the counterterms. For convenience, we collect
here the phase space parametrizations we used, relabeling the
partons where necessary, in such a way that those involved in
the singular branchings have always the smallest labels.
For the $\trrmt$ counterterm to the double-real contribution
we used eq.~(\ref{psrrmt})
\begin{equation}
d\tilde{\psp}_{n+2}^{-2}=
d\psp_{n}(\widetilde{123})\,
\frac{ds_{123}}{2\pi}\,d\psp_3(1,2,3)\,,
\label{psrrmt2}
\end{equation}
relevant to topology $\Tt$ (topology $\Topo$ was not relevant to the case
studied in this paper); we could also use eq.~(\ref{tbtemp}), which is fully
equivalent and particularly suited to integrate kernels with a structure
analogous to that of strongly-ordered limits. For the $\trrmo$ counterterm
we used eq.~(\ref{psrrmo})
\begin{eqnarray}
d\tilde{\psp}_{n+2}^{-1}&=&
d\psp_{n+1}(\widetilde{12})\,\frac{ds_{12}}{2\pi}\,
\left(1-\frac{s_{12}}{s_{12}^\Mx}\right)^{-\ep}
d\psp_2(1,2)
\label{psrrmo2}
\\*
&\equiv&d\psp_{n+1}d\psp_2^{-1}\,.
\label{psptemp}
\end{eqnarray}
Equations~(\ref{psrrmo2}) and~(\ref{psptemp}) give an explicit
expression for $d\psp_2^{-1}$, which needed not be specified
in eq.~(\ref{rvsubt}). Finally, for the counterterm to the
subtracted real-virtual contribution we used eq.~(\ref{psrvfin})
\begin{equation}
d\tilde{\psp}_{n+1}=
d\psp_n(\widetilde{12})\,\frac{ds_{12}}{2\pi}\,
d\psp_2(1,2)\,.
\label{psrvfin2}
\end{equation}
The factor that depends on ${s_{12}^\Mx}$ in eq.~(\ref{psrrmo2}) is
crucial for the correct definition of the subtracted real-virtual
contribution. The quantity ${s_{12}^\Mx}$
depends on the kinematics of the hard system whose phase space is
$d\psp_{n+1}(\widetilde{12})$. Such a dependence is in general
non trivial: the difference in the results of eqs.~(\ref{IAres})
and~(\ref{IBres}) is due to it (notice that all of the $1/\vep$ poles
are affected; thus, the failure to include the ${s_{12}^\Mx}$-dependent
regulator in eq.~(\ref{psrrmo2}) would prevent the cancellation
of singularities). The dependence of ${s_{12}^\Mx}$ on
the kinematics of the hard system which factorizes needs not be known
analytically for the computation of $\trrmo d\tilde{\psp}_{n+2}^{-1}$
and for the definition of $rv^{(s)}$. It must be known when
defining $\widetilde{rv}^{(s)}$, since this quantity has to be
integrated analytically. However, in such a case the dependence
is expected to be relatively simple (see eq.~(\ref{s12maxch})).
This is so because the subtraction term $\widetilde{rv}^{(s)}$
always corresponds to the kinematics of a strongly-ordered limit,
in which the branching of partons $1$ and $2$ will be followed by
another two-parton branching (in the case studied in sect.~\ref{sec:CFTR},
we had $q\to gq_3$ and $g\to\qp_1\bqp_2$; the case of topology
$\Topo$ will clearly be even simpler).
The phase spaces in eqs.~(\ref{psrrmo2}) and~(\ref{psrvfin2}) will
serve to integrate the $\bKtw$ kernels. This implies that, for a
given kernel, two different integrals will have to be computed,
because of the presence of the ${s_{12}^\Mx}$-dependent regulator
in $d\tilde{\psp}_{n+2}^{-1}$. This can clearly be avoided by
inserting the regulator also in eq.~(\ref{psrvfin2}). It should
however be stressed that, in doing so, we are forced to insert an
analogous regulator (${s_{123}^\Mx}$-dependent) in eq.~(\ref{psrrmt2}).
Thus, the advantage of having to compute half of the integrals relevant
to the $\bKtw$ kernels may be lost because of the additional complications
in the computations of the integrals of the $\bKth$ kernels.
\section{Conclusions\label{sec:concl}}
In this paper, we have proposed a framework for the implementation of a
subtraction formalism that allows us to cancel the soft and collinear
singularities which arise in the intermediate steps of a perturbative
computation at the next-to-next-to-leading order in QCD. The strategy is
analogous to that adopted at the next-to-leading order, which is based on the
definition of subtraction kernels whose form is both observable- and
process-independent. We have introduced a systematic way (the
$\E$-prescription) for defining the kernels, that results naturally in a
two-step procedure.
The first step is the definition of the subtraction counterterms
for the double-real contribution. They are of two different types,
denoted by $\trrmt$ and $\trrmo$; roughly speaking, the former collects
all the pure NNLO-type singularities, whereas the latter is singular
in those soft, collinear, and soft-collinear configurations that are
also relevant to NLO computations. The $\E$-prescription guarantees
the absence of double counting.
In the second step, the term $\trrmo$ is summed to the real-virtual
contribution; this sum is free of explicit $1/\vep$ poles, but is
still divergent, and a further subtraction needs be performed. This
subtraction is completely analogous to that relevant to NLO computations.
Our master formula, eq.~(\ref{jetxsecsub}), thus achieves the
cancellation of the soft and collinear singularities essentially
by two successive NLO-type subtractions; the first defines the
subtracted real-virtual contribution, and the second removes its
remaining singularities. The singularities that cannot be possibly
obtained in this way are all contained in the subtraction counterterms
$\trrmt$ (the singularities of the double-virtual contribution are
of no concern for the subtraction procedure -- we assume the
relevant matrix elements to be available).
Equation~(\ref{jetxsecsub}) is, as it stands, not sufficient for an actual
numerical computation, since suitable choices of the phase spaces involved are
necessary. We tested our subtraction formalism and the corresponding phase
space choices in the context of a simple application, the calculation of the
contribution proportional to $\CF\TR$ of the dijet cross section in $\epem$
collisions. Although this is clearly a simple example, it is, to the best of
our knowledge, the first application in which NNLO process-independent
subtraction counterterms have been constructed and integrated over the
corresponding phase spaces to achieve an explicit cancellation of soft
and collinear singularities.
We point out that, although the phase-space parametrizations adopted in this
paper are seen to induce reasonably simple analytic integrations, this
is in part due to the relative simplicity of the kernels needed in
the computation of the $\CF\TR$ part of the $\epem$ dijet cross section.
More complicated kernels may require different parametrizations,
which we did not consider in the present paper. In particular, the two
NLO-type successive subtractions are liable to be simplified with respect
to what done here. In any case, the general subtraction formula of
eq.~(\ref{jetxsecsub}) will retain its validity independently of
the specific phase-space parametrizations adopted.
There are a few difficulties that we did not address directly in this
paper; they are not difficulties of principle, but may pose technical
problems. One interesting feature will emerge when computing the
$\CF\TR$ part of the three-jet cross section in $\epem$ collisions,
namely the interplay between the $\Tt$ and $\Topo$ topologies which
need be treated separately in the definition of the subtraction
kernels. The case of collisions with one or two initial-state hadrons
will also require a more involved notation (although the subtraction
procedure will be basically unchanged). In general, it is clear that
there is a fair amount of work to be done before the subtraction scheme
proposed here is of any phenomenological use. This implies not only
the definition, and integration over the phase spaces, of the universal
subtraction kernels for all of the partonic branchings possibly occurring
at NNLO, but also the computation of more two-loop amplitudes,
which is a necessary condition for process-independent formalisms
to be more convenient than observable-specific results.
\section*{Acknowledgements}
We wish to thank Stefano Catani, Lorenzo Magnea, Fabio Maltoni, and
Michelangelo Mangano for comments on the manuscript, and CERN Theory Division
for hospitality at various stages of this work. S.F. is grateful to Zoltan
Kunszt and Adrian Signer for the many discussions they had on this matter
a while ago, and M.G. wishes to thank Stefano Catani for having introduced
him to the subject.
\newpage
|
{
"timestamp": "2005-06-08T11:50:50",
"yymm": "0411",
"arxiv_id": "hep-ph/0411399",
"language": "en",
"url": "https://arxiv.org/abs/hep-ph/0411399"
}
|
\section{Introduction}
\label{s:intro}
\begin{figure*}[htb]
\includegraphics[width=8cm,angle=-90]{aa1595f1.ps}
\caption{Images of W3 IRS5 at 43~GHz (left) and 22~GHz
(right). Contours start at 4$\sigma$ and increase by 2$\sigma$,
where $\sigma$=0.16 mJy/beam at 43~GHz and 0.15 mJy/beam at 22~GHz. }
\label{f:vla}
\end{figure*}
Stars of masses $>$8~M$_{\odot}$\ spend a significant fraction of their
lifetimes, $>$10\%, embedded in their natal molecular clouds.
Single-dish (sub)millimeter observations have clarified the structure
of high-mass protostellar envelopes on $\sim$10$^4$--10$^5$~AU scales
(see \citealt{hvdt03} and references therein). However,
the distribution and kinematics of material on ${_<\atop{^\sim}}$1000~AU scales
is poorly known, due to the large (${_>\atop{^\sim}}$1~kpc) distances involved,
and the lack of tracers at optical and near-infrared wavelengths.
These scales are of great interest to decide between formation
mechanisms for high-mass stars, and to clarify the relation with
clustered star formation \citep{evans02}. Does the distribution of
stellar masses in a star-forming region depend on the stellar density?
Also, the origin of the observed outflows and their interaction
with the environment on ${_<\atop{^\sim}}$1000~AU scales remain unclear.
Subarcsecond resolution observations are necessary to shed light on
these and other questions, which, at (sub)millimeter wavelengths, are
just coming within reach \citep{beuther04}. However,
these resolutions can already be achieved in both the infrared and
radio wavebands, where extinction is much smaller than in the optical.
In the infrared, high-resolution techniques are most advanced at
near-infrared wavelengths. Such observations probe less embedded,
more evolved phases, where a significant part of the surroundings is
already ionized. Important progress has been made with the
identification of the ionizing stars of several ultracompact H~{\sc II}\
regions (e.g., \citealt{watson+hanson}, \citealt{feldt03}). In the
case of intermediate-mass stars, the imaging of the hot inner regions of disks
is presently generating a lot of interest
(\citealt{danchi01,tuthill01}). In addition, a few more embeddded
objects have been probed (\citealt{weigelt02,preibisch02}), although
in the general case long baseline interferometers will be needed to
tackle most targets given the characteristic size scales involved
\citep{monnier02}. At mid-infrared wavelengths, pioneering work has
been done by \citet{walsh01}, but subarcsecond resolution has only
recently been achieved (\citealt{tuthill02,buizer02,greenhill04}).
In the cm-wave region, most subarcsecond-resolution studies have
concentrated on H$_2$O\ masers, which are bright
enough for Very Long Baseline Interferometry (VLBI) observations.
The excitation requirements of the masers are such that the emission
usually traces shocks associated with infalling or outflowing motions.
The VLBI data indicate that the maser emission traces moving gas
parcels, rather than shock waves propagating through an H$_2$O--rich
cloud.
In the case of outflow motions, both bipolar
and spherical flows are seen, which may represent different stages of evolution
\citep{torrelles03}.
In some cases, H$_2$O\ masers in star-forming
regions may arise in accretion shocks in infalling gas \citep{mvdt04}.
Continuum emission at centimeter wavelengths arises in ionized gas.
In stellar winds and outflows, the gas can be collisionally ionized,
and VLBI data indicate a mixture of bipolar and equatorial outflows
\citep{hoare02}. Close to hot stars, small regions of photo-ionized
gas are observed as `hypercompact' H~{\sc II}\ regions, which represent a
very early stage of high-mass star formation \citep{garay99}.
At a distance of 1.83$$\pm$$0.14~kpc \citep{imai00}, W3~IRS5 is the
nearest region of high-mass ($L$=1.2$\times$10$^5$~L$_{\odot}$:
\citealt{ladd93}) star formation after Orion. The bright mid-infrared
source has been resolved into a double (\citealt{Howell81,Neugebauer82}). Single-dish
submillimeter mapping indicate an envelope mass of 262~M$_{\odot}$\ within a
radius of 60,000~AU, with an $r^{-1.5}$ density distribution
\citep{fvdt00}. Near-infrared imaging shows a dense cluster
($\sim$3000 pc$^{-3}$: \citealt{megeath96}), mostly composed of
low-mass pre-main-sequence stars with ages 0.3--1~Myr \citep{ojha04}.
Radio continuum observations show a cluster of at least six
`hypercompact' H~{\sc II}\ regions labeled A...F (\citealt{claus94,achim97}),
at least one of which exhibits proper motions \citep{wilson03}. Water maser
mapping reveals $\sim$100 spots, grouped in two flows: one roughly
spherical and centered close to continuum source~A, and the other more
collimated and centered close to source~D (\citealt{claus94,imai00}).
Mid-infrared spectroscopy shows CO absorption features blueshifted by
4..46~km~s$^{-1}$\ relative to the systemic velocity \citep{mitch91}, which
must arise in an outflow. In millimeter-wave CO emission, blue- and
redshifted outflow lobes are detected out to $\approx$23~km~s$^{-1}$\ from
the systemic velocity \citep{claussen84}, indicating that the
highest-velocity gas is very compact.
Finally, \emph{Chandra} observations by \citet{hofner02} indicate an
X-ray luminosity of W3~IRS5 of $L_X$=9$\times$10$^{29}$ erg~s$^{-1}$
(for $d$=1.83~kpc), consistent with the typical values for T~Tauri
stars.
This paper presents new cm-wave and mid-infrared images of W3~IRS5 at
sub-arcsecond resolution. The goals are to clarify the nature of the
radio continuum sources and their relation with the infrared double,
to find which ones are self-luminous, and which ones power the region.
\section{Observations}
\begin{figure*}[t]
\includegraphics[width=16cm]{aa1595f2.ps}
\caption{Mid-infrared long-exposure images of W3 IRS5. Contours are at
0.5, 1, 2, 3, 5, 10, 20, 30 and 70\% of the peak intensity.}
\label{f:keck_long}
\end{figure*}
\begin{figure*}[t]
\includegraphics[width=16cm]{aa1595f3.ps}
\caption{Mid-infrared `speckle' images of W3 IRS5. Contours are at
0.5, 1, 2, 3, 5, 10, 20, 30 and 70\% of the peak intensity.}
\label{f:keck_spec}
\end{figure*}
\subsection{Radio Observations}
Radio observations of W3 IRS5 were carried out with the
NRAO\footnote{The National Radio Astronomy Observatory (NRAO) is
operated by Associated Universities, Inc., under a cooperative
agreement with the National Science Foundation.} Very Large Array
(VLA) on 1996 October 24, when the VLA was in its $A$-configuration.
At this time, thirteen VLA antennas were equipped with 43~GHz
receivers; the other fourteen observed at 22~GHz (respectively
known as \textit{Q}- and \textit{K}-band in radio astronomy).
Zenith opacity was
0.089 at 22.5~GHz and 0.071 at 43.3~GHz. Elevation-dependent antenna
gains were interpolated from values measured by the VLA staff. The
phase calibrator, 0228+673, was observed every 10 minutes at 43~GHz
and every 15 minutes at 22~GHz (the `fast switching' procedure was not
implemented at the time). Pointing was checked at 8.4~GHz on the same
source every 70 minutes at 43~GHz and every 5~hours at 22~GHz.
On-source integration time is 438~min at 22~GHz and 400~min at 43~GHz.
The data were edited, calibrated, and imaged with NRAO's Astronomical
Image Processing System (AIPS). Absolute calibration was obtained from
observations of 3C~286 using flux densities interpolated from the
values given by \citet{ott94}.
For 0228+673, we obtain a flux density of 1.83~Jy at 22~GHz and
1.55~Jy at 43~GHz.
Figure~\ref{f:vla} shows the 22~and 43~GHz maps, which have rms noise
levels of 0.15 and 0.16 mJy~beam$^{-1}$. These maps were obtained from
the $uv$ data by a Fourier transform with uniform weighting, and
deconvolved with the Clean algorithm. Restoring beam major and minor
axes and position angles are 44$\times$37 mas at position angle
--68$^\circ$ at 43~GHz and 89$\times$88 mas, PA --53$^\circ$ at
22~GHz.
\subsection{Infrared Observations}
Data were obtained in August 2002 with the Long Wavelength Spectrometer
(LWS) camera on the Keck~I telescope\footnote{The W.~M.\ Keck Observatory
was made possible by the support of the W.~M.\ Keck Foundation, and is
operated as a scientific partnership among the California Institute of
Technology, the University of California, and the National Aeronautics
and Space Administration.}.
Two different observing methodologies were employed.
The first set of observations utilized a standard chop-nod pattern,
with frames coadded to build up longer exposure times.
Observations of W3 IRS5 (with the calibrator $\xi$ Cyg) taken in this
mode emphasized the recovery of faint structure.
The second mode had a fast readout in addition to the chop-nod, so that
large volumes of rapid exposure data were collected in a fashion
analogous to a mid-IR `speckle' experiment, with the hope to
recover fine structure in the images. Again, these objects
were paired with similar data taken on a point source calibrator, this
time $\alpha$ Ceti. No point-source calibrator files were taken for
9.9~$\mu$m\ or 12.5~$\mu$m\ observations.
These two modes are denoted `L' and `S' for `long' and `speckle' exposure
hereafter.
Data have been analyzed with an iterative matched filter version of
the shift-and-add algorithm. This algorithm attempts to match the
shifts in the current iteration to maximise the correlation with the
output of the previous iteration. Significant gains in image resolution
were demonstrated over straight coadding, while a simple shift-and-add
strategy was foiled in this case by the presence of two nearly equal
peaks.
Figures~\ref{f:keck_long} and~\ref{f:keck_spec} present the resultant
images, except the 17.65~$\mu$m\ long-exposure image which is very
similar to the speckle image at that wavelength. Note that an artifact
due to an unwanted reflection from an optical surface affects the
4.7~$\mu$m\ image, giving a spurious feature to the North-East which is
not seen at any other wavelength. This `ghost' was also present in
point-source calibrator data in this filter. The 9.9~$\mu$m\ and (to a
lesser extent) the 10.7~$\mu$m\ data have degraded signal-to-noise due
to the much lower flux levels. There were, unfortunately, additional
experimental difficulties which were not easy to account for. The
observations were taken under conditions of variable cirrus,
increasing the errors in photometry. Compounding this was an
intermittent mechanical fault with the camera mechanism which resulted
in partial occultation of the pupil, affecting both the throughput and
the point-spread function (PSF). Although this had little effect on
the maps presented here, it did preclude our original intent of fully
deconvolving the images using the PSF from the reference star
observations.
\section{Results}
\begin{table*}
\caption{Radio emission from W3 IRS5. Numbers in brackets are
uncertainties in units of the last decimal.}
\label{t:vla}
\begin{tabular}{llllllll}
\hline
\hline
Source & $\alpha$ & $\delta$ & Peak $I_\nu$ & Total $S_\nu$ & Major axis & Minor axis & PA \\
& (J2000) & (J2000) & mJy/beam & mJy & mas & mas & deg \\
\hline
Q1 & 02 25 40.660407(635) & 62 05 51.82215(404) & 0.691(162) & 0.78(31) & 45(10)& 41(10)& 93(90) \\
Q2 & 02 25 40.676344(379) & 62 05 52.04937(261) & 1.285(158) & 1.92(36) & 59(7) & 41(5) & 47(13) \\
Q3 & 02 25 40.681511(211) & 62 05 51.53000(198) & 2.102(155) & 3.89(42) & 63(5) & 47(3) & 7(10) \\
Q4 & 02 25 40.728475(704) & 62 05 49.85180(702) & 0.666(154) & 1.43(46) & 75(17)& 46(11)& 21(18) \\
Q5 & 02 25 40.783441(212) & 62 05 52.46552(141) & 1.996(163) & 2.19(30) & 43(4) & 41(3) & 88(57) \\
& & & & \\
K1 & 02 25 40.143241(1234) & 62 05 52.28144(1080) & 0.745(142) & 0.85(27) & 139(27) & 99(19) & 25(22) \\
K2 & 02 25 40.661233(1029) & 62 05 51.86622(1174) & 0.847(140) & 1.08(29) & 175(29) & 88(15) & 160(9) \\
K3 & 02 25 40.675758(1135) & 62 05 52.04087(690) & 0.857(142) & 0.76(23) & 121(20) & 88(15) & 59(21) \\
K4 & 02 25 40.682172(526) & 62 05 51.54376(441) & 1.938(139) & 2.76(31) & 152(11) & 113(8) & 154(10) \\
K5 & 02 25 40.712459(785) & 62 05 52.44786(569) & 1.163(142) & 1.12(24) & 110(13) & 106(13)& 168(138)\\
K6 & 02 25 40.733047(1601) & 62 05 49.84977(1214) & 0.797(134) & 1.76(41) & 174(29) & 154(26)& 25(56) \\
K7 & 02 25 40.782747(737) & 62 05 52.46103(637) & 1.313(140) & 1.63(28) & 150(16) & 100(11)& 29(10) \\
K8 & 02 25 40.861211(888) & 62 05 53.46070(717) & 1.307(135) & 2.44(37) & 169(17) & 135(14)& 25(19) \\
\hline
\end{tabular}
\end{table*}
\subsection{Radio Positions}
\label{s:cm_pos}
Table~\ref{t:vla} lists the sources detected above 5$\sigma$ using the
multiple-peak-finding AIPS task SAD after trying various weighting
schemes.
For the $Q$--band data, the table uses the image obtained with uniform
weighting, which has better positional accuracy, while for the $K$--band data,
tabulated results use an image obtained with natural weighting, where more
sources are detected (rms = 142~$\mu$Jy).
Sources Q1 and Q4 are 4$\sigma$ detections and were not found by SAD,
but their detection is secure because they have $K$--band
counterparts.
The flux densities in the table are corrected for primary beam
response. Positional uncertainties are statistical errors and apply to
the relative positions of these radio sources.
Columns 6--8 of Table~\ref{t:vla} gives the source sizes. Since only
sources Q3, K6 and K8 appear marginally resolved, the sizes have not
been deconvolved. The sizes would be lower limits if extended emission
is resolved out by the interferometer (\citealt{kurtz99};
\citealt{kim01}). In the particular case of W~3 IRS5, however,
multi-configuration observations rule out extended emission down to
very low limits \citep{achim97}. Therefore, the radio sources of
W3~IRS5 belong to the class of `hypercompact' H~{\sc II}\ regions
\citep{kurtz02}.
Comparing the positions in Table~\ref{t:vla} with those from 1989 (\citealt{claus94};
\citealt{achim97}) leads to the following identifications: Q1 = K2, Q2 = K3 = A, Q3 = K4 =
B, Q4 = K6, Q5 = K7 = MD1, K5 = C, K8 = F. Source K1 is several arc seconds
away and probably unrelated. Sources Q1=K2 and Q4=K6 were not seen before and appear to be
new. On the other hand, sources E and G seem to have disappeared since 1989,
the epoch when the Tieftrunk et al data were taken. Most strikingly,
source D2 has disappeared, which is remarkable since it was
the strongest source in 1989. Perhaps D1 and D2 were not separate sources, but
merely substructure within one source, which we refer to as D hereafter.
Our sources with counterparts in the old data do not exactly lie on the
positions reported by \citet{claus94} and \citet{achim97}, but rather at
80--140 mas shifts. The shifts are 2--3 beam sizes and $>$10 times the formal
error on relative positions. The position angles of the shifts vary between
20$^\circ$ and 60$^\circ$, which argues against instrumental effects such as
pointing errors or changes in calibrator positions, which would shift all sources in
the same direction. The data thus seem to confirm the existence of proper motions
reported by \citet{wilson03}. At a distance of 1.83~kpc, a motion of 100~mas
in 7.48~yr corresponds to a transverse velocity of 116~km~s$^{-1}$. The space
velocity may be a factor $\sqrt{2}$ higher, or 164~km~s$^{-1}$. These values are
much larger than the motions of the H$_2$O\ masers of $\sim$20 km~s$^{-1}$\
\citep{imai00} and of the CO emission and absorption (\S~\ref{s:intro}).
\subsection{Infrared Positions}
\label{s:ir_pos}
\begin{table*}
\caption{Quantitative analysis of the infrared imaging: flux densities
and relative positions. The table columns are arranged as follows:
(1) observing $\lambda / \Delta \lambda$; (2) speckle `S' or long `L'
exposure; (3) FWHM of point-source reference star; (4,6,10) flux of
MIR 1--3; (5,7,11) FWHM of MIR 1--3; (8,12) separation of
sources MIR1-2 and MIR1-3; (9,13) position angle of sources MIR1-2 and MIR1-3;
(14) total flux detected. }
\label{t:ir}
\begin{tabular}{rccrrrrrrrrrrr}
\hline
\hline
Filter & Mode & PSF star &
\multicolumn{2}{c}{MIR1}&
\multicolumn{2}{c}{MIR2}& \multicolumn{2}{c}{MIR1-MIR2}&
\multicolumn{2}{c}{MIR3}& \multicolumn{2}{c}{MIR1-MIR3}& Total \\
$\lambda / \Delta \lambda$
&L/S&FWHM & flux& FWHM& flux& FWHM& Sep & PA & flux & FWHM& Sep & PA & Flux \\
($\mu$m) & &(mas)& (Jy)&(mas)& (Jy)&(mas)& (mas)&(deg)& (Jy) &(mas)& (mas)&(deg)& (Jy) \\
\hline
4.8/0.6 & L & 507 & 44 & 453 & 23 & 471 & 1217 & 215 & 0.6 & 480 & 2730 & 186 & 80 \\
8.0/0.7 & L & 384 & 140 & 432 & 108 & 480 & 1198 & 215 & 2.0 & 436 & 2741 & 187 & 301 \\
8.0/0.7 & S & 258 & 152 & 310 & 123 & 367 & 1214 & 217 & 1.8 & 304 & 2663 & 188 & 328 \\
9.9/0.8 & L & - & 4 & 576 & 4 & 568 & 1013 & 215 &$<0.5$& ~-~ & ~-~~ & ~-~ & 11 \\
10.7/1.4 & S & 291 & 13 & 460 & 12 & 552 & 1078 & 219 &$<1.2$& ~-~ & ~-~~ & ~-~ & 32 \\
11.7/1.0 & L & 364 & 60 & 519 & 50 & 607 & 1097 & 216 & 0.7 & 255 & 2739 & 187 & 142 \\
12.5/0.9 & S & - & 60 & 467 & 50 & 587 & 1145 & 218 & 1.1 & 329 & 2632 & 188 & 146 \\
17.65/0.9& L & 451 & 98 & 716 & 92 & 636 & 1049 & 217&$<10.7$& ~-~ & ~-~~ & ~-~ & 318 \\
17.65/0.9& S & 452 & 107 & 646 & 93 & 615 & 1102 & 219&$<10.3$& ~-~ & ~-~~ & ~-~ & 323 \\
\hline
\end{tabular}
\end{table*}
The infrared images (Figs.~\ref{f:keck_long} and~\ref{f:keck_spec})
show three compact sources, surrounded by diffuse emission which becomes
more pronounced toward longer wavelengths. We begin a quantitative
analysis of these images by fitting simple profiles to the data, and
by measuring flux densities in different regions. The results are
given in Table~\ref{t:ir}, which gives the fluxes, relative positions
and full-width at half-maximum (FWHM) of the various components. In
addition, the FWHM of the point-source reference stars are given,
which gives an estimate of the resultant system PSF. Examination of
these data shows that the system appears truly diffraction-limited in
either mode (`L' or `S') at 17.65~$\mu$m. At shorter wavelengths
(8.0--12.5~$\mu$m), the `S' mode delivers a significantly smaller FWHM
than `L' (8.0~$\mu$m\ gives a direct comparison) which approaches the
formal diffraction limit. The dramatic increase in size at 4.8~$\mu$m\
implies some optical problem beyond the normal effects of diffraction
and seeing, such as optical aberration or defocus.
In the absence of a wide-field image with standard stars, our only astrometric
information comes from the relative positions of the components. We refer to
the northernmost bright component as MIR1, with MIR2 being of nearly equal
brightness to the south, and MIR3 the much fainter southernmost peak.
Table~\ref{t:ir} lists the separation and the position angle of MIR2 \& MIR3
relative to MIR1 for all observations, obtained through Gaussian fits to the
emission.
The mean separation of MIR1 \& MIR2 is 1124$$\pm$$74 mas at a position angle of
36.8$$\pm$$1.7 degrees. These values are consistent with those from earlier
mid-infrared work (\citealt{Howell81,Neugebauer82}), but our data are the
first to image the mid-infrared double directly.
We compare this relative position with those of pairs of radio
sources. The best match is for pair Q3--Q5, whose separation of 1210~mas at
a position angle of 37\fdeg4 is in good agreement with the infrared peaks.
The only other radio pair with similar relative positions is K7--K8, which has
a separation of 1141 mas, but at a position angle of 28\fdeg9, inconsistent
with the infrared result. On this basis, we identify the bright mid-infrared sources
with radio sources Q5=K7=MIR1 and Q3=K4=MIR2.
The position of source MIR3, with 1--2\% of the flux density of the
main sources, then coincides with radio source Q4=K6, confirming our
identification. There is a cluster of H$_2$O\ maser spots close to this
object, at $\Delta$($\alpha$)$\approx$250 and
$\Delta$($\delta$)$\approx$2000 mas \citep{imai00}.
Table~\ref{t:id} summarizes our source identifications.
The images in Figures~\ref{f:keck_long} and~\ref{f:keck_spec} show the only
region of flux detected within the 10\farcs24 field of view of LWS, with one
exception. In a few frames (which happened to be offset from center) a faint
diffuse component was seen at the extreme edge of the field, 7\farcs3 from MIR1
at a position angle of 160$^\circ$. Using the radio identifications of MIR1 and
MIR2 as astrometric reference, this source, which we call MIR4, lies at position
$\alpha$ = 02$^h$ 25$^m$ 41\fsec1388, $\delta$ = 62$^\circ$ 05$'$ 45\farcs604
(J2000), where no radio emission is detected. The extended nature and
location at the edge of the field of view preclude measurement of its
mid-infrared flux density.
\begin{table}
\caption{Radio and infrared identifications.}
\label{t:id}
\begin{center}
\begin{tabular}{cccc}
\hline
\hline
MIR & Q & K & 1989 \\
\hline
1 & 5 & 7 & D \\
2 & 3 & 4 & B \\
3 & 4 & 6 & --\\
--& 1 & 2 & --\\
--& 2 & 3 & A \\
--& --& 5 & C \\
--& --& 8 & F \\
--& --& --& E \\
--& --& --& G \\
--& --& 1 & -- \\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Radio Brightness}
\label{s:cm_b}
The flux densities of 22~GHz sources K3, K4, K5, K7 and K8 are
significantly different from the values by \citet{achim97}, probably
due to variability. Instrumental effects, such as calibration
problems, atmospheric decorrelation, or difference in beam size, would
affect all sources in the same way. Instead, several components seem
to undergo gradual increases or decreases in 22~GHz brightness over
the available 13-year period (Figure~\ref{fig:fdevol}).
The total flux densities in Table~\ref{t:vla} indicate a spectral
index $\gamma$, defined through
$$S_\nu \propto \nu^\gamma$$
of $\gamma$=1.42$$\pm$$0.77
for source Q2, $\gamma$=0.52$$\pm$$0.33 for source Q3, and
$\gamma$=0.45$$\pm$$0.48 for source Q5. Values derived from the peak
brightness are 0.62$$\pm$$0.44, 0.12$$\pm$$0.22 and 0.64$$\pm$$0.29. These
values are consistent with thermal emission, and are not affected by
variability as the data were taken simultaneously. For the weak 43~GHz
sources Q1 and Q4, we find $\gamma$=--0.3$$\pm$$0.6, i.e., flat or
slightly nonthermal spectra. Sources K1, K5 and K8, which are not
detected at 43~GHz, have $\gamma$=--1$$\pm$$1 and may be of nonthermal
nature. The spectral indices found here are consistent with those by
\citet{wilson03}, which are also based on simultaneous measurements,
except for source K8=F which was detected at 43~GHz in 2002 but not in
1996.
In the case of Bremsstrahlung from an ionized region with a power law
distribution of the electron density with radius,
$$n_e \propto r^{-q}$$
the spectral index is
$$\gamma = (2q-3.1)/(q-0.5)$$
\citep{olnon75}. Hence the spectral index of $\gamma \approx 0.5$
measured for Q3 and Q5 corresponds to $q\approx1.9$, close to the
value of $2$ for an ionized wind. Observations of broad H~I radio
recombination lines (\citealt{achim97,sewilo04}) support this
interpretation, although higher angular resolution is needed to relate
the line emission with the continuum sources. In any case, our
measurements are not sensitive enough to rule out a flat radio
spectrum for these sources, which would indicate optically thin
emission. The objects could then be externally ionized, consistent
with the absence of mid-infrared emission.
The value $\gamma\approx 1.5$ measured for Q2 could arise in an H~{\sc II}\
region with a constant density at the center and a steep
($q\approx4.7$) outer fall-off. It is not clear which mechanism would
create such a density distribution, especially since the sound speed
of $v_S \sim$10~km~s$^{-1}$\ of H~{\sc II}~regions implies that in these compact
(\O ${_<\atop{^\sim}}$100~AU) sources, density fluctuations are washed out
within $\sim$50~yr. Therefore, Q2 is probably a uniform-density
H~{\sc II}~region which is (moderately) optically thick, but again, the data
do not rule out a wind spectrum. In either case, it is internally
ionized.
More sensitive measurements over a larger wavelength range are
necessary to constrain the emission mechanism. A constant spectral
index would support the wind model, while a bent spectrum would
indicate uniform H~{\sc II}~regions of intermediate optical depth.
Care has to be taken, however, not to include dust emission at high
frequencies, or synchrotron emission at low frequencies
(\citealt{felli93}; \citealt{reid95}).
\begin{figure}[b]
\begin{center}
\includegraphics[width=6cm,angle=-90]{aa1595f4.ps}
\caption{Flux densities (at 22 GHz) of radio sources in W3 IRS5 as a function of time.
Data are from \citet{achim97}, \citet{wilson03}, and Table~\ref{t:vla}.}
\label{fig:fdevol}
\end{center}
\end{figure}
\subsection{Infrared Brightness}
\label{s:ir_bright}
Columns 4, 6 and 10 of Table~\ref{t:ir} report the flux densities of the three
mid-infrared sources. Brightness was measured in circles of radius 600 mas,
which cover all of the diffraction and seeing patterns. For the purposes of
flux calibration, six reference stars were used, and four of these gave
consistent photometric results (the other two, presumably affected by
non-photometric conditions, were ignored). At 9.9 and 12.5~$\mu$m\ where
there were no calibrator star data taken in an identical way, the
flux calibration was derived from indirect measurements and should be
regarded as tentative. Due to these difficulties and the
variable vignetting in the camera, we quote errors on the photometry of up to
50\%.
Within these errors,
the total flux densities at 4.8--10.7~$\mu$m\ are consistent with the values
measured by \citet{willner82} in a $\sim$10$''$ beam.
However, at 9.9~and 12.5~$\mu$m, where no calibrators were observed, and
at 11.7~$\mu$m, where only one calibrator was observed, the photometric
error is probably closer to a factor of 2. Indeed, at 11.7 and
12.5~$\mu$m, our flux densities are a factor of $\sim$2 below the values
measured by Willner et al and by \citet{persi96} in a 3$''$ beam.
It is interesting to note that the relative fluxes and positional offsets
between MIR1--3 remain fairly constant across the mid-infrared and (presuming
our radio identifications) into the radio. This implies that it is unlikely
that there are large differences in the effective temperatures or the optical
depths to these 3 components. The only readily identifiable exception to this
is a systematic trend for MIR1 being brighter than MIR2 at short wavelengths,
while they are nearly equal at long wavelengths. This may imply a somewhat
hotter underlying spectrum, or there may be opacity gradients in the line of
sight.
\subsection{Contribution from PAHs}
\label{s:pah}
The observed infrared emission may be continuum emission from dust grains.
However, the mid-infrared spectra of many Galactic sources, including compact
H~{\sc II}\ regions and other star-forming regions, show strong emission
features due to Polycyclic Aromatic Hydrocarbons (PAHs).
The strongest PAH features lie at 3.3, 6.2, 7.7, 8.6, 11.2 and
12.7~$\mu$m\ \citep{peeters02}. Therefore our 8~$\mu$m\ filter contains the 7.7~$\mu$m\
feature, the 10.7 and 11.7~$\mu$m\ filters the 11.2~$\mu$m\ feature, and
the 12.5~$\mu$m\ filter the 12.7~$\mu$m\ feature.
The strength of these features reflect the ambient UV radiation field,
rather than the dust mass or temperature.
Therefore, to interpret our Keck data properly, quantifying the
contribution of PAHs to the observed emission is essential.
We have searched the ISO-SWS spectrum of W3~IRS5 (F.~Lahuis, priv.\
comm.) for PAH features. With typical widths of 0.1--0.4~$\mu$m, the
features should be easily resolved with ISO. No PAH features are
detected down to an rms noise level of $\approx$1~Jy. The flux
densities measured with ISO are 60--90\% of those measured with Keck,
so beam dilution does not play a role. We conclude that the emission
observed with Keck is thermal emission from dust grains.
\subsection{Infrared Sizes}
\label{s:ir_size}
Our fits to the mid-infrared images show that the two bright cores
MIR1 and MIR2 exhibit systematically larger sizes than the reference
stars, as measured by the Gaussian FWHM values given in Column~3 of
Table~\ref{t:ir}. In this section, we extract quantitative estimates
of the apparent angular diameters of these cores, by deconvolving with
the reference star PSF then fitting with a simple circularly-symmetric
profile (in this case a uniform disk).
However, this could only be done in a minority of cases where the
data were suitable and of sufficient quality.
The 4.8~$\mu$m\ data were affected by an unknown optical problem (as
discussed earlier), while the 9.9 \& 12.5~$\mu$m\ data had no PSF
reference star data, and were ignored here.
Furthermore, deconvolution requires the highest possible angular
resolution data, and we therefore restrict our attention to only the
rapid exposure observing mode `S', discarding `L' (e.g. all 11.7~$\mu$m\
data).
Perhaps the most difficult aspect of the deconvolution problem was
distinguishing between the resolved cores and the more extended nebula.
At the longest wavelength, 17.65~$\mu$m, this was not possible for
two reasons: firstly the extended component was relatively bright
compared to the cores, and secondly the angular resolution was not
sufficient to clearly distinguish between compact and extended flux.
The remaining datasets suitable for deconvolution and diameter fitting
were from 8.0 \& 10.7~$\mu$m. Fits were obtained with a uniform circular
disk profile, although in the partially resolved case as here any
simple model (such as a Gaussian) would serve equally well. Uniform
disk fits were obtained for MIR1 \& MIR2 at 8.0~$\mu$m, where the
diameters are 207 \& 254~mas, and at 10.7~$\mu$m, where they are 300 \&
333~mas, respectively. Although the formal errors on these quantities
are around 40~mas, the true uncertainties are hard to quantify due to
unknown seeing and optical changes between the source and calibrator
stars, and due to imperfect rejection of the extended nebula in
fitting the core.
Two systematic trends are noted here: MIR2 appears slightly larger than MIR1,
and the sizes at 10.7~$\mu$m\ are larger than those at 8.0~$\mu$m. However,
particular caution needs to be expressed over contamination from the extended
flux component which may have a role in causing these apparent extensions.
The measured mid-infrared sizes exceed the limits on the radio sizes
of $\sim$100~AU (Table~\ref{t:vla}). This result supports a model
where the radio emission comes from ionized gas very close to a star
and the mid-infrared emission from warm dust somewhat further out.
\section{Discussion}
\label{disc}
Our observational findings can be summarized as follows: three
mid-infrared point sources with radio counterparts; four radio sources
without mid-infrared counterparts that appear to change position; and
diffuse mid-infrared emission.
The following sections discuss each of these components in turn.
\subsection{Mid-infrared point sources}
\label{sec:blackbody}
\begin{figure}[tb]
\includegraphics[width=6cm,angle=-90]{aa1595f5.ps}
\caption{Mid-infrared spectral energy distributions of W3 IRS5 after
dereddening. Squares, triangles and dots indicate
MIR1, MIR2 and MIR3. Model curves are superposed.}
\label{f:dered}
\end{figure}
To estimate the physical properties of the compact mid-infrared sources, we have
compared their flux densities to a simple blackbody model. Given the lack of
strong observed colour variations (\S~\ref{s:ir_bright}), we assume that the
sources have the same temperature and foreground extinction. Based on
the results of \S~\ref{s:ir_size}, we use a radius of 250~AU for sources MIR1
and MIR2; for MIR3, $R$=30~AU is adopted based on its lower brightness. The
observed far-infrared luminosity then limits the temperature to $T<$390~K,
which appears plausible based on the measured colours and
the 2.2~$\mu$m\ photometry by \citet{ojha04}.
Using this temperature and radius, we can model the observed flux
densities if we know the foreground extinction.
The broad-band mid-infrared spectrum presented by \citet{willner82}
indicates a silicate optical depth of $\tau_S \approx$4.3--5.0
assuming pure absorption, or $\tau_S$=7.64 when correcting for
underlying silicate emission, as Willner et al do.
More recent data from ISO give consistent results (F.~Lahuis, priv.\
comm.), even though they refer to a larger beam ($\sim$20$''$) that
varies by a factor $\sim$4 over this wavelength range.
Figure~\ref{f:dered} shows the measured flux density spectrum after
de-reddening by $\tau_S$=5.0. Values for the extinction at other wavelengths
are computed using dust properties by \citet{oh94}, Model~5. This dust model
gives a good match between envelope masses derived from dust and CO
\citep{fvdt99}. The figure shows that a blackbody model with $T$=390~K
reproduces the data within a factor of 3. The largest outliers are the
9.9~and 11.7~$\mu$m\ points, at which wavelengths the calibration is the
most uncertain.
The \citet{oh94} dust model has Si:C=1.45, while values up to $\approx$2 are
observed \citep{kruegel}. Increasing the Si:C ratio would improve the
match between data and model at 9.9~$\mu$m, but would give a worse fit
at 11.7~$\mu$m. More likely, the assumption of blackbody emission is not
quite valid, so that towards shorter wavelengths, smaller radii and
higher temperatures are probed. Geometrical effects may also influence
the shape of the silicate absorption.
Temperatures below 390~K are energetically allowed, but require lower
extinctions to fit the observed flux densities. For $T<350$~K, the
required extinction drops below $\tau_S$=4.3, which we consider
unlikely based on the \citet{willner82} data. The total luminosity
for $T$=350~K is $\sim$8$\times$10$^4$~L$_{\odot}$, which leaves
$\sim$4$\times$10$^4$~L$_{\odot}$\ for a third power source, such as radio
source Q2 (\S~\ref{s:cm_b}).
This option is more likely than the case of two
power sources, because radio sources Q2, Q3 and Q5 are of similar
strength (\S~\ref{s:cm_b}).
The (tentative) size increase of MIR1~\&~2 from 8.0 to 10.7~$\mu$m\
(\S~\ref{s:ir_size}) is to be expected if a more realistic assumption of a
centrally-heated dust cloud with a thermal profile is adopted, rather than a
blackbody at a single temperature. A detailed understanding of this deeply
embedded and complex region will clearly require radiative transfer modelling,
and it is encouraging that mid-infrared imaging appears capable of placing
meaningful constraints.
The best-fit extinction of $\tau_S$=5.0 is significantly below Willners
estimate. Perhaps their simple formula to correct for silicate emission is
not valid at large extinction values. More likely, the silicate absorber is
physically decoupled from the underlying continuum emitter. One geometrical
interpretation is that of two star/disk systems surrounded by a cavity, whose
walls cause the silicate absorption. Such a cavity would also explain why
submillimeter imaging in a 15$''$ beam indicates a much higher extinction
($A_V\sim$300; \citealt{fvdt00}) than the mid-infrared data.
\subsection{Stationary radio sources}
\label{sec:ionize}
The flux densities $S_\nu$ of the radio sources (Table~\ref{t:vla})
can be used to estimate the Lyman continuum emission $N_L$ of their
ionizing sources, assuming that the H~{\sc II}~regions are uniform and
isothermal (see, e.g., \citealt{tools}). This discussion concentrates
on sources Q2, Q3 and Q5 which have positive spectral indices
(\S~\ref{s:cm_b}), and considers both optically thin and optically
thick emission as limiting cases. In the optically thin case, $N_L$ is
directly proportional to the flux density. In the optically thick
case, black body emission at $T$=10$^4$~K indicates radii of
$\sim$20~AU, consistent with the observational upper limits
(Table~\ref{t:vla}). The emission measure follows from setting the
free-free optical depth equal to unity; radii and emission measures
together indicate electron densities of 10$^6$--10$^7$~cm$^{-3}$. Balancing
photoionization with `case B' recombination \citep{osterbrock} finally
gives $N_L$. The results for both cases are
$N_L$=1...7$\times$10$^{44}$~s$^{-1}$, with a weak dependence on
electron temperature.
The similar values of $N_L$ derived for 43~GHz sources Q2, Q3 and Q5 indicate that
they have similar luminosities. Therefore it is hard to see how only
the ionizing source of Q2 could be invisible in the mid-infrared,
unless it is extremely deeply embedded. In fact, our 17.65~$\mu$m\
images may show a source about 0\farcs5 North of MIR2, but the data do
not allow to extract a flux density.
The stellar luminosities of $\approx$40,000~L$_{\odot}$\
(\S~\ref{sec:blackbody}) correspond to masses of $\approx$20~M$_{\odot}$\
and ZAMS spectral types O8 \citep{maeder89}. Their expected Lyman
continuum emissions are $\approx$6$\times$10$^{48}$~s$^{-1}$
\citep{sdk97}, which is $\sim$10$^4$ times the value just derived from
the radio continuum emission. Since dust absorption inside the
H~{\sc II}~region only accounts for factors of 2--3, this discrepancy may be
due to accretion of dust particles \citep{walmsley95}. The observed
variability (Fig.~\ref{fig:fdevol}) may then correspond to variations
in the accretion rate.
Accretion of dust would also explain why the hypercompact H~{\sc II}\
regions stay confined to a $\sim$20~AU radius. The ionization front
around an O-type star on the main sequence is a D-critical front (e.g.,
\citealt{osterbrock}) which moves at about the sound speed of
$\approx$10~km~s$^{-1}$, or somewhat less (5--7~km~s$^{-1}$) since the surrounding
H~I shell needs to be accelerated. \citet{acord98} have seen such
expanding motions in the ultracompact H~{\sc II}\ region G5.89. In the case
of W3~IRS5, expansion at 5--10~km~s$^{-1}$\ would lead to an increase in
radius from 20 to 100~AU over the observed 10-year period which is not
observed.
The accretion rate needed to confine the hypercompact H~{\sc II}\ regions may
be estimated by equating the accretion force (momentum transfer rate)
of the dust to the thermal pressure of the H~{\sc II}\ region. Using a
radius of 20~AU, a density of 3$\times$10$^6$~cm$^{-3}$, and $T$=8000~K for
the H~{\sc II}\ region, we find $\dot{M}$=1.5$\times$10$^{-8}$~M$_{\odot}$\,yr$^{-1}$
assuming that the dust is in free fall onto a 20~M$_{\odot}$\ star. In
reality, radiation pressure will slow down the dust from the free-fall
speed (42.2~km~s$^{-1}$), so that perhaps twice this $\dot{M}$ is needed.
If the stars have winds with substantial mass loss rates (e.g.,
10$^{-6}$~M$_{\odot}$\,yr$^{-1}$), even higher accretion rates may be
needed to confine the H~{\sc II}\ region.
Recent work by \citet{keto02}, however, shows that stellar gravity
prevents the hydrodynamical expansion of H~{\sc II}\ regions inside a
`gravitational radius'
$$ r_g = GM / 2c_s^2$$
where $G$ is the gravitational constant, $M$ the stellar mass, and
$c_s$ the sound speed of H~{\sc II}\ regions of $\approx$10~km~s$^{-1}$.
For the bright mid-infrared sources in W3~IRS5, $M\approx$20~M$_{\odot}$\
(\S~\ref{sec:blackbody}) so that $r_g\approx$90~AU, consistent
with the observational limits (Table~\ref{t:vla}).
In Keto's model, both the ionized region close to the star and the
surrounding molecular gas have free-fall density profiles, $n\propto
r^{-1.5}$. At $r=r_g$, the accretion flow changes from molecular to
ionized. Such a density profile was indeed found for the molecular
envelope of W3~IRS5 by \citet{fvdt00} from submillimeter continuum and
line maps.
The expected flux density of a gravitationally bound H~{\sc II}\ region only
depends on the density $n_0$ at the radius $r_m$ where the molecular gas
reaches its sound velocity \citep{keto03}.
Taking $T$=30~K for the molecular gas, $r_m\approx 0.35$~pc.
For $d$=1.83~kpc, $M$=20~M$_{\odot}$\ and $T_e$=10$^4$~K, the observed flux
density of $\approx$1~mJy at 22--43~GHz is reproduced for
$n_0\approx$1$\times$10$^5$~cm$^{-3}$.
This estimate agrees to a factor of 5 with the value of
$n_0$=2$\times$10$^4$~cm$^{-3}$\ derived by \citet{fvdt00}.
We conclude that gravitation explains the compactness of the radio
sources in W3~IRS5 which have mid-infrared counterparts.
\subsection{Transient radio sources: proper motions?}
\begin{figure}[bt]
\begin{center}
\includegraphics[width=8cm,angle=-90]{a1595f6a.ps}
\bigskip
\includegraphics[width=8cm,angle=-90]{a1595f6b.ps}
\caption{Relative positions of radio sources in W3 IRS5
before (top) and after (bottom) aligning sources B
and D. Data are from \citet{claus94}, \citet{wilson03}, and
Table~\ref{t:vla}. Arrows denote derived proper motions.
The symbol sizes represent the formal position error of 0\farcs06.}
\label{fig:motion}
\end{center}
\end{figure}
\begin{table*}[tb]
\caption{Proper motion solutions.}
\label{tab:pm}
\begin{center}
\begin{tabular}{crrrr}
\hline
\hline
\noalign{\smallskip}
Source & \multicolumn{2}{c}{Offset (mas)$^a$} & \multicolumn{2}{c}{Proper motion (mas/yr)} \\
& \multicolumn{1}{c}{$\alpha$} & \multicolumn{1}{c}{$\delta$} & \multicolumn{1}{c}{$\alpha$} & \multicolumn{1}{c}{$\delta$} \\
\noalign{\smallskip}
\hline
\multicolumn{5}{c}{\it Before Shift} \\
A & --0.093 $$\pm$$ 11.550 & 0.158 $$\pm$$ 11.223 & 5.015 $$\pm$$ 1.801 & 8.202 $$\pm$$ 1.624 \\
C & \multicolumn{1}{c}{...} & \multicolumn{1}{c}{...} & --7.813 $$\pm$$ 2.098 & --8.642 $$\pm$$ 2.112 \\
F & --2.456 $$\pm$$ \phantom{0}8.277 & --27.419 $$\pm$$ 7.630 & --5.359 $$\pm$$ 1.300 & 16.480 $$\pm$$ 0.806 \\
\multicolumn{5}{c}{\it After Shift} \\
A & 0.485 $$\pm$$ 11.550 & 16.559 $$\pm$$ 11.223 & 12.015 $$\pm$$ 1.801 & 23.176 $$\pm$$ 1.624 \\
C & \multicolumn{1}{c}{...} & \multicolumn{1}{c}{...} & --0.654 $$\pm$$ 2.098 & 10.242 $$\pm$$ 2.112 \\
F & 0.082 $$\pm$$ \phantom{0}8.277 & 26.035 $$\pm$$ 7.630 & 1.042 $$\pm$$ 1.300 & 16.340 $$\pm$$ 0.806 \\
\noalign{\smallskip}
\hline
\end{tabular}
\end{center}
$^a$ Best fit offset from the measured 1989 position; unavailable for
C where only two epochs were measured
\end{table*}
Figure~\ref{fig:motion} shows the positions of radio sources A...F derived by
us and by \citet{claus94} and \citet{wilson03}. Sources B and D appear
stationary when comparing the 1989 and 2002 data, but seem to have shifted by
$\approx$0\farcs2 in the 1996 data. This shift may be a systematic phase
error in the 1996 data.
The likely cause is an atmospheric `wedge', or $\sim$100~km-sized
parcel of dense air partially covering the interferometer and causing
a gradient in atmospheric opacity (M.~Reid, priv.\ comm.).
By aligning the positions of sources B and D in 1996 with those of 1989 and 2002,
we derive a shift of $\Delta\alpha$ = 55$$\pm$$7 and $\Delta\delta$ = 146$$\pm$$36 mas.
However, since the magnitude of the shift is uncertain, we derive
proper motion values both before and after applying the shift
(Table~\ref{tab:pm}).
Source C was only detected at two epochs, which is sufficient to
estimate the magnitude and the direction of its motion. For sources A
and F, three epochs are available, which allows us to solve
additionally for the position at the first epoch, using the least
squares technique. For source A, good fits ($\chi^2$/dof $\sim$1) to
the $\alpha$ and $\delta$ motions are obtained if no shift is applied.
However, for source F, the fits are poor ($\chi^2$/dof $>$10) whether
the shift is applied or not. One possible explanation for these poor
fits are deviations from the assumed uniform motions. The 1996 data
thus may confirm the proper motion of component F found by
\citet{wilson03} and show that components A and C may move as well,
but do not allow precise measurement of the magnitude and direction of
the motions.
If radio sources A, C and F are internally ionized, their ionizing stars
must be moving along, since at an electron density of
3$\times$10$^6$~cm$^{-3}$, the recombination time scale is
$\sim$1~month while the sources are seen over several years.
The free-fall speed in the gravitational potential of the
molecular gas ($M$=260~M$_{\odot}$, $R$=60,000~AU: \citealt{fvdt00}) is
$\approx$3~km~s$^{-1}$. The potential of the star cluster can be estimated
by $M$=20~M$_{\odot}$\ and $R$=1000~AU (the typical separation of the
radio sources: Fig.~\ref{f:vla}), which gives a free-fall speed of
$\approx$6~km~s$^{-1}$. The derived proper motions of the radio sources are much
faster than these values, and therefore would not represent
bound motions. Perhaps these stars were ejected from the
cluster in a close stellar encounter, and are very young runaway OB
stars, like the BN object in Orion, which is moving at 50~km~s$^{-1}$\ \citep{plambeck95}.
We conclude that the evidence for proper motions remains weak,
even with three epochs measured. This shows graphically in
Fig.~\ref{fig:motion}: the 1996 positions do not generally lie between
those for 1989 and 2002. Quantitatively, it shows in the large error
margins in Table~\ref{tab:pm}. The next section explores alternative
explanations for the transient radio sources in W3~IRS5.
\subsection{Transient radio sources: Shocked clumps?}
\label{sec:shock}
The transient radio sources of W3~IRS5 may also be explained by shocks
which occur when the winds from the young O-type stars hit clumps in
the surrounding molecular material. Such a picture of massive star
formation has been described by, e.g., \citet{franco90} and \citet{dyson02}.
Observational support has been found in the source Cep~A \citep{hughes01},
which is of somewhat lower luminosity and distance than W3~IRS5.
To estimate the radio emission from wind-shocked clumps, we use the
model by \citet{hollenbach89}. In the limit that the stellar wind is
much less dense than the molecular clump, the flux $F_i$ of ionizing
photons is given by
$$F_i = n_0 v_S F(v_S)$$
where $n_0$ is the density of the clump, $v_S$ the shock velocity, and
$F(v_S)$ the fractional ionization of the shocked gas. To have
$F(v_S) \sim 1$, the shock must fully dissociate the H$_2$\ clump and heat it
to $\sim$10$^5$~K, so that it emits ionizing photons.
The required shock velocity is $\sim$100~km~s$^{-1}$;
at lower velocities, $F(v_S)$ drops exponentially due to the
Boltzmann distribution.
Velocities of $\sim$100~km~s$^{-1}$\ are commonly observed for the winds of
deeply embedded high-mass stars including W3~IRS5, both in hydrogen
recombination lines \citep{bunn95} and in CO mid-infrared absorption
lines (\citealt{mitch91,fvdt99}).
The emission measure of the shock-ionized clump is
$$ n_e^2 l = \frac{F_i}{\alpha_B} \approx
10^8 \left( \frac{n_0}{10^7 {\rm cm}^{-3}} \right)
\left( \frac{v_S}{100 {\rm km\, s}^{-1}} \right)
{\rm cm}^{-6}{\rm pc} $$
where $\alpha_B$ is the Case~B recombination coefficient (\S~\ref{sec:ionize}).
The observed values of $l{_<\atop{^\sim}}$100~AU and $n_0$=10$^6$~cm$^{-3}$\ agree
with this prediction within order of magnitude. We conclude that winds
from young O-type stars shocking and ionizing clumps in the ambient
cloud provide a viable model for the transient radio emission observed
in W3~IRS5.
\subsection{Diffuse emission: Envelope structure}
\label{sec:envelope}
Flux densities for the diffuse mid-infrared emission can be obtained by
subtracting the point source contributions (columns 4, 6 and 10 of
Table~\ref{t:ir}) from the total flux density (column 14). The emission is
roughly elliptical in shape, with the major axis more or less aligned with the
line connecting MIR1 and MIR2. Going from short to long wavelengths, the axis
ratio (measured at the 1\% level) decreases from $\approx$1.6 to $\approx$1.0,
while the radius (the average of the semi-major and semi-minor axes) increases
from $\approx$2300 to $\approx$4100~AU.
The brightness distribution of the diffuse emission is consistent with
heating by sources MIR1 \& MIR2. Short wavelengths probe warm dust
close to the individual stars, so that the emission has two peaks.
Longer wavelengths probe cooler dust that is far enough away that the
distances to both stars are about the same, leading to round contour
shapes.
At a size of ${_>\atop{^\sim}}$1000~AU, the diffuse emission cannot be optically
thick: even for temperatures as unrealistically low as 100~K, the
far-infrared luminosity is exceeded, even with zero foreground
extinction.
The envelope must also have a low mid-infrared optical depth
to give us a view of the central objects.
The envelope of W3~IRS5 was modeled by \citet{campbell95}, based on
far-infrared data, and by \citet{fvdt00}, based on submillimeter data.
Assuming single power laws for the density structure,
these models have $\tau$(100~$\mu$m)$\sim$1, or $\tau$(10~$\mu$m)$\sim$40,
much higher than estimated above.
The W3~IRS5 core is embedded in a large-scale molecular cloud, and the
observed submillimeter emission could contain contibutions from the
background cloud, but not more than 50\%. Neither the mid-infrared
nor the submillimeter images suggest deviations from spherical
symmetry stronger than a modest flattening (axis ratio $<$2).
To reconcile the submillimeter and far-infrared data with the
mid-infrared data, one may explore broken power laws for the density
distribution, or a combination of dense shells and power laws.
\section{Conclusions and Outlook}
\label{concl}
\begin{figure*}[th]
\begin{center}
\includegraphics[width=12cm,angle=-90]{aa1595f7.ps}
\caption{Schematic view of the W3 IRS5 region, as projected on the
sky, with the observational characteristics of each physical
component indicated. Asterisks mark new findings of this paper.}
\label{fig:cartoon}
\end{center}
\end{figure*}
Observations at subarcsecond resolution at mid-infrared and radio
wavelengths have led to a detailed picture of the W3~IRS5 region,
shown schematically in Figure~\ref{fig:cartoon} and described below.
\begin{itemize}
\item The two bright mid-infrared sources with radio emission probably
are deeply embedded high-mass stars. They are both close to groups
of H$_2$O\ masers. The measured mid-infrared diameters are
consistent with blackbody emission at $T$=390~K and providing all of
the far-infrared luminosity of W3~IRS5.
\item A third, weaker mid-infrared and radio source with associated
H$_2$O\ masers is probably a somewhat later-type star which is energetically
unimportant.
\item Radio source A has an optically thick radio spectrum, and may
have a counterpart at long mid-infrared wavelengths
(${_>\atop{^\sim}}$17.65~$\mu$m). It may be an extremely deeply embedded
high-mass star. The three power sources of W3~IRS5 then have
$L$$\approx$40,000~L$_{\odot}$\ each, which in the models of
\citet{maeder89} makes them $\approx$20~M$_{\odot}$\ stars (ZAMS spectral
type O8).
\item The region shows several transient radio sources. Some
of these may represent runaway OB stars, but most are probably
clumps in the ambient material which are ionized and destroyed by
shocks with the winds of the O-type stars.
\item The low silicate optical depth suggests that no underlying
silicate emission is present. This is most easily explained by a
cavity separating the high-mass stars from their envelope. Perhaps
the cavity was blown by the slow spherical outflow traced by the
H$_2$O\ masers.
\item The far-infrared and submillimeter emission, as well as the
low-velocity CO mid-infrared absorption, arise in the large-scale
envelope. The dense stellar cluster visible in the near-infrared
(and X-ray) is embedded in the same envelope.
\end{itemize}
In the future, subarcsecond monitoring of W3 IRS5 at high radio
frequencies (VLA-A) is necessary to test the `proper motion' and
`shocked clump' hypotheses. If proper motions are confirmed, the
shapes of the orbits will be a test of the `runaway star' hypothesis,
and will constrain the dynamics of this young cluster. The emission
mechanism should be studied by simultaneous observations at three or
more wavelengths. If the e-VLA does not provide the sensitivity
necessary to do this, ALMA will.
Mid-infrared imaging at wavelengths ${_>\atop{^\sim}}$20~$\mu$m\ at subarcsecond
resolution is necessary to search for a mid-infrared counterpart to
radio source Q2=K3=A. The main requirements are higher sensitivity and
dynamic range than was achieved here; the higher angular resolution
offered by MIDI on the VLTI will be more useful to search for fine
structure. Spatially resolved mid-infrared spectroscopy is necessary
to assign the high-velocity CO absorption features \citep{mitch91} to
particular stellar components. This may be a good project for
VLT/CRIRES. Future radiative transfer modeling efforts should consider
broken power laws for the density distribution in the envelope of
W3~IRS5.
\begin{acknowledgement}
The authors thank David Hollenbach, Eric Keto, Ed Churchwell, Lee
Hartmann, Tom Megeath, Mark Reid,
Enrik Kr\"ugel, Karl Menten, Tom Wilson, and
Thomas Driebe for useful discussions. The staffs of the VLA
(especially Claire Chandler) and Keck telescopes were helpful in
assisting with the observations. We also thank Charles Townes, John
Monnier, and Randy Campbell for help with the Keck observations.
\end{acknowledgement}
\bibliographystyle{aa}
|
{
"timestamp": "2004-11-05T14:09:32",
"yymm": "0411",
"arxiv_id": "astro-ph/0411142",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411142"
}
|
\section{\label{sec1}Introduction}
There is a large body of work on the Feynman path
integral~\cite{Feynman48} approach to quantum mechanics on a
curved background, and the Schr\"{o}dinger equation which results
from the path integral. (See some of the references described
below.) It is widely known that, depending upon what is chosen for
the measure in the configuration space path integral, there is a
multiple of the scalar curvature to be added to the canonical
Hamiltonian in the resulting Schr\"{o}dinger equation. This is
demonstrated explicitly by Parker~\cite{Parker} who considers
including an arbitrary power of the Van
Vleck-Morette~\cite{VanVleck,Morette} determinant
$\Delta^p(x,x')$, where $p$ is an arbitrary real number, in the
measure in addition to the natural curved space volume element.
The choice $p = 0$, which corresponds to choosing the natural
volume element for the path integral measure, and $p = 1/2$, which
is motivated by the WKB approximation, were examined originally by
DeWitt~\cite{DeWitt57}. The choice $p = 1$ was shown by
Parker~\cite{Parker} to lead to no additional curvature
modification of the canonical Hamiltonian. For any value of $p$
the path integral measure is invariant under an arbitrary change
of coordinates for the curved space. General covariance alone of
the configuration space path integral does not prescribe a unique
measure.
It might be thought that this lack of uniqueness in the path
integral approach to quantum mechanics in curved space disappears
if a phase space path integral is used instead of one in
configuration space. In this case there is a unique choice of
measure; namely, the Liouville measure. However,
Kucha\v{r}~\cite{Kuchar} has shown in a very lucid paper that the
lack of uniqueness associated with the configuration space path
integral measure reappears in phase space as the lack of
uniqueness of the skeletonization of the action. (As discussed by
DeWitt~\cite{DeWitt57} Hamilton-Jacobi theory leads to a covariant
skeletonization in configuration space.) Lack of uniqueness in the
quantum theory is therefore inherent in the curved space Feynman
path integral, and cannot be eliminated solely on the grounds of
general covariance.
If this lack of uniqueness is to be eliminated from the Feynman
path integral in curved space, the only possibility is to impose
some further property in addition to general covariance under a
change of coordinates in the curved space. One example of an extra
property done within the minisuperspace approach to quantum
cosmology is to demand conformal invariance~\cite{Halliwell,Moss}.
However, this seems rather specific to quantum cosmology, and not
very compelling as a general principle, since there is no reason
to believe that conformal invariance is a fundamental symmetry of
nature. Another more general principle, based on the relationship
between the Feynman path integral and stochastic differential
equations, was given in Ref.~\cite{stochastic}. This corresponds
to the choice $p = 1$, and hence gives no curvature modification
to the canonical Schr\"{o}dinger equation. As we will see, the
approach adopted in the present paper supports the conclusion of
Ref.~\cite{stochastic}.
We will examine what happens if the Feynman path integral is
required to be equivalent to the Schwinger action
principle~\cite{SAP1,SAP2}. It was known almost from its
inception, that for flat spaces the Schwinger action principle is
completely equivalent to the path integral~\cite{Burton,Symanzik}.
This equivalence holds equally well for quantum field theory,
which is usually regarded as involving a flat configuration space.
(This is how DeWitt~\cite{DeWitt64}, for example, arrives at the
path integral.) In this paper, I wish to show how the Schwinger
action principle may be used to fix the measure in the Feynman
path integral for quantum mechanics on a curved space. If the
notation is interpreted in the spirit of DeWitt's~\cite{DeWitt64}
condensed notation, then the result holds equally well for quantum
field theory with a curved configuration space. (We will comment
briefly on this in Sec.~\ref{sec4}.) It will be shown that
equivalence between the Schwinger action principle and the Feynman
path integral is only achieved if there is a single factor of the
Van Vleck-Morette determinant in the measure, (i.e. $p = 1$
above.) This is completely consistent with the result of
Ref.~\cite{stochastic} which was based on totally different
reasoning.
\section{\label{sec2}The Schwinger action principle}
Let $q^i$ denote a set of local coordinates on some manifold $M$.
The classical motion of a particle moving on $M$ is given by
$q^i(t)$ where $q^i(t)$ is a solution to the Euler-Lagrange
equations. We will adopt a configuration space rather than a phase
space approach. In quantum mechanics we are interested in
computing the transition amplitude $\mbox{$\langle q_2,t_2|q_1,t_1\rangle$}$ where $\ket{1}$
represents the quantum state at time $t_1$, and $\ket{2}$
represents the state at time $t_2\ge t_1$. These states are chosen
to be eigenstates of the position operator $\hat{q}^i$:
\begin{equation}
\hat{q}^i\ket{\alpha}={q}^i(t_\alpha)\ket{\alpha}\ \ (\alpha=1,2)
\;.\label{2.1}
\end{equation}
The Schwinger action principle~\cite{SAP1,SAP2} states that
\begin{equation}
\delta\mbox{$\langle q_2,t_2|q_1,t_1\rangle$}=\frac{i}{\hbar}\bra{2}\delta S\ket{1}\;,\label{2.2}
\end{equation}
where $S$ represents the action obtained by the replacement of
$q^i$ in the action for the classical theory with $\hat{q}^i$,
along with an operator ordering which leads to $S$ being
self-adjoint. $\delta$ in Eq.~(\ref{2.2}) represents any possible
variation, including variations with respect to the times
$t_1,t_2$, the dynamical variables $q^i$, or the structure of the
Lagrangian. The variations of the dynamical variables $\delta q^i$
will be chosen to be c-numbers, appropriate to bosonic theories.
This choice was also made in flat space by Schwinger~\cite{SAP1},
and in curved space by Kawai~\cite{Kawai1,Kawai2}. The case of
fermionic variables will not be considered here.
Suppose that we add a source term to the action. Normally this is
done so that differentiation with respect to the source generates
the $n$-point functions of the theory. If the space is flat, this
may be accomplished by simply taking
\begin{equation}
S_J\lbrack q\rbrack = S\lbrack q\rbrack +
\int\limits_{t_1}^{t_2}dt\;J_i(t)q^i(t)\;,\label{2.3}
\end{equation}
where $S\lbrack q\rbrack$ is the original action, and $J_i(t)$ is
an external source which is turned on at time $t_1$ and off at
time $t_2$. However, on a curved space where $q^i$ are coordinates
rather than vectors, the addition of the source term in
Eq.~(\ref{2.3}) does not result in a covariant expression. This is
also the case in a flat space if the coordinates are chosen to be
curvilinear rather than Cartesian.
The covariant generalization of Eq.~(\ref{2.3}) is obtained as
follows. First, it may be noted that if the field space is flat,
the same classical theory is obtained from
\begin{equation}
S_J\lbrack q,q_\ast\rbrack = S\lbrack q\rbrack +
\int\limits_{t_1}^{t_2}dt\;J_i(t)\left(q^i(t)-
q_\ast^i(t)\right)\;,\label{2.4}
\end{equation}
as from as from Eq.~(\ref{2.3}), where the coordinates are assumed
to be Cartesian, and where $q_\ast^i$ is regarded as a fixed point
in the configuration space $M$. ($q_\ast^i$ plays the role that
the background field~\cite{DeWitt64} does in quantum field
theory.) The coordinate difference $(q^i - q_\ast^i)$ then
represents a vector which connects the fixed reference point
$q_\ast^i$ to the point $q^i$. Equivalently, $(q^i - q_\ast^i)$
represents the tangent vector to the geodesic connecting
$q_\ast^i$ to $q^i$, which for $M$ flat is just a straight line
segment. This indicates that the natural replacement for the
coordinate difference $(q^i - q_\ast^i)$ in a general space $M$ is
just the tangent vector at $q_\ast$ to the geodesic connecting
$q_\ast^i$ to $q^i$. One way of introducing this tangent vector is
by means of the geodetic interval $\sigma(q_\ast;q)$. (See
Refs.~\cite{DeWitt64,Ruse,Synge}.) By definition,
\begin{equation}
\sigma^i(q_\ast;q)=\frac{1}{2}\ell^2(q_\ast;q)\;,\label{2.5}
\end{equation}
where $\ell(q_\ast;q)$ is the length of the geodesic connecting
$q_\ast$ to $q$. The tangent vector to the geodesic at $q_\ast$ is
\begin{equation}
\sigma^i(q_\ast;q)=g^{ij}(q_\ast)\frac{\partial}{\partial
q_\ast^j} \sigma(q_\ast;q)\;,\label{2.6}
\end{equation}
where $M$ is assumed to have a metric tensor $g_{ij}$. If $M$ is
flat, and $q^i$ are Cartesian coordinates (so that
$g_{ij}=\delta_{ij}$), then
\begin{equation}
\sigma^i(q_\ast;q)=-(q^i-q_\ast^i)\;.\label{2.7}
\end{equation}
The natural replacement for $(q^i-q_\ast^i)$ in Eq.~(\ref{2.4}) is
therefore $-\sigma^i(q_\ast; q)$ resulting in
\begin{equation}
S_J\lbrack q, q_\ast\rbrack = S\lbrack q\rbrack -
\int\limits_{t_1}^{t_2}dt\;J_i(t)\sigma^i(q_\ast;q)\;.\label{2.8}
\end{equation}
$\sigma^i(q_\ast;q)$ transforms like a vector under a change of
coordinates $q_\ast$, and as a scalar under a change of
coordinates $q$. If the source $J_i(t)$ is required to transform
like a covariant vector at $q_\ast$, and be independent of $q$,
then Eq.~(\ref{2.8}) is a completely covariant definition. It is
important that $J_i(t)$ be independent of $q$ if it is to fulfill
its role as an external source. If $M$ is flat, but $q^i$ are not
Cartesian coordinates, it is possible to derive Eq.~(\ref{2.8})
from Eq.~(\ref{2.4}) using the approach of Ref.~\cite{Ellicott}.
It proves convenient to adopt condensed notation~\cite{DeWitt64}
at this stage, and to write Eq.~(\ref{2.8}) as
\begin{equation}
S_J\lbrack q,q_\ast\rbrack=S\lbrack
q\rbrack-J_i\sigma^i(q_\ast;q)\;.\label{2.9}
\end{equation}
where the index $i$ is now understood to include the time label,
and a repeated index includes integration over time. Instead of
regarding $S_J\lbrack q,q_\ast\rbrack$ as a functional of
$q,q_\ast$, it is convenient to regard it instead as a functional
$\tilde{S}_J\lbrack q_\ast;\sigma^i(q_\ast;q)\rbrack$ which is
defined using the covariant Taylor expansion~\cite{Ruse}
\begin{equation}
S\lbrack q\rbrack=\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}
S_{;(i_1,\cdots i_n)} \lbrack q_\ast\rbrack \sigma^i_1(q_\ast;q)
\cdots \sigma^i_n(q_\ast;q) \label{2.10}
\end{equation}
in Eq.~(\ref{2.9}). (The semicolon denotes the usual covariant
derivative using the Christoffel connection constructed from the
metric $G_{ij}$ on $M$.)
Let $\mbox{$\langle q_2,t_2|q_1,t_1\rangle\lbrack J\rbrack$}$ be the transition amplitude for the theory with action
$S_J\lbrack q;q_\ast\rbrack$ in Eq.~(\ref{2.9}). The Schwinger
action principle gives
\begin{equation}
\delta\mbox{$\langle q_2,t_2|q_1,t_1\rangle\lbrack J\rbrack$}=\frac{i}{\hbar}\bra{2}\delta S_J\ket{1}\mbox{$\lbrack J\rbrack$} \;.\label{2.11}
\end{equation}
If the variation is taken to be one that is with respect to the
dynamical variables $q^i$ leaving the values fixed at times $t_1$
and $t_2$, then the amplitude will not change under the variation,
and we have
\begin{equation}
0=\bra{2}\delta S_J\ket{1}\mbox{$\lbrack J\rbrack$} \;,\label{2.12}
\end{equation}
from which the equation of motion may be inferred:
\begin{equation}
\frac{\delta\tilde{S}_J}{\delta\sigma^i}-J_i=0\;.\label{2.13}
\end{equation}
Since we may regard $\mbox{$\langle q_2,t_2|q_1,t_1\rangle\lbrack J\rbrack$}$ a functional of the source $J_i$, it
may be expanded in a Taylor series about $J_i = 0$:
\begin{equation}
\mbox{$\langle q_2,t_2|q_1,t_1\rangle\lbrack J\rbrack$}=\left.\sum_{n=0}^{\infty}\frac{1}{n!}J_{i_1}\cdots J_{i_n}\,
\frac{\delta^n\mbox{$\langle q_2,t_2|q_1,t_1\rangle\lbrack J\rbrack$}}{\delta J_{i_1}\cdots \delta
J_{i_n}}\right|_{J=0} \;.\label{2.14}
\end{equation}
We will now evaluate the $n^{\rm th}$ derivative of the amplitude
which occurs in Eq.~(\ref{2.14}) using the Schwinger action
principle. The method is just that used originally by
Schwinger~\cite{SAP1}.
Suppose that the variation in Eq.~(\ref{2.11}) is one with respect
to the external source $J_i$. Since the dependence of the action
on the source is given in Eq.~(\ref{2.9}), we have
\begin{equation}
\frac{\delta\mbox{$\langle q_2,t_2|q_1,t_1\rangle\lbrack J\rbrack$}}{\delta J_i}=-\frac{i}{\hbar}\bra{2}
\sigma^i(q_\ast;q) \ket{1}\mbox{$\lbrack J\rbrack$} \;.\label{2.15}
\end{equation}
If we now perform a further variation of Eq.~(\ref{2.15}) with
respect to the source, we have
\begin{equation}
\delta\frac{\delta\mbox{$\langle q_2,t_2|q_1,t_1\rangle\lbrack J\rbrack$}}{\delta J_i}=-\frac{i}{\hbar}
\delta\bra{2} \sigma^i(q_\ast;q) \ket{1}\mbox{$\lbrack J\rbrack$} \;.\label{2.16}
\end{equation}
In order to evaluate the right hand side of this expression,
insert unity in the form $1 =\int dv'|q',t'\rangle\langle q',t'|$
where $t_1 < t' < t_2$, and $dv' = d^n q'g^{1/2}(q')$ is the
invariant volume element on $M$. If the time included in the index
$i$ of $\sigma^i(q_\ast;q)$ lies to the past of $t'$, then we will
change the source only to the future of $t'$ and the past of
$t_2$. Assuming causal boundary conditions, $\delta\bra{2}
\sigma^i(q_\ast;q) \ket{1}\mbox{$\lbrack J\rbrack$}$ cannot be affected by such a change
in the source. Thus,
\begin{eqnarray}
\delta\bra{2} \sigma^i \ket{1}\mbox{$\lbrack J\rbrack$} &=&\int dv'\delta\langle
q_2,t_2|q',t'\rangle\mbox{$\lbrack J\rbrack$} \langle q',t'|\sigma^i\ket{1}\mbox{$\lbrack J\rbrack$} \nonumber\\
&=&-\frac{i}{\hbar}\int dv'\delta J_j\bra{2}\sigma^j|q',t'\rangle\mbox{$\lbrack J\rbrack$}
\langle q',t'|\sigma^i\ket{1}\mbox{$\lbrack J\rbrack$} \nonumber\\
&=&-\frac{i}{\hbar}\delta J_j\bra{2}\sigma^j\sigma^i\ket{1}\mbox{$\lbrack J\rbrack$} \;,\label{2.17}
\end{eqnarray}
where we have dropped the argument $(q_\ast; q)$ on $\sigma^i$ and
$\sigma^j$ for brevity. Note that the time corresponding to the
condensed index $j$ lies to the future of that corresponding to
$i$.
Conversely, if $i$ lies to the future of $t'$, but to the past of
$t_2$, a similar argument shows that
\begin{equation}
\delta\bra{2} \sigma^i \ket{1}\mbox{$\lbrack J\rbrack$}= -\frac{i}{\hbar}\delta
J_j\bra{2}\sigma^i\sigma^j\ket{1}\mbox{$\lbrack J\rbrack$} \;.\label{2.18}
\end{equation}
Both situations in Eqs.~(\ref{2.17},\ref{2.18}) may be summarized
compactly by
\begin{equation}
\delta\bra{2} \sigma^i \ket{1}\mbox{$\lbrack J\rbrack$}= -\frac{i}{\hbar}
\delta J_j\bra{2}T(\sigma^i\sigma^j)\ket{1}\mbox{$\lbrack J\rbrack$} \;,\label{2.19}
\end{equation}
where $T$ is the chronological, or time ordering, symbol. It then
follows that
\begin{equation}
\frac{\delta^2\mbox{$\langle q_2,t_2|q_1,t_1\rangle\lbrack J\rbrack$}}{\delta J_{j}\delta J_{i}}=
\left(-\frac{i}{\hbar}\right)^2
\bra{2}T(\sigma^i\sigma^j)\ket{1}\mbox{$\lbrack J\rbrack$} \;.\label{2.20}
\end{equation}
It is easily established by induction that
\begin{equation}
\frac{\delta^n\mbox{$\langle q_2,t_2|q_1,t_1\rangle\lbrack J\rbrack$}}{\delta J_{i_1}\cdots\delta J_{i_n}}=
\left(-\frac{i}{\hbar}\right)^n
\bra{2}T(\sigma^{i_1}\cdots\sigma^{i_n})\ket{1}\mbox{$\lbrack J\rbrack$} \;.\label{2.21}
\end{equation}
Use of Eq.~(\ref{2.21}) in Eq.~(\ref{2.14}) leads to
\begin{eqnarray}
\mbox{$\langle q_2,t_2|q_1,t_1\rangle\lbrack J\rbrack$}&=&\sum_{n=0}^{\infty}\frac{1}{n!}
\left(-\frac{i}{\hbar}\right)^nJ_{i_1}\cdots J_{i_n}
\bra{2}T(\sigma^{i_1}\cdots\sigma^{i_n})\ket{1}
\lbrack J=0\rbrack\nonumber\\
&=&\bra{2}T\left\lbrace\exp\left(-\frac{i}{\hbar}J_i\sigma^i\right)
\right\rbrace \ket{1} \lbrack J=0\rbrack \;.\label{2.22}
\end{eqnarray}
(The exponential in the last line is understood to be defined in
terms of its Taylor series as in the preceding line.)
Define
\begin{equation}
E_i\left\lbrack q_\ast;\sigma^i(q_\ast;q)\right\rbrack=
\frac{\delta\tilde{S}}{\delta\sigma^i}\;,\label{2.23}
\end{equation}
so that the operator equation of motion Eq.~(\ref{2.13}) becomes
\begin{equation}
E_i\left\lbrack q_\ast;\sigma^i(q_\ast;q)\right
\rbrack=J_i\;.\label{2.24}
\end{equation}
We can view $E_i\left\lbrack q_\ast;\sigma^i(q_\ast;q)
\right\rbrack$ as defined in terms of the Taylor series obtained
by differentiating Eq.~(\ref{2.10}). Now consider
$\displaystyle{E_i\left\lbrack
q_\ast;-\frac{\hbar}{i}\frac{\delta}{\delta J_i}\right\rbrack}$
where $\sigma^i$ in the Taylor series for $E_i\left\lbrack
q_\ast;\sigma^i(q_\ast;q)\right\rbrack$ is replaced by
$-\frac{\hbar}{i}\frac{\delta}{\delta J_i}$. Using
Eq.~(\ref{2.22}), it is clear that
\begin{eqnarray}
E_i\left\lbrack q_\ast;-\frac{\hbar}{i}
\frac{\delta}{\delta J_i}\right\rbrack\mbox{$\langle q_2,t_2|q_1,t_1\rangle\lbrack J\rbrack$}&=&
\bra{2}T\left\lbrace E_i\left\lbrack q_\ast;
\sigma^i\right\rbrack \exp\left(-\frac{i}{\hbar}J_i\sigma^i\right
\rbrace \right) \ket{1} \lbrack J=0\rbrack\nonumber\\
&=&J_i\,\mbox{$\langle q_2,t_2|q_1,t_1\rangle\lbrack J\rbrack$}\;,\label{2.25}
\end{eqnarray}
noting Eqs.~(\ref{2.24},\ref{2.22}). This last result provides a
functional-differential equation for the amplitude $\mbox{$\langle q_2,t_2|q_1,t_1\rangle\lbrack J\rbrack$}$ which
has followed from the Schwinger action principle. Integration of
Eq.~(\ref{2.25}) will provide the link between the Schwinger
action principle and the Feynman path integral.
\section{\label{sec3}The Feynman path integral}
In order to solve Eq.~(\ref{2.25}), let
\begin{equation}
\mbox{$\langle q_2,t_2|q_1,t_1\rangle\lbrack J\rbrack$}=\int\left( \prod_i d\sigma^i( q_\ast;q)\right) F\left\lbrack
q_\ast;\sigma^i(q_\ast;q)\right\rbrack
\exp\left(-\frac{i}{\hbar}J_i\sigma^i\right) \;,\label{3.1}
\end{equation}
for some function $F$. The integration is assumed to extend over
all $\sigma^i( q_\ast;q)$ (or equivalently over all $q^i$, as will
be seen below) for which $\sigma^i( q_\ast;q) = \sigma^i(
q_\ast;q_1)$ at time $t = t_1$, and $\sigma^i( q_\ast;q) =
\sigma^i( q_\ast;q_2)$ at time $t = t_2$. If Eq.~(\ref{3.1}) is to
solve Eq.~(\ref{2.25}), we must have
\begin{eqnarray*}
0&=&\int\left(\prod_i d\sigma^i\right) \left\lbrace E_i\lbrack
q_\ast;\sigma^i(q_\ast;q) \rbrack -J_i\right\rbrace F
\left\lbrack q_\ast;\sigma^i(q_\ast;q)\right\rbrack
\exp\left(-\frac{i}{\hbar}J_i\sigma^i\right)\nonumber\\
&=&\int\left(\prod_i d\sigma^i\right) \left\lbrace E_i\lbrack
q_\ast;\sigma^i\rbrack F\left\lbrack q_\ast; \sigma^i\right\rbrack
+\frac{\hbar}{i}F\left\lbrack q_\ast;
\sigma^i\right\rbrack\frac{\delta }{\delta\sigma^i} \right\rbrace
\exp\left(-\frac{i}{\hbar}J_i\sigma^i\right) \;.
\end{eqnarray*}
If we integrate the second term in the last line by parts, then
\begin{eqnarray}
0&=&\int\left(\prod_i d\sigma^i\right) \left\lbrace E_i\lbrack
q_\ast;\sigma^i\rbrack F\left\lbrack q_\ast; \sigma^i\right\rbrack
-\frac{\hbar}{i}\frac{\delta F\left\lbrack q_\ast;
\sigma^i\right\rbrack }{\delta\sigma^i} \right\rbrace
\exp\left(-\frac{i}{\hbar}J_i\sigma^i\right)\nonumber\\
&&\left.+\frac{\hbar}{i}F\left\lbrack q_\ast;\sigma^i(q_\ast;q)
\right\rbrack \exp\left(-\frac{i}{\hbar}J_i\sigma^i\right)
\right|_{q_1}^{q_2}\;.\label{3.2}
\end{eqnarray}
Because $E_i=\delta\tilde{S}/\delta\sigma^i$, if we assume that
the surface term in Eq.~(\ref{3.2}) vanishes, then the solution to
Eq.~(\ref{3.2}) is
\begin{equation}
F\left\lbrack q_\ast;\sigma^i(q_\ast;q) \right\rbrack = f(q_\ast)
\exp\left(\frac{i}{\hbar}\tilde{S}(q_\ast;\sigma^i) \right)
\;,\label{3.3}
\end{equation}
for arbitrary $f(q_\ast)$. The condition for the surface term in
Eq.~(\ref{3.2}) to vanish is then that $S\lbrack q = q_1\rbrack =
S\lbrack q = q_2\rbrack$ (assuming that the source $J_i$ is only
non-zero for $t_1<t<t_2$.) This condition is often met in field
theory by assuming that $q^i$ is in the vacuum state at $t = t_1$
and at $t=t_2$.
We have therefore found that
\begin{equation}
\mbox{$\langle q_2,t_2|q_1,t_1\rangle\lbrack J\rbrack$}=f(q_\ast)\int\left(\prod_i d\sigma^i\right)\exp\left\lbrace
\frac{i}{\hbar}(\tilde{S}-J_i\sigma^i)\right\rbrace \;.\label{3.4}
\end{equation}
The integration in Eq.~(\ref{3.4}) may be changed to one over the
more conventional variable $q^i$ as follows. The usual rule for a
change of variable gives
\begin{equation}
\left(\prod_i d\sigma^i(q_\ast;q)\right) = \left|{\rm
det}\,\frac{\delta}{\delta q^j} \sigma^i(q_\ast;q)\right|
\left(\prod_i dq^i\right)\;.\label{3.5}
\end{equation}
Noting from Eq.~(\ref{2.6}) that
$\sigma^i(q_\ast;q)=g^{ik}(q_\ast)\delta\sigma(q_\ast;q)/\delta
q_\ast^k$, and that the Van Vleck-Morette
determinant~\cite{VanVleck,Morette} is defined by
\begin{equation}
\Delta(q_\ast;q)=|g(q_\ast)|^{-1/2} |g(q)|^{-1/2} {\rm det} \left(
-\frac{\delta^2 \sigma(q_\ast;q)}{\delta q^i\delta
q_\ast^j}\right)\;,\label{3.6}
\end{equation}
Eq.~(\ref{3.5}) becomes
\begin{equation}
\left(\prod_i d\sigma^i(q_\ast;q)\right) = \left(\prod_i
dq^i\right) |g(q)|^{1/2}|\Delta(q_\ast;q)| |g(q_\ast)|^{-1/2}
\;.\label{3.7}
\end{equation}
Here $g(q)$ denotes ${\rm det}\,g_{ij}(q)$, and the factors of
$g(q),g(q_\ast)$ have been chosen to make $\Delta(q_\ast; q)$ a
scalar in each argument.
With the change of variable described above, the expression for
the transition amplitude becomes
\begin{equation}
\mbox{$\langle q_2,t_2|q_1,t_1\rangle\lbrack J\rbrack$}= |g(q_\ast)|^{-1/2}f(q_\ast)\int \left(\prod_i dq^i\right)
|g(q)|^{1/2}|\Delta(q_\ast;q)| \exp\left\lbrace
\frac{i}{\hbar}(\tilde{S}-J_i\sigma^i)\right\rbrace \;.\label{3.8}
\end{equation}
The amplitude must be invariant under the change of coordinates
$q_\ast^i\rightarrow q_\ast^{\prime i}$. This is seen to constrain
$|g(q_\ast)|^{-1/2}f(q_\ast)$ to transform like a scalar. This
scalar is irrelevant since we typically only compare one amplitude
with another. In fact, if we require the expression to reduce to
that of Feynman when the space is flat, then
$|g(q_\ast)|^{-1/2}f(q_\ast)$ must be a constant. In any case,
because $|g(q_\ast)|^{-1/2}f(q_\ast)$ has no dependence on the
dynamical variables $q^i$, we may simply take
\begin{equation}
\mbox{$\langle q_2,t_2|q_1,t_1\rangle\lbrack J\rbrack$}= \int \left(\prod_i dq^i\right)
|g(q)|^{1/2}|\Delta(q_\ast;q)| \exp\left\lbrace
\frac{i}{\hbar}(\tilde{S}-J_i\sigma^i)\right\rbrace \;,\label{3.9}
\end{equation}
as the path integral representation for the amplitude. As promised
in the introduction, in addition to the natural volume element
$\displaystyle{\left(\prod_i dq^i\right) |g(q)|^{1/2}}$, the
additional factor of $\Delta(q_\ast;q)$ is seen to be present. As
we have already mentioned, this agrees with the conclusion of
Ref.~\cite{stochastic} and as shown by Parker~\cite{Parker}
results in a Schr\"{o}dinger equation with no explicit dependence
on the scalar curvature.
\section{\label{sec4}Discussion and conclusions}
In addition to the natural volume element in the path integral
measure, we have shown that there is an additional term which
involves the Van Vleck-Morette determinant. The origin of this
term can be traced to consistency between the Schwinger action
principle and the Feynman path integral. In the special case of a
flat space, $\Delta(q_\ast;q)=1$, so that this additional term
disappears even if curvilinear coordinates are used. As mentioned
in the introduction, the existence of the term
$|\Delta(q_\ast;q)|$ in the measure leads to the normal
Schr\"{o}dinger equation without any additional modifications due
to the curvature~\cite{Parker}.
There are of course other ways to derive the factor of
$\Delta(q_\ast;q)$ in the measure. One is the previously mentioned
method of DeWitt-Morette et al.~\cite{stochastic}. Another
approach, which is independent of the Schwinger action principle,
is to postulate that the amplitude satisfy the equation of motion
Eq.~(\ref{2.25}). This is what would be done following
Symanzik~\cite{Symanzik} for example. The steps leading up to the
end result of Eq.~(\ref{3.9}) are identical. We chose instead to
postulate the more general action principle of Schwinger, and to
derive Eq.~(\ref{2.25}) as one of its many consequences.
It is of interest to explore the consequences of the measure found
in this paper in the case of quantum field theory. A covariant
approach to quantum field theory has been advocated by
Vilkovisky~\cite{Vilkovisky}. It would be of interest to study the
implications for gauge theories, particularly in relation to the
geometrical analysis of the measure presented in
Refs.~\cite{EKT1,EKT2}.
|
{
"timestamp": "2004-11-25T11:51:57",
"yymm": "0411",
"arxiv_id": "hep-th/0411233",
"language": "en",
"url": "https://arxiv.org/abs/hep-th/0411233"
}
|
\section{Introduction}
The shell-model or independent particle model (IPM) has been very successful
in describing basic features of nuclear systems. This means that nuclei are
considered as a system of nucleons moving independently in a mean field and
the residual interaction between these particle or quasiparticles is supposed
to be weak. Therefore the response of the the system to an external
perturbation can be calculated within the Fermi Liquid theory \cite{pines} in
terms of linear response functions. These response functions are calculated
assuming a Hartree-Fock (HF) propagator for the particle-hole excitations of
the nucleons and including the residual interaction by means of the Random
Phase Approximation (RPA) approximation. In the long-wavelength limit or
external perturbations with low momentum transfer, the residual interaction
between the quasiparticles is usually parameterized in terms of Landau
parameters.
This HF plus RPA scheme is typically used to determine e.g. the neutrino
propagator in hot and dense nuclear
matter \cite{mf,reddyne,voskrn,yamada,raffelt,sedr} and it has been found
that the neutrino opacity is very sensitive to the details of these
response functions. This quantity is very crucial for the simulation of
astrophysical objects like the explosion of supernovae or the cooling mechanism
for neutron stars \cite{haensel,jerom1}.
The study of the response is also very important to determine the propagator of
mesons or a photon in the nuclear medium. Therefore such investigations have to
be performed to explore e.g. the possibility for a pion
condensation\cite{pico1,pico2} or the production and emission of mesons and
photons from the hot dense matter obtained in heavy ion reactions
\cite{Bozek:1997rv,Bozek:1998ro,eff,Cassing:2000ch,Bertsch:1996ig,Knoll:1996nz}. Last not
least the response function is also reflecting the excitation modes of nuclei.
However, the simple HF plus RPA scheme outlined above is applicable to nuclear
systems only if effective nucleon-nucleon (NN) interactions like
Skyrme\cite{skyrme} or Gogny\cite{gogny} forces are employed. The IPM fails
completely if realistic NN interactions are considered, which have been
adjusted to describe the NN scattering data. Trying to evaluate the energy of
nuclear matter from such realistic interaction within the HF approximation
yields positive energies, i.e. unbound nuclei\cite{polrep}. The reason of this
deficiency of the HF approximation in nuclear physics are the correlations
beyond the IPM approach, which are induced from the strong short-range and
tensor components of a realistic NN interaction.
These correlations have a significant effect on the single-particle propagator
for a nucleon in the nuclear medium. The spectral function still exhibits a
quasiparticle peak. A sizable fraction, however, of the strength occurs at
energies above and below the quasiparticle peak. For hole-states one typically
observes that around 15 \% of the spectral strength is shifted to energies above
the Fermi energy \cite{Bozek:2002em,Dewulf:2003nj,Frick:2003sd} which means that
the occupation probability of those states is reduced from 100 \% in the case of
the IPM approach to around 85 \%. Another fraction of the hole-strength is
shifted to energies below the quasiparticle energies, which means that it should
be found in nucleon knock-out experiments at large missing energies. These
effects of correlations on the spectral distribution are confirmed in
($e,e'p$) experiments (see e.g.\cite{rohe:2004}).
In lowest order this redistribution of the strength in the single-particle
spectral function is due to the admixture of two-hole one-particle and
two-particle one-hole contributions to the propagator in the HF field. Therefore
one may feel the temptation to use these correlated propagators and evaluate the
nuclear response function in terms of these dressed propagators. In this way one
is including two-particle two-hole admixtures to the particle-hole response
function. As we can expect from the discussion above and as we will see below,
such a procedure leads to response functions which, comparing to the HF plus RPA
response, exhibit a significant shift of the excitation strength to larger
energies.
We will also see, however, that such a significant shift of the excitation
strength, can in general not be consistent with the energy weighted sum-rules,
which are observed in the HF plus RPA scheme. It is well known that it is
rather difficult to develop a symmetry conserving approach for the evaluation
of Green's functions which accounts for correlations beyond the HF plus RPA
approximation. In the case of the response function this requires a consistent
treatment of propagator and vertex corrections. In this manuscript we will
follow the general recipe of Baym and Kadanoff\cite{kadBaym,blarip} for
calculating the in-medium coupling of an external perturbation to dressed
nucleons in a self-consistent way.
This procedure leads to an integral equation, a Bethe-Salpeter equation for the
dressed vertex. The in-medium vertex has the structure of a three-point Green's
functions. For dressed (off-shell) nucleons it is a function of two momenta and
two energies. Since such calculations are rather involved, only a few exist for
the density response function \cite{bonitz,faleev,pbpl}. In order to make these
calculation feasible, we define a simple interaction model. The nucleons are
dressed by a mean-field and a residual interaction. The residual interaction
is taken selfconsistently to the second order. The parameters of the
interaction are adjusted to reproduce the main features of single-particle
spectral functions derived from realistic NN interactions.
After this introduction we present in section 2 our interaction model and
the adjustment of its parameters to reproduce the nucleon self-energy derived
from a realistic interaction. The evaluation of the response functions with a
consistent treatment of propagator and vertex corrections is outlined in section
3. Numerical results for symmetric nuclear matter and pure neutron matter are
presented in section 4. There we also discuss the effect of multi-pair
contributions to the response functions at high excitation energies and its
relation to the spin-isospin structure of the residual interaction and the
consequences for the damping of collective modes. The final section summarizes
the main conclusions of this study.
\section{Mean-field and residual interaction}
\label{residualmf}
Calculations of the response function in nuclear matter are usually
restricted to the HF plus RPA approximation, employing parameterizations
of the effective NN interaction.
There exist many relatively simple and successful parameterization of
the mean-field Hamiltonian for nuclear systems, e.g. The Skyrme
interaction \cite{skyrme} and the Gogny interaction \cite{gogny}.
Usually such effective interactions include spin and isospin dependent
terms, and also density dependent terms.
The Skyrme interaction is a zero range interaction with velocity
dependent terms, for which a complete calculation of the RPA response is
possible
\cite{GNVS}. In general, however, the calculation of the RPA response requires
the consideration of a nontrivial sum of exchange terms\cite{pico1,depace},
which are often approximated.
Usually the response function
for finite range interactions is calculated expanding the interaction
in Landau parameters \cite{jerom2}.
In this study we want to analyze the linear response functions in a
fermionic systems when the correlations of the system
that we take into account go beyond the mean-field approximation.
We suppose that the
interactions between the nucleons are given by a mean-field
potential and a residual interaction. The mean-field interaction that we
take is based on the Gogny parameterization
\begin{equation}
\label{Vgogny}
V_{mf}(1,2)=\sum_{i}\left(W_i+B_iP^\sigma-H_iP\tau-M_i P^\sigma P^\tau\right)
e^{-(\bf{r}_1-\bf{r}_2)^2/\mu_i^2}
+ \sum_{j}t^j_3(1+x_3^j
P^\sigma)\rho^{\sigma_j}\delta^3(\bf{r}_1-\bf{r}_2) \ .
\end{equation}
The first term is a sum of two Gaussians giving a
finite range interaction and the second term is a sum of two zero-range
density dependent interactions
\cite{gognyd1p}, $P^\sigma=\frac{1}{2}(1+\sigma_1
\sigma_2)$ and $P^\tau=\frac{1}{2}(1+\tau_1 \tau_2)$.
The residual interaction is taken in a very simple form \cite{dan}
\begin{equation}
\label{vres}
V_{res}({\bf r_1}-{\bf r_2})=V_0 e^{-({\bf r_1}-{\bf r_2})^2/2\eta^2}\ ,
\end{equation}
with the parameters
$V_0=453$MeV, $\eta=0.57$fm. The single particle propagator is
calculated by taking the mean field contributions only for the
Gogny interaction (\ref{Vgogny}) and the second order direct Born term for the
residual interaction (\ref{vres}). The relevant diagrams are shown in
Fig.
\ref{selffig}.
\begin{figure}
\centering
\includegraphics*[width=0.7\textwidth]{f1.eps}
\parbox{14cm}{\it
\caption{ Diagrams for the self-energy. The first two diagrams
are the Hartree-Fock contribution for the Gogny interaction (the dashed
line). The last diagram is the contribution of the residual interaction in
the second order.}
\label{selffig}}
\end{figure}
\begin{figure}
\centering
\includegraphics*[width=0.5\textwidth]{szer.eps}
\parbox{14cm}{\it\caption{Imaginary part of the self-energy at the
quasi-particle pole from the residual interaction (solid line) and from a
self-consistent $T$-matrix calculation \cite{bcapp} (dashed line). The curves
shown correspond to symmetric nuclear matter at normal nuclear density and
temperature $T=15$MeV.}
\label{szer}}
\end{figure}
The residual interaction induces a finite width to the nucleon
excitations in the medium. Such a dressing of nucleons is expected in
any approach going beyond the simple mean-field. Calculation for the
system
including the mean-field and residual interactions are performed
in the real-time representation for the thermal Green's functions
\cite{keldysh}.
The iterated system of equations includes expressions for the
self-energies,
\begin{eqnarray}
\label{self}
\Sigma_{mf}(p)&=&V_{mf}(0)\rho - Tr \int \frac{d^3k}{(2\pi)^3}
P^\sigma P^\tau V_{mf}({\bf p}-{\bf k}) n(k) \nonumber \ , \\
\Sigma^{>(<)}({\bf p},\omega)& =& 4 i \int
\frac{d^3p_1 d\omega_1 d^3p_2 d\omega_2}
{(2 \pi)^8}
V^2_{res}({\bf p}-{\bf p_1})G^{>(<)}({\bf p_1},\omega_1)
G^{<(>)}({\bf p_2},\omega_2) \nonumber \\
& & G^{>(<)}({\bf p}-{\bf p_1}+{\bf p_2},\omega-\omega_1+\omega_2) \ ,
\end{eqnarray}
\begin{equation}
\label{dispself}
\Sigma^{r(a)}({\bf p},\omega)=\Sigma_{mf}(p)+
\int\frac{d\omega_1}{2\pi}\frac{\Sigma^<({\bf p},\omega_1)
-\Sigma^>({\bf p},\omega_1)}{\omega-\omega_1\pm i\epsilon} \ ,
\end{equation}
and the Dyson
equation for the retarded (advanced) Green's functions
\begin{equation}
\label{dyson}
G^{r(a)}({\bf p},\omega)=\frac{1}{\omega-{\bf p}^2/2m
-\Sigma^{r(a)}({\bf p},\omega)} \ .
\end{equation}
The Green's functions
\begin{eqnarray}
G^{>}({\bf p},\omega) &= & -i \left(1-f(\omega)\right)A({\bf
p},\omega) \ ,
\nonumber \\
G^{<}({\bf p},\omega) &= & i f(\omega)A({\bf p},\omega)
\end{eqnarray}
are written using the Fermi distribution $f(\omega)$ and
the spectral function
\begin{equation}
\label{spec}
A({\bf p},\omega)=-2 {\rm Im} G^r({\bf p},\omega) \ .
\end{equation}
The nucleon momentum distribution is
\begin{equation}
n(p)=\int \frac{d\omega}{2\pi} A(p,\omega)f(\omega) \
\end{equation}
and the chemical potential is adjusted at each iteration
to reproduce the assumed density
\begin{equation}
\rho=4 \int\frac{d^3p}{(2\pi)^3}n(p) \ .
\end{equation}
The self-consistent equations for the
one-body properties have been solved within similar approximations by
several groups
\cite{dan,oset,Bozek:1998ro,eff,Lehr:2000ua}.
The width obtained from a self-consistent calculation in the second
order of the residual interaction is similar to the result obtained
from a self-consistent $T$-matrix calculation using realistic bare
nucleon-nucleon interaction (Fig. \ref{szer}).
\begin{figure}
\centering
\includegraphics*[width=0.5\textwidth]{szelf.eps}
\parbox{14cm}{\it\caption{Real part of the self-energy at the quasi-particle
pole from the residual interaction (dashed-dotted line) and from the modified
mean-field interaction (dashed line). The sum of the two contributions is
shown as the solid line and compared the original Gogny potential (dotted
line).}
\label{realsf}}
\end{figure}
The real part of the self-energy $\mbox{Re }\Sigma^r(p,\omega)$ is the sum of
the mean-field (Gogny) contribution and a dispersive one obtained
from the dispersion relation in Eq. (\ref{dispself}). This means that
the real part of the self-energy at the quasiparticle pole
$\omega_p=p^2/2m+ \Sigma(p,\omega_p)$
is different from the Gogny single-particle potential.
Accordingly we have modified some parameters of the Gogny interaction
in order to have the same Fermi energy as function of density and a
similar effective mass.
In Fig. \ref{realsf} we show the real part of the self-energy at the
quasi-particle pole and compare it to the single-particle potential derived from
the original Gogny interaction.
This real part of the self-energy is the sum of the dispersive part and
the mean-field contribution originating from the modified Gogny interaction,
which are also shown.
The parameters of the mean-field interaction are given in Table
\ref{table}. We modify the parameters $M_i$ to reproduce the momentum
dependence of the self-energy and also the density dependent zero
range term, an additional term of the form $t_3^0 \delta^3({\bf
r}_1-{\bf r}_2)$ is added to the mean-field interaction.
\begin{table}
\begin{tabular}{|r|c|c|c|}
\hline
parameter & Gogny $D1P$ & Modif. I & Modif II \\
\hline
$\mu_1$ (fm) & 0.9 & - & - \\
$\mu_2$ (fm) & 1.44 & - & - \\
$W_1$ (MeV) & -372.9 & - & - \\
$W_2$ (MeV) & 34.6 & - & - \\
$B_1$ (MeV) & 62.7 & - & - \\
$B_2$ (MeV) & -14.1 & - & - \\
$H_1$ (MeV) & -464.5 & - & - \\
$H_2$ (MeV) & -70.9 & - & - \\
$M_1$ (MeV) & -31.5 & 38.5 & 38.5\\
$M_2$ (MeV) & -21 & -51 & -51 \\
$\sigma_1$ & .33 & - & -\\
$\sigma_2$ & .92 & - & -\\
$ t_3^1$ (MeV fm$ ^{3(\sigma_1+1)}$) & 1025.9 & 454.7 & -245.3 \\
$ t_3^2$ (MeV fm$ ^{3(\sigma_2+1)}$)& 1025.9 & - & - \\
$ t_3^1 x_3^1$ (MeV fm$ ^{3(\sigma_1+1)}$) & 1190 & - & -\\
$ t_3^2 x_3^2$ (MeV fm$ ^{3(\sigma_2+1)}$) & 256 & - & -\\
$ t_3^0$ (MeV fm$ ^{3}$) & 0 & 478.4 & 803.4 \\
\hline
\end{tabular}
\parbox{14cm}{\it\caption{Table of the parameters for the mean-field
interaction. The first column corresponds to the Gogny $D1P$ parameterization
\cite{gognyd1p}, the second and third columns are modifications of the
mean-field interaction used in symmetric nuclear and neutron matter,
respectively. A dash is put whenever the value of the corresponding parameter
is not changed.}\label{table}}
\end{table}
\section{Correlations and response functions}
\label{respsect}
In this section we discuss
the linear
response of a correlated system to an external perturbation. As it has already
been mentioned above the evaluation of the RPA response function requires for a
general interaction a solution of a Bethe-Salpeter (BS) equation with a
non-trivial kernel which is due to the exchange terms in the NN interaction.
Therefore one often simplifies the solution of the BS equation by a
parameterization of the particle-hole interaction in terms of
Landau parameters.
E.g. the density response function is written using the zero order Landau
parameter $f_0$ as
\begin{equation}
\Pi^r(p,\omega)=\frac{\Pi^r_0(p,\omega)}{1-f_0 \Pi_0^r(p,\omega)} \ ,
\end{equation}
where $\Pi_0^r(p,\omega)$ is the response function of the free Fermi
gas using an effective mass to describe the momentum dependence of the mean
field.
\begin{figure}
\centering
\includegraphics*[width=0.6\textwidth]{f2.eps}
\parbox{14cm}{\it\caption{The Bethe-Salpeter equation for the dressed vertex.
The particle-hole irreducible kernel $K$ is denoted by the box and the fat and
the small dots denote the dressed and the bare vertices for the coupling of
the external field to the nucleon. All the fermionic propagators are dressed by
the self-energy as displayed in Fig. \ref{selffig}.}
\label{bsfig}}
\end{figure}
When the description of the correlated systems goes beyond the
mean-field approximation the difficulty involved in a
consistent calculation of the response
function is severely increased. A naive calculation of the
polarization bubble using dressed propagators
\begin{equation}
\label{polsimp}
\Pi^{<(>)}({\bf q},\Omega)=- 4 i \int\frac{d^3p d\omega}{(2\pi)^4}
G^{<(>)}({\bf q}+{\bf p},\omega+\Omega)G^{>(<)}({\bf p},\omega) \
\end{equation}
and
\begin{equation}
\label{disppol}
\Pi^{r(a)}(p,\omega)=\int\frac{d\omega_1}{2\pi}\frac{\Pi^<({ p},\omega_1)
-\Pi^>({ p},\omega_1)}{\omega-\omega_1\pm i\epsilon} \ ,
\end{equation}
can be a rather poor estimate for the response function \cite{pbpl}. In
particular, it
severely violates the $\omega$-sum rule for finite momentum $q$.
This violation of the sum rule by the naive one-loop response function
was recently noticed by Tamm et al.~\cite{henn} in reply to an evaluation of a
one-loop response function in terms of dressed propagators for the electro gas
in metals and semiconductors\cite{schoene}.
For self-consistent approximation schemes a general recipe for
calculating the in-medium coupling of the external potential to dressed
nucleons is known \cite{kadBaym,blarip}.
The in-medium vertex describing the coupling of the external perturbation to
the nucleons nucleons is given by the solution of the BS equation
(Fig. \ref{bsfig}), where $K$ denotes the particle-hole irreducible
kernel. The kernel $K$ of the BS equation
should be taken consistently with the chosen expression for the
self-energy. It is given by the functional derivative of the
self-energy with respect to the dressed Green's function
\cite{kadBaym,blarip} $K={\delta \Sigma}/{\delta G}$.
The resulting kernel of the BS equation contains the usual mean-field
interaction (direct and exchange terms, see first and second term in the
representation of $K$ displayed in Fig. \ref{kernelfig}) and additional diagrams
which are due to the contributions of the residual interaction terms in the
self-energy and are collected as $K_{res}$ in Fig. \ref{kernelfig}.
\begin{figure}
\centering
\includegraphics*[width=0.8\textwidth]{f3.eps}
\parbox{14cm}{\it\caption{The kernel of the Bethe-Salpeter equation
corresponding to the self-energy in Fig. \ref{selffig} containing a mean field
and a residual interaction. }
\label{kernelfig}}
\end{figure}
Using the dressed vertex obtained as a solution of the BS equation the
response function in the correlated medium can be obtained from the
diagram in Fig. \ref{polver}. Only one vertex in the loop includes in-medium
modifications in order to avoid double counting.
\begin{figure}
\centering
\includegraphics*[width=0.5\textwidth]{f4.eps}
\parbox{14cm}{\it\caption{The polarization function expressed using the dressed
vertex for the coupling of the external current to the dressed nucleon.}
\label{polver}}
\end{figure}
\begin{figure}
\centering
\includegraphics*[width=0.7\textwidth]{f5.eps}
\parbox{14cm}{\it\caption{The Bethe-Salpeter equation with contributions only
from the residual interaction (Eqs. \ref{glg}, \ref{gpm}, and \ref{polext2})}
\label{bsfig2}}
\end{figure}
The three-point Green's function for the coupling of the nucleon to
an external field with
momentum ${\bf q}$ and energy $\Omega$ is
\begin{equation}
G_{(ST)}(x_1,t_1;x_2,t_2;{\bf q},\Omega)=-\int d^3x\, dt\, \exp(-i{\bf q x
}+i\Omega t) \langle{\cal T} \Psi(x_1,t_1)\rho^{(ST)}(x,t)
\Psi^\dagger(x_2,t_2) \rangle
\, ,
\end{equation}
where $\Psi^\dagger, \Psi$ are the field creation and annihilation
operators, $\rho^{(ST)}(x,t)=\Psi^\dagger(x,t)\Gamma_{(ST)}^0\Psi(x,t)$ denotes
the bare coupling to the external field with the spin-isospin operators
$\Gamma_{(ST)}^0=1, \ \sigma_3, \ \tau_3, \ \tau_3\sigma_3$ for the response
functions denoted by spin $S$ and isospin $T$ equal to $ST=00, \ 10, \
01, \ 11$, respectively. The operator ${\cal T}$ in this equation is the usual
operator for the time
ordering on the real time contour \cite{keldysh}.
This three-point Green's function for
the dressed coupling of the external field to the fermions in medium has a
complicated analytical structure and depends on the incoming
momentum ${\bf q}$ and energy $\Omega$ \cite{evans}.
Depending on the ordering of the times of the fermion operators one
can define the smaller (larger) Green's functions
$G_{(ST)}^{<(>)}(x_1,t_1;x_2,t_2;{\bf q}, \Omega)$ and also the retarded or
advanced ones.
In the momentum representation the three-point Green's function depends on
the momentum ${\bf p}$ and energy $\omega$ of the
incoming fermion and the momentum is ${\bf q}+{\bf p}$ and
energy $\omega+\Omega$ of the outgoing fermion.
We write the smaller (larger) Green's functions
\begin{equation}
G_{{(ST)}}^{<(>)}({\bf q}+{\bf p},\omega+\Omega;{\bf p},\omega)
\end{equation}
and denote the retarded (advanced) Green's functions by
\begin{equation}
G_{{(ST)}}^{r(a)}({\bf q}+{\bf p},\omega+\Omega;{\bf p},\omega) \ .
\end{equation}
The response function can be expressed using this three-point Green's
function (Fig. \ref{polver})
\begin{equation}
\Pi_{{(ST)}}^{r}({\bf q},\Omega)=-i Tr \int \frac{d^3p d\omega}{(2\pi)^4}
\Gamma_{(ST)}^0 G_{(ST)}^{<}({\bf p}+{\bf q},\omega+\Omega;{\bf p},\omega) \ .
\label{eq:respons}
\end{equation}
The three-point Green's functions $G_{{(ST)}}$ can be written in terms of
the in-medium (dressed) vertex $\Gamma_{(ST)}$
describing the in-medium coupling to the external perturbation
\begin{equation}
\label{gdlg}
G_{{(ST)}}^{r(a)}({\bf q}+{\bf p},\omega+\Omega;{\bf p},\omega)=
G^{r(a)}({\bf q}+{\bf p},\omega+\Omega)\Gamma^{r(a)}_{(ST)}({\bf q}+{\bf
p},\omega+\Omega;{\bf p},\omega)
G^{r(a)}({\bf p},\omega)
\end{equation}
and
\begin{eqnarray}
\label{gdpm}
\lefteqn{G_{{(ST)}}^{<(>)}({\bf q}+{\bf p},\omega+\Omega;{\bf p},\omega)
=}\hspace{3cm}
\nonumber\\
&&G^{r}({\bf q}+{\bf p},\omega+\Omega)\Gamma^{r}_{(ST)}({\bf q}+{\bf
p},\omega+\Omega;{\bf p},\omega)
G^{<(>)}({\bf p},\omega) \nonumber \\
&&+ G^{r}({\bf q}+{\bf p},\omega+\Omega)\Gamma^{<(>)}_{(ST)}({\bf q}+{\bf
p},\omega+\Omega;{\bf p},\omega)
G^{a}({\bf p},\omega) \nonumber \\
&&+ G^{<(>)}({\bf q}+{\bf p},\omega+\Omega)\Gamma^{a}_{(ST)}({\bf q}+{\bf
p},\omega+\Omega;{\bf p},\omega)
G^{a}({\bf p},\omega) \ .
\end{eqnarray}
The dressed vertex $\Gamma_{(ST)}$ for the coupling of
the external field to the
nucleon is the solution of the Bethe-Salpeter equation displayed in
Fig. \ref{bsfig}.
As mentioned before the
kernel of this BS equation has contributions from the mean-field
and from the residual interaction. In this work we are interested in
the role of the correlations going beyond the mean-field, which are described by
means of the residual interaction discussed above. Therefore we will also
consider the effects which are due to the mean field and the
residual interactions in the kernel of the Bethe-Salpeter in separate
steps. In a first step we will concentrate on the effects of the residual
interaction and determine a response function $\Pi_{{(ST)}
\ res}$ which includes vertex correction from the residual interaction
($K_{res}$ in Fig. \ref{kernelfig}), only. The way to evaluate $\Pi_{{(ST)}
\ res}$ will be discussed below.
\begin{figure}
\centering
\includegraphics*[width=0.5\textwidth]{polirr.eps}
\parbox{14cm}{\it\caption{The imaginary part of the response function as
function of the energy for the momentum $q=220$ MeV, at normal nuclear density
and temperature $15$ MeV, without including the RPA modifications of the
response. The dashed-dotted line is the response function for the free Fermi
gas with the same effective mass for the nucleons as in the correlated matter.
The solid line denotes the response function including the dressing of the
propagators and the vertex corrections (the density response). The dashed line
is the naive one-loop response function with dressed propagators, but without
vertex corrections. The dotted line is the response including the dressing of
the propagators and the vertex corrections from the third diagram on the right
hand side in Fig. \ref{kernelfig} (spin-isospin channels $01$, $10$, and
$11$).}
\label{polirr}}
\end{figure}
The modifications of the response function which are due to the mean-field
interaction can be taken into account by a solution of a separate
BS equation with a kernel including only this mean-field interaction
and the zero order vertex being the vertex dressed by the residual
interaction. In the following we approximate the solution of this second
BS equation with the mean-field kernel using Landau parameters for the
mean-field interaction. E.g. for the density response function we have
\begin{equation}
\label{LandauRPA}
\Pi^r_{(00)}(p,\omega)
=\frac{\Pi^r_{{(00)}\ res}(p,\omega)}{1-f_0\Pi^r_{{(00)} \ res}(p,\omega)}
\end{equation}
and analogously for other channels.
The Landau parameters are calculated using the modified Gogny
mean-field with parameters from Table \ref{table}. There is
little change of these parameters when comparing with the Landau
parameters obtained from the original Gogny interaction.
Only in the scalar channel the interaction gets less attractive.
So we now turn to the effects of the residual interaction in the response
function. In the real-time representation the Bethe-Salpeter equations
including only the residual interaction in the kernel take
the form
\begin{eqnarray}
\label{glg}
\lefteqn{\Gamma^{<(>)}_{(ST)}({\bf p}+{\bf q},\omega+\Omega;{\bf p},\omega)=}
\hspace{1cm}\nonumber \\
&&i Tr \int \frac{d^3 p_1 d\omega_1 }{(2\pi)^4} \left( V^2({\bf p}-{\bf p_1})
\Pi^{<(>)}({\bf p}-{\bf p_1},\omega-\omega_1) \right.
G_{(ST)}^{<(>)}
({\bf p_1}+{\bf q},\omega_1+\Omega;{\bf p_1},\omega_1) \nonumber \\
&&\left. + V({\bf p_1})V({\bf p_1}+{\bf q})
\widetilde{\Pi}_{(ST)}^{<(>)}({\bf p_1}+{\bf q},\omega_1+\Omega;{\bf p_1},\omega_1)
G^{<(>)}({\bf p}-{\bf p_1},\omega-\omega_1) \right)
\end{eqnarray}
and
\begin{eqnarray}
\label{gpm}
\lefteqn{\Gamma^{r(a)}_{{(ST)}}({\bf p}+{\bf q},\omega+\Omega;{\bf p},\omega)=
1+ i Tr \int \frac{d^3 p_1 d\omega_1 }{(2\pi)^4}\Bigl(}
\hspace{1cm}\nonumber \\
&& V^2({\bf p}-{\bf p_1})
\Pi^{<}({\bf p}-{\bf p_1},\omega-\omega_1) G_{(ST)}^{r(a)}
({\bf p_1}+{\bf q},\omega_1+\Omega;{\bf p_1},\omega_1) \nonumber \\
&&+V({\bf p_1})V({\bf p_1}+{\bf q})
\widetilde{\Pi}_{(ST)}^{<}({\bf p_1}+{\bf q},\omega_1+\Omega;{\bf p_1},\omega_1)
G^{r(a)}({\bf p}-{\bf p_1},\omega-\omega_1) \nonumber \\
&&+ V^2({\bf p}-{\bf p_1}) \Pi^{r(a)}({\bf p}-{\bf p_1},\omega-\omega_1)
G_{(ST)}^{>}
({\bf p_1}+{\bf q},\omega_1+\Omega;{\bf p_1},\omega_1) \nonumber \\
&& +V({\bf p_1})V({\bf p_1}+{\bf q})
\widetilde{\Pi}_{(ST)}^{r(a)}({\bf p_1}+{\bf q},\omega_1+\Omega;{\bf p_1},\omega_1)
G^{>}({\bf p}-{\bf p_1},\omega-\omega_1) \Bigr) \ .
\end{eqnarray}
The terms in these equations containing the one-loop polarization function
$\Pi^{<(>)}$ or $\Pi^{r(a)}$ of Eq.(\ref{polsimp}) and Eq.(\ref{disppol}),
respectively correspond to the second diagram on the right-hand side of the
first line in Fig. \ref{bsfig2}. This is the lowest order contribution of the
so-called induced interaction \cite{pico2,babu,pico3,friman} to the response
function and therefore we will refer to it as the induced interaction term in
the discussion below. The other terms refer to vertex corrections, which
represented by the third diagram on the right-hand side of the
first line in Fig. \ref{polext2}. They contain a three-point function which is
displayed in the second line of Fig. \ref{bsfig2} and is defined by
\begin{eqnarray}
\label{polext1}
\lefteqn{\widetilde{\Pi}_{(ST)}^{<(>)}({\bf q}+{\bf
p},\omega+\Omega;{\bf p},\omega)=- i Tr
\int\frac{d^3p_1 d\omega_1}{(2\pi)^4}
\Bigl(}\hspace{1cm}\nonumber \\ &&G_{(ST)}^{<(>)}({\bf q}+{\bf
p}+{\bf p_1},\omega+\Omega+\omega_1;{\bf p}+{\bf
p_1},\omega+\omega_1)G^{>(<)}({\bf p_1},\omega_1)\nonumber \\
&&
+ G^{<(>)}({\bf p}+{\bf p_1},\omega+\omega_1)G_{(ST)}^{>(<)}
({\bf p_1},\omega_1;
{\bf p_1}-{\bf q},\omega_1-\Omega)
\Bigr)
\end{eqnarray}
and
\begin{eqnarray}
\label{polext2}
\lefteqn{\widetilde{\Pi}_{(ST)}^{r(a)}({\bf q}+{\bf
p},\omega+\Omega;p,\omega)=- i Tr \int\frac{d^3p_1 d\omega_1}{(2\pi)^4}
\Bigl(}\hspace{1cm}\nonumber \\ && G_{(ST)}^{r(a)}({\bf q}+{\bf
p}+{\bf p_1},\omega+\Omega+\omega_1;{\bf p}+{\bf
p_1},\omega+\omega_1) G^{<(>)}({\bf p_1},\omega_1) \nonumber \\
&&+ G^{r(a)}({\bf p}+{\bf p_1},\omega+\omega_1)
G_{(ST)}^{<(>)}({\bf p_1},\omega_1;
{\bf p_1}-{\bf q},\omega_1-\Omega)
\nonumber \\
&&+ G_{(ST)}^{<(>)}({\bf q}+{\bf
p}+{\bf p_1},\omega+\Omega+\omega_1;{\bf p}+{\bf
p_1},\omega+\omega_1)G^{r(a)}({\bf p_1},\omega_1)
\nonumber \\ &&
+ G^{<(>)}({\bf p}+{\bf p_1},\omega+\omega_1)
G_{(ST)}^{r(a)}({\bf p_1},\omega_1;
{\bf p_1}-{\bf q},\omega_1-\Omega)
\Bigr) \ .
\end{eqnarray}
For a scalar residual interaction the contributions of $\widetilde{\Pi}_{(ST)}$
are nonzero only for the density response, i.e. $S=T=0$.
\section{Results and discussion}
The numerical solution of the equations for the in-medium vertex is
exorbitantly difficult \cite{bonitz,sedr,faleev,pbpl}. This is mainly due to the
complex structure of the spectral functions.
Therefore in the following we present results only at finite
temperature and for a relatively large value of the momentum transfer $q$.
In this case the spectral functions are relatively
smooth and therefore easier to handle in numerical calculations.
Eqs.
(\ref{gdlg}), (\ref{gdpm}), (\ref{glg})-(\ref{polext2})
are solved by iteration
for each given $q$ and $\Omega$,
using
the Green's functions $G$ dressed by the self-energy (\ref{self}).
Using Eq.(\ref{eq:respons}) we can then calculate the response function
$\Pi^r_{{(ST)}\ res}$ which accounts for the effects of the
residual interaction.
In Fig. \ref{polirr} we show the results for this polarization function
with vertex corrections $\Pi_{res}$ for $q=220$MeV. The results for the
imaginary part of the response function originating from the naive one-loop
polarization (\ref{polsimp}) calculated with dressed propagators are represented
by the dashed line. As compared to the Hartree-Fock response function
(dashed-dotted line) this one-loop calculation with dressed propagators yields a
significant tail at large excitation energies. As the dressed propagators include
effects of two-particle one-hole and two-hole one-particle contributions to the
propagation of particle and holes, the dressed response functions
accounts for admixtures of two-particle two-hole contributions to the response
functions. Therefore one may interprete this high-energy tail to describe a
shift of the excitation strength to higher energies due to the admixture of
these two-particle two-hole contributions.
If, however, we also account for the vertex corrections which are due to the
residual interaction, we obtain the response functions represented by the solid
line in the case of the density response and the dotted line in the case of the
response for the other spin-isospin channels. One can see that in all cases the
high-energy tail obtained in the simple one-loop result is compensated by the
vertex corrections. This means that the induced interaction term, which for our
scalar residual interaction is present in all spin-isospin channels, is
responsible for this cancellation at high energies.
The difference between the density response $\Pi_{(00)\ res}$
and the response in other spin-isospin channels is due to the sub-leading vertex
corrections represented by the second and third graph in Fig.~\ref{kernelfig}
for $K_{res}$. These vertex corrections, which are specific for the density
response, lead to an enhancement of the imaginary part of $\Pi_{(00)\ res}$ at
small energies, which makes the final result look rather similar to the free
response function without any corrections of propagator and vertex due to the
residual interaction.
\begin{figure}
\centering
\includegraphics*[width=0.7\textwidth]{polall.eps}
\parbox{14cm}{\it\caption{The imaginary part of the response functions in
different channels as function of energy for the momentum $q=220$ MeV, at
normal nuclear density and temperature $15$ MeV. The dashed-dotted line is the
response function for the Gogny interaction in the Landau parameter
approximation.The solid line denotes the response function including the
dressing of the propagators and the vertex corrections. The dashed line is the
naive one-loop response function with dressed propagators. The response
function for the system with residual interaction include the modification of
the response due to the modified Gogny mean-field, taken in Landau parameter
approximation.}
\label{polfig}}
\end{figure}
The free response and the consistently calculated response functions
in the correlated system should fulfill several sum rules.
For the scalar residual interaction the $\omega$-sum rule takes the
simple form
\begin{equation}
\label{sumreq}
-\int_{-\infty}^{\infty} \frac{\omega d \omega}{2\pi}{\rm Im}\Pi^r_{(ST)}
({\bf q},\omega)= \rho \frac{{\bf q}^2}{2m} \ ,
\end{equation}
in all the spin isospin channels ${(ST)}$.
The self-energy takes into account also the mean-field
interaction, so the sum rule is only approximate and since the response
$\Pi_{res}$ does not include the RPA corrections on the right hand
side of (\ref{sumreq})
we substitute the free mass with the effective mass. Such a modified
sum
rule is fulfilled to within a few percent by the response functions
including vertex corrections $\Pi_{(ST) \ res}$,
it is severely violated by the naive
one-loop response function (\ref{polsimp}).
The mean-field interaction contains spin
and isospin dependent terms, and the sum rule including the mean-field
part of the interaction could include a possible RPA enhancement factor
\cite{liparini}
besides the free Fermi gas sum rule (\ref{sumreq}). When restricting
the RPA response to the Landau parameter form (\ref{LandauRPA}) the
$\omega$-sum rule has the same form as in the Free Fermi gas
(\ref{sumreq})
but with the corresponding effective mass instead of the free nucleon mass.
We find that
the vertex corrections due to the first diagram
for the kernel $K_{res}$ (Fig. \ref{bsfig2}), the so-called induced
interaction terms, are the most important ones
to bring the response function close to the one for the Free Fermi
case and restore the $\omega$-sum rule.
Adding the Hartree-Fock terms in the self-energy modifies the kernel
of the equation for the dressed vertex.
As explained above we take the RPA sum
into account by means of the Landau parameter for
the mean-field part of the interaction.
The resulting response functions in different channels are shown in
Fig. \ref{polfig}. We plot also the response function for a Fermi
liquid, where the Landau parameters and the effective mass
are given by the original Gogny interaction.
For the density response the result is very close
to the response of a Fermi liquid.
In all the channels the naive one-loop polarization
with dressed propagators gives a incorrect description,
with long tail at large energies. In fact the Lindhard function, i.e.
the one-loop polarization with HF propagators, gives a much better
description of the response function \cite{pines}, similar to the one including
full dressed propagators and vertices.
In the spin isospin response some
difference to the response of a Fermi liquid is observed, which could
already be seen in Fig \ref{polirr}. However the constraint of the
$\omega$-sum rule makes the response in the correlated systems
to lie close to the Fermi liquid one also in the nonzero spin and/or
isospin channels; the overall shape of $\mbox{Im }\Pi_{(ST)}$ is similar to the RPA
response function.
\subsection{Neutron matter}
The description of weak processes in dense nuclear matter is very
important for modeling supernovae explosions and the cooling
of neutron stars.
In a hot and dense medium neutrinos have a short mean free path and
they are effectively trapped inside the proto-neutron star.
The calculation of the mean free path involves nuclear correlation
effects. The relevant hadronic part of the cross section can be
factorized in the form of the density and spin response in matter.
In this section we present a calculation the response functions in
pure neutron matter.
As for the symmetric nuclear matter, the mean-field interaction has to be
modified in order to take into account additional contributions from the
residual interaction. In this first exploratory work we opt for a
parameterization which is different in pure neutron matter and in symmetric
nuclear matter. The modifications of the mean field interaction are listed in
the third colum of parameters in Table \ref{table}. In this way we can
reproduce the same Fermi energy and similar effective mass as given by the
original Gogny interaction for a range of densities between $0.4\, \rho_0$ and
$\rho_0$. At the same time the Landau parameters are not modified drastically
from their value corresponding to the original Gogny parameters displayed in
the first column of Table \ref{table}.
\begin{figure}
\centering
\includegraphics*[width=0.5\textwidth]{polallne.eps}
\parbox{14cm}{\it\caption{The imaginary part of the response functions in
different channels as function of energy for the momentum $q=220$MeV, in
neutron matter at normal density. Symbols are the same as in Fig.
\ref{polfig}.}
\label{polne}}
\end{figure}
The formulas for the density and spin response in neutron matter can
be written in the same way as outlined in section \ref{respsect}.
We find that
for the density response the whole kernel $K_{res}$ displayed in
Fig.~\ref{bsfig2} must be considered in the
BS equation, while for the spin response only the first graph in the
kernel $K_{res}$, the induced interaction term, is nonzero
($\widetilde{\Pi}_{{(S)}}=0$ for $S=1$).
The results are very similar to what we found for the symmetric
nuclear matter. When
both propagator and vertex modification in the medium are taken into
account the response function in the correlated system is very similar
to the one obtained in the Fermi liquid theory
(Fig \ref{polne}). It is not surprising,
since the $\omega$-sum rule has the same form as in the
noninteracting system.
The naive one-loop response function with dressed propagators
cannot be trusted and violates the sum rule.
\subsection{Multi-pair contributions to the response function}
Both for the symmetric and pure neutron matter we find that the
response function in a correlated system is very close to
response function in free Fermi gas, or when the mean field is taken
into account the response function is similar as in the Fermi liquid
theory. This means that the cancellation of propagator dressing and vertex
correction effectively drives the response of the system to the
response given by the excitation of a single particle-hole pair.
However, this result is not general. This cancellation is
due to the particular form of
the residual interaction, which we have considered to be scalar in spin and
isospin. This also leads to
the simple form (\ref{sumreq}) of the $\omega$-sum rule in all the channels.
\begin{figure}
\centering
\includegraphics*[width=0.7\textwidth]{poliso.eps}
\parbox{14cm}{\it\caption{The imaginary part of the response functions in
different channels as function of energy and $q=220$MeV for the isospin
dependent residual interaction (\ref{newres}). Symbols are the same as in Fig.
\ref{polfig}. The $ST=11$ response function with vertex and propagator
dressing is the same as the naive one-loop result with only propagator
dressing. }
\label{poliso}}
\end{figure}
Therefore, for the discussion in this section we modify the residual
interaction and assume it to be isospin dependent in the form
\begin{equation}
\label{newres}
V_{res}({\bf r_1}-{\bf r_2})P_\tau
=V_0 e^{-({\bf r_1}-{\bf r_2})^2/2\eta^2}P_\tau
\ .
\end{equation}
In symmetric nuclear matter the single-particle self-energy
is the same as obtained for the scalar residual interaction.
Therefore the same modified Gogny parameterization of the mean field
interaction is used as
in section \ref{residualmf}.
The kernel $K_{res}$ of the BS equation, however, is different than in the case
of a scalar residual interaction.
For the $ST=00$ channel all the three graphs for $K_{res}$ in
Fig. \ref{kernelfig}
contribute. However, for the response function in channels with isospin
$T=1$ the first diagram in $K_{res}$, the induced interaction term does not
contribute. We have found previously that this induced interaction diagram in
the vertex dressing is crucial for the suppression of the high-energy tail in
the response function. This led to the restoration of the
$\omega$-sum rule and made the response function
similar to the one in the free Fermi gas.
So if this induced interaction contribution to the
vertex dressing is absent (channels $ST=01, \ 11$) we expect a
strong modification of the response function by the residual
interaction.
The vertex and propagator dressing do no longer cancel. In fact,
in the channel $ST=11$ there are no
vertex corrections at all ($K_{res}=0$), and the response function has
the same form as a naive one-loop calculation with dressed propagators.
In Fig. \ref{poliso} the response functions obtained with the isospin
dependent interaction (\ref{newres}) are compared to
the response functions from the Fermi liquid theory.
For $T=0$ channels the correlated response function is similar to the
one particle-one hole response function. On the other hand,
the isovector response is closer to the naive
one-loop result.
For the $T=1$ channels the $\omega$-sum rule is different than in the
free Fermi gas.
The residual interaction in the Hamiltonian
gives a modification factor in the $\omega$- sum rule at finite
momentum $q$
\begin{eqnarray}
\label{sumr2}
\lefteqn{-\int_{-\infty}^{\infty} \frac{\omega d \omega}{2\pi}{\rm Im}\Pi^r_{(ST)\ res}({\bf q},\omega)=
\rho \frac{{\bf q}^2}{2m}}\hspace{2 cm} \nonumber \\
&&+ \frac{2}{3}\int d{\bf r}_1 d{\bf r}_2
V_{res}({\bf r_1}-{\bf r_2}) \tau_1\tau_2
\Psi^\dagger(r_1)\Psi^\dagger(r_2)\Psi(r_2)\Psi(r_1)\left(
e^{i{\bf q}({\bf r}_1-{\bf r}_2)}-1\right) \ .
\end{eqnarray}
This enhancement of the sum rule in the $T=1$ response is consistent
with observed long tail in the response function $\mbox{Im }\Pi$ at large
energies. A nonzero value of the imaginary part of the
response function at large energies is not kinematically
allowed by one particle-one hole configuration with on shell
propagation. Nonzero contribution do appear due to the dressing of the
single-particle propagator by the self-energy from the residual
interaction. Such a dressed propagation involves nucleons which are put
off shell by the scattering on other particles in the medium.
For the isospin dependent interaction and $T=1$ response
these off-shell propagation
effects are not canceled by vertex corrections. In the case of the
residual interaction of the form (\ref{newres}) off-shell nucleons
couple in the same way as free nucleons to isovector potentials.
For a general residual interaction containing scalar, spin, and isospin
dependent terms, we expect that the spin and isospin responses in a
correlated system lie in between the naive one-loop result and the
Fermi liquid theory result.
\subsection{Collective modes}
The response function may show a pronounced peak at a certain excitation
energy. This is a collective mode, which corresponds to the excitation of a
single collective state in the interacting system. Depending on the
spin-isospin character of the response these are the zero sound mode, spin or
isospin waves. In nuclear physics the isovector
response is of particular importance \cite{braghin}.
In finite nuclei it shows up as the giant dipole resonance, which has
extensively been studied.
Within the Fermi Liquid theory a collective excitation at zero temperature is a
discrete peak in the imaginary part of the response function. The state
corresponding to the collective excitation cannot couple to the incoherent one
particle-one hole excitations. At finite temperature such a coupling is
possible, it can be calculated and the width of the collective state at finite
temperature is usually small. The collective state can acquire a finite width
(also at zero temperature) due to a coupling to multi pair configurations
\cite{pines}. The description of this damping of the collective states
from such admixtures goes beyond the usual Fermi liquid theory. In the
preceeding subsection we have seen that a isospin dependent residual
interaction can produce correlations in the response function which correspond
to the admixture of multi pair configurations.
For the chosen temperature and kinematics, however, the Hamiltonian considered
here does not lead to strong collective modes in any of the response functions.
To study the role of the multi pair configurations on the collective
modes we increase the value of the Landau parameters.
In Fig. \ref{collfig} we present the density response function assuming a Landau
parameter
$F_0=4$. In this case the RPA response function shows a well defined
peak.
The relatively high
temperature yields a collective zero sound mode with a finite width due to
the coupling to thermally excited one particle-one hole states.
The calculation including multi-pair correlations in the system from
the residual interaction does also show a collective state in the
density response. The position of this collective mode is almost at
the same place as for the Fermi liquid (Fig. \ref{collfig}).
\begin{figure}
\centering
\includegraphics*[width=0.5\textwidth]{coll.eps}
\parbox{14cm}{\it\caption{The imaginary part of the response functions as
function of energy and $q=220$MeV for the isospin dependent residual
interaction (\ref{newres}) (solid lines) and for the scalar residual
interaction (\ref{vres}) (dotted line). Dashed-dotted line are results for a
Fermi liquid at finite temperature. In order to get a collective mode for the
chosen momentum and temperature the Landau parameters are set by hand to $F_0,\
F_0^{'}=4$.}
\label{collfig}}
\end{figure}
The $ST=00$ response function with propagator and vertex
corrections is almost the same as in the free Fermi gas.
The difference is that
at high energy the response function $ \mbox{Im } \Pi_{ (00)\ res}$ is slightly
larger than the finite temperature Lindhard function.
This causes the collective mode to
have a larger width in the system with residual interactions. The
damping of the zero
sound has two origins in a system interacting with a residual
interaction:
a finite temperature width and a width due to the coupling
to multi-pair states.
In the lower panel of Fig. \ref{collfig} the isovector
response function is shown for the Landau parameter $F_0^{'}=4$. The Fermi
liquid theory predicts the presence of a well defined collective state.
The finite width is due of course to the finite temperature.
For the scalar residual interaction the response function in the
correlated system shows a collective state at similar energy.
It has a larger width due to the contribution of multi-pair
configurations, analogously as in the density response.
The isospin dependent residual interaction leads to a response
$\mbox{Im } \Pi_{(01) \ res}$ with a long tail at large energies.
Due to the large contribution of these configurations the
width of the collective
states in the isovector response is very large. In fact, the
collective mode disappears. The disappearance of the collective mode is
an extreme case where the coupling to multi-pair state is not reduced
by vertex corrections (special case of the interaction
(\ref{newres})).
For a general interaction we expect a whole range
of behavior depending on the energy of the collective state and on
the strength of isospin dependent terms in the residual interaction.
The collective state would be generally broader than in the Fermi
liquid theory, due to the coupling to multi-pair configurations. In some
cases this coupling can lead to a disappearance of the collective state.
The same phenomena are expected also for the spin wave
collective
state in the presence of spin dependent residual interactions.
\section{Conclusions}
The aim of this paper has been a consistent study of correlation effects on the
response function going beyond the usual HF plus RPA approach. For that purpose
we consider a mean-field interaction and a
residual interaction. This residual interaction generates contributions to the
self-energy of the nucleons, which describe the admixture of two-hole
one-particle and two-particle one-hole configurations to the single-particle
propagator.
The response function using these dressed propagators in a one-loop
approximation will in general violate the energy weighted sum
rule for the excitation function. These sum-rules are fulfilled only if the
response functions are calculated employing a consistent treatment of propagator
and vertex corrections following the recipe of Baym and Kadanoff
\cite{kadBaym,blarip}. A scheme for such a consistent treatment of correlation
effects in the response function of nuclear matter is outlined and numerical
results are presented for symmetric nuclear matter and pure neutron matter at
finite temperature.
Assuming a residual interaction of scalar-isoscalar form it turns out that the
effects originating from propagator
corrections are to a large extent compensated
by vertex corrections in the response function for all spin-isospin channels.
The induced interaction terms in particular are responsible for the
compensation
of the correlation effects in the single-particle propagator.
If, however, a residual interaction with non-trivial spin isospin structure is
considered this cancellation of correlation effects is removed in specific
spin-isospin channels. Consequences for the damping of collective excitation
modes due to these admixtures of multi-particle multi-hole contributions are
discussed.
The present investigation employs residual interaction with a rather simple
spin-isospin structure. More realistic interaction models should be
investigated in extended kinematical regions of $q$ and $\Omega$
to obtain detailed information on the importance of correlation effects on the
nuclear response for the various excitation modes.
{\bf Acknowledgments}
\vskip .3cm
We thank Armen Sedrakian for useful and interesting discussions.
P.B. would like to acknowledge the support of the KBN
under Grant No. 2P03B05925 (Poland) and of the Humboldt Stiftung (Germany).
|
{
"timestamp": "2004-11-12T09:15:14",
"yymm": "0411",
"arxiv_id": "nucl-th/0411048",
"language": "en",
"url": "https://arxiv.org/abs/nucl-th/0411048"
}
|
\section{Appendixes}
|
{
"timestamp": "2004-11-25T23:05:27",
"yymm": "0411",
"arxiv_id": "cond-mat/0411660",
"language": "ja",
"url": "https://arxiv.org/abs/cond-mat/0411660"
}
|
\section{Introduction
Studies of $B$ meson decays to three-body charmless hadronic final states are
a natural extension of studies of decays to two-body charmless final states.
Some of the final states considered so far as two-body (for example, $\rho \pi$,
$K^*\pi$, etc.) proceed via quasi-two-body processes involving a wide
resonance state that immediately decays in the simplest case to two particles,
thereby producing a three-body final state. $B$ meson decays to
three-body charmless hadronic final states may
provide new possibilities for CP violation searches.
Observation of $B$ meson decays to various three-body charmless hadronic final
states has already been reported by the Belle~\cite{garmash,chang,garmash2},
CLEO~\cite{eckhart} and BaBar~\cite{aubert} Collaborations. First results on
the distribution of signal events over the Dalitz plot in the three-body
$\ensuremath{B^+\to K^+\pi^+\pi^-}$ and $\ensuremath{B^+\to K^+K^+K^-}$ decays are described in Ref.~\cite{garmash}. With a data
sample of $29.1$~fb$^{-1}$ a simplified analysis technique was used
because of lack of
statistics. Using a similar technique, the BaBar collaboration has
reported results of their analysis of the Dalitz plot for the decay $\ensuremath{B^+\to K^+\pi^+\pi^-}$
with a $56.4$~fb$^{-1}$ data sample~\cite{babar-dalitz}. With the large data
sample that is now available, we can perform a full amplitude analysis. The
analysis described in this paper is based on a 140\,fb$^{-1}$ data sample
containing 152 million $B\bar{B}$ pairs, collected with the Belle detector
operating at the KEKB asymmetric-energy $e^+e^-$ collider.
\section{Amplitude Analysis}
\label{sec:aa}
Analysis of two-body mass spectra shows that a
significant fraction of the signals
observed in $\ensuremath{B^+\to K^+\pi^+\pi^-}$ and $\ensuremath{B^+\to K^+K^+K^-}$ decays can be assigned to quasi-two-body
intermediate states. These resonances will cause a non-uniform distribution
of events in phase space that can be analyzed using the technique pioneered by
Dalitz. Multiple resonances that occur nearby in phase space will
interfere and provide an opportunity to measure both the amplitudes and
relative phases of the intermediate states. This in turn allows us to deduce
their relative branching fractions. Details of the event selection
and amplitude analysis could be found in the Belle contributed
paper~\cite{Belle_cont}. Here we can present the main
results only. The examples of the two-body invariant mass
distributions and their description by a fit of $\ensuremath{B^+\to K^+\pi^+\pi^-}$ and
$\ensuremath{B^+\to K^+K^+K^-}$ decays are presented in Fig.~\ref{fig:kpp-mods}. Results on
the branching fractions for quasi-two-body decays are summarized
in Table~\ref{tab:branch}.
\begin{figure*}[th]
\centering
\includegraphics[width=0.32\textwidth]{fig-09d.eps}
\includegraphics[width=0.32\textwidth]{fig-09e.eps}
\includegraphics[width=0.32\textwidth]{fig-13d.eps} \\
\caption{Results of the fit to $\ensuremath{K^+\pi^+\pi^-}$ and $\ensuremath{K^+K^+K^-}$ events. Points with
error bars are data, the open histograms
are the fit result and hatched histograms are the background
components. Insets show: (a) the $K^*(892)-K_0^*(1430)$
mass region in 20~MeV/$c^2$~ bins; (b) the
$\ensuremath{\chi_{c0}}$ mass region in 25~MeV/$c^2$~ bins; (d) the
$\phi(1020)$ mass region in 2~MeV/$c^2$~ bins.}
\label{fig:kpp-mods}
\end{figure*}
\begin{table*}[!ht]
\caption{Summary of branching fraction results. The first quoted error is
statistical, the second is systematic and the third is the model
error. The
charmless total fractions in this table exclude the $\ensuremath{\chi_{c0}}$
contribution. The value given in brackets for the $K^*_0(1430)\pi^+$
and $\ensuremath{\chi_{c0}} K^+$
channels corresponds to the second solution
(see~$^7$ for details).}
\medskip
\label{tab:branch}
\centering
\begin{tabular}{lcr} \hline \hline
\hspace*{2mm}Mode\hspace*{3mm} &
\hspace*{0mm}$\ensuremath{{\cal{B}}}(B^+\to Rh^+)$ &
\hspace*{1mm}$\ensuremath{{\cal{B}}}(B^+\to Rh^+)\times10^{6}$ \\
&
\hspace*{0mm}$\times\ensuremath{{\cal{B}}}(R\to h^+h^-)\times10^{6}$ &
\\ \hline \hline
$\ensuremath{K^+\pi^+\pi^-}$ charmless total & $-$
& $46.6\pm2.1\pm4.3$ \\
$K^*(892)^0\pi^+$, $K^*(892)^0\to K^+\pi^-$
& $6.55\pm0.60\pm0.60^{+0.38}_{-0.57}$
& $9.83\pm0.90\pm0.90^{+0.57}_{-0.86}$ \\
$K^*_0(1430)\pi^+$, $K^*_0(1430)\to K^+\pi^-$
& $27.9\pm1.8\pm2.6^{+8.5}_{-5.4}$
& $45.0\pm2.9\pm6.2^{+13.7}_{-~8.7}$ \\
& ($5.12\pm1.36\pm0.49^{+1.91}_{-0.51}$)
& ($8.26\pm2.20\pm1.19^{+3.08}_{-0.82}$) \\
$K^*(1410)\pi^+$, $K^*(1410)\to K^+\pi^-$
& $<2.0$ & $-$ \\
$K^*(1680)\pi^+$, $K^*(1680)\to K^+\pi^-$
& $<3.1$ & $-$ \\
$K^*_2(1430)\pi^+$, $K^*_2(1430)\to K^+\pi^-$
& $<2.3$ & $-$ \\
$\rho^0(770)K^+$, $\rho^0(770)\to\pi^+\pi^-$
& $4.78\pm0.75\pm0.44^{+0.91}_{-0.87}$
& $4.78\pm0.75\pm0.44^{+0.91}_{-0.87}$ \\
$f_0(980)K^+$, $f_0(980)\to\pi^+\pi^-$
& $7.55\pm1.24\pm0.69^{+1.48}_{-0.96}$
& $-$ \\
$f_2(1270)K^+$, $f_2(1270)\to\pi^+\pi^-$
& $<1.3$ & $-$ \\
Non-resonant
& $-$
& $17.3\pm1.7\pm1.6^{+17.1}_{-7.8}$ \\
\hline
$\ensuremath{K^+K^+K^-}$ charmless total & $-$
& $30.6\pm1.2\pm2.3$ \\
$\phi K^+$, $\phi\to K^+K^-$
& $4.72\pm0.45\pm0.35^{+0.39}_{-0.22}$
& $9.60\pm0.92\pm0.71^{+0.78}_{-0.46}$ \\
$\phi(1680)K^+$, $\phi(1680)\to K^+K^-$
& $<0.8$ \\
$f_0(980)K^+$, $f_0(980)\to K^+K^-$
& $<2.9$ \\
$f'_2(1525)K^+$, $f'_2(1525)\to K^+K^-$
& $<2.1$ \\
$a_2(1320)K^+$, $a_2(1320)\to K^+K^-$
& $<1.1$ \\
Non-resonant
& $-$
& $24.0\pm1.5\pm1.8^{+1.9}_{-5.7}$ \\
\hline
$\ensuremath{\chi_{c0}} K^+$, $\ensuremath{\chi_{c0}}\to\pi^+\pi^-$
& $1.37\pm0.28\pm0.12^{+0.34}_{-0.35}$
& $-$ \\
$\ensuremath{\chi_{c0}} K^+$, $\ensuremath{\chi_{c0}}\to K^+K^-$
& $0.86\pm0.26\pm0.06^{+0.20}_{-0.05}$
& $-$ \\
& ($2.58\pm0.43\pm0.19^{+0.20}_{-0.05}$)
& $-$ \\
$\ensuremath{\chi_{c0}} K^+$ combined & $-$
& $196\pm35\pm33^{+197}_{-26}$ \\
\hline \hline
\end{tabular}
\end{table*}
\section{Discussion}
\label{sec:discussion}
With a 140~fb$^{-1}$ data sample collected with the Belle detector, we
have performed the first amplitude analysis~\footnote {At this
Conference Babar presented the Dalits plot analysis for $\ensuremath{K^+\pi^+\pi^-}$~\cite{Babar_kpp}.} of
$B$ meson decays to the three-body charmless $\ensuremath{K^+\pi^+\pi^-}$ and $\ensuremath{K^+K^+K^-}$ final
states. Such analysis often suffers from uncertainties related to the non-unique
description of the decay amplitude. In our case the uncertainty
comes mainly from the parametrization of the non-resonant amplitude.
It is worth noting that fractions of the non-resonant decay in both
$\ensuremath{B^+\to K^+\pi^+\pi^-}$ and $\ensuremath{B^+\to K^+K^+K^-}$ decays are comparable in size and comprise a
significant fraction of the total three-body signal, which may indicate the
common nature of the amplitudes.
Despite the large model uncertainty discussed above, there is a set of
quasi-two-body signals whose branching fractions can be measured with a
relatively small model error. In particular, clear signals are observed in
the $B^+\to K^*(892)^0\pi^+$, $B^+\to\rho^0(770)K^+$, $B^+\to f_0(980)K^+$
and $B^+\to\phi K^+$ decay channels. The model uncertainty for these channels
is small due to the narrow width of the resonances and in vector-pseudoscalar
decays due to the clear signature of the vector meson polarization.
The branching fraction value measured for the decay $B^+\to K^*(892)^0\pi^+$
is significantly lower than that reported earlier~\cite{garmash,babar-dalitz}.
The simplified technique used for Dalitz analysis of the $\ensuremath{B^+\to K^+\pi^+\pi^-}$ decay
described in~\cite{garmash,babar-dalitz} has no sensitivity to the relative
phases between different resonances, resulting in a large model error.
The full amplitude analysis presented in this paper consistently treats
effects of interference between different quasi-two-body amplitudes thus
reducing the model error.
The analysis suggests the presence of a large \mbox{non-$K^*(892)^0\pi^+$}
(presumably non-resonant) amplitude in the mass region of the $K^*(892)^0$ that
absorbs a significant fraction of the $B$ signal.
The $B^+\to K^*(892)^0\pi^+$ branching fraction measured in our analysis is in
better agreement with theoretical predictions based on the QCD factorization
approach.
The decay mode $B^+\to f_0(980)K^+$ is the first observed example of a $B$
decay to a charmless scalar-pseudoscalar final state. The mass
$M(f_0(980))=976\pm4^{+2}_{-3}$~MeV/$c^2$ ~and width
$\Gamma(f_0(980))=61\pm9^{+14}_{-8}$~MeV/$c^2$ ~obtained from the fit are in
agreement with previous measurements.
The sensitivity to the $B^+\to f_0(980)K^+$ decay in the $\ensuremath{K^+K^+K^-}$
final state is greatly reduced by the large $B^+\to\phi K^+$ signal and by
the scalar non-resonant amplitude. No statistically significant
contribution from this channel to the $\ensuremath{K^+K^+K^-}$ three-body final state is
observed, thus only a 90\% confidence level upper limit for the
product of the corresponding branching fractions is reported.
We report the first observation of the decay
$B^+\to\rho^0(770)K^+$. This is one of the
channels where large direct CP violation is expected.
Due to the very narrow width of the $\phi$ meson, the branching fraction for
the decay $B^+\to\phi K^+$ is determined with a small model uncertainty.
A clear signal is also observed for the decay $B^+\to \ensuremath{\chi_{c0}} K^+$ in both
$\ensuremath{\chi_{c0}}\to \pi^+\pi^-$ and $\ensuremath{\chi_{c0}}\to K^+K^-$ channels.
Although quite significant statistically, the $B^+\to\ensuremath{\chi_{c0}} K^+$ signal
constitutes only a small fraction of the total three-body signal and thus
suffers from a large model error, especially in the $\ensuremath{K^+K^+K^-}$ final state. For
this decay mode, the charmless non-resonant amplitude in the $\ensuremath{\chi_{c0}}$ mass
region is enhanced compared to the $\ensuremath{K^+\pi^+\pi^-}$ final state due to the interference
caused by the presence of the two identical kaons.
For other quasi-two-body channels the interpretation of fit results is less
certain. Although the $B^+\to K^*_0(1430)\pi^+$ signal is observed with a high
statistical significance, its branching fraction is determined with a large
model error. Two solutions with significantly different fractions of the
$B^+\to K^*_0(1430)\pi^+$ signal but similar likelihood values are obtained
from the fit to $\ensuremath{K^+\pi^+\pi^-}$ events. A study with MC simulation confirms the
presence of the second solution. This
may indicate that in order to choose a unique solution additional external
information is required. In this sense, the most useful piece of information
seems to be the phenomenological estimation of the $B^+\to K^*_0(1430)\pi^+$
branching fraction. The analysis of $B$ meson decays to scalar-pseudoscalar
final states
described in Ref.~\cite{b2ps} suggests that the branching fraction for the
$B^+\to K^*_0(1430)\pi^+$ decay can be as large as $40\times 10^{-6}$.
Unfortunately, the predicted value suffers from a large systematic error
that is mainly due to uncertainty in the $K^*_0(1430)$ decay constant
$f_{K^*_0(1430)}$. Different techniques used to estimate
$f_{K^*_0(1430)}$~\cite{b2ps,maltman} give significantly different results.
Further improvement in this field would be useful.
We also check possible contributions from $B$ decays to various
pseudoscalar-tensor ($PT$) states.
In the factorization approximation, charmless $B$ decays
to $PT$ final states are expected to occur at
the level of $10^{-7}$ or less.
We find no statistically significant signal in any of these
channels. As a result, we set 90\% confidence level upper limits for their
branching fractions.
We cannot identify unambiguously the broad structures observed in the
$M(\ensuremath{\pi^+\pi^-})\simeq1.3$~GeV/$c^2$ mass region in the $\ensuremath{K^+\pi^+\pi^-}$ final state
denoted as $f_X(1300)$ in our analysis and at $M(\ensuremath{K^+K^-})\simeq1.5$~GeV/$c^2$
in the $\ensuremath{K^+K^+K^-}$ final state denoted as $f_X(1500)$. If approximated by a
single resonant state, $f_X(1300)$ is equally well described by a
scalar or vector amplitude. Analysis with higher statistics might allow
a more definite conclusion. The best description of the $f_X(1500)$
is achieved with a scalar amplitude with mass and width from the fit
consistent with $f_0(1500)$ states.
Results of the $\ensuremath{B^+\to K^+K^+K^-}$ Dalitz analysis can be also useful in connection
with the measurement of CP violation in $B^0\to K^0_SK^+K^-$ decay reported
recently by the Belle~\cite{kskk-cp-belle} and BaBar~\cite{kskk-cp-babar}
collaborations. An isospin analysis of $B$ decays to
three-kaon final states suggests the dominance of the CP-even component in the
$B^0\to K^0_SK^+K^-$ decay (after the $B^0\to\phi K^0_S$ signal is excluded).
This conclusion can be checked independently by an amplitude analysis of the
$K^0_SK^+K^-$ final state, where the fraction of
CP-odd states can be obtained as a fraction of states with odd orbital momenta.
Unfortunately, such an analysis is not feasible with the current experimental
data set. Nevertheless, the fact that we do not observe any vector amplitude
other than $B^+\to\phi K^+$ in the decay $\ensuremath{B^+\to K^+K^+K^-}$ confirms the conclusion.
|
{
"timestamp": "2004-11-01T06:11:43",
"yymm": "0411",
"arxiv_id": "hep-ex/0411004",
"language": "en",
"url": "https://arxiv.org/abs/hep-ex/0411004"
}
|
\section{Introduction}
Lorentz symmetry is a fundamental feature of relativity theory.
In special relativity,
it is a global symmetry relating the laws of
physics in different inertial frames under boosts and rotations.
It is also linked by a general theorem to the combined
discrete symmetry CPT formed from the product of charge conjugation C,
parity P, and time reversal T.
In general relativity,
Lorentz symmetry becomes a local symmetry that relates
the physics in different freely falling frames in
a gravitational field.
The Standard Model (SM) of particle physics does not include
gravity as a fundamental interaction at the quantum level.
It is therefore expected that the SM and gravity
will merge in the context of a fundamental unified theory.
The relevant energy scale for quantum gravity is the Planck scale
$M_{\rm Pl} = \sqrt{\hbar c/G} \simeq 10^{19}$ GeV.
Much current work in theoretical high-energy physics is
aimed at finding a unified fundamental theory that describes
physical interactions at the Planck scale.
Promising candidates include string theory,
D-branes, and theories of quantum gravity.
Many of these include effects that violate
assumptions of the SM,
including higher dimensions of spacetime,
unusual geometries, nonpointlike interactions,
and new forms of symmetry breaking.
In particular,
it is possible that small violations of Lorentz
symmetry might occur in theories of quantum gravity.
For example,
it is known that there are mechanisms in string theory that can
lead to spontaneous violations of Lorentz and CPT symmetry.\cite{kskp}
This is due to certain types of interactions in string theory
among Lorentz-tensor fields that can destabilize the vacuum
and generate nonzero vacuum expectation values for Lorentz tensors.
These vacuum expectation values fill the true vacuum and cause
spontaneous Lorentz violation.
It is also known that geometries with noncommutative
coordinates can arise naturally in string theory
and that Lorentz violation is intrinsic to noncommutative field
theories.\cite{chlkO01}
One method of searching for signals of Planck-scale physics is
to look for highly suppressed effects involving inverse
powers of the Planck scale.
In this approach, Lorentz violation becomes an ideal
signal since all of the interactions in the SM
preserve Lorentz symmetry and therefore no conventional
signal could mimic the effects of Lorentz violation.
To observe a signal of Lorentz violation experimentally,
one needs to perform experiments with exceptional sensitivity.
Experiments in QED systems provide many of the best
oppportunities for testing Lorentz symmetry.
One example is provided by measurements of photons that have
traveled over cosmological distances.
Any small phase effect would be amplified during
the long transit time.
Other examples with photons include high-precision
laboratory-based experiments with resonance cavities.
Experiments in atomic physics can also be performed at
low energy with extremely high precision.
For example, some atomic experiments are routinely sensitive
to small frequency shifts at the level of 1 mHz or less.
Interpreting this as being due to an energy shift expressed in GeV,
it would correspond to a sensitivity of approximately
$4 \times 10^{-27}$ GeV.
Such a value is well within the range of energy one
might associate with suppression factors originating
from the Planck scale.
The main focus of this work is to investigate tests of Lorentz and
CPT symmetry performed in QED systems.
The general goals are to analyze the sensitivity
of QED systems to possible Lorentz and CPT violation,
to uncover possible new signals that can be tested in experiments,
and to express experimental sensitivities in the context of a common
framework that permits comparisons across different experiments.
To this end,
we use the Standard-Model Extension (SME)
as our theoretical framework.\cite{sme}
The SME permits detailed investigations of Lorentz and CPT tests
in all particle sectors of the SM.
Our analysis here focuses on the QED sector of the SME.
This is presented in the following section and is then used
to examine a number of experiments involving photons,
trapped particles, atomic clocks, muons, and a spin-polarized pendulum.
Additional details about many of these experiments can be found as
well in several of the other articles in this volume.
\section{QED Sector of the SME}
The subset of the SME lagrangian relevant to experiments in QED systems
can be written as
\begin{equation}
{\mathcal L}_{\rm QED} = {\mathcal L}_0 + {\mathcal L}_{\rm int}
\quad .
\label{lag}
\end{equation}
The lagrangian ${\mathcal L}_0$ contains the usual Lorentz-invariant
terms in QED that describe photons, massive charged fermions,
and their conventional couplings.
If we restrict our investigation to the remormalizable
and gauge-invariant terms in the SME in flat spacetime,
then the Lorentz-violating part of the lagrangian is given by\cite{note1}
\begin{eqnarray}
{\mathcal L}_{\rm int} &=& - a_\mu \bar \psi \gamma^\mu \psi
- b_\mu \bar \psi \gamma _5 \gamma^\mu \psi
+ ic_{\mu \nu} \bar \psi \gamma^\mu D^\nu \psi
\nonumber \\
&& \quad\quad
+ id_{\mu \nu} \bar \psi \gamma_5 \gamma^\mu D^\nu \psi
- {\textstyle{1\over 2}} H_{\mu \nu} \bar \psi \sigma^{\mu \nu} \psi
\nonumber \\
&&
-\frac14 (k_F)_{\kappa\lambda\mu\nu} F^{\kappa\lambda}F^{\mu\nu}
+ \frac 12 (k_{AF})^\kappa \epsilon_{\kappa\lambda\mu\nu} A^{\lambda} F^{\mu\nu}
\quad .
\label{qedsme}
\end{eqnarray}
Here,
natural units with $\hbar = c = 1$ are used,
and $i D_\mu \equiv i \partial_\mu - q A_\mu$.
The terms with coefficients $a_\mu$, $b_\mu$ and $(k_{AF})_\mu$
violate CPT,
while those with
$H_{\mu \nu}$, $c_{\mu \nu}$, $d_{\mu \nu}$, and $(k_F)_{\kappa\lambda\mu\nu}$
preserve CPT.
All seven terms break Lorentz symmetry.
Sensitivities to Lorentz and CPT violation can be
expressed in terms of the SME coefficients.
This provides a straightforward way of making comparisons across different
types of experiments.
Each different particle sector in the QED extension
has an independent set of Lorentz-violating coefficients.
These are distinguished using superscript labels.
A thorough investigation of Lorentz and CPT violation
requires looking at as many different particle sectors as possible.
\section{Photon Experiments}
The relevant part of the lagrangian for a freely propagating
photon in the presence of Lorentz violation is given by
\begin{equation}
{\mathcal L} = -\frac14 F_{\mu\nu}F^{\mu\nu}
-\frac14 (k_F)_{\kappa\lambda\mu\nu}
F^{\kappa\lambda}F^{\mu\nu}\
+ \frac12 (k_{AF})^\kappa \epsilon_{\kappa\lambda\mu\nu} A^{\lambda} F^{\mu\nu} ,
\end{equation}
where $F_{\mu\nu}$ is the field strength,
$F_{\mu\nu} \equiv \partial_\mu A_\nu -\partial_\nu A_\mu$.
The CPT-odd term with coefficient $k_{AF}$ has been investigated
extensively both theoretically and experimentally.\cite{cfj,jk}
It is found theoretically that this term leads to negative-energy contributions
and is a potential source of instability.
One solution is to set $k_{AF} \approx 0$,
which is consistent with radiative corrections in the SME.
Stringent experimental constraints consistent with $k_{AF} \approx 0$
have also been determined by studying the polarization of radiation
from distant radio galaxies.\cite{cfj}
In the following,
we will therefore ignore the effects of $k_{AF}$.
The CPT-even term with coefficients $k_{F}$ have been
investigated more recently.\cite{km}
It provides positive-energy contributions.
The set of coefficients $k_{F}$ has 19 independent components.
It is useful to make a decomposition of these in terms of a new set:
$\tilde\kappa_{e+}$, $\tilde\kappa_{e-}$,
$\tilde\kappa_{o+}$, $\tilde\kappa_{o-}$, and $\tilde\kappa_{\rm tr}$.
Here, $\tilde\kappa_{e+}$, $\tilde\kappa_{e-}$, and $\tilde\kappa_{o-}$
are $3\times3$ traceless symmetric matrices
(with 5 independent components each),
while $\tilde\kappa_{o+}$ is a
$3\times3$ antisymmetric matrix
(with 3 independent components),
and the remaining coefficient $\tilde\kappa_{\rm tr}$
is the only rotationally invariant component.
The lagrangian in terms of this
decomposition becomes
\begin{eqnarray}
{\mathcal L} &=& {\textstyle{1\over 2}} [(1+\tilde\kappa_{\rm tr})\vec E^2
-(1-\tilde\kappa_{\rm tr})\vec B^2]
+{\textstyle{1\over 2}} \vec E\cdot(\tilde\kappa_{e+}
+\tilde\kappa_{e-})\cdot\vec E
\nonumber \\
&&
-{\textstyle{1\over 2}}\vec B\cdot(\tilde\kappa_{e+}
-\tilde\kappa_{e-})\cdot\vec B
+\vec E\cdot(\tilde\kappa_{o+}
+\tilde\kappa_{o-})\cdot\vec B\
\quad .
\label{modL}
\end{eqnarray}
Here, $\vec E$ and $\vec B$ are
the usual electric and magnetic fields.
The equations of motion following from this lagrangian
give rise to modifications of Maxwell's equations,
which have been explored in several recent astrophysical
and laboratory experiments.
The ten coefficients $\tilde\kappa_{e+}$ and $\tilde\kappa_{o-}$ lead to
birefrigence of light.
Spectropolarimetry of light from distant galaxies leads to
bounds on these parameters of order $2 \times 10^{-32}$.\cite{km}
Seven of the eight coefficients $\tilde\kappa_{e-}$ and $\tilde\kappa_{o+}$ are bounded
in experiments using optical and microwave cavities.\cite{photonsexpts}
Sensitivities on the order of $\tilde\kappa_{e-} \mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$} 10^{-15}$
and $\tilde\kappa_{o+} \mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$} 10^{-11}$ have been attained,
and it is expected that future experiments in boosted frames will be sensitive
to the remaining two parameters as well.
\section{Atomic Experiments}
In recent years,
a number of atomic experiments have been performed which
have very sharp sensitivity to Lorentz and CPT violation.
Bounds from these experiments can be expressed in terms of the coefficients
$a_\mu$, $b_\mu$, $c_{\mu \nu}$, $d_{\mu \nu}$, and $H_{\mu \nu}$
in the QED sector of the SME.
Comparisons across different types of experiments can then be made which avoid
the problems that can arise when different physical quantities
($g$ factors, charge-to-mass ratios, masses, frequencies, etc.)
are used in different experiments.
In the following,
a number of atomic experiments involving the
proton, neutron, electron, and muon are examined.
\subsection{Penning-Trap Experiments}
Two recent sets of experiments with electrons and positrons in Penning traps
provide sharp tests of Lorentz and CPT symmetry.\cite{bkr9798}
Both involve measurements of the
anomaly frequency $\omega_a$ and the cyclotron frequency $\omega_c$.
The first consists of a reanalysis by Dehmelt's group of existing data
for electrons and positrons in a Penning trap.\cite{dehmelt99}
The signal involves looking for an instantaneous difference in the
anomaly frequencies of electrons and positrons,
which can be nonzero when Lorentz and CPT symmetry are broken.
In contrast the instantaneous cyclotron frequencies remain approximately equal
at leading order in the Lorentz-violation corrections.
Dehmelt's original measurements of $g-2$
did not involve looking for possible instantaneous variations in $\omega_a$.
Instead,
the ratio $\omega_a/\omega_c$ was computed using averaged values.
It is important to realize that the Lorentz-violating corrections to the
anomaly frequency $\omega_a$ can occur even though the $g$ factor remains unchanged.
The new analysis looks for an
instantaneous difference in the electron and positron anomaly frequencies.
A bound on this difference can be expressed in terms of the parameter $b^e_3$,
which is the component of $b^e_\mu$ along the quantization
axis in the laboratory frame.
It is given as $|b^e_3| \mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$} 3 \times 10^{-25}$ GeV.
A second signal for Lorentz and CPT violation in the electron
sector has been obtained using data for the electron alone.\cite{mittleman99}
In this case,
the idea is that the Lorentz-violating interactions depend on
the orientation of the quantization axis in the laboratory frame,
which changes as the Earth turns on its axis.
As a result,
both the cyclotron and anomaly frequencies have small corrections which
cause them to exhibit sidereal time variations.
These variations can be measured using electrons alone,
eliminating the need for comparison with positrons.
The bounds in this case are given with respect to a
nonrotating coordinate frame such as celestial equatorial coordinates.
The interactions involve a combination of laboratory-frame components
that couple to the spin of the electron.
This combination is denoted using tildes as
$\tilde b_3^e \equiv b_3^e - m d_{30}^e - H_{12}^e$.
When expressed in terms of components $X$, $Y$, $Z$ in the nonrotating
frame,
the obtained bound is
$|\tilde b_J^e| \mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$} 5 \times 10^{-25} {\rm GeV}$ for $J=X,Y$.
\subsection{Clock-Comparison Experiments}
The Hughes-Drever experiments are
classic tests of Lorentz invariance.\cite{kl99,cctests}
There have been a number of different types of these
experiments performed over the years,
with steady improvements in their sensitivity.
They all involve making high-precision comparisons of
atomic clock signals as the Earth rotates.
The clock frequencies are typically hyperfine or Zeeman transitions.
Many of the sharpest Lorentz bounds for the proton, neutron, and electron
stem from atomic clock-comparison experiments.
For example,
Bear {\it et al.} use a two-species noble-gas maser to
test for Lorentz and CPT violation in the neutron sector.\cite{dualmaser}
They obtain a bound
$|\tilde b_J^n| \mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$} 10^{-31} {\rm GeV}$ for $J=X,Y$,
which is currently the best bound for the neutron sector.
Note that these Earth-based laboratory experiments are not
sensitive to Lorentz-violation coefficients along the $J=Z$
direction parallel to Earth's rotation axis.
They also neglect the velocity effects due to Earth's
motion around the sun,
which would lead to bounds on the timelike components along $J=T$.
These limitations can be overcome by performing experiments in space\cite{space}
or by using a rotation platform.
The earth's motion can also be taken into account.
A recent boosted-frame analysis of the dual noble-gas maser
experiment yields bounds on the order of $10^{-27}$ GeV
on many boost-dependent SME coefficients for the neutron
that were previously unbounded.\cite{cane}
It should also be pointed out that certain assumptions about the nuclear
configurations must be made to obtain bounds in
clock-comparison experiments.
For this reason,
these bounds should be viewed as good to within about
an order of magnitude.
To obtain cleaner bounds it is necessary to consider
simpler atoms or to perform more sophisticated nuclear modeling.
\subsection{Hydrogen-Antihydrogen Experiments}
The simplest atom in the periodic table is hydrogen,
and the simplest antiatom is antihydrogen.
There are three experiments underway at CERN that
can perform high-precision Lorentz and CPT tests in antihydrogen.
Two of the experiments (ATRAP and ATHENA) intend to
make high-precision spectroscopic measurements of the 1S-2S
transitions in hydrogen and antihydrogen.
These are forbidden (two-photon) transitions that have a relative linewidth
of approximately $10^{-15}$.
The ultimate goal is to measure the line center of this
transition to a part in $10^3$ yielding a frequency comparison
between hydrogen and antihydrogen at a level of $10^{-18}$.
An analysis of the 1S-2S transition in the context of the
SME shows that the magnetic field plays an important role
in the attainable sensitivity to Lorentz and CPT violation.\cite{bkr99}
For instance,
in free hydrogen in the absence of a magnetic field,
the 1S and 2S levels are shifted by equal amounts at leading order.
As a result,
in free H or $\bar {\rm H}$ there are no leading-order corrections
to the 1S-2S transition frequency.
In a magnetic trap,
however,
there are fields that can mix the spin states in the
four different hyperfine levels.
Since the Lorentz-violating interactions depend on the spin orientation,
there will be leading-order sensitivity
to Lorentz and CPT violation in comparisons of 1S-2S transitions in
trapped hydrogen and antihydrogen.
At the same time,
however,
these transitions are field-dependent,
which creates additional experimental challenges
that would need to be overcome.
An alternative to 1S-2S transitions is to consider the sensitivity
to Lorentz violation in ground-state Zeeman hyperfine transitions.
It is found that there are leading-order corrections in these levels
in both hydrogen and antihydrogen.\cite{bkr99}
The ASACUSA group at CERN is planning to measure the Zeeman hyperfine
transitions in antihydrogen.
Such measurements will provide a direct CPT test.
Experiments with hydrogen alone have been performed using a maser.\cite{Hmaser}
They attain exceptionally sharp sensitivity to Lorentz and CPT
violation in the electron and proton sectors of the SME.
These experiments use a double-resonance technique that does
not depend on there being a field-independent point for the transistion.
The sensitivity for the proton attained in these experiments
is $|\tilde b_J^p| \mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$} 10^{-27}$ GeV.
Due to the simplicity of hydrogen,
this is an extremely clean bound and is currently the most stringent test
of Lorentz and CPT violation for the proton.
\subsection{Spin-Polarized Matter}
Experiments at the University of Washington using a spin-polarized
torsion pendulum\cite{eotwash}
are able to achieve very high sensitivity to
Lorentz violation in the electron sector.\cite{bk00}
The sensitivity arises because the pendulum has a huge
number of aligned electron spins but a negligible magnetic field.
The pendulum is built out of a stack of toroidal magnets,
which in one version of the experiment achieved
a net electron spin $S \simeq 8 \times 10^{22}$.
The apparatus is suspended on a rotating turntable and
the time variations of the twisting pendulum are measured.
An analysis of this system shows that in addition to a signal having the
period of the rotating turntable,
the effects due to Lorentz and CPT violation also cause additional
time variations with a sidereal period caused by the rotation
of the Earth.
The group at the University of Washington has analyzed their data
and find that thay have sensitivity to the electron coefficients
at the levels of $|\tilde b_J^e| \mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$} 10^{-29}$ GeV for $J=X,Y$ and
$|\tilde b_Z^e| \mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$} 10^{-28}$ GeV.\cite{eotwash}
These are currently the best Lorentz and CPT bounds for the electron.
More recently,
a new pendulum has been built,
and it is expected that improved sensitivities will
be attained.
\subsection{Muon Experiments}
Muons are second-generation leptons.
Lorentz tests performed with muons are therefore independent
of the tests involving electrons.
There are two main classes of experiments with muons
that have been conducted recently.
These are experiments with muonium\cite{muonium99}
and $g-2$ experiments with muons at Brookhaven.
In muonium,
measurements of the frequencies
of ground-state Zeeman hyperfine transitions
in a strong magnetic field have the greatest sensitivity
to Lorentz and CPT violation.
An analysis searching for sidereal time variations
in these transitions was able to attain
sensitivity to Lorentz violation at the level of
$| \tilde b^\mu_J| \le 2 \times 10^{-23}$ GeV.
At Brookhaven,
relativistic $g-2$ experiments with positive muons
have been conducted using muons with boost parameter $\delta = 29.3$.
An analysis of the obtained data as a test of Lorentz symmetry
is still forthcoming.
We estimate that a sensitivity to Lorentz violation is
possible in these experiments at a level of $10^{-25}$ GeV.\cite{bkl00}
\section{Conclusions}
Experiments in QED systems continue to provide many of the
sharpest tests of Lorentz and CPT symmetry.
In recent years,
a number of new astrophysical and laboratory
tests have been performed that have lead
to substantially improved sensitivities for the photon.
Similarly,
atomic experimentalists continue to find ways of improving
the sensitivity to Lorentz violation in many of the matter sectors of the SME.
In particular,
experiments in boosted frames are providing sensitivity
to many of the previously unprobed SME coefficients.
All of the bounds obtained are within the range of sensitivity associated
with suppression factors arising from the Planck scale.
The coming years are likely to remain productive.
QED experiments will continue to provide
increasingly sharp new tests of Lorentz and CPT symmetry.
|
{
"timestamp": "2004-11-10T19:16:42",
"yymm": "0411",
"arxiv_id": "hep-ph/0411149",
"language": "en",
"url": "https://arxiv.org/abs/hep-ph/0411149"
}
|
\section{Introduction}
One of the most useful and efficient ways to study the properties
of black holes is by scattering matter waves off them
\cite{futterman}. From the more practical point of view, the study
of wave scattering in black hole spacetimes is crucial to the
understanding of the signals expected to be received by the new
generation of gravitational-wave detectors in the near future
\cite{hough}. Since the linear perturbations of black holes are
represented by fields of integral spins, the study of the
scattering of wave fields are concentrated on these cases while
that of the Dirac fields are thus less common, especially for the
massive ones \cite{futterman,unruh}. Recently, Finster and
collaborators \cite{finster} have renewed the interests in that of
the massive Dirac fields by investigating in details their
evolutions in various black hole spacetimes. Here we would like to
work in this direction by considering the scattering solutions of
the radial equations of massive Dirac fields in spherically
symmetric Schwarzschild black hole spacetimes.
There are both numerical and analytical methods in solving the
various wave equations in black hole scattering \cite{futterman}.
In this work we shall use the semi-analytic WKB approximation
\cite{berry,bender}, which has been proven to be very useful and
accurate in many cases like, for example, the evaluation of the
quasinormal mode frequencies \cite{schutz,iyer}. For the radial
Dirac equations we consider here, the effective potential can
change from a barrier to a step and vice versa when the mass $m$
or the angular momentum parameter $\kappa$ are varied. The WKB
scheme can be accommodated in various ways to consider all these
situations with different values of the energy $E$ of the field,
whether it is above or below the maximum value $V_{m}$ of the
potential.
In \cite{mukhopadhyay}, Mukhopadhyay and Chakrabarti have studied
in details the radial equations of a massive Dirac field in the
Schwarzschild black hole spacetime using a modified WKB (they
called it the "instantaneous WKB") approximation. However, they
only looked at the case with $m=M/2$, where $M$ is the mass of the
black hole, $\kappa=1$, and $E\approx m$. In this paper we use the
standard WKB scheme instead, and we shall extend the consideration
to cases with different values of $m$ and $\kappa$. In addition,
we shall consider all possible values of $E$ with $E\geq m$. In
this manner we can discuss the variations of the transmission
probabilities with respect to these various parameters.
In the next section, we consider the reduction of the massive
Dirac equation in Schwarzschild spacetimes into a set of
Schr\"odinger-like equations. We also discuss briefly the
properties of the corresponding effective potential \cite{cho}. In
Section III, we look at the three different WKB schemes for the
cases with $E^{2}\gg V_{m}$, $E^{2}\approx V_{m}$, and $E^{2}\ll
V_{m}$. Explicit approximated formulae for the transmission
probability are given. In Section IV, we apply the formulation to
calculate the transmission probabilities of the massive radial
Dirac equations for various values of $E$, $m$, and $\kappa$.
Conclusions and discussions are given in Section V.
\section{Dirac Equation in the Schwarzschild Spacetime}
In this section we discuss briefly the massive Dirac equation in
the Schwarzschild black hole spacetime, including its reduction to
a set of Schr\"odinger-like radial equations and the properties of
the corresponding effective potentials.
\subsection{Radial equations}
In the Schwarzschild spacetime,
\begin{equation}
ds^{2}=-\frac{\Delta}{r^2}dt^{2}+\frac{r^2}{\Delta}dr^{2} +r^{2}
d\theta^{2}+r^{2}{\rm sin}^{2}\theta\ d\phi^{2},
\end{equation}
where $\Delta=r(r-2M)$ and $M$ is the mass of the black hole.
Consider the Dirac equation in this background spacetime
\cite{brill},
\begin{equation}
[\gamma^{a}{e_{a}}^{\mu}(\partial_{\mu}+\Gamma_{\mu})+m]\Psi=0,
\label{diraceq}
\end{equation}
where $m$ is the mass of the Dirac field, and ${e_{a}}^{\mu}$ is
the inverse of the vierbein ${e_{\mu}}^{a}$ defined by the metric
$g_{\mu\nu}$,
\begin{equation}
g_{\mu\nu}=\eta_{ab}{e_{\mu}}^{a}{e_{\nu}}^{b},
\end{equation}
with $\eta_{ab}={\rm diag}(-1,1,1,1)$ being the Minkowski metric.
$\gamma^{a}$ are the Dirac matrices
\begin{equation}
\gamma^{0}= \left(
\begin{array}{cc}
-i&0\\0&i
\end{array}\right),\ \ \
\gamma^{i}=\left(
\begin{array}{cc}
0&-i\sigma^{i}\\i\sigma^{i}&0
\end{array}\right),\ i=1,2,3,
\end{equation}
where $\sigma^{i}$ are the Pauli matrices. $\Gamma_{\mu}$ is the
spin connection given by
\begin{equation}
\Gamma_{\mu}=\frac{1}{8}[\gamma^{a},\gamma^{b}]{e_{a}}^{\nu}e_{b\nu;\mu}\
,\label{spinconnection}
\end{equation}
where
$e_{b\nu;\mu}=\partial_{\mu}e_{b\nu}-\Gamma^{\alpha}_{\mu\nu}e_{b\alpha}$
is the covariant derivative of $e_{b\nu}$ with
$\Gamma^{\alpha}_{\mu\nu}$ being the Christoffel symbols.
Here it is convenient to choose the vierbein
\begin{equation}
{e_{\mu}}^{a} = \left(
\begin{array}{cccc}
\Delta^{1/2}/r&0&0&0\\ 0&r{\rm sin}\theta\ \!{\rm
cos}\phi/\Delta^{1/2} & r{\rm sin}\theta\ \!{\rm
sin}\phi/\Delta^{1/2} & r{\rm cos}\theta/\Delta^{1/2} \\ 0 & r{\rm
cos}\theta\ \!{\rm cos}\phi & r{\rm cos}\theta\ \!{\rm sin}\phi &
-r{\rm sin}\theta
\\ 0 & -r{\rm sin}\theta\ \!{\rm sin}\phi & r{\rm
sin}\theta\ \!{\rm cos}\phi & 0
\\
\end{array}\right).
\end{equation}
Since the spacetime is spherically symmetric, one can, after some
algebra, simplify the Dirac equation to \cite{brill,groves}
\begin{equation}
\frac{\gamma^{0}r}{\Delta^{1/2}}\frac{\partial\Psi}{\partial
t}+\frac{\tilde{\gamma}\Delta^{1/4}}{r^{3/2}}\frac{\partial}{\partial
r}(r^{1/2}\Delta^{1/4}\Psi)
-\frac{\tilde{\gamma}}{r}(\vec{\Sigma}\cdot\vec{L}+1)\Psi+m\Psi=0,
\end{equation}
where $\tilde{\gamma}$ is defined as
\begin{equation}
\tilde{\gamma}=\gamma^{1}{\rm sin}\theta\ \!{\rm cos}\phi\
\!+\gamma^{2}{\rm sin}\theta\ \!{\rm sin}\phi+\gamma^{3}{\rm
cos}\theta.
\end{equation}
Also
\begin{equation}
\vec{\Sigma}=\left(
\begin{array}{cc}
\vec{\sigma}&0\\0&\vec{\sigma}
\end{array}\right),
\end{equation}
and $\vec{L}=\vec{r}\times\vec{p}$ are the ordinary angular
momentum operators. The wavefunction $\Psi$ can be separated into
its radial and angular parts by writing
\begin{equation}
\Psi=\frac{1}{r^{1/2}\Delta^{1/4}}e^{-iEt}\Phi,
\end{equation}
where \cite{bjorken}
\begin{equation}
\Phi(r,\theta,\phi)= \left(\begin{array}{c}
\frac{iG^{(\pm)}(r)}{r}\varphi^{(\pm)}_{jm}(\theta,\phi)\\
\frac{F^{(\pm)}(r)}{r}\varphi^{(\mp)}_{jm}(\theta,\phi)\end{array}
\right),
\end{equation}
with the angular parts of the wavefunction
\begin{equation}
\varphi^{(+)}_{jm}=\left(
\begin{array}{c}
\sqrt{\frac{l+1/2+m}{2l+1}}{Y_{l}}^{m-1/2}\\
\sqrt{\frac{l+1/2-m}{2l+1}}{Y_{l}}^{m+1/2}
\end{array}\right),
\end{equation}
for $j=l+1/2$, and
\begin{equation}
\varphi^{(-)}_{jm}=\left(
\begin{array}{c}
\sqrt{\frac{l+1/2-m}{2l+1}}{Y_{l}}^{m-1/2}\\
-\sqrt{\frac{l+1/2+m}{2l+1}}{Y_{l}}^{m+1/2}
\end{array}\right),
\end{equation}
for $j=l-1/2$.
Then the radial equations \cite{chandrasekhar,cho} can be written
as
\begin{eqnarray}
\left(-\frac{d^{2}}{dx^{2}}+V_{1}\right)\hat{F}
&=&E^{2}\hat{F},\label{V1}\\
\left(-\frac{d^{2}}{dx^{2}}+V_{2}\right)\hat{G}
&=&E^{2}\hat{G},\label{V2}
\end{eqnarray}
where
\begin{equation}
V_{1,2}=\pm\frac{dW}{dx}+W^{2},\label{V12}
\end{equation}
with
\begin{equation}
W=\frac{\Delta^{1/2}(\kappa^{2}+m^{2}r^{2})^{3/2}}{r^{2}(\kappa^{2}+m^{2}r^{2})+m\kappa
\Delta/2E}, \label{W}
\end{equation}
where
\begin{equation}
x=r+2M{\rm ln}\left(\frac{r}{2M}-1\right)+\frac{1}{2E}{\rm
tan}^{-1}\left(\frac{mr}{\kappa}\right).
\end{equation}
Here we have combined the $(+)$ and the $(-)$ cases, with $\kappa$
going over all positive and negative integers. Positive integers
represent the $(+)$ cases with
\begin{equation}
\kappa=j+\frac{1}{2}\ \ \ {\rm and}\ \ \ j=l+\frac{1}{2},
\end{equation}
and
\begin{equation}
\left(
\begin{array}{c}
\hat{F}\\ \hat{G} \end{array} \right) =\left(
\begin{array}{cc}
{\rm sin}(\theta/2)&{\rm cos}(\theta/2)\\ {\rm
cos}(\theta/2)&-{\rm sin}(\theta/2)
\end{array}\right)
\left(
\begin{array}{c}
F\\G
\end{array}
\right),
\end{equation}
where
\begin{equation}
\theta={\rm tan}^{-1}(mr/\kappa).
\end{equation}
While negative integers represent the $(-)$ cases with
\begin{equation}
\kappa=-\left(j+\frac{1}{2}\right)\ \ \ {\rm and}\ \ \
j=l-\frac{1}{2},
\end{equation}
and
\begin{equation}
\left(
\begin{array}{c}
\hat{F}\\ \hat{G} \end{array} \right) =\left(
\begin{array}{cc}
{\rm cos}(\theta/2)&-{\rm sin}(\theta/2)\\ {\rm
sin}(\theta/2)&{\rm cos}(\theta/2)
\end{array}\right)
\left(
\begin{array}{c}
F\\G
\end{array}
\right).
\end{equation}
From the Schr\"odinger-like equations in Eqs.~(\ref{V1}) and
(\ref{V2}), we shall consider the scattering of the massive Dirac
fields. Note that $V_{1}$ and $V_{2}$, which are related as shown
in Eq.~(\ref{V12}), are supersymmetric partners derived from the
same superpotential $W$ \cite{cooper}. It has been shown that
potentials related in this way possess the same spectra, discrete
as well as continuous \cite{anderson}. Physically this just
indicates that Dirac particles and antiparticles will scatter in
the same manner around the Schwarzshild black hole. We shall
therefore concentrate just on Eq.~(\ref{V1}) with potential
$V_{1}$ in the following sections.
\subsection{Properties of the effective potential}
Here we discuss briefly the dependence of the effective potential
\begin{eqnarray}
&&V(r,\kappa,m,E)\nonumber\\
&=&\frac{\Delta^{1/2}(\kappa^{2}+m^{2}r^{2})^{3/2}}
{(r^{2}(\kappa^{2}+m^{2}r^{2})+m\kappa\Delta/2E)^{2}}
\left[\Delta^{1/2}(\kappa^{2}+m^{2}r^{2})^{3/2}+((r-1)(\kappa^{2}+m^{2}r^{2})+
3m^{2}r\Delta)\right]\nonumber\\ &&\ \ \
-\frac{\Delta^{3/2}(\kappa^{2}+m^{2}r^{2})^{5/2}}{(r^2(\kappa^{2}+m^{2}r^{2})+m\kappa\Delta/2E)^{3}}
\left[2r(\kappa^{2}+m^{2}r^{2})+2m^{2}r^{3}+m\kappa(r-1)/E\right].
\label{massiveV}
\end{eqnarray}
on the parameters $m$, $\kappa$, and $E$. Since we shall only work
with $V_{1}$, but not $V_{2}$, we have dropped the subscript of
$V$.
\begin{figure}[!]
\includegraphics{fig1}
\caption{\label{potentialwithm} Variation of the effective
potential with the mass $m$ (in units of $M$) of the Dirac field
for $\kappa=1$ and $E=M$.}
\end{figure}
First, its dependence on $m$ is showed in
Fig.~\ref{potentialwithm} with energy $E=M$ and with $\kappa=1$.
Note that we have shown $V$ as a function of $x$ in the figure.
For small values of $m$, the potential is in the form of a
barrier, with the asymptotic value
\begin{equation}
V(x\rightarrow\infty)=m^2.
\end{equation}
As $m$ is increased, the peak of the potential also increases but
does so very slowly. Eventually, the height of the peak is lower
than the asymptotic value $m^2$. When $m$ is increased further,
the peak disappears altogether, and the potential barrier turns
effectively into a potential step.
The effective potential $V$ also depends on the energy $E$.
However, as shown in \cite{cho}, the general behaviors of the
potential remain the same as $E$ is increased from its minimum
value $m$. Indeed, from the form of the potential in
Eq.~(\ref{massiveV}), $E$ appears all in the denominators. The
terms involving $E$ can never get large enough to change the
general behaviors of the potential since $E$ cannot be smaller
than $m$.
\begin{figure}[!]
\includegraphics{fig2}
\caption{\label{potentialwithk} Variation of the effective
potential with $\kappa$ for the Dirac fields of $m=0.2M$ and
$0.4M$.}
\end{figure}
Finally, in Fig.~\ref{potentialwithk} we show the dependence of
the effective potential with the angular momentum quantum number
$\kappa$. We see that as $\kappa$ increases, the behaviors of the
potential approach to that of the massless one, regardless of the
mass of the field. This can be seen by taking the limit
$|\kappa|\rightarrow\infty$ in Eq.~(\ref{massiveV}),
\begin{equation}
V(|\kappa|\rightarrow\infty)\approx\frac{\Delta\kappa^{2}}{r^{4}},
\end{equation}
which is independent of $m$.
\section{WKB approximations}
In this section we consider the radial equations obtained in the
previous section as a general one-dimensional quantum mechanical
problem,
\begin{equation}
\left(-\frac{d^{2}}{dx^{2}}+V\right)\psi
=E^{2}\psi,\label{schrodinger}
\end{equation}
with the asymptotic values of the potential
\begin{equation}
V(x\rightarrow\infty)=m^{2}\ \ \ \ \ {\rm and}\ \ \ \ \
V(x\rightarrow -\infty)=0.
\end{equation}
Suppose an incident wave comes from the right ($x=\infty$). Then
the boundary conditions for $\psi$ are
\begin{eqnarray}
\psi(x\rightarrow\infty)&=&e^{-i\sqrt{E^{2}-m^{2}}\ \!
x}+Re^{i\sqrt{E^{2}-m^{2}}\ \!x}\label{bcinf}\\ \psi(x\rightarrow
-\infty)&=&Te^{-iEx},\label{bcneginf}
\end{eqnarray}
where $R$ and $T$ represent the reflection and transmission
coefficients, respectively.
Since exact solutions for Eq.~(\ref{schrodinger}) are usually hard
to find, we have to resort to approximations. Here we adopt the
WKB approximation, which has been proved to be extremely useful
and sometimes to be more accurate than expected. We try to develop
approximate expressions for the transmission probability
\begin{equation}
{\cal T}=\sqrt{\frac{E^{2}}{E^{2}-m^{2}}}\
|T|^{2},\label{transdef}
\end{equation}
for the whole range of the energy $E$, including the cases with
$E^{2}\gg V_{m}$, $E^{2}\approx V_{m}$, and $E^{2}\ll V_{m}$,
where $V_{m}$ is the maximum value of the potential. In the case
of a barrier, $V_{m}$ will be the peak value of $V$, while in the
case of a step, $V_{m}=m^{2}$.
\subsection{$E^{2}\gg V_{m}$}
For $E^{2}\gg V_{m}$, the standard WKB form of the wavefunction
can be given by
\begin{equation}
\psi(x)=C_{+}W_{+}(x)+C_{-}W_{-}(x),\label{wkb}
\end{equation}
for $-\infty<x<\infty$, with
\begin{equation}
W_{\pm}(x)=\frac{1}{\sqrt{p(x)}}e^{\pm i\int^{x}_{x_{0}}dx'p(x')}
\end{equation}
being the WKB wavefunctions. Here $p(x)=\sqrt{E^{2}-V(x)}$,
$x_{0}$ is some fixed reference point, and $C_{+}$ and $C_{-}$ are
constants to be determined from the boundary conditions.
Eq.~(\ref{wkb}) is a useful approximation for the wavefunction as
long as the validity condition
\begin{equation}
\left\vert\frac{V'}{2p^{3}}\right\vert\ll 1,\label{validity}
\end{equation}
where $V'=dV/dx$, is fulfilled.
From the boundary condition at $x\rightarrow -\infty$
(Eq.~(\ref{bcneginf})), we have $C_{+}=0$, while from the
condition at $x\rightarrow\infty$ (Eq.~(\ref{bcinf})), we have
\begin{equation}
C_{-}=(E^{2}-m^{2})^{1/4}e^{-i\sqrt{E^{2}-m^{2}}\ \!x_{0}}
e^{i\int^{\infty}_{x_{0}}dx'(p(x')-\sqrt{E^{2}-m^{2}})}.
\end{equation}
Therefore, the transmission coefficient
\begin{eqnarray}
T&=&\frac{C_{-}}{\sqrt{E}}e^{iEx_{0}}e^{i\int^{x_{0}}_{-\infty}dx'(p(x')-E)}\nonumber\\
&=&\left(\frac{E^{2}-m^{2}}{E^{2}}\right)^{1/4}e^{i(E-\sqrt{E^{2}-m^{2}})x_{0}}
e^{i\int^{\infty}_{x_{0}}dx'(p(x')-\sqrt{E^{2}-m^{2}})}e^{i\int^{x_{0}}_{-\infty}dx'
(p(x')-E)},
\end{eqnarray}
and the transmission probability in Eq.~(\ref{transdef}) becomes
\begin{equation}
{\cal T}=1.
\end{equation}
This is of course consistent with the classical result in which
the particle moves to the left without rebounding. However,
quantum mechanically ${\cal T}$ is not exactly equal to 1, there
is always a small probability for reflection. Thus to get a better
approximation on ${\cal T}$ in this quantum mechanical situation,
we need to extend the WKB scheme used here.
Suppose we write the general solution of Eq.~(\ref{schrodinger})
as \cite{berry}
\begin{equation}
\psi=C_{+}(x)W_{+}(x)+C_{-}(x)W_{-}(x),\label{general}
\end{equation}
where $C_{+}(x)$ and $C_{-}(x)$ are now functions of $x$. If we
also take
\begin{equation}
C_{+}'(x)W_{+}(x)+C_{-}'(x)W_{-}(x)=0,
\end{equation}
and together with Eq.~(\ref{general}), we can obtain the
differential equations for $C_{\pm}$,
\begin{equation}
C_{\pm}'(x)=\mp
i\left(\frac{p''(x)}{4p(x)^2}-\frac{3(p'(x))^{2}}{8p(x)^{3}}\right)\left[C_{\pm}(x)+C_{\mp}(x)e^{\mp
2i\int^{x}_{x_{0}}dx' p(x')}\right]\label{deforc}
\end{equation}
From the boundary conditions in Eqs.~(\ref{bcinf}) and
(\ref{bcneginf}), we have
\begin{eqnarray}
C_{+}(-\infty)&=&0,\\
C_{-}(\infty)&=&(E^{2}-m^{2})^{1/4}e^{-i\sqrt{E^{2}-m^{2}}\
\!x_{0}}e^{i\int^{\infty}_{x_{0}}dx'
(\sqrt{E^{2}-V(x')}-\sqrt{E^{2}-m^{2}})}.
\end{eqnarray}
Finally, with these boundary values, the differential equations in
Eq.~(\ref{deforc}) can be turned into integral equations,
\begin{eqnarray}
C_{+}(x)&=&-i\int^{x}_{-\infty}dx'
\left(\frac{p''(x')}{4p(x')^2}-\frac{3(p'(x'))^{2}}{8p(x')^{3}}\right)\left[C_{+}(x')+C_{-}(x')e^{-
2i\int^{x'}_{x_{0}}dx'' p(x'')}\right],\\
C_{-}(x)&=&C_{-}(\infty)-
i\int^{\infty}_{x'}dx'\left(\frac{p''(x')}{4p(x')^2}-\frac{3(p'(x'))^{2}}{8p(x')^{3}}\right)
\left[C_{-}(x')+C_{+}(x')e^{ 2i\int^{x'}_{x_{0}}dx''
p(x'')}\right].\nonumber\\
\end{eqnarray}
To the lowest order, we just substitute the boundary values of
$C_{\pm}$ into the right hand side of the above equations,
\begin{eqnarray}
C_{+}(x)&\approx&-iC_{-}(\infty)\int^{x}_{-\infty}dx'
\left(\frac{p''(x')}{4p(x')^2}-\frac{3(p'(x'))^{2}}{8p(x')^{3}}\right)e^{-
2i\int^{x'}_{x_{0}}dx'' p(x'')},\label{cplus}\\
C_{-}(x)&\approx&C_{-}(\infty)\left[1-
i\int^{\infty}_{x'}dx'\left(\frac{p''(x')}{4p(x')^2}
-\frac{3(p'(x'))^{2}}{8p(x')^{3}}\right)\right].\label{cminus}
\end{eqnarray}
Under this approximation, the reflection coefficient becomes
\begin{equation}
R=\frac{C_{+}(\infty)}{(E^{2}-m^{2})^{1/4}}e^{-i\sqrt{E^{2}-m^{2}}\
\!x_{0}}e^{i\int^{\infty}_{x_{0}}dx'(\sqrt{E^{2}-V(x')}-\sqrt{E^{2}-m^{2}})},
\end{equation}
with
\begin{equation}
C_{+}(\infty)=-iC_{-}(\infty)\int^{\infty}_{-\infty}dx'
\left(\frac{p''(x')}{4p(x')^2}-\frac{3(p'(x'))^{2}}{8p(x')^{3}}\right)e^{-
2i\int^{x'}_{x_{0}}dx'' p(x'')},
\end{equation}
as given in Eq.~(\ref{cplus}). The reflection probability in this
approximation becomes
\begin{eqnarray}
{\cal R}&=&|R|^{2}\nonumber\\
&=&\left\vert\int^{\infty}_{-\infty}dx'
\left(\frac{p''(x')}{4p(x')^2}-\frac{3(p'(x'))^{2}}{8p(x')^{3}}\right)e^{-
2i\int^{x'}_{x_{0}}dx'' p(x'')}\right\vert^{2},
\end{eqnarray}
while the transmission coefficient is given by
\begin{eqnarray}
{\cal T}&=&1-{\cal R}\nonumber\\
&=&1-\left\vert\int^{\infty}_{-\infty}dx'
\left(\frac{p''(x')}{4p(x')^2}-\frac{3(p'(x'))^{2}}{8p(x')^{3}}\right)e^{-
2i\int^{x'}_{x_{0}}dx'' p(x'')}\right\vert^{2}.\label{e2ggvm}
\end{eqnarray}
This constitutes our WKB approximation for $E^{2}\gg V_{m}$ as
long as the validity condition in Eq.~(\ref{validity}) is
satisfied. Note that we can also obtain, in this approximation,
the wavefunction by substituting $C_{+}(x)$ and $C_{-}(x)$ in
Eqs.~(\ref{cplus}) and (\ref{cminus}), respectively, into
Eq.~(\ref{general}).
\subsection{$E^{2}\approx V_{m}$}
In the case of a potential barrier, when the energy $E$ of the
field is decreased to such an extend that $E^{2}$ is close to the
peak of the potential, the validity condition
(Eq.~(\ref{validity})) will no longer be satisfied near $x_{m}$,
the position of the peak. This situation can be remedied by
representing the part of the potential near $x_{m}$ as a parabola,
while maintaining the WKB solutions on either side of it. Exact
solutions can be found for the parabolic potential, and then the
approximate solution for the whole range, $-\infty<x<\infty$, can
be obtained by matching the three solutions across the
intertwining regions \cite{berry,bender}.
For the part of the potential near $x_{m}$, we can write, in the
parabolic approximation,
\begin{equation}
V(x)\approx V_{m}+\frac{1}{2}V''(x_{m})(x-x_{m})^{2},
\end{equation}
and
\begin{eqnarray}
E^{2}-V(x)&\approx& (E^{2}-V_{m})+\lambda(x-x_{m})^{2}\nonumber\\
&\approx& \left\{
\begin{array}{c}
\sqrt{\lambda}(z^{2}+\xi^{2})\ \ \ \ \ E^{2}>V_{m}\\
\sqrt{\lambda}(z^{2}-\xi^{2})\ \ \ \ \ E^{2}<V_{m}
\end{array}\right.
\end{eqnarray}
where
\begin{equation}
\lambda=-V''(x_{m})/2,\ \ \ \ \ z=\lambda^{1/4}(x-x_{m}),\ \ \ \ \
{\rm and} \ \ \ \ \ \xi=\frac{|E^{2}-V_{m}|^{1/2}}{\lambda^{1/4}}.
\end{equation}
Note that for
\begin{equation}
z=\pm\xi\Rightarrow
x_{1,2}=x_{m}\pm\frac{|E^{2}-V_{m}|^{1/2}}{\sqrt{\lambda}}.
\end{equation}
When $E^2$ is smaller than $V_{m}$, $x_{1}$ and $x_{2}$ are just
the turning points with $E^2=V(x_{1,2})$. In any case for
$E^{2}\approx V_{m}$, we can divide $x$ into three regions: (I)
$x>x_{1}$, (II) $x_{1}>x>x_{2}$, and (III) $x<x_{2}$. For regions
(I) and (III), we still use the WKB form of the wavefunction,
\begin{eqnarray}
\psi_{I}(x)&=&A_{+}W_{+}(x) +A_{-}W_{-}(x),\nonumber\\
\psi_{III}(x)&=&BW_{+}(x).
\end{eqnarray}
For region II, we have, in the parabolic approximation, the
Schr\"odinger equation in Eq.~(\ref{general}) can be written as
\begin{equation}
\frac{d^{2}\psi}{dz^{2}}+(z^{2}\pm\xi^{2})\psi=0,
\end{equation}
with the general solution
\begin{equation}
\psi_{II}(z)=\alpha
D_{-\frac{1}{2}\mp\frac{i\xi^{2}}{2}}(\sqrt{2}e^{i\pi/4}z)+\beta
D_{-\frac{1}{2}\mp\frac{i\xi^{2}}{2}}(-\sqrt{2}e^{i\pi/4}z),
\end{equation}
where $D_{\nu}(t)$ are the parabolic cylinder functions.
Taking into account of the boundary conditions in
Eqs.~(\ref{bcinf}) and (\ref{bcneginf}), and matching the
wavefunctions in different regions across $x=x_{1}$ and $x_{2}$,
we can solve for the constants $A_{+}$, $A_{-}$, $\alpha$,
$\beta$, and $B$ \cite{bender,iyer}. One can then obtain
\begin{eqnarray}
{\cal T}=\left\{
\begin{array}{l}
1/(1+e^{-\pi\xi^{2}})\ \ \ \ \ E^{2}>V_{m}\\ 1/(1+e^{\pi\xi^{2}})\
\ \ \ \ \ E^{2}<V_{m}
\end{array}\right.
\end{eqnarray}
These two cases can be combined by writing
\begin{equation}
{\cal
T}=\frac{1}{1+e^{\pi(V_{m}-E^{2})/\sqrt{\lambda}}},\label{e2appvm}
\end{equation}
which is the WKB approximation we shall use for $E^{2}\approx
V_{m}$.
\subsection{$E^{2}\ll V_{m}$}
When the energy $E$ of the field is lowered further, the turning
points $x_{1}$ and $x_{2}$ will move far apart in such a way that
the parabolic approximation is no longer valid. In this case, one
can divide the $x$-axis into five regions: (I) $x>x_{1}$, (II)
$x\approx x_{1}$, (III) $x_{1}>x>x_{2}$, (IV) $x\approx x_{2}$,
and (V) $x<x_{2}$. For regions (I), (III), and (V), we still use
the standard WKB wavefunctions,
\begin{eqnarray}
\psi_{I}(x)&=&A_{+}W_{+}(x) +A_{-}W_{-}(x),\nonumber\\
\psi_{III}(x)&=&B_{+}W_{+}(x) +B_{-}W_{-}(x),\nonumber\\
\psi_{V}(x)&=&CW_{+}(x).
\end{eqnarray}
For region (II), we use the linear approximation \cite{bender},
\begin{equation}
V(x)\approx V(x_{1})+V'(x_{1})(x-x_{1}),
\end{equation}
and
\begin{equation}
E^{2}-V(x)\approx \mu_{1}^{2/3}z_{1},
\end{equation}
where
\begin{equation}
\mu_{1}=-V'(x_{1}),\ \ \ \ \ {\rm and}\ \ \ \ \
z_{1}=\mu_{1}^{1/3}(x-x_{1}).
\end{equation}
Then the Schr\"odinger equation becomes
\begin{equation}
\frac{d^{2}\psi}{dz_{1}^{2}}+z_{1}\psi=0,
\end{equation}
with the general solution
\begin{equation}
\psi_{II}(z_{1})=\alpha{\rm Ai}(-z_{1})+\beta{\rm Bi}(-z_{1}),
\end{equation}
where Ai($t$) and Bi($t$) are Airy functions. One can consider
region (IV) in a similar way to obtain
\begin{equation}
\psi_{IV}(z_{2})=\gamma{\rm Ai}(z_{2})+\delta{\rm Bi}(z_{2}),
\end{equation}
where
\begin{equation}
\mu_{2}=-V'(x_{2}),\ \ \ \ \ {\rm and}\ \ \ \ \
z_{2}=\mu_{2}^{1/3}(x-x_{2}).
\end{equation}
Again matching the boundary conditions at $x=\pm\infty$ and the
wavefunctions across $x=x_{1}$ and $x_{2}$, one can obtain the
constants $A_{+}$, $A_{-}$, $B_{+}$, $B_{-}$, $C$, $\alpha$,
$\beta$, $\gamma$, and $\delta$. The transmission probability is
then given by
\begin{equation}
{\cal
T}=e^{-2\int^{x_{1}}_{x_{2}}dx'\sqrt{V(x')-E^{2}}}.\label{e2llvm}
\end{equation}
This is our WKB approximation for the cases with $E^{2}\ll V_{m}$.
\section{Transmission Probabilities for the Dirac Field}
Using the WKB approximations outlined above we shall calculate in
this section the transmission probabilities for the Dirac field in
the radial equation (Eq.~(\ref{V1})) with the potential as given
by Eq.~(\ref{massiveV}).
\subsection{Massless cases}
First, we consider the massless cases, with the potential
\begin{equation}
V(r,\kappa)=\frac{|\kappa|\Delta^{1/2}}{r^{4}}[|\kappa|\Delta^{1/2}-(r-3)],
\label{m0V}
\end{equation}
where we have used the mass $M$ of the black hole as a unit of
mass and length to simplify the notation so that $\Delta=r(r-2)$.
To proceed we first consider the case with $\kappa=1$. Then we
have
\begin{equation}
\sqrt{V_{m}}=0.216.
\end{equation}
Hence, for $E\gg 0.216$, we can use the formula in
Eq.~(\ref{e2ggvm}) to evaluate the transmission probability. To
check the validity condition, we plot $\left\vert
V'/2p^{3}\right\vert$ for various energy $E$ in
Fig.~\ref{validityfig}. From it, we see that for $E=0.35$, the
maximum value of this quantity,
\begin{equation}
\left\vert\frac{V'}{2p^{3}}\right\vert_{max}\approx 0.1.
\end{equation}
The formula in Eq.~(\ref{e2ggvm}) can therefore be a good
approximation for energy $E$ larger than around 0.35. The
transmission probabilities ${\cal T}$ for $0.35\leq E\leq 1$
calculated from this formula is plotted in Fig.~\ref{kappa1}.
\begin{figure}[!]
\includegraphics{fig3}
\caption{\label{validityfig} Validity condition for various values
of $E$ of the massless Dirac fields with $\kappa=1$.}
\end{figure}
\begin{figure}[!]
\includegraphics{fig4}
\caption{\label{kappa1} Transmission probabilities of the massless
Dirac field with $\kappa=1$ in the various WKB approximations for
$E^{2}\gg V_{m}$ (dotted line), $E^{2}\approx V_{m}$ (solid line),
and $E^{2}\ll V_{m}$ (dashed line).}
\end{figure}
For $E$ smaller than 0.35, one can use the parabolic approximation
for the transmission probability as given in Eq.~(\ref{e2appvm}).
The transmission probabilities for $0\leq E\leq 0.35$ calculated
from this approximation is also plotted in Fig.~\ref{kappa1}. Near
the base of the barrier, the parabolic approximation is no longer
valid because the turning points are far apart. This is apparent
from the fact that the curve tends to around $0.0377$ instead of
zero as $E$ goes to zero.
When the turning points are isolated, one can use the
approximation given in Eq.~(\ref{e2llvm}) for the transmission
probability ${\cal T}$. The transmission probabilities for $0\leq
E\leq 0.216$ calculated from Eq.~(\ref{e2appvm}) are also plotted
in Fig.~\ref{kappa1}. When $E$ is close to $\sqrt{V_{m}}=0.216$,
the formula in Eq.~(\ref{e2appvm}) cannot be trusted as we can see
that ${\cal T}\rightarrow 1$ as $E\rightarrow V_{m}$ in this
approximation.
In Fig.~\ref{kappa1}, we see that the solid and dashed lines
calculated from these two approximations overlap at around
$E=0.154$. Thus for $E>0.154$, we should take the result with the
parabolic approximation (solid line) and for $E<0.154$, we should
take that of the tunneling approximation (dashed line) instead.
While in the region with $E\approx 0.154$, we can extrapolate
smoothly between these two curves. Combining with the result for
$E>0.35$, we can obtain a curve for the transmission probabilities
in the entire region, $0<E<1$. This is plotted in
Fig.~\ref{kappa1to5}.
\begin{figure}[!]
\includegraphics{fig5}
\caption{\label{kappa1to5} Transmission probabilities ${\cal T}$
of the massless Dirac field with $\kappa=1$ to $5$.}
\end{figure}
In addition to the curve for $\kappa=1$, we also plot the
transmission probabilities for $\kappa=2$, 3, 4, and 5 in
Fig.~\ref{kappa1to5}. These curves are obtained in the same
procedure as outlined above for $\kappa=1$. They are similar to
each other and shift to the right as $\kappa$ increases. The
energy at which ${\cal T}=1/2$ occurs is $\sqrt{V_{m}}$ which
increases as the peak of the barrier gets higher and higher when
$\kappa$ is increased.
\subsection{Massive cases}
The situations with nonzero $m$ are more complicated because the
potentials can change from a barrier to a step or vice versa as
the parameters $m$ and $\kappa$ are varied as shown in
Figs.~\ref{potentialwithm} and \ref{potentialwithk}. With
$\kappa=1$, the potentials are barriers for $m=0$, 0.1, and 0.2.
One can use the relevant approximations for different values of
the energy $E$, along the same lines as in the massless cases in
the last subsection. The results are plotted in
Fig.~\ref{massive}. However, for $m=0.3$, 0.4, and 0.5, the
potentials are steps. The only approximation one needs is for
$E\gg m$. We also check the validity condition and the result
indicates that the approximation is useful all the way down to
energy value very close to $m$. The results for these masses are
also plotted in Fig.~\ref{massive}. For the potential steps, the
transmission probabilities are almost equal to 1, which are
consistent with the classical results.
\begin{figure}[!]
\includegraphics{fig6}
\caption{\label{massive} Transmission probabilities ${\cal T}$ of
the Dirac field with $\kappa=1$ and $m=0$ to $0.5$.}
\end{figure}
\begin{figure}[!]
\includegraphics{fig7}
\caption{\label{logTvsm} Variation of the logarithmic of the
transmission probabilities of the Dirac field ($E=1$ and
$\kappa=1$) with $m$.}
\end{figure}
We also see from the various diagrams in Fig.~\ref{massive} that
the variations of the transmission probabilities ${\cal T}$ with
$m$ are numerically very small. In order to see the changes in
more details, we plotted ${\cal T}$ versus $m$ in
Fig.~\ref{logTvsm} for $E=1$ and $\kappa=1$. Since the
transmission probabilities in these cases are very close to 1, we
take the logarithmic of ${\cal T}$ in the plot. From this figure
we see that ${\cal T}$ first decreases from $m=0$, attends a
minimum around $m=0.14$, and then increases as $m$ is further
increased.
The variations of ${\cal T}$ between $m=0.3$ to $0.5$ are quite
unexpected. This is because with these values of $m$, the
potentials are in the form of steps with larger and larger step
height when $m$ is increased. For simple potential steps, it is
known that the transmission probabilities decrease as the steps
get higher and higher for fixed energy. We thus see that for the
black hole effective potentials, the transmission probabilities
are not determined only by the height of the potential but also
the overall shape of it.
\begin{figure}[!]
\includegraphics{fig8}
\caption{\label{TvsmsmallE} Variation of the transmission
probabilities of the Dirac field with $m$ for $E=0.1$ and $0.2$.}
\end{figure}
In Fig.~\ref{logTvsm}, $E=1$ and is much larger than
$\sqrt{V_{m}}$. For energy values closer to $\sqrt{V_{m}}$, the
variations can be quite different. As shown in
Fig.~\ref{TvsmsmallE}, for $E=0.2$, that is, when $E^{2}$ is near
the peak of the potential, ${\cal T}$ increases first as $m$ is
increased from $0$, attends a maximum value around $m=0.09$, and
then decreases. The same trend is found for $E=0.1$ where it is
well below the peak of the potential. However, the maximum value
in this case is at around $m=0.05$.
As we can see from the above discussions, the transmission
probability ${\cal T}$ in general increases with the mass $m$ when
$m$ is large enough that the effective potential is in the form of
a step. However, when $m$ is smaller and the effective potential
is in the form of a barrier, the variations of ${\cal T}$ with $m$
can be quite complicated. When $E$ is large and well above the
peak, ${\cal T}$ first decreases and then increases when $m$ is
increased from $m=0$. When $E$ is small with its value near or
well below the peak of the potential, the variation is reversed,
that is, ${\cal T}$ first increases and then decreases when $m$ is
increased from $m=0$.
\section{Conclusions and Discussions}
We study the radial equations of the massive Dirac field in the
spherically symmetric Schwarzschild spacetime. Using the WKB
approximations and the appropriate connection formulae, we are
able to give semianalytic formulae for the transmission
probability ${\cal T}$ of the radial wavefunction with the energy
$E^{2}\gg V_{m}$, $E^{2}\approx V_{m}$, and $E^{2}\ll V_{m}$,
$V_{m}$ is the maximum value of the effective potential. For the
massless cases, we find that, as shown in Fig.~\ref{kappa1to5},
the variations of ${\cal T}$ with the energy $E$ and the angular
momentum number $\kappa$ are similar to that for the scalar case
\cite{futterman,sanchez}. Since the potentials are in the form of
barriers and the heights of the peaks of these barriers increase
with $\kappa$, ${\cal T}$ for fixed $E$ will thus decrease as
expected. This means that waves with lower angular momenta but
with fixed energy will be absorbed more easily by the black hole.
The massive cases are more complicated. The effective potentials
in these cases change from barriers to steps when the mass of the
field is increased. When the potential is still in the form of a
step and the energy $E$ of the field is well above the maximum
value of the potential, the transmission probability ${\cal T}$
decreases with $m$. Thus higher mass fields will get absorbed by
the black hole more easily. However, when $m$ is decreased to such
an extend that the potentials become barriers, the variation trend
changes as shown in Fig.~\ref{logTvsm}. At some value of $m$
(around 0.14 for $E=1$ and $\kappa=1$), ${\cal T}$ turns around
and increases when $m$ is further decreased to $0$. Therefore, we
see that the variations of ${\cal T}$ with $m$ is complicated when
the potentials are in the form of barriers. This is also true when
$E$ is smaller with its value near or well below the peak of the
potential as shown in Fig.~\ref{TvsmsmallE}.
After calculating the transmission probabilities for the radial
wavefunctions, one can evaluate the corresponding phase shifts and
cross sections in various scattering situations \cite{futterman}.
In the semiclassical limit one can also study the interesting
phenomena such as black hole glories \cite{matzner}, orbiting and
spiraling scatterings \cite{anninos} by deriving the semiclassical
deflection function \cite{ford}. We hope to further investigate
these issues in our future publications.
Quite peculiarly for black holes, one can also study the
absorption cross sections for various wave fields. There has been
quite a lot of interest in these cross sections in relation to the
higher-dimensional black holes in string theory, especially the
low-energy absorption cross sections \cite{das}. However, the WKB
approximations that we use in this work is not adequate at low
energy, that is, when $E\approx m$. One can nevertheless improve
the WKB approximations in these threshold situations when $E$ is
near the top of the step or when it is near the base of the
barrier \cite{eltschka,moritz}. This improved approximations may
therefore provide an alternative to the usual method \cite{unruh}
in obtaining these low-energy absorption cross sections.
\begin{acknowledgments}
This work is supported by the National Science Council of the
Republic of China under contract number NSC 91-2112-M-032-011.
\end{acknowledgments}
|
{
"timestamp": "2006-01-03T15:19:03",
"yymm": "0411",
"arxiv_id": "gr-qc/0411090",
"language": "en",
"url": "https://arxiv.org/abs/gr-qc/0411090"
}
|
\section{Introduction}
\label{introduction}
The best way to simultaneously measure $(\theta_{13},\delta)$ is the (golden)
$\ensuremath{\nu_{e}}\ \!\!\rightarrow \ensuremath{\nu_{\mu}}\ $ appearance channel \cite{Cervera:2000kp} (and its T and CP
conjugate ones). Unfortunately this measure is severely affected by the presence
of an eightfold degeneracy \cite{degeneracies}.
Various methods have been considered to get rid of degeneracies: spectral
analysis, combination of experiments and/or different channels. In principle,
the eightfold degeneracy can be completely solved if a sufficient large set of
independent informations is added. At the cost, of course, of increasing the
number of detectors and/or beams and, consequently, budget needs.
We try to understand here if the effect of degeneracies can be reduced adding
informations from both the appearance and the disappearance channels at a single
experiment. We consider as a reference two proposed CERN-based facilities: the
standard-$\gamma$ $\beta$-Beam\ \cite{zucchelli} and the 4 MWatt SPL Super-Beam\ \cite{jjsb}.
Both neutrino beams are directed from CERN toward a 1 Mton water Cerenkov detector
placed in the underground Fr\'ejus laboratory. The considered baseline is $L=130$ km.
The average neutrino energy for both beams is $\approx 250$ MeV.
The eightfold degeneracy for these two facilities has been comprehensively studied
in \cite{Donini:2004hu,Donini:2004xx} and we refer to those papers for all the
technical details and a complete set of bibliographic references.
\section{$\beta$-Beam Appearance and Disappearance Channels}
The considered $\beta$-Beam\ setup consists of a $\bar \nu_e$-beam produced by the decay of
$^6$He ions boosted at $\gamma = 60$ and of a $\nu_e$-beam produced in the decay of
$^{18}$Ne ions boosted at $\gamma = 100$. The average neutrino energies of the
$\nu_e,\bar \nu_e$ beams are 0.37 GeV and 0.23 GeV, respectively.
The measurement of $(\theta_{13},\delta)$ at this facility has been already actively
discussed in the literature \cite{allBB,Donini:2004hu}. In Fig.~\ref{fig:appBB} we plot
our results for $\theta_{13} = 8^\circ$ and two different CP phases: $\delta = 45^\circ$
and $- 90^\circ$. The input value used in the fit is always shown as a filled
black box. Throughout the paper we are using the following reference values for the
atmospheric and solar parameters: $\Delta m^2_{atm}=2.5 \times 10^{-3}$ eV$^2$,
$\theta_{23}=40^\circ$, $\theta_{12}=33^\circ$ and $\Delta m^2_{sol}=8.2 \times 10^{-5}$
eV$^2$. The $90$\% CL contours for each of the degenerate solutions are depicted in the
plot assuming a $5$\% systematic error and are fully explained in the caption.
\begin{figure}[t!]
\epsfig{file=FIG/BBapp_comb28.eps,width=7.5cm,angle=0}
\caption{\it
$90$\% CL contours in the ($\theta_{13},\delta$) plane using the appearance channel
after a $10$ years run at the considered $\beta$-Beam\ for $\theta_{13}= 2^\circ,8^\circ$, and
$\delta= 45^\circ, -90^\circ$. A $5$\% systematic error is assumed. Continuous, dotted,
dashed and dot-dashed lines stand respectively for the intrinsic, sign, octant and mixed
degeneracy.}
\label{fig:appBB}
\end{figure}
In Fig.~\ref{fig:appBB} it can be seen the dramatic impact that degeneracies have
in the precision of the measure of $(\theta_{13},\delta)$:
(1) the error in the $\theta_{13}$ measurement is increased by a factor four (two)
for large (small) values of $\theta_{13}$\cite{Donini:2003vz}. The presence of
degeneracies has a small impact on the ultimate $\theta_{13}$ sensitivity;
(2) the error in the $\delta$ measurement grows in a significant way in presence of the
clones, almost spanning half of the parameter space for small values of $\theta_{13}$.
These facts are well understood: being the standard $\beta$-Beam\ a (short distance) counting
experiment there are not enough independent informations to cancel any of the degeneracies.
At the $\beta$-Beam\ the \ensuremath{\nu_{e}}\ disappearance channel is also available. The \ensuremath{\nu_{e}}\ disappearance
probability does not depend on the CP violating phase $\delta$ and the atmospheric
mixing angle $\theta_{23}$. Thus, the $\theta_{13}$ measurement, in this channel, is
not affected by the intrinsic, octant and mixed ambiguities.
\begin{figure}[t]
\epsfig{file=FIG/BBdis8_0-2-5.eps,width=7.5cm,angle=0}
\caption{\it
$90$\% CL contours in the ($\theta_{13},\Delta m^2_{atm}$) plane using the disappearance
channel after a $10$ years run at the considered $\beta$-Beam\ for $\theta_{13}= 8^\circ$.
Three different values of the systematic errors have been considered: $0$\%, $2$\% and
$5$\%. Continuous (dotted) lines stand for the true solution (sign degeneracy).}
\label{fig:disBB}
\end{figure}
In Fig.~\ref{fig:disBB} we present the \ensuremath{\nu_{e}}\ disappearance measure for three different
systematic errors, namely $0$\% (``theoretical-unrealistic'' scenario), $2$\%
(``optimistic'' scenario) and $5$\% (``pessimistic'' scenario). The $0$\% systematic
line represent the ultimate (error free) reach of this experiment. The $2$\% and $5$\%
lines will cover the optimistic and pessimistic feelings about future improvements in
understanding a Mton water detector. The 90\% CL contours in the $(\theta_{13},
\Delta m^2_{atm})$ plane are shown for the input values $\theta_{13}=8^\circ$ and
$\Delta m^2_{atm}=2.5 \times 10^{-3}$ eV$^2$. The \ensuremath{\nu_{e}}\ disappearance channel is only
slightly sensitive to the sign clone. In fact, the dependence on the sign of the
atmospheric mass difference arises only at ${\cal O}(\theta_{13}^2)$. As a consequence
the \ensuremath{\nu_{e}}\ disappearance channel is an almost ``clone-free'' environment for the $\beta$-Beam\ ,
as it is for reactors experiments. However, even in the case of an optimistic 2\%
systematic error, no improvement is obtained adding the disappearance channel
informations to the results of Fig.~\ref{fig:appBB} for the appearance channel.
The resulting 90\% CL contours practically coincide with the previous ones, and for
this reason we do not consider to present them in a separate figure. The $\theta_{13}$
indetermination coming from the clone presence in the appearance channel is smaller
than the disappearance error itself. Only considering an unrealistic 0\%
systematics the disappearance channel starts to be useful in eliminating clones.
\section{Super-Beam Appearance and Disappearance Channels}
\begin{figure}[t]
\epsfig{file=FIG/SBapp_comb28.eps,width=7.5cm,angle=0}
\caption{\it
$90$\% CL contours using the appearance channel after a $2+8$ years run at the
Super-Beam\ for two different values of $\theta_{13} = 8^\circ$, and two values of $\delta$:
$45^\circ$ and $-90^\circ$. A $5$\% systematic error is assumed.
The legend is the same as in Fig.~\ref{fig:appBB}.}
\label{fig:appSB}
\end{figure}
\begin{figure}[t!]
\epsfig{file=FIG/SBdis2_80_40.eps,width=7.5cm,angle=0}
\caption{\it
$90$\% CL contours in the $(\theta_{23},\Delta m^2_{atm})$ plane using the disappearance
channel after a $2+8$ years run at the Super-Beam\ for $\theta_{23} = 40^\circ$. A $2$\% systematic
error is assumed. Continuous (dotted) lines stand for true (wrong) atmospheric mass sign
assignment.}
\label{fig:disSB}
\end{figure}
The considered Super-Beam\ setup is a conventional neutrino beam based on the 4 MWatt
CERN SPL 2.2 GeV proton driver \cite{jjsb}. The average neutrino energies of the
$\nu_\mu$, $\bar \nu_\mu$ beams are 0.27 GeV and 0.25 GeV, respectively.
The possibility to measure $(\theta_{13},\delta)$ at a standard Super-Beam facility
has been already widely discussed \cite{allSB,Donini:2004hu}.
In Fig.~\ref{fig:appSB} we plot our results for $\theta_{13}=8^\circ$ and two different
CP phases: $\delta = 45^\circ$ and $-90^\circ$. The input value used in the fit is
shown as a filled black box. The $90$\% CL contours for each of the degenerate
solutions are depicted in the plot assuming a $5$\% systematic error and are
explained in the caption.
As it appears from comparison of Fig. \ref{fig:appSB} with Fig. \ref{fig:appBB},
the ``figures of merit'' of a standard $\beta$-Beam and the SPL Super-Beam are
very similar. Also the Super-Beam appearance channel is severely affected by
proliferation of clones. The precision in measuring $(\theta_{13},\delta)$ is
practically identical in the two cases. This is well explained by the comparable
statistics in the golden channel ($\ensuremath{\nu_{e}}\ \!\!\rightarrow \ensuremath{\nu_{\mu}}\ \!\!$ vs $\ensuremath{\nu_{\mu}}\ \!\!\rightarrow
\ensuremath{\nu_{e}}\ \!\!$) and an almost equal $L/E$ ratio for the two experiments. There is no
real synergy between this two setups and the only effect in summing these two
experiments (concerning the $(\theta_{13},\delta)$ measure) is to double the
statistics.
\begin{figure}[t!]
\epsfig{file=FIG/SBtot2_comb28.eps,width=7.5cm,angle=0}
\caption{\it
$90$\% CL contours using the appearance and the disappearance channels after a $2+8$ years
run at the Super-Beam\ , for $\theta_{13} = 2^\circ, 8^\circ$ and $\delta=45^\circ, -90^\circ$.
A $5$\% ($2$\%) systematic error is assumed for the appearance (disappearance) channel.
The legend is the same as in Fig.~\ref{fig:appBB}.}
\label{fig:totSB}
\end{figure}
Nevertheless, the great advantage of the Super-Beam\ facility compared with the $\beta$-Beam\ one
is the possibility to measure directly the atmospheric parameters using the
$\ensuremath{\nu_{\mu}}\ \!\!$ disappearance channel reducing, in particular, the atmospheric mass
difference error to less than 10\%. In Fig. \ref{fig:disSB} we show the measure
of $(\theta_{23},\Delta m^2_{atm})$ at the SPL Super-Beam\ with a 2\% systematic error
for a non-maximal atmospheric mixing, $\theta_{23} = 40^\circ$ and $\Delta m^2_{atm}
= 2.5 \times 10^{-3}$ eV$^2$. The continuous (dotted) contour represents the fit
to the right (wrong) choice of the sign of the atmospheric mass difference. Since
we plot the results in the full $\theta_{23} \in [35^\circ-55^\circ]$ parameter
space, the octant and mixed clones are automatically taken into account and do
not appear as separate regions (within the considered errors). As one can notice
the sign ambiguity implies that the errors on the atmospheric mass difference are
roughly doubled with respect to what expected in the absence of degeneracies.
The presence of degeneracies in the \ensuremath{\nu_{\mu}}\ disappearance channel can be easily
understood looking at the the $\ensuremath{\nu_{\mu}}\ \!\! \rightarrow \ensuremath{\nu_{\mu}}\ \!\!$ vacuum oscillation
probability expanded to the second order $\theta_{13}$ and $(\Delta m^2_{sol} L/E)$
(see for example \cite{Akhmedov:2004ny,Donini:2004xx}).
In Fig.~\ref{fig:totSB} we present the simultaneous measurement of $(\theta_{13},\delta)$
using both the appearance (with a 5\% systematic error) and the disappearance (with a
2\% systematic error) channels at the Super-Beam. Contrary to the $\beta$-Beam case,
the disappearance Super-Beam\ channel introduces in the fit significant changes. Notably enough,
the sign clone has disappeared in any case considered. This is not a surprise as these
fits are performed at a fixed $\Delta m^2_{atm}$: since in the disappearance channel
the sign clone manifests itself at a larger value of $\Delta m^2_{atm}$ (see
Fig.~\ref{fig:disSB}), in the combination with the appearance channel the tension
between the two suffices to remove the unwanted clone in the $(\theta_{13},\delta)$
plane. Notice, moreover, that in some cases the octant clone is considerably reduced
or even solved, due to the octant-asymmetric contributions in the \ensuremath{\nu_{\mu}}\ disappearance
probability. Nonetheless, this does not mean that thanks to the combination of the
appearance and the disappearance channels we are indeed able to measure the sign of
the atmospheric mass difference. The mixed clones are generally still present for
large values of $\theta_{13}$, thus preventing us from measuring the atmospheric mass
difference sign if the $\theta_{23}$-octant is not known at the time the experiment
takes place. It is clear that these results should be confirmed by a complete
multi-dimensional analysis is underway.
\section{Conclusions}
We have tried to understand the impact of disappearance measurements on the
$(\theta_{13},\delta$) eightfold degeneracy for the standard-$\gamma$ $\beta$-Beam\ and the
SPL Super-Beam. We presented a complete analysis of degenerations in the \ensuremath{\nu_{e}}\ and
\ensuremath{\nu_{\mu}}\ disappearance channels: the \ensuremath{\nu_{e}}\ disappearance is affected only by a twofold
degeneracy, since the \ensuremath{\nu_{e}}\ probability does not depend on $\delta$ and $\theta_{23}$.
The \ensuremath{\nu_{\mu}}\ disappearance is affected by a fourfold-degeneracy. The inclusion of
degeneracies almost double the error on the measure of $\Delta m^2_{atm}$.
The standard-$\gamma$ $\beta$-Beam setup looks somewhat limited, being the neutrino
energy too low for using energy resolution techniques and the combination with the
\ensuremath{\nu_{e}}\ disappearance, potentially of interest, is in practice useless once a realistic
systematic error is taken into account. The $\beta$-Beam\ idea should be certainly pursued
further, but using for example higher $\gamma$ options, where should be possible to
take advantage of energy resolution and, possibly, the silver channel.
The SPL Super-Beam\ appearance channel is also severely affected by degeneracies. However, in
this case the complementarity between the appearance and disappearance channels can be
fully exploited even when a realistic systematic error is taken into account. In
particular, the sign ambiguity can be strongly reduced as the disappearance sign
clone is located at a different $\Delta m^2_{atm}$.
|
{
"timestamp": "2004-11-30T18:38:03",
"yymm": "0411",
"arxiv_id": "hep-ph/0411403",
"language": "en",
"url": "https://arxiv.org/abs/hep-ph/0411403"
}
|
\section{Introduction} \label{sec_intro}
\noindent
Consider a smooth, projective,
geometrically irreducible curve $X$ over $\F_q$, and fix some point
$\infty$ on this curve. Let $$A:= \Gamma(X - \infty, \O_X)$$ be
the ring of functions on $X$ which are regular outside
$\infty$. Let $f \in A\bs \F_q$ be a non-constant element, and let
$$A_f := A[f^{-1}].$$ The moduli
scheme $M^2(f)$ plays an important role in this paper. It represents
the functor which associates to every $A_f$-scheme $S$ the set of
isomorphy classes of Drinfeld modules with a level $f$-structure over
$S$.\\ In this paper we will address the following problems:
\begin{itemize}
\item[$(i)$] Construct a morphism $$w_f: M^r(f) \longrightarrow} \def\ll{\longleftarrow M^1(f).$$ This morphism
is induced by the Weil pairing for Drinfeld modules. The Weil
pairing is defined in \cite{Hei02}.
\item[$(ii)$] Define the Tate-Drinfeld module and describe its universal
property, using ideas of G. B\"ockle in Chapter $2$ of \cite{Boe02}
and ideas of M. van der Put and J. Top in \cite{PT1} and \cite{PT2}.
\item[$(iii)$] Describe a compactification $\ov M^2(f)$ of $M^2(f)$.
The Tate-Drinfeld module will enable us to describe the scheme of cusps
$${\it Cusps} = (\ov M^2(f) - M^2(f))^{\rm red}.$$
\item[$(iv)$] Compute the number of components of $M^2(f)$
and describe the cusps of the analogue of the classical curve $X_0(N)$.
\end{itemize}
\noindent
We would like to point out that another description of the compactification
of $M^2(f)$ can be found in T. Lehmkuhl's `Habilitation' \cite{Leh00}.
Our treatment uses the Weil pairing and gives an explicit description of
the Tate-Drinfeld module. In a forthcoming
paper this enables us to develop the N\'eron model of the Tate-Drinfeld
module, analogous to the Deligne and Rapoport's construction of the N\'eron
model of the Tate-elliptic curve in \cite{DR73}. This will probably give
rise to an extension of the functor represented by $M^r(f)$ to a functor
represented by the compactification $\ov M^r(f)$.
\par\bigskip\noindent
Sections \ref{sec_defs} and \ref{sec_moduli} give a brief introduction
to the moduli problem and to the moduli schemes. At the end of Section
\ref{sec_moduli} we state the assumptions which are used throughout
this paper. In Section \ref{sec_defs} we recall the definition of
Drinfeld modules over schemes and level structures. In Section
\ref{sec_moduli} we describe the moduli problem that Drinfeld
considers in his original paper \cite{Drin74}. The goal of Section
\ref{sec_weil-map} is to prove Theorem \ref{thm_map}, i.e., to
construct the morphism $w_f$ considered in problem $(i)$. In Sections
\ref{sec_redth}, \ref{sec_TD} and \ref{sec_UP} we discuss problem
$(ii)$. In Section \ref{sec_redth} we classify the Drinfeld modules of
rank $2$ with level $f$-structure over the quotient field of some
complete discrete valuation ring $V$ which have stable reduction of
rank $1$. The main result of this section is Theorem \ref{thm_reduct}.
In Section \ref{sec_TD} we define the Tate-Drinfeld module of type
$\mf m$ with level $f$-structure. In Section \ref{sec_UP} we define
the universal Tate-Drinfeld module $\Cal Z$; cf. Theorem
\ref{thm_STD_up}. For the application to problem $(iii)$ we need a
weak version of the universal property of the scheme $\Cal Z$ as
stated in Theorem \ref{thm_univTD}. In Sections \ref{sec_compact} and
\ref{sec_Cusp_TD} we study problem $(iii)$. In Section
\ref{sec_compact} we give a compactification of $M^2(f)$. The
treatment given here is analogously to the compactification of the
classical modular curve given by N.M. Katz and B. Mazur in Chapter 8
of their book \cite{KM85}. This enables us in Section
\ref{sec_Cusp_TD} to identify the formal neighbourhood of the scheme
of cusps $\ov M^2(f) - M^2(f)$ with the universal Tate-Drinfeld
module. The main results are Theorem \ref{thm_CompToTD} and
Corollary \ref{cor_cusp}. In Section \ref{sec_compon} we compute, using
the scheme of cusps, the number of components for every geometric fibre;
cf. Theorem \ref{thm_geom_comp}.
\section{Drinfeld modules over schemes}
\label{sec_defs}
\noindent
Throughout this paper we will denote the quotient field of any integral
domain $D$ by $K_D$. We recall the definition of Drinfeld modules over
schemes and level structures. There are many texts available for a more
extensive account of these definitions; cf. \cite{Drin74}, \cite{Del87},
\cite{Sai96}, \cite{Leh00} and \cite{Tae02}.
\subsection{Line bundles and morphisms}
\noindent
Let $B$ be a commutative $\F_q$-algebra with $1$ and let ${\mathbb{G}}} \def\O{{\mathcal O}} \def\Q{{\mathbb{Q}}_{a, B}$
denote the additive group over $B$. The ring of
$\F_q$-linear endomorphisms $\operatorname{End}_{\F_q}({\mathbb{G}}} \def\O{{\mathcal O}} \def\Q{{\mathbb{Q}}_{a, B})$ of ${\mathbb{G}}} \def\O{{\mathcal O}} \def\Q{{\mathbb{Q}}_{a, B}$ is
isomorphic to the skew polynomial ring $B\{\t\}$. In this skew polynomial
ring multiplication is determined by the rule $\t b = b^q \t$ for all
$b\in B$.
\par\bigskip\noindent
This can be generalized to schemes. Let $S$ be an $\F_q$-scheme, and let
$L \longrightarrow} \def\ll{\longleftarrow S$ be a line bundle. As usual, $L$ is also a group scheme
due to its additive group scheme structure.
A {\em trivialization of $L$} is a covering ${\rm Spec}(B_i)$ of open affines
of $S$ together with isomorphisms $L_{\mid {\rm Spec}(B_i)} \cong
{\mathbb{G}}} \def\O{{\mathcal O}} \def\Q{{\mathbb{Q}}_{a, B_i}$. By $\operatorname{End}_{\F_q}(L)$ we denote the
$\F_q$-linear $S$-group scheme endomorphisms of $L$. Let $\Cal L$ be
the invertible $\O_S$-sheaf corresponding to $L$, and let
$$\t^i: \Cal L \longrightarrow} \def\ll{\longleftarrow \Cal L^{q^i} \quad \mbox{by} \quad s
\mapsto s\otimes \ldots \otimes s.$$ The ring $\operatorname{End}_{\F_q}(L)$ is
isomorphic to the ring of all formal expressions $\sum_i \alpha_i \t^i$
which are locally finite. Here $\alpha_i: \Cal L^{q^i} \longrightarrow} \def\ll{\longleftarrow \Cal L$ is
an $\O_S$-module homomorphism for every $i$. Multiplication in the ring of
formal expressions is given by $\alpha_i\t^i \beta_j \t^j = \alpha_i \otimes
\beta_j^{q^i} \t^{i+j}$. If $\{ {\rm Spec}(B_i) \}_{i\in I}$ is a
trivialization of $L$, then the restriction of $\operatorname{End}_{\F_q}(L)$ to
$B_i$ is simply $B_i\{ \t \}$.\\
Furthermore, we denote by $\partial} \def\s{\sigma} \def\t{\tau_0$ the point derivation at $0$:
$$\partial} \def\s{\sigma} \def\t{\tau_0: \operatorname{End}_{\F_q}(L) \longrightarrow} \def\ll{\longleftarrow \Gamma (S, \Cal O_S) \quad \mbox{by} \quad
\sum_i \alpha_i \t^i \mapsto \alpha_0.$$
\subsection{Drinfeld modules over a scheme}
\begin{definition} \label{def_DMField}
Let $K$ be an $A$-field equipped with an $A$-algebra structure
given by $\g: A\longrightarrow} \def\ll{\longleftarrow K$. Let $L = {\mathbb{G}}} \def\O{{\mathcal O}} \def\Q{{\mathbb{Q}}_{a, K}$.
A {\em Drinfeld module over $K$} is a ring homomorphism $$\varphi} \def\g{\gamma:
A \longrightarrow} \def\ll{\longleftarrow \operatorname{End}_{\F_q}({\mathbb{G}}} \def\O{{\mathcal O}} \def\Q{{\mathbb{Q}}_{a, K})$$ such that
\begin{itemize}
\item[$(i)$] $\partial} \def\s{\sigma} \def\t{\tau_0 \circ \varphi} \def\g{\gamma = \g$;
\item[$(ii)$] there is an $a \in A$ such that $\varphi} \def\g{\gamma_a \neq \g(a)$.
\end{itemize}
\end{definition}
\noindent
A Drinfeld module over a field $K$ has a {\it rank}, i.e., there is an
integer $r > 1$ such that $\deg_{\t} \varphi} \def\g{\gamma_a = r \deg(a)$ for all $a
\in A$.
\begin{definition}
Let $S$ be an $A$-scheme via the morphism $$\g_S: S \longrightarrow} \def\ll{\longleftarrow {\rm Spec}(A).$$
A {\em Drinfeld module of rank $r$ over $S$} is a pair $(L, \varphi} \def\g{\gamma)_S$
of a line bundle $L\longrightarrow} \def\ll{\longleftarrow S$ and a ring homomorphism $\varphi} \def\g{\gamma: A \longrightarrow} \def\ll{\longleftarrow
\operatorname{End}_{\F_q}(L)$ such that
\begin{itemize}
\item[$(i)$] $\partial} \def\s{\sigma} \def\t{\tau_0 \circ \varphi} \def\g{\gamma = \g_S^{\#}$;
\item[$(ii)$] For all $a \in A$ the morphism $\varphi} \def\g{\gamma_a$ is finite
of degree $q^{r \deg(a)}$.
\end{itemize}
\end{definition}
\begin{remark}
The pull-back of a Drinfeld module $(L, \varphi} \def\g{\gamma, S)$ along a morphism
$${\rm Spec}(K) \longrightarrow} \def\ll{\longleftarrow S$$ for some field $K$ is a Drinfeld module over
$K$ in the sense of Definition \ref{def_DMField}.
\end{remark}
\noindent
If $S = {\rm Spec}(B)$ and $L$ is isomorphic to ${\mathbb{G}}} \def\O{{\mathcal O}} \def\Q{{\mathbb{Q}}_{a, B}$, then we simply
write $\varphi} \def\g{\gamma$ instead of $(L, \varphi} \def\g{\gamma)_S$. The morphism $\g_S$ is called the
{\it characteristic} of $(L, \varphi} \def\g{\gamma)_S$. An ideal $\mf n \subset A$ is called
{\it away from the characteristic} if $V(\mf n)$ is disjoint with the
image of $\g_S$.
\begin{definition}
A {\em morphism $\xi$ of Drinfeld modules over $S$} $$\xi: (L,
\varphi} \def\g{\gamma)_S \longrightarrow} \def\ll{\longleftarrow (M, \psi)_S$$ is a map $\xi \in \operatorname{Hom}_{\F_q}(L, M)$ such
that $\xi \circ \varphi} \def\g{\gamma_a = \psi_a \circ \xi$ for all $a \in A$. A
morphism $\xi$ is called an {\it isomorphism}, if it gives an isomorphism
between the line bundles $L$ and $M$ over $S$.\\
An {\it isogeny} of Drinfeld modules is a finite morphism.
\end{definition}
\begin{remark}
An isogeny exists only between Drinfeld modules of the same rank.
\end{remark}
\begin{remark}
A Drinfeld module $(L, \varphi} \def\g{\gamma)_S$ of rank $r$ and a morphism $$f: T
\longrightarrow} \def\ll{\longleftarrow S$$ give by pull-back rise to a Drinfeld module $(f^* L, f^*
\varphi} \def\g{\gamma)_T$ over $T$ of rank $r$.
\end{remark}
\subsection{Level structures}
\noindent
For any non-zero $f \in A$, let $$\varphi} \def\g{\gamma [f] := \ker(\varphi} \def\g{\gamma_f: L \longrightarrow} \def\ll{\longleftarrow L).$$ This is a
finite, flat group scheme over $S$, namely $$\ker(\varphi} \def\g{\gamma_f: L \longrightarrow} \def\ll{\longleftarrow L) = L
\times_L S$$ where the fibre product is taken over $\varphi} \def\g{\gamma_f: L \longrightarrow} \def\ll{\longleftarrow L$ and
the unit-section $e: S \longrightarrow} \def\ll{\longleftarrow L$ of the group scheme $L$. If $\mf n \subset
A$ is any non-zero ideal, then $$\varphi} \def\g{\gamma[\mf n] = \prod_{f \in \mf n} \varphi} \def\g{\gamma[f]$$
where the product is the fibre product over $L$. The scheme $\varphi} \def\g{\gamma[\mf n]$ is
\'etale over $S$ if and only if $\mf n$ is away from the characteristic.
\begin{remark}
Let $K$ be an algebraically closed $A$-field.
If $S = {\rm Spec}(K)$, then
$$\varphi} \def\g{\gamma[\mf n] = {\rm Spec}(K[X]/(\varphi} \def\g{\gamma_{f}(X), \varphi} \def\g{\gamma_g(X)))$$ with $\mf n =
(f, g)$.
\end{remark}
\begin{definition} \label{def_level}
Suppose that $\mf n$ is away from the characteristic, then a level
$\mf n$-structure $\lambda$ over $S$ of $(L, \varphi} \def\g{\gamma)_S$
is an $A$-isomorphism $$\lambda : (A/\mf n)^r
\stackrel{\sim}{\longrightarrow} \def\ll{\longleftarrow} \varphi} \def\g{\gamma[\mf n](S).$$
\end{definition}
\begin{remark}
If $\mf n$ is not away from the characteristic, then
the definition of a level $\mf n$-structure is more involved.
One can view $\varphi} \def\g{\gamma[\mf n]$ as an
effective Cartier divisor on $L$. By definition a level
$\mf n$-structure of $(L, \varphi} \def\g{\gamma)_S$ is an $A$-homomorphism $$\lambda: (A/\mf
n)^r \longrightarrow} \def\ll{\longleftarrow L(S)$$ which induces an equality of Cartier-divisors:
$$\sum_{\alpha \in (A/\mf n)^r} \lambda(\alpha) = \varphi} \def\g{\gamma[\mf n].$$ We
will not expand on this. In this paper we restrict to the cases for
which Definition \ref{def_level} is enough.
\end{remark}
\noindent
Let the triple $(L, \varphi} \def\g{\gamma, \lambda)_S$ denote a Drinfeld module of rank
$r$ over $S$ with level $\mf n$-structure $\lambda$.
\begin{definition}
A {\em morphism} between two triples $(L, \varphi} \def\g{\gamma, \lambda)_S$ and $(M,
\psi, \mu)_S$ is a morphism $\xi: (L, \varphi} \def\g{\gamma)_S \longrightarrow} \def\ll{\longleftarrow (M, \psi)_S$ of
Drinfeld modules over $S$ such that $\xi(S) \circ \lambda = \mu$
where $\xi(S): L(S) \longrightarrow} \def\ll{\longleftarrow M(S)$ is induced by $\xi$. A
morphism is called an {\em isomorphism} if $\xi$ is an isomorphism
of Drinfeld modules.
\end{definition}
\section{The moduli problem}
\label{sec_moduli}
\noindent
Let $\mf n\subset A$ be a non-zero, proper ideal.
In his original paper, Drinfeld considers the following moduli problem
for Drinfeld modules. Let $$\Cal F^r(\mf n): A-\mbox{\sc Schemes} \longrightarrow} \def\ll{\longleftarrow
\mbox{\sc Sets}$$ be the functor which associates to each $A$-scheme $S$
the set of isomorphy classes of Drinfeld modules over $S$ of rank $r$ with
level $\mf n$-structure over $S$. Drinfeld showed the following;
cf. Proposition 5.3 and its corollary in \cite{Drin74}.
\begin{theorem}[V.G. Drinfeld] \label{thm_drin}
If $\mf n \subset A$ is an ideal divisible by at least
$2$ distinct primes, then there exists a fine moduli space $$M^r(\mf
n) \longrightarrow} \def\ll{\longleftarrow {\rm Spec}(A)$$
representing the moduli problem $\Cal F^r(\mf n)$. Moreover, this
scheme has the following properties:
\begin{itemize}
\item[$(i)$] $M^r(\mf n)$ is affine and smooth of dimension $r$;
\item[$(ii)$] $M^r(\mf n) \longrightarrow} \def\ll{\longleftarrow {\rm Spec}(A)$ is smooth of relative
dimension $r-1$ over ${\rm Spec}(A) - V(\mf n)$.
\end{itemize}
\end{theorem}
\noindent
For arbitrary non-zero ideals $\mf n \subset A$, the functor $\Cal
F^r(\mf n)$ has in general only a coarse moduli scheme. This coarse
moduli scheme will also be denoted by $M^r(\mf n)$. We recall here
briefly the construction of this scheme; cf. \cite{Tae02} for a nice
exposition of this. Let $\mf n, \mf m \subset A$ be ideals such that $\mf n
\mf m$ is divisible by at least two distinct prime ideals, then by Theorem
\ref{thm_drin}
there exists an affine scheme $M^r(\mf n \mf m)$ representing the moduli
functor $\Cal F^r(\mf n \mf m)$. Let $(L, \varphi} \def\g{\gamma, \lambda)_S$ be a Drinfeld
module of rank $r$ with full level $\mf m \mf n$-structure $\lambda$ over
an $A$-scheme $S$. On these triples the group $\Gl_r(A/\mf m \mf n)$
acts by $$\s (L, \varphi} \def\g{\gamma, \lambda) :=
(L, \varphi} \def\g{\gamma, \lambda \circ \s) \quad \mbox{for all} \quad \s \in
\Gl_r(A/\mf n).$$
Note that $\Gl_r(A/\mf n)$ is isomorphic to the kernel of the
$~{\rm mod}~\mf m$ reduction map $$\Gl_r(\mf m \mf n) \longrightarrow} \def\ll{\longleftarrow \Gl_r(\mf m).$$
This induces an action of $\Gl_r(A/\mf n)$ on $M^r(\mf m \mf n)$. The
coarse moduli scheme of $\Cal F^r(\mf m)$ is defined as $$M^r(\mf m)
:= M^r(\mf m \mf n) / \Gl_r(A/\mf n).$$ This quotient exists,
because $\Gl_r(A/\mf n)$ is finite. It is, however, not obvious that
this scheme is coarse for the given moduli problem. See \cite{Tae02}
for a proof of the coarseness of the scheme.
The scheme $M^r(\mf m)$ does not depend on the choice of $\mf n$.
\subsection{Actions on $M^r(\mf n)$}
\noindent
Let $\hat A = \underset{\ll}{\lim}~A/\mf n$, and let $\A_f = \hat A
\otimes_A K_A$. In this subsection we describe the natural action of
$\A_f \cdot
\Gl_r(\hat A)$ on $M^r(\mf n)$; cf. \cite[5D]{Drin74}. Using this action,
we can define the action of $\Gl_r(A/\mf n)$ and ${\rm Cl}(A)$ on
$M^r(\mf n)$.
To keep this paper self-contained, we recall here the treatment given in
Section 3.5 of \cite{Leh00}, where the reader can find proofs and details.
\par\bigskip\noindent
A {\it total level structure} of a Drinfeld module $(L, \varphi} \def\g{\gamma)_S$ is a
homomorphism $$\kappa: (K_A/A)^r \longrightarrow} \def\ll{\longleftarrow L(S)$$ such that its restriction to
$(\mf n^{-1} A/A)^r$ defines a level $\mf n$-structure.
Let $$M^r := \underset{\ll}{\lim} ~M^r(\mf n)$$ where $\mf n$ runs through
the non-zero ideals of $A$. This is an affine scheme, and $M^r$ represents
the functor which associates to each $A$-scheme $S$ the set of isomorphy
classes of Drinfeld modules with a total level structure over $S$.
\par\bigskip\noindent
There is a natural action of $\Gl_r(\A_f)$ on $M^r$, which is defined as
follows. Let $S$ be an $A$-scheme, and let $(L, \varphi} \def\g{\gamma, \kappa)_S$ be a
Drinfeld module with a total level structure over $S$.
Let $\s \in \Gl_r(\A_f)$ such that the entries
of $\s$ are elements of $\hat A$, then $\s$ gives rise to a map
$$\s: (K_A/A)^r \longrightarrow} \def\ll{\longleftarrow (K_A/A)^r.$$ Let $H_{\s}$ denote the kernel of $\s$.
The kernel $H_{\s}$ gives rise to a finite subgroup scheme
of $L$. We can divide out the pair $(L, \varphi} \def\g{\gamma)_S$ by this subgroup scheme.
This gives us a pair $(L', \varphi} \def\g{\gamma')_S$. The following diagram equips
the pair $(L', \varphi} \def\g{\gamma')_S$ with a total level structure $\kappa'$:
\begin{equation} \label{diag_ClGpAct}
\begin{CD}
0 @>>> H_{\s} @>>> (K_A/A)^r @> \s >> (K_A/A)^r @>>> 0\\
& & @VVV @V \kappa VV @VV \kappa' V\\
0 @>>> H_{\s}(S) @>>> L(S) @>>> L'(S).
\end{CD}
\end{equation}
\noindent
If $\s$ comes from an element in $A\bs \{ 0 \}$, then its action is
trivial. This implies that we get an action of $\Gl_r(\A_f)/K_A^*$
on $M^r(S)$. As this action is functorial in $S$, this defines
an action of $\Gl_r(\A_f)/K_A^*$ on $M^r$.
\par\bigskip\noindent
For the moduli scheme $M^r(\mf n)$ we have $M^r(\mf n) = \Gamma(\mf n)
\bs M^r$ with $$\Gamma(\mf n) := \ker(\Gl_r(\hat A) \longrightarrow} \def\ll{\longleftarrow \Gl_r(A/\mf n)).$$
The restriction of the universal triple $(L, \varphi} \def\g{\gamma, \kappa)$ on $M^r$
to $M^r(\mf n)$ gives the universal pair $(\varphi} \def\g{\gamma, \lambda)$ on $M^r(\mf n)$
(Recall that the line bundle of the universal Drinfeld module on $M^r(\mf n)$
is trivial.)
As $\A_f^* \cdot \Gl_r(\hat A)$ commutes with $\Gamma(\mf n)$ in
$\Gl_r(\A_f)$, it
follows that the action of $\Gl_r(\A_f)$ on $M^r$ defines an
action of $\A_f^* \cdot \Gl_r(\hat A)$ on $M^r(\mf n)$.
The normal subgroup $K_A^* \cdot \Gamma(\mf n) \subset
\A_f^* \cdot \Gl_r(\hat A)$ acts trivially on $M^r(\mf n)$.
Let $$G := \A_f^* \cdot \Gl_r(\hat A)/K_A^* \cdot \Gamma(\mf n).$$
As $\A_f^*/K_A^* \cdot \hat A^* \cong {\rm Cl}(A)$, it is not
difficult to see that we have the following exact sequence
$$0 \longrightarrow} \def\ll{\longleftarrow \Gl_r(A/\mf n)/\F_q^* \longrightarrow} \def\ll{\longleftarrow G \longrightarrow} \def\ll{\longleftarrow {\rm Cl}(A) \longrightarrow} \def\ll{\longleftarrow 0.$$
\par\bigskip\noindent
To describe the action of $\Gl_r(A/\mf n)/\F_q^*$, let $\s \in
\Gl_r(\hat A)$ and let $\tilde \s$ be the image of $\s$ under
the reduction map $\Gl_r(\hat A) \longrightarrow} \def\ll{\longleftarrow \Gl_r(A/\mf n)/\F_q^*$.
Then $$\s: (\varphi} \def\g{\gamma, \lambda) \mapsto (\varphi} \def\g{\gamma, \lambda \circ \tilde \s^{-1}).$$
\begin{remark}
However, in the sequel we prefer to drop the inverse. If we talk about
the action of $\s \in \Gl_r(A/\mf n)/\F_q^*$ on $(\varphi} \def\g{\gamma, \lambda)$, then we
mean the action given by
$$\s: (\varphi} \def\g{\gamma, \lambda) \mapsto (\varphi} \def\g{\gamma, \lambda \circ \s).$$
Consequently, $\Gl_r(A/\mf n)/\F_q^*$ acts on the right of
$M^r(\mf n)$ and not on the left.
\end{remark}
\par\bigskip\noindent
Let $m \in \hat A \cap \A_f^*$, then $m$ defines a unique ideal
$\mf m = (m) \cap A \subset A$. We suppose that $\mf m$ is a non-zero,
proper ideal which is relatively prime to $\mf n$.
Let ${\rm I}_r$ denote the identity element in
$\Gl_r(\A_f)$, and let $\s = m \cdot {\rm I}_r$. We describe the
action of $\s$ on $M^r$. Clearly, $H_{\s} = (\mf m^{-1} A/A)^r$, and
$H_{\s}$ maps to $\varphi} \def\g{\gamma[\mf m](M^r)$ under $\kappa$. This means that the isogeny
$$\xi_{\mf m}: (L, \varphi} \def\g{\gamma) \longrightarrow} \def\ll{\longleftarrow (L', \varphi} \def\g{\gamma')$$ defined by $\s$ has kernel
$\varphi} \def\g{\gamma[\mf m]$. The total level structure $\kappa'$ is given by
$$\kappa' = \xi_{\mf m} \circ \kappa \circ m^{-1}.$$
Let $\varphi} \def\g{\gamma'$ denote the restriction of $(L', \varphi} \def\g{\gamma')$ to $M^r(\mf n)$.
Let $\ov m$ denote the image of $m$ under the reduction map $\hat A
\longrightarrow} \def\ll{\longleftarrow A/\mf n$. As $\mf m + \mf n = A$, we see that $\ov m \in (A/\mf
n)^*$. Let $\varphi} \def\g{\gamma$ be the restriction of $(L', \varphi} \def\g{\gamma')$ to $M^r(\mf n)$,
then the action of $m$ on the universal pair $(\varphi} \def\g{\gamma, \lambda)$ on
$M^r(\mf n)$ is given by $$m: (\varphi} \def\g{\gamma, \lambda) \mapsto (\varphi} \def\g{\gamma',
\xi_{\mf m} \circ \lambda \circ \ov m^{-1} ).$$
This describes the action of $m$ on $M^r(\mf n)$.
\par\bigskip\noindent
Let $\mf m \subset A$ be a non-zero ideal relatively prime to
$\mf n$, i.e., $\mf m + \mf n = A$. Choose $m \in \hat A$ such that
$(m) = \mf m \hat A$ and $m \equiv 1 ~{\rm mod}~ \mf n$.
We define the action of $\mf m$ on
$(\varphi} \def\g{\gamma, \lambda)$ to be the action of $(m)$:
$$\mf m: (\varphi} \def\g{\gamma, \lambda) \mapsto (\varphi} \def\g{\gamma', \xi_{\mf m} \circ \lambda).$$
This action is well-defined: the chosen element $m$ is unique up to an
element $\alpha \in \hat A^*$ with $\alpha \equiv 1 ~{\rm mod}~ \mf
n$. For such an element $\alpha$ we have $\alpha {\rm I}_r \in
\Gamma(\mf n)$. Consequently, $m \cdot {\rm I}_r$ and $\alpha m \cdot
{\rm I}_r$ give the same element in $G$.\\
Using this, we can define the {\it action of ${\rm Cl}(A)$ on $M^r(\mf n)$}.
First note that by Lemma \ref{lem_choice-m} we can represent every class
in ${\rm Cl}(A)$ by a non-zero ideal $\mf m$ with $\mf m + \mf n = A$.
Namely, suppose that $\mf m$ and $\mf m'$ are both non-zero
ideals relatively prime to $\mf n$ which represent the same class in
${\rm Cl}(A)$, then there is an element $x \in K_A^*$ with $\mf m = x
\mf m'$. Let $m, m' \in \hat A$ be elements which define the action of
$\mf m$ and $\mf m'$, respectively. Then there is an element
$\alpha \in \hat A^*$ with $\alpha \equiv 1 ~{\rm mod}~\mf n$ such that
$m' = x\alpha m$ with $x \alpha \cdot {\rm I}_r \in K_A^* \cdot
\Gamma(\mf n)$. Therefore, $(m)$ and $(m')$ give the same element in $G$.
\begin{remark}
These considerations also imply that
$$G\cong {\rm Cl}(A) \times \Gl_r(A/\mf n)/\F_q^*.$$
\end{remark}
\begin{remark}
Let us once more stretch that we use the convention that the action
of $\s \in \Gl_r(A/\mf n)$ on
$M^r(\mf n)$ is given by $$\s: (\varphi} \def\g{\gamma, \lambda) \longrightarrow} \def\ll{\longleftarrow (\varphi} \def\g{\gamma, \lambda \circ \s).$$
So $G = {\rm Cl}(A)\times \Gl_r(A/\mf n)/\F_q^*$ acts on $M^r(\mf n)$
on the right.
\end{remark}
\subsection{Assumptions in this paper}
\noindent
In this paper we will make the following two assumptions.
\begin{enumerate}
\item Throughout this paper we will assume that $\mf n = (f)$ is a
non-zero, proper, principal ideal. This simplifies the description of the
Tate-Drinfeld module. Dropping this assumption does not seem to
give rise to different results.\\
If $\mf m\subset A$ is a non-zero
proper ideal containing $f$, then by the previous, we see that
$M^r(\mf n) = M^r(f)/G$ where $G$ is given by dividing out the action
of the kernel $$\ker(\Gl_r(A/fA) \longrightarrow} \def\ll{\longleftarrow \Gl_r(A/\mf n)).$$
\item We will not consider the moduli problem over $A$-schemes, but
over $A_f$-schemes $S$, i.e., $f$ is invertible in $S$. This implies
that $f$ is away from the characteristic of $(L, \varphi} \def\g{\gamma)_S$.
This assumption is used because the Weil pairing plays an important
role in our description, and the Weil pairing is in \cite{Hei02} only
defined for $f$-torsion which is away from the
characteristic.\\ This assumption implies that a level $f$-structure
is an isomorphism $$\lambda: (A/fA)^r \stackrel{\sim}{\longrightarrow} \def\ll{\longleftarrow} \varphi} \def\g{\gamma[f](S).$$
\end{enumerate}
\noindent
So in this paper we will be considering the moduli problem
$$\Cal F^r(f): A_f-\mbox{\sc Schemes} \longrightarrow} \def\ll{\longleftarrow \mbox{\sc Sets}$$ which
associates to each $A_f$-scheme $S$, the set of isomorphy classes of
Drinfeld modules of rank $r$ with full level $f$-structure
over $S$. We write $M^r(f)$ for the scheme which represents
$\Cal F^r(f)$. It follows from the proof of Theorem \ref{thm_drin}
that the moduli scheme $M^r(f)$ is a fine moduli scheme if $f \neq 1$.
\par\bigskip\noindent
Throughout this paper we will write ${\rm Spec}(R) = M^1(f)$. The ring $R$
is regular and $M^1(f)$ is connected. In fact, $R$ is the integral
closure of $A_f$ in a field extension of $K_A$. The Galois group of
$K_R/K_A$ is the group $$G \cong (A/fA)^*/\F_q^* \times {\rm Cl}(A)$$
that we discussed above; cf. Section 8 in \cite{Drin74}.
\section{The Weil pairing on the modular schemes.} \label{sec_weil-map}
\noindent
In this section we will show that the Weil pairing for
Drinfeld modules over an $A_f$-field $K$ as defined in the
previous paper gives rise to the following theorem:
\begin{theorem} \label{thm_map}
The Weil pairing induces an $A_f$-morphism $$w_f: M^r(f)
\longrightarrow} \def\ll{\longleftarrow M^1(f).$$ The Weil pairing is ${\rm Cl}(A) \times
\Gl_r(A/fA)$-equivariant.
\end{theorem}
\noindent
Let $(\varphi} \def\g{\gamma, \lambda)$ be a Drinfeld module $\varphi} \def\g{\gamma$ of rank $r$ over $K$ with
level $f$-structure. The Weil pairing is an $A/fA$-isomorphism $$w_f:
\wedge^r \varphi} \def\g{\gamma[f](K) \stackrel{\sim}{\longrightarrow} \def\ll{\longleftarrow} \psi[f](K) \otimes_A
\Omega_f^{\otimes r-1}.$$ It is unique up to a unique isomorphism of
$\psi$. Once and
for all we fix a generator $\omega$ of the $A/fA$-module
$\Omega_f$. This gives an $A/fA$-isomorphism $$w_f: \wedge^r
\varphi} \def\g{\gamma[f](K) \stackrel{\sim}{\longrightarrow} \def\ll{\longleftarrow} \psi[f](K).$$
\par\smallskip\noindent
The level $f$-structure $\lambda$ induces a canonical isomorphism
$$\wedge^r \lambda: \wedge^r (A/fA)^r \stackrel{\sim}{\longrightarrow} \def\ll{\longleftarrow}
\wedge^r \varphi} \def\g{\gamma[f](K).$$ Because $\wedge^r (A/fA)^r$ is canonically
isomorphic to $A/fA$, $\psi$ comes equipped with a level $f$-structure
$\mu$ over $K$ via the following commutative diagram:
$$\begin{CD}
\wedge^r (A/fA)^r @> \wedge^r \lambda >> \wedge^r \varphi} \def\g{\gamma[f](K)\\
@AAA @V w_f VV\\
A/fA @> \mu >> \psi[f](K).
\end{CD}$$
\noindent
Note that if $\xi$ is an isomorphism between $\psi$ and $\psi'$, then
also the pairs $(\psi, \mu)$ and $(\psi', \mu')$ are isomorphic via
$\xi$. Here $\mu$ and $\mu'$ are defined by the previous diagram by
$\psi$ and $\psi'$ respectively. So the pair $(\psi, \mu)$
is unique up to isomorphy. These considerations show
the following:
\begin{lemma} \label{lem_isolev}
The Weil pairing gives for all $A_f$-fields $K$ rise to a map
$$w_K: M^r(f)(K) \longrightarrow} \def\ll{\longleftarrow M^1(f)(K).$$
This map depends functorially on $K$.
\end{lemma}
\begin{proof}
The construction of the map $w_K$ is described above. It
associates to each isomorphy class $(\varphi} \def\g{\gamma, \lambda)$ of rank $r$ over
$K$ a unique isomorphy class $(\psi, \mu)$ of rank $1$ over $K$.
That this map depends functorially on $K$ follows immediately from
the construction in terms of $A$-motives.
\end{proof}
\noindent
From this lemma, we proceed as follows to prove the existence of the map
$w_f$.
\begin{proof}[Proof of Theorem \ref{thm_map}]
Let $(\varphi} \def\g{\gamma, \lambda)$ be the universal pair over $M^r(f)$.
The moduli scheme $M^r(f)$ is affine and regular over
${\rm Spec}(A_f)$. So we may write $$M^r(f) \cong \coprod_{i=0}^n
{\rm Spec}(S_i)$$ such that
each $S_i$ is an integrally closed domain of relative dimension $r-1$
over $A_f$. Moreover, from Drinfeld's description we know that
$$M^1(f) = {\rm Spec}(R)$$ where $R$ is an integrally closed domain.
Let $K_{S_i}$ be the quotient field of
$S_i$. Lemma \ref{lem_isolev} gives rise to a unique
isomorphy class $$(\psi, \mu) \in M^1(f)(\prod_i K_{S_i}).$$ This means
that there exists a unique $A_f$-ring homomorphism $$h: R
\longrightarrow} \def\ll{\longleftarrow \prod_j K_{S_j}.$$ Hence, the theorem
follows if we can show that $h(R) \subset
\prod_j S_j$.\\ Let $\mf p \in {\rm Spec}(S_j)$
be a closed point of height $1$. By Lemma \ref{lem_reg} it follows
that there is a map ${\rm Spec}(S_{j, \mf p}) \longrightarrow} \def\ll{\longleftarrow M^1(f)$, inducing
the map $${\rm Spec}(K_{S_j}) \longrightarrow} \def\ll{\longleftarrow M^1(f).$$ Consequently,
$$h(R) \cap K_{S_j} \subset S_{j, \mf p}.$$ As
$S_j = \cap_{\mf p} S_{j, \mf p}$
where the intersection runs over all primes of height $1$ in $S_j$, one has
$h(R) \cap K_{S_j} \subset S_j$.\\
For the ${\rm Cl}(A) \times \Gl_r(A/fA)$-equivariance of $w_f$, note
that the $\Gl_r(A/fA)$-equivariance is obvious. Let $\mf m \subset
A$ be a non-zero, proper ideal with $f \not \in \mf m$ representing
an element in ${\rm Cl}(A)$. For the
${\rm Cl}(A)$-equivariance we recall its definition in the
previous section. Using the notations
of the previous section, we see that the action of $\mf m$
on $(\varphi} \def\g{\gamma, \lambda)$ is given by $$(\varphi} \def\g{\gamma, \lambda) \mapsto (\varphi} \def\g{\gamma',
\xi_{\mf m} \circ \lambda),$$ with $\xi_{\mf m} \varphi} \def\g{\gamma_a = \varphi} \def\g{\gamma'_a \xi_{\mf m}$
for all $a \in A$.
Let $(\psi, \mu)$ be the image of $(\varphi} \def\g{\gamma, \lambda)$ under the
Weil pairing, and let $(\psi', \mu')$ be the image of $(\varphi} \def\g{\gamma',
\xi_{\mf m} \circ \lambda)$ under $w_f$.
Let $F = \prod_i \ov K_{S_i}$. The isogeny $\xi_{\mf m}$ induces an
isogeny $\zeta_{\mf m}: \psi \longrightarrow} \def\ll{\longleftarrow \psi'$. The kernel of $\zeta_{\mf m}$
is
$$\ker(\zeta_{\mf m})(F) = \wedge^r \ker(\xi_{\mf m})(F)
= \wedge^r \varphi} \def\g{\gamma[\mf m](F) \cong \psi[\mf m](F).$$ Therefore, the
action of $\zeta_{\mf m}$ on $\psi$ coincides with the action of
$\mf m$ on $\psi$. So we have $$\mf m: (\psi, \mu) \mapsto
(\psi', \zeta_{\mf m} \circ \mu) = (\psi', \mu').$$
\end{proof}
\begin{lemma} \label{lem_reg}
Let $S$ be a regular local $A_f$-ring, let $K_S$ be its quotient
field, and let $(\varphi} \def\g{\gamma, \lambda) \in M^r(f)(S)$. The unique class
$(\psi, \mu) \in M^1(f)(K_S)$ associated to $(\varphi} \def\g{\gamma, \lambda)$ by the
Weil pairing comes from a unique class $(\psi', \mu') \in M^1(f)(S)$
via the canonical embedding $S \longrightarrow} \def\ll{\longleftarrow K_S$.
\end{lemma}
\begin{proof}
Via the ring homomorphism $S \longrightarrow} \def\ll{\longleftarrow K_S$ we may view the pair
$(\varphi} \def\g{\gamma, \lambda)$ over $K_S$, and we can associate via the Weil pairing
a pair $(\psi, \mu)$ of rank $1$ over $K_S$. We want to prove
that there is a representing pair in the isomorphy class of
$(\psi, \mu)$ which is defined over $S$.\\
Let $V$ be the set of all height one primes of $S$.
The ring $S$ is a UFD; cf. Theorem 20.3 in \cite{Mat80}.
Consequently, every $\mf p \in V$ is of the form $\mf p = (h_{\mf p})$
for some irreducible $h_{\mf p} \in S$; cf. Theorem 20.1 in
\cite{Mat80}. Let $v_{\mf p}$ denote the valuation at $\mf p$.
The invertible elements of $S$ are given by $$S^* = \{ s \in S
\mid v_{\mf p}(s) = 0 ~\mbox{for all $\mf p \in V$} \}.$$
Let $S_{\mf p}$ denote the local ring of $S$ at $\mf p$. This is
a discrete valuation ring.\\
As the $f$-torsion of $\psi$ is $K_S$-rational, it follows that
$\psi$ has good reduction at every $\mf p \in V$. This implies
that for all $\mf p \in V$ there is an element $x_{\mf p}
\in K_S$ such that $x_{\mf p} \psi x_{\mf p}^{-1}$ is a Drinfeld module
of rank $1$ defined over $S_{\mf p}$. In fact, we may assume
$x_{\mf p} = h_{\mf p}^{m_{\mf p}}$ for some $m_{\mf p} \in \Z$.\\
There are only finitely many $m_{\mf p} \neq 0$, as we show below.
So we can define $x = \prod_{\mf p \in V}
h_{\mf p}^{m_{\mf p}}$. The pair $(x \psi x^{-1}, x \mu)$ is defined
over $S$. So this is the pair that we are looking for.\\
To see that there are only finitely many $m_{\mf p} \neq 0$,
let $a \in A \bs \F_q$ and consider the leading coefficient $c$
of $\psi_a$. Under the isomorphism $x_{\mf p}$, the leading coefficient
becomes $x_{\mf p}^{1 - q^{\deg(a)}} c$. As $v_{\mf p}(c) = 0$ for
all but finitely many $\mf p$'s, it follows that $m_{\mf p} = 0$ for
all but finitely many $\mf p$'s.
\end{proof}
\begin{prop}
The morphism $w_f$ induces the maps $w_K$ for any $A_f$-field $K$,
where $w_K$ is as in Lemma \ref{lem_isolev}.
\end{prop}
\begin{proof}
By the functoriality in $K$ of the maps $w_K$, it suffices to prove
the statement for algebraically closed fields $\ov K$. So let $\ov K$ be
algebraically closed and let $$\zeta : {\rm Spec}(\ov K) \longrightarrow} \def\ll{\longleftarrow M^r(f)$$ be a
geometric point of $M^r(f)$. If $\zeta$ is a generic point of one of the
connected components of $M^r(f)$, then $w_f$ induces $w_{\ov K}$ by
construction. If $\zeta$ is a closed point we have to do something more.
Clearly, $\zeta$ factors over ${\rm Spec}(k_{\zeta})$ where
$k_{\zeta}$ is the residue field at (the image of) $\zeta$. To see that
$w_{k_{\zeta}}$ and consequently $w_{\ov K}$ is induced by $w_f$, we have
to dive into the language of $A$-motives a little; cf. \cite{Hei02}.\\
Let $V = \O_{\zeta}$, then $V$ is a regular local $A_f$-ring and
let $(\varphi} \def\g{\gamma, \lambda)$ be defined over $V$ of rank $r$. Let $K_V$ be the
quotient field of $V$. Then the
construction of $A$-motives associates to $\varphi} \def\g{\gamma$ the Drinfeld module
$\psi$ of rank $1$ for which $$M(\psi) \cong \wedge^r M(\varphi} \def\g{\gamma).$$ By
Lemma \ref{lem_reg}, the pair $(\psi, \mu)$ is also defined over
$V$. Because $\varphi} \def\g{\gamma$ is defined over $V$, it makes sense to consider
$M^0(\varphi} \def\g{\gamma) = V\{ \t \}$ with the obvious $V\{ \t \}\otimes_{\F_q} A$-action
such that $$M^0(\varphi} \def\g{\gamma) \otimes_V K_V = M(\varphi} \def\g{\gamma).$$ Similarly, we can define
$M^0(\psi)$ because $\psi$ is defined over $V$. Clearly,
$$\wedge_{K_V \otimes A}^r M(\varphi} \def\g{\gamma) \cong K_V \otimes_V \wedge_{V\otimes
A}^r M^0(\varphi} \def\g{\gamma).$$ Consequently, the $K_V\{\t\} \otimes A$-isomorphism
$$M(\psi) \cong \wedge^r M(\varphi} \def\g{\gamma)$$ comes from a $V\{ \t \}\otimes
A$-isomorphism $$M^0(\psi) \cong \wedge_{V\otimes A} M^0(\varphi} \def\g{\gamma).$$
This construction can be reduced modulo the maximal of $V$. This
gives us the construction of $w_{k_{\zeta}}$. Therefore, $w_f$
induces $w_{\ov K}$.
\end{proof}
\section{Drinfeld modules of rank $2$ with stable reduction of rank $1$}
\label{sec_redth}
\noindent
For this section we fix the following notation. Let $V$ be a complete
discrete valuation ring which is also an $A_f$-algebra. Let $K_V$ be the
quotient field of $V$, and let $\pi \in V$ be a generator of the maximal
ideal of $V$. Let $v(x)$ denote the $\pi$-valuation of $x$ for every
$x \in V$.
\begin{definition}
Let $\varphi} \def\g{\gamma$ be a Drinfeld module of rank $r$ over $K_V$, then $\varphi} \def\g{\gamma$
has {\em stable reduction at $v$ of rank $r'$} if
$\varphi} \def\g{\gamma$ is isomorphic over $K_V$ to a Drinfeld module $\varphi} \def\g{\gamma'$ over $K_V$
such that for all $a \in A$ each coefficient $\beta_i(a)$ of the sum
$\varphi} \def\g{\gamma'_a = \sum \beta_i(a) \t^i$ is an element of $V$ and the
reduction $\varphi} \def\g{\gamma' ~{\rm mod}~\pi V$ is a Drinfeld module of rank $r'$ over
$V/\pi V$.\\
The Drinfeld module $\varphi} \def\g{\gamma$ has {\em potentially
stable reduction at $v$ of rank $r'$} if there is a finite field
extension $L$ of $K_V$ and a valuation $w$ of $L$ extending $v$ such
that $\varphi} \def\g{\gamma$ has stable reduction at $w$ of rank $r'$.
\end{definition}
\noindent
Let $\varphi} \def\g{\gamma$ be a Drinfeld module of rank $2$ over $K_V$
with full level $f$-structure $\lambda$ over $K_V$ such that $\varphi} \def\g{\gamma$
has potentially stable reduction of rank $1$.
The goal of this section is Theorem \ref{thm_reduct} which describes
the pairs $(\varphi} \def\g{\gamma, \lambda)$ in terms of Drinfeld modules of rank $1$
and lattices.
\begin{lemma} \label{lem_stab_red}
Let $\varphi} \def\g{\gamma$ be a Drinfeld module with $K_V$-rational $f$-torsion.
If $\varphi} \def\g{\gamma$ has potentially stable reduction at $(\pi)$, then $\varphi} \def\g{\gamma$
has stable reduction at $(\pi)$.
\end{lemma}
\begin{proof}
As $\varphi} \def\g{\gamma$ has potentially stable reduction, there is an element
$k \in \Q$ such that for all $x \in K_V^{\rm sep}$ with $v(x) = k$
we have the following: $\tilde \varphi} \def\g{\gamma := x \varphi} \def\g{\gamma x^{-1}$ is a Drinfeld
module, $\tilde \varphi} \def\g{\gamma_a$ has all coefficients in $V$ for all $a \in A$,
and $\tilde \varphi} \def\g{\gamma ~{\rm mod}~ \pi V$ is a Drinfeld module over $V/\pi V$.
Cf. Section 4.10 in \cite{Gos96}.
As $f \in V^*$, it is not difficult to see that $k$ is the smallest
slope of the Newton polygon of $\frac{1}{X} \varphi} \def\g{\gamma_f(X)$. Therefore,
$$k = \max \{ v(\alpha) \mid \alpha \in \varphi} \def\g{\gamma[f](\ov K_V)\bs \{ 0 \} \}.$$
As the $f$-torsion of $\varphi} \def\g{\gamma$ is $K_V$-rational, we have $k \in \Z$.
Therefore, we may choose $x \in K_V$.
\end{proof}
\noindent
To abbreviate notation, we introduce the following two properties $P$
and $P'$. Let $\chi$ be a Drinfeld module of rank $r$ over $K_V$.
\begin{itemize}
\item[$P(\chi)~:$] $\chi$ has stable reduction of rank $1$.
\item[$P'(\chi):$] $\chi$ has stable reduction of rank $1$, $\chi_a$
has all coefficients in $V$ for all $a \in A$ and $\chi[f](V) \cong A/fA$.
\end{itemize}
\noindent
Suppose that $\kappa$ is a level $f$-structure of $\chi$ over $K_V$.
If $r = 1$, then $P'(\chi)$ implies that $(\chi, \kappa)$ is defined
over $V$.
\par\bigskip\noindent
By Lemma \ref{lem_stab_red} we have $P(\varphi} \def\g{\gamma)$ for the pair $(\varphi} \def\g{\gamma, \lambda)$.
Therefore, we may assume that $\varphi} \def\g{\gamma_a$ has all its coefficients in $V$
for all $a \in A$. Moreover, we have that the smallest slope
of the Newton polygon of $\varphi} \def\g{\gamma_f$ equals $v(\varphi} \def\g{\gamma_f) = 0$.
Consequently, $\varphi} \def\g{\gamma[f](V) \cong A/fA$.
So in the isomorphy class of $(\varphi} \def\g{\gamma, \lambda)$, there is a
representing element $(\varphi} \def\g{\gamma, \lambda)$ with $P'(\varphi} \def\g{\gamma)$ and this element
is unique up to $V^*$. In fact,
\begin{equation} \label{eq_iso1}
\{ (\varphi} \def\g{\gamma, \lambda)_{K_V} \mbox{with $P'(\varphi} \def\g{\gamma)$} \} /V^*
\stackrel{\rm bij}{\cong} \{ (\varphi} \def\g{\gamma, \lambda)_{K_V} \mbox{with $P(\varphi} \def\g{\gamma)$}
\} / K_V^*.
\end{equation}
\par\bigskip\noindent
Note that the Weil pairing equips $V$ with
an $R$-structure. The isomorphy class of $(\varphi} \def\g{\gamma, \lambda)$ is induced by an
$A_f$-morphism ${\rm Spec}(K_V) \longrightarrow} \def\ll{\longleftarrow M^2(f)$. Composing this morphism with
$w_f$ gives rise to an $A_f$-linear ring homomorphism $R \longrightarrow} \def\ll{\longleftarrow K_V$.
As $R$ is integral over $A_f$, it follows that this ring homomorphism
gives an $A_f$-linear ring homomorphism $h: R \longrightarrow} \def\ll{\longleftarrow V$.
\subsection{Drinfeld's bijection without level structure}
\noindent
To classify the isomorphy classes $(\varphi} \def\g{\gamma, \lambda)$ with stable reduction
of rank $1$, we recall Drinfeld's classification of
Drinfeld modules of rank $2$ with potentially stable
reduction of rank $1$; cf. Proposition 7.2 in \cite{Drin74}.\\
An {\it $A$-lattice of rank $1$ in $K_V^{\rm sep}$} is a
projective $A$-module of rank $1$ which lies discretely in $K_V^{\rm
sep}$ and which is invariant under the action of $G_{K_V}:=
\operatorname{Gal}(K_V^{\rm sep}/K_V)$. Two $A$-lattices $\Lambda_1$ and
$\Lambda_2$ are called {\it isomorphic} if there is an element
$x \in (K_V^{\rm sep})^*$
such that $x \Lambda_1 = \Lambda_2$. Then Drinfeld's result states the
following:
\begin{theorem}[V.G. Drinfeld] \label{thm_Dri}
There is a bijection between the set of isomorphy classes over $K_V$
of Drinfeld modules of rank $2$ over $V$ with potentially stable
reduction of rank $1$ and the set of isomorphy classes over $K_V$ of
pairs $(\psi, \Lambda)$, where $\psi$ is a Drinfeld module of rank $1$
over $V$ and $\Lambda$ is an $A$-lattice of rank $1$ inside
$K_V^{\rm sep}$.
\end{theorem}
\begin{proof}[Sketch of the proof.]
Applying Proposition 5.2 in \cite{Drin74} to the ring $V_n = V/(v)^n$
with $n\in \Z_{\geq 1}$ gives us the existence of unique elements
$s_n \in V_n\{\t\}$ such that $s_n \varphi} \def\g{\gamma_a s_n^{-1}$ is in standard
rank $1$ form over $V_n$. Moreover, each $s_n$ has the form $$s_n = 1
+ \sum_{i=1}^{k_n} v_i \t^i \quad \mbox{with} \quad v_i \in (v).$$
Let $s = \underset{\ll}{\lim} ~s_n$, then $s$ is an element in
$V\{\! \{\t\}\! \}$, the set of skew formal power series in $\t$ over $V$.
In the proof of Proposition 7.2 in \cite{Drin74}, Drinfeld shows that the
homomorphism $s$ is in fact analytic. One has by construction
$$s = 1 + \underset{i \geq 1}{\sum} v_i \t^i \quad \mbox{with}
\quad v_i \in (v).$$ This implies both that $\Lambda:= \ker(s)$ is
contained in $K_V^{\rm sep}$. Moreover, each element in
$\Lambda \backslash \{ 0 \}$ has strictly negative valuation.\\ Let
$\psi' = s \varphi} \def\g{\gamma s^{-1}$, then $\psi' \mod (v) = \psi$.
We get the following diagram, which is commutative for all $a \in A$:
\begin{equation} \label{diag1}
\begin{CD}
0 @>>> \Lambda @>>> K_V^{\rm sep} @> e_{\Lambda} >> K_V^{\rm sep}\\
& & @VV \psi_a V @VV \psi_a V @VV \varphi} \def\g{\gamma_a V\\
0 @>>> \Lambda @>>> K_V^{\rm sep} @> e_{\Lambda} >> K_V^{\rm sep},
\end{CD}
\end{equation}
\noindent
where
$$e_{\Lambda}(z) = z \prod_{\alpha \in \Lambda \backslash \{0\}}
\left( 1 - \frac{z}{\alpha} \right) = s(z).$$
Let $a \in A \backslash \F_q$, then $\psi_a^{-1}\Lambda$ is mapped
surjectively to $\varphi} \def\g{\gamma[a]$, hence $\psi_a^{-1} \Lambda/\Lambda \cong
(A/aA)^2$ as $A$-module. On the other hand, the kernel of the
surjective map $$\psi_a^{-1} \Lambda/\Lambda \longrightarrow} \def\ll{\longleftarrow \Lambda/\psi_a
\Lambda$$ is isomorphic to $\psi[a](K_V^{\rm sep})$. So we may
conclude that $\Lambda/\psi_a \Lambda \cong A/aA$. This implies that
$\Lambda$ is a projective $A$-module of rank $1$. We already saw that
$\Lambda$ consists of elements with strictly negative valuation, hence
$\Lambda$ lies discretely in $K_V^{\rm sep}$. Finally, for any element
$\s \in G_{K_V}$, we have $\s \circ s(z) = s \circ \s(z)$, hence
$\Lambda$ is $G_{K_V}$-invariant. We conclude that $\Lambda$ is an
$A$-lattice of rank $1$ in $K_V^{\rm sep}$.
\end{proof}
\subsection{The bijection with level $f$-structure}
\noindent
Let $(\psi^{\rm un}, \mu^{\rm un})$ be the universal pair of rank $1$
defined over $R$. We will assume that $\mu^{\rm un}(1) = 1$, which is
possible because $f$ is invertible in $R$.
Drinfeld's construction in Theorem \ref{thm_Dri} lifts the rank
$1$ Drinfeld module
$\varphi} \def\g{\gamma~{\rm mod}~\pi V$ to a unique Drinfeld module $\psi$ of
rank $1$ defined over $V$. Also, the $f$-torsion of $\psi$
is $V$-rational. We would like to equip $\psi$ with a natural
level $f$-structure $\mu$ which comes from $\lambda$. For this
we use the ring homomorphism $$h: R \longrightarrow} \def\ll{\longleftarrow V$$ which arises from the
Weil pairing.\\
Suppose that $\psi$ is equipped with a level $f$-structure $\tilde \mu$,
then the isomorphy class of $(\psi, \tilde \mu)$ comes from an $A_f$-linear
ring homomorphism
$$\tilde h: R \longrightarrow} \def\ll{\longleftarrow V,$$ i.e., there is a unique element $v \in V^*$ such that
$$\tilde h( (\psi^{\rm un}, \mu^{\rm un}) ) = (v \psi v^{-1}, v \tilde \mu).$$
The pair $(\tilde h, v)$ uniquely determines $(\psi, \tilde \mu)$.
\par\bigskip\noindent
As $R$ is integral over $A_f$, there exists an $A_f$-automorphism
$$g_{\mu}: R \longrightarrow} \def\ll{\longleftarrow R$$ with $g_{\tilde \mu} \in G = \operatorname{Gal}(R/A_f)$ such
that $\tilde h = h \circ g_{\tilde \mu}.$ We
described this Galois group in Section \ref{sec_moduli}:
$$G \cong {\rm Cl}(A) \times (A/fA)^*/\F_q^*.$$
So the element $g_{\tilde \mu}$ is given by a pair
$(\mf m, \s)\in {\rm Cl}(A) \times (A/fA)^*/\F_q^*$.\\
If we have another level $f$-structure $\mu'$, then $\mu' = \alpha \tilde \mu$
for some $\alpha \in (A/fA)^*$ and we see that $g_{\mu'}$ corresponds
to the pair $(\mf m, \alpha \s)$.
\par\bigskip\noindent
We equip $\psi$ with the level $f$-structure $\mu$ such that
$g_{\mu}$ is given by the pair $(\mf m, 1)$. More specifically, let
$h_{\mu} := h \circ g_{\mu}$. Let $v \in V^*$ be an element with
$v \psi v^{-1} = \tilde h(\psi^{\rm un})$, then this element $v$ is unique
up to $\F_q^*$. Let $(\psi, \mu)$ be the pair determined by $(\tilde h, v)$.\\
The pair $(\psi, \mu)$ that we obtain in this way is unique up to $\F_q^*$.
In particular, the isomorphy class of $(\psi, \mu)$
is uniquely determined in this way.
\begin{remark} \label{rem_vis}
To make the choice of an element in $\F_q^*$ in the above
construction `visible', we consider the set of elements $z$ in
$\varphi} \def\g{\gamma[f](K_V)\bs \varphi} \def\g{\gamma[f](V)$ with $$w_f(z, e_{\Lambda}(\mu(1))) \in
\F_q^* \cdot w_f(\lambda(1,0), \lambda(0,1)).$$ The set of these elements
equals $\F_q^* \cdot z'$ where $z'$ is any element in this set.
Fix one of those
elements $z$. We can now equip $\psi$ with the unique $\mu$ such
that $$w_f(z, e_{\Lambda}(\mu(1))) = w_f(\lambda(1,0), \lambda(0,1)).$$
In this way, we can associate to a triple $(\varphi} \def\g{\gamma, \lambda, z)$ a unique
triple $(\psi, \mu, \Lambda)$.
\end{remark}
\par\bigskip\noindent
Our next goal is to prove Theorem \ref{thm_reduct}.
Drinfeld's construction gives an $A$-lattice $\Lambda$ of rank $1$
such that the pair $(\psi, \Lambda)$ determines $\varphi} \def\g{\gamma$ and vice versa.\\
As in Drinfeld's proof, we can consider $\psi_a$ as a map
$$\psi_a: K_V^{\rm sep} \longrightarrow} \def\ll{\longleftarrow K_V^{\rm sep}$$ for every $a \in A$.
Because $f \not \in \ker(A_f \longrightarrow} \def\ll{\longleftarrow K_V)$, it follows that all roots of
the equation $\psi_a(X) = \alpha$ lie in $K_V^{\rm sep}$ for all $\alpha\in
\Lambda$. And thus we have $$(\psi_f)^{-1} \Lambda \subset K_V^{\rm sep},$$ as
in Drinfeld's proof.
\begin{lemma} \label{lem_rat-lat}
If the $f$-torsion of $\varphi} \def\g{\gamma$ is $K_V$-rational, then $(\psi_f)^{-1} \Lambda
\subset K_V$. In particular $\Lambda \subset K_V$.
\end{lemma}
\begin{proof}
Cf. Lemma 2.4 in \cite{Boe02}. By definition $\Lambda$ is
invariant under $G_{K_V}$. Consequently, also $(\psi_f)^{-1}
\Lambda$ is invariant under $G_{K_V}$ as the coefficients of
$\psi_f$ lie in $V$. In fact, the action of $G_{K_V}$ on
$(\psi_f)^{-1}\Lambda$ splits according to the following splitting exact
sequence of $A$-modules: $$0 \longrightarrow} \def\ll{\longleftarrow \psi[f](K_V) \longrightarrow} \def\ll{\longleftarrow (\psi_f)^{-1}
\Lambda \longrightarrow} \def\ll{\longleftarrow ((\psi_f)^{-1} \Lambda)_{\rm proj} \longrightarrow} \def\ll{\longleftarrow 0.$$ The $A$-module
$((\psi_f)^{-1} \Lambda)_{\rm proj}$ is the projective part of
$(\psi_f)^{-1}\Lambda$, which is isomorphic to $\Lambda$.\\
By Drinfeld's construction there is an analytic map $e_{\Lambda}$
with the commuting property as in diagram (\ref{diag1}), and this map
commutes with the action of $G_{K_V}$. Moreover, via this map
$\varphi} \def\g{\gamma[f]$ is isomorphic to $(\psi_f)^{-1}\Lambda/\Lambda$. By the
assumption that $\varphi} \def\g{\gamma[f]$ is $K_V$-rational, it follows that
$G_{K_V}$ acts trivially on $(\psi_f)^{-1}\Lambda/\Lambda$ and thus on
$\psi[f]$. The latter fact implies that the only action of
$G_{K_V}$ on $(\psi_f)^{-1}\Lambda$ is on the projective part of
$(\psi_f)^{-1}\Lambda$, hence this action gives a subgroup $G$ of $\Gl_1(A)
= \F_q^*$. On the other hand, this subgroup $G$ maps injectively
into $\Aut_A((\psi_f)^{-1}\Lambda/\Lambda)$. But $G_{K_V}$ acts
trivially on $(\psi_f)^{-1}\Lambda/\Lambda$, hence $G$ is trivial.
\end{proof}
\noindent
So we see that we can associate to a pair $(\varphi} \def\g{\gamma, \lambda)$ with
$P'(\varphi} \def\g{\gamma)$ a triple $(\psi, \mu, \Lambda)$ such that $(\psi, \mu)$ is defined
over $V$ and $\Lambda$ is an $A$-lattice with $(\psi_f)^{-1}
\Lambda \subset K_V$. This triple is unique up to $\F_q^*$.
I.e., the pair $(\varphi} \def\g{\gamma, \lambda)$ determines a unique element in
$\{ (\psi, \Lambda, \mu) \} /\F_q^*$.
\par\bigskip\noindent
Note, however, that the level $f$-structure $\lambda$ is not the
unique level structure such that $(\varphi} \def\g{\gamma, \lambda)$
is mapped to this unique element in
$\{ (\psi, \mu, \Lambda \} /\F_q^*$. In fact, if $\s \in
\Gl_2(A/fA)$, then $(\varphi} \def\g{\gamma, \lambda \circ \s) \mapsto \{ (\psi, \mu,
\Lambda) \} /\F_q^*$ if and only if $\det(\s) \in \F_q^*$.
This is due to the fact that the Weil pairing is $\Gl_2(A/fA)$-equivariant.
Therefore, if one changes $\lambda$ by $\s$, then the morphism $h$
induced by the Weil pairing $w_f$ changes by $\det(\s)$. Consequently,
the triple $(\psi, \Lambda, \mu)$ changes by $\det(\s)$.
Let $\Sigma$ denote the subgroup of $\Gl_2(A/fA)$ given by
$$\Sigma = \{ \s \in \Gl_2(A/fA) \mid \det(\s) \in \F_q^* \}.$$
The previous shows that the map
$$\{ (\varphi} \def\g{\gamma, \lambda) \} / \Sigma \longrightarrow} \def\ll{\longleftarrow \{ (\psi, \mu, \Lambda) \} /\F_q^*$$
is injective.
\begin{remark} \label{rem_injpair}
Again, once we have chosen an element $z$ as in Remark \ref{rem_vis},
we get an injective map
$$\{ (\varphi} \def\g{\gamma, \lambda, z) \} /{\rm Sl}} \def\Gl{{\rm Gl}} \def\Aut{{\rm Aut}_2(A/fA) \longrightarrow} \def\ll{\longleftarrow \{ (\psi, \mu, \Lambda) \}.$$
\end{remark}
\par\bigskip\noindent
On the other hand, let a triple $(\psi, \mu, \Lambda)$ be given.
We show that there exists a pair $(\varphi} \def\g{\gamma, \lambda)$ such that
under the above construction $(\varphi} \def\g{\gamma, \lambda)$ is mapped to
the class $\{ (\psi, \mu, \Lambda) \}/\F_q^*$.\\
The triple $(\psi, \mu, \Lambda)$ gives rise to the following:
\begin{enumerate}
\item a morphism $\tilde h: R \longrightarrow} \def\ll{\longleftarrow V$ which induces $(\psi, \mu)$ on $V$;
\item the pair $(\psi, \Lambda)$ gives a Drinfeld module $\varphi} \def\g{\gamma$ of
rank $2$;
\item as the $f$-torsion of $\varphi} \def\g{\gamma$ comes from $(\psi_f)^{-1} \Lambda/\Lambda$,
the $f$-torsion of $\varphi} \def\g{\gamma$ is $K_V$-rational, and thus $\varphi} \def\g{\gamma$ has $P'(\varphi} \def\g{\gamma)$.
\end{enumerate}
\noindent
We equip $\varphi} \def\g{\gamma$ with a level $f$-structure as follows:
define $\lambda(0,1) := e_{\Lambda}(\mu(1))$ and let
$\lambda(1,0) = z$ for some element $z \in \varphi} \def\g{\gamma[f](K_V)\bs \varphi} \def\g{\gamma[f](V)$.
Any $z$ gives rise to a ring homomorphism $h_z: R \longrightarrow} \def\ll{\longleftarrow V$ induced by
the Weil pairing $w_f$. As before, there exists an $A_f$-automorphism
$g_z$ of $R$ with $\tilde h \circ g_z = h$, and
$g_z$ is given by a pair $(\mf m, \s) \in {\rm Cl}(A) \times (A/fA)^*$.
We choose an element $z$ for which $\s$ is the identity.
As before, $z$ is unique up to a choice of $\F_q^*$.
Clearly, the pair $(\varphi} \def\g{\gamma, \lambda)$ is mapped to the class
$\{ (\psi, \mu, \Lambda) \} /\F_q^*$.\\
This shows that we have a bijection between the set $$\{ \mbox{all pairs
$(\varphi} \def\g{\gamma, \lambda)$ with $P'(\varphi} \def\g{\gamma)$} \} / \Sigma$$ and the set
$$\{ \mbox{all triples $(\psi, \mu, \Lambda)$ with $(\psi, \mu)$ over $V$
and $(\psi_f)^{-1}\Lambda \subset K_V$} \}/\F_q^*.$$
\begin{remark}
Similarly, the above argument shows that the injective map of Remark
\ref{rem_injpair} is a bijection.
\end{remark}
\par\bigskip\noindent
This bijection can be rephrased in terms of isomorphy classes of
pairs $(\varphi} \def\g{\gamma, \lambda)$ and triples $(\psi, \mu, \Lambda)$ as follows.
We say that a triple $(\psi, \mu, \Lambda)$ is defined over $V$ if the
pair $(\psi, \mu)$ is defined over $V$. Two triples $(\psi, \mu,
\Lambda)$ and $(\psi', \mu', \Lambda')$ over $V$ are called {\it
isomorphic} if there exists an element $v \in V^*$ with $$(v \psi
v^{-1}, v \mu, v\Lambda) = (\psi', \mu', \Lambda').$$ Note that
\begin{equation} \label{eq_iso2}
\{(\psi, \mu, \Lambda) \mbox{~over $V$} \}/V^* \stackrel{\rm bij}{\cong}
\{ (\psi, \mu, \Lambda)_{K_V} \}/ K_V^*.
\end{equation}
\noindent
Moreover, if $v \in V^*$, then $$v: (\varphi} \def\g{\gamma, \lambda) \mapsto (v
\varphi} \def\g{\gamma v^{-1}, v \lambda)$$ and $$v: (\psi, \mu, \Lambda) \mapsto (v
\psi v^{-1}, v \mu, v \Lambda).$$ Dividing out the action of $V^*$ and
considering the bijections (\ref{eq_iso1}) and (\ref{eq_iso2}) gives
the following theorem.
\begin{theorem} \label{thm_reduct}
Let $\Sigma = \{ \s \in \Gl_2(A/fA) \mid \det(\s) \in \F_q^* \}.$
There is a bijection between the following two sets:
\begin{enumerate}
\item Isomorphy classes of pairs $(\varphi} \def\g{\gamma, \lambda)$ over
$K_V$ modulo $\Sigma$ where $\varphi} \def\g{\gamma$ is a Drinfeld module
of rank $2$ with stable reduction of rank $1$ and $\lambda$ is
a full level $f$-structure over $K_V$.
\item Isomorphy classes of triples $(\psi, \mu, \Lambda)$
over $K_V$, where $\psi$ is a rank $1$ Drinfeld module over
$K_V$, $\mu$ is a full level $f$-structure over $K_V$ and
$\Lambda$ is an $A$-lattice of rank $1$, such that
$(\psi_f)^{-1}\Lambda \subset K_V$.
\end{enumerate}
\end{theorem}
\begin{proof}
This theorem follows from the bijection that we have given above.
\end{proof}
\section{Tate-Drinfeld modules}
\label{sec_TD}
\noindent
We follow the approach of \cite[2.2]{Boe02} and \cite{PT2} to
construct the Tate-Drinfeld module of type $\mf m$. The Tate-Drinfeld
module describes the formal neighbourhood of the cusps of the moduli scheme.
At the cusps the universal Drinfeld module with level structure degenerates
into a Drinfeld module with stable reduction. Therefore, to define
the Tate-Drinfeld module, we use the description of the stable reduction
modules as given in the previous section.\\
Let $(\psi, \mu)$ be the universal Drinfeld module of rank $1$ with
level $f$-structure over $R$. Because $f$ is invertible in $R$,
we may assume that the generator $\mu(1)$ of the $f$-torsion of $\psi$
is an invertible element in $R$. So we may and will assume that
$\mu(1) = 1$. Then by push-forward via the embeddings
$$R \longrightarrow} \def\ll{\longleftarrow R[\![x]\!] \longrightarrow} \def\ll{\longleftarrow R(\!(x)\!)$$ one has a Drinfeld module of rank $1$ with
level $f$-structure on both $R[\![x]\!]$ and $R(\!(x)\!)$.
For an element $y = \sum_{i
\geq k} r_i x^i \in R(\!(x)\!)$ with $r_k \neq 0, k \in \Z,$ we define its
valuation $v_x(y)$ to be the $x$-valuation considered as element in
$K_R(\!(x)\!)$, and $v_x(y) = k$.
\par\bigskip\noindent
To construct a Drinfeld module of rank $2$ over $R(\!(x)\!)$, we first
construct a lattice $\Lambda_{\mf m} \subset K_R(\!(x)\!)$,
where $\mf m \subset A$ is an ideal. This lattice turns out to depend
only on the class of $\mf m$ in the class group of $A$. As before, we can
consider $\psi_f$ as a map $$\psi_f: K_R(\!(x)\!)^{\rm sep} \longrightarrow} \def\ll{\longleftarrow
K_R(\!(x)\!)^{\rm sep}.$$ The
lattice $\Lambda_{\mf m}$ will be constructed in such a way that
$(\psi_f)^{-1} \Lambda_{\mf m} \subset K_R(\!(x)\!)$. Applying Theorem
\ref{thm_reduct} to $(\psi, \mu, \Lambda_{\mf m})$ will give us the
Tate-Drinfeld module $\varphi} \def\g{\gamma$.
\subsection{The construction of the lattice}
\noindent
Let $\mf m \subset A$ be an ideal. To prepare the construction of the lattice,
note the following:\\
1. There is a unique monic skew polynomial $P \in K_R \{ \t \}$ with
minimal degree such that $$\ker(P)(K_R^{\rm sep}) =
\psi[\mf m](K_R^{\rm sep}).$$ In fact, $P \in R \{ \t \}$
because the elements in $\ker(P)(K_R^{\rm sep})$ are integral over $R$
and $R$ is integrally closed.\\
2. Because $A_f \hookrightarrow K_R$, the extension
$K_R(\psi[\mf m])/K_R$ is Galois. Moreover, because
$\psi[\mf m](K_R^{\rm sep}) \cong A/\mf m$, there is an injective
representation $$\operatorname{Gal}(K_R(\psi[\mf m])/K_R) \longrightarrow} \def\ll{\longleftarrow (A/\mf m)^*.$$ So the Galois
group of this extension is a subgroup of $(A/\mf m)^*$.\\
3. The field $K_R(\psi[\mf m])(\!(y)\!)$ is the splitting field of the equation
$P(\frac{1}{y}) = \frac{1}{x}$ over $K_R(\!(x)\!)$. Then $$K_R(\psi[\mf
m])(\!(y)\!)/K_R(\psi[\mf m])(\!(x)\!)$$ is a Galois extension which is totally
ramified and its Galois group is isomorphic to $A/\mf m$. The
Galois action is given by $y \mapsto y + \alpha$ with $\alpha \in
\psi[\mf m](K_R(\psi[\mf m]))$.
\par\bigskip\noindent
Let $l$ be the following $A$-module homomorphism:
$$l: f^{-1} A \longrightarrow} \def\ll{\longleftarrow K_R(\psi[\mf m])(\!(y)\!) \quad {\rm by} \quad
f^{-1} a \mapsto \psi_a \left(\frac{1}{y} \right).$$
We use $l$ to define the lattice $\Lambda_{\mf m}$.
\begin{lemma}
The $A$-module $l(f^{-1} \mf m)$ lies inside $R(\!(x)\!)$.
\end{lemma}
\begin{proof}
For every $m \in \mf m$, there exists a skew polynomial $Q \in
R \{ \t \}$ such that $\psi_m = Q \cdot P$. Note that we use here that
one has division with remainder in the skew ring $R\{ \t \}$,
because the leading coefficients of both $P$ and $\psi_m$ are in $R^*$.
Consequently, $\psi_m(\frac{1}{y}) = Q(\frac{1}{x}) \in R(\!(x)\!)$.
\end{proof}
\begin{remark}
Let $m_1, m_2$ generate $\mf m$, then there
are elements $Q_i \in R\{ \t \}$ with $Q_i \circ P = \psi_{m_i}$. We will
use this in the following a few times without further mentioning it.
\end{remark}
\noindent
Define the lattice $\Lambda$ as follows:
$$\Lambda := l(\mf m), \quad
{\rm then} \quad (\psi_f)^{-1}\Lambda = l(f^{-1}\mf m) + \psi[f](R)
\subset R(\!(x)\!).$$
\begin{lemma} \label{lem_latcl}
The lattice $\Lambda$ only depends on the class of $\mf m$ in ${\rm
Cl}(A)$.
\end{lemma}
\begin{proof}
Suppose that $\mf m' \subset A$ is another ideal representing the
same class as $\mf m$ in ${\rm Cl}(A)$. Then there exist elements
$b, b' \in A$ with $b' \mf m' = b \mf m$. So we may reduce to
case where $\mf m' = b \mf m$ for some $b \in A$. The extension
$K_R(\psi[\mf m'])/K_R(\psi[\mf m])$ is given by adding the roots of
the polynomial $\psi_b(X)$ to $K_R(\psi[\mf m])$.
The extension $$K_R(\psi[\mf m'])(\!(y')\!)/K_R(\psi[\mf m'])(\!(y)\!)$$
is given
by adding the roots of the equation $\psi_b(\frac{1}{y'}) = \frac{1}{y}$.
It is not difficult to see that $\mf m$ and $\mf m'$ give the same
$\Lambda$.
\end{proof}
\noindent
By this lemma it makes sense to talk about the type $\mf m$ in ${\rm Cl}(A)$
of the lattice. We will denote $\mf m$ for both the ideal in $A$ as for the
ideal class in ${\rm Cl}(A)$. Moreover, this gives us some freedom in choosing
$\mf m$ to construct a lattice of type $\mf m$:
\begin{lemma} \label{lem_choice-m}
Let $\mf a, \mf m \subset A$ be non-zero ideals, then there is an
element $x \in K_A$, the quotient field of $A$, such that
$x \mf m + \mf a = A$.
\end{lemma}
\begin{proof}
By Proposition VII.5.9 in \cite{Bou}, there is an element $x \in K_A$ with
$v_{\mf p}(x) = - v_{\mf p}(\mf m)$ for all primes $\mf p \subset A$
dividing $\mf a$ and $v_{\mf p}(x) \geq 0$ for all other primes $\mf p$ of
$A$. Consequently, $x \mf m \subset A$, and there is no prime ideal $\mf p$
of $A$ dividing both $x \mf m$ and $\mf a$.
\end{proof}
\noindent
This lemma shows that we can choose a representative $\mf m$ of the class
type $[\mf m]$ of the lattice such that this representative is
relatively prime to some chosen ideal $\mf a \subset A$. This gives some
help in technical parts of some proofs later on.
\par\bigskip\noindent
We write $\Lambda_{\mf m}$ for the lattice $\Lambda$ of type $\mf m$
that we constructed above. Note that $K_R(\!(x)\!)$ is the quotient
field of the complete discrete valuation ring $K_R[\![x]\!]$. By the
constructions in Section \ref{sec_redth} and Theorem \ref{thm_reduct},
we may associate to the triple $(\psi, \mu, \Lambda_{\mf m})$ a unique Drinfeld
module $\varphi} \def\g{\gamma$ of rank $2$ over $K_R(\!(x)\!)$ with stable reduction of rank
$1$. Moreover, the $f$-torsion of $\varphi} \def\g{\gamma$ is $K_R(\!(x)\!)$-rational.
\par\bigskip\noindent
In fact, $\varphi} \def\g{\gamma$ is a Drinfeld module over $R(\!(x)\!)$, as we will now
show.
Using Theorem \ref{thm_Dri}, the corresponding exponential map is
$$e_{\Lambda_{\mf m}}(z) := z
\prod_{\alpha \in \Lambda_{\mf m} \bs \{ 0 \}} \left( 1- \frac{z}{\alpha}
\right),$$ and $\varphi} \def\g{\gamma$ is determined by the following diagram, which
commutes for all $a \in A$:
\begin{equation} \label{diag2}
\begin{CD}
0 @>>> \Lambda_{\mf m} @>>> K_R(\!(x)\!) @> e_{\Lambda_{\mf m}} >>
K_R(\!(x)\!) \\
& & @VV \psi_a V @VV \psi_a V @VV \varphi} \def\g{\gamma_a V\\
0 @>>> \Lambda_{\mf m} @>>> K_R(\!(x)\!) @> e_{\Lambda_{\mf m}} >>
K_R(\!(x)\!).\\
\end{CD}
\end{equation}
\noindent
Note that by construction $\Lambda_{\mf m} \subset R(\!(x)\!)$, each
non-zero element of $\Lambda_{\mf m}$ has negative $x$-valuation
and the leading coefficient of this non-zero element is in $R^*$.
Consequently, the map $e_{\Lambda_{\mf m}}(z) = 1 + \sum_{i\geq 1} s_i z^i$
has all its coefficients $s_i \in R[\![x]\!]$. So its inverse exists in
$R[\![x]\!][\![z]\!]$ and $$\varphi} \def\g{\gamma_a = e_{\Lambda_{\mf m}} \circ \psi_a \circ
e_{\Lambda_{\mf m}}^{-1}.$$ Therefore, $\varphi} \def\g{\gamma_a$ has its coefficients in
$R[\![x]\!]$ for all $a \in A$.
\begin{lemma}
The ring homomorphism $\varphi} \def\g{\gamma$ is a Drinfeld module over $R(\!(x)\!)$.
\end{lemma}
\begin{proof}
We only need to prove that the leading coefficient of $\varphi} \def\g{\gamma_a$ is an
element of $R(\!(x)\!)^*$. To prove this, we simply copy the computation
of Lemma 2.10 in \cite{Boe02}.
Note that $$\varphi} \def\g{\gamma_a(z) = a z \prod_{\alpha \in ((\psi_a)^{-1} \Lambda_{\mf m}
/\Lambda_{\mf m}) \bs \{ 0 \} } \left( 1 - \frac{z}{e_{\Lambda_{\mf m}}
(\alpha)} \right).$$ So we need to show that
$$(*) \quad \prod_{\alpha \in ((\psi_a)^{-1} \Lambda_{\mf m}
/\Lambda_{\mf m}) \bs \{ 0 \} } e_{\Lambda_{\mf m}}(\alpha) =
a \cdot u$$ with $u \in R(\!(x)\!)^*$\\
Every
$\alpha \in ((\psi_a)^{-1} \Lambda_{\mf m} /\Lambda_{\mf m}) \bs \{ 0 \}$
which is not in $\psi[a]$ can be written uniquely as $\alpha = \alpha_1
+ \alpha_2$ where $\alpha_1$ runs through $\psi[a]$ and $\alpha_2$ runs
through a set of representatives in
$\psi_a^{-1}\Lambda_{\mf m}/\Lambda_{\mf m}$ of
the non-zero elements of $(\psi_a)^{-1} \Lambda_{\mf m}
/(\Lambda_{\mf m}+\psi[a])$. This set is denoted by $S_1$. The remaining
elements $\alpha$ can be written as $\alpha = \alpha_1$ where
$\alpha_1$ runs through $\psi[\mf m] \bs \{ 0 \}$. This set is denoted
by $S_2$.\\
By definition $$e_{\Lambda_{\mf m}}(\alpha) = \alpha \prod_{\beta \in
\Lambda_{\mf m} \bs \{ 0 \}} \left( 1- \frac{\alpha}{\beta} \right).$$
Using the rule $$\psi_a(z) = \prod_{\alpha_1 \in \psi[a]} (z - \alpha_1),$$
we see that $$\prod_{S_1} e_{\Lambda_{\mf m}}(\alpha) =
\prod_{\alpha_2 \neq 0} \left( \psi_a(\alpha_2) \prod_{\beta \in
\Lambda_{\mf m} \bs \{ 0 \}} \left( \frac{\psi_a (\beta - \alpha_2)}
{\beta^{\# A/(a)}} \right) \right).$$ This is in fact an element in
$R(\!(x)\!)^*$, which can be seen as follows. Clearly, $0 \neq
\psi_a(\alpha_2) \in \Lambda_{\mf m}$, so this element is in $R(\!(x)\!)^*$.
Moreover, the element $\psi_a (\beta - \alpha_2) \in \Lambda_{\mf m}$
cannot be $0$: if it were $0$, then any representative
of $\alpha_2$ would lie in $\beta + \psi[a] \subset \Lambda_{\mf m} +
\psi[a]$, i.e., the class $\alpha_2 = 0$,
contradicting the definition of the set $S_2$. So also
$\psi_a (\beta - \alpha_2) \in R(\!(x)\!)^*$. Finally, for almost all
$\beta$ we have that
$$\frac{\psi_a (\beta - \alpha_2)}{\beta^{\# A/(a)}} \in
R[\![x]\!]^*.$$ So the product exists, and $$\prod_{S_1}
e_{\Lambda_{\mf m}}(\alpha) \in R(\!(x)\!)^*.$$
On the other hand, using the rule $$h(z):= \frac{\psi_a(z)}{z} =
\prod_{\alpha_1 \in S_2} (z - \alpha_1)$$ and recalling that $h(0) = a$,
we see that $$\prod_{S_2} e_{\Lambda_{\mf m}}(\alpha) =
a \cdot \prod_{\beta \in \Lambda_{\mf m} \bs \{ 0 \}}
\left( \frac{h(\beta)}{\beta^{\# A/(a) - 1}} \right),$$ with each
$\frac{h(\beta)}{\beta^{\# A/(a) - 1}}$ is in $R[\![x]\!]^*$.\\
Finally, $$\prod_{\alpha \in ((\psi_a)^{-1} \Lambda_{\mf m}
/\Lambda_{\mf m}) \bs \{ 0 \} } e_{\Lambda_{\mf m}}(\alpha) =
\prod_{S_1} e_{\Lambda_{\mf m}}(\alpha) \cdot \prod_{S_2}
e_{\Lambda_{\mf m}}(\alpha).$$ This finishes the proof.
\end{proof}
\par\bigskip\noindent
From the construction of the Drinfeld module $\varphi} \def\g{\gamma$ with $\theta$ over
$R(\!(x)\!)$, coming from the triple $(\psi, \mu, \Lambda_{\mf m})$, we
can deduce at once the following list of properties:
\begin{enumerate} \label{list_tdmod}
\item $\varphi} \def\g{\gamma: A \longrightarrow} \def\ll{\longleftarrow R[\![x]\!]\{ \t \}$ is a ring homomorphism;
\item the $f$-torsion of $\varphi} \def\g{\gamma$ is $R(\!(x)\!)$-rational;
\item there is an isomorphism $$A/fA \stackrel{\sim}{\longrightarrow} \def\ll{\longleftarrow}
\varphi} \def\g{\gamma[f](R[\![x]\!])$$ given by $1 \mapsto e_{\Lambda_{\mf m}}(\mu(1))$.
\item $\varphi} \def\g{\gamma~{\rm mod}~xR[\![x]\!] = \psi$, because
$e_{\Lambda_{\mf m}}(z)~{\rm mod}~xR[\![x]\!]$ is the identity map.
\end{enumerate}
\noindent
As in Section \ref{sec_redth}, the triple $(\psi, \mu, \Lambda_{\mf m})$
induces on $\varphi} \def\g{\gamma$ a level $f$-structure $\lambda$ with
$\lambda(0,1) = e_{\Lambda_{\mf m}}(\mu(1))$ and $\lambda(1,0)$ is
determined up to $\F_q^*$ by the Weil pairing.
\par\bigskip\noindent
In this way, we get for every element $\mf m \in {\rm Cl}(A)$ a pair
$(\varphi} \def\g{\gamma^{\mf m}, \lambda^{\mf m})$. The action of $\s \in \Gl_2(A/fA)$ on
this pair is given by $$\s: (\varphi} \def\g{\gamma^{\mf m}, \lambda^{\mf m}) \longrightarrow} \def\ll{\longleftarrow
(\varphi} \def\g{\gamma^{\mf m}, \lambda^{\mf m} \circ \s).$$
The action of the class group of $A$ is described in the following lemma.
Let $\mf n \in {\rm Cl}(A)$ and let $g_{\mf n}$ denote the $A_f$-linear
automorphism of $R$ which describes the action of $\mf n$ on $R$.
Let $\Lambda_{\mf n^{-1} \mf m}$ denote the lattice of type
$\mf n^{-1}\mf m$.
\begin{lemma} \label{lem_actClA}
The element $\mf n$ maps the triple $(\psi, \mu, \Lambda_{\mf m})$
to the triple $$(g_{\mf n}(\psi), g_{\mf n}(\mu),
g_{\mf n}(\Lambda_{\mf n^{-1}\mf m})).$$
\end{lemma}
\begin{proof}
We choose representatives $\mf m, \mf n \subset A$ of the classes
of $\mf m$ and $\mf n$ in ${\rm Cl}(A)$ such that
$\mf n^{-1} \mf m \subset A$, and $\mf n$ and $\mf m$ are relatively
prime to $f$.\\
Write $\varphi} \def\g{\gamma = \varphi} \def\g{\gamma^{\mf m}$.
The action of $\mf n$ on $\varphi} \def\g{\gamma$ is given by a unique monic skew
polynomial $Q$ with minimal degree such that
$$\ker(Q)(K_R(\!(x)\!)^{\rm sep}) = \varphi} \def\g{\gamma[\mf n](K_R(\!(x)\!)^{\rm
sep}).$$ Let $\varphi} \def\g{\gamma'$ be the image under $Q$: $$\varphi} \def\g{\gamma
\stackrel{Q}{\longrightarrow} \def\ll{\longleftarrow} \varphi} \def\g{\gamma'.$$ Writing $\Lambda_{\mf m} = l(\mf m)$
as before, it is not difficult to see that
$$\ker(Q\circ e_{\Lambda_{\mf m}})(K_R(\!(x)\!)^{\rm sep}) =
l(\mf n^{-1} \mf m) + \psi[\mf n](K_R(\!(x)\!)^{\rm sep}).$$
The action of $\mf n$ on $R$, denoted by $g_{\mf n}$, corresponds to
a skew polynomial $P \in R\{ \t \}$ with $\ker(P)(K_R^{\rm sep}) =
\psi[\mf n](K_R^{\rm sep})$. Let $\Lambda'$ be the
lattice given by $$\Lambda':= P\circ
\ker(Q\circ e_{\Lambda_{\mf m}})(K_R(\!(x)\!)^{\rm sep}) =
P \circ l(\mf n^{-1} \mf m) $$ $$= \{ g_{\mf n}(\psi_{fa})
P(\frac{1}{y}) \mid a \in \mf n^{-1} \mf m \}.$$ Then $\Lambda' =
g_{\mf n}(\Lambda_{\mf n^{-1}\mf m})$, and it is not difficult to see
that $\varphi} \def\g{\gamma'$ corresponds to the pair $(g_{\mf n}(\psi),
g_{\mf n}(\Lambda_{\mf n^{-1}\mf m}))$.
\end{proof}
\par\bigskip\noindent
As a corollary to this lemma, we see that $\mf n$ maps $\varphi} \def\g{\gamma^{\mf m}$ to
$g_{\mf n}(\varphi} \def\g{\gamma^{\mf n^{-1} \mf m})$. As the definition of $\lambda^{\mf m}$
and $\lambda^{\mf n^{-1} \mf m}$ depend on a choice of $\F_q^*$, we cannot
say that $\mf n$ maps $(\varphi} \def\g{\gamma^{\mf m}, \lambda^{\mf m})$ to
$(\varphi} \def\g{\gamma^{\mf n^{-1} \mf m}, \lambda^{\mf n^{-1} \mf m})$.
Bearing this in mind, we propose for every type $\mf m$ the following
definition of the pairs $(\tdm, \ldm)$. For $\mf m = A$ we define
$(\tdm, \ldm) := (\varphi} \def\g{\gamma^{\mf m}, \lambda^{\mf m}).$
This pair is unique up to $\F_q^*$. For the rest of this paper we keep
this choice is fixed. In general, $\mf m^{-1}$ gives rise
to an automorphism $g_{\mf m^{-1}}$ of $R$. We extend this to an automorphism
of $R[\![x]\!]$ by letting it act trivially on $x$. We define
$(\tdm, \ldm)$ to be the image of $(\ph^{\rm td}} \def\tdm{\ph^{\rm td}_{\mf m}_{A}, \lambda^{\rm td}} \def\ldm{\lambda^{\rm td}_{\mf m}_{A})$ under $\mf m^{-1}$.
\begin{definition} \label{def_TDlev}
The pair $(\tdm, \ldm)$ is called the {\em standard Tate-Drinfeld
module of rank $2$ and type $\mf m$ with level $f$-structure}.
A pair $(\tdm, \lambda)$, where $\lambda$ is any level $f$-structure
over $R(\!(x)\!)$ is called a {\it Tate-Drinfeld module with level
$f$-structure.}
\end{definition}
\noindent
The pairs $(\xi \tdm \xi^{-1}, \xi \lambda)$ with $\xi \in R[\![x]\!]^*$
all belong to the same isomorphy class $(\tdm, \lambda)$.
Every isomorphy class of Tate-Drinfeld modules with level $f$-structure
comes from a unique morphism $${\rm Spec}(R(\!(x)\!)) \longrightarrow} \def\ll{\longleftarrow M^2(f).$$
Because we always assume that $\mu(1) = 1$, the pair $(\tdm, \ldm)$ is fixed
in its isomorphy class. For every $\lambda$ there is a unique $\s \in
\Gl_2(A/fA)$ with $\lambda = \ldm \circ \s$. Therefore, the pair
$(\tdm, \lambda)$ is fixed in its isomorphy class.
\section{The universal Tate-Drinfeld module}
\label{sec_UP}
\noindent
In this section we introduce the universal Tate-Drinfeld module.
This is a scheme $\Cal Z$ consisting of the coproduct of a number
of copies of ${\rm Spec}(R[\![x]\!])$ and which is equipped with a
Tate-Drinfeld structure $(\ph^{\rm td}} \def\tdm{\ph^{\rm td}_{\mf m}, \lambda^{\rm td}} \def\ldm{\lambda^{\rm td}_{\mf m})$. Let $N \subset
\Gl_2(A/fA)$ be the subgroup $$N = \left( \begin{array}{cc} \F_q^* & A/fA \\
0 & (A/fA)^* \end{array} \right).$$ Choose representatives
$\s_1, \ldots, \s_n$ of the cosets of $N \s_i$ in
$\Gl_2(A/fA)$ and let $\s_1$ be the identity. As $\det(N) = (A/fA)^*$,
we may assume that the representatives $\s_i$ are elements of ${\rm Sl}} \def\Gl{{\rm Gl}} \def\Aut{{\rm Aut}_2(A/fA)$.
Set $$\Cal Z = {\rm Spec} \left( \underset{(\mf m, \s_i)}{\oplus}
R[\![x]\!]_{(\mf m, \s_i)} \right),$$
The pair $(\ph^{\rm td}} \def\tdm{\ph^{\rm td}_{\mf m}, \lambda^{\rm td}} \def\ldm{\lambda^{\rm td}_{\mf m})$ is the Drinfeld module $\ph^{\rm td}} \def\tdm{\ph^{\rm td}_{\mf m}$ of rank $2$ with
level $f$-structure $\lambda^{\rm td}} \def\ldm{\lambda^{\rm td}_{\mf m}$ over ${\rm Spec} \left( \underset{(\mf m, \s_i)}{\oplus}
R(\!(x)\!)_{(\mf m, \s_i)} \right)$ such that the
restriction of $(\ph^{\rm td}} \def\tdm{\ph^{\rm td}_{\mf m}, \lambda^{\rm td}} \def\ldm{\lambda^{\rm td}_{\mf m})$ to $R(\!(x)\!)_{(\mf m, \s_i)}$ is equal to
$(\tdm, \ldm \circ \s_i)$ for every pair $(\mf m, \s_i)$.
\begin{definition}
The {\it universal Tate-Drinfeld module of rank $2$ with level
$f$-structure} is the scheme $\Cal Z$ together with the pair $(\ph^{\rm td}} \def\tdm{\ph^{\rm td}_{\mf m}, \lambda^{\rm td}} \def\ldm{\lambda^{\rm td}_{\mf m})$.
\end{definition}
\begin{remark} \label{rem_Act}
As before, a pair $(\mf m, \s_i) \in {\rm Cl}(A)
\times \Gl_2(A/fA)$ determines a unique action on $(\ph^{\rm td}} \def\tdm{\ph^{\rm td}_{\mf m}_{A}, \lambda^{\rm td}} \def\ldm{\lambda^{\rm td}_{\mf m}_{A})$.
And by definition we have $$(\tdm, \ldm \circ \s_i) = (\mf m^{-1}, \s_i)
(\ph^{\rm td}} \def\tdm{\ph^{\rm td}_{\mf m}_{A}, \lambda^{\rm td}} \def\ldm{\lambda^{\rm td}_{\mf m}_{A}).$$
The action of ${\rm Cl}(A) \times \Gl_2(A/fA)$ on the universal
Tate-Drinfeld module
is given by this action of $(\mf m, \s_i)$ and the fact that $N$ acts
trivially.
\end{remark}
\par\bigskip\noindent
Clearly, we would like to have a certain universal property for this universal
Tate-Drinfeld module. The weak versions of this universal property that
we need can be found in Theorems \ref{thm_univTD} and
\ref{thm_STD_up}. In Proposition \ref{prop_STD_up} the main work is done
for Theorem \ref{thm_STD_up}. This proposition explains the
subgroup $N$.
\begin{prop} \label{prop_STD_up}
Let $\s \in \Gl_2(A/fA)$. There exists an $A_f$-linear ring homomorphism
$h_{\s}: R[\![x]\!] \longrightarrow} \def\ll{\longleftarrow R[\![x]\!]$ such that $$h_{\s}(\tdm, \ldm) \cong
(\tdm, \ldm \circ \s)$$ if and only if $\s \in N$.
If $\s \in N$, then $h_{\s}$ is given by $\det(\s): R \longrightarrow} \def\ll{\longleftarrow R$ and
$x \mapsto \delta x$ for some $\delta \in R[\![x]\!]^*$.
Moreover, for any $\s \in N$ the image of $(\tdm, \ldm \circ \s_i)$
under $h_{\s}$ is isomorphic to $(\tdm, \ldm \circ \s \circ \s_i)$.
\end{prop}
\begin{proof}
We write $\s = (\s_{i,j})$. First suppose that $h_{\s}$ exists. Because
$h_{\s}$ is determined by $\det(\s)$ and $h_{\s}(x)$, it must respect
the ordering of the $x$-valuation. This implies that
$h_{\s} (\lambda(1,0))$ must
have minimal negative valuation and $h_{\s} (\lambda(0,1))$ must
have valuation $0$. Hence the only $\s$'s whose action may come from an
$A_f$-linear ring homomorphism $h_{\s}$ are $\s \in N$. This shows the
'only if'.\\
We prove the 'if'-part in two steps. Let $\s = (\s_{i,j}) \in N$.\\
1. First suppose that $\s = \left( \begin{array}{cc} 1 & 0 \\ 0 & \alpha
\end{array} \right)$ with $\alpha \in (A/fA)^*$. The action of $\s$
induces an action $$\alpha = \det(\s): M^1(f) \longrightarrow} \def\ll{\longleftarrow M^1(f),$$ i.e., there
is an $A_f$-linear ring homomorphism $h_{\alpha}: R \longrightarrow} \def\ll{\longleftarrow R$ such that
$$(h_{\alpha}(\psi), h_{\alpha}(\mu)) = (\xi \psi \xi^{-1}, \xi \mu
\alpha) \cong (\psi, \mu \alpha),$$ for some $\xi \in R^*$.
We use the notations of Section \ref{sec_TD}. Recall that
the element $\frac{1}{y}$ was used to define the lattice
$\Lambda^{\rm td}_{\mf m}$ and that $\frac{1}{x} = P(\frac{1}{y})$.
$P \in R\{ \t \}$ is the skew polynomial of minimal degree with $$\ker(P)
(K_R^{\rm sep}) = \psi[\mf m](K_R^{\rm sep}).$$ The map $\frac{1}{y}
\mapsto \xi \frac{1}{y}$ induces, by applying $P$, $$\frac{1}{x} \mapsto
h_{\alpha}(P)(\xi \frac{1}{y}) = \delta^{-1} \frac{1}{x} \quad \mbox{for
some $\delta^{-1} \in R[\![x]\!]^*$.}$$ To see this, note that $h_{\alpha}(P)
= \zeta P \xi^{-1}$ for some $\zeta \in R[\![x]\!]^*$. (In fact, $\zeta$
is determined by the fact that $h_{\alpha}(P)$ is monic.)\\
The map $h_{\alpha}$ is extended to a ring homomorphism
$$h_{\s}: R[\![x]\!] \longrightarrow} \def\ll{\longleftarrow R[\![x]\!], \quad x \mapsto \delta x,
\quad h_{\s}(r) = h_{\alpha}(r) \quad \forall r \in R.$$
An easy computation shows that $$h_{\s}(\Lambda^{\rm td}_{\mf m}) = \xi
\Lambda^{\rm td}_{\mf m},\quad \mbox{and thus} \quad h_{\s}(\tdm) = \xi
\tdm \xi^{-1}.$$ Using that there is an element $m \in \mf m$ with
$\lambda(1,0) = e_{\Lambda^{\rm td}_{\mf m}}(\psi_m(\frac{1}{y}))$, we see
that
$$h_{\s}
\left( \begin{array}{c} \lambda(1,0) \\ \lambda(0,1) \end{array} \right) =
\left( \begin{array}{c} e_{\xi \Lambda^{\rm td}_{\mf m}}(\xi
\psi_m(\frac{1}{y})) \\ e_{\xi \Lambda^{\rm td}_{\mf m}}(\xi \mu(1) \alpha)
\end{array} \right) =
\left( \begin{array}{c} \xi \lambda(1,0) \\ \xi \lambda(0,1) \alpha
\end{array} \right).$$
And so indeed, $$h_{\s}(\tdm, \lambda) = (\xi \tdm \xi^{-1},
\xi \lambda \circ \s) \cong (\tdm, \lambda \circ \s).$$
2. Having dealt with the first case, we may assume that $\s \in N$ and
$\det(\s) \in \F_q^*$. As a consequence, the map $h_{\s}$ that we are looking
for is $R$-linear and in particular, if $(\psi, \mu, \Lambda)$ is the
triple associated to $(\tdm, \lambda)$, then $h_{\s}(\psi) = \psi$
and $h_{\s}(\mu) = \mu$.\\ As always, the action of $\F_q^* \cdot \left(
\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)$ is trivial, so we
may assume that $\s_{2,2} = 1$.
We first give a proof in the simple case $\Lambda \cong A$ and then prove
it for general $\Lambda$.\\
i. Suppose $\Lambda \cong A$. Note that in that case, the elements
$e_{\Lambda}(\frac{1}{x})$ and $e_{\Lambda}(\mu(1))$ generate the
$f$-torsion
of $\ph^{\rm td}} \def\tdm{\ph^{\rm td}_{\mf m}$. In fact, there is a basis transformation $$\alpha = \left(
\begin{array}{cc} \alpha_{1,1} & \alpha_{1,2} \\ 0 & 1 \end{array} \right)
\quad \mbox{with} \quad \alpha
\left( \begin{array}{c} \lambda(1,0) \\ \lambda(0,1) \end{array} \right) =
\left( \begin{array}{c} e_{\Lambda}(\frac{1}{x}) \\ e_{\Lambda}(\mu(1))
\end{array} \right)$$ and with $\alpha_{1,1} \in (A/fA)^*$,
$\alpha_{1,2} \in A/fA$.\\
Let $\rho = \alpha \s \alpha^{-1}$, then $\rho = \left(
\begin{array}{cc} \rho_{1,1} & \rho_{1,2} \\ 0 & 1 \end{array} \right)$
with $\rho_{1,1} \in \F_q^*$, $\rho_{1,2} \in A/fA$.
We set $$\delta^{-1} = \rho_{1,1} + \rho_{1,2} (\mu(1)) \cdot x \in
R[\![x]\!]^*,$$ and we define $h_{\s}$ to be the $R$-linear map given by
$$h_{\s}: x \mapsto \delta \cdot x.$$ Because $$\psi_f(\delta^{-1}
\frac{1}{x}) = \rho_{1,1} \psi_f(\frac{1}{x}),$$ it follows that
$h_{\s}(\Lambda) = \Lambda$, and thus $h_{\s}$ commutes with $e_{\Lambda}$.
Moreover $$h_{\s} (\psi, \mu, \Lambda) = (\psi, \mu, \Lambda).$$ Therefore
$$h_{\s}(\tdm) = \tdm.$$
So $h_{\s}$ commutes with the $A$-action and with $e_{\Lambda}$. To see
what happens on the level structure $\lambda$ is now an easy computation.
$$h_{\s}
\left( \begin{array}{c} \lambda(1,0) \\ \lambda(0,1) \end{array} \right) =
h_{\s} \alpha^{-1}
\left( \begin{array}{c} e_{\Lambda}(\frac{1}{x}) \\ e_{\Lambda}(\mu(1))
\end{array} \right) =$$
$$\alpha^{-1} \rho \left( \begin{array}{c} e_{\Lambda}(\frac{1}{x}) \\
e_{\Lambda}(\mu(1)) \end{array} \right) =
\s \left( \begin{array}{c} \lambda(1,0) \\ \lambda(0,1) \end{array}
\right).$$
We conclude that $$h_{\s}(\varphi} \def\g{\gamma^{\rm td}, \lambda^{\rm td}) =
(\varphi} \def\g{\gamma^{\rm td}, \lambda^{\rm td} \circ \s).$$
ii. In general, let $\Lambda \cong \mf m$ and recall the construction of
$\Lambda$ in Section \ref{sec_TD}. Using the same notations as in Section
\ref{sec_TD}, let $m \in \mf m$ be an element such that the image of
$\frac{1}{z} := \psi_m(\frac{1}{y})$ and $\mu(1)$ under $e_{\Lambda}$
generate the $f$-torsion of $\tdm$. We let $\alpha$ be as in i,
i.e., $$\alpha \left( \begin{array}{c} \lambda(1,0) \\ \lambda(0,1)
\end{array} \right) = \left( \begin{array}{c} e_{\Lambda}(\frac{1}{z}) \\
e_{\Lambda}(\mu(1)) \end{array} \right)$$ and $\rho = \alpha \s \alpha^{-1}$.
As in the previous case, we look for an $h_{\s}$ such that
$$\frac{1}{z} \mapsto (\rho_{1,1} + \rho_{1,2}
\mu(1) z) \cdot \frac{1}{z}.$$ This can be done as follows.
We start by assuming that $m \in (A/fA)^*$ - we will show below that we
may assume this in general. Let $b \in A$ such that $b \equiv m^{-1}
\rho_{1,2} \in A/fA$. Let $$\frac{1}{y} \mapsto
\rho_{1, 1} \frac{1}{y} + \psi_{b} \mu(1).$$ Applying $P$ gives our
candidate for $h_{\s}$: $$\frac{1}{x} \mapsto
\delta^{-1} \frac{1}{x} \quad ~{\rm with}~
\delta^{-1} = \rho_{1, 1} + P(\psi_{b} (\mu(1))) \cdot x.$$
By construction
$$h_{\s}: \frac{1}{z} \mapsto \rho_{1,1} \frac{1}{z} +
\psi_m(\psi_b (\mu(1))) = \rho_{1,1} \frac{1}{z} + \rho_{1,2} \mu(1).$$
Note that all elements of $\Lambda$ have the form $\psi_{f \tilde m}
(\frac{1}{y})$, with $\tilde m \in \mf m$. And clearly,
$$h_{\s} (\psi_{f \tilde m}(\frac{1}{y})) = \rho_{1,1} \psi_{f \tilde m}
(\frac{1}{y}).$$ Because $\rho_{1,1} \in \F_q^*$, we see that
$h_{\s}(\Lambda) = \Lambda$.\\
We can conclude the proof in the same way as in the case of $\Lambda
\cong A$.\\
Finally, it remains to be shown that $m$, which
we used to define $\frac{1}{z} = \psi_m(\frac{1}{y})$, is an element of
$(A/fA)^*$. By Lemma \ref{lem_choice-m} we may assume that $\mf m$ and
$(f)$ are relatively prime. Furthermore, because $\frac{1}{z}$ generates
one direct summand of the $f$-torsion, one has $\psi_b(\frac{1}{z})
\in \Lambda$ if and only if $b \in fA$, i.e., $b m \in f \mf m$ if and only
if $b \in fA$. Consequently, $(f) + (m) = A$.\\
The `moreover'-part of the proposition is obvious.
\end{proof}
\noindent
Using Proposition \ref{prop_STD_up}, we can immediately prove one weak form
of the universal property of the universal Tate-Drinfeld module.
\begin{theorem} \label{thm_STD_up}
For every Tate-Drinfeld module $(\tdm, \lambda)$ there is a unique
ring homomorphism $$h: \underset{(\mf m', \s_i)}{\oplus} R[\![x]\!]_{(\mf m',
\s_i)} \longrightarrow} \def\ll{\longleftarrow R[\![x]\!]$$
such that $$h(\ph^{\rm td}} \def\tdm{\ph^{\rm td}_{\mf m}, \lambda^{\rm td}} \def\ldm{\lambda^{\rm td}_{\mf m}) \cong (\tdm, \lambda).$$
\end{theorem}
\begin{proof}
Let $\s \in \Gl_2(A/fA)$ such that $\ldm \circ \s = \lambda$
and let $\s_k \in \{ \s_1, \ldots, \s_n \}$ such that $\s \in \s_k N$.
The map $h$ is defined as follows: $h$ is the zero-map on
$R[\![x]\!]_{(\mf m', \s_i)}$ if $\mf m' \neq \mf m$ in the class group
of $A$ or if
$\s_k \neq \s_i$. On $R[\![x]\!]_{(\mf m, \s_k)}$ it is the map defined
in Proposition \ref{prop_STD_up}.
To show uniqueness, note that any $A_f$-linear ring homomorphism $$h:
R[\![x]\!]_{(\mf m', \s_i)} \longrightarrow} \def\ll{\longleftarrow R[\![x]\!]$$ induces on $R[\![x]\!]$ a
Tate-Drinfeld
module whose corresponding lattice has type $\mf m'$. Hence,
any such ring homomorphism keeps the type of the Tate-Drinfeld module fixed.
Moreover, if there is an $A_f$-morphism
$$R[\![x]\!]_{(\mf m, \s_i)} \longrightarrow} \def\ll{\longleftarrow R[\![x]\!]_{(\mf m, \s_j)}$$ which
induces the Tate-Drinfeld structure, then there is a morphism
$$R[\![x]\!]_{(\mf m, \s_1)} \longrightarrow} \def\ll{\longleftarrow R[\![x]\!]_{(\mf m, \s_j \s_i^{-1})},$$
and thus by Proposition \ref{prop_STD_up} we see that
$\s_j \s_i^{-1} \in N$. So $\s_i = \s_j$.
\end{proof}
\subsection{The weak version of the universal property of $\Cal Z$}
\noindent
Let, as in Section \ref{sec_redth}, $V$ be a complete discrete valuation
$A_f$-ring, let $\pi$ be a generator of its maximal ideal, and let $K_V$
be its field of
fractions. Let $(\varphi} \def\g{\gamma, \lambda)$ be a Drinfeld module $\varphi} \def\g{\gamma$ of rank $2$ over
$K_V$ with level $f$-structure $\lambda$ such that $\varphi} \def\g{\gamma$ has
stable reduction of rank $1$ at $\pi$. In this subsection we
discuss the other weak version of the universal property of the
universal Tate-Drinfeld module, which we
need in the next section. We prove that there exists a unique ring
homomorphism
$$h_{\varphi} \def\g{\gamma, \lambda}: \underset{(\mf m, \s_i)}{\oplus} R[\![x]\!] \longrightarrow} \def\ll{\longleftarrow V$$
such that $$h_{\varphi} \def\g{\gamma, \lambda}(\ph^{\rm td}} \def\tdm{\ph^{\rm td}_{\mf m}, \lambda^{\rm td}} \def\ldm{\lambda^{\rm td}_{\mf m}) \cong (\varphi} \def\g{\gamma, \lambda).$$
\par\smallskip\noindent
In Theorem \ref{thm_reduct}
we showed that each triple $(\varphi} \def\g{\gamma, \lambda, z)$ corresponds modulo
$\Sigma$ to a unique triple $(\psi, \mu, \Lambda)$. Let $\mf m$
be the type of $\Lambda$. In trying to avoid confusion, we will write
$(\psi^{\rm un}, \mu^{\rm un}, \Lambda_{\mf m}^{\rm td})$ for the triple
used in defining the Tate-Drinfeld module of type $\mf m$ over $R(\!(x)\!)$,
where indeed $(\psi^{\rm un}, \mu^{\rm un})$ is the universal Drinfeld
module with level $f$-structure of rank $1$ over $R$.\\
The pair $(\psi, \mu)$ over $V$ comes from an $A_f$-linear ring
homomorphism, which we called $\tilde h$: $$\tilde h: R \longrightarrow} \def\ll{\longleftarrow V.$$ We have
$\tilde h(\psi^{\rm un}, \mu^{\rm un}) =(\psi, \mu)$.
\par\bigskip\noindent
We show that there exists an extension of $\tilde h$ to $R[\![x]\!]$
$$h_{\psi, \mu}: R[\![x]\!] \longrightarrow} \def\ll{\longleftarrow V,$$ where $R[\![x]\!]$ comes equipped
with $\tdm$, such that $h_{\psi, \mu}(\tdm) = \varphi} \def\g{\gamma$; cf. Proposition
\ref{prop_stab-from-td}. The main point is showing that there exists a
ring homomorphism $h_{\psi, \mu}$ such that
$$h_{\psi, \mu}((\psi^{\rm un}_f)^{-1}
\Lambda^{\rm td}_{\mf m}) = (\psi_f)^{-1} \Lambda.$$
\begin{lemma} \label{lem_gen-elm-lat}
Let $m_1$ and $m_2$ generate the ideal $\mf m$. Then there
exists an element $\zeta \in \ov K_V$ such that the projective part of
$(\psi_f)^{-1} \Lambda$ is generated as $A$-module by the elements
$\psi_{m_1}(\zeta)$ and $\psi_{m_1}(\zeta)$.
\end{lemma}
\begin{proof}
Let $M = \ov K_V$ be the algebraic closure of $K_V$, and let $M_{\rm
tor}$ be the set of $A$-torsion points in $M$; the $A$-action
is given by $\psi$.\\
1. The $A$-module $M_{\rm tor}$ is divisible, i.e., for all $a \in A
\backslash \{ 0 \}$ the map $$\psi_a: M_{\rm tor} \longrightarrow} \def\ll{\longleftarrow M_{\rm tor}$$
is surjective. Namely, if $x \in M_{\rm tor}$, then the equation
$\psi_a(z) = x$ has solutions in $M_{\rm tor}$. Consequently,
$M_{\rm tor}$ is an
injective module. Cf. Theorem 7.1 in \cite{HS97}], where it is shown that
divisibility is the same as injectivity for modules over a PID; it
is not difficult to extend this to a theorem over Dedekind domains.\\
2. $M/M_{\rm tor}$ has a natural $K_A$-module structure.
The following sequence of $A$-modules is
exact: $$0 \longrightarrow} \def\ll{\longleftarrow M_{\rm tor} \longrightarrow} \def\ll{\longleftarrow M \longrightarrow} \def\ll{\longleftarrow M/M_{\rm tor} \longrightarrow} \def\ll{\longleftarrow 0.$$ Note
that $\Lambda$ is torsion-free, hence $\Lambda \oplus
M_{\rm tor} \hookrightarrow M$. Consider the projection map $$s:
\Lambda \oplus M_{\rm tor} \longrightarrow} \def\ll{\longleftarrow M_{\rm tor}, \quad {\rm by}~\alpha
\oplus m \mapsto m.$$ Because $M_{\rm tor}$ is an injective module, it
follows that $s$ extends to a map $$s: M \longrightarrow} \def\ll{\longleftarrow M_{\rm tor}.$$ So the
exact sequence splits according to $s$ and $M \cong M_{\rm tor}
\oplus M/M_{\rm tor}$.\\
3. According to $2.$, we may write $(\psi_f)^{-1} \Lambda =
N_1 \oplus N_2$ with $N_1 = \psi[f](V)$ being the torsion part
and $N_2 \cong \mf m$ being the projective part of $(\psi_f)^{-1}\Lambda$.
Let $e_1, e_2$ be generators of $N_2$ such that $\psi_{m_2} e_1 =
\psi_{m_1} e_2$. Then $$(\psi_{m_1})^{-1} e_1 \equiv (\psi_{m_2})^{-1}
e_2 \mod M_{\rm tor}.$$ Let $\zeta_i \in M$ be the unique
element with $\zeta_i \mapsto (\psi_{m_i})^{-1} e_i \in M/M_{\rm
tor}$ and $s(\zeta_i) = 0$. Then $\zeta_1 - \zeta_2 \mapsto 0 \in
M/M_{\rm tor}$ and $s(\zeta_1 - \zeta_2) = 0$. Consequently,
$\zeta:= \zeta_1 = \zeta_2$ is the element we are looking for.
\end{proof}
\begin{prop} \label{prop_stab-from-td}
Every rank $2$ Drinfeld module $\varphi} \def\g{\gamma$ over $V$ with $K_V$-rational
$f$-torsion and with stable reduction of rank $1$ and type $\mf m$ is
induced by $\tdm$ via the ring homomorphism
$$h_{\psi, \mu}: R[\![x]\!] \longrightarrow} \def\ll{\longleftarrow V,$$
i.e., $h_{\psi, \mu}(\tdm) \cong_V \varphi} \def\g{\gamma$.
Extending $h_{\psi, \mu}$ to $$R(\!(x)\!) \longrightarrow} \def\ll{\longleftarrow K_V$$ maps the $f$-torsion
of $\tdm$ isomorphically to the $f$-torsion of $\varphi} \def\g{\gamma$.
\end{prop}
\begin{proof}
\noindent
In Section \ref{sec_TD} we introduced skew polynomials
$P, Q_1, Q_2 \in R\{ \t \}$ with $Q_i \circ P = \psi^{\rm un}_{m_i}$.
In particular, we see that $\tilde h(P)$ divides $P':= {\rm gcd}(
\psi_{m_1}, \psi_{m_2}) \in K_V\{ \t \}$. We may assume
that $\mf m$ is not contained in the kernel of $A_f \longrightarrow} \def\ll{\longleftarrow V$. Therefore,
$\deg_{\t} P' = \deg_{\t} h_{\psi, \mu}(P)$. Consequently, there exist
elements $\beta_i \in K_V \{ \t \}$ with
$$\tilde h(P) = \beta_1 \psi_{m_1} + \beta_2 \psi_{m_2}.$$
Let $\zeta$ be the element from Lemma \ref{lem_gen-elm-lat}, and
define $\frac{1}{z} := \tilde h(P)(\zeta).$
As $\psi_{m_1}(\zeta)$ and $\psi_{m_1}(\zeta)$ generate the projective
part of $(\psi_f)^{-1} \Lambda \subset K_V$, we see that $\frac{1}{z} \in
K_V$ and $z \in V$.\\
We extend $\tilde h$ to
$$h_{\psi, \mu}: R[\![x]\!] \longrightarrow} \def\ll{\longleftarrow V \quad \mbox{by} \quad x \mapsto z.$$
One can easily verify that $$h_{\psi, \mu}((\psi^{\rm un}_f)^{-1}
\Lambda^{\rm td}_{\mf m}) = (\psi_f)^{-1} \Lambda.$$
\end{proof}
\begin{remark}
The proof of Proposition 2.5 in \cite{PT2} is not entirely complete.
One way of completing it, is by adding the construction of the lattice as
is done here. This makes sure that the morphism $h$ in Proposition 2.5 in
\cite{PT2} indeed exists.
\end{remark}
\begin{theorem} \label{thm_univTD}
Every pair $(\varphi} \def\g{\gamma, \lambda)$ consisting of a rank $2$ Drinfeld module
$\varphi} \def\g{\gamma$ over $V$ with stable reduction of rank $1$ with level $f$-structure
$\lambda$ is induced by $(\ph^{\rm td}} \def\tdm{\ph^{\rm td}_{\mf m}, \lambda^{\rm td}} \def\ldm{\lambda^{\rm td}_{\mf m})$ via the unique ring homomorphism
$h_{\varphi} \def\g{\gamma, \lambda}$.
\end{theorem}
\begin{proof}
Let $(\psi, \mu, \Lambda)$ be the triple associated to $(\varphi} \def\g{\gamma, \lambda)$,
and let $\mf m$ be the type of $\Lambda$. Let $h_{\psi, \mu}: R[\![x]\!]
\longrightarrow} \def\ll{\longleftarrow V$ be the morphism such that $h_{\psi, \mu}(\tdm) = \varphi} \def\g{\gamma$.
Let $\s \in \Gl_2(A/fA)$ such that $h_{\psi, \mu}(\ldm \circ \s) = \lambda$.
The element $\s$ lies in a unique class $N \s_i$. Write $\s = \t \s_i$
with $\t \in N$. Let $h_{\t}$ be the ring homomorphism which is defined in
Proposition \ref{prop_STD_up}.
Define $$h_{\varphi} \def\g{\gamma, \lambda}: \oplus R[\![x]\!]_{(\mf m', \s_j)} \longrightarrow} \def\ll{\longleftarrow V$$ as
follows: $h_{\varphi} \def\g{\gamma, \lambda}$ equals $h_{\psi, \mu} \circ h_{\t}$ on
$R[\![x]\!]_{(\mf m, \s_i)}$ and is zero on the other copies of
$R[\![x]\!]$.\\
The uniqueness follows from the construction and from Theorem
\ref{thm_STD_up}.
\end{proof}
\section{The compactification of $M^2(f)$} \label{sec_compact}
\noindent
In this section we describe a compactification $\ov M^2(f)$ of
$M^2(f)$, which is analogous to the compactification of the classical
modular curves given by Katz and Mazur in Chapter 8 of \cite{KM85}. We
define the scheme of cusps, which we call {\it Cusps}. This is a
closed subscheme of $\ov M^2(f)$. Moreover, we consider the formal
scheme $\widehat{\it Cusps}$, which is the completion of $\ov M^2(f)$
along {\it Cusps}. In the following section we will use the universal
Tate-Drinfeld module as defined in the previous sections to describe
the scheme of cusps.
\subsection{The morphism $j_a$}
\noindent
Let $a \in A \bs \F_q$.
Let $(\varphi} \def\g{\gamma, \lambda)$ be the universal Drinfeld module of rank $2$ with
level $f$-structure over $M^2(f)$. Let $B$ be the ring with ${\rm Spec}(B)
= M^2(f)$ and write $$\varphi} \def\g{\gamma_a = \sum_{i=0}^{2\deg(a)} b_i \t^i \quad
\mbox{with $b_i \in B$ for all $i$ and $b_{2\deg(a)} \in B^*$}.$$
Let $j_a: M^2(f) \longrightarrow} \def\ll{\longleftarrow \A^1_{A_f}$ be the morphism given by
$$j_a^{\#}: A_f[j] \longrightarrow} \def\ll{\longleftarrow B, \quad j \mapsto b_{\deg(a)}^{q^{\deg(a)}
+ 1}/b_{2\deg(a)};$$ cf. \cite[4.2]{Leh00}. Clearly, $j_a$ factors
over $M^2(1)$.
\begin{lemma} \label{lem_ja}
The morphism $j_a$ is finite and flat.
\end{lemma}
\begin{proof}
(This is the proof of Proposition 4.2.3 in \cite{Leh00}.) The morphism
$j_a$ is of finite type, hence we may use the valuative criterion to
prove properness. Suppose that $V$ is a discrete valuation ring, let
$K_V$ be its quotient field, and suppose that there are morphisms given
that make the following diagram commutative.
$$\begin{CD} {\rm Spec}(K_V)
@>>> M^2(f)\\ @VVV @VV j_a V\\ {\rm Spec}(V) @>>> \A^1_{A_f}.
\end{CD}$$
Note that the upper horizontal map gives rise to a (unique) map
$${\rm Spec}(V) \longrightarrow} \def\ll{\longleftarrow M^2(f)$$ if and only if the pull-back of the pair $(\varphi} \def\g{\gamma,
\lambda)$ via the upper horizontal map has good reduction at the
maximal ideal of $V$. So $j_a$ is proper if and only if $j_a(x)
\in V$ implies that the pull-back $(\varphi} \def\g{\gamma', \lambda')$ over $K_V$ has
good reduction. If this
pull-back does not have good reduction, it has stable reduction of
rank $1$. Suppose that $(\varphi} \def\g{\gamma, \lambda)$ has stable reduction of
rank $1$, then there exists an element $s\in K_V^*$ such that $s
\varphi} \def\g{\gamma_a s^{-1}$ has all coefficients in $V$, the
$\deg(a)$'th coefficient has valuation $0$, and the $2\deg(a)$'th
coefficient has strictly positive valuation. This means that the image of
$j_a(x)$ is not in $V$. We conclude that $j_a$ is proper.\\
Because each connected component of $M^2(f)$ is an affine variety
over $\F_q$, it follows by \cite[ex. II.4.6]{Har77} that $j_a$
restricted to such a connected component is finite. And thus $j_a$
is finite.\\ The finite ring homomorphism $j_a^{\#}$ is
injective. Let $\mf P \subset B$ be a prime ideal lying above $\mf p
\subset A_f[j]$. Then both local rings $B_{\mf P}$ and $A_f[j]_{\mf
p}$ are regular and of equal dimension. By the finiteness of $j_a$
it follows that $B_{\mf P}$ is a free $A_f[j]_{\mf p}$-module, cf.
Corollary IV.22 in \cite{Ser65}. Hence, $j_a$ is flat.
\end{proof}
\subsection{The compactification}
\noindent
The ring $B$ is a finite $A_f[j]$-algebra via $j_a^{\#}$.
Let $C$ denote the normalization of $A_f[\frac{1}{j}]$ inside the
quotient ring of $B$. Then $C$ is finite over $A_f[\frac{1}{j}]$;
cf. Corollary 13.13 in \cite{Eis95}.\\
The compactification $\ov M^2(f)$ of $M^2(f)$ is defined as the
scheme obtained by glueing ${\rm Spec}(B)$ and ${\rm Spec}(C)$ along their
intersection. We obtain a finite morphism
$$\ov j_a: \ov M^2(f) \longrightarrow} \def\ll{\longleftarrow \P^1_{A_f}.$$ The following diagram is cartesian
$$\begin{CD}
M^2(f) @>>> \ov M^2(f)\\
@V j_a VV @VV \ov j_a V\\
\A^1_{A_f} @>>> \P^1_{A_f}.
\end{CD}.$$
\begin{remark}
By Theorem \ref{thm_CompToTD} it follows that $\ov j_a$ is flat
in the points of the boundary of $\ov M^2(f)$; therefore, $\ov j_a$ is
flat.
\end{remark}
\begin{lemma} \label{lem_ind_comp}
The scheme $\ov M^2(f)$ is independent of the chosen element $a$.
\end{lemma}
\begin{proof}
Let $B'$ be a connected component of $B$.
Let $a_1, a_2 \in A \bs \F_q$ and consider the maps
$j_{a_i}: A_f[j_i] \longrightarrow} \def\ll{\longleftarrow B'$. Let $C_i$ be the integral closure
of $A_f[\frac{1}{j_i}]$ inside $K_{B'}$. Let $X_i$ be the scheme
obtained by glueing ${\rm Spec}(C_i)$ and ${\rm Spec}(B')$ along their
intersection.\\
Let $\mf p \in X_1 \bs {\rm Spec}(B')$ be any prime of height one of
$C_1$, then $\frac{1}{j_1} \in \mf p$. Let $v_{\mf p}$ be the valuation
of $K_{B'}$ given by $\mf p$. If $v_{\mf p}(\frac{1}{j_2}) \leq 0$, then
$\mf p$ would correspond to a valuation of $K_A[j_2]$ and therefore to
a valuation of $B'$; cf. Section VII.9 in \cite{Bou}.
Consequently, $v_{\mf p}(\frac{1}{j_2}) > 0$.
The same is true if we interchange $j_1$ and $j_2$. We conclude
that the set of valuations $v$ of $K_B'$ with $v(j_i) > 0$ does not
depend on $i$. Therefore, $X_1 = X_2$.
\end{proof}
\subsection{The scheme of cusps}
\noindent
To describe the boundary of $\ov M^2(f)$, we introduce the {\it scheme
of cusps}, which we call {\it Cusps}. Let $\mf r$ be the intersection
of all height $1$ primes $\mf p$ containing $\frac{1}{j}$, i.e.,
$\mf r = {\rm rad}(\frac{1}{j})$. And $V(\mf r) = \ov M^2(f)
\backslash M^2(f)$. Let $\widehat} \def\bs{\backslash C := \underset{\ll}{\lim}~C/\mf r^n$.
\begin{lemma} \label{lem_Chat}
The ring $\hat C$ is normal and a finite $A_f[\![\frac{1}{j}]\!]$-algebra.
\end{lemma}
\begin{proof}
The ring $B$ is regular. So $C = \oplus_i C_i$ where each
$C_i$ is an integrally closed domain. The ring $C$ is excellent.
By \cite[7.8.3.vii]{EGA42} it follows that $\widehat} \def\bs{\backslash C$ is normal.
As $C'$ is a finite $A_f[\frac{1}{j}]$-algebra, it follows that
$\widehat} \def\bs{\backslash C$ is a finite $A_f[\![\frac{1}{j}]\!]$-algebra.
\end{proof}
\noindent
We denote $A_f(\!(\frac{1}{j})\!) := A_f[\![\frac{1}{j}]\!][j]$.
Furthermore, we define the formal scheme
$$\widehat} \def\bs{\backslash {\it Cusps} := {\rm Spf}(\widehat} \def\bs{\backslash C),$$ which is the formal neighbourhood
of {\it Cusps}. Let $\O$ denote the structure
sheaf of $\widehat} \def\bs{\backslash {\it Cusps}$. The scheme of cusps is defined as
$${\it Cusps} := (\widehat} \def\bs{\backslash {\it Cusps}, \O/ \mf r) = {\rm Spec}(C/\mf r).$$
\begin{theorem} \label{thm_extWP}
The $A_f$-morphism $w_f$ given by Theorem \ref{thm_map}
can be extended to an $A_f$-morphism $$w_f: \ov M^2(f) \longrightarrow} \def\ll{\longleftarrow M^1(f).$$
Its restriction to the scheme of cusps gives a finite $A_f$-morphism
$$w_f: {\it Cusps} \longrightarrow} \def\ll{\longleftarrow M^1(f).$$
\end{theorem}
\begin{proof}
The Weil pairing gives an $R$-algebra structure $R \longrightarrow} \def\ll{\longleftarrow B$. Because $R$ is
integral over $A_f$, $B$ is the integral closure of $A_f[j]$ in the quotient
ring of $B$ and $C$ is the integral closure of $A_f[\frac{1}{j}]$ in this
quotient ring, it follows immediately, that the ring
homomorphism $R \longrightarrow} \def\ll{\longleftarrow B$ gives a ring homomorphism $R \longrightarrow} \def\ll{\longleftarrow C$. These
two maps glue to $$w_f: \ov M^2(f) \longrightarrow} \def\ll{\longleftarrow M^1(f).$$
The restriction of $w_f$ to ${\it Cusps}$ is given by $R \longrightarrow} \def\ll{\longleftarrow C \longrightarrow} \def\ll{\longleftarrow C/\mf r$.
As $C$ is finite over $A_f[\frac{1}{j}]$, it follows that $C/\mf r$ is
finite over $A_f$. As $R$ is finite over $A_f$, we may conclude that
$w_f$ restricted to ${\it Cusps}$ is finite.
\end{proof}
\section{The cusps and the Tate-Drinfeld module}
\label{sec_Cusp_TD}
\noindent
In the previous section, we defined the scheme of cusps
and the formal scheme $\widehat} \def\bs{\backslash {\it Cusps} = {\rm Spf}(\widehat} \def\bs{\backslash C)$. In this section,
we will relate these schemes to the universal Tate-Drinfeld module, which we
introduced in Section \ref{sec_UP}. In fact, using the universal property
of the universal Tate-Drinfeld module and the ${\rm Cl}(A) \times
\Gl_2(A/fA)$-equivariance of the Weil-pairing, we will be able to
prove the following theorem. If we write `$\oplus_{\mf p}$', we mean
the direct sum over all minimal primes $\mf p$ containing $\frac{1}{j}$.
\begin{theorem} \label{thm_CompToTD}
There exists an $R[\![\frac{1}{j}]\!]$-linear isomorphism
$$\widehat} \def\bs{\backslash C \cong \oplus_{\mf p} \lim_{\ll} C/\mf p^n
\stackrel{\sim}{\longrightarrow} \def\ll{\longleftarrow} {\oplus} R[\![x]\!]_{(\mf m, \s_i)},$$
such that $${\it Cusps} \stackrel{\sim}{\longrightarrow} \def\ll{\longleftarrow} \oplus R_{(\mf m, \s_i)}.$$
\end{theorem}
\noindent
From this theorem we can derive the following important corollary:
\begin{cor} \label{cor_cusp}
The compactification $\ov M^2(f)$ of $M^2(f)$ is regular, and even smooth
over ${\rm Spec}(A_f)$. Furthermore, the scheme of cusps is isomorphic to
$${\it Cusps} \cong \coprod_{(\mf m, \s_i)} M^1(f),$$ where $\mf m$ runs
through ${\rm Cl}(A)$ and $\s_i$ runs through the cosets of $N\bs
\Gl_2(A/fA)$ where $$N = \left( \begin{array}{cc}
\F_q^* & A/fA \\ 0 & (A/fA)^* \end{array} \right) \subset \Gl_2(A/fA).$$
Consequently, the scheme ${\it Cusps}$ consists of $\frac{h(A) \cdot \#
{\rm Sl}} \def\Gl{{\rm Gl}} \def\Aut{{\rm Aut}_2(A/fA)}{\# (A/fA) \cdot (q-1)}$ copies of $M^1(f)$.
\end{cor}
\begin{proof}
By Theorem \ref{thm_CompToTD} the ring $C$ is regular in the points
above $\frac{1}{j}$ and thus $C$ is regular. Consequently, $\ov M^2(f)$
is regular. The description of ${\it Cusps}$ and the number of
its components follows from Theorem \ref{thm_CompToTD}.\\
To prove smoothness over ${\rm Spec}(A_f)$, note that by the corollary to
Proposition 5.4 in \cite{Drin74}, the morphism $M^2(f) \longrightarrow} \def\ll{\longleftarrow {\rm Spec}(A_f)$
is smooth. So we only need to prove smoothness in the closed points
of ${\it Cusps}$. We have $$\widehat} \def\bs{\backslash C \cong \oplus R[\![x]\!]_{(\mf m, \s_i)},$$
so $\widehat} \def\bs{\backslash C$ is formally smooth over $A_f$. This implies by 17.5.1 and
17.5.3 in \cite{EGA44} that the morphism $\ov M^2(f) \longrightarrow} \def\ll{\longleftarrow
{\rm Spec}(A_f)$ is smooth in the closed points of ${\it Cusps}$.
\end{proof}
\subsection{The proof of Theorem \ref{thm_CompToTD}}
\noindent
The rest of this section is devoted to proving Theorem \ref{thm_CompToTD}.
The universal Tate-Drinfeld module over $\Cal Z$ gives rise
to an $A_f$-morphism $$\Cal Z_{\rm open} \longrightarrow} \def\ll{\longleftarrow M^2(f),$$ where $\Cal Z_{\rm
open}$ denotes the localization of $\Cal Z$ at $(x)$, i.e.,
$$\Cal Z_{\rm open} = {\rm Spec}(\oplus R(\!(x)\!)_{(\mf m, \s_i)}).$$
It follows from Remark \ref{rem_Act} that this morphism is ${\rm Cl}(A)\times
\Gl_2(A/fA)$-equivariant.
\par\bigskip\noindent
Let $\Cal Z_{x=0}$ denote the scheme $${\rm Spec}\left(
\oplus R[\![x]\!]_{(\mf m, \s_i)} /(x) \right).$$
The line of argument is as follows: in Lemma \ref{lem_ClAequiv} we
show how to relate the Tate-Drinfeld module to the study of the
cusps, and in Lemma \ref{lem_isoCusps} we describe the
scheme ${\it Cusps}$. This latter lemma enables us to lift
the isomorphism ${\it Cusps} \longrightarrow} \def\ll{\longleftarrow \Cal Z_{x = 0}$ to
an isomorphism $\widehat} \def\bs{\backslash C \longrightarrow} \def\ll{\longleftarrow \Cal Z$. Let $\widehat} \def\bs{\backslash C_{\mf p}$ denote the
completion of the local ring of $C$ at $\mf p$.
\begin{lemma} \label{lem_ClAequiv}
Every $R[\![x]\!]_{(\mf m, \s_i)}$ is a finite
$A_f[\![\frac{1}{j}]\!]$-algebra. Consequently, the morphism $\Cal
Z_{\rm open} \longrightarrow} \def\ll{\longleftarrow M^2(f)$ comes from a morphism
$$h_1: \Cal Z \longrightarrow} \def\ll{\longleftarrow {\rm Spec}(C) \subset \ov M^2(f).$$
Moreover, the universal property of the Tate-Drinfeld module
gives rise to a morphism $$h_2: {\rm Spec}(\oplus_{\mf p} \widehat} \def\bs{\backslash C_{\mf p})
\longrightarrow} \def\ll{\longleftarrow \Cal Z.$$ The composition $h_1 \circ h_2$ is the natural
morphism.
\end{lemma}
\begin{proof}
Write $R[\![x]\!] = R[\![x]\!]_{(\mf m, \s_i)}$ for some pair
$(\mf m, \s_i)$ and write $(\tdm)_a = \sum_i c_i \t^i$ for the element
$a \in A$ which is used to define $j_a$. Then $c_{2\deg(a)}
\in R(\!(x)\!)^*$, because $\tdm$ is a Drinfeld module over $R(\!(x)\!)$.
Moreover, $$\tdm \mod (x) = \psi.$$ So the coefficient $c_{\deg(a)}
\in R[\![x]\!]^*$.
This implies by definition, that $\frac{1}{j}$ is mapped to
$\alpha \cdot x^k \in R[\![x]\!]$, with $\alpha \in R[\![x]\!]^*$ and $k \in
\Z_{>0}$. From this it follows that $R[\![x]\!]$ is a finite
$R[\![\frac{1}{j}]\!]$-module.
The morphism $$\Cal Z_{\rm open} \longrightarrow} \def\ll{\longleftarrow M^2(f)$$ comes from a ring homomorphism
$$C[j] \longrightarrow} \def\ll{\longleftarrow \left( \oplus R[\![x]\!]_{(\mf m, \s_i)} \right)
\otimes_{A_f[\![\frac{1}{j}]\!]} A_f(\!(\frac{1}{j})\!),$$
Because $C$ is finite over $A_f[\frac{1}{j}]$ and $R[\![x]\!]$ is
finite over $A_f[\![\frac{1}{j}]\!]$, it follows that the image of
$C$ under this ring homomorphism lies in $\oplus R[\![x]\!]_{(\mf m,
\s_i)}$.\\
For the `moreover'-part, let $\mf p \subset C$ be a minimal prime
ideal containing $\frac{1}{j}$. The ring $\widehat} \def\bs{\backslash C_{\mf p}$
is a complete discrete valuation ring and comes equipped with a
Tate-Drinfeld structure via the morphism $${\rm Spec}(K_{\widehat} \def\bs{\backslash C_{\mf p}})
\longrightarrow} \def\ll{\longleftarrow M^2(f).$$ By Theorem \ref{thm_univTD}
there exists a unique ring homomorphism $$\oplus
R[\![x]\!]_{(\mf m, \s_i)} \longrightarrow} \def\ll{\longleftarrow \widehat} \def\bs{\backslash C_{\mf p}$$ which induces on
$\widehat} \def\bs{\backslash C_{\mf p}$ this Tate-Drinfeld structure.
This can be done for every minimal prime $\mf p$ containing $\frac{1}{j}$.
\end{proof}
\begin{lemma} \label{lem_isoCusps}
The morphism $h_1$ induces an isomorphism
$$\Cal Z_{x = 0} \stackrel{\sim}{\longrightarrow} \def\ll{\longleftarrow} {\it Cusps}.$$
Every pair $(\mf m, \s_i)$ corresponds via this isomorphism to one and
only one minimal prime $\mf p \subset C$ containing $\frac{1}{j}$.
Consequently, $$\widehat} \def\bs{\backslash C = \oplus_{\mf p} \lim_{\ll} C/\mf p^n.$$
\end{lemma}
\begin{proof}
We will first prove that the number of irreducible components of
${\it Cusps}$ equals the number of irreducible components of
$\Cal Z_{x = 0}$. Subsequently, we will show that these components
of ${\it Cusps}$ intersect nowhere.\\
Because ${\it Cusps} \cong {\rm Spec}(C/(\cap \mf p))$,
the irreducible components of ${\it Cusps}$ are in a one-to-one
corresponcende to the minimal primes containing $\frac{1}{j}$.
The morphisms $h_1$ and $h_2$ introduced in Lemma \ref{lem_ClAequiv}
give rise to the following maps on the sets of irreducible components:
{\small
$$\{ \mbox{irr. comp. of ${\it Cusps}$} \} \stackrel{h_2}{\longrightarrow} \def\ll{\longleftarrow}
\{ \mbox{irr. comp. of $\Cal Z_{x = 0}$} \} \stackrel{h_1}{\longrightarrow} \def\ll{\longleftarrow}
\{ \mbox{irr. comp. of ${\it Cusps}$} \}.$$}
\noindent
As a morphism of schemes $h_1 \circ h_2$ is the natural map. Therefore,
the composition $h_1 \circ h_2$
on the set of irreducible components is the identity. Consequently,
$h_2$ on the irreducible components is injective.\\
Moreover, the set of irreducible components of $\Cal Z_{x = 0}$ is by
definition one orbit under the elements $(\mf m, \s_i)$.
Clearly, the map $h_1$ on the set of connected components is equivariant
under this group action,
and as the image of $h_2$ is not empty, it follows that the first
map on the irreducible components is also surjective. So we may conclude
that the number of irreducible components of ${\it Cusps}$ equals the
number of irreducible components of $\Cal Z_{x = 0}$.\\
The irreducible components of $\Cal Z_{x = 0}$ intersect nowhere.
We will prove that this is also the case for the irreducible components of
${\it Cusps}$. By the extension of the Weil pairing to
$\ov M^2(f)$, the ring $C$ comes equipped with an $R$-algebra structure.
Let $$\zeta: R \longrightarrow} \def\ll{\longleftarrow \oplus R[\![x]\!]_{(\mf m, \s_i)}$$ denote the
composition $R \stackrel{w_f^{\#}}{\longrightarrow} \def\ll{\longleftarrow} C \stackrel{h_1^{\#}}{\longrightarrow} \def\ll{\longleftarrow}
\oplus R[\![x]\!]_{(\mf m, \s_i)}.$\\
Choose any maximal ideal $\mf n \subset R$ and let $\mf q$ run over all
minimal primes of $C/\mf n C$ containing $\frac{1}{j}$. We write
$$\widehat} \def\bs{\backslash{C/\mf n C} = \lim_{\ll} (C/\mf n C)/ (\frac{1}{j})^n,$$
and $\widehat} \def\bs{\backslash{(C/\mf n C)}_{\mf q}$ for the completion along $\mf q$ of
the local ring $(C/\mf n C)_{\mf q}$.\\
In this case we have analogues of the morphisms $h_1, h_2$, namely,
$R$-algebra homomorphisms
$$\begin{CD} \widehat} \def\bs{\backslash{C/\mf n C} @> \tilde h_1 >>
\oplus R/\zeta(\mf n)[\![x]\!]_{(\mf m, \s_i)} @> \tilde h_2 >>
\oplus \widehat} \def\bs{\backslash{C/\mf n C}_{\mf q}.
\end{CD}$$
And now, as before, we define maps on the sets of irreducible components:
{\small
$$\tilde h_2: \{ \mbox{irr. comp. of ${\it Cusps} \times {\rm Spec}(R/\mf n)$}
\}
\longrightarrow} \def\ll{\longleftarrow \{ \mbox{irr. comp. of $\Cal Z_{x = 0} \times {\rm Spec}(R/\mf n)$} \},$$
$$\tilde h_1: \{ \mbox{irr. comp. of $\Cal Z_{x = 0}\times
{\rm Spec}(R/\mf n)$} \} \longrightarrow} \def\ll{\longleftarrow \{ \mbox{irr. comp. of
${\it Cusps}\times {\rm Spec}(R/\mf n)$} \}.$$}
\noindent
The composition of these two maps on the set of irreducible components
is the identity. Namely, the composition $\tilde h_1 \circ
\tilde h_2$ is the natural map on the rings. Using the same argument
as above shows that $\tilde h_2$ on the irreducible components is a
bijection.\\
We conclude that for every prime $\mf n \subset R$ the number of
irreducible components of $${\it Cusps} \times {\rm Spec}(R/\mf n)$$ equals the
number of irreducible components of ${\it Cusps}$. Recall that by Theorem
\ref{thm_extWP} the morphism $w_f: {\it Cusps} \longrightarrow} \def\ll{\longleftarrow M^1(f)$ is finite.
Therefore, if the irreducible components
would intersect above some prime ideal $\mf n \subset R$, then
${\it Cusps}\times {\rm Spec}(R/\mf n)$ would have less irreducible components.
As this is not the case, we conclude that the irreducible components
of ${\it Cusps}$ intersect nowhere.\\
We write $${\it Cusps} = {\rm Spec}(\oplus S_{(\mf m, \s_i)})$$ where
${\rm Spec}(S_{(\mf m, \s_i)})$ are the connected components of ${\it Cusps}$.
For every
pair $(\mf m, \s_i)$, we get $R$-linear ring homomorphisms on the
connected components:
$$S_{(\mf m, \s_i)} \stackrel{h_1^{\#}}{\longrightarrow} \def\ll{\longleftarrow} R \stackrel{h_2^{\#}}
\longrightarrow} \def\ll{\longleftarrow S_{(\mf m, \s_i)}.$$
Because the composition is the identity and $S_{(\mf m, \s_i)}$ is
finite over $R$, it follows that $$S_{(\mf m, \s_i)} \cong R.$$
For the latter two statements of the lemma, note that the isomorphism
implies that the minimal primes $\mf p$ are relatively prime and,
consequently, $\mf r = \prod \mf p.$
\end{proof}
\noindent
The next step is to lift the isomorphism from Lemma
\ref{lem_isoCusps} to an isomorphism $$\widehat} \def\bs{\backslash C \stackrel{\sim}{\longrightarrow} \def\ll{\longleftarrow}
\oplus R[\![x]\!]_{(\mf m, \s_i)}.$$ Let
$$\Cal W := \underset{\ll}{\lim}~ C/\mf p^n$$ for some minimal prime
$\mf p$ containing $\frac{1}{j}$.
\begin{lemma} \label{lem_isoP}
The ring $\Cal W$ is isomorphic to $R[\![x]\!]$.
\end{lemma}
\begin{proof}
By Lemma \ref{lem_Chat} the ring $\Cal W$ is integrally closed and a finite
$A_f[\![\frac{1}{j}]\!]$-algebra, and by the isomorphism of Lemma
\ref{lem_isoCusps}, the ring $\Cal W$ is a finite
$R[\![\frac{1}{j}]\!]$-algebra.\\
The completion of the local ring $\Cal W_{\mf p}$ is isomorphic
to $\widehat} \def\bs{\backslash C_{\mf p}$. The morphisms $h_1$ and $h_2$ give on the
completions of the local rings injective maps
$$\widehat} \def\bs{\backslash C_{\mf p} \stackrel{h_1^{\#}}{\longrightarrow} \def\ll{\longleftarrow} K_R[\![x]\!]
\stackrel{h_2^{\#}}{\longrightarrow} \def\ll{\longleftarrow} \widehat} \def\bs{\backslash C_{\mf p}.$$
As $h_2^{\#} \circ h_1^{\#}$ is the identity, there exists an isomorphism
$K_R[\![x]\!] \cong \widehat} \def\bs{\backslash C_{\mf p}.$\\
We conclude that $\Cal W$ is regular. Therefore, we may asssume that
$x$ is an element of $\Cal W$ and that $\Cal W$ is a finite
$R[\![x]\!]$-algebra.
So we get injective $R[\![x]\!]$-linear ring homomorphisms
$$R[\![x]\!] \longrightarrow} \def\ll{\longleftarrow \Cal W \longrightarrow} \def\ll{\longleftarrow R[\![x]\!]$$ where the first
map is the $R[\![x]\!]$-structure morphism of $\Cal W$ and the second
map is $h_1^{\#}$. We conclude that $\Cal W \cong R[\![x]\!]$.
\end{proof}
\noindent
This enables us to prove Theorem \ref{thm_CompToTD}:
\begin{proof}[Proof of Theorem \ref{thm_CompToTD}]
By the previous lemma, it follows that $$\widehat} \def\bs{\backslash C \cong \oplus
R[\![x]\!]_{(\mf m, \s_i)}.$$ Together with Lemma \ref{lem_isoCusps}
the theorem follows.
\end{proof}
\section{Components of $M^2(f)$}
\label{sec_compon}
\noindent
In this section we describe the geometric components of $\ov M^2(f)$ and prove
the connectedness of $M^2(f)$. For a non-zero prime $\mf P \subset R$ we
write $\kappa(\mf P) := R/\mf P$. The first result is the following:
\begin{theorem} \label{thm_geom_comp}
The scheme $$\ov M^2(f) \underset{A_f}{\times} M^1(f)$$
consists of $h(A)\cdot [(A/fA)^*:\F_q^*]$ connected components, which
are all geometrically connected. Moreover, for every non-zero prime
ideal $\mf P \subset R$ the fibre at $\mf P$
$$\ov M^2(f) \underset{A_f}{\times} {\rm Spec}(\kappa(\mf P)),$$
consists of $h(A) \cdot [(A/fA)^*: \F_q^*]$ connected components, which are
all geometrically connected.
\end{theorem}
\begin{proof}
Let $K_{\infty}$ be the completion of the quotient field of $A_f$
along the point $\infty$, and let $\C_{\infty}$ denote the
completion of the algebraic closure of $K_{\infty}$. By the analytic
theory, as is shown in \cite{PT1}, we know that $$\ov M^2(f)
\times_{A_f} {\rm Spec}(\C_{\infty})$$ consists of $h(A) \cdot
[(A/fA)^*:\F_q^*]$ components. Because $R$ is a Galois extension of
$A_f$ with Galois group $G$, we have $R\otimes_{A_f} R \cong
\oplus_G R.$ By the Weil pairing one sees that $$\ov M^2(f)
\underset{A_f}{\times} M^1(f) \stackrel{w_f}{\longrightarrow} \def\ll{\longleftarrow} M^1(f)
\times_{A_f} M^1(f)$$ consists of $h(A)\cdot \# G$ connected components.
As $\# G = [(A/fA)^*:\F_q^*]$, these components are geometrically
connected components.\\
Consider the fibres over $R$. Let $\mf P \subset R$ be a non-zero prime
ideal and let $V$ be the
completion along $\mf P$ of the local ring $R_{\mf P}$. Suppose
$\ov M^2(f) \times_R {\rm Spec}(\kappa(\mf P))$ has more than one
connected component, then also $\ov M^2(f) \times_R {\rm Spec}(V/\mf
P^n)$ has more than one connected component for every $n$ and
consequently, both $\ov M^2(f) \times {\rm Spec}(V)$ and $\ov M^2(f)
\times_R {\rm Spec}(K_V)$ consist of more than one component. This,
however, contradicts the fact that $\ov M^2(f) \times_R M^1(f)$ is
geometrically connected. So we conclude that
$\ov M^2(f) \times_R {\rm Spec}(\kappa(\mf P))$ is geometrically connected.
\end{proof}
\noindent
This theorem enables us to say something about the Drinfeld modular curves.
Let, as before, $N = \left( \begin{array}{cc} \F_q^* & A/fA \\ 0 & (A/fA)^*
\end{array} \right)$.
\begin{theorem}
For every $R$-field $K$ the curve $\ov M^2(f) \times_R {\rm Spec}(K)$ is
a smooth, irreducible curve containing $h(A) \cdot [\Gl_2(A/fA): N]$
cusps.
\end{theorem}
\begin{proof}
Clearly, the scheme {\it Cusps} consists of $h(A) \cdot [\Gl_2(A/fA): N]$
copies of $R$. Consequently, $${\it Cusps} \times {\rm Spec}(K)$$ consists
of $h(A) \cdot [\Gl_2(A/fA): N]$ points. The irreducibility follows
immediately from the proof of Theorem \ref{thm_geom_comp}.
\end{proof}
\subsection{The analogue of $X_0(N)$}
\noindent
The analogue in the setting of Drinfeld modular curves of the modular
curve $X_0(N)$ is the curve $$X_0(f):= \ov M^2(f)/H, \quad {\rm where}~
H = \left( \begin{array}{cc}
(A/fA)^* & A/fA\\ 0 & (A/fA)^* \end{array} \right) \subset \Gl_2(A/fA).$$
One may deduce from Theorem \ref{thm_geom_comp} the following
theorem concerning the cusps and the geometric components of $\ov M^2(f)/H$.
Define $R_0 = R^{(A/fA)^*/\F_q^*}$, i.e., ${\rm Spec}(R_0) = M^1(1)$.
Write ${\it Cusps}_0$ for the scheme of cusps of $X_0(f)$.
\begin{theorem}
The Weil pairing induces an isomorphism
$${\it Cusps}_0 \stackrel{\sim}{\longrightarrow} \def\ll{\longleftarrow} \coprod_{(\mf m, \rho)} M^1(1)$$ where
$\rho$ runs through the double cosets $N \bs \Gl_2(A/fA) / H$.\\
The scheme $X_0(f)$ is connected, and for any $R_0$-field $K$ the
scheme $X_0(f) \times {\rm Spec}(K)$ consists of $h(A)$ geometrically
connected components.
\end{theorem}
\begin{proof}
The morphism $w_f$ gives an isomorphism between $M^1(f)$ and
any connected component of ${\it Cusps}$. Consequently, $w_f$ gives
an isomorphism ${\it Cusps} \longrightarrow} \def\ll{\longleftarrow \oplus_{(\mf m, \s_i)} M^1(f)$.
Recall that $N$ acts trivially on each copy of $M^1(f)$. Furthermore,
the action of $\s = \left( \begin{array}{cc} \alpha & 0 \\ 0 &
\alpha \end{array} \right)$ with $\alpha \in (A/fA)^*$ on ${\it Cusps}$
is as follows. Let $i,j\in \N$ such that $\s_i \circ \s \in N \s_j$,
and consider $\alpha$ as an element of the Galois group
$\operatorname{Gal}(K_R/K_{R_0}) \cong (A/fA)^*/\F_q^*$,
then $\s$ acts as $$M^1(f)_{(\mf m, \s_i)} \stackrel{\alpha}{\longrightarrow} \def\ll{\longleftarrow}
M^1(f)_{(\mf m, \s_j)}.$$
Because $H$ contains the subgroup $\left\{ \left( \begin{array}{cc} \alpha
& 0 \\ 0 & \alpha \end{array} \right) \mid \alpha \in (A/fA)^* \right\}$,
we see that dividing out ${\it Cusps}$ by $H$ gives an isomorphism
$${\it Cusps}_0 \stackrel{\sim}{\longrightarrow} \def\ll{\longleftarrow}
\oplus_{(\mf m, \rho)} M^1(1)$$ where
$\rho$ runs through the cosets of $N\bs \Gl_2(A/fA) / H$.\\
The number of components follows immediately from Theorem
\ref{thm_geom_comp}.
\end{proof}
\def$'$} \def\cprime{$'${$'$} \def$'$} \def\cprime{$'${$'$}
\def\polhk#1{\setbox0=\hbox{#1}{\ooalign{\hidewidth
\lower1.5ex\hbox{`}\hidewidth\crcr\unhbox0}}}
|
{
"timestamp": "2004-11-22T17:37:55",
"yymm": "0411",
"arxiv_id": "math/0411490",
"language": "en",
"url": "https://arxiv.org/abs/math/0411490"
}
|
\section{INTRODUCTION}
The Lattice Quantum Chromodynamics calculations predict the existence of a transition between a hadronic gas
and a quark gluon plasma at a temperature around 170MeV in a domain close to the net-baryon free region \cite{Fodor03}.
This deconfined state of quarks and gluons is expected to be formed in ultra-relativistic heavy ion collisions. The study of the final hadronic state properties of such
collisions essentially dominated by the low $p_{T}$ part of the particle spectra (the bulk) via multi-strange particles may provide information on its dynamics from the early stage to the chemical and thermal freeze-out (FO). \\
Strange quarks whose mass is comparable to the temperature of the QGP formation are expected to be abundantly produced in the high temperature QGP phase because of parton rescattering and should achieve equilibration \cite{Raf03}. Hadronization process has been very well described in the framework of statistical model \cite{Raf03,PBM99,Bec04} by adjusting four free parameters, the chemical temperature $T_{ch}$, the baryon and strange chemical potential $\mu_{B}$ and $\mu_{s}$ and the strangeness phase space occupancy factor $\gamma_{s}$. This latter provides a measurement of the degree of strangeness equilibration in the system. Its evolution with centrality as well as $T_{ch}$ evolution gives a quantitative measurement of strangeness evolution in the bulk matter. \\
A study of the collective motion of the collision in the framework of a hydrodynamically-inspired model has previously shown that $\pi$, K, p particles seem to take part to the same transverse collective flow and freeze-out kinetically at a temperature smaller than $T_{ch}$ suggesting an expansion and the cooling of the system between chemical and thermal FO. Concerning multi-strange baryons, it has been suggested that these particles should not develop such significant transverse radial flow due to their presumably small cross section so that they should decouple much earlier in the collision \cite{nxu98,cheng03}. Their observed transverse radial flow would then primarily reflects partonic flow behaviour. Elliptic flow due to the initial asymmetry of the system in non-central collisions has also proven to be a good tool for understanding the properties of the early stage of the collisions \cite{Olli92}. Thus multi-strange baryon elliptic flow could be a valuable probe of the initial partonic system. As flow is an additive quantity, we present both radial transverse flow and elliptic flow measurements of multi-strange baryons in order to disentangle its hadronic and partonic contributions.
\section{RESULTS AND DISCUSSION}
\subsection{Bulk chemical properties}
\vspace*{2mm}
\begin{floatingfigure}[r]{8cm}%
\begin{spacing}{0.8}
\hspace*{-0.3in}
\vspace*{-0.3in}
\includegraphics*[ width=10cm,height=9cm,
keepaspectratio]{gammas.eps}
\vspace*{-.15in}
\caption{Evolution of chemical temperature $T_{ch}$ (top) and strangeness phase space occupancy factor $\gamma_s$ (bottom) as a function of the number of participants. $T_{ch}$ and $\gamma_s$ have been calculated from statistical fits including $\pi$, $K$, $p$ (squares) and $\pi$, $K$, $p$, $\Lambda$, $\Phi$, $\Xi$, $\Omega$ particles (circles).}
\label{fig:gammas}
\vspace*{.20in}
\end{spacing}
\end{floatingfigure}%
All the data presented in this contribution have been collected by the STAR detector described in \cite{Star03}. Multi-strange particles are identified via the topology of their decay $\Xi$$\rightarrow$$\Lambda+\pi$ and $\Omega$$\rightarrow$$\Lambda+K$ then the subsequent decay $\Lambda$$\rightarrow$$p+\pi$ with the respective branching ratios 100$\%$, 68$\%$ and 64$\%$. For more details see \cite{Effi02}. Corrections for tracking efficiency and detector acceptance were applied. The final corrected transverse momentum distributions are fitted in order to extract yields and inverse slope parameters.
The results of a statistical fit of particle ratios including $\pi^{\pm}$, $K^{\pm}$, $p$, $\overline{p}$, $\Lambda$, $\overline{\Lambda}$, $\Phi$, $\Xi^{\pm}$ and $\Omega^{\pm}$ are presented on Figure~\ref{fig:gammas}. Very good agreement between our data and the model are achieved for each centrality range at $\sqrt{s_{NN}}=200GeV$ at RHIC \cite{PBM03}. A fit has been performed first including $\pi^{\pm}$, $K^{\pm}$, $p$, $\overline{p}$. For the most central collisions, the four free parameters of the fit are $T_{ch}=157\pm6MeV$, $\mu_{B}=22\pm4MeV$, $\mu_{s}=3.8\pm2.6MeV$ and $\gamma_{s}=0.86\pm0.11$. The evolution of $T_{ch}$ and $\gamma_{s}$ with centrality are represented as square symbols on Figure~\ref{fig:gammas}. An other fit has also been performed including then all the hadrons $\pi^{\pm}$, $K^{\pm}$, $p$, $\overline{p}$, $\Lambda$, $\overline{\Lambda}$, $\Phi$, $\Xi^{\pm}$ and $\Omega^{\pm}$. The parameters obtained for the most central collision are $T_{ch}=160\pm5MeV$, $\mu_{B}=24\pm4MeV$, $\mu_{s}=1.4\pm1.6MeV$ and $\gamma_{s}=0.99\pm0.07$. $T_{ch}$ and $\gamma_{s}$ evolutions with centrality are represented as circles. We note no dependence of $T_{ch}$ with centrality. All the particles seem to chemically freeze-out at a temperature of 160$\pm$5MeV close to LQCD predictions, $T_{ch}$ seems to be essentially fixed by the most numerous $\pi$, $K$, $p$ particles and does not seem to be dependent on the initial system size. From peripheral to central collisions, including (multi-)strange hadrons in the fit, $\gamma_{s}$ increases from 0.8 and saturates at 1. This value suggests that in most central collisions at top RHIC energy, the phase space is saturated in strange quarks so that the system is close to strangeness equilibration. This increase of $\gamma_{s}$ also signs the existence of significant $s$$\overline{s}$ production processes at a partonic level such as gluon fusions.
\vspace*{3mm}
\subsection{Collision dynamics}
\vspace*{2mm}
A hydrodynamically-inspired fit known as ``blastwave fit'' \cite{Blas93} assuming all particles are emitted from a thermal expanding source with a transverse flow velocity $<$$\beta_{T}$$>$ at the thermal freeze-out temperature $T_{fo}$ has been performed on $\pi$, $K$, $p$ spectra together and on $\Xi$ and $\Omega$ separately. A velocity profile $\beta_{T}(r)$=$\beta_{s}(r/R)^{n}$ was used, where $R$ is the radius of the source and $n$ was determined from the fit to the $\pi$, $K$ and $p$ spectra ranging from $n$=0.81 for the most central bin to $n$=1.42 for the most peripheral. For ($\pi$, $K$, $p$), 9 bins of centrality indexed from 1 (most central) to 9 (most peripheral) have been considered while 5 centrality bins for $\Xi^{-}+\overline{\Xi}^{+}$ have been studied. For $\Omega^{-}+\overline{\Omega}^{+}$, only the most central bin has been investigated. The results of the fits are presented on Figure~\ref{fig:blastwave}.
\begin{floatingfigure}[r]{9.cm}%
\begin{spacing}{0.8}
\hspace*{-0.2in}
\includegraphics*[ width=9.5cm,
keepaspectratio]{msbV2Fig2_5.eps}
\vspace*{-.15in}
\hspace*{.5in}
\caption{Kinetic freeze-out temperature, $T_{fo}$, as a function of the transverse flow velocity $<\beta_{T}>$ extracted from a hydro-inspired model of blastwave type from transverse momentum distributions.}
\label{fig:blastwave}
\vspace*{.20in}
\end{spacing}
\end{floatingfigure}%
The one and two sigma contours are represented for the best fit values ($T_{fo}$,$<$$\beta_{T}$$>$). For the most central bin, we note : 1) there is no overlap of the contours for ($\pi$,$K$,$p$) and ($\Xi$) suggesting that ($\pi$,K,p) take part to the same collective transverse radial flow different from the one developed by the $\Xi$ ; 2) concerning $\Xi$, they seem to thermally freeze-out at a temperature of $T_{fo}\sim153MeV$, close to the chemical FO temperature previously obtained from statistical fits to particle ratios whereas $T_{fo}$ for ($\pi$,$K$,$p$) amounts $90MeV$. It suggests that multi-strange particles should have decoupled earlier in the collision close to chemical FO ; 3) furthermore, the fact that they develop an as significant flow as $\Lambda$ and that their interaction cross-section is presumably very small, suggests that their flow has been developed prior to chemical FO so prior to the hadronization, probably at a partonic stage of the system and not in the hadronic phase as for ($\pi$,$K$,$p$). Otherwise, it is corroborated by the fact that the thermal FO parameters of the multi-strange baryons do not depend on the centrality and that $T_{ch}$ is close to $T_{fo}$. Concerning ($\pi$,$K$,$p$), results show that $T_{ch}$$>$$T_{fo}$ and that the difference between these two temperatures increases with centrality. It suggests a longer duration time between these two FO for the lightest particles ($\pi$,$K$,$p$) essentially due to their rescattering in the hadron phase while the system is cooling down.
These results indicate that Au+Au collisions with different initial conditions evolve always to the same chemical FO temperature, and then cool down further to a kinetic FO dependent on centrality.
So this radial flow scenario suggests that for multi-strange baryons, a significant fraction (if not all) of the transverse flow has been developed probably in a partonic phase of the system so that multi-strange baryons should develop elliptic flow. Figure~\ref{fig:v2Pt} shows the measurement of the elliptic flow $v_{2}$ of $\Xi^{-}+\bar{\Xi}^{+}$ and $\Omega^{-}+\bar{\Omega}^{+}$ as a function of $p_{T}$ for the minimum bias data. $v_{2}$ of $K_{s}^{0}$ and $\Lambda+\bar{\Lambda}$ previously measured \cite{Soer04} are also represented for comparison.
\vspace*{-6mm}
\begin{figure}[htb]
\begin{minipage}[t]{80mm}
{\rule[10mm]{-2mm}{52mm}\epsfig{figure=XiOmV2VsPt.eps,height=6cm}
\hspace{-1cm}
\vspace*{-10mm}
\caption{Elliptic flow $v_{2}$ of $K_{s}^{0}$, $\Lambda+\bar{\Lambda}$, $\Xi^{-}+\bar{\Xi}^{+}$ and $\Omega^{-}+\bar{\Omega}^{+}$ from 200GeV Au+Au minimum bias collisions. Hydrodynamic model calculations are shown (colored zone) as well as hydro-inspired model calculations (thick lines) but are not commented in this letter \cite{QM04}.}
\label{fig:v2Pt}
\vspace*{-6mm}
\end{minipage}
\hspace{\fill}
\begin{minipage}[t]{75mm}
{\rule[10mm]{-2mm}{52mm}\epsfig{figure=V2nVsPtn.eps,height=6cm}}
\vspace*{-10mm}
\caption{Elliptic flow $v_{2}$ of $K_{s}^{0}$, $\Lambda+\bar{\Lambda}$ and $\Xi^{-}+\bar{\Xi}^{+}$ normalized to the number of constituent quarks (n) as a function of $p_{T}/n$.}
\label{fig:v2nPtn}
\vspace*{-6mm}
\end{minipage}
\end{figure}
First we observe that $v_{2}$ of multi-strange baryons is different from zero and seems to follow the same behaviour as $\Lambda$ $v_{2}$. That means same shape (saturation at a $p_{T}\sim3GeV/c$) and same amplitude (saturation at $v_{2}\sim20\%$). In the low $p_{T}$ region, $\Xi$ $v_{2}$ is in agreement with hydrodynamic model calculations (colored zone) which predict its mass ordering in this $p_{T}$ region. However, for a $p_{T}>2GeV/c$, $v_{2}$ deviates from Hydrodynamic model prediction and shows different behaviour for $K_{s}^{0}$ which saturates at a $p_{T}=2GeV/c$, at a value around $14\%$ compared to the strange baryons $v_{2}$.
It confirms a previously established baryon to meson dependence of the elliptic flow parameter from a particle mass dependence in the intermediate $p_{T}$ region \cite{Soer04}. This particle type dependence is well and ``simply'' explained by quark coalescence or recombination models \cite{Moln03,Frie03} in which hadrons are dominantly produced by the coalescence of constituent quarks from a partonic system supporting the idea of a collectivity between partons. These models predict a universal scaling of transverse momentum $p_{T}$ and elliptic flow to the number of constituent quarks ($n$). Previously, such scaling has been demonstrated for the mesons $K^{0}_{s}$ and the baryons $\Lambda$ at intermediate $p_{T}$ \cite{Soer04}. Figure~\ref{fig:v2nPtn} shows the superposition of the scaled elliptic flows $v_2/n=f(p_{T})/n$ for $K^{0}_{s}$, $\Lambda$ as well as for $\Xi^{-}+\overline{\Xi}^{+}$, supporting that the flow of $s$ quarks is close to that of $u$ and $d$ quarks within error bars.
\section{CONCLUSION}
We have presented the evolution of freeze-out parameters with Au+Au collision centrality. The increase of the strange quark phase space saturation factor, $\gamma_{s}$, up to 1 for the most central collision suggests that strangeness equilibration is achieved at top RHIC energy. $T_{ch}$, common for all particles, appears to be independent of the system size. The kinetic FO parameters obtained from blastwave fit to $\pi$, $K$, $p$ spectra suggest that they are taking part to a same collective flow behaviour with an increasing duration time between chemical and thermal FO with centrality essentially due to hadron rescatterings. For multi-strange baryons, the collective behaviour seems to be quite different. Since $T_{ch}$ and $T_{fo}$ are close to each other and show no dependence with centrality, it indicates that multi-strange baryons take less part in the evolution dynamics in the hadronic phase and should have decoupled much earlier in the collision than ($\pi$, $K$, $p$), carrying with them an important partonic flow contribution. This idea is emphasized by the measurement of their elliptic flow whose scaling by the constituent quarks in the intermediate $p_{T}$ region is well described by coalescence and recombination models.
|
{
"timestamp": "2004-11-18T11:30:50",
"yymm": "0411",
"arxiv_id": "nucl-ex/0411034",
"language": "en",
"url": "https://arxiv.org/abs/nucl-ex/0411034"
}
|
\section{Introduction}
\label{intro}
Magnetic fields are pervasive throughout the universe on all
scales, from the fields surrounding the Earth up to fields in the
intracluster medium. Recently,
the r\^{o}le of cosmic magnetic fields has gained prominence across
many astrophysical disciplines where the fields present
are a key factor in understanding a variety phenomena such
as large-scale structure formation, galaxy and star formation,
and cosmic ray acceleration.
While astrophysical magnetic fields, on all scales, have been investigated
since the late seventies, the mostly indirect measurement
techniques have meant that it has been difficult to address many basic questions.
Thus, investigations into the seeding and amplification mechanisms,
strength and uniformity of magnetics fields produce a plethora of results and remain
topics of interest and vigorous debate within the community.
In particular, the notion of significant extra-galactic magnetic fields,
specifically those in clusters of galaxies was first discussed by \citet{Burbidge}.
However, it was not until recently that the existence
of such intracluster fields outside the lobes of radio galaxies
could be statistically confirmed \citep{ktk} and still later that convincing
numerical values could be ascribed to them \citep{Clarkephd}.
In this paper we will examine the magnetic field in the cores of southern,
X-ray luminous galaxy clusters via a statistical analysis of a sample
of Faraday rotation measures obtained from background and embedded radio sources.
To begin with we will describe the process of Faraday rotation in Section \ref{frm}.
In Section \ref{prev} we will discuss previous attempts to measure cluster
magnetic fields using a statistical analysis of rotation measures. Section
\ref{current} presents the rationale and selection criteria used in the
current study and Section \ref{obs} gives details of the observations. In Section
\ref{results} the results are discussed and then comparisons to other datasets
and conclusions are given in Section \ref{conclusions}.
\subsection{Faraday Rotation}
\label{frm}
Faraday rotation is a process where the application of an external
magnetic field will, in certain circumstances, produce a measurable change
to an electromagnetic wave. Incident polarised electromagnetic radiation
passing through a magnetised plasma will have its plane of polarisation
rotated by an amount determined by the properties of the plasma and the magnetic
field strength. The amount of rotation experienced is strongly dependent on the
frequency of the radiation. Comparisons of the amount of rotation at different
frequencies produces a metric known as the rotation measure or RM. Rotation
measures along the line of sight from a background object, such as a distant
radio galaxy or quasar, are used as probes of the foreground magnetic fields
through which the radiation has passed. If the properties of the plasma are
known (electron density and path length) it is possible to use the RM to
compute an averaged field strength along that line of sight.
Faraday Rotation,
was first proposed for astronomical sources by \citet{Cooper}, who used
it to explain the observed wavelength dependence of the polarisation position
angle seen in Centaurus A. The phenomenon can be described by:
\begin{equation}
\label{eqn_padef}
\chi = \chi_{in} + {(RM)} \lambda^2,
\end{equation}
\noindent
where $\chi_{in}$ is the intrinsic position angle of the radiation in
radians, and $\lambda$ is the wavelength in metres. If the measurable
quantities $\chi$ and $\lambda^2$ are plotted against each other,
the y-intercept will correspond to the
intrinsic position angle of the source and
the slope of the line will give the RM which
is defined as $RM=\chi/\lambda^{2}$. The degree of rotation is given
by the relation between the RM, line of sight magnetic field and the
electron density as described by the standard RM equation given below.
\begin{equation}
\label{eq_bint}
{\langle RM \rangle} = 8.1 \times 10^5 \int_0^L B_{\parallel} n_e {d}l
\hspace{0.2cm} \mbox{rad m}^{-2},
\end{equation}
\noindent
where $B_{\parallel}$ is the line of sight component of the
magnetic field in Gauss, $n_e$ is the electron density in cm$^{-3}$ and
$dl$ is the path length in pc.
Thus by obtaining
measurements of the position angle of the electric vector at a number of
different wavelengths it is possible to determine the RM.
It can be seen from equation \ref{eqn_padef} that the integral can be
separated for different Faraday screens, implying that the position angle
measured will be the linear sum of all rotations along the line of sight.
Thus, a more general form of equation \ref{eqn_padef} is:
\begin{equation}
\label{eqn_pasum}
\chi = \chi_{in} + \sum_{i}{(RM)}_{i} \lambda^2,
\end{equation}
\noindent
where $\sum_{i}{(RM)}_{i}$ denotes the linear sum of all RM contributions
along the line of site.
For extra-galactic radio sources there are at least four RM
terms to consider; the RM due to internal conditions in the probe
source, the RM of the objects along the line of sight,
the RM due to our own Galaxy and the RM caused by the Earth's ionosphere.
A further complication is the consideration of redshift, which gives a
correction to the wavelength emitted. In general, the observed RM, which
is the sum of all RM components, must also be corrected for redshift such that:
\begin{equation}
\label{eqn_rmred}
{(RM)}_{obs} = \sum_{i}^{F} [\frac{{(RM)}_{i}^{F}} {(1 + {z}_{F})^{2}}],
\end{equation}
\noindent
where ${RM}_{obs}$ is the measure RM and
$\frac{{RM}_{i}^{F}} {(1 + {z}_{F})^{2}}$ denotes each RM
component along the line of sight as a function of redshift. As the clusters
examined in this study were all at redshifts less than 0.06, the redshift
correction will be minimal (around ten percent).
\section{Previous Galaxy Cluster RM Studies}
\label{prev}
Part of the difficulty of investigating cluster magnetic fields through
Faraday rotation is that at present such a study may only be undertaken statistically.
This is due to the addition of all contributing Faraday screens along
the line of sight, making it impossible to disentangle the cluster rotation
measure components from either internal rotation in the source, a
Galactic rotation measure component, or an ionospheric component. However,
comparison of a sample of sources with lines
of sight through the intra-cluster medium, as compared to a control
sample, provides a statistically valid approach for the confirmation of an
enhancement of the rotation measure in cluster regions. Several analyses of
this kind have been performed with increasing degrees of success.
\citet{d} first attempted a statistical study of cluster
rotation measures by comparing a sample of 16 cluster radio sources to 16
controls.
Unfortunately, the sample size was too small for the result to be conclusive.
Following this, \citet{ld} reported a broadening of the scatter in their
RM data of 50 rad~m$^{-2}$ in a cluster sample of 24 sources when compared with only
10 rad~m$^{-2}$ in their control. They interpreted this as evidence of a cluster field with
B of the order of 1$\mu$G with a scale length greater than 20 kpc.
This result is only marginal as there was no
account taken of error broadening of the sample. \citet{hoe}
conducted the first statistical RM study to perform fits at four
different wavelengths, which largely removed the problem of the
$n\pi$-ambiguity in the RM. Their study found no
significant difference between their
cluster (16 sources) and control samples and they reported an upper limit on
the RM width excess in clusters (as compared to the control population)
of 55 rad~m$^{-2}$. This led to an upper limit on the cluster
magnetic field of B = 0.07 $\mu$G
for a uniform untangled field with core radius of 500 kpc.
\citet{Goldshmidt93} questioned this result and
recalculated it, assuming a scale length of 20 kpc over the
500 kpc core radius. They assumed an electron density of
$3 \times 10^{-3}$ cm$^{-3}$, which gave a net field strength of less than
0.2 $\mu$G. All three studies are considered too small to be
statistically significant on their own \citep{Goldshmidt93}.
\citet{Kim90} investigated the magnetic field in the
Coma cluster using 18 sources in the cluster field, 11 comprising the cluster
sample and 7 in the control. The authors themselves drew attention to the
problem of the small source sample, but nevertheless concluded a
``first order result'' of B less that 2 $\mu$G derived from an
{\it {average}} RM width excess of 30 rad~m$^{-2}$. Unfortunately, on closer
examination, a question arises as to the validity of those points used in the
\citet{Kim90} analysis for which only two wavelengths were used to
calculate the RM. Only RMs of at least four
(or a very carefully chosen three) wavelengths may produce a unique fit.
Given the small sample size and lack of uniqueness of a two-frequency fit
for half of the sample, it is not possible to draw any significant conclusion
from these data.
Following these observations \citet{ktk}
improved their source statistics by examining a large number of rotation
measures from the literature \citep{SN, Broten, hoe, Kimphd, ld, Vallee86}
supplemented by unpublished data from Kronberg.
152 source RMs were obtained and compared with the positions of Abell clusters.
This produced a catalogue with 53 sources comprising the cluster sample and
99 sources in the control. This study contains the largest cluster sample to
date.
The \citet{ktk} study, hereafter denoted KTK, divided both the cluster and
control sample into various subsamples. The cluster sample was divided into
two sections based on whether a source fell within one sixth of the
Abell radius.
The control sample contained a subsample of isolated giant elliptical
galaxies with properties similar to central dominant (cD) galaxies. The study
found that the distribution of RMs for the cluster sample was broader than
the control at the 99.9\% confidence level. Further, there was seemingly no
difference between the elliptical galaxy sample and the
rest of the control, from which it was inferred that the excess RM width in the
cluster sample was due to the intracluster medium and not some bias
due to preferential observation of cluster galaxies. At the 99\% confidence
level the excess RM width was in the range 51--84 rad~m$^{-2}$. Assuming a field
which was uniform through the cluster core but locally tangled on scales of
10 kpc, a B value of between 0.5 and 1.25 $\mu$G was obtained; this
was in contrast to a value derived using the model of B(r) $\propto$ n(r)
\citep{Jaffe}, which gave 1--2.25 $\mu$G for the inner cluster sample and
1.9--4.7 $\mu$G for the outer cluster sample. These results have been
examined in detail for robustness and it was found that there
is a significant statistical difference between the cluster and core samples
\citep{Goldshmidt93}. However, there is some concern as to the validity of the
RM fits in this sample also. Questions have also arisen as to the numerical
validity of the KTK results in light of the fact that some of
the ``clusters'' examined in their study were undetected at X-ray wavelengths
by the
$\it {Einstein}$ satellite
\citep{Clarkephd}. The lack of X-ray flux indicates that the clusters are
either quite poor, or are not gravitationally bound systems with significant
intracluster gas. RMs along the line of sight towards these objects will
have a considerably lower value than those directed at X-ray bright clusters
and thus will introduce a lowering numerical bias to the result. This implies
that the KTK result is likely to be {\em more} significant, both in terms of
detection and implied B value, than first thought and therefore should
be re-examined.
All of the afore-mentioned studies, with the exception of \citet{hoe},
suffer from lack of well-defined source selection criteria, which may lead
to bias in the sample. This was the impulse for \citet{Goldshmidt93}
to re-examine the KTK sample. \citet{Clarkephd} attempted to address the
lack of a large well-defined cluster RM sample by observing radio sources
toward 24 X-ray luminous Abell clusters using a strict set of selection
criteria. The \citet{Clarkephd} data comprised of both a cluster and
control sample
with 27 and 89 sources respectively. The cluster sample was chosen from 24 Abell
clusters with X-ray luminosity greater than 1 $\times$ 10$^{44}$ erg s$^{-1}$ in the 0.1 to
2.4 KeV band \citep{Ebeling96}.
In order to reduce Galactic contamination of the RM sample, clusters were
selected no closer than 13 degrees from the Galactic Plane and an averaging
technique was employed in an attempt to correct for this effect. A statistically
significant width difference between the RM distributions for the (Galactic field
corrected) cluster and control samples was observed
with the standard deviations of each distribution differing by almost an order of
magnitude ( $\sigma_{cluster}$ = 113 rad~m$^{-2}$ to $\sigma_{control}$ =
15 rad~m$^{-2}$). Further, the two samples were found to be drawn from
different populations at the 99.4\% confidence level. Using, electron densities
obtained from ROSAT X-ray observations, \citet{Clarkephd} also statistically examined
the strength of the cluster magnetic field. Two magnetic field
models were investigated. The first was a simple ``slab'' model, wherein the
magnetic field is assumed to be uniform in both magnitude and direction throughout
the cluster. This predicted field strengths of around 0.5 $\mu$G. The second, and
more sophisticated model, used tangled magnetic fields with particular cell
sizes; this produced field strengths of the order of 1 -- 1.5 $\mu$G.
A further investigation of cell sizes via RM mapping of sources in three of the
sample clusters suggested that the field had large scale uniformity at around 100 kpc
with smaller 10 kpc features. The structure observed in the RM mapping suggested
that a tangled cell model was more likely, and the uniform slab values were rejected.
In conclusion, \citet{Clarkephd} and \citet{Clarke01} asserted that field
strengths of $\geq$ 1 $\mu$G
were unlikely in rich X-ray luminous galaxy clusters.
\section{Current Study}
\label{current}
The samples of all previous studies have steered away from investigating radio
probes within the very cores of clusters (the median distance from the cluster
centre for sources in the \citet{Clarkephd} sample is 445 kpc). This is due, in
part, to the difficulty of find sufficiently polarised sources in a reasonable
amount of observing time with current instruments and in part because
radio sources embedded in so-called ``cooling core''
clusters have extreme RM values \citep{Taylor}, which are not indicative of
the overall cluster magnetic field. This work investigates a sample
of radio background and embedded probes directed toward seven rich X-ray-luminous, southern
galaxy clusters which do not exhibit a cooling core X-ray profile. The aim of
this investigation was to determine the extent to which the RM excess observed
by \citet{Clarkephd} is enhanced in the cores of clusters.
\subsection{Source Selection}
\label{source_sel}
The Australia Telescope Compact Array (ATCA) was selected for the task of measuring
the RMs due to both its southern locale and excellent polarisation properties. An initial
declination cut-off was imposed to select only clusters south of
$-30^{\circ}$ as these could reasonably be imaged by the ATCA.
A further concern was to select those clusters for which there was a high
probability of finding a polarised background source projected through
the cluster core. If one assumes that the density of polarised background
radio sources, which is a function of the telescope sensitivity, is constant
across the sky, then the probability of finding a suitable background source
behind the cluster core goes as the angular size of the core projected on
the sky. In order to give maximum probability of detecting a polarised probe,
a low redshift cut-off for the sample was established with only clusters
with z less than 0.06 examined.
As demonstrated in Section \ref{frm}, the measured RM will be
the linear combination of all RMs along the line of sight. If we
assume that the gas density in the intercluster medium is sufficiently low
so as to render the resultant RMs from any possible magnetic fields in this
region close to zero, equation \ref{eqn_pasum} can be rewritten as:
\begin{equation}
\label{eqn_pasum2}
\chi = \chi_{in} + \left[ (RM)_{cluster} + (RM)_{gal} +
(RM)_{ion}\right] \lambda^2
\end{equation}
where $(RM)_{cluster}$ is the cluster contribution to the RM, $(RM)_{gal}$ is the
galactic contribution and $(RM)_{ion}$ is the ionospheric contribution. This is not an
unreasonable assumption even for sources at
high redshift which have long path lengths through intercluster space, as
no redshift-RM correlation has ever been observed \citep{Kronberg76, Welter84, Vallee90}.
This leaves the observed RMs as the linear combination of the source
intrinsic, cluster, Galactic and ionospheric Faraday rotation contributions.
Ionospheric Faraday rotation has been studied in some detail \citep{IonFRM}
and is believed to contribute less than 5 rad~m$^{-2}$ at the ATCA observing
frequencies (Whiteoak, J., private communication 2001). The ionospheric
contribution will of course vary depending on the Solar Cycle and the value of less
than 5 rad~m$^{-2}$ is a average over a long time period. As these observations were
carried out just after a minimum in the 11 year Solar Cycle, this estimate should
be sufficient. The presence of a Galactic contribution to the RM error may be
minimised by appropriate selection criteria. This leaves the intrinsic RM which cannot
be removed.
In order to ensure that the RM sample was minimally contaminated by the
Galactic magnetic field an extensive study of the effect of the Galaxy on the RM sky was
undertaken \citep{mjh03} and only clusters more than $30^{\circ}$ from the Galactic plane
were considered. At these Galactic latitudes it was found that the
RM$_{galactic}$ $\sim$ 10 rad m$^{-2}$.
To further reduce the effect of Galactic contamination, a third
selection criterion was used in this study: that the clusters studied
must be X-ray luminous. The rationale identified here is that clusters with
high X-ray luminosity will have a large gas content (high n$_e$), which,
even in the
presence of weak magnetic fields, will give rise to relatively large RMs. Large cluster
RM contributions will then tend to dominate over the smaller
Galactic RMs at these Galactic latitudes. Clusters with X-ray luminosities
greater than $2 \times 10^{44}$ ergs~s$^{-1}$ in the 0.1-- 2.4 keV band were
selected from the XBAC sample \citep{Ebeling96}. This is a brighter
cut-off than that chosen by \citet{Clarkephd} who selected all clusters
down to $1 \times 10^{44}$ erg~s$^{-1}$ from the same sample.
Table \ref{tab_selection_crit} re-iterates these criteria.
\begin{table}
\caption[Cluster Selection Criteria for Statistical RM Study]{Selection
criteria for the Southern, rich Abell clusters used in the
statistical study of rotation measure in cores of
non-cooling-flow clusters.}
\begin{center}
\begin{tabular}{l|c|c}
\hline
Criteria & Parameter & Range \\
\hline
Southern Sample & Declination & dec $\leq -30^{\circ}$ \\
Low $RM_{gal}$ & Galactic Latitude & $|b| \geq 30^{\circ}$ \\
Angular Size & Redshift & z $\leq$ 0.06\\
Independent $n_{e}$ & X-ray Luminosity & $L_{x} \geq 2 \times 10^{44}$ erg s$^{-1}$ \\
\hline
\end{tabular}
\end{center}
\label{tab_selection_crit}
\end{table}
\subsection{Candidate Sources}
Applying the selection criteria outlined in Section \ref{source_sel} a
list of nine suitable rich, Southern, X-ray luminous clusters was obtained.
One cluster, A3532, which straddled the border for both the $-30^{\circ}$
declination and Galactic latitude cut-off was discarded.
Properties of the selected clusters are outlined in Table \ref{tab_clust}.
\begin{table*}
\caption[Clusters used in Statistical RM Study]{Southern, rich Abell
clusters used in the statistical study on rotation measures in
cores of non-cooling flow clusters. Col 1 is the ACO cluster name; col 2 the J2000 Right Accession; col 3 the J2000 Declination; col 4 the redshift; col 5 the X-ray flux in the
0.1--2.4 KeV band from \citet{Ebeling96} and col 6 shows if 1.4 GHz data were present in the ATCA archive.}
\begin{center}
\begin{tabular}{cccccc}
\hline
Name & RA & Dec & z & L$_{x} \times 10^{+44}$ & Archival\\
& J2000 & J2000 & & erg s$^{-1}$ &\\
\hline
A3667 & 20 12 30.1 & -56 49 00 & 0.0555 & 8.76 & yes\\
A3571 & 13 47 28.9 & -32 51 57 & 0.0397 & 7.36 & yes\\
A3558 & 13 27 54.8 & -31 29 32 & 0.0477 & 6.27 & yes\\
A3266 & 04 31 11.9 & -61 24 23 & 0.0545 & 6.15 & yes\\
A3562 & 13 33 31.8 & -31 40 23 & 0.0502 & 3.33 & yes\\
A3128 & 03 30 12.4 & -52 33 48 & 0.0590 & 2.12 & yes\\
A3158 & 03 42 39.6 & -53 37 50 & 0.0590 & 5.31 & no\\
A3395 & 06 27 31.1 & -54 23 58 & 0.0506 & 2.80 & yes\\
\hline
\end{tabular}
\end{center}
\label{tab_clust}
\end{table*}
Archival total intensity ATCA data at 1.4 GHz were available for seven of these
clusters. These data were examined to obtain a list of sources
which might act as suitable probes to the cluster magnetic field. As this
study was to focus on the cluster core, initially only sources which fell
within the fifty percentile contour of the X-ray emission and had a peak
flux density at 1.4 GHz greater than 12 mJy per beam (using a 6 arcsecond beam)
were selected. However, as this gave only twelve potential sources the criteria
were relaxed to 4 mJy per beam in the fifty percentile X-ray contour and to also include
sources which were
projected through any part of the X-ray emission region and that had a peak
flux density greater than 12 mJy per beam (using a six arcsecond beam).
Two additional sources located
behind the diffuse radio emission in A3667 was also included.
This generated a list of 39 candidate sources.
The double cluster A3395 was removed from the sample as the available
X-ray data shows clear signs of substructure, making it difficult to
determine a suitable X-ray centre.
The cluster, A3158, for which there were no archival radio
data was also observed at 20 and 13 cm for use in a future study.
In order to have sufficient points in the $\chi-\lambda^{2}$ plane to give
an unique fit to the RM, sources were required to have at least 5$\sigma$ detection of
polarisation at four frequencies.
Unlike the study of \citet{Clarkephd} who was able to make use
of the polarimetric data from the NVSS for selection of suitable polarised probe
sources, this study had out of necessity to begin with a polarimetric pilot
survey of the selected sources.
\section{Observations}
\label{obs}
In order to establish percentage polarisation the 39
candidate sources were targeted for ATCA observation in continuum mode at
1.4, 2.4, 4.7 and 6.7 GHz. The sources were first observed for a total period of 1 hour
each in ``cuts'' mode at 4.7 and 6.7 GHz. Data were examined using the UVFLUX routine in MIRIAD. This routine
provides information on the amplitude and associated noise in each of the Stokes values.
The results of UVFLUX were then used to determined the total polarised flux and percentage
polarisation observed for all sources at each frequency (the full set of results
are given in Table A.2 in \citet{mjh03}). It should be noted that UVFLUX is most
useful for determining characteristics of point sources.
This was not the optimal way to investigate the extended sources as, in general, while
the core maybe the brightest part of the source in these images, it is likely to be
less polarised then the surrounding low surface brightness material. However, it was
felt this was an acceptable procedure for the initial survey.
Unfortunately, the observations centred at 6.7 GHz were in a region of the
band where only spectral line observing is usually performed as the system
temperatures are quite high. As a result, these data were of very poor quality
and it was not possible to determine reliable source characteristics at this
frequency. For all subsequent observations the frequency was shifted to 6.2
GHz to avoid the high system temperatures at the very edge of the band.
Thus, only the 4.7 GHz data were used to obtain a list of
18 suitable cluster probes for re-observation. Sources were selected if they had a 5$\sigma$
detection in both Stokes I and one of either Stokes Q or U and the total linear percentage polarisation
was less than 45\%. As these data are not corrected for Ricean noise bias it is important to
select regions of high signal-to-noise ratios in order to reduce this effect. Thus, the
5$\sigma$ cut off in either Q or U was selected.\\
This gave a list of 15 sources. However, for the cluster A3562, this
gave only one source (A3562$\_$4e). So the last criterion was relaxed and this
gave an additional 3 sources for this cluster. Two anomalies occurred in the sources
selection here; the first was that
the source A3571$\_$8, which did fulfil the selection criteria, was accidentally omitted from
the follow-up source selection, the second is that the sources A3667$\_$28 and
A3667$\_$17 were accidentally switched and
A3667$\_$17 was observed in the follow-up study.
This turned out to be fortuitous as A3667$\_$17 turned out to be
sufficiently polarised for a RM
fit to be obtained and as the source was seen in projection
through the Mpc-scaled region of
diffuse radio emission in A3667 \citep{mjh03b}.
Thus, from 39 potential targets 15 were selected and an additional 3 added. This is a similar
attrition rate to that experienced by Clarke who began with roughly 250 sources and
obtained less than 60 useable sources (Clarke 2000, private communication).
\begin{table*}
\label{tab_surv3}
\caption{The 20cm flux of the sources
selected for the statistical RM study. Col 1 is the source identifications used in the ATCA observing program
and ATCA archive; col 2 is the J2000 right
accension; col 3 is the J2000 declination; col 4 is the peak flux density at 1.4 GHz in mJy; col 5 is
the known optical identification and col 6 is the redshift; col 7 gives the location of the source as
either embedded or background to the cluster (sources for which there was no optical counterpart
found in current sky surveys are assumed to be quite distant and hence background sources).}
$$
\begin{array}{lllllll}
\noalign{\smallskip}
\noalign{\hrule}
\noalign{\smallskip}
{\rm Source} & {\rm RA} & {\rm DEC} & {\rm S_1._4(peak)} & {\rm Optical}{\rm ID} & {\rm z} & {\rm Location}\\
& {\rm J2000} & {\rm J2000} & {\rm mJy} & & & \\
\noalign{\smallskip}
\noalign{\hrule}
A3128\_5 & 03\:51\:10.070 & -52\:28\:46.71 & 186.0 & & &{\rm background} \\
A3128\_10 & 03\:31\:15.000 & -52\:41\:47.98 & 44.0 & {\rm APMBGC 155-096-118}& 0.0665 &{\rm background} \\
A3266\_3 & 04\:30\:42.130 & -61\:27\:18.13 & 9.7 & {\rm J0430419-612716}& 0.0632 & {\rm background} \\
A3266\_4{\rm e} & 04\:30\:21.950 & -61\:31\:59.90 & 165.9 & {\rm J0430219-613201} & & {\rm background} \\
A3558\_1{\rm e} & 13\:28\:29.880 & -31\:19\:31.75 & 22.0 & & & {\rm background} \\
A3558\_7 & 13\:29\:04.510 & -31\:31\:10.09 & 76.5 & & & {\rm background} \\
A3558\_8 & 13\:28\:31.530 & -31\:35\:06.04 & 98.6 & & & {\rm background} \\
A3558\_10 & 13\:92\:13.300 & -31\:21\:54.60 & 14.7 & & & {\rm background} \\
A3558\_13 & 13\:28\:02.580 & -31\:45\:21.77 & 17.7 & {\rm J1328026-314520} & 0.0429 & {\rm embedded}\\
A3562\_3 & 13\:33\:37.370 & -31\:30\:47.18 & 27.1 & & & {\rm background} \\
A3562\_4 & 13\:43\:37.450 & -31\:32\:52.74 & 13.9 & & & {\rm background} \\
A3562\_5 & 13\:34\:22.480 & -31\:39\:08.31 & 13.2 & & & {\rm background} \\
A3562\_6{\rm e} & 13\:33\:31.566 & -31\:41\:02.77 & 20.0 & {\rm A3558:[MGP94] 4108}& 0.0482 & {\rm embedded}\\
A3571\_1 & 13\:47\:54.100 & -32\:37\:00.60 & 11.7 & & & {\rm background} \\
A3571\_3{\rm e} & 13\:48\:07.620 & -32\:46\:15.00 & 101.7 & & & {\rm background} \\
A3667\_A & 20\:11\:09.272 & -56\:26\:59.59 & 35.1 & & & {\rm background} \\
A3667\_26{\rm e}& 20\:11\:27.540 & -56\:44\:06.60 & 93.4 & {\rm SC 2008-565:[PMS88] 037}& 0.0552 & {\rm embedded}\\
A3667\_17 & 20\:09\:25.368 & -56\:33\:26.76 & 48.5 & & & {\rm background} \\
\\
\noalign{\smallskip}
\noalign{\hrule}
\end{array}
$$
\end{table*}
The 18 sources given in Table \ref{tab_surv3}
were re-observed in the short observation mode on the ATCA
at 1.4, 2.4, 4.7 and 6.2 GHz over the period from Feb 1999 to November 2000.
This was to both improve the signal-to-noise ratio and thus reduce the effect of
Ricean noise bias and to obtain the 4 frequencies for the RM fitting.
Observations at 1.4 and 2.4 GHz were carried out
simultaneously using a 6 km configuration while the observations at 4.7 and
6.2 GHz were performed using a 1.5 km configuration so as to match the
resolution of the lower frequency observations. Data were then reduced in
the MIRIAD \citep{Sault95} suite using standard calibration procedures.
Tapering in the $uv-$plane was applied in
order to lower the resolution of each observing frequency to that of the 1.4 GHz
images. Images of all four Stokes parameters were then made at all four
frequencies and total polarisation and position angle images were calculated.
In all cases the images were used to measure the polarisation and position angle of
the brightest part of each source. For the extended sources this meant that the
core was used initially. Treatment of the low surface brightness components of
the extended sources will follow in second paper.
\subsection{Polarisation Data}
It has been well established that while the total intensity
of radio sources usually decreases inversely with the observing frequency, the
percentage polarisation increases. Thus, there was always some risk
that sources selected with sufficient polarised flux at
the one of the higher frequencies (e.g. 4.7 GHz) would not be detectably polarised at the lower
frequencies. This turned out to be the case for some of the
sources in the final sample. There were also other problems with five sources
not well detected in polarisation at any of the frequencies used in the final
sample. Of these 1 was only just detected in the pilot survey above the 5$\sigma$ level
in the Ricean bias un-corrected data and it is thus not surprising that it
turned out to be unpolarised. However, the other 4 sources all had greater
than 10$\sigma$ detections for polarisation at 4.7 GHz in the pilot survey, and
it is a puzzle as to why they were not at least detected again at this frequency.
Table 4 gives the measurable position
angles for each source at each frequency. As only sources which had a
measurable position angle for at least three frequencies could be used for reliable RM
fitting, this reduced the sample to 11. The source ``A3558$\_$1e'' was partly
resolved into a double radio galaxy with considerable polarisation
in both lobes at three of the observing frequencies. Position angle
measurements were taken separately from each lobe, this provided an
additional line of sight through the ICM bring the total number of
RM obtained to 12, of which 9 are background and 3 are embedded cluster sources.
\begin{table*}
\label{tab:theta}
\caption{Observed Position angle results for the RM
Source Sample. Col 1 gives the source name used in the ATCA observing program and data archive; col 2 is the measured position angle at 1.4 GHz in degrees;
col 3 is the measured
position angle at 2.4 GHz in degrees; col 4 is the measured position angle at 4.7 GHz in degrees; col 5 is
the measured position angle at 6.2 GHz in degrees; col 6 gives notes on the source morphology ;col 7 is the
distance from the cluster centre, often called the impact parameter, in kiloparsecs and col 8 is the sources
rotation measure in rad~m$^{-2}$.}
$$
\begin{array}{lcccclcr}
\noalign{\smallskip}
\noalign{\hrule}
\noalign{\smallskip}
{\rm Source} & {\rm \psi_1._4} & {\rm \psi_2._4} & {\rm \psi_4._7} & {\rm \psi_6._2}& {\rm Notes} & {\rm Dist (kpc)} & {\rm RM} \\
\noalign{\smallskip}
\noalign{\hrule}
{\rm A}3128\_5 & 57 \pm 4 & 9 \pm 4 & -49 \pm 1 & -51 \pm 1 & {\rm extended} & 626\pm4 & 43.7\pm 6.4\\
{\rm A}3128\_10 & 23 \pm 10 & 16 \pm 4 & 22 \pm 5 & 35 \pm 10 & {\rm point} & 769\pm6 & -75.9\pm 11.7\\
{\rm A}3266\_3 & & & & & {\rm extended} & & \\
{\rm A}3266\_4e & -13 \pm 8 & -2 \pm 0.8 & -72 \pm 2 & -82 \pm 4 & {\rm extended} & 557\pm5 & 99.7\pm 8.3\\
{\rm A}3558\_1en & 4 \pm 1 & -42 \pm 3 & -55 \pm 7 & & {\rm extended} & 637\pm6 & 25.0 \pm10.0\\
{\rm A}3558\_1es & -48 \pm 1 & 16 \pm 3 & -27 \pm 8 & & {\rm extended} & 637\pm6 & 66.4 \pm8.6\\
{\rm A}3558\_7 & & & -12\pm6 & & {\rm point} & & \\
{\rm A}3558\_8 & -56 \pm0.4 & -74 \pm 1 & -86 \pm 1 & -85 \pm 3 & {\rm point} & 490\pm5 & -61.4 \pm3.3\\
{\rm A}3558\_10 & & & & & {\rm point} & & \\
{\rm A}3558\_13 & & & & & {\rm point} & & \\
{\rm A}3562\_3 & -25 \pm 2 & 74\pm 6 & -84 \pm3 & & {\rm pointish} & 517\pm5 & 250.7 \pm7.4\\
{\rm A}3562\_4 & & & & & {\rm point} & & \\
{\rm A}3562\_5 & -21 \pm 3 & & -68 \pm3 & -70 \pm 2 & {\rm point} & 580\pm5 & 19.6\pm 7.2\\
{\rm A}3562\_6e & & & 54 \pm 4 & & {\rm headtail} & & \\
{\rm A}3571\_1 & & & & & {\rm point} & & \\
{\rm A}3571\_3e & -48 \pm 2 & -58 \pm 2 & -75 \pm 3 & -89 \pm 3& {\rm double} & 427\pm5 & 161.0 \pm7.0\\
{\rm A}3667\_17 & -22 \pm 1 & 70\pm 2 & 39 \pm2 & 52 \pm 4 & {\rm point} & 1743\pm15 & -174.6 \pm6.8\\
{\rm A}3667\_26e & -79 \pm 3 & 51 \pm 2 & -53 \pm 5 & -43 \pm 3 & {\rm headtail} & 578\pm5 & -86.2 \pm7.4\\
{\rm A}3667\_a & -47 \pm 1 & 68 \pm 9 & 32 \pm 3 & 41 \pm 4& {\rm double} & 1444\pm12 & -107.7\pm 6.8\\
\\
\noalign{\smallskip}
\noalign{\hrule}
\end{array}
$$
\end{table*}
\section{Analysis: RM Fitting}
Plots of the position angle and frequency were examined. Due to the small
number of points to consider it was not necessary to write a complicated
fitting routine to find the best fit. Data were adjusted by hand under
the assumption that there would be no rotation between the two closely spaced points at 4.7 and
6.2 GHz and that a maximum of 360 degrees ambiguity was likely to
have occurred in the 1.4 GHz value. Assuming no rotation between the closely
space values at 4.7 and 6.2 GHz allows for $|$RM$|$ $\leq$ 1350 rad~m$^{-2}$ to
be fitted unambiguously. The resultant points were then passed through a standard linear
least-squares fitting routine. It was found in many cases that the 2.4 GHz points
were difficult to reconcile with the other three measurements, giving slightly
discrepant values. Despite efforts to minimise errors this is likely to be a
result of the polarisation response of the 13cm feed on the ATCA. Though
sources were observed near the beam centre in order to have the best polarisation
characteristics at all frequencies it appears that some effect is still evident in the
13 cm data. Thus, the 13cm points were given less weighting
during the fitting procedure.
Figure \ref{fig:rmfit} shows the resultant plots,
while the RMs obtained are listed in Table \ref{tab:theta}.
Surprisingly the fitting worked extremely well with the
worst case fit to the data still giving a 99 \% confidence to a straight line.
\begin{figure}[htbp]
\centering
\resizebox{\hsize}{!}{\includegraphics{rmfits1v2.eps}}
\resizebox{\hsize}{!}{\includegraphics{rmfits2v2.eps}}
\caption{RM fits for the cluster sample examined here. The graph shows the
equation of the line where the RM corresponds to the slope, the R$^{2}$ statistic
which is a measure of the goodness of fit to a straight line is also
given (an R$^{2}$ value of 1 corresponds to a perfect fit, the 13 cm data is given
less weight in the fitting due to the poor ATCA off-axis response) .}
\label{fig:rmfit}
\end{figure}
It should be noted that the use of observations from a period
spanning 18 months gives some concern as the data are not all from the same
epoch. Nevertheless the quality of the RM fits is excellent and it is
therefore assumed that there is no significant variation in the observed source RMs over
this time scale.
\section{Results: Comparison to Other Data}
\label{results}
The RMs obtained from the fitting procedure were then corrected for
the contribution from the Galactic rotation measure, G$_{RM}$. This was done by
using an interpolated all-sky rotation measure map generated from published
RM catalogues \citep{mjh04}. The Galactic contribution was subtracted
from the measured RM to give a residual RM, (RRM) which represents a
combination of the cluster RM and the intrinsic source RM.
Previous studies have used the standard deviation of
the distribution of extra-galactic RMs at high galactic latitudes,
beyond the influence of the Galaxy, to argue that the contribution from
internal RMs is small. The standard deviation of around 400 extra-galactic RMs
at greater than 30 degrees from the Galactic plane was found to be
10 rad~m$^{-2}$. This suggests that
the intrinsic RM component should be small
and that RRM should adequately represent the Cluster contribution to the
measured RM. Previously it was assumed that the distribution was Gaussian and
thus the likelihood of encountering a moderate to high intrinsic RM
was very low. However, further analysis of the high Galactic
latitude extra-galactic RM population has shown the distribution
is exponential at above the 99.9\% confidence level. This means that it is
more likely to observe a background source with a
significant internal contribution to the measured RM than previously thought.
This reinforces the requirement to examine cluster magnetic fields statistically.
\begin{figure}[htbp]
\centering
\resizebox{\hsize}{!}{\includegraphics{RM1.eps}}
\resizebox{\hsize}{!}{\includegraphics{RM3.eps}}
\caption{Impact Parameter versus Residual Rotation Measure. The labelled points
correspond to sources seen in projection through a region of diffuse synchrotron
emission in A3667.}
\label{fig:RMplot}
\end{figure}
\begin{figure}[htbp]
\centering
\resizebox{\hsize}{!}{\includegraphics{RM4.eps}}
\resizebox{\hsize}{!}{\includegraphics{RM2.eps}}
\caption{Impact Parameter versus Residual Modulus Rotation Measure. The labelled points
correspond to sources seen in projection through a region of diffuse synchrotron
emission in A3667.}
\label{fig:RMplot2}
\end{figure}
\citet{Clarkephd} and \citet{Clarke01} also corrected for the Galactic
contribution via examining published RMs in a 15 degree radius about each
cluster. In order to directly compare the two datasets, the RRMs from the
\citet{Clarkephd} sample were recalculated using the interpolated map. In most
cases this made a 5-10\% in the RRM values. The
two samples were then plotted together on two graphs showing distance from the
cluster centre (the so-called impact parameter) versus RRM and $|$RRM$|$.
These plots are shown in Figures \ref{fig:RMplot} and \ref{fig:RMplot2}
respectively.
Each plot shows the entire combined dataset out to an impact parameter of
9000 kpc in the top panel and restricted impact parameter range of 0 to 3000
kpc in the lower panel. The RRM of the southern cluster
sample presented here (see Table \ref{tab_RRM}) agrees well with the
northern sample of Clarke which drops to a background level at around
800 kpc from the cluster core.
The two labelled points are both from A3667 and are background sources to
the largest and brightest diffuse radio emission region yet discovered
\citep{Rottgering97, mjh03}. These two points are significantly above the RM level
suggested by the other data. They are further beyond the region of X-ray
emission in A3667 and should fall at a background level, the fact that they do
not strongly suggests that these RMs are probing the magnetic field of the
diffuse radio emission.
The data for both lobes of the source A3558.1e show quite different RRMs
(-0.5 and 41.9) despite being closely spaced. This may either be interpreted
as due to tangling of the cluster magnetic field on scales of order of 10 kpc,
or as a difference in the internal properties of the radio lobes. Observations
of radio jets in low power radio sources have shown that the magnetic
field is aligned along the jet and becomes tangled in the resultant lobes due
to entrainment. Thus, it is likely here that we are seeing a combination internal
and environmental effects.
\begin{table}
\label{tab_RRM}
\caption{Residual RM corrections. Column 1 gives
the source name used in the ATCA observing program and archive; col 2
is the J2000 galactic longitude; col 3 is the J2000
galactic latitude; col 4 is the Galactic contribution to the rotation measure
at these co-ordinates as calculated with an interpolated all-sky rotation
measure map \citep{mjh03b} and col 5 is the residual rotation measure for
each source once the Galactic contribution is subtracted.}
$$
\begin{array}{lllll}
\noalign{\smallskip}
\noalign{\hrule}
\noalign{\smallskip}
{\rm Source} & {\rm l} & {\rm b} & {\rm G_{RM}} & {\rm RRM} \\
\noalign{\smallskip}
\noalign{\hrule}
{\rm A}3266\_4{\rm e} & 272.29 & -40.23 & -44.3 & 144.0\\
{\rm A}3128\_5 & 262.75 & -48.20 & -4.9 & 48.6\\
{\rm A}3128\_10 & 264.82 & -50.91 & -3.2 & -72.7\\
{\rm A}3558\_1{\rm en} & 312.15 & 30.88 & 24.5 & -0.5\\
{\rm A}3558\_1{\rm es} & 312.15 & 30.88 & 24.5 & 41.9 \\
{\rm A}3558\_8 & 312.11 & 30.62 & 24.5 &-85.9 \\
{\rm A}3562\_3 & 313.37 & 30.50 & 29.1 & 221.6\\
{\rm A}3562\_5 & 313.52 & 30.34 & 26.3 & -6.7\\
{\rm A}3571\_3{\rm e} & 316.49 & 28.60 & 22.2 & 138.8\\
{\rm A}3667\_{\rm A} & 341.32 & -33.21 & -9.5 & -98.2\\
{\rm A}3667\_26{\rm e}& 340.98 & -33.25 & -9.5 & -76.7\\
{\rm A}3667\_17 & 341.19 & -32.97 & -9.5 & -165.1\\
\noalign{\hrule}
\end{array}
$$
\end{table}
A Kolmogorov-Smirnoff test to assess the likelihood that the combined cluster and the control
samples are drawn from the same population was preformed. The test rejected the null hypothesis
at greater than 99 precent.
\subsection{Embedded versus Background Sources}
Recently it has been claimed that the rotation measures of radio sources
embedded in clusters are not useful as probes of the global cluster magnetic
fields but rather only probe the field local to the source \citep{larry}.
As a counter argument to this we have taken the subsample of those
galaxies presented here and in Clarke (2000) which are background to
the clusters and performed a Kolmogorov-Smirnoff (KS) test to assess if
these data are drawn from the sample population as the control galaxies of Clarke (2000).
The KS test rejects the null hypothesis at greater than the 99\% confidence level
demonstrating that the two samples are not drawn from the same population. We also added the
data from \citet{hoe} to the sample increasing both the cluster and control sample and again per
formed a
KS test to see if the two samples are drawn from the same population. Again the null hypothesis
can be
rejected at over 99\% confidence. In addition, we also
considered the hypothesis that the RMs derived from embedded and background sources might be
drawn from the same population. In this case the null hypothesis could not be rejected and it
seems, that statistically speaking at least, there is no difference in the value of RMs derived
from
background or embedded cluster sources seen in project through galaxy clusters. Thus,
despite concerns over the validity of the embedded
galaxies as probes there is no statistical evidence this class of source gives rise to
significantly different
RMs to background galaxies though both a combined embedded and cluster background sample gives
rises to a significantly different RMs than the control sample. Further, even using only
background sources there is a still statistically
significant excess RM detected along lines of sight through X-ray luminous clusters when
compared with other lines of sight. With the exception
of the points in A3667, our data agree well with the previous samples and support the
finding by Clarke (2000) and show an excess RM toward the centre of galaxy clusters. We find
the combined cluster RM sample of Clarke (2000), \citet{hoe} and these data has a standard deviation of
$\sigma_{RM}$ = 125 rad~m$^{-2}$. In comparison, the standard deviation of a sample of 474
extra-galactic sources at least 30 degrees from the galactic plane gave $\sigma$ =10 rad~m$^{-2}$ \citep{mjh03}.
\section{Conclusion}
\label{conclusions}
We have presented the results of a search to detected excess Faraday rotation toward the
cores of several southern, non-cooling flow clusters. We find that the population of RMs derived
from combined sample of data from this work and the literature through lines of sight through
galaxy clusters is statistically different to RMs from other lines of sight. Further, we find that
this holds even for a sample of only background galaxies seen in projection through clusters.
Additionally, a comparison of data from embedded and background sources can not reject the null
hypothesis that the values are drawn from the same population. In conclusion we argue that the
results of this study agree well with those of Clarke (2000), supporting the notion that a
statistically significant broadening of the RM distribution is measured out to
around 800 kpc for nearby galaxy clusters. This suggests cluster magnetic
fields to be of the order of 1--2 $\mu$G assuming a tangled cell model.
\section*{Acknowledgements}
We thank Dr Tracy Clarke for providing updated information relating to her thesis work and her useful
discussions on the topic. In addition, we thank Dr Larry Rudnick for interesting discussions in
particular those on sources of error in statistical RM analyses. MJH extends her thanks to the
staff of ATNF for their support during all stages of data collection and analysis.
The Australia Telescope Compact Array telescope is part of the Australia Telescope
which is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO.
This research has made use of the NASA/IPAC Extragalactic Database (NED) which is
operated by the Jet Propulsion Laboratory, California Institute of Technology, under
contract with the National Aeronautics and Space Administration.
\bibliographystyle{aa}
|
{
"timestamp": "2004-11-02T06:07:35",
"yymm": "0411",
"arxiv_id": "astro-ph/0411045",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411045"
}
|
\section*{Appendix \thesection\protect\indent \parbox[t]{11.715cm} {#1}
}
\addcontentsline{toc}{section}{Appendix \thesection\ \ \ #1}
}
\begin{document}
\begin{titlepage}
\hfill\hbox to 3cm {\parbox{4cm}{
NSF-KITP-04-120 \\
}\hss}
\vspace{.5cm}
\begin{center}
\mbox{\large\bf Geometric Transitions,}
\vspace{.15cm}
\mbox{\large\bf Non-Kahler Geometries and String
Vacua}
\vspace{1cm}
{Katrin Becker$^a$,~ Melanie Becker$^b$,~ Keshav Dasgupta$^c$,~ Radu
Tatar$^d$}
\vspace{1cm}
{\it ${}^a$ Department of Physics, University of Utah\\}
{\it Salt Lake City, UT 84112, USA}
\vspace{.5cm}
{\it ${}^b$ Department of Physics, University of Maryland \\}
{\it College Park, MD 20742, USA}
\vspace{.5cm}
{\it ${}^c$ Department of Physics, Stanford University \\}
{\it Stanford, CA 94305, USA}
\vspace{.5cm}
{\it ${}^d$ Theoretical Physics Group, LBL Berkeley \\}
{\it Berkeley, CA 94720, USA}
\vspace{.5cm}
{\it ${}^d$Kavli Institute for Theoretical Physics \\}
{\it University of California \\}
{Santa Barbara , CA 93106-4030, USA}
\end{center}
\vspace{1.5cm}
\begin{abstract}
\noindent We summarize an explicit construction of a duality cycle for
geometric transitions in type II and heterotic theories. We
emphasize that the manifolds with torsion constructed with this
duality cycle are crucial for understanding different phenomena
appearing in effective field theories.
\end{abstract}
\end{titlepage}
\section{Introduction}
The connections between string theory and realistic supersymmetric
gauge theories have been extensively studied in the last years.
One of the approaches is the one taken by Vafa \cite{vafa1} that
is based on the duality between open topological strings on a
Lagrangian submanifold and closed topological strings on a
resolved conifold. This has been extended to the type IIB theory
in \cite{civ,eot} where the open string side is described by D5
branes wrapping a two cycle of a resolved conifold and the closed
string side is a warped deformed conifold with fluxes. The open
string side captures the far IR behavior of the gauge theory. The
full picture which studies the UV as well as IR behavior was
described in reference \cite{katz} where the cascading from the UV
to the IR is shown to arrive from an infinite sequence of flop
transitions.
The first goal of this note is to describe the type IIA
transition in detail, by considering D6 branes wrapped on three
cycles inside a non-K\"ahler deformation of the deformed conifold
\cite{trans1}. This corresponds to the open string picture. The
closed string dual is a compactification with RR fluxes on
another non-K\"ahler manifold with $dJ \ne 0$ and $d \Omega \ne
0$ and with a superpotential
\begin{equation}
W_{IIA} = \int (J+i B) \wedge d \Omega.
\end{equation}
The second goal is to extend the cycle of geometric transitions
to type I and heterotic strings. The dual corresponds to either a
type I string compactified on a non-K\"ahler but complex manifold
(observe that the type IIA manifolds were non-complex) with the
superpotential \cite{prok,ccdl}
\begin{equation}
W_{I} = \int (H_{RR} + i d J) \wedge \Omega,
\end{equation}
or to the heterotic string compactified on a non-K\"ahler but
again complex manifold with the superpotential \cite{prok,ccdl}
\begin{equation}
W_{het}=\int(H+i d J)\wedge \Omega.
\end{equation}
Here $H$ is the usual three form of the heterotic theory that
satisfies $dH = {\rm tr} ~R \wedge R - \frac{1}{30} \mbox{tr}~F
\wedge F$ and $H_{RR}$ is the RR three form that is S-dual to $H$.
One important result of our work is that we identify a geometric
transition for both the type I and heterotic string theories. We
will be able to specify the backgrounds in the type I and
heterotic theories on both sides of the duality. This suggests
the possibility of having a gravity description for wrapped D5
branes (for type I) or wrapped NS5 branes (for heterotic). Our
work also provides an alternative picture to the Landscape
distribution of string vacua \cite{kklt,douglas}.
\section{Type IIA Superstrings and Non-Calabi-Yau Manifolds}
Geometric transitions are examples of generalized AdS/CFT
correspondences which relate D-branes in the open string picture
and fluxes in the closed string picture. There are several types
of geometric transitions depending on the framework in which we
formulate them. The type IIB geometric transition starts with
$D5$ branes wrapping a $P^1$ of a resolved conifold. The
corresponding metric is given in \cite{pando} as
\begin{equation}
\label{2b} ds^2 = (dz + \Delta_1~{\rm cot}~\theta_1 ~dx +
\Delta_2~{\rm cot}~\theta_2 ~dy)^2 + \vert dz_1\vert^2 + \vert
dz_2\vert^2,
\end{equation}
where we have
replaced the metric of two spheres of the resolved conifold by two
tori with complex structures $\tau_1$ and $\tau_2$. The complex one forms
$dz_i,~ i = 1, 2$
are therefore defined as
\begin{equation}
dz_1 = dx - \tau_1~d\theta_1, ~~~~~~ dz_2 = dy - \tau_2~d\theta_2.
\end{equation}
We now want to apply the result of \cite{syz} which tells us that
for a manifold admitting a $T^3$ structure, and in the limit of a
large complex structure the mirror manifold is obtained by
performing three T-dualities on the $T^3$ torus \footnote{This is
a different type of T-duality than the one considered in
\cite{dot,rr,llt} where the result of a single T-duality was a
brane configuration in type IIA.}. In our case, the large complex
structure is obtained by boosting \footnote{This can also be
viewed as if we had introduced new complex structures on the two
tori. There is a subtlety related to whether these complex
structures are integrable or not. Considering only the integrable
complex structures leads us very close to the right mirror metric,
which can nevertheless be obtained by choosing non-integrable
complex structures. This discussion has appeared in
\cite{trans1}.}
\begin{equation}
dz_i \rightarrow dz_i + f_i~d\theta_i,~~~f_i \rightarrow \infty,~~i=1,2.
\end{equation}
In the presence of a NS field with components $b_{x\theta_1}$ and
$b_{y\theta_2}$, we have explicitly performed the mirror
tranformation in \cite{trans1}. The outcome after the mirror
transformation
is the following metric \footnote{There is a subtlety that we should mention here.
The mirror rules of \cite{syz} tell
us that we should take a limit of large complex structures. On the other hand
geometric transitions occur in exactly the opposite limit. Therefore naively applying
\cite{syz} we do not get the right answer. The correct answer was
derived in \cite{trans1} by performing a set of coordinate
transformations.}:
\begin{eqnarray}
\label{before1} ds_{IIA}^2 =& g_1~[dz +
\Delta_1~\mbox{cot}~\hat{\theta}_1~(dx - b_{x\theta_1}~d\theta_1)
+ \Delta_2~\mbox{cot}~ \hat{\theta}_2~ (dy -
b_{y\theta_2}~d\theta_2)]^2 \\ \nonumber & + g_2~[d\theta_1^2 +
(dx - b_{x\theta_1} d\theta_1)^2] + g_3~ [d\theta_2^2 + (dy -
b_{y\theta_2} d\theta_2)^2{]}+ \\ \nonumber & + g_4~{\rm sin}~\psi~ {[} (dx -
b_{x\theta_1}d\theta_1)d \theta_2 + (dy -
b_{y\theta_2} d\theta_2) d\theta_1 {]} + \\ \nonumber & + g_4~\mbox{cos}~\psi~ [
d\theta_1 d\theta_2 - (dx - b_{x\theta_1} d\theta_1) (dy -
b_{y\theta_2} d\theta_2)].
\end{eqnarray}
This metric (\ref{before1}) is exactly a non-K\"ahler deformation
of the metric for D6 branes on the three cycle of a deformed
conifold. The non-K\"ahler deformation can be seen from the
presence of the fields $b_{x\theta_1}$ in $d\hat{x} = dx - b_{x\theta_1}~d\theta_1$
and of $b_{y\theta_2}$ in $d \hat{y}=dy - b_{y\theta_2}~d\theta_2$. This
implies that the K\"ahler form $J$ and the 3-form $\Omega$ are
not-closed. Even though the manifold does not have an SU(3)
holonomy it has an SU(3) structure, so that supersymmetry is
preserved.
Furthermore one can easily show that any $B_{NS}$ field appearing
on the type IIA side after mirror is a gauge artifact. This is
most transparent if one chooses integrable complex structures for
the two tori in type IIB theory from the very beginning. There are
also non-trivial one form fluxes from the D6 branes sources. The
three form field vanishes and the coupling constant is equal to
the type IIB coupling constant.
This is the starting point of the type IIA transition. In order
to go to the closed string side, we need to first lift the
geometry to M theory to perform a flop and then dimensionally
reduce again to the type IIA theory \cite{amv}. The fact that the
type IIA theory is compactified on an SU(3)-structure manifold
implies that M theory is compactified on a $G_2$ structure
manifold. The absence of a $G_2$ holonomy is due to the
non-closure of $\Phi =J \wedge e^7 + \Omega_+ $ and its Hodge
dual. It was shown in \cite{trans1} that the identification of the
one forms and the performance of a flop can be done using the
methods of \cite{amv} \footnote{As expected, the one forms that we
would now require will be different from the ones chosen by
\cite{amv}. Indeed this is what we get. The one forms that we use
to specify the M-theory manifolds reduce to the one forms of
\cite{amv} when we turn off the non-K\"ahlerity of the type IIA
theory.}. After doing so and descending to the type IIA theory,
the result we get is
\begin{eqnarray}
ds^2 = h_1~ [d\theta_1^2 + (dx - b_{x\theta_1}~d\theta_1)^2] +
h_2~[d\theta_2^2 + (dy - b_{y\theta_2}~d\theta_2)^2] \\ \nonumber
+ h_3 ~[dz + \Delta_1~\mbox{cot}~\hat{\theta}_1~(dx -
b_{x\theta_1}~d\theta_1) + \Delta_2
~\mbox{cot}~\hat{\theta}_2~(dy - b_{y\theta_2}~d\theta_2)]^2,
\end{eqnarray}
which is precisely the metric
of a resolved conifold when we switch off $b_{x\theta_1}$ and
$b_{y\theta_2}$. This is thus the closed string background with
no D6 branes but only sources. The manifold is non-K\"ahler as
well as non-complex. The identification between the open string
side and the closed string side was made by mapping the
expectation value of the gluino condensate on the stack of D6
branes and the volume of the resolution two cycle on the resolved
conifold side. This map requires the term $J \wedge B_{(4)}$ on
the flux side \cite{vafa1}. Our proposal is that the presence of
$B_{(4)}$ is due to the fact that $d\Omega \ne 0$ and that the
type IIA superpotential contains a term $J \wedge d\Omega$
\footnote{For half-flat manifolds this has also been proposed in
\cite{wal}.}.
Making a further mirror transformation to this background we
obtain the closed string side of the type IIB geometric transition
\cite{trans2}. The type IIB manifold turns out to be a K\"ahler
deformed conifold with RR and NS three forms. This is exactly
what was expected from the results of \cite{vafa1}.
To summarize, non-K\"ahler manifolds play a crucial in the gauge theory/
string theory duality. They provide
important contributions to the superpotentials
which are crucial for a correct description of
the corresponding effective field theories.
\section{Type I/Heterotic Strings and Non-Calabi-Yau Manifolds}
Even though non-K\"ahler manifolds were never studied in the
traditional string theory literature in much detail, their
importance has become evident in recent times due to the large
amount of new results in the area of string compactifications
with fluxes. We will now extend the calculation done in the
previous section to other type of models.
To do so, we start again from the type IIB compactification with
NS and RR fluxes and go to its orientifold limit which will
contain D7 branes and O7 planes. In order to obtain a metric, we
consider the metric of (\ref{2b}) and analyze which terms are
invariant under the O7 action. We consider the directions $x$ and
$y$ to be transverse to the O7 planes. The metric should then be
invariant under the orientifold action, should preserve some
number of supersymetries (i.e ${\cal N} = 1$), should have a form
close to the original type IIB metric and should allow wrapped D5
branes along with some number of D7 branes and O7 planes. Finally,
after two T-dualities, it's form should closely resemble the
metric obtained after T-dualizing the resolved conifold.
A metric which satisfies these conditions was computed in
\cite{trans2}
\begin{equation}
ds^2 = a_1(dx^2 + \vert \tilde\tau_1 \vert^2 ~dy^2 + 2
\mbox{Re}~\tilde\tau_1~dx~dy) + a_2 (d\theta_1^2 + \vert
\tilde\tau_2\vert^2~d\theta_2^2) + a_3~dz^2 + a_4~dr^2.
\end{equation}
After two T-dualities, this metric becomes a type I metric which takes the form
\begin{eqnarray}
ds^2 & = \alpha(1 + A^2)(dy - b_{y\theta_2}~d\theta_2)^2 +
\alpha (1 + B_1^2)(dx + b_{x\theta_1}~ d\theta_1)^2 +
\gamma'\sqrt{H} dr^2 \\ \nonumber ~& + 2~ \alpha A B ~(dx +
b_{x\theta_1}~d\theta_1)(dy - b_{y\theta_2}~d\theta_2) + \alpha
(1 + A^2) dz^2 + a_2 \vert d\chi \vert^2.
\end{eqnarray}
The T-dualities were performed along the $x$ and $y$ directions.
As $y$ is the angular direction of the $P^1$ cycle on which the D5
branes are wrapped on, the D5 branes loose the $y$ direction and
gain the direction $x$, thus the final configuration is again with
D5 branes wrapped on a $P^1$ cycle with the angular direction now
being the $x$ direction. Therefore, the starting point of the
geometric transition is given by D5 branes wrapping on a two cycle
inside a non-K\"ahler manifold.
After the transition we are again in the orientifold limit of some type IIB
configuration. If we impose the same conditions as we did before the
metric will take the form:
\begin{equation}
ds^2 = {b}_1~\vert d\chi_1 \vert^2 + {b}_2 ~\vert d\chi_2 \vert^2 + {b}_3 ~dz^2 +
{b}_4~ dr^2,
\end{equation}
but now the complex structures will be different. We have
$\mbox{Re}~\tilde\tau_1 = 0$ and $\mbox{Re}~\tilde\tau_2 \ne 0$
for this solution while earlier the complex structures satisfied
$\mbox{Re}~\tilde\tau_1 \ne 0$ and $\mbox{Re}~\tilde\tau_2 = 0$.
The final type I manifold can be shown to be another non-K\"ahler
manifold \cite{trans2} whose explicit metric is given by:
\begin{eqnarray}
ds^2 & = \frac{1}{h_2 + a_2^2 h_1} ~(dy - b_{y\theta_2} ~d\theta_2)^2 +
\frac{1}{h_4 + a_1^2 h_1} ~(dx - b_{x\theta_1}~ d\theta_1)^2 \\ \nonumber &
+ h_1~ dz^2 + h_3~h_4 ~\vert d\chi_2\vert^2 + \gamma'\sqrt{H}~ dr^2.
\end{eqnarray}
One important aspect of these type I compactifications is that
the metrics are all non-K\"ahler but complex. The integrability
of the complex structures is related to the torsional constraints
the metrics are required to satisfy, as well as the DUY equations
for the vector bundles.
We can go one step further by performing an S-duality to go from
the type I theory to the heterotic string. This means trading the
RR flux of the type I theory with the NS flux of heterotic. A
geometric transition will then take place between NS branes
wrapped on some two cycles of a non-K\"ahler complex manifold and
NS flux on another non-K\"ahler complex manifold.
One important check of our approach is the fact that there are two
conditions which should be mapped into one another.
These two conditions are the self-duality of the type IIB fluxes in the
orientifold limit
\begin{equation}
H_{RR} = * H_{NS},
\end{equation}
and the torsional equation for the heterotic string compactifications
\begin{equation}
H_{NS}= * dJ \equiv i~(\partial - \bar{\partial}) J.
\end{equation}
In \cite{trans2} it was shown that this mapping holds up to some
conjectured identification between constants which enter in the
flux definitions. This gives a strong check on the consistencies
of these results.
\section{Conclusions}
In this short note we summarized the complete duality cycle of the
geometric transitions taking place in all string theories and
M-theory. For the type II and M-theory transitions, these
transitions are well accounted for with various direct and
indirect checks. On the other hand the type I and heterotic cases
are relatively new. Our conjecture here is that the world volume dynamics
of wrapped branes in the type I and heterotic theories on
non-K\"ahler complex manifolds will have a description purely in terms of
another non-K\"ahler complex manifold with branes replaced by
fluxes. The manifolds predicted in the type I and heterotic
theories are examples of new non-K\"ahler complex manifolds that
complement the existing examples in the literature
\cite{prok,bbdgs,serone}.
\section*{Acknowledgments}
We thank S.~Alexander and A.~Knauf for their collaborations.
M.B. is supported by NSF
grant PHY-01-5-23911 and an Alfred Sloan Fellowship. K.B is
supported by NSF grant PHY-0244722, an Alfred Sloan Fellowship and
the University of Utah. K.D. is supported by
a Lucile and Packard Foundation Fellowship 2000-13856. R.T. is
supported by DOE Contract DE-AC03-76SF0098 and NSF grant
PHY-0098840. R.T. would like to thank KITP for hospitality during the
completion of this work. This research was supported in part by
National Science Foundation Grant No. PHY99-07949.
|
{
"timestamp": "2004-11-08T08:16:40",
"yymm": "0411",
"arxiv_id": "hep-th/0411039",
"language": "en",
"url": "https://arxiv.org/abs/hep-th/0411039"
}
|
\section{Introduction}
The Seesaw Mechanism\cite{SEESAW,MINK}, which we are here to celebrate, must be viewed in the context of the intellectual turmoil generated by the Standard Model. The renormalizability of massive Yang-Mills theories\cite{HOOFT}, the emergence of a common description of Weak and Electromagnetic Interactions\cite{STANDARD}, and the realization that the Strong Interactions weaken at shorter distances\cite{AF} established the Standard Model as the paradigm for all Fundamental Interactions except Gravity. Like all such paradigms, the Standard Model is (thankfully) incomplete, has suggested new puzzles of it own, and elicited many questions. None has been more dominating than Pati and Salam's\cite{PS} proposal that quarks and leptons are equal partners in one mathematical structure at very short distances, the idea of Grand-Unification.
To appreciate the significance of the Seesaw mechanism as the link between small neutrino masses and Physics near the Planck scale, one must first describe the great theoretical speculations which led to its creation.
\section{TRIUMPHS OF THE STANDARD MODEL}
The Fundamental Interactions (save for Gravity) are described by the Standard Model. It has withstood, practically unscathed, almost four decades of experiments, confirming {\it inter alia} its radiative structure. All of its quarks and leptons have been discovered. Its main features are
\vskip .2cm
\noindent -- Interactions stem from three {\it weakly coupled } Yang-Mills theories based on $SU(3)$, $SU(2)$ and $U(1)$.
\noindent -- Quarks and leptons are needed for quantum consistency: gauge anomalies cancel between quarks and leptons.
\noindent -- There are three chiral families of quarks and leptons, each with a massless neutrino.
\noindent -- The gauge symmetries are spontaneously broken: the shorter the distance, the {\it more} the symmetry.
\noindent -- It predicts a fundamental scalar particle, the Higgs boson.
\vskip .2cm
Only one of these predictions has been proved wrong by experiments: neutrinos have masses. Today, only few of its parameters await measurement, the mass of the elusive Higgs particle, the strong CP-violating phase, and the mass of any of the three neutrinos.
\section{ OLD \& NEW PUZZLES }
Although the successes of the Standard Model have exceeded expectations, it has a dark side:
\vskip .2cm
\noindent -- It predicts CP-violation in the Strong interaction, albeit with unknown strength.
\noindent -- It requires Yukawa interactions without any organizing principle.
\noindent -- It fails to {\it explain} the values of masses and mixing patterns of quarks and charged leptons.
\noindent -- It contains too many parameters to be truly fundamental.
\noindent -- Without Gravitation it only describes the matter side of Einstein's equation, {\em sans} cosmological constant.
\noindent -- It fails to account for neutrino masses.
\vskip .2cm
The Standard Model presents an unfinished picture of Nature. It reminds one of the shards of a once beautifull pottery, shattered in the course of cosmological evolution.
\section{ GRAND UNIFICATION}
The quantum numbers of the three chiral families of quarks and leptons strongly suggest a more unified picture.
Pati and Salam's original idea is, remarkably enough, realized by unifying the three gauge groups of the Standard Model into one. In the simplest, $SU(5)$\cite{GG}, each family appears in two representations. In $SO(10)$\cite{FM}, they are grouped in its fundamental spinor representation, by adding a right-handed neutrino for each family. At the next level, we find $E_6$\cite{ESIX} where each family contains several right-handed neutrinos as well as vector-like matter. Organizing the elementary particles into these beautiful structures
\vskip .2cm
\noindent -- Unifies the three gauge groups.
\noindent -- Relates Quarks and Leptons.
\noindent -- Explains anomaly cancellations.
\vskip .2cm
There are indications that this idea ``wants to work". When last seen, the three coupling constants of the Standard Model are perturbative. Using the renormalization group equations to continue them deep into the ultraviolet, they get closer to one another, but fail to meet at one scale: the quantum number patterns did not quite match the dynamical information. This near (thought at the time to be exact) unification introduced Planck scale physics into the realm of particle physics.
One by-product of Grand Unification is violation of baryon number. Hitherto unobserved, proton decay remains one of the most important consequences
from these ideas. In a serendipitous twist, proton decay detectors now serve as the telescopes of neutrino astronomy! Other global symmetries also bite the dust: the relative lepton numbers are violated in $SU(5)$ and $SO(10)$ violated the total lepton number as well, and the extraordinary limits on these processes are consistent with the grand-unified scale.
\section{GRAND-UNIFIED LEGACIES}
Grand Unification by itself does not yet have any direct experimental vindication; it is an incubator of new ideas that, even today, drive speculations on the Physics at extra-short distances.
\vskip .2cm
\noindent -- It linked the large grand-unified scale to tiny neutrino masses\cite{SEESAW}.
\noindent -- It suggested relations between quark and charged lepton masses, although the flavor riddles of the Standard Model remain unexplained.
\noindent -- It created the ``gauge hierarchy" problem: why quantum corrections keep the ratio of the Higgs mass to the Unification scale small.
\vskip .2cm
Moreover, two of its predictions have linked particle physics to pre-Nucleosynthesis Cosmology:
\vskip .2cm
\noindent -- The possibility of monopoles in our universe led to the idea of Inflationary Cosmology\cite{GUTH}, which solves many long standing puzzles and whose prediction of a flat universe has been recently verified.
\noindent -- Proton decay. This offered a framework for understanding the baryon asymmetry\cite{YOSHIMURA} of the Universe.
\vskip .2cm
Today, only one of these predictions, tiny neutrino masses, has been borne out by experiment. On the conceptual side, it has also provided an alternative mechanism for the generation of Baryon asymmetry of the Universe through a primordial lepton asymmetry\cite{LEPTOGENESIS}. Still, Grand Unification is at most a partial theory of Nature, since it does not address Gravity (space-time is either flat or a fixed background ), nor the origin of the three chiral families and its associated flavor puzzles.
\section{ SUPERSTRINGS}
At the 1973 London conference, David Olive declared Superstring Theories to be candidate ``Theories of Everything", since they reproduce Einstein's gravity at large distances with no ultraviolet divergences, and also contain (some) gauge theories. This view has since gained much credence and notoriety. The matter content has gotten much closer to reality\cite{HET}, although this unification of the gravitational and gauge forces takes place in a somewhat unsettling background:
\vskip .2cm
\noindent -- Fermions and Bosons are related by a new type of symmetry: {\it Supersymmetry}\cite{PMR}.
\noindent -- Ultimate Unification takes place in nine or ten space dimensions!
\vskip .2cm
Nature at the millifermi displays neither Supersymmetry nor extra space dimensions. Yet, the lesson of the Standard Model of more symmetries at shorter distances provide an argument for these to be fabrics of the Ultimate Theory; these symmetries are somehow destroyed in the process of cosmological evolution. To compare the highly symmetric superstring theories to Nature, a dynamical understanding of their breakdown is required, an understanding that still eludes us.
To relate to Nature, experiments at energies at which these symmetries appear must be carried out. All could be just around the energy corner, although circumstantial evidence lends more credence to low- energy Supersymmetry than to low-energy extra dimensions. The collapse of the extra space dimensions occurs first, while Supersymmetry hangs on to later times (lower energies). It is a challenge to theory to find a dynamical reason which triggers the breakdown of higher-dimensional space (perhaps through brane formation), while leaving Supersymmetry nearly intact.
\section{ SUPERSYMMETRY}
Supersymmetry is an attractive theoretical concept; it is required by the unification of gravity and gauge interactions, and links fermions and bosons. Also, the mass of the spinless superpartner of a Weyl fermion, inherits quantum-naturality\cite{GILDENER} through the chiral symmetry of its partner.
Morever, when applied to the Standard Model, it yields quantitative predictions that fit remarkably well with Gauge Unification. With Supersymmetry,
\vskip .2cm
\noindent -- The Gauge hierarchy problem is managed: the Higgs mass is stabilized even in the presence of a large (grand-unification) scale
\noindent -- The three gauge couplings of the Standard Model run to a single value in the deep ultraviolet with the addition of superpartners in the TeV range. Thus naturally emerges a new scale using the renormalization group, a scale that matches the quantum number patterns of the elementary particles.
\noindent -- With supersymmetry the renormalization group displays an infrared fixed point that predicts\cite{PENDLETON} the top quark mass, in agreement with experiment.
\noindent -- Under a large class of ultraviolet initial conditions, the same renormalization group shows that the breaking of supersymmetry triggers electroweak breaking\cite{EWBREAK}.
\vskip .2cm
Supersymmetry at low energy is the leading theory for physics beyond the Standard Model, although many puzzles remain unanswered and new ones are created as well.
For one, there are almost as many theories of supersymmetry breaking as there are theorists, and none, theories and theorists alike, are convincing. It is an experimental question.
In addition, Supersymmetry deepens the flavor riddles of the Standard Model by predicting new scalar particles which generically produce flavor-changing neutral processes. Even if the breaking mechanism is flavor-blind (tasteless), non-trivial effects are expected: supersymmetry-breaking is already highly constrained by the existing data set.
The existence of low-energy Supersymmetry will soon be tested at the LHC. May the supersymmetry-breaking mechanism parameters prove to be so unique as to allow intellectually-challenged theorists (the author included) to infer its origin from the LHC data alone!
\section{ MINUTE NEUTRINO MASSES}
The only solid experimental evidence to date for physics beyond the Standard Model is the observation of oscillation among neutrino species. Thirty five years of experiments on solar neutrinos, Homestake\cite{Homestake}, GALLEX\cite{GALLEX}, SAGE\cite{SAGE}, SUPERK\cite{SKsol} and SNO\cite{SNO}, yield
$$\Delta m^2_\odot~=~\vert~m^2_{\nu_1}-m^2_{\nu_2}~\vert~\sim~7.\times 10^{-5}_{}~{\rm eV^2}\ ,$$
with corroborating evidence on antineutrinos\cite{Kamland}. Neutrinos born in Cosmic ray collisions\cite{SKatm}, and on earth\cite{K2K} give
$$\Delta m^2_\oplus~=~\vert~m^2_{\nu_2}-m^2_{\nu_3}~\vert~\sim~3.\times 10^{-3}_{}~{\rm eV^2}\ .$$
The best bound to their absolute value of the masses comes from WMAP\cite{WMAP}
$$\sum_i~m^{}_{\nu_i}~<~.71~{\rm eV}\ .$$
These experimental findings are not sufficient to determine fully the mass patterns. One oscillates between three patterns, {\it hierarchy},
$$|m_{\nu _1}| < |m_{\nu _2}| \ll |m_{\nu _3}|\ ,$$
{\it inverse hierarchy}
$$|m_{\nu _1}|
\simeq |m_{\nu _2}| \gg |m_{\nu _3}|\ ,$$
and {\it hyperfine}
$$|m_{\nu _1}| \simeq |m_{\nu _2}| \simeq |m_{\nu _3}|\ .$$
The mixing patterns provide some surprises, since it contains one small angle and two large angles. In terms of the MNS mixing matrix,
$${\begin{pmatrix}\cos\theta^{}_\odot&\sin\theta^{}_\odot&\epsilon\cr
-\cos\theta^{}_\oplus~\sin\theta^{}_\odot&\cos\theta^{}_\oplus~\cos\theta^{}_\odot&\sin\theta^{}_\oplus\cr
\sin\theta^{}_\oplus~\sin\theta^{}_\odot&-\sin\theta^{}_\oplus~\cos\theta^{}_\odot&\cos\theta^{}_\oplus\end{pmatrix}}\ ,$$
the various experiments yield
$$ \sin^2 2\theta^{}_\oplus~>~0.85\ ,\qquad 0.30~<~ \tan^2\theta^{}_\odot~< ~0.65 \ ,$$
while there is a only a limit\cite{CHOOZ} on the third angle
$$ \vert~\epsilon~\vert^2_{}~<~0.05\ .$$
Spectacular as they are, these results generate new questions for experimenters:
\vskip .2cm
\begin{itemize}
\item Are the masses Majorana-like (i.e. lepton number violating)?
\item What are their absolute values?
\item Can one measure the sign of $\Delta m^2$?
\item What is the value of the CHOOZ angle?
\item Is CP-violation in the lepton sector observable?
\end{itemize}
\noindent They also generate new theoretical questions
\begin{itemize}
\item Are there right-handed neutrinos?
\item How many? How heavy, and with what hierarchy?
\item Where do they live? Brane or bulk?
\item Do their decays trigger leptogenesis\cite{LEPTOGENESIS}?
\end{itemize}
\section{ Standard Model Analysis}
In the context of Grand Unification, one needs to discuss both quark and lepton mass matrices. To that effect, recall that the masses and mixings of the quarks are determined from the diagonalization of Yukawa matrices generated by the $\Delta I_{\rm W}=\frac{1}{2}$ breaking
of electroweak symmetry, for charge $2/3$
$$
{\mathcal U}^{}_{2/3}\,
{\begin{pmatrix}m^{}_u&0&0\cr 0&m^{}_c&0\cr 0&0&m_t^{}\end{pmatrix}}
\,{\mathcal V}^{\dagger}_{2/3}\ ,$$
and charge $-1/3$
$$
{\mathcal U}^{}_{-1/3}\,
{\begin{pmatrix}m^{}_d&0&0\cr 0&m^{}_s&0\cr 0&0&m_b^{}\end{pmatrix}}
\,{\mathcal V}^{\dagger}_{-1/3}\ ,
$$
resulting in the observable CKM matrix
$$
{\mathcal U}^{}_{CKM}~\equiv~{\mathcal U}^{\dagger}_{2/3}\,{\mathcal U}^{}_{-1/3}\ ,
$$
which, up to corrections of the order of the Cabibbo angle, $\theta_c\sim 13^\circ$, is equal to the unit matrix. This implies similar family mixings
for up-like and down-like quarks. Their masses are of course highly hierarchical. The charged lepton Yukawa matrix
$${\mathcal U}^{}_{-1}\,
{\begin{pmatrix}m^{}_e&0&0\cr 0&m^{}_\mu&0\cr 0&0&m_\tau^{}\end{pmatrix}}
\,{\mathcal V}^{\dagger}_{-1}$$
also stems from $\Delta I_{\rm W}=\frac{1}{2}$ electroweak breaking, and has hierarchical eigenvalues.
To obtain neutrino masses in the Standard Model, it is simplest to add one right-handed neutrino for each family. This yields another $\Delta I_{\rm W}=\frac{1}{2}$ Yukawa matrix
$${\mathcal U}^{}_{0}\,{\begin{pmatrix}m^{}_1&0&0\cr 0&m^{}_2&0\cr 0&0&m_3^{}\end{pmatrix}}\,{\mathcal V}^{\dagger}_{0}\ ,$$
but does not explain the extraordinary gap between charged and neutral leptons.
The right-handed neutrino masses are of Majorana type, since they have no gauge quantum numbers to forbid it (unlike electrons, say), and necessarily violate total lepton number.
In the context of effective field theories, one expects their masses to be of the order of lepton number breaking. Total lepton number-violating processes have never been seen resulting in a bound from neutrinoless double $\beta$ decay experiments. So either they are very large or zero.
If they are zero, the analysis proceeds as in the quark sector, and the observable MNS lepton mixing matrix is just
$${\mathcal U}^{}_{MNS}~\equiv~ {\mathcal U}^{\dagger}_{-1}\,{\mathcal U}^{}_{0}\ .$$
As for the quarks, it would be generated solely from the isospinor breaking of electroweak symmetry, even though the mixing patterns are so different.
In the belief that global symmetries are an endangered species (for one, black holes eat them up), we expect their masses to set the scale of the Standard model's cut-off, since they are unprotected by gauge symmetries. This yields the seesaw where large right-handed masses engender tiny neutrino masses, the latter being suppressed over that of the charged particles by the ratio of the two scales
$$\frac{\Delta I_{\rm W}=\frac{1}{2}}{\Delta I_{\rm W}=0}\ ,$$
thus introducing a large electroweak-singlet scale in the Standard Model. The neutrino mass matrix is then
$${\mathcal M}^{(0)}_{Seesaw}~=~{\mathcal M}^{(0)}_{ {Dirac}}\,
\frac{1}{{\mathcal M}^{(0)}_{ {Majorana}}}\,{\mathcal M}^{(0)\,T}_{ {Dirac}}\ ,$$
which we can rewrite as
$${\mathcal M}^{(0)}_{ {Seesaw}}~=~{\mathcal U}^{}_{0}\,\,
{\bf{\mathcal C}}\,\,{\mathcal U}^{T}_{0}\ ,$$
in terms of the central matrix\cite{DLR}
$$ {\mathcal C}~=~{\mathcal D}_0^{}\,{\mathcal V}^{\dagger}_{0}\,\frac{1}{{\mathcal M}^{(0)}_{ {Majorana}}}\,
{\mathcal V}^{*}_{0}\,{\mathcal D}_0^{}\ .$$
It is diagonalized by the unitary matrix ${\mathcal F}$
$$~~~ {\mathcal C}~=~ {\mathcal F}\,{\mathcal D}^{}_\nu\, {\mathcal F^{\,T}_{}}\ ,$$
where the mass eigenstates produced in $\beta$-decay are (unimaginatively labelled as ``1", ``2", ``3")
$$
{\mathcal D}_\nu^{}~=~{\begin{pmatrix}m^{}_{\nu_1}&0&0\cr 0&m^{}_{\nu_2}&0\cr 0&0&m_{\nu_3}^{}\end{pmatrix}}\ .$$
The effect of the Seesaw is to add the unitary ${\mathcal F}$ matrix to the MNS lepton matrix
$${\mathcal U}^{}_{MNS}~=~ {\mathcal U}^{\dagger}_{-1}\,{\mathcal U}^{}_{0}\,\, {\mathcal F}\ .$$
This framework enables us to recast theoretical questions in terms of $\mathcal F$. In particular, where do the large angles come from? We
catalog models in terms of the number of large angles contained in $\mathcal F$, none, one or two?
\section{A Modicum of Grand Unification}
To answer that question, we must turn to Grand Unification ideas for guidance, where relations between the $\Delta I^{}_{\rm W}=\frac{1}{2}$ quark and lepton Yukawa matrices appear naturally.
In $SU(5)$, the charge $-1/3$ and charge $-1$ Yukawa matrices are family-transposes of one another.
$$ {\mathcal M}^{(-1/3)}_{}~\sim~{\mathcal M}^{(-1)\,T}_{}\ .$$
In $SO(10)$, it is the charge $2/3$ Yukawa matrix that is related to the Dirac charge $0$ matrix
$$ {\mathcal M}^{(2/3)}_{}~\sim~{\mathcal M}^{(0)}_{ {Dirac}}\ .$$
These result in naive expectations for the unitary matrices that yield observable mixings
$${\mathcal U}^{}_{-1/3}~\sim~{\mathcal V}^{*}_{-1}\ ;\qquad {\mathcal U}^{}_{2/3}~\sim~{\mathcal U}^{}_{0}\ .$$
Assuming this pinch of grand-unification, we can relate the CKM and MNS matrices
\begin{eqnarray}
\nonumber{\mathcal U}^{}_{MNS}&=& {\mathcal U}^{\dagger}_{-1}\,{\mathcal U}^{}_{0}\, {\mathcal F}\cr
\nonumber&\sim&{\mathcal U}^{\dagger}_{-1}\,{\mathcal U}^{}_{-1/3}\,{\mathcal U}^{\dagger}_{CKM}\, {\mathcal F} \cr
&\sim& \Big({{\mathcal V}^T_{-1/3}\,{\mathcal U}^{}_{-1/3}}\,\Big)\,{\mathcal U}^{\dagger}_{CKM}\,\,{\mathcal F}\end{eqnarray}
Hence two wide classes of models:
\vskip .2cm
\noindent I-) Family-Symmetric ${\mathcal M}^{}_{-1/3}$ Yukawa matrices.
In these we have
$${{\mathcal U}^{}_{-1/3}}~=~{{\mathcal V}^*_{-1/3}}\ ,$$
so that
$$
{{{\mathcal U}^{}_{MNS}~=~ {\mathcal U}^{\dagger}_{CKM}\,\, {\mathcal F}
}}\ .$$
In these models, ${\mathcal F}$ necessarily contains two large angles. In the absence of any symmetry acting on $\mathcal F$, these models require a highly structured $\mathcal F$ matrix, which could even be non-Abelian.
Interestingly, these models provide a testable prediction for the size of the CHOOZ angle. With a family-symmetric charge $-1/3$ matrix, the MNS matrix reads
\begin{eqnarray}
& &{\mathcal U}^{}_{MNS}~=~{\mathcal U}^{\dagger}_{CKM}\,\times \cr & &\cr
& &{\begin{pmatrix}\cos\theta^{}_\odot&\sin\theta^{}_\odot& {\lambda^\gamma}\cr
-\cos\theta^{}_\oplus~\sin\theta^{}_\odot&\cos\theta^{}_\oplus~\cos\theta^{}_\odot&\sin\theta^{}_\oplus\cr
\sin\theta^{}_\oplus~\sin\theta^{}_\odot&-\sin\theta^{}_\oplus~\cos\theta^{}_\odot&\cos\theta^{}_\oplus\end{pmatrix}}
\ ,\nonumber\end{eqnarray}
where we have chosen to fill the zero in the $\mathcal F$ matrix by a Cabibbo effect, with $\gamma$ presumably greater than one. It follows that
$$
\theta^{}_{13}~\sim~\lambda\sin\theta_\oplus~\sim~\frac{1}{\sqrt 2}\,\lambda\ .
$$
It will be interesting to see if this definite prediction of type I models, $\theta_{13}\sim 7-9^\circ$, is borne out in future experiments.
\vskip .2cm
\noindent II-) Family-Skewed ${\mathcal M}^{}_{-1/3}$ Yukawa matrices. One can make a compelling arguments for at least one large angle to reside in $\mathcal U_{-1}$. If we extend the Wolfenstein\cite{WOLF} expansion of the CKM matrix in powers of the Cabibbo angle $\lambda$ to include quark mass ratios
$$\frac{m^{}_s}{m^{}_b}~\sim~{\lambda^2_{}}\qquad \frac{m^{}_d}{m^{}_b}~\sim~{\lambda^4_{}}\ ,$$
we find the charge $-1/3$ Yukawa matrix
$${\mathcal M}^{(-1/3)}_{}~=~ {\begin{pmatrix} \lambda^4_{}& \lambda^3_{}& \lambda^3_{}\cr
\lambda^?_{}& \lambda^2_{}& \lambda^2_{}\cr
\lambda^?_{}& \lambda^{?}_{}& 1\end{pmatrix}}\ .$$
If the exponents are related to charges, as in the Froggatt-Nielsen\cite{FN} schemes, the lower diagonal exponents are known, and we get the orders of magnitude
$${{\mathcal M}^{(-1/3)}_{}}~=~
{\begin{pmatrix} {\lambda^4_{}}& {\lambda^3_{}}& {\lambda^3_{}}\cr
{\lambda^3_{}}& {\lambda^2_{}}& {\lambda^2_{}}\cr
{\lambda^1_{}}& 1&1\end{pmatrix}}\ ,$$
which is not family-symmetric. In the limit of no Cabibbo mixing,
$${\mathcal M}^{(-1/3)}_{}~\approx~{\begin{pmatrix}0&0&0\cr 0&0&0\cr 0& {a}& b\end{pmatrix}}
+{\mathcal O}({ \lambda})\ ,$$
and
$${\mathcal U}^{}_{MNS}~=~
{\begin{pmatrix}1&0&0\cr 0&\cos\theta^{}_\oplus&\sin\theta^{}_\oplus\cr 0&-\sin\theta^{}_\oplus&\cos\theta^{}_\oplus\end{pmatrix}}\, {\mathcal F}\ ,$$
where
$$\tan\theta^{}_\oplus~=~\frac{a}{b}\ , $$
is of order one\cite{ILR}. In these models, ${\mathcal F}$ need contain only one large angle, which is very natural, although they give no generic prediction for the CHOOZ angle.
\section{Right-Handed Hierarchy}
In most models, $\mathcal F$ must contain at least one large angle to accomodate the data. This presents a puzzle since $\mathcal F$ diagonalizes a matrix which contains the neutral Dirac Yukawa matrix which is presumably hierarchical, coming from the isospinor electroweak breaking.
This suggests special restrictions put upon the Majorana mass matrix of the right-handed neutrinos. We want to illustrate this point by looking at a $2\times 2$ two-families case\cite{DLR}, and write
$$
{\mathcal D}^{}_0~=~m{\begin{pmatrix}a\,{\lambda^\beta_{}}&0\cr 0&1\end{pmatrix}}\ , $$
and define $M^{}_1\ , M^{}_2$ to be the eigenvalues of the right-handed neutrino's Majorana mass matrix. This matrix can be diagonalized by a large mixing angle in one of two cases:
\vskip .2cm
\noindent -- Its matrix elements have similar orders of magnitude ${\mathcal C}_{11} ~\sim~{{\mathcal C}_{22}} ~\sim~{\mathcal C}_{12}$, in which case we find that
$$
\frac{M_1}{M_2}~\sim~ {\lambda^{2\beta}_{}}\ ,$$
suggesting a doubly {\it correlated hierarchy} betwen the $\Delta I_{\rm W}=0$ and $\Delta I_{\rm W}=\frac{1}{2}$ Sectors. This agrees well with grand-unified models such as $SO(10)$ and $E_6$, where each right-handed neutrinos is part of a family.
\vskip .2cm
\noindent --A large mixing angle can occur if the diagonal elements are much smaller than the diagonal ones, that is
${\mathcal C}_{11}\, , \,{\mathcal C}_{22} ~\ll~ {\mathcal C}_{12}$. Then we find
$$\frac{\lambda^\alpha~m^2}{\sqrt{-M_1 M_2}}\,
{\begin{pmatrix}0&a\cr a&0\end{pmatrix}}\ .$$
Hence maximal mixing may infer that some of the right-handed neutrinos are Dirac partners of one another, leading to conservation in the right-handed mass matrix, of a relative lepton number $L_1-L_2$.
\section{Cabibbo Flop}
As we have seen, Grand-Unification, even in its simplest form, implies Cabibbo-sized effects in the MNS matrix. In the quark sector, Cabibbo mixing is the strongest between the first and second family. Applied to the lepton sector, the solar angle may be maximal, with a Cabibbo correction of $13^\circ$\cite{FUJIHARA,SMIRNOV}.
Recently, we\cite{DER} have been exploring possible Wolfenstein parametrizations of the MNS matrix, in the hope that some regularity might emerge from the data, once Cabibbo effects are taken into account.
Since we do not know how the Cabibbo angle is generated in flavor theories. So we start by asking if the limit ${\theta_c^{}\rightarrow 0}$ makes any theoretical sense. To simplify matters, assume there is only one small parameter in the flavor sector; then the quark and charged lepton masses of the first two families are zero. In the same limit, ${\mathcal U}_{CKM}=1$, and there are no neutral flavor changes. Of course the mixing between the first two families is undetermined.
We do not know ${\mathcal U}_{MNS}$ in that limit, the starting point of a Wolfenstein parametrization for the lepton mixing matrix.
The measured values of the lepton mixing angles are
$$\theta_\odot^{}={32.5^\circ_{}}^{\,+\,2.4^\circ}_{ \,-\,2.3^\circ} \ ;\quad \theta^{}_\oplus~=~ {45.00^\circ_{}}^{\,+10^\circ}_{\,-10^\circ} \ ;\quad \theta_{CHOOZ}< 13^\circ_{}\ .$$
The solar angle is well measured, but the atmospheric angle is not, and could very well be non-maximal. Furthermore, their values could be affected by Cabibbo flop of $\pm ~13^\circ_{}$, and the CHOOZ angle could well be a Cabibbo effect. Our starting point is
$${\mathcal U}^{}_{MNS}= {\begin{pmatrix}\cos{\eta_\odot} &\sin{\eta_\odot} &0\cr -\cos{\eta_\oplus}\sin{\eta_\odot}&\cos{\eta_\oplus}\cos{\eta_\odot}
&\sin{\eta_\oplus}\cr
\sin{\eta_\oplus}\sin{\eta_\odot}&-\sin{\eta_\oplus}\cos{\eta_\odot}&\cos{\eta_\oplus}\end{pmatrix}}~+~\cdots\ ,$$
with a range of initial angles
$$15^\circ_{}~<~ \eta^{}_\odot~<~ 45^\circ_{}\ ;\qquad 30^\circ_{}~<~ \eta^{}_\oplus~<~ 60^\circ_{}\ .$$
We write the Wolfenstein expansion of the MNS Matrix in the form
$${\mathcal U}^{}_{MNS}~\equiv ~{\mathcal W}~+~{{\mathcal O}(\lambda)}\ ,$$
where the starting matrix is split in two parts, showing the large angles
$${\mathcal W}~=~{\mathcal W}_\oplus\,{\mathcal W}_\odot \ ,$$
with
$${\mathcal W}_\oplus~=~{\begin{pmatrix}1&0&0\cr 0&\cos{\eta_\oplus}&- \sin{\eta_\oplus}\cr 0&\sin{\eta_\oplus}&\cos{\eta_\oplus}\end{pmatrix}}\ ,$$
$${\mathcal W}_\odot~=~{\begin{pmatrix}\cos{\eta_\odot} & \sin{\eta_\odot} &0\cr -\sin{\eta_\odot} &\cos{\eta_\odot} &0\cr 0&0&1\end{pmatrix}}\ .$$
We introduce Cabibbo flop through the unitary matrix
$${\mathcal V}~=~I+\Delta(\lambda)\ ,$$
with ${\Delta(0)~=~1}$. Unlike the quark sector it does not commute with the starting matrix
$$ [\,{\mathcal W}\,,\,{{\mathcal V}(\lambda)}\,]~\neq~0\ .$$
This means that Cabibbo effects from the left and from the right or even in between the two starting matrices are not equivalent. Hence we consider basic flops
\begin{itemize}
\item {{ Left }} ~~~~~\,${\mathcal U}_{MNS}={{\mathcal V}(\lambda)}\,{\mathcal W}\,$
\item {{ Right }}~~~\, ${\mathcal U}_{MNS}={\mathcal W}\,{{\mathcal V}(\lambda)}$
\item {{ Middle }}~~ ${\mathcal U}_{MNS}={\mathcal W}_\oplus{{\mathcal V}(\lambda)}\,{\mathcal W}_\odot\,$
\end{itemize}
and we can have one ${\mathcal O}(\lambda)$ correction (single flop), or two (double flop). The present data is not sufficient to single out a particular
Wolfenstein parametrization, but the hope is that by considering possible Cabibbo effects on various starting matrices, generic features suggestive of flavor patterns might become obvious. In particular, they would restrict the size of the CHOOZ angle and of the CP-violation.
To illustrate these points, consider the effect of flop matrices, shown here to ${\mathcal O}(\lambda^3)$,
$${\mathcal V}^{}_{12}(\lambda)~=~{\begin{pmatrix} 1-\frac{a^2}{2}\lambda^2&a\,\lambda&b\,\lambda^2\cr -a\,\lambda&1-\frac{a^2}{2}\lambda^2&0\cr -b\,\lambda^2&0&1\end{pmatrix}}$$
$${\mathcal V}^{}_{23}(\lambda)~=~{\begin{pmatrix} 1&0&b\,\lambda^2\cr 0&1-\frac{a^2}{2}\lambda^2&a\,\lambda\cr -b\,\lambda^2&-a\,\lambda&1-\frac{a^2}{2}\lambda^2\end{pmatrix}}$$
$${\mathcal V}_{\rm double}(\lambda)~=~{\begin{pmatrix} 1-\frac{a^2}{2}\lambda^2&a\,\lambda&(b+\frac{aa'}{2})\,\lambda^2\cr -a\,\lambda&1-\frac{a^2+a'^2}{2}\lambda^2&a'\,\lambda\cr (\frac{aa'}{2}-b)\,\lambda^2&-a'\,\lambda&1-\frac{a'^2}{2}\lambda^2\end{pmatrix}}\ ,$$
where we have limited ourselves to $a=\pm 1\ ;\quad a'=\pm 1\ ;\quad 0.8<~b~<~ 1.2$. For instance, a left single flop ${\mathcal V}_{23}$, yields values for the starting angles that are different from the data,
\vskip .5cm
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$~{\eta^\circ_\odot} ~$& $~{\eta^\circ_\oplus} ~$&${\theta^\circ_\odot}$&${\theta^\circ_\oplus}$ &${\theta^\circ_{13}}$ \\
\hline \hline
$~ 30~ $&$ ~30~$ &$\sim 31~$& $43$&$.06-.4$\\
\hline
$~30~$& $~60~$ &$31-32$&$\sim 47$&$.6-2.5$\\
\hline
\end{tabular}\end{center}
\vskip .5cm
Right single flops with ${\mathcal V}_{23}$ and ${\mathcal V}_{12}$ produce:
\vskip .5cm
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$~{\eta^\circ_\odot} ~$& $~{\eta^\circ_\oplus} ~$&${\theta^\circ_\odot}$&${\theta^\circ_\oplus}$ &${\theta^\circ_{13}}$ \\
\hline \hline
$~30~$& $~60~$ &$30-31$&$ 48-50$&$3-10$\\
\hline
$~15~$& $~45~$& $\sim 30.3$ &$44-45$&$2-4$\\
\hline
$~45~$&$~45~$&$~\sim 32$&$44-46$&$1-3$\\
\hline
\end{tabular}\end{center}
\vskip .5cm
A right double flop with ${\mathcal V}_{\rm double}$:
\vskip .5cm
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$~{\eta^\circ_\odot} ~$& $~{\eta^\circ_\oplus} ~$&${\theta^\circ_\odot}$&${\theta^\circ_\oplus}$ &${\theta^\circ_{13}}$ \\
\hline \hline
$~15~$& $~60~$ &$ \sim 30.3$&$ 45-51$&$1-6$\\
\hline
$~45~$& $~60~$& $32$ &$48-53$&$6-12$\\
\hline
\end{tabular}\end{center}
\vskip .5cm
Finally a left double flop with ${\mathcal V}_{\rm double}$:
\vskip 1cm
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$~{\eta^\circ_\odot} ~$& $~{\eta^\circ_\oplus} ~$&${\theta^\circ_\odot}$&${\theta^\circ_\oplus}$ &${\theta^\circ_{13}}$ \\
\hline \hline
$~45~$& $~30~$ &$ \sim 34$&$ 40-46$&$5-8$\\
\hline
\end{tabular}\end{center}
We see that double flops can produce a larger CHOOZ angle. Also a left flop from a family-symmetric Yukawa,
$${\mathcal U}^{}_{MNS}={\begin{pmatrix}1&{\lambda}&{\lambda^3}\cr
{\lambda}&
1&{\lambda^2}\cr {\lambda^3}& {\lambda^2}&1\end{pmatrix}}
{\begin{pmatrix}\cos\eta^{}_\odot&\sin\eta^{}_\odot&0\cr
-\cos\eta^{}_\oplus~\sin\eta^{}_\odot&\cos\eta^{}_\oplus~\cos\eta^{}_\odot&\sin\eta^{}_\oplus\cr
\sin\eta^{}_\oplus~\sin\eta^{}_\odot&-\sin\eta^{}_\oplus~\cos\eta^{}_\odot&\cos\eta^{}_\oplus\end{pmatrix}}\ ,
$$
yields $\eta_\odot~\sim~ 40^\circ\ ;\quad \eta_\oplus~ \sim~ 45^\circ\ ; \quad \theta^{}_{13}~\sim~0.7\,{\lambda}~\sim~9^\circ $, which we have already seen. Finally we note that CP-violation effects can be much larger than in the quark sector. This is because the CP-violating lepton invariant\cite{JARLSKOG,GREENBERG} is
$$J~\sim~(\lambda-\lambda^3)\,\sin\delta\ ,$$
to be compared with that in the quark sector which is of order $\lambda^6$. If the limit of zero Cabibbo mixing is meaningful for theory, analyses of the type we have just presented will assume some importance. One important remark emerges: precision measurements of the MNS matrix is quite important for theory.
\section{Conclusions}
We are beginning to read the new lepton data, but there is much work to do before a credible theory of flavor is proposed. The Seesaw Mechanism links static neutrino to physics that can never be reached by accelerators, creating a new era of the physics which centers around right-handed neutrinos. With no electroweak quantum numbers, they could hold the key to the flavor puzzles. The second large neutrino mixing angle suggests that hierarchy is independent of electroweak breaking, and occurs at grand-unified scales.
I would like to express my gratitude to P. Bin\'etruy, M. Cribier, J. Orloff, S. Lavignac and D. Vignaud for organizing this conference {\it tr\`es sympathique} in the heart of Paris, at the Institut Henri Poincar\'e. I also wish to thank my collaborators A. Datta and L. Everett for many useful insights.
|
{
"timestamp": "2004-10-31T17:49:14",
"yymm": "0411",
"arxiv_id": "hep-ph/0411010",
"language": "en",
"url": "https://arxiv.org/abs/hep-ph/0411010"
}
|
\section{Introduction}
\label{introduction}
The correlation in galaxies between the star formation rate
and the average gas surface density over several orders of magnitude
(Kennicutt 1998) suggests a simple,
deterministic prescription (Schmidt 1959) for star formation.
Yet the finding that, at least in the Milky Way,
all star formation occurs in dense, cold clouds of
molecular hydrogen and dust raises the question of how information
about the average gas density of a galaxy reaches the small scale
on which star formation occurs.
Furthermore, observations of our own interstellar
medium (ISM) as well as that of other galaxies reveal that far from being
well described by a global quantity like the average gas density, the ISM
has a spectacularly complex structure on many scales.
Diffuse ionized gas in edge-on spirals is concentrated in webs of
filaments and shells (Rand, Kulkarni \& Hester 1990, Dettmar 1992,
Ferguson, Wyse, \& Gallagher 1996). Atomic gas detected by 21 cm
emission in our Galaxy (Heiles 1979, 1984) as well as in several
other spirals (Irwin 1994, Rand \& van der Hulst 1993, Lee \& Irwin 1997,
King \& Irwin 1997) resides in ``supershells'' and ``worms''.
In maps of the nearby spirals M31 and M33 (Brinks \& Bajaja 1986,
Deul \& den Hartog 1990), it is
also found to be depleted in numerous 100 pc -- 1 kpc ``holes''.
Attempts to quantify this elaborate
ISM structure are confronted with questions of identification.
Structures are interconnected, with, for example, denser regions of gas
embedded within filaments. Hence for example, potential sites of star formation
cannot be picked out, without introducing a density threshold and thereby
a bias to separate them from the underlying density field.
An alternative way to analyze the ISM is with Fourier transform power spectra.
Applied to HI emission maps of the Large and Small Magellanic Clouds,
power laws over $\sim$ 2 orders of magnitude are found
(Stanimirovic et al. 1999, Elmegreen, Kim, Stavely-Smith 2001),
providing another insight into the structure of the ISM, namely
that as other observations have already suggested, it is likely to be
turbulent.
Clues about the energy sources for the stirring of the ISM
come from measurements of the sizes and velocities of
shells. In some cases stellar winds and supernovae are found
to be adequate for creating the supershells, and HI holes. In other
cases larger quantities of energy are demanded and then collisions of
external clouds with the galaxies are invoked (Tenorio-Tagle 1981).
As for the diffuse ionized medium, although the energy available from
O stars would be sufficient to account for its photoionization,
a well-known problem is that photons from the O stars cannot travel
far from their origin without being absorbed by the molecular clouds
and HI halos surrounding them.
In that case the photons either reach larger distances by traveling
through photoionized conduits carved out by earlier supernovae or as suggested
by an alternative model they are additionally generated in
turbulent mixing layers at the interfaces
between hot and cold gas. These are ubiquitous in the ISM, and have been
invoked as an efficient means to
convert the thermal energy generated by shear flows to ionizing radiation
(Begelman \& Fabian 1990, Slavin, Shull, \& Begelman 1993).
Ultimately the energy source in the latter model
is again the supernovae which create the hot gas. Recent X-ray images
from Chandra map out this hot, tenuous gas, predicted
by Spitzer (1956), above and below the galactic plane of
disk galaxies (Wang et al. 2001).
Even without a heat source due to its long cooling time,
once it is generated by supernovae, such gas can persist for
millions of years. Cox \& Smith (1974) reasoned that
given that OB stars occur in associations, it is likely that
a supernovae will go off inside the hot cavity generated by a previous
supernovae, thereby rejuvenating it and creating an even larger cavity.
In this way, successive supernovae can overlap creating a network
of tunnels. Expanding at high speed within these tunnels,
the hot gas can move above the galactic plane where it is either halted
by insufficient speed to escape the galactic potential, or by an encounter
with a large mass of cold, high density gas, or by efficient mixing
with cooler gas which increases its density thereby accelerating its
radiative energy losses.
In light of this complex environment in which star formation occurs,
it is even more surprising that the Schmidt law is so successful.
It is in the context of this complexity, that we undertake a study of the
star formation rate in a multiphase ISM. We restrict ourselves
to a local study of the ISM, namely that of a $\sim$ 1 kpc$^3$
region. The earliest local study which included supernovae feedback was
done by Rosen \& Bregman (1995) in two dimensions. They considered
a segment of a galactic disk, taking into account a fixed
external gravitational potential, but neglecting rotational effects,
self-gravity, and magnetic fields.
In a three-dimensional model
which included the effects of an external gravitational potential, rotation,
and magnetic fields, Korpi et al. (1999a,b) studied a
supernova driven galactic dynamo. Meanwhile, to investigate the disk
halo interaction, Avillez (2000) followed the evolution of a segment of a
galactic disk with an adaptive mesh refinement code. Unlike these studies,
ours follows self-consistently and in three dimensions both the gas and the
stars, treating the latter as a system
of collisionless particles subject to gravity. Rosen \& Bregman (1995)
followed the stellar component but treated the stars with the same fluid
equations used for the gas thereby making their flow more viscous than that expected for
a collisionless system of particles. Without star
particles tagged with their ages, Rosen \& Bregman (1995)
decided upon a supernovae rate for their simulation, then proceeded to
set off supernovae with a probability of occurrence correlated to the stellar density.
Avillez (2000) approached the issue by constructing an algorithm to
distinguish between isolated and clustered supernovae.
For isolated supernovae events, Avillez (2000) randomly determined the positions
of supernovae in the disk plane with
rates based on observed ones. To mimic clustered
supernovae, a percentage of the supernovae sites were chosen to coincide with
locations where there was a previous supernova. In the Korpi et al. (1999a,b)
implementation there was a density criteria to determine the locations of
isolated supernovae. In both Avillez (2000) and Korpi et
al. (1999a,b), supernovae occuring above the disk plane were placed
in random locations with an exponential distribution characterized by a scale
height also adopted from observations.
Given that we are interested in the impact of supernovae feedback on
star formation, we cannot rely on these methods of modeling the supernovae
locations. Instead we require that the locations, ages, and masses of the
star particles self-consistently determine the supernovae events.
A simple calculation shows that a star with a velocity of 10 km/s will travel
$\sim$ 100 pc (e.g. the average size of a molecular cloud) in 10 Myr. The latter
corresponds to a typical time delay between the
birth and death of a star with M $\sim$ 80 M$_\odot$. In a follow-up paper
we explore how our results change when we neglect this time delay
and instead allow the stars to explode as supernovae
immediately after their birth (Slyz, Devriendt, Bryan, \& Silk,
{\em in preparation}).
Obviously a local model such as ours
is of limited relevance for quantitative comparisons to the ISM in galaxies.
As later detailed in section~\ref{discussion}, the limitations of our idealized
boundary conditions and the absence in our models of an external gravitational
potential as well as of a shear flow arising from rotation
means that there are many fundamental questions that we cannot
address. Nevertheless we believe that for the purposes of studying
the non-linear interplay between star formation and stellar feedback,
our simple model yields important insights.
The question we address is what physical processes regulate the rate at which
gas turns into stars in a multiphase ISM. In section~\ref{method}
we describe the numerical method we use as well as the
ingredients of our simulation. To model the large dynamic range in
densities and temperatures of a gaseous medium compressed by self-gravity and
by shocks maintained by supernovae and stellar winds, a robust high-resolution
hydrodynamical scheme proves essential.
To get a qualitative idea about the phenomena involved,
section~\ref{generalpics} presents the general morphological,
thermodynamical, and dynamical features of our simulations.
A more quantitative
analysis of the gas structure and dynamics is presented
in section~\ref{detailedpics} where we explore changes in the gas probability
density function and energy spectra with the addition of more and more
physics thought to be relevant for star formation.
Section~\ref{comparesilk} compares the star formation rates we measure
in our simulations to simple analytic prescriptions
and section~\ref{discussion} discusses the limitations of our
simulations.
Finally our main conclusions are summarized in section~\ref{conclusion}.
\section{Method and Ingredients of the Simulations}
\label{method}
Traditionally the problem with numerical simulations trying to model
star formation and feedback processes is that the radiative losses of
the hot component generated by supernovae are enourmous, even though
in the absence of any interaction of the hot gas with the cold gas the
cooling time of the hot gas is on the order of 100 Myr. In many cases
the culprit is numerical diffusion which mixes cold gas into
the hot gas more than it physically should. As a result, since the
density of the cold gas is high, mixing even a small fraction of it
with the low
density hot gas increases the density of the hot component sufficiently
for it to cool more efficiently than it should. For this reason,
high resolution grid codes are better suited for studies of the
multiphase interstellar medium than more diffusive particle based methods
which require carefully constructed algorithms to circumvent artificial
cooling (e.g. Marri \& White 2003).
With this in mind,
we model the evolution of gas and stars in a three--dimensional periodic box
which is 1.28 kpc on a side with a grid-based scheme for the gas and
a particle-mesh method for the stars. More specifically we have incorporated the BGK hydrocode
(Prendergast \& Xu 1993, Slyz \& Prendergast 1999)
into Bryan's {{ENZO}} code (Bryan \& Norman 1997, 1999) which uses a
Lagrangean particle-mesh (PM) algorithm to follow the
collisionless stars moving in the gravitational potential
the gas and the stars themselves generate. Based on gas-kinetic theory, BGK
computes time-dependent hydrodynamical
fluxes from velocity moments of a distribution function which is a
local solution to a model of the collisional Boltzmann equation, namely the
BGK equation (Bhatnagar et al. 1954). The hydrodynamics code has been extensively tested
on discontinuous non-equilibrium flows (see Xu 1998 for a review) and performs
well both at flow discontinuities and strongly rarefied regions, a criterion
which is mandatory for ISM simulations.
Initially the gas has constant density ($\rho_{\mathrm {gas}} =$ 1 atom/cm$^3$)
and temperature (T$_{\mathrm {gas}}=$ 10$^5$ Kelvin) and
similar to the initialization in MacLow et al. (1998),
its velocity field is drawn from a gaussian random field characterized by
a power spectrum scaling like k$^{-4}$. We truncate this velocity
power spectrum so that the field only has power on large scales, i.e. in modes
up to k = 4. The initial v$_{\mathrm{rms}}$ is
$\sim$ 50 km/s.
Contrary to MacLow et al. 1998,
we do not add velocity perturbations at each time step to
drive the `turbulence`. We only impose the velocity
perturbations once at the beginning of the simulation. We assume radiative
cooling of an optically thin gas which is
in collisional ionization equilibrium. More specifically,
our cooling function, displayed in figure~\ref{cool_function},
is an extension of the cooling curve of Sarazin \& White (1987)
down to temperatures of ${\mathrm {T}}_{\mathrm {min}} = 310$ K to
account for ${\mathrm {H}}_{2}$ cooling using the rates given in
Rosen \& Bregman (1995). The extension to lower temperatures assumes a solar
metallicity, a completely ionized gas at 8000 K and an
ionization fraction that gradually drops to 10$^{-3}$ below 8000 K.
Fitting a piecewise power law to our cooling curve
gives:
\begin{displaymath}
\Lambda(T) = \left\{ \begin{array}{ll} 0 & \mbox{if $T < 310 K$,} \\
(2.2380 \times 10^{-32}) T^{2} & \mbox{if $310 K \leq T < 2000 K$}\\
(1.0012 \times 10^{-30}) T^{1.5} & \mbox{if $2000 K \leq T < 8000 K$}\\
(4.6240 \times 10^{-36}) T^{2.867} & \mbox{if $8000 K \leq T < 39811 K$}\\
(3.1620 \times 10^{-30}) T^{1.6} & \mbox{if $39811 K \leq T < 10^5 K$}\\
(3.1620 \times 10^{-21}) T^{-0.2} & \mbox{if $10^5 K \leq T < 2.884 \times 10^5 K$}\\
(6.3100 \times 10^{-6}) T^{-3} & \mbox{if $2.884 \times 10^5 K \leq T < 4.732 \times 10^5 K$}\\
(1.047 \times 10^{-21}) T^{-0.22} & \mbox{if $4.732 \times 10^5 K \leq T < 2.113 \times 10^6 K$}\\
(3.981 \times 10^{-4}) T^{-3} & \mbox{if $2.113 \times 10^6 K \leq T < 3.981 \times 10^6 K$}\\
(4.169 \times 10^{-26}) T^{0.33} & \mbox{if $3.981 \times 10^6 K \leq T < 1.995 \times 10^7 K$}\\
(2.399 \times 10^{-27}) T^{0.5} & \mbox{if $T \geq 1.995 \times 10^7 K$}
\end{array}
\right.
\end{displaymath}
The lower temperature cutoff of the cooling
function at 310 K is unphysical, although Rosen, Bregman, \& Norman
(1993) argue that truncating it there is a way to model the
contribution to the ISM pressure from sources such as magnetic fields
and cosmic rays, which do not decrease as the gas radiatively cools.
\begin{figure}
\centerline{\psfig{file=cool_wphases_bw.ps,width=.9\hsize}}
\caption{Cooling curve with vertical dotted lines overplotted to delineate
several different temperature regimes which we consider in section~\ref{generalpics}. }
\label{cool_function}
\end{figure}
\subsection{Implementation of star formation and feedback}
Following Cen and Ostriker (1992), we assume that star formation
is inevitable if a region is
contracting ($\nabla \cdot v < 0$), cooling rapidly
($\mathrm t_{\mathrm {cool}} < \mathrm t_{\mathrm {dyn}}$ and
$T_{\mathrm {gas}} \leq T_{\mathrm {min}}$),
and is overdense ($\rho > \rho_{\mathrm {crit}}$). Since we check the grid on a cell
by cell basis to see if these conditions are met, each timescale is computed
for each grid cell. Here $t_{\mathrm {dyn}}$ is the
dynamical collapse timescale, i.e. $t_{\mathrm {dyn}} = \sqrt{3.0 \pi/(32 G
\rho_{\mathrm{tot}})}$ where $\rho_{\mathrm{tot}}$ is the sum of the gas
density, $\rho$,
and the stellar density. $\;t_{\mathrm {cool}}$ is the cooling timescale,
i.e. $t_{\mathrm {cool}} = \mathrm{k T / n} \Lambda$, where $n$ is the
gas particle number density.
${\mathrm {T}}_{\mathrm {min}}$ is the minimum of our cooling curve, 310 K,
and $\rho_{\mathrm {crit}}$ for the different simulations is specified in
table~\ref{sims}.
If all our star forming criteria
are met within a grid cell then we convert the following amount of gas,
$\Delta m_{\mathrm {gas}} = \epsilon \frac{\Delta t}{t_{\mathrm{dyn}}}
\rho_{\mathrm{gas}} \Delta x^3$ into a ``star particle'', where $\epsilon$
is a star formation efficiency
whose value is given in table~ref{sims}, and $\Delta t$ is the updating timestep.
We only allow at maximum 90\% of the gas in a cell to be converted to stars in
one timestep.
In practice however, once supernovae inject hot gas into the
medium, the updating timestep is short as it is
set by the hot, low density gas. As a result $\Delta t < t_{\mathrm{dyn}}$, and
this 90\% threshold is never reached.
We give the new star particle the same velocity as the gas out of which
it formed and we follow the stars dynamically.
The star particle is labeled with its mass, $m_{\star}$, its formation time,
$t_{\mathrm{SF}}$, and the dynamical time, $t_{\mathrm{dyn}}$, of the gas out
of which it formed.
For the purposes of the feedback however, rather than assume that
the ``star particle'' formed instantaneously at $t_{\mathrm{SF}}$, we
spread the star formation over several dynamical times by
computing the amount of gas mass that would
form stars after time $t_{\mathrm{SF}}$ to be:
\begin{equation}
\Delta \mathrm{m}_{\mathrm{stars}}(t) = m_{\star}
\frac{(t - t_{\mathrm{SF}})}{\tau^{2}} {\mathrm{exp}}\frac{-(t - t_{\mathrm{SF}})}{\tau}
\label{expsfr}
\end{equation}
where $\tau = \mathrm{max}(t_{\mathrm{dyn}},10 \;{\mathrm {Myr}})$.
With this time-dependent star formation rate, stars form at an exponentially
decreasing rate after a dynamical time. If the dynamical timescale
of the gas in a
star-forming cell is shorter than the typical lifespan of a massive star,
i.e. 10 Myr, then 10 Myr is used in place of $t_{\mathrm{dyn}}$ in
equation~\ref{expsfr} for the value of $\tau$.
Then, as a crude model of a stellar wind, we return
25\% of $\Delta \mathrm{m}_{\mathrm{stars}}$
to the gas, and since this returned mass
has the velocity of the ``star particle'' we alter the momentum
of the gas appropriately.
Finally assuming only the occurence of Type II supernovae,
we add 10$^{-5}$ of the rest-mass energy of
$\Delta \mathrm{m}_{\mathrm{stars}}$
to the gas' thermal energy (Ostriker \& Cowie 1981). The
supernovae input is added locally into one cell. We explore the limitations
of our supernovae implementation in future work.
As we do not have the resolution to follow every individual
star and to therefore sample a realistic Initial Mass Function (IMF)
for them, each star particle is more like a small star cluster
with a typical mass in the range $\sim 120 - 220 {\mathrm M_{\odot}}$.
Table~\ref{sims} presents the simulations we ran, listing the values
of the parameters for star formation and feedback.
Although we performed several simulations with a density threshold for star
formation, $\rho_{\mathrm {crit}}$, set to 1 atom/cm$^{3}$ (runs B5, C5
and C6),
for the remainder of the paper we focus only on the runs with
$\rho_{\mathrm {crit}}$ = 10 atom/cm$^{3}$. This is because we found that
dropping the density threshold by one order of magnitude to
1 atom/cm$^{3}$ did not change the SFR by a factor ten, but merely by
about 10\% at the peak of star formation. As Table~\ref{sims} indicates,
we also experimented with the value of $\epsilon$ and found that taking
a value of $\epsilon$ = 0.01 (ten times smaller than our fiducial value)
left the conclusions presented in this paper unchanged, i.e. the medium
became porous and the SFR peaked at roughly the same value although
with a slight time delay compared to the run with $\epsilon$ = 0.1 .
\begin{table*}
\caption{Summary of the performed runs. All of the runs have radiative cooling.
The first three columns indicate
whether self-gravity, star formation and/or feedback are activated.
$\rho_{\mathrm {crit}}$ is the density threshold for star formation,
$\epsilon$ is the star formation efficiency and the final column
indicates the grid resolution. Each simulation cube is 1.28 kpc per side.
}
\label{sims}
\begin{tabular}{l||cc|cc|cc|cc}
& self--gravity & stars & feedback & $\rho_{\mathrm {crit}}$
(at/${\mathrm {cm}}^3$) & $\epsilon$ & grid resolution (pc) \\ \hline \hline
A & -- & -- & -- & -- & -- & 5 \\
B1 & -- & yes & -- & 10. & 0.1 & 10 \\
B2 & yes & yes & -- & 10. & 0.1 & 10 \\
B3 & -- & yes & yes & 10. & 0.1 & 10 \\
B4 & yes & yes & yes & 10. & 0.1 & 10 \\
B5 & yes & yes & yes & 1. & 0.1 & 10 \\
B6 & yes & yes & yes & 10. & 0.01 & 10 \\
C1 & -- & yes & -- & 10. & 0.1 & 20 \\
C2 & yes & yes & -- & 10. & 0.1 & 20 \\
C3 & -- & yes & yes & 10. & 0.1 & 20 \\
C4 & yes & yes & yes & 10. & 0.1 & 20 \\
C5 & yes & yes & yes & 1. & 0.1 & 20 \\
C6 & yes & yes & yes & 1. & 0.01 & 20 \\
\\
\\
\end{tabular}
\end{table*}
\section{General Features of the Multiphase Medium}
\label{generalpics}
We begin by showing the time evolution of one of our simulations, namely B4,
which includes all the physical processes we considered, namely
``turbulent'' initial conditions (as defined in section~\ref{method}), radiative cooling, self-gravity,
star formation and feedback. In figure~\ref{dTp_128}
we show the gas density, temperature and pressure
in a 12.8 pc $\times$ 1.28 kpc $\times$ 1.28 kpc slice
of this run. Due to the compression caused by
turbulence and self-gravity, the gas in certain regions,
satisfies our criteria for star formation. Following their formation,
this first generation of stars soon explodes as
supernovae, releasing hot gas into the interstellar medium.
The morphologies of the hot bubbles are extremely non-spherical
due to the fact that the supernovae are releasing their thermal energy
into a spatially inhomogeneous and non-stationary medium.
Because this hot, low density gas has a long cooling time and because
the star formation rate is sufficiently high, subsequent generations
of supernovae bubbles overlap, filling more and more of the volume.
Ultimately the density and temperature span more than six orders of
magnitude in such a simulation and are anti-correlated:
high density regions are cold, and low density regions are hot.
As the third column in figure~\ref{dTp_128} shows, this anti-correlation results
in near pressure equilibrium between these two phases of the gas.
Nevertheless the dense gas is about one order of magnitude lower in pressure
than the low density gas indicating that a thermal instability is active.
Other regions which are out of pressure equilibrium by 1 -- 2 orders of
magnitude are those which have just experienced
thermal energy input from supernovae. Self-gravitating gas would also appear
out of pressure equilibrium, something we see in
later stages of the simulation.
\begin{figure*}
\centerline{\psfig{file=snapshot_dTp_early_new.ps,width=\hsize,angle=270}}
\caption{Time evolution of the logarithm of the gas density (first column),
temperature (second column) and pressure (third column) in a 12.8 pc
$\times$ 1.28 kpc $\times$ 1.28 kpc slice for run B4.}
\label{dTp_128}
\end{figure*}
The dynamical state of the stars and of the gas in different temperature
regimes in the simulation is summarized by a plot of the average velocity
dispersions (fig.~\ref{sigma}). Guided by some of the features in the cooling
curve (see figure~\ref{cool_function}), we divide the temperature into
the following four categories: (I) T $<$ 2000 K,
(II) 2000 K $<$ T $<$ $10^5$ K,
(III)$10^5$ K $<$ T $<$ $4 \times 10^6$ K, (IV) $4 \times 10^6$ K $<$ T.
We compute the average velocity dispersion of the gas in each of these
4 regimes, and in addition, we calculate the mass-weighted velocity
dispersion of the gas, as well as the mass-weighted velocity dispersion
of the stars.
As the stars are assigned the velocity of their progenitor gas at formation,
their velocity dispersion closely follows the velocity dispersion of the cold
gas. Furthermore, we find that with the exception of the hottest phase (IV),
the velocity dispersion of the other phases approximately settles to
the following values: (I) 15 km/s, (II) 30 km/s and (III) 75 km/s.
What is very striking in the plot of
the velocity dispersions, is the high velocities ($\sim$ 500 km/s)
attained by the hot, low density component of the gas.
The densest structures which provide the raw material for star formation,
collide and break apart, but are also subject to stripping via
hydrodynamical and thermal instabilities when
this hot, low density material flows rapidly past them.
The picture of a ``violent interstellar medium'' (McCray \& Snow 1979) emerges.
\begin{figure}
\centerline{\psfig{file=avgs_ylog_128_fbk_sg_pt1_rcr10.ps,width=.9\hsize}}
\caption{Time evolution of the logarithm of the velocity
dispersion in run B4 for the gas in different temperature
regimes: (I) T $<$ 2000 K (triangles), (II) 2000 K $<$ T $<$ $10^5$ K
(plus signs), (III)$10^5$ K $<$ T $<$ $4 \times 10^6$ K (squares),
(IV) $4 \times 10^6$ K $<$ T (diamonds). The crosses mark the
average mass--weighted velocity dispersion of the gas and the asterices that
of the stars.}
\label{sigma}
\end{figure}
\begin{figure*}
\centerline{\psfig{file=pdf_phase_evolution_fbk_wgrav_bw.ps,width=.6\hsize,angle=270}}
\caption{Time evolution of the density PDF (top row) and phase diagrams
(bottom row) for run B4 (128$^3$ run with star formation, feedback and
self-gravity).
In the phase diagrams, the dotted vertical (horizontal) line marks the critical density, $\rho_{\mathrm {crit}}$, (temperature, $T_{\mathrm {crit}}$)
for star formation. Dotted diagonal lines mark lines
of constant pressure, and are labeled for the ${\mathrm t} = 0$ Myr frame:
${\mathrm P} = 10^{6}$, $10^{5}$, $10^{4}$,
and $10^{3}\;{\mathrm k}_{\mathrm B} \;{\mathrm {cm}}^{-3}\; {\mathrm K}$.
Dashed diagonal
lines (labeled for the $t=0$ Myr frame) mark the Jeans length:
$\lambda_J = $10 pc, 34 pc, 113 pc, 380 pc and 1.28 kpc. }
\label{thermalevolution_4}
\end{figure*}
Regarding the evolution of the thermal state of the gas, this is
well portrayed in phase diagrams of the gas (bottom row of
figure~\ref{thermalevolution_4}) which show the distribution of the mass
fraction of the gas as a function of its temperature and density.
Given our initial conditions of uniform density and temperature, if we were
to plot a phase diagram of the gas at time $t = 0$ Myr, all the
gas would occupy a single point.
Because the initial temperature ($10^5$ K)
of the gas coincides with the peak of the cooling curve, by
9 Myr (first panel of bottom row of figure~\ref{thermalevolution_4}) the
majority of the gas quickly radiatively cools to an approximately isothermal
state at a temperature corresponding to the minimum of the cooling curve,
i.e. 310 K. As we instantaneously imprint a spectrum of velocity perturbations
at the beginning of the simulation, the gas acquires a range of density values
and therefore has a spread in densities by this time.
Thereafter, with the injection of hot gas into the medium, a tail of low density, hot
gas appears.
However as gas with temperatures $10^5$ K $<$ T $<$ $4 \times 10^6$ K (phase
III) is thermally unstable, it gradually vanishes from the medium,
dividing the gas into two parts in the phase diagram.
The majority of the coldest (T $\sim$ 300 K) gas
differs by approximately a one order
of magnitude pressure jump from gas with T $\geq$ 5 $\times$ $10^5$ K.
Finally the pressure of both the hot and cold gas changes with time.
It rises as more and more hot gas fills the simulation volume,
a situation that would probably be different if hot gas were allowed
to escape the box.
\begin{figure*}
\centerline{\psfig{file=gas3d.ps,width=.7\hsize,angle=180}}
\caption{Isodensity surfaces of the gas for run B4 and $\rho = 10^{-3}, 1,
10,$ and 50 atoms/${\mathrm {cm}^3}$.}
\label{gas3d}
\end{figure*}
Although complex, pictures of the gas density and temperature distribution
in a two-dimensional slice through the simulation volume, do not
capture the intricacy of the three-dimensional structure. In an attempt
to display this structure, in figure~\ref{gas3d} we plot isodensity
surfaces of the gas for $\rho = 10^{-3}, 1, 10,$ and 50
atoms/${\mathrm {cm}^3}$ at 50 Myrs. It is clear from these figures that
the hot, low density
component fills most of the volume, while the densest regions fill the
smallest fraction of the space, and are scattered throughout the box.
A three-dimensional rendering of the stellar density at the same time
instant (fig.~\ref{star3d}), reveals traces of the imprint of the high density gas
distribution and encouragingly bears some qualitative resemblance to the
distribution of H$\alpha$ emission in disk galaxies
(e.g. NGC 4631, Wang et al. 2001).
\begin{figure}
\centerline{\psfig{file=stellardensity3d.ps,width=.75\hsize,angle=270}}
\caption{Isodensity surface of the stellar density for run B4 and
$\rho_{\star} = 0.1$ M${_{\odot}/\mathrm {pc}^3}$.}
\label{star3d}
\end{figure}
\section{Quantifying the Structure and Energetics of the Multiphase Medium}
\label{detailedpics}
In an effort to assess what determines star formation rates,
we systematically examine how different physical processes change
the structure and the energetics of the interstellar medium.
The sequence of runs listed in table~\ref{sims} are designed to
isolate the effects of successively more complicated physical processes.
A plot comparing the star formation rates for this sequence of runs
(figure~\ref{sfrs})
invites us to study what keeps star formation at a minimum and alternatively
what is necessary to drive it to high values.
Resolution effects immediately manifest themselves in figure~\ref{sfrs}.
The $64^3$ and $128^3$ runs start from the same initial conditions. Preceeding
star formation, feedback is non-existent, but self-gravity plays a larger
role in the $64^3$ run where a grid cell of equivalent density to that in the
$128^3$ grid is 8 times more massive. Therefore in the $64^3$ run with only
self-gravity (run C2), the SFR rises more rapidly at
earlier times than for the comparable run performed on the $128^3$ grid (run
B2). Once feedback occurs, a mechanism
supplementary to turbulence exists for creating high density contrasts which
are stronger in the higher resolution runs. This causes higher peaks of SFR
in the $128^3$ runs with feedback (runs B3 and B4) as compared to
the equivalent $64^3$ runs (C3 and C4). On the other hand, feedback also
creates an extra source of pressure to fight self-gravity which explains
why the C2 run leads to higher SFRs at earlier times than the C3 and C4 runs.
What remains unclear without performing a simulation at
still higher resolution is whether the indistinguishability between the
$128^3$ runs
with feedback regardless of whether or not there is self-gravity (runs B3 and
B4) are a manifestation of convergence or coincidence. However, we believe
convergence is the more probable explanation as increasing the resolution tends to increase
the dominance of feedback processes over self-gravity. More specifically,
in the case of the $64^3$ runs a rise in the SFR is driven more rapidly when
self-gravity is included. In contrast, star formation increases at similar rates
regardless of whether self-gravity is included in the $128^3$
runs. Therefore we do not see any reason why this trend should be inverted by
further increasing the resolution.
\begin{figure*}
\centerline{\psfig{file=sfr_comp_bw_newaug.ps,width=.5\hsize,angle=270}}
\caption{Time evolution of the star formation rate
for a series of runs (see table~\ref{sims}) differing in their physics.
The left panel displays the results from runs C1 (diamonds), C2 (squares),
C3 (triangles) and C4 (asterices). The right panel displays the results
from runs B1 (diamonds), B2 (squares), B3 (triangles) and B4 (asterices).
Symbols are the measured SFRs and the dotted and dashed lines are
analytic models from Silk (2001).}
\label{sfrs}
\end{figure*}
Before proceeding, we calculate roughly the supernovae rate
corresponding to the measured star formation rates in our
simulations. In our $1.28^3$ ${\mathrm {kpc}^3}$ box, typical star
formation rates are SFR $\sim 0.1 - 0.8 \,{\mathrm M}_\odot/{\mathrm {yr}}$. Scaling
these values to a Milky Way type galaxy, where $M_{\mathrm {MW}}$ is the
mass of gas in the Milky Way, and $M_{\mathrm {box}}$ is the mass of
gas in our simulation cube,
\begin{equation}
{\mathrm {SFR}} \, (M_{\mathrm {MW}}/M_{\mathrm {box}}) \approx 100 - 800
\,{\mathrm M}_\odot/{\mathrm yr}.
\end{equation}
For a Salpeter IMF there is approximately 1 SN/200 ${\mathrm M}_\odot$,
implying that the typical supernovae rates in our simulation volume
are $\sim 0.5 - 4$ SN/yr. Furthermore, with this scaling to higher mass
the projected gas surface density increases
by about 4 orders of magnitude bringing both the SFRs and
surface densities to values representative of the starburst regime in the Kennicutt relation
(Kennicutt 1998).
\begin{figure*}
\centerline{\psfig{file=snapshot_labels.ps,width=.85\hsize,angle=270}}
\caption{The gas density, temperature and pressure at time
${\mathrm t = 45}$ Myrs in a
12.8 pc $\times$ 1.28 kpc $\times$ 1.28 kpc slice for runs A, B1, B2 and B4
(see table~\ref{sims} for the specifications of each of these runs). }
\label{compare_rhoTp}
\end{figure*}
A visual examination of a 2D snapshot of the gas density,
temperature and pressure taken at the same time (${\mathrm t = 45}$ Myrs)
for runs including different physics is useful for comparing
some of the consequences of the different processes.
Figure~\ref{compare_rhoTp} clearly shows how self-gravity, which
is a radially directed force towards regions of locally high density,
causes high density regions to take on a more spherical
appearance. Furthermore, all the runs without feedback have
gas with pressure spanning over $\sim$ 6 orders of magnitude, and
a small range in temperatures compared to the run with feedback.
The low density gas regions in the runs without feedback, are cold (T $\sim$ 300 K)
and are created by adiabatic cooling during extreme expansion
in certain regions of the ``turbulent'' medium. Another feature
that appears in this sequence of simulations is that the dense structures
in the run with feedback are sharper due to destruction of intermediate
density material by thermal and hydrodynamical instabilities.
\subsection{Probability Density Function of Mass Density}
A density probability distribution function (PDF) is a simple
one-dimensional statistical
measure of the structure of a medium. In practice for simulations performed
on a grid, PDFs are instantaneous
histograms tallying the number of grid cells of a certain density in the
simulation. Under the premise that stars form in high density regions, the
statistical properties of the density field, itself nonlinearly coupled
to the velocity field, might give clues to the process of star formation.
Efforts to uncover how the gas density organizes itself in media
structured by different dynamical processes are ongoing.
V\'{a}zquez-Semadeni (1994) presented a statistical argument to show that
turbulent (random), supersonic, compressible flows
naturally generate hierarchical structure without necessitating an appeal
to things like fragmentation in a gravitationally unstable system (Hoyle 1953).
In the limit of very high Mach numbers these flows have a pressureless
behaviour and if, in addition, self-gravity is negligible then the
hydrodynamical equations are scale-invariant. Consequently, whatever the
density in a given region, that region has the same probablity of producing
a relative fluctation with respect to its normalizing density, as any other
region in the flow. Assuming that in a random flow successive density steps are
independent, the central limit theorem dictates that the density distribution
should be log-normal. And indeed, V\'{a}zquez-Semadeni's (1994)
two-dimensional, essentially isothermal
($\gamma$ = 1.0001) simulations of a weakly compressible (M $\sim$ 1),
turbulent flow without self-gravity developed log-normal density PDFs both on
the large scale of the simulation and in subregions within the simulation.
Subsequently, numerical experiments of three-dimensional, isothermal,
randomly forced, supersonic turbulence by Padoan, Nordlund \& Jones (1997)
also found that the gas density follows a log-normal distribution,
\begin{equation}
{\mathrm {PDF}} = \, \frac{1}{\sigma \sqrt{2 \pi}} \, {\mathrm e}^{-({\mathrm {ln}}\rho - <{\mathrm {ln}}\rho>)^2/2\sigma^2}.
\end{equation}
Furthermore they observed empirically that
the dispersion, $\sigma$, of the log--normal scales with the
root--mean--squared Mach number, $M_{\mathrm {rms}}$, as follows:
\begin{equation}
\sigma^2 = \, {\mathrm {ln}}(1 + (\frac{M_{\mathrm {rms}}}{2})^2)
\end{equation}
or, for the case of the linear dispersion
\begin{equation}
\sigma_{\mathrm{linear}} = \, \frac{M_{\mathrm {rms}}}{2}.
\end{equation}
These dispersion relations reflect the fact that in a medium with
higher $M_{\mathrm {rms}}$, the gas achieves greater density contrasts.
Passot \& V\'{a}zquez-Semadeni (1998) found the same linear scaling
relation for the isothermal case. A formal proof for the lognormal
PDF in the case of isothermal, supersonic turbulence was provided
by Nordlund \& Padoan (1999) based on the formalism given in
Pope \& Ching (1993).
\begin{figure*}
\centerline{\psfig{file=pdf_phase_evolution_nofbk_nograv_nostars_bw.ps,width=.7\hsize,angle=270}}
\caption{Time evolution of the density PDF (top row) and phase diagrams
(bottom row) for run A (128$^3$ run with no star formation, no feedback
and no self-gravity). The thick dashed line overplotted on the measured
PDFs (symbols) is the log--normal PDF predicted by Padoan, Nordlund \& Jones (1997).
In the phase diagrams, the dotted vertical (horizontal) line marks the critical density, $\rho_{\mathrm {crit}}$, (temperature, $T_{\mathrm {crit}}$)
for star formation. Dotted diagonal lines mark lines
of constant pressure, and are labeled for the ${\mathrm t} = 0$ Myr frame:
${\mathrm P} = 10^{6}$, $10^{5}$, $10^{4}$,
and $10^{3}\;{\mathrm k}_{\mathrm B} \;{\mathrm {cm}}^{-3}\; {\mathrm K}$.}
\label{pdf_nofbknosg}
\end{figure*}
Scalo et al. (1998) and Passot \& V\'{a}zquez-Semadeni (1998)
extended this work on isothermal flows by considering the polytropic case.
Having conducted two-dimensional simulations including various combinations
of physical processes (e.g. self-gravity, magnetohydrodynamics, Burgers
turbulence), Scalo et al. (1998) found PDFs that were more consistent
with power laws than with log-normal distributions. Seeking to understand
this result and its discrepancy with previous work on isothermal
flows which consistently found lognormal distributions,
Scalo et al. (1998) performed one-dimensional simulations of forced,
supersonic, polytropic turbulence
and uncovered a lognormal PDF for the cases where either the gas was isothermal
($\gamma$ = 1) or where the Mach number was small (M$\ll$ 1).
Otherwise, when $\gamma <$ 1, power laws developed for densities
larger than the mean. Alternatively,
Nordlund \& Padoan (1999) interpreted Scalo et al.'s results for
the PDFs occuring in the $\gamma \neq$ 1 case as skewed log-normals
and Passot \& V\'{a}zquez-Semadeni (1998) provided a mathematical framework
for understanding why these distributions arose.
Our work extends these investigations on the PDF in the direction
of the cases where the ISM is constrained neither to be isothermal
nor polytropic.
As a result our local temperature and pressure are not simple functions of the
density but arise from the evolution of the thermal energy.
Because we consider processes (e.g. radiative cooling, self--gravity,
star formation) whose effectiveness depends on the density, the hydrodynamic
equations are no longer scale--invariant. Therefore the condition of
randomness between subsequent density fluctuations is violated and
one cannot expect a log--normal density PDF
(e.g. V\'{a}zquez-Semadeni's (1994)). In our series of experiments of
increasing complexity (see Table~\ref{sims}), the simplest
simulation we performed was of non--isothermal supersonic turbulence (run A).
Despite the inclusion of density--dependent cooling processes, we found
that the structure of the gas quickly evolved to a density PDF consistent
with a log--normal. This is not a surprising result since without a heat source
the majority of the gas quickly cools to a nearly isothermal state (see bottom
row of figure~\ref{pdf_nofbknosg}) with an average temperature corresponding
to the minimum of the cooling curve (horizontal dashed line in bottom row of
figure~\ref{pdf_nofbknosg}). Furthermore, the scaling for the
dispersion of the PDF given by Padoan, Nordlund \& Jones (1997) continued
to hold. In fact, rather than fit log--normal functions to our density PDFs,
we measured the average of the logarithm of the gas density,
$<{\mathrm {log}}_{10} \rho>$, and
the $M_{\mathrm {rms}}$ of the gas at different
time instances and then overplotted Padoan, Nordlund \& Jones' (1997)
prediction for the log--normal distribution.
For the runs where we formed stars (without self--gravity or feedback)
in addition to having radiative cooling (runs B1 and C1),
the gas density PDF continued to have the same behavior:
the $M_{\mathrm {rms}}$ of the system progressively declined with time, while
the density PDF remained consistent with a log--normal distribution
(fig.~\ref{pdf_nofbknosgstars}).
\begin{figure*}
\centerline{\psfig{file=pdf_phase_evolution_bw_nofbk_nograv.ps,width=.7\hsize,angle=270}}
\caption{Time evolution of the density PDF (top row) and phase diagrams
(bottom row) for run B1 (128$^3$ run with star formation, no self-gravity, and
no feedback). The thick dashed line overplotted on the measured
PDFs (symbols) is the log--normal PDF predicted by Padoan, Nordlund \& Jones (1997).
In the phase diagrams, the dotted vertical (horizontal) line marks the
critical density, $\rho_{\mathrm {crit}}$, (temperature, $T_{\mathrm {crit}}$)
for star formation. Dotted diagonal lines mark lines
of constant pressure, and are labeled for the ${\mathrm t} = 0$ Myr frame:
${\mathrm P} = 10^{6}$, $10^{5}$, $10^{4}$,
and $10^{3}\;{\mathrm k}_{\mathrm B} \;{\mathrm {cm}}^{-3}\; {\mathrm K}$}
\label{pdf_nofbknosgstars}
\end{figure*}
\begin{figure*}
\centerline{\psfig{file=pdf_phase_evolution_nofbk_wgrav_bw.ps,width=.7\hsize,angle=270}}
\caption{Time evolution of the density PDF (top row) and phase diagrams
(bottom row) for run B2 (128$^3$ run with star formation and self-gravity but
no feedback). The thick dashed line overplotted on the measured
PDFs (symbols) is the log--normal PDF predicted by Padoan, Nordlund \& Jones (1997).
The solid line with a slope of -1.5 plotted at ${\mathrm t = 85}$ Myr
is a fit to the high density end of the PDF. In the phase diagrams, the
dotted vertical (horizontal) line marks the
critical density, $\rho_{\mathrm {crit}}$, (temperature, $T_{\mathrm {crit}}$)
for star formation. Dotted diagonal lines mark lines
of constant pressure, and are labeled for the ${\mathrm t} = 0$ Myr frame:
${\mathrm P} = 10^{6}$, $10^{5}$, $10^{4}$,
and $10^{3}\;{\mathrm k}_{\mathrm B} \;{\mathrm {cm}}^{-3}\; {\mathrm K}$}
\label{pdf_nofbksg}
\end{figure*}
The runs which showed the first departure from log--normal density
PDFs were the runs which included self-gravity (runs B2 and C2)
but still no feedback (figure~\ref{pdf_nofbksg}). Repeating the exercise
of measuring the average of the logarithm of the gas density,
$<{\mathrm {log}}_{10} \rho>$, and
the $M_{\mathrm {rms}}$ of the gas at different times, we found
two differences: (a) the $M_{\mathrm {rms}}$ initially declined
but then stabilized
at a value higher than that seen in the runs without self--gravity, and
(b) the log--normal PDF predicted by Padoan, Nordlund \& Jones (1997)
consistently underpredicted the distribution at high gas density.
A power--law fit the high density tail well. In one-dimensional simulations
of Burgers flows, i.e. infinitely compressible flows, power--laws
were also found to be good fits to the density PDFs
(Gotoh \& Kraichnan 1993). We therefore interpret the power--law behavior
for the run with self--gravity, as reflecting the added possibility of
the gas, once it has a high density, to compress to even higher density,
reminiscent of the behavior in Burgers flows. Klessen (2000) also
explored the form of the density PDF for the cases of decaying and driven
self-gravitating turbulence. Although he found a departure from log-normal
at high densities, the departure could not be characterized by a power law.
\begin{figure}
\centerline{\psfig{file=pdf_comp128_bw_new.ps,width=.9\hsize}}
\caption{Comparison of the PDFs at 110 Myrs for runs with different physics.}
\label{pdfcomp}
\end{figure}
When we add feedback to the list of simulated processes,
either with self--gravity (runs B4 and C4) or without (runs B3 and C3),
the density PDF becomes markedly bimodal (figure~\ref{pdfcomp}),
illustrating that the majority of the simulation volume is occupied
by low density gas. A bimodal
density distribution is also a sign of a thermal instability
(V\'{a}zquez-Semadeni, Gazol \& Scalo (2000)) the
consequences of which we will discuss in a future paper
(Slyz, Devriendt, Bryan \& Silk, {\em in preparation}). For the
runs with self--gravity, the high density power--law tail disappears.
Perhaps it can be argued that the high density part of the density PDF
may be fit with a log--normal distribution (figure~\ref{pdf_b4_xlnfit}).
The exercise of overplotting the log--normal given by
Padoan, Nordlund \& Jones (1997) is not possible because the
$M_{\mathrm {rms}}$ measured for the entire simulation box does not
correspond to the $M_{\mathrm {rms}}$ of the high density gas for
which the log--normal function may be a good description.
Hence we can only {\em fit} log--normals to the high density gas,
similar to what others, e.g. Wada \& Norman (2001), Kravstov (2003),
do in their global simulations of the ISM.
\begin{figure}
\centerline{\psfig{file=pdf_fbk_sg_fit_128_85myr.ps,width=.95\hsize}}
\caption{Log--normal fit to high density end of the PDF for run B4 at
${\mathrm {t = 85} \,\mathrm{Myr}}$. The scaling we use for the fit is a
log--normal with average density of $10^{1.7}$ atoms/cm$^3$ and with a
dispersion of $\sim 10^{1.22}$ atoms/cm$^3$.}
\label{pdf_b4_xlnfit}
\end{figure}
The interest of describing the density structure of the ISM
with a single function, such as the log--normal, lies in finding
a link between the gas density averaged over kiloparsec sized regions
and the high density regions which might form stars.
This is precisely the link required for an explanation of the Schmidt law.
Rewriting the Schmidt law in a form where the star formation rate is
equal to some constants multiplied by
the fraction of gas in high density regions and by the gas density averaged
over large scales (his equation 7), Elmegreen (2002) emphasized that star
formation rates depend on the geometry of the density field, i.e. the PDF.
If the shape of the density PDF is universal,
then the fraction of gas in high density regions is known.
Consequently, if the high density
regions are also self--gravitating, then the fraction of gas available for
star formation is also known. Admittedly, the density PDF contains no
spatial information, hence there is no reason for which the
high density regions should find themselves to be spatially contiguous,
so that they comprise regions of mass greater than the Jeans mass.
In fact, figure~\ref{pdf_nofbksg} clearly shows that at least some of the
dense gas regions are not contiguous because if they were they would
simply not persist as all the gas would be converted to stars on a
dynamical timescale since these regions are well above $\rho_{\mathrm {crit}}$
and cold. We therefore have to identify these regions with divergent gas flows.
A two-dimensional study of the ISM in a galactic disk by Wada \& Norman (2001)
has claimed that the log--normal distribution is a robust description of
the ISM density distribution over many orders of magnitude in density,
regardless of the simulated physics. More specifically, in their simulations
the presence of stellar feedback does not change the shape of the
PDF but increases the dispersion of the lognormal.
In three-dimensional simulations
of a high-redshift galaxy performed in a cosmological context, Kravtsov (2003)
finds a density distribution similar to Wada \& Norman's (2001). Its
shape at every redshift epoch has a flat region at
$\rho_{\mathrm{gas}} \leq$ 1 -- 10 $\mathrm{M}_{\odot}$ $\mathrm{pc}^{-3}$ and
a power law distribution at high densities. He claims that the log--normal
distribution is a fair description of the high density tail of the PDF and
agrees with Wada \& Norman (2001) on the insensitivity of the distribution
to feedback, except at the low density end, where the simulation with feedback
produces more low density gas.
As figure~\ref{pdfcomp} shows, our less realistic study of star formation
occuring in a periodic box without the global gravitational galactic
potential or the shear instabilities present in a self--gravitating
rotating disk, appears to be more sensitive to the input physics.
Only the runs which include stellar feedback are nearly equivalent,
regardless of whether there is self-gravity. When log--normals are
overplotted for the runs without feedback, the position of the maximum
of the log--normal is shifted to lower densities by more than one
order of magnitude from the position of the maximum of the log--normal
fit to the high density part of the PDF for the runs with feedback.
Indeed the densities in certain cells for the run with only self--gravity
reach the same high values as the runs with feedback, but a much smaller
fraction of the simulation volume has these high densities.
Another blatant difference between the PDFs we find in our runs with feedback
and the PDFs found by Wada \& Norman (2001) and Kravtsov (2003)
is that their runs do not show as high a peak at low densities.
The smaller quantity of low density gas in their simulations is likely due
to the much lower supernovae rates in Wada \& Norman's (2001) simulations
(0.01 SN/yr as compared to 0.5 -- 4 SN/yr in our simulations) and
in Kravtsov's (2003) case, to the
more realistic boundary conditions, which allow tenuous, hot gas to escape
the disk.
\subsection{Energy Spectra}
\begin{figure*}
\centerline{\psfig{file=energyspec_labels.eps,width=\hsize}}
\caption{Time evolution of the compressible ($E_c$) and solenoidal
($E_s$) components of the energy spectra for runs B1, B2, B3 and B4. Symbols
denote energy spectra at time intervals separated by 30 Myrs. Solid line
represents time $t=0$ Myr. Plus signs: 30 Myr, asterices: 60 Myr, filled
diamonds: 90 Myr, open diamonds: 120 Myr, open triangles: 150 Myr,
crosses: 180 Myr, open squares: 210 Myr. We also draw a solid line
through the symbols when they represent the final timestep that we
are displaying. Thick dashed lines indicate power laws with slopes
similar to that of the last curve shown.}
\label{energy_spectra}
\end{figure*}
Energy spectra of the ISM carry complementary
information to that given by a study of its density structure. With the
density PDFs, we confirmed that in many cases there exists a
clear relationship between the density contrast achieved and the
$\mathrm{M}_{\mathrm{rms}}$ of a system
(i.e. $\sigma_{\mathrm{linear}} \sim {\mathrm{M}}_{\mathrm{rms}}$). But the
${\mathrm{M}}_{\mathrm {rms}}$ of a system is
only a global measurement of its kinetic energy content.
With measurements of the kinetic energy spectra, we expect to learn how
the energy is distributed on different spatial scales and how the different
physical processes we considered influence the time evolution of this
distribution.
The Kolmogorov theory of incompressible, subsonic turbulence predicts
that energy fed on large scales progressively cascades to smaller scales
until it is dissipated by molecular viscosity on the smallest scales in vortex
rings. The transfer of energy is a local process and
the spectra of the velocity
field is a power law with $E_k \sim k^{-5/3}$ (Kolmogorov 1941).
With supersonic, compressible turbulence, strong shocks come into play.
They allow energy to be transferred over widely separated scales and it
is possible that rather than being dissipated in vortex rings, the
energy is ultimately dissipated in sheets, filaments and cores
(Boldyrev 2002). Given the analogy between highly supersonic and
pressureless flows, one might
expect the compressible, supersonic flows to have the same behavior
as Burgers turbulence with power spectra in the inertial regime of the
form, $E_k \sim k^{-2}$ (Burgers 1974, Gotoh \& Kraichnan 1993). However
this appears to only be true in one and two dimensions. In three dimensions,
compressible, supersonic flows differ from Burgers flows because they generate
vorticity (Boldyrev 2002). In three-dimensional simulations of
compressible, supersonic,
magnetized forced turbulence with Mach number initially $\sim$ 10,
Boldyrev, Nordlund \& Padoan (2002) find energy power spectra in the
inertial range to be $E_k \sim k^{-1.74}$, i.e. close to the Kolmogorov
value.
As we lack the grid resolution to ascertain if the energy spectra
in our simulations are tending towards power laws, we cannot make any
credible statements
about the values of the power law slopes.
Furthermore in incompressible turbulence, the energy spectrum is a power
law in the inertial regime (at $k$ wavenumbers below the energy injection
scale but above the energy dissipation scale). In our simulations
the feedback energy is injected on scales equivalent to the grid resolution,
i.e. the smallest scales, but it can propagate to larger scales depending
on the ISM dynamics. Therefore for the runs with feedback
the inertial regime has a more complicated meaning.
Instead in figure~\ref{energy_spectra} we focus on the time development
of the energy spectra, and the presence of characteristic features.
The standard approach involves dividing the kinetic energy
into two components: a compressible one for which
$\nabla \times \; v_{\mathrm {comp}} = 0$, and a solenoidal one with
$\nabla \cdot \; v_{\mathrm {sol}} = 0$. In words, the compressible component
measures the strength of the shocks in the system, while the solenoidal
component measures the degree of rotation. Typically, the compressible
component is expected to decay faster than the solenoidal component
as the shock energy is transformed into vortical eddy motions.
Because we remove gas from the system to form stars, the kinetic
energy whose spectra we measure, is rather a specific kinetic energy,
i.e. we divide the instantaneous total kinetic energy by the
total gas mass present at that moment. In all our runs, the
kinetic energy which is initially imprinted only on large scales
quickly (within $\sim$ 30 Myr) redistributes itself to smaller
scales as well. Following this redistribution, for the run with
neither self--gravity nor feedback (run B1), the compressible and solenoidal
components of the energy spectra progressively decay all the while
maintaining approximately the same form. The ratio $E_c/E_s$ is always
less than 1, i.e. the compressible component decays faster
than the solenoidal one, but increases towards the dissipative regime.
In high resolution simulations ($512^{3}$, $1024^{3}$) of decaying
compressible turbulence with Mach number initially on the order of 1 (an
order of magnitude lower than the initial Mach number in our simulations),
Porter, Woodward \& Pouquet (1998) find a similar result with
$E_c/E_s \sim 0.1$.
In contrast to these runs in which the kinetic energy decays, the runs
with self--gravity (run B2) and/or feedback (runs B3, B4),
show energy spectra which
climb to higher amplitudes with time and have shallower slopes than the
decaying run (B1). Furthermore in plots of the ratios of the compressible to
solenoidal components, between 90 and 150 Myrs the runs with
feedback show a peak at $\sim$ 65 pc consistent with what one
would predict for the characteristic lengthscale for a simulation
with supernovae expanding into a medium with ambient pressure of
$P = 10^{6} cm^{-3} \; K \; k_{B}$. More explicitly,
ignoring adiabatic and radiative losses,
a supernovae with $10^{51}$ ergs of energy will be halted by an ambient
medium at this pressure when it has expanded to a radius,
$r \sim (E/P)^{1/3} \sim$ 65 pc. This signature in $E_c/E_s$ for
the run with feedback points to a way to understand SFRs, which
we explore below.
\section{Numerical versus analytical star formation rates}
\label{comparesilk}
An alternative to searching for a generic density PDF
as an explanation for star formation rates, is to
consider arguments concerning the competition between the expansion
of supernovae remnants and the pressure which halts them.
In this vain, Silk (1997, 2001) developed porosity models
of a regulated ISM. Introduced by Cox and Smith (1974),
porosity, ${\mathrm Q}$, is proportional to the product of the
supernovae rate per unit volume and the maximum extent of the
4--volume of the supernovae remnants. In other terms, the porosity
measures the fraction of hot gas, ${\mathrm f_h}$,
in the ISM through the relation ${\mathrm Q} =
-{\mathrm {ln}}\,(1 - {\mathrm f_h})$.
Silk reasoned that since the supernovae production rate is proportional to the
star formation rate (SFR), and the maximum extent of
a supernovae remnant is limited by the ambient pressure, the
following expression arises:
\begin{equation}
{\mathrm Q} = {\mathrm {SFR}} \; {\mathrm G}^{-1/2}\; \rho_{\mathrm {gas}}^{-3/2} \; (\sigma_{\mathrm {gas}}/\sigma_{\mathrm f})^{-2.72}
\label{sfrequation}
\end{equation}
where $\rho_{\mathrm {gas}}$ is the gas density, $\sigma_{\mathrm {gas}}$ is
the gas velocity dispersion, and $\sigma_{\mathrm f}$ is a fiducial
velocity dispersion that is proportional to
$E_{\mathrm {SN}}^{1.27}\,m_{\mathrm {SN}}^{-1}\,\zeta_{\mathrm g}^{-0.2}$.
Here $E_{\mathrm {SN}}$ is the energy of a single supernova,
$\zeta_{\mathrm g}$ is the metallicity relative to solar of the ambient gas,
and $m_{\mathrm {SN}}$ is the mean mass in newly formed stars required to produce
a supernovae. For $E_{\mathrm {SN}}$ = $10^{51}$ erg, $\zeta_{\mathrm g} = 1$,
and $m_{\mathrm {SN}} = 250 M_{\odot}$ i.e. the case where we assume
only the occurrence of type II supernovae with a Miller--Scalo IMF,
the fiducial velocity dispersion is $\sim$ 22 km ${\mathrm s}^{-1}$.
Our simulations with feedback provided a laboratory to test
this analytic description of the SFR. For the purpose of computing
the porosity of the medium, we measured the fraction of hot gas in our
volume, defining hot to be gas with temperature T $\geq$ 4 $\times 10^{6}$ K.
For $\rho_{\mathrm {gas}}$
in equation~\ref{sfrequation} we took the average gas density in
our simulation volume, and for $\sigma_{\mathrm {gas}}$ we took the average
mass--weighted velocity dispersion of the gas. We kept the value for
$\sigma_{\mathrm f}$ at 22 km ${\mathrm s}^{-1}$. Given these values as
functions of time, we plotted as dotted and dashed lines the expectation
from eq.~\ref{sfrequation} for the SFRs in figure~\ref{sfrs}.
Computing the actual star formation rates in the box by defining the
mass of newly formed stars to be the mass of stars formed in the past
3 Myr, we overplotted the results as symbols in the same figure. Astonishingly,
the analytic values match the measured rates to better than
a factor 2.
Given the simplifications in the derivation of the analytic model,
there was no {\em a priori} reason for the fit to be a good
description of the star formation rate in an inhomogenous, non--stationary
model of the ISM. For example,
Silk takes the expression for the 4--volume of the SNR remnant in its
cooling phase from Cioffi, Mckee \& Bertschinger
(1988). They derive it under the assumptions that the supernovae expands
in a spherical manner, the ISM is homogenous and uniform
(i.e. no density gradients), there is no dust cooling or thermal conduction,
and the ambient ISM pressure is negligible until the
last stage of supernovae evolution when the remnant merges with the ambient
ISM. In contrast, we find that at least in the initial stages of our
simulations,
the supernovae remnants are highly non--spherical, the ISM is inhomogeneous
with ubiquitous density gradients and the ambient ISM gas pressure is
highly non-negligible ($P = 10^{5}-10^{6} cm^{-3} \; K \; k_{B}$).
However, as more of the gas turns into
stars, and the hot phase fills the majority of the simulation volume, the
ISM does start to resemble something more in line with the Cioffi et al.
assumptions.
\begin{figure}
\centerline{\psfig{file=massfprhosigma64+128.ps,width=1.5\hsize,angle=270}}
\caption{Plots comparing the time evolution of the cold mass fraction,
porosity, average gas density, and mass-weighted velocity dispersion for runs
containing different physics, on the $64^3$ and $128^3$ grid.}
\label{massfprhosigma64and128}
\end{figure}
When we examine in figure~\ref{massfprhosigma64and128} the time
evolution of each of the
physical quantities entering into the analytic model
for the SFR, we find the following. The runs (B1 and C1) which
produced the lowest star formation rates have zero porosity
and high fractions of cold gas ($f_{\mathrm{cold}} \sim$ 0.8--0.9),
but a continuously declining velocity dispersion. The runs reaching
a peak (runs B2, B3, B4, C2, C3)
or multiple peaks of high star formation (run C4) all
displayed depleted cold gas fractions after their final star formation
peak, a rise to a maximum in its velocity dispersion at the peak, and
either zero porosity for the case of the runs with self--gravity but
no feedback (runs B2 and C2) or a porosity that levels off to a constant
value around the time of the SFR peak (Q $\sim$ 4--5 for the $64^3$ case
(runs C3 and C4), and Q $\sim$ 4 for the $128^3$ case (runs B3 and B4)
after the SFR peak). We interpret the behavior in these parameters
as reflecting the importance of a high velocity dispersion for
generating high SFRs. Indeed in the analytic model for the SFR
(eq.~\ref{sfrequation}),
the gas velocity dispersion, $\sigma_{\mathrm {gas}}$, plays the most
important role, as it is raised to the highest power in the expression.
However even with velocity dispersions sustained at high values
($\sigma_{\mathrm{gas}} \sim$ 20 km/s),
SFRs will drop if the reserves of cold gas decline.
\section{Discussion}
\label{discussion}
Given the simplicity of our simulations, we examine their relevance
for representing true star formation processes in real galaxies.
The first issue we address is whether the star formation rates we obtain
are consistent with the Kennicutt relation. In section~\ref{detailedpics}
we scaled the mass in our simulation volume to that of the Milky Way,
finding that our star formation rates and surface densities were consistent
with star formation occuring in the starburst regime.
If we do not scale our SFRs and gas densities to a Milky Way type galaxy but
instead take them at face value we find that our initial 1 atom/cm$^3$ gas
density in a (1.28 kpc)$^3$ volume yields in projection about a 30 M$_\odot$/pc$^2$ column
density which lies at the boundary between Kennicutt's normal disks and
centers of normal disks (Kennicutt 1998). Transforming our average star formation
rate of 0.2-0.3 M$_\odot$/yr into a star formation rate per unit volume
leads us to an average star formation
rate density of about 0.1 M$_\odot$/yr/kpc$^2$, on the high side but in fair
agreement with Kennicutt's measurements for our computed surface density
(Kennicutt 1998, figure 6). We note
that Kennicutt's law is a static relation as it
concerns space averaged quantities in local galaxies, and a moment in the
history of these galaxies is bound to exist when their main progenitor
will be entirely gaseous (i.e. with no stars yet formed) and the Kennicutt relation
will break. As our simulations start from an exclusively gaseous medium, we do
not expect our simulation to follow the Kennicutt
relation from the very beginning, but to move towards it as it does.
We nevertheless consider our simulations to be in a starburst mode because
the duration of the star formation episode is much shorter than that of what
one expects in either a disk or spheroidal
galaxy. But this is not unusual since we are only modeling a chunk of
a galaxy and are therefore neglecting effects on larger length and therefore
timescales.
The second issue we address is whether periodic
boundary conditions drive the high star formation rates seen in our simulations.
When hot gas starts to fill the bulk of the simulation volume, because
the boundary conditions trap the hot gas,
conditions in the simulation may be viewed as a pressure cooker
and the increased pressure may drive higher star formation rates.
In our simulation by the time the
pressure cooker is operative, the SFRs are already at starburst levels as seen
when one
scales the SFRs and gas densities to a Milky Way type galaxy as we do in section~\ref{detailedpics}.
To be more specific, for the pressure cooker to be operative we have to
wait $\sim$ 10 Myr for the first supernovae to go off
and then we have to wait for the volume to
become significantly filled by this supernovae generated hot gas for the
hot gas to be able to
traverse the volume unobstructed by cold, dense gas. According to figure 15,
it takes on the order of 50 Myr for the hot gas filling fraction to be
approximately 50\%, corresponding to a porosity of about 0.7. Hence boundary
effects are not dominant in shaping the star formation rate until after that
time.
We also point out that the limitations of the boundary conditions should not
obfuscate the point that the manner in which we implement supernovae is a more important
factor leading to the build up of large quantities of hot gas in the medium.
When we perform simulations in all points identical to those presented in this
paper but with
supernovae going off instantaneously, as opposed to exploding with a more realistic 10
Myr time delay used in the work presented in this paper, we get extremely low
star formation rates (a few hundred times smaller than those we get in our
simulation here), because the hot gas never fills a significant fraction of
the simulation cube. In other words, the periodic boundary conditions cannot
dominate the physics of star formation driven by hot gas pressure until the
hot gas has already been generated, and we find that this depends strongly on
the way the
supernovae are implemented. As mentioned in section~\ref{introduction}, we leave the
discussion of this to a future paper.
The limitations of our closed, periodic box, and the
absence of a stratified external gravitational potential certainly keep our simulations far
from being representative of realistic galactic systems. For example, a credible simulation
of a disk galaxy, would have to be performed in a realistic cosmological
context to capture such effects as tidal encounters and stripping from
neighbours. Excluding these external stellar heating processes as well
as spiral waves, results in the neglect of processes that
would increase the velocity dispersion of the stars in real
galaxies. Therefore our simulations certainly have a higher
fraction of cold ISM and cold stars after a gas consumption time which may
prolong and strengthen star formation in our simulations.
We also emphasize that with our crude assumption of a
closed box not only can no material escape the box, affecting star formation
rates once hot gas permeates the simulation volume, but no material can
enter the simulation volume either. It could well be that accretion of cold
material is more relevant for star formation in real disks than either the
external star heating processes missing from the simulations discussed above or
the fact that hot gas cannot leave the simulation volume.
One can argue that perhaps the simulations presented in this paper
are more representative of what happens in the central
kiloparsec of a spheroidal starburst galaxy. In that case the potential well
might indeed trap a fraction of the hot gas and the pressure cooker
environment which comes into play after high star formation rates occur
in the simulation, if not as drastic as in our simulations might well be fairly
realistic.
\section{Summary and conclusions}
\label{conclusion}
To unravel which global parameters control star formation, we
have examined star formation occurring in media whose dynamics
are structured by various combinations of physical processes (e.g.
``turbulence'', radiative cooling, self-gravity, feedback from
supernovae and stellar winds). We sought to understand our models
of the ISM from structural and dynamical perspectives, finding
that in some cases there was a well-defined link between the two.
In particular, measurements of the density PDFs confirmed that
for the simulations without feedback, lognormals were an adequate
description of the structure of the medium, and that the density contrasts
achieved in the media were directly correlated to their
${\mathrm{M}}_{\mathrm{rms}}$. Lognormals consistently underpredicted
the high density end of the runs with self-gravity which appeared to
be well-fit by a power law. For the runs with feedback, the
dense gas reached higher densities than those reached by the
runs without feedback implying that in these simulations, feedback
was positive in the sense that it encouraged higher star formation rates.
However the PDF for the runs with feedback had a distinctly bimodal shape
with the majority of the volume filled by low density gas. In summary, we
did not find a universal PDF. Most markedly, runs with feedback had a different
PDF from the runs without feedback, although arguably, the high density
end might be fit by a lognormal.
Measurements of the energy spectra in our simulations were consistent
with the information provided by the density PDFs. Self--gravity alone
was sufficient to sustain the kinetic energy of the medium, and hence
maintain the high density contrast we observed in the PDFs. Feedback
also succeeded in keeping high quantities of kinetic energy in the media
and inspection of ratios of compressible to solenoidal energy revealed
that supernovae were pumping energy into the system at a characteristic
scale consistent with the ambient pressure in the hot, low density
component of the medium.
For the runs with feedback, comparing Silk's (2001) star formation model to the measured
values of the SFRs in our simulations, revealed a good match that
led us to inspect the parameters involved in Silk's prescription.
They showed clearly that the SFR depends strongly on the
underlying velocity field which we saw could be energized
by self--gravity and/or feedback to produce high density contrasts. Without
a means to create these high densities, star formation rates decline
even in the presence of a large reservoir of cold gas.
In light of the issues neglected in our simulations, we
stress that the simplifying assumptions made in this paper facilitated our
choice to start from as strong as possible a local physical basis as possible
before trying to tackle star formation in a more global context. As such
we neglect numerous physical processes which may invalidate partially or completely our current
results, but this remains to be addressed in future work.
Nevertheless we hope that the present work sheds some light on the local
physics that should be included in future realistic simulations of star
formation.
\section*{Acknowledgments}
The authors thank Fabian Heitsch for a careful reading of the manuscript.
A. Slyz acknowledges the support of a Fellowship from the UK
Astrophysical Fluids Facility (UKAFF) where some of the computations reported
here were performed.
|
{
"timestamp": "2004-11-14T19:41:41",
"yymm": "0411",
"arxiv_id": "astro-ph/0411383",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411383"
}
|
\section{Introduction}
Having become extremely fashionable for their role as standardisable
cosmological candles, Type Ia Supernovae (SNe Ia) are becoming the centre of a
kind of scientific industry, with large programmes, both observational and
theoretical, being launched to explore their properties, detect them at all
redshifts, and understand the physical properties of the explosion and the
mechanism that holds the key to their apparent predictability. Since the main
observational feature -- the brightness--decline rate relation, is a
one-parameter relation \citep{phil93}, and because most modelling work has
been performed in 1D, it was typically assumed that the ejecta of SNe Ia are
homogeneous, with little or no deviation from smooth density profiles and
spherical symmetry. Recently, however, some suggestion that these assumptions
may not always be correct has come from polarisation measurements (Wang et~al.\
2003). Another, possibly related finding, is the detection of detached,
high-velocity components in near-maximum spectra.
The first suggestion of high-velocity components was made by \citet{hatano99},
who noticed features that they interpreted as high-velocity Ca~{\sc ii}\ and
Fe~{\sc ii}\ in the spectra of SN~1994D. They showed through spectral modelling that
Ca~{\sc ii}\ and Fe~{\sc ii}\ are present at $v > 25000$\,km~s$^{-1}$, and reaching $v =
40000$\,km~s$^{-1}$, detached from the photospheric component ($v < 16000$\,km~s$^{-1}$).
Further evidence came from SN~2000cx \citep{li01}, which shows two strong
and well separated components of the Ca~{\sc ii}\ IR triplet at high velocities.
\citet{thomas04} analysed the spectra of this rather peculiar SN~Ia, using a
simple 3D spectrum synthesis model. They confirmed that detached Ca~{\sc ii}\ is
present at high velocities, this time $v > 16000$\,km~s$^{-1}$. They also noticed a
corresponding broadening of the Ca~{\sc ii}\ H\&K doublet, which however did not show
detached features. Recently, \citet{bra04} confirmed these findings, and
additionally reported that high-velocity Ti~{\sc ii}\ features are also present.
Detached features were also observed in SN~2001el \citep{wang03} at
$v \sim 22-26000$\,km~s$^{-1}$, in SN~2003du at $v \sim 18000$\,km~s$^{-1}$
\citep{ger04}, and in SN~2004dt (F. Patat, priv. comm.).
In this paper we present and discuss evidence for a high-velocity component
that is present in the spectra of another otherwise normal SN~Ia, SN~1999ee
\citep{hamuy02,stritz02}. We argue in this case that not only Ca~{\sc ii}\ and
possibly Fe~{\sc ii}\ show high-velocity features, but that a sudden change in the
shape of the profile and the position of Si~{\sc ii}\ 6355\AA, the characterising
line of SNe~Ia, is due to the presence (and progressive thinning out) of a
high-velocity Si component, which is identified for the first time in a SN~Ia.
Since the presence of Si has major implications for the properties of the
explosion, we model the time-evolution of the spectra in order to identify the
nature of the discontinuity giving rise to the high-velocity feature
(abundance, ionisation, density, interaction with circumstellar material), and
to quantify the amount of high-velocity material, in particular Si, required to
reproduce the observations.
\section{Evidence for high-velocity features in the spectra of SN~1999ee}
Optical and infrared spectra of SN~1999ee covering the period from -9d to +42
days relative to $B$ maximum were presented by \citet{hamuy02}. SN~1999ee
is a rather slow-decliner ($\Delta m_{15}(B) = 0.91$), but it does not show the
spectroscopic peculiarities of SN~1991T and similar SNe. The spectral evolution
and line identification are discussed by \citet{hamuy02}.
Although the spectra of SN~1999ee look like those of a typical SN~Ia, the
Ca~{\sc ii}\ IR triplet shows two small notches, separated by exactly the line
separation of the two strongest components ($\lambda 8542$ and 8662\AA). The
narrow features are first visible on day --2, and persist until day +3, during
which time their position does not change significantly, indicating that they
are formed in a layer that is detached above the photosphere, with central
velocity $\sim 22000$\,km~s$^{-1}$. This is similar to the SNe~Ia discussed above,
although in SN~1999ee the narrow features are not clearly separated from the
main broad absorption, which is of photospheric origin. If these features are
correctly identified, their weakness in the two earlier spectra (day --9 and
--7) could result from the fact that the photosphere at those early epochs was
located at velocities comparable to that of the narrow features. These would
therefore blend almost completely with the broad component, resulting in a very
broad Ca~{\sc ii}\ IR triplet, as is indeed observed. The later disappearance of the
narrow features, going from day +3 to day +8, is then the combined consequence
of the decreasing density in the high-velocity zone caused by expansion and of
the inward motion of the photosphere, which becomes removed from the detached
Ca~{\sc ii}\ zone. Fig. 1 shows how the Ca~{\sc ii}\ IR triplet can be decomposed into
three different components, one broad and two narrow. Fig. 2 shows the time
evolution of the central velocity of each component. The broad photospheric
absorption drops rather smoothly from 20000 to 15000\,km~s$^{-1}$, but the two
detached features are clearly measured at velocities between 24000 and
20000\,km~s$^{-1}$\ and then disappear.
\begin{figure}
\includegraphics[width=89mm]{f1.eps}
\caption{Decomposition of the absorption part of the Ca~{\sc ii}\ IR
triplet into three gaussian components. The broad component is the blended
photospheric feature, while the narrow ones are the high-velocity absorptions
of the two strongest lines in the triplet.}
\label{CaIIdec}
\end{figure}
There is however another piece of observational evidence, not as clear perhaps
as that of the Ca~{\sc ii}\ IR triplet but certainly at least as rich in physical
implications: the sudden change in the shape of the Si~{\sc ii}\ 6355\AA\ line
between day --7 and day --2. In the two earliest spectra, the line appears
unusually blue (with a central wavelength of $\sim 6000$\,\AA, indicating a
velocity of $\sim 16000$\,km~s$^{-1}$). Additionally, it displays a P-Cygni profile
which increases in strength towards the highest velocities, both in absorption
and in emission, which is unusual for SNe~Ia lines. After this phase, the line
moves suddenly to the red, and it looks like a perfectly normal SN line on day
+3. The velocity evolution of the Si~{\sc ii}\ line is also shown in Fig.2. After
the sudden drop, the velocity of the line continues to decrease, but at a much
lower rate.
\begin{figure}
\includegraphics[width=89mm]{f2.eps}
\caption{Time evolution of the observed velocity of the Ca~{\sc ii}\ IR
triplet components, the Si~{\sc ii}\ 6355\AA\ line, and the photospheric velocity
adopted in model calculations.}
\label{Vel_CaIR}
\end{figure}
The coincidence in time of the appearing of the Ca~{\sc ii}\ narrow features and the
redward shift of the Si~{\sc ii}\ line, and then of the disappearing of the Ca~{\sc ii}\
features and the return to normal of the Si~{\sc ii}\ line suggests that these events
may be correlated. The behaviour of the Si~{\sc ii}\ line may also be due to the
presence of high-velocity material, which makes the line appear at bluer
wavelengths than usual at first. Later, as this material becomes optically
thin, the line recovers its typical profile.
\section{Modelling the spectra and identifying the detached features}
In order to locate accurately the regions responsible for the detached features
and to describe realistically their physical properties, we modelled the
sequence of spectra of SN~1999ee with our Montecarlo code
\citep{m&l93,lucy99,maz00}. As a first step, we tried to obtain reasonable
matches to each spectrum across the observed wavelength range. We used a
standard explosion model \citep[W7,][]{nom84}, and adjusted its 1D abundances
to achieve a good match to the observations. We then modified the density and
abundance distributions to reproduce the narrow Ca~{\sc ii}\ features and the
behaviour of the Si~{\sc ii}\ line.
We have modelled all 6 available spectra, starting from the first one, at day
--9, and until day +8. This covers the time when the spectral anomalies are
present. We tried to reproduce the global properties of the spectra, neglecting
the narrow Ca~{\sc ii}\ features or the fast Si~{\sc ii}\ line when present. This step was
necessary in order to define quantities such as the luminosity and the
photospheric velocity, which we can compare to the observed velocities plotted
in Fig.2.
The series of synthetic spectra is shown in Fig.3. The earliest spectra are
sufficiently well reproduced, but then the synthetic spectra become rapidly
worse. Even at the earliest epochs, close inspection reveals defects caused by
neglecting of the high-velocity components. In particular, looking at the first
spectrum, the Ca~{\sc ii}\ IR triplet and H\&K doublet, and the Si~{\sc ii}\ 6355\AA\ line
are significantly too red in the model, while other features such as the
absorptions near 4250 and 4900\AA, which are dominated by Fe~{\sc iii}\ lines at this
epoch, are correctly reproduced. This behaviour repeats at other epochs,
including those where the narrow components are present.
\begin{figure}
\includegraphics[width=89mm]{f3.eps}
\caption{Synthetic spectral sequence (dotted lines) using W7 densities and
abundances.}
\label{3}
\end{figure}
\section{An abundance enhancement?}
Based on the models above, we tried to reproduce the various spectral anomalies
by modifying the distribution in velocity space of the elements involved.
Guided by the observed spectra, we introduced regions of increased abundances
(relative to W7) of Si, Ca, and Fe at well defined velocities, trying to
improve the synthetic spectra.
The reason we experimented with the Fe abundance, as well as those of Si and Ca,
is that \citet{hatano99} attribute a feature near 4700\AA\ seen in SN~1994D to
high-velocity Fe~{\sc ii}\ (multiplet 48) absorption. A similar feature is observed
in SN~1999ee, although the simple W7-based models seem to reproduce that region
reasonably well, at least compared to the Fe-dominated region near 4300\AA.
Starting from the first spectrum, we increased the abundance of Si by factors
between 5 and 20 at velocities between about 16000 and 22000\,km~s$^{-1}$, and that of
Ca by a factor of about 20 at velocities larger than about 20000\,km~s$^{-1}$. It is
however not possible to reproduce the narrow features near 4700\AA\ by simply
increasing the Fe abundance. This in fact generates both Fe~{\sc iii}\ and Fe~{\sc ii}\
lines at high velocity, but there are Fe~{\sc iii}-dominated absorptions, like that
at 4250\AA, that do not show high-velocity components. Therefore, we increased
only the opacity of the Fe~{\sc ii}\ in order to get a qualitative assessment of the
possible role of high-velocity Fe~{\sc ii}. The opacity of Fe~{\sc ii}\ was increased by
factors of $\sim 10^4$ at $v > 28000$\,km~s$^{-1}$. The results for the earliest
spectrum are shown in Fig.4. The position of the Si~{\sc ii}\ and Ca~{\sc ii}\ lines is now
reproduced much better, and the absorption near 4700\AA\ is now modelled as
high-velocity Fe~{\sc ii}, demonstrating the plausibility of the hypothesis that
high-velocity components are responsible.
\begin{figure}
\includegraphics[width=89mm]{f4.eps}
\caption{Model for the Oct. 9 spectrum with increased abundances of Si,
Ca, and increased optical depth of Fe~{\sc ii}\ lines (thin line).}
\label{4}
\end{figure}
If high-velocity features are to be taken seriously it should be possible to
reproduce their effect on the spectra consistently at all epochs. Therefore, we
applied the enhancements to the abundances of Si, Ca, and the Fe~{\sc ii}\ opacity in
the velocity shells defined above, at all other epochs. The results are shown
in Fig.5, while blow-ups of the Si~{\sc ii}\ and Ca~{\sc ii}\ IR regions are shown in
Figures 6 and 7, respectively. Remarkably, the adopted high-velocity
distribution seems to give an excellent description of all observed
peculiarities. The Si~{\sc ii}\ line is still blue on day --7 ($\lambda \sim
6000$\AA), but in the next epoch, day --2, it has shifted by at least 150\AA\
to the red. This is due to the reduced opacity of the high-velocity Si region,
as the photosphere moves futher inwards. The line is then reproduced well at
all later epochs. As for Ca~{\sc ii}, in the two earliest epochs the introduction of
the high-abundance region leads only to a blueward shift of the absorption,
since $v(ph) \sim v(det)$. At later epochs, however, the Ca~{\sc ii}\ high-abundance
region becomes detached. Since the velocity separation of the high-abundance
region from the photosphere is much greater than the velocity separation of the
two strongest lines in the triplet, which is in turn larger than the velocity
width of the high-velocity region, the two narrow absorptions are formed. They
persist - at the correct wavelength - until the last epoch, when they disappear
as their optical depth becomes too small. As we remarked earlier, Ca~{\sc ii}\ H\&K
does not show distinctly detached features. However, the introduction of the
high-velocity component leads to significant line optical depth at high
velocity, causing the line to shift bluewards. This is the case for all the six
spectra modelled. Finally, the feature we tentatively attributed to Fe~{\sc ii}\ is
also reproduced very accurately in all except possibly the last epoch.
\begin{figure}
\includegraphics[width=89mm]{f5.eps}
\caption{Synthetic spectral series using the abundance distribution discussed
in Sect.4 (dotted lines).}
\label{5}
\end{figure}
\begin{figure}
\includegraphics[width=89mm]{f6.eps}
\caption{A blow-up of the Ca~{\sc ii}\ IR triplet region from the series of spectra
shown in Fig.6. The models are shown here as thin continuous lines to highlight
the line profiles.}
\label{6}
\end{figure}
\begin{figure}
\includegraphics[width=89mm]{f7.eps}
\caption{A blow-up of the Si~{\sc ii}\ line region from the series of spectra shown
in Fig.6. }
\label{7}
\end{figure}
The modified abundances imply that the regions involved are dominated by Si
($\sim 90$\% by mass), and have a high abundance of Ca ($\sim 10$\% by mass).
The Fe abundance can be low ($\sim 1$\% by mass), but the ionisation degree of
Fe must favour Fe~{\sc ii}\ over Fe~{\sc iii}. \citet{wang03} obtained rather similar
values for SN~2001el. These abundances would imply significant burning of the
outer layers of progenitor white dwarf (WD).
Burned material such as Si may be produced at very high (but not the highest)
velocities if the explosion mechanism was a delayed detonation
\citep[e.g.\ ][]{hof96,iwa99}. Alternatively, in the deflagration model, a thin He
envelope ($0.01 M_{\odot}$) could be located in the outermost part of the WD and be
burned by a precursor shock during the explosion; for a layer with densities as
low as $\sim 10^5$ and $\sim 10^6$\,g\,cm$^{-3}$, Si-rich and Ca-rich elements
are synthesized, respectively \citep{hash83,nom82b}.
An alternative possibility might be that the outermost shells have a higher
abundance of these species, reflecting the metallicity of the progenitor
\citep{Len00}. The required abundance ratios seem however to high for this
scenario: even in the most metal-rich situation \citet{Len00} consider, the
abundance of Ca would be much less than the 10\% by mass which is required in
our models.
Another relatively unexplored possibility is that He shell flashes during the
pre-SN evolution might produce elements such as Mg, Si, and Ca. The He flash is
stronger for slower accretion, becoming stronger as the white dwarf approaches
the Chandrasekhar mass \citep{nom82a}.
\section{A density enhancement?}
A different possibility to obtain high-velocity features is an overall increase
in density above what is predicted by W7. This may also help explaining the low
ionisation degree of Fe at high velocity. Therefore, in the next set of models,
we increased the density in a few shells at high velocity, leaving the original
W7 abundances unchanged.
In these models, there is much less freedom to change parameters to obtain a
good fit, since the test is to determine whether changing the density in a
number of shells can lead to all high-velocity features being reproduced
consistently, and to verify that models with this density change reproduce the
observed spectra at all epochs.
Since the high-velocity features appear at $v > 16000$\,km~s$^{-1}$, we increased the
density of the corresponding shells with respect to W7, without changing the
abundances. We selected a set of density changes which allowed us to get a good
match of the day --9 spectrum. In this model, shown in Fig. 8, the density was
increased by a factor 1.5 at $16750 < v < 20750$\,km~s$^{-1}$, by a factor 8 at
$20750 < v < 22500$\,km~s$^{-1}$, and by a factor 5 at $v > 22500$\,km~s$^{-1}$. The extra
mass contained in the `bump' is $0.1M_{\odot}$. This is a large increase of the
mass at the highest velocities. Model W7, in fact, has only $\sim 0.07M_{\odot}$
of material above 16750\,km~s$^{-1}$.
\begin{figure}
\includegraphics[width=89mm]{f8.eps}
\caption{Model for the Oct. 9 spectrum with increased density at high
velocity (thin line).}
\label{8}
\end{figure}
Increasing the density leads to much broader and bluer Ca~{\sc ii}\ and Si~{\sc ii}\ lines.
The spectrum is very well reproduced, but the 4700\AA\ feature is not, although
the ionisation of Fe is indeed reduced at the velocities where the density is
enhanced. The Ca~{\sc ii}\ IR triplet becomes broad, but it is still narrower than the
observed profile.
We used this modified density distribution to compute spectra at all observed
epochs. The results are shown in the sequence of Fig.9. The change in density
is able to explain both the sudden redward shift of the Si~{\sc ii}\ line and the
appearance of the narrow components in the Ca~{\sc ii}\ IR triplet. The fact that the
timing of the change is correctly reproduced confirms that both the position
and the amount of the modification are correctly estimated.
\begin{figure}
\includegraphics[width=89mm]{f9.eps}
\caption{Synthetic spectral series using the increased density discussed in
Sect.5 (dotted lines).}
\label{9}
\end{figure}
A weak high-velocity Fe~{\sc ii}\ absorption appears at 4700\AA\ in the spectra near
maximum light. An increase in the Fe abundance as well as in density seems to
be necessary to reproduce the feature as Fe~{\sc ii}, so we cannot confirm this
identification.
The quality of the synthetic spectra when compared to the observed ones suggests
that an increase in density is a possibility that must be considered
seriously. An overall variation of the density by this amount may occur if the
explosion is not spherically symmetric, so that parts of the ejecta may be
affected by burning differently from others. If these regions have sufficiently
large angular scale, they may give rise to the observed spectral peculiarities.
Polarisation measures in SN~2001el \citep{wang03} may support this
possibility.
\section{Interaction with a CSM?}
In the models in the previous section, the full width of the Ca~{\sc ii}\ IR triplet
in the earliest spactrum was not reproduced, even though the density was
significantly enhanced. One possibility would be to add even more mass at the
highest velocities. However, it is probably not reasonable to expect that the
deviation from the spherically symmetric density structure can be much larger
than what we have used. A different possibility to add mass at the highest
velocities is the accumulation of circumstellar material.
If the SN ejecta interact with a circumstellar environment, it is most likely
that the CSM composition is dominated by hydrogen. We tested different ways of
adding hydrogen in the spectrum on day --9, taking only thermal effects into
account. In each test, the limiting value of the H mass was constrained by the
strength of the synthetic H$\alpha$ line. H$\alpha$ is in fact not visible in
the observed spectra.
First, starting from the modified W7 density distribution that we derived in
the previous section, we introduced H uniformly in the ejecta at $v >
11250$~km~s$^{-1}$. This was done by simultaneously reducing the abundances of all
other elements. With this method we obtained an upper limit for the H mass of
$0.021 M_{\odot}$, corresponding to a H abundance of 4\% H by mass, which is
actually $\sim 50$\% by number. Although H$\alpha$ is produced at this point,
no changes are seen in either the Ca~{\sc ii}\ or the Si~{\sc ii}\ line profiles.
Then we assumed that only the outermost parts of the modified density structure
contains hydrogen, 50\% by mass, to simulate the piling up of CSM material, and
increased this H shell inwards. It is sufficient to introduce H above
25000\,km~s$^{-1}$\ to see a change in the synthetic Ca~{\sc ii}\ IR triplet. Although
H$\alpha$ is not seen, the presence of H has an indirect influence on the
spectra, making the Ca~{\sc ii}\ IR triplet significantly broader. The overall
spectrum is shown in Fig.10, and a blow-up of the Ca~{\sc ii}\ IR region is shown in
Fig. 11. The total H mass in this model is only $0.004 M_{\odot}$.
\begin{figure}
\includegraphics[width=89mm]{f10.eps}
\caption{Model for the Oct. 9 spectrum using the increased density of
Sect.5 but with an outer $0.004 M_{\odot}$ of Hydrogen (thin line). The model
without H shown in Fig.8 is also shown here as a dotted line for comparison.}
\label{10}
\end{figure}
\begin{figure}
\includegraphics[width=89mm]{f11.eps}
\caption{A blow-up of the Ca~{\sc ii}\ IR triplet region from Fig.10. }
\label{11}
\end{figure}
This can be explained as follows: since the electron density is significantly
increased when even a relatively small amount of H is added, recombination is
favoured. Since at the highest velocities Ca is mostly doubly ionised, once H
is introduced the fraction of Ca~{\sc ii}\ increases (by factors between 3 and 6 in
the zones affected for the particular model we used). This leads to an
increased strength of the Ca~{\sc ii}\ lines and to the observed broader absorption.
A second-order effect is also at play. A higher electron density means that
photons have a higher probability of scattering off electrons. This results in
a longer residence time of photons in the H-rich shells, and thus in turn in a
higher probability that photons can interact with spectral lines. The effect of
this is an increased line absorption - in all lines - at the highest
velocities, since that is where the electron density effect is at play. Since
the Ca~{\sc ii}\ lines are the strongest in the optical spectrum, they are also the
most affected. Electron scattering opacity is also responsible for the partial
suppression of the peak near 4000~\AA.
At later epochs, the photosphere becomes further removed from the region where
the hydrogen was added, and the spectra are therefore not affected: the results
are similar to those of Figure 9.
A necessary condition for hydrogen to affect the Ca~{\sc ii}\ IR triplet is that the
CSM and the SN ejecta are well mixed, at least in a narrow region between 25000
and 28000\,km~s$^{-1}$. The abundance of Ca in the CSM (taking solar as a typical
value) is in fact too small to give rise to the line opacity which is required
to reproduce the observed line broadening, so the line must be due to Ca from
the SN ejecta. Thorough mixing may not be easy to achieve, but what is required
here affects only a very small mass $(\sim 0.006M_{\odot})$, which may represent
the interface between SN ejecta and CSM. In our model, this region contains
$\sim 5$\% Ca by mass.
These calculations cannot by themselves prove that the spectral peculiarities
of SN~1999ee are -- at least partially -- due to an outer shell of H,
especially since the H mass we used is smaller than the mass added by modifying
the density profile (\citet{ger04} use $0.02 M_{\odot}$ of H-rich material to
broaden the Ca~{\sc ii}\ IR triplet in SN~2003du, but those spectra do not show a
broad Si~{\sc ii}\ line, and in any case the effect thay predict on that line is the
opposite, namely a narrowing of the line). However, our results suggest that
the presence of H-rich material can have far-reaching effects on the spectra,
even if the Balmer lines are not themselves visible. An accurate study of the
effect of H at high velocity will be the topic of future work.
\section{Discussion}
We have shown that a modification of the abundances or an increase of the mass
at the highest ejecta velocities can explain the high velocity features
observed in the spectrum of SN~1999ee. The former situation might result from
the nuclear burning reaching the outermost layers of the white dwarf. The outer
layers must contain mostly products of incomplete burning, in particular Si,
and some Fe. A change of the density structure could be due to either a
deviation, possibly not spherical, from the average properties of the
explosion, or to the accumulation of CSM material, or perhaps to both factors.
If the bump in density is due to SN material, it should contain $\sim
0.1\,M_{\odot}$ of material. This is a rather large change. However, it is quite
possible that what we can reproduce as a `density bump' in one-dimensional
models may actually be just a `density blob' in three dimensions. Attempts have
been made to model detached components as blobs in 3D \citep{kasen04}.
Unfortunately, spectropolarimetric data are not available for SN~1999ee, and so
the geometry cannot be constrained, resulting in a degeneracy of solutions.
Qualitatively, in order to reproduce the observed spectral signatures, any blob
must not be too small in size compared to the size of the photosphere. If one
such blob is observed because it happens to lie along our line-of-sight,
chances are that at least a few others are ejected as well. To support this, we
note that two separate sets of high-velocity features were observed in
SN~2000cx. It is a matter of probability to determine the optimal number and
size of such blobs so that they are only observed in a few SNe. Unfortunately,
the blobs become too thin to be visible as narrow emissions in the nebular
spectra, where a head count would be much easier since radiative transport is
not an issue. The question whether a global density enhancement or a blob are
to be preferred could be resolved with more spectropolarimetric observations.
Both SN~2001el \citep{wang03} and SN~2004dt show polarisation at early epochs,
which may support the blob hypothesis, as might the fact that most well-observed
SNe~Ia near maximum do not show high-velocity features. These seem to be a much
more common property of SNe~Ia at earlier epochs ($\sim 1$ week before maximum
or earlier, Mazzali et~al.\ , 2005, in preparation). Perhaps, while broad early
absorptions reaching high velocities may be the result of ejecta-CSM
interaction, narrow high-velocity absorptions near maximum could be the
signature of blobs.
If we assume that H-dominated CSM material is piled up at the highest
velocities, the enhanced electron density causes the ionisation degree to be
lower in those regions. Line broadening is then caused by the increased
optical depth of the Ca~{\sc ii}\ IR triplet.
In view of the mounting evidence that SNe~Ia can indeed interact with CSM (e.g.\
\citet{hamuy03,deng04,kotak04}). It will be interesting to verify whether
models where the only density enhancement is due to H-rich material can also
reproduce the high-velocity features.
\section*{Acknowledgments}
This work was partly supported by the European Research and Training Network
2002-2006 "The Physics of Type Ia Supernovae" (contract HPRN-CT-2002-00303),
and by the grant-in-Aid for Scientific Research (15204010, 16042201, 16540229)
and the 21st Century COE Program (Quests) of the MEXT, Japan. We thank the
anonymous referee for useful remarks that helped improving the presentation of
the paper.
|
{
"timestamp": "2004-11-19T13:52:13",
"yymm": "0411",
"arxiv_id": "astro-ph/0411566",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411566"
}
|
\section{Introduction}
ESA has decided to flight-test the technology that its Laser
Interferometer Space Antenna (LISA) will use to detect gravitational
waves (Irion 2002, Danzmann 2003, Hechler and Folkner 2003). The main
goal in this test is to find out whether non-gravitational
disturbances can be removed below the sensitivity limit by careful
design and a drag-free and attitude control system (DFACS). The DFACS
uses an inertial sensor to detect non-gravitational accelerations
acting on the spacecraft hull, and then compensates them using
micro-propulsion. The second mission in the frame of the ESA programme
for Small Missions for Advanced Research in Technology (SMART-2) was
initiated to perform the required space-borne tests. Here we describe
the mission design for SMART-2/LISA Pathfinder. As the interplanetary
target orbit for LISA requires substantial on-board communication
equipment, which is not suitable for a small mission, alternative
options for the operational orbit of SMART-2/LISA Pathfinder are
considered. The driving requirement for the selection of the
operatinal orbit is to find an environment, which does not exceed the
capabilities of the DFACS. This poses mainly requirements on the power
spectrum of spurious accelerations in the frequency band around
$1\:{\rm mHz}$, strength of gravity gradients, thermal variations, and
electro-static charging currents. In combination these requirements
essentially rule out geo-centric orbits.
\section{Selection of Operational Orbit}
The requirements for the operational orbit are:
\begin{enumerate}
\item differential gravity $<2.5\times 10^{-10}\:{\rm m}\:{\rm
s}^{-2}$
\item high thermal stability
\item maximum $\Delta v$ capacity: $3,170\:{\rm m}\:{\rm s}^{-1}$
\item full-year launch window
\item daily visibility from the ESA ground-station in Villafranca
$\geq 8\:{\rm hours}$
\end{enumerate}
\begin{figure}
\begin{center}
\epsfxsize=.5\hsize
\epsfbox[170 440 415 725]{zvc.ps}
\end{center}
\caption{\label{fig_zvc} Curves of zero velocity (ZVC) in the RC3P of
Sun, Earth, and spacecraft (not to scale). The near-Earth Lagrange
points ${\rm L}_1$ and ${\rm L}_2$ are located at the interstection of the ZVCs on
the line connecting Sun and Earth.}
\end{figure}
In order to achieve the required flatness of the gravity field
simultaneously with the thermal stability, an orbit far away from the
Earth must be found without violating the maximum $\Delta v$
capability. Libration orbits around the near-Earth Lagrange points of
the restricted circular three-body problem (RC3P) are located $\approx
1.5\times 10^6\:{\rm km}$ from Earth and can be reached without an
insertion manoeuvre (Farquhar 1973, G{\'o}mez et al 2001).
For LISA Pathfinder the Sun-ward Lagrange point ${\rm L}_1$\ is selected in
order to facilitate spacecraft design with the solar panel on one side
and the antenna on the other. The two types of free transfer (no
insertion) libration orbits around ${\rm L}_1$\ are distinguished by the
sequence whith which they arrive at the southernmost point, when the
daily coverage from a ground-station at northern geo-centric latitudes
is worst. For type-1 orbits this is the case $102\:{\rm days}$, and
for type-2 orbits $205\:{\rm days}$ after the escape manoeuvre.
\begin{figure}
\begin{center}
\epsfxsize=.45\hsize
\epsfbox{reference_ops_type1_rot.ps}
\epsfxsize=.45\hsize
\epsfbox{reference_ops_type2_rot.ps}
\end{center}
\caption{\label{fig_reference_ops} Three-dimensional representation of
the type-1 (left) and type-2 (right) trajectories after the escape
manoeuvre. The Earth-centred synodic (rotating) $x$-$y$-$z$ system is
chosen such that the $x$-$y$ plane is the ecliptic plane and the
$x$-axis points towards the Sun.}
\end{figure}
The relatively large libration orbits shown in
figure~\ref{fig_reference_ops} can be reached without insertion
manoeuvre and minimise the variation of the Sun-Earth-spacecraft
angle, which allows to install a fixed horn antenna with a $22^\circ$
beam width ($3\:{\rm dB}$) and thus reasonable gain. The antenna will
be installed such that it points with a cant angle of $20^\circ$ to
$30^\circ$ with respect to the main spacecraft axis, so that the Earth
will always be in the field of view of the antenna if the spacecraft
is rotated around its main axis (which is aligned with the Sun-Earth
line) by $2^\circ$ per day.
The downside of these orbits is the large out-of-plane amplitude,
which decreases the daily visibility from a ground-station on the
northern hemisphere of the Earth. As the $15\:{\rm m}$ antenna in
Villafranca ($40^\circ 26'\:{\rm N}$) will be used as the prime
station in the operational phase, type-1 orbits must be selected for
launches around the solstices and type-2 orbits for launches around
the equinoxes. Even when combining type-1 and type-2 orbits, the
launch window does not cover a full year, as illustrated by
figure~\ref{fig_vis}.
\begin{figure}
\begin{center}
\epsfxsize=.9\hsize
\epsfbox{s2vis_all.ps}
\caption{\label{fig_vis} Daily visibility of a spacecraft on type-1
and type-2 libration orbits as shown in figure~\ref{fig_reference_ops}
from the ESA ground stations in Villafranca, Perth, and Kourou as a
function of the date of the escape manoeuvre.}
\end{center}
\end{figure}
\section{Launch and Transfer from Low Earth Orbit}
As LISA Pathfinder will be launched by a small launcher into a
low-Earth orbit (LEO), the total $\Delta v$ to reach the parabolic
transfer is substantial ($>3\:{\rm km}\:{\rm s}^{-1}$). With a small
chemical propulsion module (thrust $400\:{\rm N}$), the escape
manoeuvre must be split in order to avoid an excessive burn duration and
thus gravity loss.
\begin{figure}
\begin{center}
\epsfxsize=.5\hsize
\epsfbox{sequence_3D.ps}
\end{center}
\caption{\label{fig_sequence_3D} Sequence of orbits after release into
a $900\times 200\:{\rm km}$ LEO by the Launcher. The propulsion module
raises the apogee in $10$ steps to $66,000\:{\rm km}$, before the
eleventh burn puts the spacecraft on an escape trajectory towards
${\rm L}_1$.}
\end{figure}
For the initial phase of the transfer the visibility by
ground-stations is critical, as shown in
figures~\ref{fig_groundtrack0} and~\ref{fig_groundtrack1}. Only short
passages over the stations occur. The sequence illustrated in figure 4
is optimised so that each burn targets for an apogee pass over a
ground station in Villafranca, Kourou, or Perth. During the pass
ranging and Doppler measurements are taken by analysing the received
signal repeated by the on-board transponder. The accuracy of a single
measurement is $3\:{\rm mm}\:{\rm s}^{-1}$ in Doppler and less than
$1\:{\rm km}$ in ranging. This first pass is to be followed by a
second one, where more orbit determination measurements are
performed. The second pass is needed in order to achieve the required
precision of orbit information before the size of the next manoeuvre
is calculated. The total time allocated for the data processing, orbit
calculation, planning of the next manoeuvre, and preparation of the
commands is 8 hours. After this period, the next pass over one of the
ground stations is used to uplink the commands for manoeuvre execution
during the next perigee. This sequence is repeated for all
apogee-raise manoeuvres during the early transfer phase.
\begin{figure}
\begin{center}
\epsfxsize=.8\hsize
\epsfbox{phase0_gt.ps}
\end{center}
\caption{\label{fig_groundtrack0} Ground-track of the first
orbit of the LISA Pathfinder mission. The coverage circles show the
area where the spacecraft is visible to the ground-station in
Kourou (K) and Perth (P). Assuming a launch in
northern Russia, the powered phase of the launch (shown in red) ends
over China. Then, the spacecraft follows a track that leads it to the
injection into the $900\times 200\:{\rm km}$ orbit, which is achieved
by the launcher upper stage (left).}
\end{figure}
\begin{figure}
\begin{center}
\epsfxsize=.8\hsize
\epsfbox{phase1_gt.ps}
\end{center}
\caption{\label{fig_groundtrack1} Ground-track of the second and third
orbits of the LISA Pathfinder mission. The coverage circles show the
area where the spacecraft is visible to the ground-station in
Villafranca (V), Kourou (K) and Perth (P). First acquisition of the
spacecraft after injection by a ground-station will be in Perth.}
\end{figure}
\section{Summary and Discussion}
The challenging mission design for LISA Pathfinder is in an advanced
state, where optimised reference trajectories are established, and
transfer strategies are defined. The environmental requirements set by
the technology demonstration payloads can all be met by placing the
spacecraft in a large amplitude Lissajous orbit around the first
co-linear Lagrange point of the Sun-Earth system. There, the solar
illumination is constant in magnitude as well as direction, minimising
the variability of spurious accelerations and the thermal
equilibrium.
The transfer sequence puts the spacecraft from an initially low orbit
onto the escape parabola, which leads to a free transfer towards the
large amplitude Lissajous orbit, without violating visibility
constraints from the ESA ground-stations. In theory, the propellant
allocation for the transfer could be reduced by using a lunar swing-by
technique. The saving in $\Delta v$ is however only $50\:{\rm m}\:{\rm
s}^{-1}$, which is negligible compared to the total $\Delta v$ budget
and thus does not justify the increase in operational risk and
complexity.
The current mission design presented above demonstrates how low-cost
missions can be implemented using advanced astrodynamic methods in
ordert to fulfill demanding science requirements under strong
constraints in the total payload mass and launcher performance. With
the LISA Pathfinder mission implemented in this way, the first step
towards detecting gravitational waves in the mHz regime will be taken.
\References
\item[] Danzmann K 2003 {\it Adv. Space Res.} {\bf 32} 1233
\item[] Farquhar R W 1973 {\it Celest. Mech.} {\bf 7} 458
\item[] G{\'o}mez, Jorba {\`A}, Masdemont J, and Sim{\'o} C 2001 {\it
Dynamics and Mission Design Near Libration Points}
\item[] Hechler F and Folkner W M 2003 {\it Adv. Space Res.} {\bf 32} 1277
\item[] Irion R 2002 {\it Science} {\bf 297} 1113
\endrefs
\end{document}
|
{
"timestamp": "2004-11-15T08:59:54",
"yymm": "0411",
"arxiv_id": "gr-qc/0411071",
"language": "en",
"url": "https://arxiv.org/abs/gr-qc/0411071"
}
|
\section{Introduction}
Given $x,y\in \bits n$ one way to measure how much they
differ is the Hamming distance.
\begin{definition}
If $x,y\in \bits n$ then
${\rm HAM}(x,y)$ is the number of bits on which $x$ and $y$ differ.
\end{definition}
If Alice has $x$ and Bob has $y$ then how many bits
do they need to communicate such that they both know ${\rm HAM}(x,y)$?
The trivial algorithm is to have Alice send $x$ (which takes $n$ bits)
and have Bob send ${\rm HAM}(x,y)$ (which takes $\ceil{\lg (n+1)}$ bits)
back to Alice. This takes $n+\ceil{\lg (n+1)}$ bits.
Pang and El Gamal~\cite{pang}
showed that this is essentially optimal.
In particular they showed that ${\rm HAM}$ requires
at least $n +\lg(n+1-\sqrt n)$ bits to be communicated.
(See~\cite{abdel,commdoc,metzner,orlit} for more on
the communication complexity of ${\rm HAM}$.
See~\cite{securemulti} for how Alice and Bob can approximate
${\rm HAM}$ without giving away too much information.)
What if Alice and Bob just want to know if ${\rm HAM}(x,y)\le a$?
\begin{definition}
Let $n\in{\sf N}$. Let $a$ be such that $0\le a\le n-1$.
$HAM_n^{(a)}:\bits n \times \bits n \rightarrow \{0,1\}$ is the function
$$HAM_n^{(a)}(x,y) = \cases { 1 & if ${\rm HAM}(x,y)\le a$\cr
0 & otherwise.\cr
}
$$
\end{definition}
The problem $HAM_n^{(a)}$ has been studied by
Yao~\cite{qfingerprinting} and Gavinsky et al~\cite{qpublic}.
Yao showed that there is an $O(a^2)$ public coin simultaneous
protocol for $HAM_n^{(a)}$ which yields (by Newman~\cite{Newman}, see also \cite{commcomp}) an
$O(a^2+\log n)$ private coin protocol and also
an $O(2^{a^2}\log n)$ quantum simulataneous message protocol with bounded error~\cite{qfingerprinting}.
Gavinsky et al. give an $O(a\log n)$ public coin
simultaneous protocol, which yields an $O(a\log n)$ private coin protocol.
For $a \gg \log n$ this is better than Yao's protocol.
All of the protocols mentioned have a small probability of error.
How much communication is needed for this problem if we demand no error?
There is, of course, the trivial $(n+1)$-bit protocol.
Is there a better one?
In this paper we show the following; in the list of results below, the
``$c$'' (in the ``$c \sqrt{n}$'' terms) is some positive absolute
constant.
\begin{enumerate}
\item\label{det:n-2}
For any $0 \leq a \leq n-1$, $HAM_n^{(a)}$ requires at least $n-2$ bits
in the deterministic model.
\item\label{det:n}
For $a\le c \sqrt{n}$, $HAM_n^{(a)}$ requires at least $n$ bits
in the deterministic model.
\item\label{quant:n-2}
For any $0 \leq a \leq n-1$, $HAM_n^{(a)}$ requires at least $n-2$ bits
in the quantum model with Alice and Bob share an infinite number of EPR pairs,
using a classical channel, and always obtain the correct answer.
\item
For $a\le c \sqrt{n}$, $HAM_n^{(a)}$ requires at least $n$ bits
in the quantum model in item \ref{quant:n-2}.
\item\label{quant:n/2minus1}
For any $0 \leq a \leq n-1$, $HAM_n^{(a)}$ requires at least $\frac{n}{2}-1$ bits
in the quantum model with Alice and Bob share an infinite number of EPR pairs,
using a quantum channel, and always obtain the correct answer.
\item\label{quant:n/2}
For $a\le c \sqrt{n}$, $HAM_n^{(a)}$ requires at least $n/2$ bits
in the quantum model in item \ref{quant:n/2minus1}.
\end{enumerate}
Note that if $a=n$ then $(\forall x,y)[HAM_n^{(a)}(x,y)=1$,
hence we do not include that case.
What if Alice and Bob need to determine if ${\rm HAM}(x,y)=a$ or not?
\begin{definition}
Let $n\in{\sf N}$. Let $a$ be such that $0\le a\le n$.
$HAM_n^{(=a)}:\bits n \times \bits n \rightarrow \{0,1\}$ is the function
$$HAM_n^{(=a)}(x,y) = \cases { 1 & if ${\rm HAM}(x,y)\le a$\cr
0 & otherwise.\cr
}
$$
\end{definition}
We show the exact same results for $HAM_n^{(=a)}$ as we do for $HAM_n^{(a)}$.
There is one minor difference: for $HAM_n^{(a)}$ the $a=n$ case had complexity 0
since all pairs of strings differ on at most $n$ bits;
however, for $HAM_n^{(=a)}$ the $a=n$ case has complexity $n+1$ as it is
equivalent to equality.
All our results use the known ``log rank'' lower bounds on
classical and quantum communication complexity: Lemmas~\ref{le:rank}
and \ref{le:qrank}. Our approach is to lower-bound the ranks of the
appropriate matrices, and then to invoke these known lower bounds.
\section{Definitions, Notations, and Useful Lemmas}
We give brief definitions of both classical and quantum communication
complexity. See~\cite{commcomp} for more details on classical,
and~\cite{qsurvey} for more details on quantum.
\begin{definition}
Let $f$ be any function from $\bits n \times \bits n$ to $\{0,1\} $.
\begin{enumerate}
\item
A \textit{protocol } for computing $f(x,y)$, where Alice has $x$ and Bob has $y$,
is defined in the usual way (formally using decision trees).
At the end of the protocol both Alice and Bob know $f(x,y)$.
\item
$D(f)$ is the number of bits transmitted in the optimal deterministic protocol for $f$.
\item
$Q^*(f)$ is the number of bits transmitted in the optimal quantum protocol where
we allow Alice and Bob to share an infinite number of EPR pairs
and communicate over a quantum channel.
\item
$C^*(f)$ is the number of bits transmitted in the optimal quantum protocol where
we allow Alice and Bob to share an infinite number of EPR pairs
and communicate over a classical channel.
\item
$M_f$ is the $2^n \times 2^n$ matrix where the rows and columns are indexed
by $\bits n$ and the $(x,y)$-entry is $f(x,y)$.
\end{enumerate}
\end{definition}
Let $\lg$ denote the logarithm to the base two.
Also, as usual, if $x < y$, then ${x \choose y}$ is taken to be zero.
The following theorem is due to
Mehlhorn and Schmidt~\cite{ranklower};
see also \cite{commcomp}.
\begin{lemma}\label{le:rank}
If $f:\bits n \times \bits n \rightarrow \{0,1\} $
then
$D(f)\ge \lg({\rm rank}(M_f))$.
\end{lemma}
Buhrman and de Wolf~\cite{qlogrank} proved a
similar theorem for quantum communication complexity.
\begin{lemma}\label{le:qrank}
If $f:\bits n \times \bits n \rightarrow \{0,1\} $
then the following hold.
\begin{enumerate}
\item
$Q^*(f)\ge \frac{1}{2}\lg({\rm rank}(M_f))$.
\item
$C^*(f)\ge \lg({\rm rank}(M_f))$.
\end{enumerate}
\end{lemma}
\section{The Complexity $HAM_n^{(a)}$ for $a\le O(\sqrt n)$}\label{se:hamasq}
We start by presenting results for general $a$, and then specialize
to the case where $a \leq c \sqrt{n}$.
\begin{definition}
Let $M_a$ be $M_{HAM_n^{(a)}}$, the $2^n\times 2^n$ matrix representing $HAM_n^{(a)}$.
\end{definition}
\begin{lemma}
$M_a$ has $2^n$ orthogonal eigenvectors.
\end{lemma}
\begin{proof}
This follows from $M_a$ being symmetric.
\end{proof}
We know that $M_a$ has $2^n$ eigenvalues; however, some of them may be 0.
We prove that $M_a$ has few 0-eigenvalues.
This leads to a lower bound on $D(HAM_n^{(a)})$ by Lemma~\ref{le:rank}.
\begin{definition}\label{de:vz}
Let $z\in \bits n$.
\begin{enumerate}
\item
$v_z \in R^{2^n}$ is defined by, for all $x\in \bits n$, $v_z(x) = (-1)^{\sum_i x_i z_i }$. The entries $v_z(x)$ of $v_z$ are ordered in the natural
way: in the same order as the order of the index $x$
in the rows (and columns) of $M_a$.
\item
We show that $v_z$ is an eigenvector of $M_a$. Once that is done
we let $eig(z)$ be the eigenvalue of $M_a$ associated with $v_z$.
\end{enumerate}
\end{definition}
\begin{lemma}\label{le:main}~
\begin{enumerate}
\item The vectors $\{v_z: ~z\in \bits n\}$ are orthogonal.
\item
For all $z\in \bits n$, $v_z$ is an eigenvector of $M_a$.
\item
If $z$ has exactly $m$ 1's in it,
then
$$eig(z)=\sum_{j=0}^a \ \ \sum_{k=\max\{ 0,j+m-n \} }^{\min\{j,m\}}\ \ {m\choose k} {n-m \choose j-k} (-1)^k.$$
\end{enumerate}
\end{lemma}
\begin{proof}
The first assertion (orthogonality) follows by simple counting.
We now prove the final two assertions together.
Let $z\in \bits n$ have exactly $m$ ones in it.
Fix a row in $M_a$ that is indexed by $x\in \bits n$.
Denote this row by $R_x$.
We need the following notation:
\[
\begin{array}{rl}
L_a = & \{ y \mid {\rm HAM}(x,y) \le a \}\cr
E_j = & \{ y \mid {\rm HAM}(x,y) = j \}\cr
\end{array}
\]
We will show that $R_x \cdot v_z$
is a constant multiple (independent of $x$) times $v_z(x)$.
Now,
$$R_x\cdot v_z = \sum_{y\in \bits n} HAM_n^{(a)}(x,y) v_z(y) = \sum_{y\in L_a} v_z(y) = \sum_{y\in L_a} (-1)^{\sum_i y_iz_i }.$$
We would like to have this equal $b \times v_z(x)$ for some constant $b$.
We set it equal to $b\times v_z(x)$ and deduce what $b$ works.
So, suppose
$$b \times v_z(x) = \sum_{y\in L_a} (-1)^{\sum_i y_iz_i }.$$
We have
\begin{eqnarray}
b & = & \frac{1}{v_z(x)} \sum_{y\in L_a} (-1)^{\sum_i y_iz_i } \nonumber \\
& = & v_z(x) \sum_{y\in L_a} (-1)^{\sum_i y_iz_i} \nonumber \\
& = & (-1)^{\sum_i x_iz_i} \sum_{y\in L_a} (-1)^{\sum_i y_iz_i } \hbox{\ \ \ (by the definition of $v_z(x)$)} \nonumber \\
& = & \sum_{y\in L_a} (-1)^{\sum_i (x_i+y_i)z_i} \nonumber \\
& = & \sum_{y\in L_a} (-1)^{\sum_i |x_i-y_i|z_i } \hbox{\ \ \ (since $x_i+y_i \equiv |x_i-y_i| \pmod 2$) } \nonumber \\
& = & \sum_{j=0}^a \sum_{y\in E_j} (-1)^{\sum_i |x_i-y_i|z_i } \hbox{\ \ \ (since $L_a = \bigcup_{j=0}^a E_j$)}. \label{eqn:b}
\end{eqnarray}
We partition $E_j$.
If $y\in E_j$ then
$x$ and $y$ differ in exactly $j$ places.
Some of those places $i$ are such that $z_i=1$.
Let $k$ be such that the number of places where $x_i\ne y_i$ and $z_i=1$.
\begin{description}
\item{Upper Bound on $k$:}
Since there are exactly $m$ places where $z_i=1$ we have $k\le m$.
Since there are exactly $j$ places where $x_i\ne y_i$ we have $k\le j$.
Hence $k\le\min\{j,m\}$.
\item{Lower Bound on $k$:}
Since there are exactly $n-m$ places where $z_i=0$, we have $j-k\le n-m$.
Hence $k\ge \max\{0,j+m-n\}$.
\end{description}
In summary, the only relevant $k$ are
$\max\{0,j+m-n\} \le k \le \min\{j,m\}$.
Fix $j$. For
\noindent
$\max\{0,j+m-n\}\le k\le \min\{j,m\}$,
let $D_{j,k}$ be defined as follows:
$$D_{j,k} = \{ y \mid ((y \in E_j) \wedge
(\hbox{on exactly $k$ of the coordinates
where $x_i\ne y_i$, we have $z_i = 1$})) \}.$$
Note that
$$E_j = \bigcup_{k=0}^{\min\{j,m\}} D_{j,k}$$
and $|D_{j,k}|= {m\choose k} {n-m \choose j-k}$. So, by (\ref{eqn:b}),
$$
b = \sum_{j=0}^a \sum_{y\in E_j} (-1)^{\sum_i |x_i-y_i|z_i }
= \sum_{j=0}^a \ \ \sum_{k=\max\{ 0,j+m-n \} }^{\min\{j,m\}}\ \ \sum_{y\in D_{j,k}} (-1)^{\sum_i |x_i-y_i|z_i }.
$$
By the definition of $D_{j,k}$ we know that for exactly $k$ of the values of $i$
we have both $|x_i-y_i|=1$ and $z_i=1$. On all other values one of the two
quantities is 0. Hence we have the following:
\begin{eqnarray*}
b & = & \sum_{j=0}^a \ \ \sum_{k=\max\{ 0,j+m-n \} }^{\min\{j,m\}}\ \ \sum_{y\in D_{j,k}} (-1)^k \\
& = & \sum_{j=0}^a \ \ \sum_{k=\max\{ 0,j+m-n \} }^{\min\{j,m\}}\ \ |D_{j,k}| (-1)^k \\
& = & \sum_{j=0}^a \ \ \sum_{k=\max\{ 0,j+m-n \} }^{\min\{j,m\}}\ \ {m\choose k} {n-m \choose j-k} (-1)^k.
\end{eqnarray*}
Notice that $b$ is independent of $x$ and is of the form required.
\end{proof}
\begin{definition}
Let
$$F(a,n,m)= \sum_{j=0}^a \ \ \sum_{k=\max\{ 0,j+m-n \} }^{\min\{j,m\}}\ \ {m\choose k} {n-m \choose j-k} (-1)^k.$$
\end{definition}
The following lemma will be used in this section to obtain a lower bound
when $a=O(\sqrt n)$, and in Section~\ref{se:gen} to obtain a lower bound for general $a$.
\begin{lemma}\label{le:uslea}~
\begin{enumerate}
\item
$D(HAM_n^{(a)}) \ge \lg \sum_{m: F(a,n,m)\ne 0 } { n \choose m }.$
\item
$Q^*(HAM_n^{(a)}) \ge \frac{1}{2}\lg \sum_{m: F(a,n,m)\ne 0 } { n \choose m }.$
\item
$C^*(HAM_n^{(a)}) \ge \lg \sum_{m: F(a,n,m)\ne 0 } { n \choose m }.$
\end{enumerate}
\end{lemma}
\begin{proof}
By Lemma~\ref{le:main}, the eigenvector $v_z$ has a nonzero eigenvalue
if $v_z$ has $m$ 1's and $F(a,n,m)\ne0$.
The rank of $M_a$ is the number of nonzero eigenvalues that correspond
to linearly independent eigenvectors.
This is $\sum_{m: F(a,n,m)\ne 0 } { n \choose m }.$
The theorem follows from Lemmas~\ref{le:rank} and \ref{le:qrank}.
\end{proof}
\begin{lemma}\label{le:lea}
The number of values of $m$ for which $F(a,n,m)=0$ is
$\le a$.
\end{lemma}
\begin{proof}
View the double summation $F(a,n,m)$
as a polynomial in $m$. The $j$th summand has degree $k+(j-k)=j$.
Since $j\le a$ the entire sum can be written as a polynomial in
$m$ of degree $a$. This has at most $a$ roots.
\end{proof}
\begin{theorem}\label{th:main}
There is a constant $c > 0$ such that if $a\le c\sqrt n$
then the following hold.
\begin{enumerate}
\item
$D(HAM_n^{(a)}) \ge n$.
\item
$Q^*(HAM_n^{(a)}) \ge n/2$.
\item
$C^*(HAM_n^{(a)})\ge n$.
\end{enumerate}
\end{theorem}
\begin{proof}
By Lemma~\ref{le:uslea}
$D(f),Q^*(f) \ge \lg (\sum_{m: F(a,n,m)\ne 0 } { n \choose m })$ and
$C^*(f) \ge \frac{1}{2}\lg (\sum_{m: F(a,n,m)\ne 0 } { n \choose m })$.
Note that
$$2^n = \sum_{m: F(a,n,m)\ne 0 } { n \choose m } +\sum_{m: F(a,n,m)=0 } { n\choose m}.$$
By Lemma~\ref{le:lea} $|\{ m : F(a,n,m)=0 \}|\le a$. Hence,
$$\sum_{m: F(a,n,m)=0 } { n\choose m} \le |\{ m : F(a,n,m)=0 \}|\cdot\max_{0\le m\le n} {n\choose m}
\le a{ n \choose {n/2} } \le \frac{a2^n}{\sqrt n}.$$
So, if $a\le \frac{1}{4}\sqrt n$, then
$$\sum_{m: F(a,n,m)\ne 0 } { n \choose m } \ge 2^n - \frac{a2^n}{\sqrt n}
\geq 2^n - 2^{n-2}.$$
Hence,
$$\lg \left(\sum_{m: F(a,n,m)\ne 0 } { n \choose m }\right) \ge
\lg(2^n - 2^{n-2}); ~~
\mathrm{i.e.}, ~
\left\lceil \lg \left(\sum_{m: F(a,n,m)\ne 0 } { n \choose m }\right)
\right\rceil \ge n.$$
\end{proof}
\section{The Complexity of $HAM_n^{(=a)}$ for $a\le O(\sqrt{n})$}\label{se:hameasq}
We again start by deducing results for general $a$, and then specialize
to the case where $a \leq c \sqrt{n}$.
\begin{definition}
Let $M_{=a}$ be $M_{HAM_n^{(=a)}}$, the $2^n\times 2^n$ matrix representing $HAM_n^{(=a)}$.
\end{definition}
The vectors $v_z$ are the same ones defined in Definition~\ref{de:vz}.
We show that $v_z$ is an eigenvector of $M$. Once that is done
we let $eig(z)$ be the eigenvalue of $M$ associated to $z$.
The lemmas needed, and the final theorem,
are very similar (in fact easier) to those
in the prior section. Hence we just
state the needed lemmas and final theorem.
\begin{lemma}\label{le:maina}~
\begin{enumerate}
\item
For all $z\in \bits n$ $v_z$ is an eigenvector of $M_{=a}$.
\item
If $z$ has exactly $m$ 1's in it
then
$$eig(z)=\ \ \sum_{k=\max\{ 0,a+m-n \} }^{\min\{a,m\}}\ \ {m\choose k} {n-m \choose a-k} (-1)^k.$$
\end{enumerate}
\end{lemma}
\begin{definition}
\label{defn:f}
$$f(a,n,m)= \ \ \sum_{k=\max\{ 0,a+m-n \} }^{\min\{a,m\}}\ \ {m\choose k} {n-m \choose a-k} (-1)^k.$$
\end{definition}
Note, from our convention that
``if $x < y$, then ${x \choose y}$ is taken to be zero'', that
we can also write
\[ f(a,n,m)= \sum_{k=0}^{a} {m\choose k} {n-m \choose a-k} (-1)^k. \]
The following lemma will be used in this section to obtain a lower bound
when $a=O(\sqrt n)$, and in Section~\ref{se:gen} to obtain a lower bound for general $a$.
\begin{lemma}\label{le:useqa}~
\begin{enumerate}
\item
$D(HAM_n^{(=a)}) \ge \lg \sum_{m: f(a,n,m)\ne 0 } { n \choose m }.$
\item
$Q^*(HAM_n^{(=a)}) \ge \lg \sum_{m: f(a,n,m)\ne 0 } { n \choose m }.$
\item
$C^*(HAM_n^{(=a)}) \ge \frac{1}{2} \cdot \lg \sum_{m: f(a,n,m)\ne 0 } { n \choose m }.$
\end{enumerate}
\end{lemma}
\begin{lemma}\label{le:ea}
The number of values of $m$ for which $f(a,n,m)=0$ is
$\le a$.
\end{lemma}
\begin{theorem}\label{th:maine}
There is a constant $c > 0$ such that if $a\le c\sqrt n$
then the following hold.
\begin{enumerate}
\item
$D(HAM_n^{(=a)}) \ge n$.
\item
$Q^*(HAM_n^{(=a)}) \ge n/2$.
\item
$C^*(HAM_n^{(=a)})\ge n$.
\end{enumerate}
\end{theorem}
\section{The Complexity of $HAM_n^{(a)}$ and $HAM_n^{(=a)}$ for General $a$}\label{se:gen}
We now consider the case of general $a$. As above,
we will show that $F(a,m,n)$ and $f(a,m,n)$ are nonzero for many values of
$m$. This will imply that the matrices $M_a$
and $M_{=a}$ have high rank, hence $HAM_n^{(a)}$ and $HAM_n^{(=a)}$
have high communication complexity.
We will use
general generating-function methods to derive facts about these sums. A good
source on generating functions is~\cite{wilfgen}.
One of our main
results will be Lemma~\ref{le:singlesum}, which states that if
$0 \leq a \leq m < n$, then ``$f(a,m,n)=0$'' implies
``$f(a,m+1,n) \neq 0$''.
The idea behind our proof of Lemma~\ref{le:singlesum} will
be the following: we will show a relationship
between the sum $f(a,m,n)$ and a certain new sum $h(a,m,n)$. Then we will
derive generating functions for $f$ and $h$, and translate this relationship
into a relation between their generating functions. Finally, we will show
that this relation cannot hold under the assumption that
$f(a,m,n)=f(a,m+1,n)=0$, thus reaching a contradiction. Some auxiliary
results needed for this are now developed in Section~\ref{sec:aux}.
\subsection{Auxiliary Notation and Results}
\label{sec:aux}
\begin{notation}
$[x^b]g(x)$ is the coefficient of $x^b$ in the power series
expansion of $g(x)$ around $x_0=0$.
\end{notation}
\begin{notation}
$t^{(i)}(x)$ is the $i$'th derivative of $t(x)$.
\end{notation}
We will make use of the following lemma, which follows by an easy induction
on $i$:
\begin{lemma}
\label{lem:deriv}
Let $t(x)$ be an infinitely differentiable function.
Let $T_1(x)=(x-1)t(x)$, and
\noindent
$T_2(x)=(x+1)t(x)$. Then for any $i\geq 1$:\\
$T_1^{(i)}(x)=(x-1)t^{(i)} + i\cdot t^{(i-1)}(x)$\\
$T_2^{(i)}(x)=(x+1)t^{(i)} + i\cdot t^{(i-1)}(x)$
\end{lemma}
For the rest of Section~\ref{sec:aux}, the integers $a, m, n$ are
arbitrary subject to the constraint $0 \leq a \leq m \leq n$,
unless specified otherwise.
\begin{definition}~
\begin{enumerate}
\item
$h(a,m,n)=\sum_{i=0}^{a} {m \choose i}{n-m \choose a-i}\frac{(-1)^i}{m-i+1}$.
\item
$g(x)=\frac{x^{m+1} - (x-1)^{m+1}}{m+1}\cdot (x+1)^{n-m}$.
\end{enumerate}
\end{definition}
We will show an interesting connection between $h$ and $f$.
\begin{claim}\label{cl:one}
Suppose $f(a,m,n)=0$. Then $f(a,m+1,n)=0$ iff $h(a,m,n)=0$.
\end{claim}
\begin{proof}
\[
\begin{array}{rl}
f(a,m+1,n)=&\sum_{i=0}^{a} {m+1 \choose i}{n-m-1 \choose a-i}(-1)^i\cr
=&\frac{m+1}{n-m}\sum_{i=0}^{a} {m \choose i}{n-m \choose a-i}(-1)^i
\cdot \frac{n-m-a+i}{m-i+1}\cr
=&\frac{m+1}{n-m}((n+1-a)\sum_{i=0}^{a} {m \choose i}{n-m \choose a-i}\frac{(-1)^i}{m-i+1})-\sum_{i=0}^{a} {m \choose i}{n-m \choose a-i}(-1)^i)\cr
=&\frac{m+1}{n-m}((n+1-a)h(a,m,n)-f(a,m,n))\cr
\end{array}
\]
Thus, if $f(a,m,n)=0$,
then $f(a,m+1,n)=0$ iff $h(a,m,n)=0$.
\end{proof}
We next show a connection between $g(x)$ and $h$.
\begin{claim}\label{cl:two}
$h(a,m,n)=(-1)^m\cdot [x^a]g(x)$.
\end{claim}
\begin{proof}
\[
\begin{array}{rl}
g(x)=&\frac{x^{m+1} - (x-1)^{m+1}}{m+1}\cdot (x+1)^{n-m}\cr
=&\frac{x^{m+1} - \sum_{i=0}^{m+1}{m+1 \choose i}x^i(-1)^{m+1-i}}{m+1}\cdot (x+1)^{n-m}\cr
=&(-1)^m\sum_{i=0}^{m}{m \choose i}x^i\frac{(-1)^i}{m+1-i}\cdot (x+1)^{n-m}\cr
=&(-1)^m\sum_{i=0}^{m}{m \choose i}x^i\frac{(-1)^i}{m+1-i}\cdot\sum_{j=0}^{n-m}{n-m \choose j}x^j\cr
\end{array}
\]
Therefore, $h(a,m,n)=(-1)^m\cdot [x^a]g(x)$.
\end{proof}
Next, define an auxiliary function $\phi(u,v,w)$ as the $w$'th derivative
of the function $(x+1)^u(x-1)^v$ evaluated at $x=0$. We now relate
$\phi$ and $h$.
\begin{claim}\label{cl:three}
$h(a,m,n)=0$ iff $\phi(n-m, m+1, a) = 0$.
\end{claim}
\begin{proof}
By Claim~\ref{cl:two}
\[
\begin{array}{rl}
h(a,m,n)=&(-1)^m\cdot [x^a]g(x)\cr
=&\frac{(-1)^m}{m+1}([x^a](x^{m+1}\cdot (x+1)^{n-m})-[x^a]((x-1)^{m+1}\cdot (x+1)^{n-m})).\cr
\end{array}
\]
But $[x^a](x^{m+1}\cdot (x+1)^{n-m})=0$, since $a<m+1$. So
\[
\begin{array}{rl}
h(a,m,n)=&\frac{(-1)^{m+1}}{m+1}[x^a]((x-1)^{m+1}\cdot (x+1)^{n-m})\cr
=&\frac{(-1)^{m+1}}{m+1}\cdot\frac{\phi(n-m, m+1, a)}{a!}.\cr
\end{array}
\]
Thus, $h(a,m,n)=0$ iff $\phi(n-m, m+1, a) = 0$.
\end{proof}
Now we can relate the zeroes of $f$ with those of $\phi$:
\begin{claim}\label{cl:four}
$f(a,m,n)=0$ iff $\phi(n-m, m, a)=0$.
\end{claim}
\begin{proof}
\[
\begin{array}{rl}
(x-1)^m(x+1)^{n-m}=&\sum_{i=0}^m {m \choose i}x^i(-1)^{m-i} \cdot \sum_{j=0}^{n-m}{n-m \choose j}x^j\cr
=&(-1)^m\sum_{i=0}^m {m \choose i}x^i(-1)^i \cdot \sum_{j=0}^{n-m}{n-m \choose j}x^j\cr
=&(-1)^m\sum_{b=0}^{n}\sum_{k=0}^b {m \choose k}{n-m \choose b-k}(-1)^k x^b\cr
=&(-1)^m\sum_{b=0}^{n} f(b,m,n)\cdot x^b.\cr
\end{array}
\]
So $f(a,m,n)=\frac{(-1)^m}{a!}\cdot\phi(n-m, m, a)$, thus
$f(a,m,n)=0$ iff $\phi(n-m, m, a)=0$.
\end{proof}
\begin{claim}\label{cl:five}
Suppose $m < n$ and
$\phi(n-m, m, a)=0$. Then
$$\phi(n-m-1, m+1, a)=0 \hbox{ iff } \phi(n-m, m+1, a)=0.$$
\end{claim}
\begin{proof}
This claim follows from
Claims~\ref{cl:one}, \ref{cl:three}, and \ref{cl:four}.
\end{proof}
\medskip
We are now able to prove a recursive relation between values of $\phi$:
\begin{claim}\label{cl:six}
If $k>0$, $a > 0$, and $\phi(k, m, a)=\phi(k,m,a-1)=0$, then
\noindent
$\phi(k-1, m, a)=\phi(k-1,m,a-1)=0$.
\end{claim}
\begin{proof}
Suppose $\phi(k, m, a)=\phi(k,m,a-1)=0$. By Lemma~\ref{lem:deriv},
\begin{equation}
\label{eqn:phi-k-m+1}
\phi(k,m+1,a)=-\phi(k,m,a)+a\cdot\phi(k,m,a-1)=0.
\end{equation}
By Claim~\ref{cl:five}, since $\phi(k, m, a)=0$,
we know that
$$\phi(k-1,m+1,a)=0 \hbox{ iff }\phi(k,m+1,a)=0.$$
Now, (\ref{eqn:phi-k-m+1}) yields
$\phi(k-1,m+1,a)=0$. Applying Lemma~\ref{lem:deriv} again, we obtain:
\[
\begin{array}{rl}
0 = & \phi(k-1,m+1,a)=-\phi(k-1,m,a)+ a \cdot\phi(k-1,m,a-1); \cr
0 =& \phi(k,m,a)=\phi(k-1,m,a)+ a \cdot \phi(k-1,m,a-1)\cr
\end{array}
\]
Solving the equations, we get
$$\phi(k-1,m,a) = \phi(k-1,m,a-1)=0.$$
Thus the claim is proved.
\end{proof}
\subsection{The main results}
\label{sec:main-results}
We are now ready to prove our main lemma.
\begin{lemma}
\label{le:singlesum}
Let $0 \leq a \leq m < n$, and suppose $f(a,m,n)=0$. Then $f(a,m+1,n) \neq 0$.
\end{lemma}
\begin{proof}
The lemma holds trivially for $a = 0$, since both
$f(a,m,n)$ and $f(a,m+1,n)$ are nonzero if $a = 0$.
So suppose $a \geq 1$.
Suppose $f(a,m,n)=f(a,m+1,n)=0$.
Then by Claims~\ref{cl:four} and \ref{cl:five}, we know that
$$\phi(n-m, m, a)=\phi(n-m-1, m+1, a)=\phi(n-m, m+1, a)=0.$$
By Lemma~\ref{lem:deriv},
$$\phi(n-m, m+1, a)= -\phi(n-m,m,a)+a\cdot\phi(n-m,m,a-1),$$
i.e., $\phi(n-m,m,a-1) = 0$.
Hence $\phi(n-m,m,a-1)= \phi(n-m,m,a)=0$. Now, an iterative
application of Claim~\ref{cl:six} eventually yields $\phi(0,m,a)=\phi(0,m,a-1)=0$. By
definition, $\phi(0,m,a)$ is the $a$'th derivative of
$$(x-1)^m=\sum_{i=0}^{m} {m \choose i}x^i(-1)^{m-i}$$
evaluated
at $x=0$. But $m \geq a$, so this is clearly not zero. Thus we have
reached a contradiction, and Lemma~\ref{le:singlesum} is proved.
\end{proof}
\begin{theorem}
\label{thm:eqcomplexity}
For large enough $n$ and all $0 \leq a \leq n$ the following hold.
\begin{enumerate}
\item
$D(HAM_n^{(=a)}) \geq n-2$.
\item
$Q^*(HAM_n^{(=a)})\geq \frac{n}{2}-1$.
\item
$C^*(HAM_n^{(=a)})\ge n-2$.
\end{enumerate}
\end{theorem}
\begin{proof}
By Lemma~\ref{le:useqa},
$$D(f), C^*(f) \ge \lg(\sum_{m: f(a,m,n)\ne 0} { n \choose m })$$
and
$$Q^*(f) \ge \frac{1}{2}\lg(\sum_{m: f(a,m,n)\ne 0} { n \choose m }).$$
First suppose $a \leq n/2$. We have
\begin{equation}
\label{eqn:small-a-large-m}
\sum_{m: f(a,m,n)\ne 0} {n \choose m} \geq
\sum_{m \geq n/2: f(a,m,n)\ne 0} {n \choose m}.
\end{equation}
Let us lower-bound the r.h.s.\ of (\ref{eqn:small-a-large-m}). First
of all, since the r.h.s.\ of (\ref{eqn:small-a-large-m}) works in the
regime where $m \geq n/2 \geq a$, Lemma~\ref{le:singlesum} shows
that no two consecutive values of $m$ in this range satisfy the
condition ``$f(a,m,n) = 0$''. Also, for $m \geq n/2$,
${n \choose m}$ is a non-increasing function of $m$.
Thus, if we imagine an adversary whose
task is to keep the r.h.s.\ of (\ref{eqn:small-a-large-m}) as small
as possible, the adversary's best strategy, in our regime
where $m \geq n/2$, is to make $f(a,m,n) = 0$ exactly when
$m \in S$, where
\begin{equation}
\label{eqn:S}
S \doteq \{\lceil n/2 \rceil, \lceil n/2 \rceil + 2,
\lceil n/2 \rceil + 4, \ldots \}.
\end{equation}
Now,
\begin{equation}
\label{eqn:m-large}
2^{n-1} \leq \sum_{m \geq n/2} {n \choose m} \leq
2^{n-1} + O(2^n / \sqrt{n}).
\end{equation}
(We need the second inequality to handle the case where $n$ is even.)
Also, recall that an $(1 - o(1))$ fraction of the
sum $\sum_{m \geq n/2} {n \choose m}$ is obtained from the range
$n/2 \leq m \leq n/2 + \sqrt{n \log n}$, for instance.
(Here and in what follows, ``$o(1)$'' denotes a function of $n$ that
goes to zero as $n$ increases.) In this range,
the values of ${n \choose m}$ for any two consecutive values of $m$ are
within $(1 + o(1))$ of each other. In conjunction with (\ref{eqn:m-large}),
this shows that
\[ \sum_{m \geq n/2: f(a,m,n)\ne 0} {n \choose m}
\geq \sum_{m \geq n/2: m \not\in S} {n \choose m} \geq
(1/2 - o(1)) 2^{n-1}. \]
Thus,
\[ \left\lceil \lg\left(\sum_{m \geq n/2: f(a,m,n)\ne 0} {n \choose m}\right)
\right\rceil \geq n-2, \]
completing the proof for the case where $a \leq n/2$.
Now we apply symmetry to the case $a> n/2$: note that Alice can
reduce the problem with parameter $a$ to the problem with parameter
$n - a$, simply by complementing each bit of her input $x$. Thus,
the same communication complexity results hold for the case $a > n/2$.
\end{proof}
\begin{lemma}\label{le:zero}
Let $0 \leq a < m < n$, and suppose $F(a,m,n)=0$. Then $F(a, m+1, n) \neq 0$.
\end{lemma}
\begin{proof}
We have
$f(j,m,n)=(-1)^m[x^j]((x-1)^m(x+1)^{n-m})$. By definition,
\[
\begin{array}{rl}
F(a,m,n)=&\sum_{j=0}^a f(j,m,n)\cr
=&(-1)^m \sum_{j=0}^a [x^j]((x-1)^m(x+1)^{n-m})\cr
=&(-1)^m [x^a]((x-1)^m(x+1)^{n-m}\cdot\sum_{j=0}^{\infty}x^j)\cr
=&(-1)^m [x^a]((x-1)^m(x+1)^{n-m}\cdot\frac{1}{1-x})\cr
=&(-1)^{m-1} [x^a]((x-1)^{m-1}(x+1)^{n-m})=f(a,m-1,n-1).\cr
\end{array}
\]
So $F(a,m,n)=F(a,m+1,n)=0$ iff
$f(a,m-1,n-1)=f(a,m,n-1)=0$. But the latter is impossible
by Lemma~\ref{le:singlesum}, thus the lemma is proved.
\end{proof}
\begin{theorem}
For large enough $n$ and all $0 \leq a \leq n-1$,
the following hold.
\begin{enumerate}
\item
$D(HAM_n^{(a)}) \geq n-2$.
\item
$Q^*(HAM_n^{(a)}) \geq \frac{n}{2}-1$.
\item
$C^*(HAM_n^{(a)}) \geq n-2$.
\end{enumerate}
\end{theorem}
\begin{proof}
The proof is identical to that of Theorem~\ref{thm:eqcomplexity}
except for one point. In that proof we obtained the $a> n/2$
case easily from the $a\le n/2$ case. Here it is also easy
but needs a different proof. Let $a> n/2$ and, for all $x\in \bits n$,
let $\overline{x}$ be obtained from $x$ by flipping every single bit.
Note that
$HAM_n^{(a)}(x,y)=1$ iff ${\rm HAM}(x,y)\le a$ iff ${\rm HAM}(\overline{x},y)\ge n-a$
iff NOT(${\rm HAM}(\overline{x},y)\le (n-a)-1$ iff
${\rm HAM}_{n-a-1}(\overline{x},y)=1$.
Since $n-a-1 \le n/2$ we have that a lower bound for the $a\le n/2$
case implies a lower bound for the $a>n/2$ case.
\end{proof}
\section{Open Problems}
We make the following conjectures.
\begin{enumerate}
\item
For all $n$, for all $a$, $0\le a\le n-1$, $D(HAM_n^{(a)}), C^*(HAM_n^{(a)}), Q^*(HAM_n^{(a)}) \ge n+1$.
\item
For all $n$, for all $a$, $0\le a\le n$, $D(HAM_n^{(=a)}), C^*(HAM_n^{(=a)}), Q^*(HAM_n^{(=a)}) \ge n+1$.
\end{enumerate}
|
{
"timestamp": "2004-11-20T20:55:19",
"yymm": "0411",
"arxiv_id": "cs/0411076",
"language": "en",
"url": "https://arxiv.org/abs/cs/0411076"
}
|
\section{The statements}
\label{secsta}
\setcounter{equation}{0}
We review the Kontsevich-Kuperberg-Thurston construction of an invariant $Z$ of rational homology spheres in \cite{kt,ko}. (See \cite{as1,as2,bc1,bc2,cat} for another construction.) This invariant is constructed by means of configuration space integrals, it is valued in the algebra ${\cal A}(\emptyset)$ of Jacobi diagrams.
Its main property, that was proved by Greg Kuperberg and Dylan Thurston, is that it is a universal real finite type invariant for homology spheres in the sense of
\cite{ggp,hab,oht}. A generalization of this property is proved in \cite{sumgen}. Here, we provide detailed
and elementary proofs for the invariance of $Z$, and for the
properties of $Z$ that are needed in \cite{sumgen}.
All the main ideas here are due to Witten, Axelrod, Singer, Kontsevich, Bott, Taubes, Cattaneo, G.~Kuperberg and D.~Thurston among others.
I thank Dylan Thurston for explaining them to me.
The invariant $Z$ is a powerful generalization of the Casson invariant for integral homology $3$-spheres.
In this setting, the {\em Casson invariant\/} normalized as in \cite{akmc,mar} may be described as \index{N}{lambda@$\lambda$}
$$\lambda(M)=\frac{1}{6}\int_{C_2(M)}\omega_M^3$$
for any 2-form $\omega_M$ that satisfies the hypotheses stated in Subsections~\ref{subfundform}, \ref{subapseckkt} and \ref{subpont}
on the configuration space $C_2(M)$ defined in Subsection~\ref{subconfap}.
\subsection{The configuration space $C_2(M)$}
\label{subconfap}
\index{N}{Ctwo@$C_2(M)$}
When $A$ is a subset of $B$, $(B \setminus A)$ denotes its complement in $B$, when $x \in B$, $(B \setminus \{x\})$ is also denoted by $(B \setminus x)$.
Let $X=X^d$ be a smooth $d$-dimensional (real) manifold, and let $Y^k$ be a smooth $k$-dimensional submanifold of $X$.
If $TY$ denotes (the total space of) the tangent bundle of $Y$, then
$TX/TY$ is the {\em normal bundle\/} of $Y$. When $V$ is a real vector space, the group $]0,\infty[$ acts on $V$ by multiplication, and we set
$$SV=S(V)=(V \setminus 0)/]0,\infty[.$$
\index{N}{SV@$S(V)$}
The {\em unit normal bundle\/} $SN_XY$ of $Y$ in $X$ is the bundle over $Y$ whose fiber over $y$ is $S(T_yX/T_yY)$.
In this article, {\em to blow-up\/} a submanifold $Y^k$ in $X^d$ amounts to replace $Y$ by its unit normal bundle.
For example, if $Y \times \mathbb{R}^{d-k}$ is a tubular neighborhood of $Y=Y \times \{0\}$ in $X$, the blow-up is equivalent to the sequence of operations:
$$Y \times \mathbb{R}^{d-k} \longrightarrow [(Y \times \mathbb{R}^{d-k}) \setminus Y]=Y \times
]0,\infty[ \times S^{d-k-1} \longrightarrow Y \times
[0,\infty[ \times S^{d-k-1}.$$
In general, this provides a local definition. See Definition~\ref{defblodif}
for the general definition.
The blown-up manifold inherits a smooth structure of a manifold with corners
from the smooth structure of $X$. See Proposition~\ref{propblodifun}.
Note that the blown-up manifold has the homotopy type of $(X \setminus Y)$.
Let $M$ be a closed oriented 3-manifold. Fix $\infty \in M$.
Let $C_1(M)$
\index{N}{Cone@$C_1(M)$}
denote the manifold obtained from $M$ by blowing-up $\infty$.
The boundary of $C_1(M)$ is $ST_{\infty}(M)$. It is homeomorphic to $S^2$.
Let $M^2(\infty,\infty)$
\index{N}{Mtwoinfty@$M^2(\infty,\infty)$}
denote the manifold obtained from $M^2$ by blowing up
$(\infty,\infty)$ that becomes $S(T_{(\infty,\infty)}M^2) \cong S^5$.
In $M^2(\infty,\infty)$, the closures of the three submanifolds
of $M^2 \setminus (\infty,\infty)$, $\infty \times (M \setminus \infty)$,
$(M \setminus \infty) \times \infty$ and $\mbox{diag}((M \setminus \infty)^2)$
are three disjoint submanifolds of $M^2(\infty,\infty)$ which intersect
$S(T_{(\infty,\infty)}M^2)$ along $S(0 \times T_{\infty}M)$, $S( T_{\infty}M \times 0)$ and $S(\mbox{diag}((T_{\infty}M)^2))$, respectively.
The three of them are canonically diffeomorphic to $C_1(M)$. They will be denoted
by $\infty \times C_1(M)$,
$C_1(M) \times \infty$ and $\mbox{diag}(C_1(M)^2)$, respectively.
The normal bundle of $\mbox{diag}((M \setminus \infty)^2)$ in $(M \setminus \infty)^2$ is identified with the tangent bundle of $(M \setminus \infty)$
through $$(u,v) \in (T_xM)^2/\mbox{diag}((T_xM)^2) \mapsto (v-u) \in T_xM.$$
The {\em configuration space\/} $C_2(M)$ is
the {\em compactification\/} of $$\breve{C}_2(M)=(M \setminus \infty)^2 \setminus \mbox{diagonal}$$
obtained from $M^2(\infty,\infty)$
by blowing-up $\infty \times C_1(M)$,
$C_1(M) \times \infty$ and $\mbox{diag}(C_1(M)^2)$.
\subsection{Fundamental forms on $C_2(M)$}
\label{subfundform}
For $S^3=\mathbb{R}^3 \cup \infty$, we have a homotopy equivalence $p_{S^3}$ that makes
the following square commute:
$$\diagram{\breve{C}_2(S^3)&\hfl{p_{S^3}}&S^2 \cr\vfl{=}&&\uvfl{\mbox{projection}}
\cr (\mathbb{R}^3)^2 \setminus \mbox{diag}&\hfl{\cong}&\mathbb{R}^3 \times ]0,\infty[ \times S^2 \cr (x,y)& \mapsto & (x,\parallel y-x \parallel , \frac{y-x }{\parallel y-x \parallel})}.
$$
The following lemma is proved at the end of Subsection~\ref{subsdifblowup}.
The natural projection onto the $X$-factor of a product is denoted by $\pi_X$.
\begin{lemma}
\label{lemextproj}
The map $p_{S^3}$ smoothly extends to $C_2(S^3)$, and its extension $p_{S^3}$
satisfies:
$$p_{S^3}=\left\{\begin{array}{ll} -\pi_{S^2} \;\;& \mbox{on} \;ST_{\infty}(S^3) \times (S^3 \setminus \infty)=S^2 \times (S^3 \setminus \infty)\\ \pi_{S^2} \;\;& \mbox{on} \; (S^3 \setminus \infty) \times ST_{\infty}(S^3)=(S^3 \setminus \infty) \times S^2\\ \pi_{S^2} \;\;& \mbox{on} \;ST(\mathbb{R}^3)
{=} \mathbb{R}^3 \times S^2\end{array}\right.$$
\end{lemma}
Let $B^3(r)$ \index{N}{bt@$B^3(r)$} be the ball of $\mathbb{R}^3$ centered at $0$ with radius $r$.
Let $\phi$ be an orientation-preserving embedding of $(S^3 \setminus \mbox{Int}(B^3(1)))$ into $M$.
Then
$$M=\left(\mathbb{R}^3 \cup \infty \setminus \mbox{Int}(B^3(1))\right) \cup_{ ]1,3] \times S^2} B_M,$$ where $$B_M=M \setminus \phi\left(S^3 \setminus \mbox{Int}(B^3(3))\right),$$ and $ (]1,3]\times S^2 = \phi(]1,3]\times S^2))$ is a collar of $\partial B_M$ in $B_M$. Fix $(\infty \in M)=\phi(\infty)$.
This identifies $ST_{\infty}M$ to $(ST_{\infty}(S^3)=S(\mathbb{R}^3)=S^2)$.
\begin{definition}
A {\em trivialisation of \/$T(M \setminus \infty)$ that is standard near $\infty$\/}
is a trivialization \index{T}{trivialisation standard near $\infty$}
$$\tau: T(M \setminus \infty) \longrightarrow (M \setminus \infty) \times \mathbb{R}^3$$
of the tangent bundle $T(M \setminus \infty)$ of $(M \setminus \infty)$ that agrees with the standard trivialization $\tau_{S^3}$ of $\mathbb{R}^3$ outside $B_M(1)= B_M \setminus ( ]1,3] \times S^2 )$.
\end{definition}
Let $\tau_M$ be such a trivialisation (that exists by Lemma~\ref{lemtrivexist}). Note that $\tau_M$ identifies $S(T(M \setminus \infty))$ to $(M \setminus \infty) \times S^2$.
\begin{remark}
In the sequel, we define an invariant under orientation-preserving diffeomorphisms for pairs $(M, \phi)$ where
$M$ is an oriented $\mathbb{Q}$-sphere, and $\phi$ is an orientation-preserving embedding of $(S^3 \setminus \mbox{Int}(B^3(1)))$ into $M$. Since all such embeddings $\phi$
are isotopic in $M$, the choice of $\phi$ will not matter. This allows us to fix our decomposition
$$M=\left(\mathbb{R}^3 \cup \infty \setminus \mbox{Int}(B^3(1))\right) \cup_{ ]1,3] \times S^2} B_M,$$
once for all. This will not be discussed anymore.
\end{remark}
Let
$$P:C_2(M) \longrightarrow M^2$$
be the natural projection map.
The identification of $M$
and $S^3$ in a neighborhood of $\infty$ provides identifications of neighborhoods of $P^{-1}(\infty,\infty)$ in $\partial C_2(S^3)$ and in $\partial C_2(M)$.
Define $p_M(\tau_M): \partial C_2(M) \longrightarrow S^2$
\index{N}{pM@$p_M(\tau_M)$}
by carrying the definition of $p_{S^3}$ in the neighborhood above and by mimicking the definition of $p_{S^3}$ elsewhere on $\partial C_2(M)$. Recall that $ST_{\infty}(M)=S^2$
$$p_M(\tau_M)=\left\{\begin{array}{ll} -\pi_{S^2} \;\;& \mbox{on} \;ST_{\infty}(M) \times (M \setminus \infty)=S^2 \times (M \setminus \infty)\\ \pi_{S^2} \;\;& \mbox{on} \; (M \setminus \infty) \times ST_{\infty}(M)=(M \setminus \infty) \times S^2\\ \pi_{S^2}(\tau_M) \;\;& \mbox{on} \;ST(M \setminus \infty) \stackrel{\tau_M}{=} (M \setminus \infty) \times S^2\end{array}\right.$$
Let $\iota$
\index{N}{iota@$\iota$}
be the involution of $C_2(M)$ that extends ($(x,y) \mapsto (y,x)$)
and let $\overline{\iota}$
\index{N}{iotab@$\overline{\iota}$}
be the antipode of $S^2$.\\
Let $\omega_{S^2}$ be a volume form on $S^2$ such that
$\int_{S^2}\omega_{S^2}=1$.
We say that $\omega_{S^2}$ is {\em antisymmetric\/} if $\overline{\iota}^{\ast}(\omega_{S^2})=-\omega_{S^2}$.
Let $\tau_M$ be a trivialisation of $T(M \setminus \infty)$ that is standard near $\infty$.
\begin{definition}
\label{defformfund}
\index{T}{form!fundamental}
\noindent A two-form $\omega_M$ on $C_2(M)$ is {\em fundamental with respect to \/$\tau_M$ and \/} $\omega_{S^2}$ if:\\
$\bullet$ its restriction to $\partial C_2(M)$ is $p_M(\tau_M)^{\ast}(\omega_{S^2})$, and,\\
$\bullet$ it is closed.\\
Such a two-form is {\em antisymmetric\/} if
\index{T}{form!antisymmetric}
$\iota^{\ast}(\omega_M)=-\omega_M$.
\end{definition}
It will be easily shown (Lemma~\ref{lemexisfunap}) that such forms exist for any trivialization $\tau_M$ when $M$ is a $\mathbb{Q}$-sphere.
\subsection{Jacobi diagrams}
Here, a {\em \indexT{Jacobi diagram}\/} $\Gamma$ is a trivalent graph $\Gamma$ without simple loop like $\begin{pspicture}[.2](0,0)(.6,.4)
\psline{-*}(0.05,.2)(.25,.2)
\pscurve{-}(.25,.2)(.4,.05)(.55,.2)(.4,.35)(.25,.2)
\end{pspicture}$. The set of vertices of such a $\Gamma$ will be denoted by $V(\Gamma)$,
\index{N}{VGamma@$V(\Gamma)$}
its set of edges will be denoted by $E(\Gamma)$.
\index{N}{EGamma@$E(\Gamma)$}
A {\em \indexT{half-edge}\/} $c$ of $\Gamma$ is an element of
$$H(\Gamma)=\{c=(v(c);e(c)) | v(c) \in V(\Gamma); e(c) \in E(\Gamma);v(c) \in e(c)\}.$$
\index{N}{HGamma@$H(\Gamma)$}
An {\em automorphism\/} of $\Gamma$
\index{T}{Jacobi diagram!automorphism of}
is a permutation $b$ of $H(\Gamma)$
such that for any $c,c^{\prime} \in H(\Gamma)$,
$$v(c)=v(c^{\prime}) \Longrightarrow v(b(c))=v(b(c^{\prime}))\;\;\mbox{and}\;\;e(c)=e(c^{\prime}) \Longrightarrow e(b(c))=e(b(c^{\prime})).$$
The number of automorphisms of $\Gamma$ will be
denoted by $\sharp \mbox{Aut}(\Gamma)$.
\index{N}{AutGamma@$\sharp \mbox{Aut}(\Gamma)$}
For example, $ \sharp \mbox{Aut}(\tata)=12$.
{\em An orientation\/} of a vertex of such a diagram $\Gamma$
is a cyclic order of the three
half-edges that meet at that vertex.
\index{T}{Jacobi diagram!orientation of}
A Jacobi diagram $\Gamma$ is {\em oriented\/} if all its vertices are oriented (equipped with an orientation).
The {\em degree} of such a diagram is
half the number of its vertices.
Let ${\cal A}_n(\emptyset)$
\index{N}{An@${\cal A}_n(\emptyset)$}
denote the real vector space generated by the degree $n$ oriented Jacobi diagrams, quotiented out by the following relations AS and IHX:
$$ {\rm AS :} \begin{pspicture}[.2](0,-.2)(.8,1)
\psset{xunit=.7cm,yunit=.7cm}
\psarc[linewidth=.5pt](.5,.5){.2}{-70}{15}
\psarc[linewidth=.5pt](.5,.5){.2}{70}{110}
\psarc[linewidth=.5pt]{->}(.5,.5){.2}{165}{250}
\psline{*-}(.5,.5)(.5,0)
\psline{-}(.1,.9)(.5,.5)
\psline{-}(.9,.9)(.5,.5)
\end{pspicture}
+
\begin{pspicture}[.2](0,-.2)(.8,1)
\psset{xunit=.7cm,yunit=.7cm}
\pscurve{-}(.9,.9)(.3,.7)(.5,.5)
\pscurve[border=2pt]{-}(.1,.9)(.7,.7)(.5,.5)
\psline{*-}(.5,.5)(.5,0)
\end{pspicture}=0,\;\;\mbox{and IHX :}
\begin{pspicture}[.2](0,-.2)(.8,1)
\psset{xunit=.7cm,yunit=.7cm}
\psline{-*}(.1,1)(.35,.2)
\psline{*-}(.5,.5)(.5,1)
\psline{-}(.75,0)(.5,.5)
\psline{-}(.25,0)(.5,.5)
\end{pspicture}
+
\begin{pspicture}[.2](0,-.2)(.8,1)
\psset{xunit=.7cm,yunit=.7cm}
\psline{*-}(.5,.6)(.5,1)
\psline{-}(.8,0)(.5,.6)
\psline{-}(.2,0)(.5,.6)
\pscurve[border=2pt]{-*}(.1,1)(.3,.3)(.7,.2)
\end{pspicture}
+
\begin{pspicture}[.2](0,-.2)(.8,1)
\psset{xunit=.7cm,yunit=.7cm}
\psline{*-}(.5,.35)(.5,1)
\psline{-}(.75,0)(.5,.35)
\psline{-}(.25,0)(.5,.35)
\pscurve[border=2pt]{-*}(.1,1)(.2,.75)(.7,.75)(.5,.85)
\end{pspicture}
=0.
$$
\index{N}{AS}
\index{N}{IHX}
Each of these relations relate diagrams which can be represented by planar immersions that are identical outside the part of them represented in the pictures. Here, the orientation of vertices is induced by the counterclockwise order of the half-edges. For example,
AS identifies the sum of two diagrams which only differ by the orientation
at one vertex to zero.
${\cal A}_0(\emptyset)$ is equal to $\mathbb{R}$ generated by the empty diagram.
\subsection{The invariants $Z_n$}
\label{subapseckkt}
\begin{definition}
Let $V$ be a finite set. An {\em orientation\/} of $V$
\index{T}{orientation of a finite set}
is a bijection from $V$ to $\{1,2,\dots,\sharp V\}$ (or a total order on $V$) up to an even permutation.
\end{definition}
When $M$ is an odd-dimensional oriented manifold, an orientation of $V$ provides an ordering of the factors of $M^V$ (up to an even permutation), and therefore induces an orientation of $M^V$. Thus, the datum of an orientation of $V$ is equivalent to the datum of an orientation of $M^V$.
Let $\Gamma$ be a Jacobi diagram. Let $H(\Gamma)$ be its set of half-edges.
When the edges of $\Gamma$ are oriented, the orientations of the edges
induce an orientation of $H(\Gamma)$ that is called the {\em \indexT{edge-orientation}\/} of
$H(\Gamma)$ and that is represented by a total order of $H(\Gamma)$ of the following form. Fix an arbitrary order on the set of edges, then take the two halves of the first edge ordered from origin to the end, next the two halves of the second edge, and so on.
When the set $V(\Gamma)$ of vertices of $\Gamma$ is oriented and when the vertices of $\Gamma$ are oriented (as the sets of their three half-edges),
these data induce an orientation of $H(\Gamma)$ that is called the {\em \indexT{vertex-orientation}\/} of
$H(\Gamma)$ and that is defined as follows.
Number the vertices of $\Gamma$ from $1$ to $\sharp V(\Gamma)$ by a bijection that induces
the given orientation of $V(\Gamma)$.
The wanted order of $H(\Gamma)$ is given by taking first the half-edges of the first vertex
with an order that agrees with the vertex-orientation, then the half-edges that contain the second vertex, and so on.
\\
Let $M$ be a $\mathbb{Q}$-sphere. Set $$\breve{C}_{V(\Gamma)}(M)=(M \setminus \infty)^{V(\Gamma)} \setminus \{\mbox{all diagonals}\}.$$
\index{N}{CbreveV@$\breve{C}_{V(\Gamma)}(M)$}
The set $\breve{C}_{V(\Gamma)}(M)$ is the set of injective maps from $V(\Gamma)$ to $(M\setminus \infty)$. It is an open submanifold of $(M \setminus \infty)^{V(\Gamma)}$ that is oriented as soon as $V(\Gamma)$ is oriented.
An edge $e$ of $\Gamma$ defines a pair $P(e)$ of elements of $V(\Gamma)$.
Then the restriction of maps induces a canonical map from $\breve{C}_{V(\Gamma)}(M)$
to $\breve{C}_{P(e)}(M)$. An orientation of $e$ orders the pair $P(e)$
and produces a canonical identification of $\breve{C}_{P(e)}(M)$ with $\breve{C}_{\{1,2\}}(M) \subset C_2(M)$. (The origin of $e$ is mapped to $1$.)
For any oriented edge $e$ of $\Gamma$, the composition of these maps will be denoted by $$p_e:\breve{C}_{V(\Gamma)}(M)\longrightarrow C_2(M).$$
\index{N}{pe@$p_e$}
Let $\omega_M$ be an antisymmetric two-form that is fundamental with respect to $\tau_M$ and $\omega_{S^2}$.
Let $\Gamma$ be an oriented Jacobi diagram.
Orient the edges of $\Gamma$, and orient $V(\Gamma)$ so that the
edge-orientation of $H(\Gamma)$ coincides with the vertex-orientation
of $H(\Gamma)$.
Set $$I_{\Gamma}(\omega_M)=\int_{\breve{C}_{V(\Gamma)}(M)}\bigwedge_{e \in E(\Gamma)}p_e^{\ast}(\omega_M).$$
\index{N}{IGammaomegaM@$I_{\Gamma}(\omega_M)$}
This integral is convergent thanks to Proposition~\ref{propconfunc} below. It is easy to see that its sign only depends on the vertex-orientation of $\Gamma$ up to an
even number of changes. In particular, the product $I_{\Gamma}(\omega_M)[\Gamma]$ only depends on the (unoriented) Jacobi diagram $\Gamma$.
\begin{proposition}[\cite{kt}]
\label{propthkktun}
Let $M$ be a $\mathbb{Q}$-sphere. Let $\omega_M$ be an antisymmetric two-form that is fundamental with respect to a trivialization $\tau_M$ standard near $\infty$ and to a form $\omega_{S^2}$ such that $\int_{S^2}\omega_{S^2}=1$. Then with the notation above \index{N}{ZnMtauM@$Z_n(M;\tau_M)$}
$$Z_n(M;\tau_M)=\sum_{\Gamma \;\mbox{\small Jacobi diagram with $2n$ vertices}}\frac{I_{\Gamma}(\omega_M)}{\sharp \mbox{Aut}(\Gamma)}[\Gamma] \in {\cal A}_n(\emptyset)$$
only depends on the oriented diffeomorphism type of $M$ and on the homotopy class of $\tau_M$.
(Here the sum runs over Jacobi diagrams without vertex-orientations.)
\end{proposition}
In the next subsection, we shall see that any $\mathbb{Z}$-sphere $M$ has a preferred homotopy class $[\tau^0_M]$ of trivialisations that are standard near $\infty$, and this
will allow us to define the invariants of $\mathbb{Z}$-spheres by
$$Z_n(M)=Z_n(M;\tau^0_M).$$
\index{N}{ZnM@$Z_n(M)$}
In general, we shall need a correction term, called the {\em framing correction\/} that is described in Subsection~\ref{subfra}.
\subsection{Homotopy classes of trivialisations of $\mathbb{Q}$-spheres.}
\label{subpont}
Recall that $GL^+(\mathbb{R}^3)$ is homotopy equivalent to the pathwise connected group $SO(3)$, that $\pi_1(SO(3))\cong \mathbb{Z}/2\mathbb{Z}$, $\pi_2(SO(3)) \cong 0$ and
$\pi_3(SO(3)) \cong \mathbb{Z}[\rho]$ where the generator $ [\rho]$ of $\pi_3(SO(3))$
is represented by the following covering map
$$\rho:S^3 \longrightarrow SO(3).$$
\index{N}{rho@$\rho$}
See $S^3$ as the unit sphere of the quaternionic field $({\mathbb{H}}=\mathbb{R} \oplus \mathbb{R} i \oplus \mathbb{R} j \oplus \mathbb{R} k)$.
\index{N}{H@${\mathbb{H}}$}
Then, for any
element $\gamma$ of $S^3$, $\rho(\gamma)$ is the restriction of the conjugacy $(x \mapsto \gamma x\gamma^{-1})$ to the euclidean space $\mathbb{R}^3$ of the pure quaternions.
Boundaries of oriented manifolds are oriented
with the outward normal first convention. Unit spheres of oriented euclidean
vector spaces are oriented as the boundaries of unit balls. In particular,
the sphere $S^3$ is the oriented boundary of the unit ball of ${\mathbb{H}}$.
The group $SO(3)$ is locally oriented as $S^2 \times S^1$ (oriented rotation axis in $S^2$, rotation angle with respect to the previous axis) (outside its center). With these orientations, $\mbox{deg}(\rho)=2$.
\index{N}{rho@$\rho$}
\begin{lemma}
\label{lemtrivexist}
The trivialisation $\tau_M$ defined on $]1,3] \times S^2$
extends to $B_M$.
\end{lemma}
\noindent {\sc Proof: } Choose a cell decomposition of $B_M$ with respect to its boundary.
Since $GL^+(\mathbb{R}^3)$ is pathwise connected, we may extend the trivialisation to the one-skeleton of $B_M$. If there were an obstruction in
$$H^2(B_M,\partial B_M=S^2;\mathbb{Z}/2\mathbb{Z})=H^2(B_M;\mathbb{Z}/2\mathbb{Z})$$ to extend $\tau_M$
on the two-skeleton of $B_M$, there would exist a surface $\Sigma$ immersed in $M$ such that the pull-back of $TM$ under this immersion is not trivialisable on $\Sigma$. But this
pull-back is isomorphic to the sum of the tangent space $T\Sigma$ of $\Sigma$, and the unique one-dimensional fibered bundle $\eta$ over $\Sigma$ that makes $T\Sigma \oplus \eta$ orientable. Therefore, the pull-back of $TM$ under this immersion is isomorphic to the pull-back of $T\mathbb{R}^3$ under any immersion of $\Sigma$ into $\mathbb{R}^3$ and is trivialisable on $\Sigma$. Thus, $\tau_M$ extends to the two-skeleton of $\Sigma$. Since $\pi_2(GL^+(\mathbb{R}^3)) = 0$, $\tau_M$ also extends to the three-skeleton.
\eop
The above proof also shows that any oriented 3-manifold is parallelisable.
Recall that the signature of a $4$-manifold is the signature of the intersection form on its $H_2$. Also recall that any closed oriented three-manifold bounds a compact oriented $4$-dimensional manifold whose signature may be arbitrarily changed by connected
sums with copies of $\mathbb{C} P^2$ or $-\mathbb{C} P^2$.
Let $W=W^4$ be a signature $0$ cobordism between $B^3(3)$ and $B_M$, that is a compact oriented $4$-dimensional manifold with corners such that
$$\partial W= B_M \cup (-[0,1] \times S^2) \cup -B^3(3)$$
where $\partial B_M= \partial B^3(3)=S^2$.
$$\begin{pspicture}[.4](-3,-.7)(5.5,2.5)
\psline(0,0)(4,0)
\psline(4,2)(0,2)
\psline[linewidth=2pt](0,0)(0,2)
\psline[linewidth=2pt](4,0)(4,2)
\rput(2,1){$W^4$}
\rput[r](-.1,1){$\{0\} \times B^3(3)=B^3(3) $}
\rput[l](4.1,1){$\{1\} \times B_M=B_M$}
\psline[linestyle=dashed,dash=3pt 2pt](1.2,0)(.8,-.4)(1.2,2)
\rput[r](.7,-.4){$[0,1] \times S^2$}
\rput[b](4,.1){$ \rightarrow $}
\rput[b](3.5,.1){$ \rightarrow $}
\rput[b](3,.1){$ \rightarrow $}
\rput[t](3,-.1){$ \vec{N}$}
\end{pspicture}$$
Let $\tau_M$ be a trivialisation of $(M \setminus \infty)$ that is standard near $\infty$.
Define the {\em \indexT{Pontryagin number}\/} of $\tau_M$ $$p_1(\tau_M) \in \mathbb{Z}$$ \index{N}{pone@$p_1$} as follows.
Consider the complex $4$-bundle $TW \otimes \mathbb{C}$ over $W$.
Near $\partial W$, $W$ may be identified to an open subspace of one of the products $[0,1] \times B^3(3)$ or $[0,1] \times B_M$.
Let $\vec{N}$ be the tangent vector to $[0,1] \times \{\mbox{pt}\}$ (under these identifications), and let
$\tau(\tau_M)$ denote the trivialization of $TW \otimes \mathbb{C}$ over $\partial W$ that is obtained by stabilizing either $\tau_{S^3}$ or $\tau_M$ into $\vec{N} \oplus \tau_M$ or $\vec{N} \oplus \tau_{S^3}$. Then the obstruction to extend this trivialization to $W$ is the relative first \indexT{Pontryagin class} $$p_1(W;\tau(\tau_M))=p_1(\tau_M)[W,\partial W] \in H^4(W,\partial W;\mathbb{Z})=\mathbb{Z}[W,\partial W]$$ of the trivialisation.
Now, we specify our sign conventions for this Pontryagin class. They are the same as in \cite{milnorsta}.
In particular, $p_1$ is the opposite of the second Chern class $c_2$ of the complexified tangent bundle. See \cite[p. 174]{milnorsta}.
More precisely, equip $M$ with a riemannian metric that coincides with the standard metric
of $\mathbb{R}^3$ outside $B^3(1)$, and equip $W$ with a riemannian metric that coincides with the orthogonal product metric of one of the products $[0,1] \times B^3(3)$ or $[0,1] \times B_M$ near $\partial W$.
Equip $TW \otimes \mathbb{C}$ with the associated hermitian structure. The determinant bundle of $TW$
is trivial because $W$ is oriented and $det(TW \otimes \mathbb{C})$ is also trivial.
We only consider the trivialisations that
are unitary with respect to the hermitian structure of $TW \otimes \mathbb{C}$ and the standard hermitian form of $\mathbb{C}^4$, and that are special with respect to
the trivialisation of $det(TW \otimes \mathbb{C})$.
Since $\pi_i(SU(4))=\{0\}$ when $i<3$, the trivialisation $\tau(\tau_M)$
extends to a special unitary trivialisation $\tau$ outside the interior of a $4$-ball $B^4$
and defines
$$\tau: (TW \otimes \mathbb{C})_{|S^3} \longrightarrow S^3 \times \mathbb{C}^4$$
over the boundary $S^3=\partial B^4$ of this $4$-ball $B^4$.
Over this $4$-ball $B^4$, the bundle is trivial and admits a trivialisation
$$\tau_B: (TW \otimes \mathbb{C})_{|B^4} \longrightarrow B^4 \times \mathbb{C}^4.$$
Then $\tau_B \circ \tau^{-1}(v \in S^3, w \in \mathbb{C}^4)=(v, \phi(v)(w))$
where $\phi(v) \in SU(4)$.
Let $i^2(m^{\mathbb{C}}_r)$
\index{N}{itwo@$i^2(m^{\mathbb{C}}_r)$}
be the following map
$$\begin{array}{llll} i^2(m^{\mathbb{C}}_r): &(S^3 \subset \mathbb{C}^2) & \longrightarrow & SU(4)\\
& (z_1,z_2) & \mapsto & \left[\begin{array}{cccc} 1&0&0&0\\0&1&0&0\\0&0&z_1&-\overline{z}_2\\0&0&z_2& \overline{z}_1\end{array} \right] \end{array}.$$
When $(e_1,e_2,e_3,e_4)$ is the standard basis of $\mathbb{C}^4$, the columns of the matrix contain the coordinates of the images of the $e_i$ with respect to $(e_1,e_2,e_3,e_4)$.
Then the homotopy class $[i^2(m^{\mathbb{C}}_r)]$ of $i^2(m^{\mathbb{C}}_r)$ generates $\pi_3(SU(4))=\mathbb{Z}[i^2(m^{\mathbb{C}}_r)]$
and the homotopy class of $\phi: S^3 \longrightarrow SU(4)$ satisfies
$$[\phi]=-p_1(\tau_M)[i^2(m^{\mathbb{C}}_r)] \in \pi_3(SU(4)).$$
\begin{proposition}
\label{proppont}
The first \indexT{Pontryagin number} $p_1(\tau_M)$ \index{N}{pone@$p_1$} is well-defined by the above
conditions. (It is independent of the choices that were made.)
It only depends on the homotopy class of the trivialisation $\tau_M$
among the trivialisations that are standard near $\infty$.
For any closed 3-manifold $M$, for any
trivialisation $\tau_M$ of $T(M \setminus \infty)$ that is standard near $\infty$, and for any $$g: (B_M, ]1,3] \times S^2) \longrightarrow (SO(3),1),$$
let $\mbox{deg}(g)$ denote the degree of $g$ and let
$$\begin{array}{llll}
\psi(g): &B_M \times \mathbb{R}^3 &\longrightarrow &B_M \times \mathbb{R}^3\\
&(x,y) & \mapsto &(x,g(x)(y))\end{array}$$
\index{N}{psig@$\psi(g)$}
then
$$p_1(\psi(g) \circ \tau_M)-p_1(\tau_M)=-2\mbox{deg}(g).$$
If $M$ is a given $\mathbb{Z}$-sphere, then $p_1$ defines a bijection from the set of homotopy classes of trivialisations of $M$ that are standard near $\infty$ to $4 \mathbb{Z}$.
\end{proposition}
This proposition will be proved in Subsection~\ref{subproofpont}.
Of course, for a given $\mathbb{Z}$-sphere, our preferred class of trivialisations will be $p_1^{-1}(0)$. By definition, the standard trivialisation of $\mathbb{R}^3$ is in this class when $M=S^3$.
\subsection{The framing correction}
\label{subfra}
Let $X$ be a 3-dimensional vector space. Let $V$ be a finite set. Then
$\breve{S}_V(X)$
\index{N}{SbreveV@$\breve{S}_V(X)$}
denotes the set of injective maps from $V$ to $X$ up to translations and dilations. It is an open subset of the smooth manifold $S(X^V/
\mbox{diag}(X^V))$. Set $\breve{S}_n(X)=\breve{S}_{\{1,2,\dots, n\}}(X)$.
\index{N}{Sbreven@$\breve{S}_n(X)$}
When $V$ and $X$ are oriented, $X^V$ and $(\mbox{diag}(X^V) \cong X)$ are oriented,
then the quotient $X^V/\mbox{diag}(X^V)$ is oriented so that $X^V$ has the
(fiber $\mbox{diag}(X^V)$ $\oplus$ quotient $X^V/\mbox{diag}(X^V)$) orientation.
When $W$ is a vector space, $S(W)$ is oriented as the boundary of a unit ball
of $W$ equipped with an arbitrary norm, that is so that the multiplication from $]0,\infty[ \times S(W)$ to $W$
preserves the orientation. This orients $S(X^V/
\mbox{diag}(X^V))$ and hence $\breve{S}_V(X)$.
For an $\mathbb{R}^3$ vector bundle, $p:E \longrightarrow B$, $\breve{S}_{V}(E)$ denotes the fibered space over $B$ where the fiber over $(g \in B)$ is $\breve{S}_{V}(p^{-1}(g))$.
When $B$ is an oriented manifold, $\breve{S}_{V}(E)$ is next oriented with the (base $B$ $\oplus$ fiber) orientation.
Let $p:E_1 \longrightarrow S^4$ be the $\mathbb{R}^3$ vector bundle over $S^4=B^4 \cup_{S^3} (-B^4)$ whose total space is \index{N}{Eone@$E_1$}
$$E_1= B^4 \times \mathbb{R}^3 \cup_{S^3 \times \mathbb{R}^3} (-B^4) \times \mathbb{R}^3$$ \index{N}{rho@$\rho$}
where the two parts are glued by identifying $(g,x) \in S^3 \times \mathbb{R}^3$ of the first factor to $$\left((g,\rho(g)(x)) \in (-B^4) \times \mathbb{R}^3\right).$$
The vector bundle $E_1$ is equipped with the involutive bundle isomorphism $\iota$
\index{N}{iota@$\iota$}
over $\mbox{Id}_{S^4}$ that is the multiplication by $(-1)$ over each fiber.
In particular, $\breve{S}_2(E_1)$ is a (compact) $S^2$-bundle that is denoted by $S_2(E_1)$. \index{N}{StwoEone@$S_2(E_1)$}
Let $\omega_T$ \index{N}{omegaT@$\omega_T$} be a closed $2$-form on $S_2(E_1)$ that represents the Thom class of this $S^2$-bundle such that $\overline{\iota}^{\ast}(\omega_T)=-\omega_T$. ($[\omega_T]$ is dual to
a $4$-dimensional manifold that intersects the "left-hand side part" $B^4 \times S^2$ of $S_2(E_1)$ as $(B^4 \times \{\mbox{point}\})$.)
Let $\Gamma$ be an oriented Jacobi diagram.
Each edge $e$ of $\Gamma$ again defines a pair $P(e) \subset V(\Gamma)$
that induces a projection
$$\breve{S}_{V(\Gamma)}(E_1) \longrightarrow S_{P(e)}(E_1)$$
by restriction on the fibers.
An orientation of the edge $e$ induces an order on $P(e)$ that identifies
$S_{P(e)}(E_1)$ to $S_2(E_1)$, and this again defines
$$p_e: \breve{S}_{V(\Gamma)}(E_1) \longrightarrow S_2(E_1).$$
\index{N}{pe@$p_e$}
Orient the vertices, the edges of $\Gamma$, and orient $V(\Gamma)$ so that the
edge-orientation of $H(\Gamma)$ coincides with the vertex-orientation
of $H(\Gamma)$.
Set \index{N}{IGammaomegaT@$I_{\Gamma}(\omega_T)$} \index{N}{Eone@$E_1$}
$$I_{\Gamma}(\omega_T)[\Gamma]=\int_{\breve{S}_{V(\Gamma)}(E_1)}\bigwedge_{e \; \mbox{\small edge of}\; \Gamma}p_e^{\ast}(\omega_T)[\Gamma],$$
and define
$$\xi_n=\sum_{\Gamma \;\mbox{\small connected Jacobi diagram with $2n$ vertices}}\frac{I_{\Gamma}(\omega_T)}{\sharp \mbox{Aut}(\Gamma)}[\Gamma] \in {\cal A}_n.$$ \index{N}{ksin@$\xi_n$}
Define $${\cal A}(\emptyset)=\prod_{n \in \mathbb{N}}{\cal A}_n(\emptyset)$$
\index{N}{Aemptyset@${\cal A}(\emptyset)$}as the topological product
of the vector spaces ${\cal A}_n(\emptyset)$.
Set \index{N}{ZMtauM@$Z(M;\tau_M)$} $$Z(M;\tau_M)=(Z_n(M;\tau_M))_{n\in \mathbb{N}} \in {\cal A}(\emptyset)$$
where $Z_0(M;\tau_M)=1[\emptyset]$.
Similarly, with $\xi_0=0$,
$$\xi=(\xi_n)_{n\in \mathbb{N}}.$$\index{N}{ksi@$\xi$}
Equip ${\cal A}(\emptyset)$ with the continuous product that maps
two (classes of) graphs to (the class of) their disjoint union.
This product turns ${\cal A}(\emptyset)$ into a commutative algebra.
\begin{theorem}[\cite{kt}]
\label{thkktfra}
The obtained $\xi_n$ \index{N}{ksin@$\xi_n$}\index{N}{omegaT@$\omega_T$} does not depend on the closed form $\omega_T$ that represents the Thom class of $S_2(E_1)$ such that $\iota^{\ast}(\omega_T)=-\omega_T$.
For any $\mathbb{Q}$-sphere $M$, set
$$Z(M)=Z(M;\tau_M) \exp(\frac{p_1(\tau_M)}{4}\xi).$$ \index{N}{ksi@$\xi$}
\index{N}{ZM@$Z(M)$}
Then $Z$ is a topological invariant of $M$.
\end{theorem}
Note that Theorem~\ref{thkktfra} obviously implies Proposition~\ref{propthkktun}. Thus, we are left with the proof of Theorem~\ref{thkktfra}.
Equip ${\cal A}(\emptyset)$ with the involution that maps $(x_n \in {\cal A}_n(\emptyset))_{n \in \mathbb{N}}$ to $\overline{(x_n)_{n \in \mathbb{N}}}=((-1)^nx_n)_{n \in \mathbb{N}}$.
Then, we have the following proposition.
\begin{proposition}
\label{proporrev}
For any integer $k$, $\xi_{2k}=0$. \index{N}{ksi@$\xi$} \\
For any $\mathbb{Q}$-sphere $M$, let $(-M)$ denotes the manifold obtained from $M$ by reversing its orientation, then \index{N}{ZM@$Z(M)$}
$$Z(-M)=\overline{Z(M)}.$$
\end{proposition}
\noindent {\sc Proof: }
The involution $\iota$ still makes sense on $S_{2n}(E_1)$, it
reverses the orientation and it commutes with the projections $p_e$. Therefore, $$\begin{array}{ll}I_{\Gamma}(\omega_T)[\Gamma]&=-\int_{\breve{S}_{V(\Gamma)}(E_1)}
\iota^{\ast}\left(\bigwedge_{e \; \mbox{\small edge of}\; \Gamma}p_e^{\ast}(\omega_T)\right)[\Gamma]\\
&=-\int_{\breve{S}_{V(\Gamma)}(E_1)}\left(\bigwedge_{e \; \mbox{\small edge of}\; \Gamma}p_e^{\ast}(-\omega_T)\right)[\Gamma]\\
&=(1)^{\sharp E(\Gamma)+1}I_{\Gamma}(\omega_T)[\Gamma].\end{array}$$
Since $3\sharp V(\Gamma)=2\sharp E(\Gamma)=12k$, when the degree of $\Gamma$ is $2k$, we conclude that $\xi_{2k}=0$.
Consider a trivialisation
$\tau_M:T(M \setminus \infty) \rightarrow (M \setminus \infty) \times \mathbb{R}^3$
of $M$ that is standard near $\infty$. Its composition $\tau_{-M}$ by $(\mbox{Id}_{M \setminus \infty} \times (-1)\mbox{Id}_{\mathbb{R}^3})$ is a trivialisation of $T(-M \setminus \infty)$ that is standard near $\infty$, with respect to the composition of the previous embedding of $(S^3 \setminus B^3(1))$
into $(M \setminus B_M(1))$ by the multiplication by $(-1)$.
On $\partial C_2(M)$, $p_{-M}(\tau_{-M})=\overline{\iota} \circ p_M(\tau_M)$. Therefore if $\omega(\tau_M)$ is an antisymmetric form that is fundamental with respect to $\tau_{M}$, $\iota^{\ast}(\omega(\tau_M))=-\omega(\tau_M)$ is an antisymmetric form that is fundamental with respect to $\tau_{-M}$.
Since changing the orientation of $M$, does not change the orientation of $C_{2n}(M)$, we see as before that $Z_{n}(-M;\tau_{-M})=Z_{n}(-M;-\omega(\tau_M))=(-1)^nZ_n(M;\tau_M)$.
Therefore, we are left with the proof that $p_1(\tau_{-M})=-p_1(\tau_M)$.
In order to prove it, note that changing the orientation of the cobordism $W$
between $B^3$ and $B_M$
tranforms it into a cobordism between $-B^3 \cong B^3$ and $(B_{-M}=-B_M)$.
Furthermore, changing the trivialisation by preserving its first vector and reversing the other ones
equips $B^3$ with its standard trivialisation.
The latter trivialisation extends to the complement of a $4$-ball $B^4$ as the composition of the previous one by the above symmetry. Therefore the induced change of basis on $\partial B^4$ is conjugate through this symmetry of the connected group $U(4)$, and hence homotopic. Since the orientation of $\partial B^4$ is the opposite to the one used in the computation of
$p_1(\tau_M)$, $p_1(\tau_{-M})=-p_1(\tau_M)$. \eop
We shall also prove that $\xi_1=-\frac{1}{12}[\tata]$ in Proposition~\ref{propxiun}.
Dylan Thurston and Greg Kuperberg also proved that $Z$ is a universal finite type invariant of integral homology $3$-spheres, that $Z$ is multiplicative
under the connected sum of $3$-manifolds, and that \index{N}{lambda@$\lambda$}
$$Z_1(M)=\frac{\lambda(M)}2[\tata]$$
for any integral homology sphere $M$ where $\lambda$ denotes the {\em Casson invariant\/} normalized as in \cite{akmc,mar}.
The article \cite{sumgen} contains splitting formulae for $Z$ that generalize
the formulae used in the Thurston and Kuperberg proof of $Z$'s universality.
It also contains a proof that $Z_1(M)=\frac{\lambda_W(M)}4[\tata]$
\index{N}{lambdaW@$\lambda_W$}
for any rational homology sphere $M$ where $\lambda_W$ denotes the Walker
extension of the Casson invariant normalized as in \cite{wal}.
Since the current article has been written in order to provide the detailed background
for \cite{sumgen}, the Thurston and Kuperberg proof of $Z$'s universality will not be discussed here. The multiplicativity of $Z$ under connected sum that is not
needed in \cite{sumgen} is not proved here either. This article is only a partial
detailed presentation of the properties of $Z$ that were discovered by Dylan Thurston and Greg Kuperberg in \cite{kt}, or by Maxim Kontsevich.
\newpage
\section{Proof of Theorem~\ref{thkktfra}}
\setcounter{equation}{0}
\label{secprothkkt}
\subsection{More on the topology of $C_2(M)$.}
\index{N}{Ctwo@$C_2(M)$}
Since the map $p_{S^3}:C_2(S^3) \longrightarrow S^2$ is a homotopy equivalence, $C_2(S^3)$ has the homotopy type of $S^2$.
In general, $C_2(M)$ has the homotopy type of $\left[(M \setminus \infty)^2 \setminus \mbox{diagonal}\right]$. Indeed, it has the homotopy type of
$$M^2(\infty,\infty) \setminus \left( (\infty \times C_1(M)) \cup
(C_1(M) \times \infty) \cup \mbox{diag}(C_1(M))\right)$$ that has the homotopy
type of $\left[(M \setminus \infty)^2 \setminus \mbox{diagonal}\right]$.
Therefore, we have the following lemma.
\begin{lemma}
\label{lemhctwo}
Let $\Lambda= \mathbb{Z}$ or $\mathbb{Q}$.
If $M$ is a $\Lambda$-sphere, then
$$H_{\ast}(C_2(M);\Lambda)=H_{\ast}(S^2;\Lambda).$$
and if \/$[S(T_xM)]$ denotes the homology class of a fiber of $(ST(M \setminus \infty) \subset C_2(M))$, then $H_{2}(C_2(M);\Lambda)=\Lambda[S(T_xM)]$.
\end{lemma}
\noindent {\sc Proof: }
In this proof, the homology coefficients are in $\Lambda$. Since $(M \setminus \infty)$ has the homology of a point, the K\"unneth Formula implies that
$(M \setminus \infty)^2$ has the homology of a point. Now, by excision,
$$H_{\ast}((M \setminus \infty)^2,(M \setminus \infty)^2 \setminus \mbox{diag}) \cong H_{\ast}((M \setminus \infty) \times \mathbb{R}^3,(M \setminus \infty)\times (\mathbb{R}^3 \setminus 0))$$
$$ \cong H_{\ast}( \mathbb{R}^3, S^2) \cong \left\{\begin{array}{ll} \Lambda \;\;&\;\mbox{if} \;\ast =3,\\
0\;&\;\mbox{otherwise.} \end{array} \right.$$
Of course, $(M \setminus \infty) \times \mathbb{R}^3$ denotes a tubular neighborhood of the diagonal in $(M \setminus \infty)^2$. Note that such a neighborhood can be easily obtained by integrating the vector fields given by a trivialisation of $T(M \setminus \infty)$ standard near $\infty$. With $(m, \lambda (v \in S^2))$, associate $(m,\gamma_{\lambda}(m,v))$ where $\gamma_{0}(m,v)=m$ and $\frac{\partial}{\partial t}(\gamma_{t}(m,v))(t_0)=\tau_M^{-1}((\gamma_{t_0}(m,v),v))$. When $\varepsilon$ is a small enough positive number, this defines an embedding of $(M \setminus \infty) \times (\{ x \in \mathbb{R}^3; \norm{x}< \varepsilon\} \cong \mathbb{R}^3)$.
Using the long exact sequence associated to the pair $((M \setminus \infty)^2,(M \setminus \infty)^2 \setminus \mbox{diag})$, we get that
$$H_{\ast}(C_2(M))=H_{\ast}(S^2)$$
and that $H_{2}(C_2(M);\Lambda)=\Lambda[S(T_xM)]$.
\eop
Therefore, there is a preferred generator $L_M$
\index{N}{LM@$L_M$}
of $H^2(C_2(M);\mathbb{Q})$ such that when $B$ is a 3-ball embedded in $M$ equipped with the orientation of $M$ and when $x$ is a point in the interior of $B$, the evaluation of $L_M$ on the homology class of $(\{x\} \times \partial B) \subset C_2(M)$ is one.
If $(K_1 \sqcup K_2) \subset M \setminus \infty$ is a two-component link of $M$, then the evaluation of $L_M$ on the homology class of the torus
$(K_1 \times K_2 \subset C_2(M))$ is the {\em \indexT{linking number}\/}
of $K_1$ and $K_2$ in $M$ that is denoted by $\ell(K_1,K_2)$. Here, it will be our definition for the linking number.
Let us now prove the existence of fundamental forms.
For this, we first recall the following standard consequence of the definition of the De Rham cohomology.
\begin{lemma}
\label{lemdr}
Let $A$ be a compact submanifold of a compact manifold $B$, let $\omega_A$ be a closed $n$-form on $A$, and let $i:A \longrightarrow B$ denote the inclusion.
Then the three following assertions are equivalent:
\begin{enumerate}
\item The form $\omega_A$ extends to $B$ as a closed $n$-form.
\item The cohomology class of $\omega_A$ belongs to $i^{\ast}(H^n(B;\mathbb{R}))$.
\item The integral of $\omega_A$ vanishes on $\mbox{Ker}(i_{\ast}:H_n(A;\mathbb{R})
\longrightarrow H_n(B;\mathbb{R}))$.
\end{enumerate}
\end{lemma}
\begin{lemma}
\label{lemhombordc}
The restriction map
$$H^2(C_2(M)) \longrightarrow H^2(\partial C_2(M))$$ is an isomorphism.
\end{lemma}
\noindent {\sc Proof: } Write the exact sequence (with real coefficients)
$$H^2(C_2(M),\partial C_2(M)) \cong H_4(C_2(M))= 0 \longrightarrow $$
$$ \longrightarrow H^2(C_2(M)) \longrightarrow H^2(\partial C_2(M)) \longrightarrow $$
$$ \longrightarrow H^3(C_2(M),\partial C_2(M)) \cong H_3(C_2(M))= 0.$$
\eop
\begin{lemma}
\label{lemexisfunap}
Any closed two-form $\omega_{M}$ on $\partial C_2(M)$ extends on $C_2(M)$ as a closed two-form. If $\omega_{M}$ is antisymmetric with respect to the involution $\iota$ on $C_2(M)$, then we can demand that the extension is antisymmetric, too.
\end{lemma}
\noindent {\sc Proof: }
The first assertion is a direct consequence of the two previous lemmas.
When $\omega_{M}$ is antisymmetric, let $\omega$ denote one of its closed extensions, then the average
$$\tilde{\omega}=\frac{\omega - \iota^{\ast}(\omega)}{2}$$
is an extension of $\omega_{M}$ that is closed and antisymmetric.
\eop
In particular fundamental forms exist. Note that the cohomology class of a fundamental form is $L_M$, since its integral along the generator $S(T_xM)$ of $H_2(C_2(M))$ is one.
\subsection{Needed statements about configuration spaces}
\label{substaconf}
Compactifications of $\breve{C}_{V(\Gamma)}(M)$ are useful to study the behaviour of our integrals
$$I_{\Gamma}(\omega_M)=\int_{\breve{C}_{V(\Gamma)}(M)}\bigwedge_{e \in E(\Gamma)}p_e^{\ast}(\omega_M)$$
near $\infty$, and their dependence on the choice of $\omega_M$.
Indeed, in order to prove the convergence, it is sufficient to find a smooth
compactification (that will have corners) where the form $\bigwedge p_e^{\ast}(\omega_M)$ smoothly extends.
The variation of this integral when adding an exact form $d\eta$, will be the integral of $\eta$ on the codimension one faces of the boundary that needs to be precisely identified.
Therefore, the proof of Theorem~\ref{thkktfra} will require a deeper knowledge of configuration spaces.
We give all the needed statements in this subsection. All of them will be proved in Section~\ref{seccomp}.
Recall that a map from $[0,\infty[^d \times \mathbb{R}^{n-d}$ to $\mathbb{R}^k$ is $C^{\infty}$ or smooth at $0$
if it can be extended to a $C^{\infty}$ map in a neighborhood of $0$ in $\mathbb{R}^{n}$. A {\em smooth manifold with corners\/} is a manifold where every point has a neighborhood that is diffeomorphic to a neighborhood of $0$ in
$[0,\infty[^d \times \mathbb{R}^{n-d}$.
The {\em codimension $d$ faces\/} of a smooth manifold $C$ with corners are the connected components of the set of points that are mapped to $0$ under a diffeomorphism from one of their neighborhoods to a neighborhood of $0$ in
$[0,\infty[^d \times \mathbb{R}^{n-d}$. The union of the codimension $0$ faces of such a $C$ is called the {\em interior\/} of $C$.
Let $V$ denote a finite set, let $M$ be a closed oriented three-manifold and let $X$ be a 3-dimensional vector space.
We shall study the open submanifold
$$\breve{C}_V(M)= (M \setminus \infty)^V \setminus\mbox{all diagonals}$$
of $(M \setminus \infty)^V$. It will be seen as the space of injective maps from $V$ to $(M \setminus \infty)$.
We shall also study the open submanifold $\breve{S}_V(X)$ of the smooth manifold $S(X^V/
\mbox{diag}(X^V))$
made of injective maps from $V$ to $X$ up to translations and dilations.
These manifolds are our {\em configuration spaces.\/} $\breve{S}_n(X)=\breve{S}_{\{1,2,\dots, n\}}(X)$
\index{N}{Sbreven@$\breve{S}_n(X)$}
and $\breve{C}_n(M)=\breve{C}_{\{1,2,\dots, n\}}(M)$.
\index{N}{Cbreven@$\breve{C}_n(M)$}
Note that $\breve{S}_2(X)$ may be seen as the set of maps from $\{2\}$ to $X \setminus 0$ (when choosing to map $\{1\}$ to $0$). This provides a diffeomorphism
from $\breve{S}_2(X)$ to $S(X)$ that is diffeomorphic to $S^2$.
For any subset $B$ of $V$, the restriction of maps provides well-defined
projections $p_B$ from $\breve{C}_V(M)$ to $\breve{C}_B(M)$, and from $\breve{S}_V(X)$
to $\breve{S}_B(X)$. A total order on $B$ identifies $B$ to $\{1, \dots, , \sharp B\}$ and therefore identifies $\breve{C}_B(M)$ to $\breve{C}_{\sharp B}(M)$
and $\breve{S}_B(X)$ to $\breve{S}_{\sharp B}(X)$.
In particular, by composition, any ordered pair $e$ of $V$ induces canonical
maps
$$p_e:\breve{C}_{V}(M) \longrightarrow \breve{C}_{2}(M)$$
and
$$p_e:\breve{S}_{V}(X) \longrightarrow \breve{S}_{2}(X).$$
We are going to define suitable compactifications for these spaces. Namely, we shall prove the following propositions.
\begin{proposition}
\label{propconfunc}
There exists a well-defined smooth compact manifold with corners $C_{V}(M)$ \index{N}{CV@$C_V(M)$}
whose interior is canonically diffeomorphic to $\breve{C}_{V}(M)$
such that
\begin{itemize}
\item $C_{\{1\}}(M)$ and $C_{\{1,2\}}(M)$ coincide with the compactifications $C_1(M)$ and $C_2(M)$
\index{N}{Ctwo@$C_2(M)$}
defined in Section~\ref{subconfap}.
\item For any ordered pair $e$ of $V$,
the projection
$$p_e:\breve{C}_{V}(M) \longrightarrow {C}_{2}(M) $$
smoothly extends to $C_{V}(M)$.
\end{itemize}
\end{proposition}
\begin{proposition}
\label{propconfuns}
There exists a well-defined smooth compact manifold with corners $S_{V}(X)$
\index{N}{SVX@$S_{V}(X)$}
whose interior is canonically diffeomorphic to $\breve{S}_{V}(X)$
such that, for any ordered pair $e$ of $V$,
the projection
$$p_e:\breve{S}_{V}(X) \longrightarrow {S}_{2}(X) $$ smoothly extends to $S_{V}(X)$.
\end{proposition}
For our purposes, it will be important to know the codimension one faces
of these compactifications. For $C_{V}(M)$, they will be the configuration spaces $F(\infty;B)$ and $F(B)$ defined below, for some subsets $B$ of $V$, where $F(\infty;B)$ will contain
limit configurations that map $B$ to $\infty$, and $F(B)$ will contain
limit configurations that map $B$ to a point of $(M \setminus \infty)$.
Let $B$ be a non-empty subset of $V$.
Let $S_i(T_{\infty}M^B)$
\index{N}{SiTinfty@$S_i(T_{\infty}M^B)$}
denote the set of injective maps from $B$ to $(T_{\infty}M \setminus 0)$ up to dilation. Note that $S_i(T_{\infty}M^B)$ is an open submanifold of $S((T_{\infty}M)^B)$. Define \index{N}{FinftyB@$F(\infty;B)$}
$$F(\infty;B)=\breve{C}_{(V \setminus B)}(M) \times S_i(T_{\infty}M^B)$$
where $\breve{C}_{\emptyset}(M)$ has one element.
Any ordered pair $e$ of $V$ defines a canonical map
$p_e$ from $F(\infty;B)$ to $C_2(M)$ in the following way.
\begin{itemize}
\item If $e \subseteq V \setminus B$, then $p_e$ is the composition of the natural
projections
$$F(\infty;B) \longrightarrow \breve{C}_{(V \setminus B)}(M) \longrightarrow C_e(M)=C_2(M).$$
\item If $e \subseteq B$, then $p_e$ is the composition of the natural
maps
$$F(\infty;B) \longrightarrow S_i(T_{\infty}M^B) \longrightarrow S_i(T_{\infty}M^e) \hookrightarrow C_e(M)=C_2(M).$$
\item If $e \cap B =\{b^{\prime}\}$, then $p_e$ is the composition of the natural
maps
$$F(\infty;B) \longrightarrow \breve{C}_{(e \setminus \{b^{\prime}\})}(M) \times S_i(T_{\infty}M^{\{b^{\prime}\}}) \longrightarrow $$
$$ \longrightarrow (M \setminus \infty)^{(e \setminus \{b^{\prime}\})} \times S(T_{\infty}M^{\{b^{\prime}\}}) \hookrightarrow C_e(M)=C_2(M).$$
\end{itemize}
Let $B$ be a subset of $V$ of cardinality $(\geq 2)$.
Let $b \in B$. Let $F(B)$ \index{N}{FB@$F(B)$}
denote the total space of the fibration over $\left(\breve{C}_{\{b\} \cup(V \setminus B)}(M)\right)$
where the fiber over an element $c$ is $\breve{S}_B(T_{c(b)}M)$.
Again, any ordered pair $e$ of $V$ defines a canonical map
$p_e$ from $F(B)$ to $C_2(M)$.
\begin{itemize}
\item If $e \subseteq (V \setminus B) \cup \{b\}$, then $p_e$ is the composition of the natural
projections
$$F(B) \longrightarrow \breve{C}_{\{b\} \cup(V \setminus B)}(M) \longrightarrow C_e(M)=C_2(M).$$
\item If $e \subseteq B$, then $p_e$ is the composition of the natural
projections
$$F(B) \longrightarrow \breve{S}_B(T_{c(b)}M) \longrightarrow \breve{S}_e(T_{c(b)}M) \longrightarrow C_e(M)=C_2(M).$$
\item If $e \cap B =\{b^{\prime}\}$, let $\tilde{e}$ be obtained from $e$ by replacing $b^{\prime}$ by $b$, then $p_e=p_{\tilde{e}}$.
\end{itemize}
Set
$$\partial^{\infty}_1(C_{V}(M))= \{F(\infty;B); B \subseteq V; B \neq \emptyset\},$$
$$\partial^d_1(C_{V}(M))= \{F(B); B \subseteq V; \sharp B \geq 2\},$$
and
$$\partial_1(C_{V}(M))=\partial^{\infty}_1(C_{V}(M)) \cup \partial^d_1(C_{V}(M)).$$
\index{N}{deloneC@$\partial_1(C_{V}(M))$}
The following proposition is proved in Subsection~\ref{subsketchdifcv}.
\begin{proposition}
\label{propconffaceun}
Any $F \in \partial_1(C_{V}(M))$ embeds canonically into $C_{V}(M)$, and its image is a codimension one face of $C_{V}(M)$. Therefore, any such $F$ will be identified
to its image. Then $\partial_1(C_{V}(M))$
\index{N}{deloneC@$\partial_1(C_{V}(M))$}
is the set of codimension one faces of $C_{V}(M)$. Furthermore, for any ordered pair $e$ of $V$, for any $F \in \partial_1(C_{V}(M))$, the restriction to $F$ of the canonical map $p_e$ defined from $C_V(M)$ to $C_2(M)$ is the map $p_e$ defined above.
\end{proposition}
Let $B$ be a strict subset of $V$ of cardinality $(\geq 2)$. Let $b \in B$.
Let $X$ be a $3$-dimensional vector space.
Let \index{N}{fBX@$f(B)(X)$}
$$f(B)(X)=\breve{S}_B(X) \times \breve{S}_{\{b\} \cup (V \setminus B)}(X)$$
be a space of limit configurations where $B$ collapses.
Any ordered pair $e$ of $B$ provides the following canonical projection $p_e$ from $f(B)(X)$ to $S_2(X)$ as follows.
\begin{itemize}
\item If $e \subseteq (V \setminus B) \cup \{b\}$, then $p_e$ is the composition of the natural
projections
$$f(B)(X) \longrightarrow \breve{S}_{\{b\} \cup(V \setminus B)}(X) \longrightarrow S_e(X)=S_2(X).$$
\item If $e \subseteq B$, then $p_e$ is the composition of the natural
projections
$$f(B)(X) \longrightarrow \breve{S}_B(X) \longrightarrow S_e(X)=S_2(X).$$
\item If $e \cap B =\{b^{\prime}\}$, let $\tilde{e}$ be obtained from $e$ by replacing $b^{\prime}$ by $b$, then $p_e=p_{\tilde{e}}$.
\end{itemize}
In this article, the sign $\subset$ stands for "$\subseteq$ and $\neq$". Set \index{N}{deloneS@$\partial_1(S_{V}(X))$}
$$\partial_1(S_{V}(X))= \{f(B)(X); B \subset V; \sharp B \geq 2\}.$$
The following proposition is proved in Subsection~\ref{subsketchdifcv}.
\begin{proposition}
\label{propconffacedeux}
Any $F \in \partial_1(S_{V}(X))$
\index{N}{deloneS@$\partial_1(S_{V}(X))$}
embeds canonically into $S_{V}(X)$, and its image is a codimension one face of $S_{V}(X)$. Therefore, any such $F$ will be identified
to its image. Then $\partial_1(S_{V}(X))$ is the set of codimension one faces of $S_{V}(X)$.
Furthermore, for any ordered pair $e$ of $V$, for any $F \in \partial_1(S_{V}(X))$, the restriction to $F$ of the canonical map $p_e$ defined from $S_V(X)$ to $S_2(X)$ is the map $p_e$ defined above.
\end{proposition}
\subsection{Sketch of the proof of Theorem~\ref{thkktfra}}
\label{subsketchpkkt}
We shall first see that the wanted invariant $Z$ is the exponential of
a simpler series in ${\cal A}(\emptyset)$, that we are going to present
in another way by means of {\em labelled diagrams\/} that will make the proofs clearer.
A degree $n$ {\em labelled\/} Jacobi diagram
\index{T}{Jacobi diagram!labelled}
is a Jacobi diagram whose vertices are numbered from $1$ to $2n$, and whose edges are numbered from $1$ to $3n$.
Let $\overline{\Gamma}$ be a labelled Jacobi diagram with underlying Jacobi diagram $\Gamma$. The automorphisms of $\Gamma$
\index{T}{Jacobi diagram!automorphism of}
act on the labelling of $\overline{\Gamma}$. In particular, there are exactly $\sharp \mbox{Aut}(\Gamma)$
\index{N}{AutGamma@$\sharp \mbox{Aut}(\Gamma)$}
labellings of $\overline{\Gamma}$ that give rise to a labelled Jacobi diagram isomorphic to $\overline{\Gamma}$ as a labelled Jacobi diagram, and the number of labelled Jacobi diagrams with underlying Jacobi diagram $\Gamma$ is $\frac{(2n)!(3n)!}{\sharp \mbox{Aut}(\Gamma)}$.
A Jacobi diagram is {\em edge-oriented\/}
\index{T}{Jacobi diagram!edge-oriented}
when its edges are oriented. Any labelled Jacobi diagram has $2^{3n}$ such edge-orientations.
A labelled edge-oriented Jacobi diagram inherits a canonical vertex-orientation (up to an even
number of changes), namely the vertex-orientation that together with the orientation of $V(\Gamma)$ induced by the vertex labels provides a vertex-orientation of $H(\Gamma)$ equivalent to its edge-orientation.
Therefore, an edge-oriented labelled graph $\overline{\Gamma}$ has a well-determined class $[\overline{\Gamma}]$ in ${\cal A}(\emptyset)$.
Furthermore, an edge-oriented labelled graph $\overline{\Gamma}$ defines a map \index{N}{PGamma@$P(\Gamma)$}
$$P(\overline{\Gamma}):\breve{C}_{2n}(M) \longrightarrow C_{2}(M)^{3n}$$
whose projection $p_i \circ P(\overline{\Gamma})=P_i(\overline{\Gamma})$
\index{N}{PiGamma@$P_i(\Gamma)$}
onto the $i^{th}$ factor of $C_{2}(M)^{3n}$ is $p_{e(i)}$
where $e(i)$ denotes the edge labelled by $i$.
Let $\tau_M$ be a trivialisation of $T(M \setminus \infty)$ standard near $\infty$.
For any $i \in \{1, \dots, 3n\}$, let $\omega^{(i)}_M$ be a two-form that is fundamental with respect to $\tau_M$ and to a form $\omega^{(i)}_{S^2}$ such that $\int_{S^2}\omega^{(i)}_{S^2}=1$. Define the $6n$-form on $C_{2}(M)^{3n}$ \index{N}{Omega@$\Omega$}
$$\Omega=\bigwedge_{i=1}^{3n}p_i^{\ast}(\omega^{(i)}_M).$$
Proposition~\ref{propconfunc} allows us to define \index{N}{IGammaMOmega@$I_{\Gamma}(M;\Omega)$} $$I_{\overline{\Gamma}}(M;\Omega)=\int_{{C}_{2n}(M)}P(\overline{\Gamma})^{\ast}(\Omega)=\int_{{C}_{2n}(M)}\bigwedge_{i=1}^{3n}P_i(\overline{\Gamma})^{\ast}(\omega^{(i)}_M).$$
Let ${\cal E}_n$
\index{N}{Ecaln@${\cal E}_n$}
denote the set of all connected edge-oriented labelled Jacobi diagrams with $2n$ vertices.
We are going to prove the following propositions.
\begin{proposition}
\label{propun}
Under the above assumptions, \index{N}{zntauM@$z_n(\tau_M)$}
$$z_n(\tau_M) = \sum_{\Gamma \in {\cal E}_n} I_{{\Gamma}}(M;\Omega)[{\Gamma}]$$
only depends on $M$ and on $\tau_M$.
\end{proposition}
\begin{proposition}
\label{propdeux}
Let $\omega_T$ be a closed two-form \index{N}{omegaT@$\omega_T$}
that represents the Thom class of $S_2(E_1)$. \index{N}{StwoEone@$S_2(E_1)$}
Then \index{N}{deltan@$\delta_n$}
$$\delta_n= \sum_{\Gamma \in {\cal E}_n} \int_{{S}_{2n}(E_1)}\bigwedge_{i=1}^{3n}P_i({\Gamma})^{\ast}(\omega_T)[{\Gamma}]$$
does not depend on the choice of $\omega_T$.
\end{proposition}
\begin{proposition}
\label{proptrois}
Under the above assumptions, \index{N}{zntauM@$z_n(\tau_M)$}
$z_n(\tau_M)$
only depends on $M$ and on the homotopy class of $\tau_M$ among the trivialisations that are standard near $\infty$.
For any closed 3-manifold $M$, for any
trivialisation $\tau_M$ of $T(M \setminus \infty)$ that is standard near $\infty$, and for any $$g: (M \setminus \infty, M \setminus B_M(1)) \longrightarrow (SO(3),1),$$
define $$\begin{array}{llll}
\psi(g): &(M \setminus \infty) \times \mathbb{R}^3 &\longrightarrow &(M \setminus \infty) \times \mathbb{R}^3\\
&(x,y) & \mapsto &(x,g(x)(y))\end{array}$$ then \index{N}{deltan@$\delta_n$}
$$z_n(\psi(g) \circ \tau_M)-z_n(\tau_M)=\frac{1}{2}\mbox{deg}(g)\delta_n.$$
\end{proposition}
Note that Propositions~\ref{propconfunc} and \ref{propconfuns}
ensure that all the mentioned integrals
are well-defined and that all the previous ones are convergent.
Let us now show that Propositions~\ref{propun}, \ref{propdeux}, \ref{proptrois}, \ref{proppont}
prove Theorem~\ref{thkktfra}.
First note that for an antisymmetric $\omega_M$ that is fundamental with respect to $\tau_M$ and to a two-form $\omega_{S^2}$ such that $\int_{S^2}\omega_{S^2}=1$, $\int_{{C}_{2n}(M)}\bigwedge_{i=1}^{3n}P_i(\overline{\Gamma})^{\ast}(\omega_M)[\overline{\Gamma}]$
is independent of the labelling and is equal to $I_{\Gamma}(\omega_M)[\Gamma]$.
Therefore, \index{N}{zntauM@$z_n(\tau_M)$}
$$z_n(\tau_M)=2^{3n}(3n)!(2n)!\sum_{\Gamma \;\mbox{\small connected Jacobi diagrams with 2n vertices}}\frac{I_{\Gamma}(\omega_M)}{\sharp\mbox{Aut}(\Gamma)}
[\Gamma],$$
and $$\tilde{Z}(\tau_M)=\exp\left(\left(\frac{1}{2^{3n}(3n)!(2n)!}z_n(\tau_M)\right)_n\right)$$ only depends on $\tau_M$, according to Proposition~\ref{propun}.
\begin{lemma}
$Z(M;\tau_M)=\tilde{Z}(\tau_M)$. \index{N}{ZMtauM@$Z(M;\tau_M)$}
\end{lemma}
These are two series of combinations of diagrams and it suffices to compare the coefficients
of $[\Gamma]$, for a diagram $\Gamma$ which is a disjoint union of $k_1$ copies of $\Gamma_1$,
$k_2$ copies of $\Gamma_2$, \dots, $k_r$ copies of $\Gamma_r$, where $\Gamma_1$, $\Gamma_2$ and $\Gamma_r$ are non-isomorphic connected Jacobi diagrams. The coefficient of $[\Gamma]=\prod_{i=1}^r[\Gamma_i]^{k_i}$ in $Z(M;\tau_M)$ is
$\frac{I_{\Gamma}(\omega_M)}{\sharp\mbox{Aut}(\Gamma)}$
where $I_{\Gamma}(\omega_M) = \prod_{i=1}^r I_{\Gamma_i}(\omega_M)^{k_i}$, and
$\sharp\mbox{Aut}(\Gamma)=\prod_{i=1}^r[(\sharp\mbox{Aut}(\Gamma_i))^{k_i}(k_i)!]$. Therefore, the coefficient in $Z(M;\tau_M)$ is
$$\prod_{i=1}^r\frac{I_{\Gamma_i}(\omega_M)^{k_i}}{\sharp\mbox{Aut}(\Gamma_i)^{k_i}(k_i)!}.$$
Let $k=\sum_{i=1}^rk_i$. The coefficient of $[\Gamma]$ in $\tilde{Z}(\tau_M)$
is
its coefficient in the product
$$\frac{1}{k!}\left(\left(\frac{1}{2^{3n}(3n)!(2n)!}z_n(\tau_M)\right)_n\right)^k$$ where $\prod_{i=1}^r[\Gamma_i]^{k_i}$ occurs $\frac{k!}{ \prod_{i=1}^r (k_i)!}$ times
with the coefficient $\frac{1}{k!}\prod_{i=1}^r\frac{I_{\Gamma_i}(\omega_M)^{k_i}}{\sharp\mbox{Aut}(\Gamma_i)^{k_i}}$.
Thus, the two coefficients coincide.
\eop
Of course, Proposition~\ref{propdeux} implies that \index{N}{ksin@$\xi_n$} \index{N}{deltan@$\delta_n$}
$$\xi_n= \frac{1}{2^{3n}(3n)!(2n)!}\delta_n$$ is independent on the used $\omega_T$.
Now, Propositions~\ref{proptrois}
and \ref{proppont}
clearly imply that
$(\frac{1}{2^{3n}(3n)!(2n)!}z_n(\tau_M) + \frac{p_1(\tau_M)}{4}\xi_n)$ \index{N}{ksin@$\xi_n$} is
independent of $\tau_M$, and this in turn implies Theorem~\ref{thkktfra}.
Propositions~\ref{propun}, \ref{propdeux} and \ref{proptrois}
and \ref{proppont}
will be proved in Subsections~\ref{subproofpropun}, \ref{subsproofdeux}, \ref{submoretriv}
and \ref{subproofpont}, respectively.
\subsection{Proof of Proposition~\ref{propun}, the dependence on the forms.}
\label{subproofpropun}
In this subsection, we prove Proposition~\ref{propun}.
Of course, the only choice in the expression of $z_n(\tau_M)$ is the choice of the $\omega^{(i)}_M$, and it is enough to prove that changing an $\omega^{(i)}_M$ into an $\hat{\omega}^{(i)}_M$ that is fundamental with respect to
$\tau_M$ and to a form $\hat{\omega}^{(i)}_{S^2}$ such that $\int_{S^2}\hat{\omega}^{(i)}_{S^2}=1$ does not change $z_n=z_n(\tau_M)=z_n(\Omega)$.
For later use in \cite{sumgen}, we shall rather study how $z_n$ varies when $\omega^{(i)}_M$
varies within a class of forms that is more general than the fundamental forms.
\begin{definition}
\label{defformad}
\index{T}{form!admissible}
\noindent A two-form $\omega_M$ on $C_2(M)$ or on $\partial C_2(M)$ is {\em admissible\/} if:\\
$\bullet$ its restriction to $\partial C_2(M) \setminus ST(B_M)$ is $p_M(\tau_M)^{\ast}(\omega_{S^2})$ for some trivialisation $\tau_M$ of $T(M \setminus \infty)$ standard near $\infty$ and for some two-form $\omega_{S^2}$ on $S^2$ with total volume one, and,\\
$\bullet$ it is closed.\\
Such a two-form is {\em antisymmetric\/} if $\iota^{\ast}(\omega_M)=-\omega_M$.
\end{definition}
According to Lemma~\ref{lemexisfunap}, an admissible two-form on $\partial C_2(M)$ extends as an admissible two-form on $C_2(M)$, and an admissible antisymmetric two-form on $\partial C_2(M)$ extends as an admissible antisymmetric two-form on $C_2(M)$.
We are going to prove the following proposition.
\begin{proposition}
\label{propzad}
Let \/ $\omega_M$ be an antisymmetric admissible two-form on $C_2(M)$, then
with the notation before Proposition~\ref{propthkktun} \index{N}{ZnomegaM@$Z_n(\omega_M)$}
$$Z_n(\omega_M) =\sum_{\Gamma \;\mbox{\small Jacobi diagram with $2n$ vertices}}\frac{I_{\Gamma}(\omega_M)}{\sharp \mbox{Aut}(\Gamma)}[\Gamma]$$
only depends on $M$ and on the restriction of $\omega_M$ to $ST(B_M)$.
\end{proposition}
In what follows, all the two forms $\omega^{(j)}_M$, for $j \in \{1,2,\dots,2n\}$, and $\hat{\omega}^{(i)}_M$ are admissible with respect to two-forms on $S^2$ denoted by $\omega^{(j)}_{S^2}$, for $j \in \{1,2,\dots,2n\}$ and $\hat{\omega}^{(i)}_{S^2}$, respectively. Note that the restriction of $\omega^{(j)}_M$ on $ST(B_M)$ determines $\omega^{(j)}_{S^2}$, and hence determines $\omega^{(j)}_M$ on $\partial C_2(M)$.
We fix a trivialisation $\tau_M$ of $T(M \setminus \infty)$ standard near $\infty$.
\begin{lemma}
\label{lemnolosseta}
There
exists a one-form $\eta_{S^2}$ on $S^2$ such that $d \eta_{S^2} = \hat{\omega}^{(i)}_{S^2}-{\omega}^{(i)}_{S^2}$, and
a one-form $\eta$ on $C_2(M)$ such that
\begin{enumerate}
\item $d\eta=\hat{\omega}^{(i)}_M-\omega^{(i)}_M$,
\item the restriction of $\eta$ on $\partial C_2(M) \setminus ST(B_M)$ is $p_M(\tau_M)^{\ast}(\eta_{S^2})$,
\item if $\hat{\omega}^{(i)}_M$ and $\omega^{(i)}_M$ are fundamental with respect to $\tau_M$, then the restriction of $\eta$ on the whole $ \partial C_2(M)$ is $p_M(\tau_M)^{\ast}(\eta_{S^2})$,
\item if $\hat{\omega}^{(i)}_M$ and $\omega^{(i)}_M$ coincide on $ST(B_M)$, then the restriction of $\eta$ on $ \partial C_2(M)$ is zero.
\end{enumerate}
\end{lemma}
\noindent {\sc Proof: }
Since $\hat{\omega}^{(i)}_M$ and $\omega^{(i)}_M$ are cohomologous there exists
$\eta$ such that $d\eta=\hat{\omega}^{(i)}_M-\omega^{(i)}_M$ on $C_2(M)$.
Similarly, there exists $\eta_{S^2}$ such that $d \eta_{S^2} = \hat{\omega}^{(i)}_{S^2}-{\omega}^{(i)}_{S^2}$. If $\hat{\omega}^{(i)}_M$ and $\omega^{(i)}_M$ coincide on $ST(B_M)$, then $\hat{\omega}^{(i)}_{S^2}={\omega}^{(i)}_{S^2}$, and we choose $ \eta_{S^2}=0$.
Now, $ d(\eta -p_M(\tau_M)^{\ast}(\eta_{S^2}))=0$ on $\partial C_2(M) \setminus ST(B_M)$, and on $ \partial C_2(M)$ if $\hat{\omega}^{(i)}_M$ and $\omega^{(i)}_M$ are fundamental with respect to $\tau_M$, or if $\hat{\omega}^{(i)}_M$ and $\omega^{(i)}_M$ coincide on $ST(B_M)$.
Thanks to the exact sequence
$$ 0=H^1(C_2(M)) \longrightarrow H^1(\partial C_2(M)) \longrightarrow H^2(C_2(M), \partial C_2(M)) \cong H_4(C_2(M))=0, $$
$H^1(\partial C_2(M))=0$. \index{N}{Ctwo@$C_2(M)$}
It is easy to see that $H^1(\partial C_2(M)\setminus ST(B_M))=0$, too.
Therefore, there exists a function $f$ from $\partial C_2(M)$ to $\mathbb{R}$ such that $$df =\eta -p_M(\tau_M)^{\ast}(\eta_{S^2})$$
on $\partial C_2(M) \setminus ST(B_M)$, and on $ \partial C_2(M)$ if $\hat{\omega}^{(i)}_M$ and $\omega^{(i)}_M$ are fundamental with respect to $\tau_M$ or if $\hat{\omega}^{(i)}_M$ and $\omega^{(i)}_M$ coincide on $ST(B_M)$.
Extend $f$ to a $C^{\infty}$ map on $C_2(M)$ and change $\eta$ into $(\eta -df)$.
\eop
Set $$z_n=\sum_{\Gamma \in {\cal E}_n} \int_{{C}_{2n}(M)}\bigwedge_{i=1}^{3n}P_i({\Gamma})^{\ast}(\omega^{(i)}_M)[{\Gamma}].$$
Set $\hat{\omega}^{(j)}_M={\omega}^{(j)}_M$ for $j \neq i$, and
let $\hat{z}_n=\sum_{\Gamma \in {\cal E}_n} \int_{{C}_{2n}(M)}\bigwedge_{i=1}^{3n}P_i({\Gamma})^{\ast}(\hat{\omega}^{(i)}_M)[{\Gamma}]$.
Set $$\tilde{\omega}^{(j)}_M= \left\{
\begin{array}{ll} {\omega}^{(j)}_M\;&\mbox{if} \; j \neq i\\
\eta \;& \mbox{if} \; j = i, \end{array}
\right.$$
and define the $(6n-1)$-form $\tilde{\Omega}=
\bigwedge_{j=1}^{3n}p_j^{\ast}(\tilde{\omega}^{(j)}_M)$ on $C_{2}(M)^{3n}$.
Then $d\tilde{\Omega}= \bigwedge_{j=1}^{3n}p_j^{\ast}(\hat{\omega}^{(j)}_M)-\bigwedge_{j=1}^{3n}p_j^{\ast}({\omega}^{(j)}_M)$.
For an element $F$ of the set $\partial_1(C_{2n}(M))$ of codimension $1$ faces of $C_{2n}(M)$ described before Proposition~\ref{propconffaceun}, set \index{N}{IGammaF@$I_{\Gamma,F}$}
$$I_{{\Gamma},F}=\int_{F}P({\Gamma})^{\ast}(\tilde{\Omega})=
\int_{F}\bigwedge_{j=1}^{3n}P_j({\Gamma})^{\ast}(\tilde{\omega}^{(j)}_M),$$
where $F$ is oriented as a part of the boundary of the oriented manifold $C_{2n}(M)$,
$$I_{{\Gamma},\partial}=\sum_{F \in \partial_1(C_{2n}(M))}I_{{\Gamma},F}.$$
Then according to the Stokes theorem, $$\hat{z}_n-z_n= \sum_{\Gamma \in {\cal E}_n} I_{{\Gamma},\partial}[\Gamma].$$
We are going to prove that several terms cancel in this sum. More precisely, we shall prove Proposition~\ref{propunb} that obviously implies Proposition~\ref{propun}, and Proposition~\ref{propzad} since
$$Z(\omega_M)=\exp\left(\left(\frac{1}{2^{3n}(3n)!(2n)!}z_n(\omega_M)\right)_n\right)$$
with $$z_n(\omega_M)=\sum_{\Gamma \in {\cal E}_n} \int_{{C}_{2n}(M)}\bigwedge_{i=1}^{3n}P_i({\Gamma})^{\ast}(\omega_M)[{\Gamma}].$$
\begin{proposition}
\label{propunb}
With the notation above,
$$\hat{z}_n-z_n= \sum_{\Gamma \in {\cal E}_n} I_{{\Gamma},F(V)}[\Gamma]$$
where $I_{{\Gamma},F(V)}=0$ for any $\Gamma \in {\cal E}_n$ if $\hat{\omega}^{(i)}_M$ and $\omega^{(i)}_M$ are fundamental with respect to $\tau_M$, or if $\hat{\omega}^{(i)}_M$ and $\omega^{(i)}_M$ coincide on $ST(B_M)$.
\end{proposition}
When $B$ is a subset of $V$, and when a graph $\Gamma$ is given, $E_B$ denotes the set of edges of $\Gamma$ that contain two elements of $B$,
and $\Gamma_B$ \index{N}{GammaB@$\Gamma_B$}
is the subgraph of $\Gamma$ made of the vertices of $B$ and the edges of $E_B$.
\begin{lemma}
\label{lemfaceinf}
For any non-empty subset $B$ of $V$, for any $\Gamma \in {\cal E}_n$,
$I_{{\Gamma},F(\infty;B)}=0$.
\end{lemma}
\noindent {\sc Proof: }
Set $A = V \setminus B$.
Let $E_C$ be the set of the edges of $\Gamma$ that contain an element of $A$ and an element of $B$.
Let $p_2$ denote the projection of $F(\infty;B)$ onto $S_i(T_{\infty}M^B)$.
For $e \in E^B \cup E^C$, $P_e:(S^2)^{E^B \cup E^C} \longrightarrow S^2$ is the projection onto the factor indexed by $e$.
We show that there exists a smooth map
$$g : S_i(T_{\infty}M^B) \longrightarrow (S^2)^{E^B \cup E^C}$$
such that
$$\bigwedge_{e \in E^B \cup E^C}p_e^{\ast}(\tilde{\omega}^{(i(e))}_M)
=(g \circ p_2)^{\ast}\left( \bigwedge_{e \in E^B \cup E^C}P_e^{\ast}(\tilde{\omega}^{(i(e))}_{S^2}) \right)$$
where $i(e) \in \{1,2, \dots, 3n\}$ is the label of the edge $e$, and
$$\tilde{\omega}^{(i(e))}_{S^2}= \left\{
\begin{array}{ll} {\omega}^{(i(e))}_{S^2}\;&\mbox{if} \; i(e) \neq i\\
\eta_{S^2}\;& \mbox{if} \; i(e) = i. \end{array}
\right.$$
Indeed, if $e \in E^B \cup E^C$, $p_e(F(\infty;B)) \subset \partial C_2(M)$, $$p_e^{\ast}(\tilde{\omega}^{(i(e))}_M)=(p_M(\tau_M) \circ p_e)^{\ast}(\tilde{\omega}^{(i(e))}_{S^2}),$$
and $p_M(\tau_M) \circ p_e$ factors through $S_i(T_{\infty}M^B)$ (and therefore reads
$((P_e \circ g) \circ p_2)$). Indeed,
if $e \in E^C$, $p_M(\tau_M) \circ p_e$ only depends on the projection on $S(T_{\infty}M)$ of the vertex at $\infty$ (of $B$),
while, if $e \in E^B$, $p_M(\tau_M) \circ p_e$ factors through $S_i(T_{\infty}M^e)$.
Therefore if the degree of the form $\left( \bigwedge_{e \in E^B \cup E^C}p_e^{\ast}(\tilde{\omega}^{(i(e))}_{S^2}) \right)$ is bigger than the dimension $(3 \sharp B-1)$ of $S_i(T_{\infty}M^B)$, this form vanishes on $F(\infty;B)$.
The degree of the form
is $( 2\sharp E^B + 2\sharp E^C)$ or $( 2\sharp E^B + 2\sharp E^C-1)$, while
$$(3 \sharp B-1)= 2\sharp E^B + \sharp E^C-1.$$
Therefore, the integral vanishes unless $E^C$ is empty.
In this case, since $\Gamma$ is connected, $B=V$,
$F(\infty;V)= S_i(T_{\infty}M^V)$, all the $p_M(\tau_M) \circ p_e$
locally factor through the conjugates under the inversion $(x \mapsto x/\norm{x}^2)$ of the translations that make sense, and the form vanishes, too.
\eop
As soon as there exists a smooth map from $F(B)$ to a manifold of strictly smaller
dimension that factorizes $P(\Gamma)$, then $I_{{\Gamma},F(B)}=0$.
We shall use this principle to get rid of some faces.
\begin{lemma}
\label{lemdiscon}
Let $\Gamma \in {\cal E}_n$. For any subset $B$ of $V$ such that $\Gamma_B$ is not connected,
$I_{{\Gamma},F(B)}=0$.
\end{lemma}
\noindent {\sc Proof: } Indeed, in the fiber $\breve{S}_B(T_{c(b)}M)$ we may translate
one connected component of $\Gamma_B$ whose set of vertices is $C$ independently. This amounts to factorize the $p_e$ through $\breve{C}_{\{b\} \cup(V \setminus B)}(M)$ if $\sharp B=2$, or through the fibered space over $\breve{C}_{\{b\} \cup(V \setminus B)}(M)$ whose fiber is an open subspace of
$$S\left(\frac{T_{c(b)}M^B}
{\mbox{diag}(T_{c(b)}M^B) \oplus (0^{B \setminus C} \times \mbox{diag}(T_{c(b)}M^C))}
\right).$$ In both cases, all the $p_e$ factor through
a space with smaller dimension.
\eop
\begin{lemma}
\label{lemedge}
Let $\Gamma \in {\cal E}_n$. Let $B$ be a subset of $V$ such that $\sharp B \geq 3$. If some element of $B$ belongs to exactly one edge of $\Gamma_B$, then
$I_{{\Gamma},F(B)}=0$.
\end{lemma}
\noindent {\sc Proof: } Let $b$ be the mentioned element, and let $e$ be its edge in $\Gamma_B$, let $d \in B$ be the other element of $e$. The group $]0,\infty[$ acts on the map $t$ from $B$ to $T_{c(b)}M$ by moving $t(b)$ on the half-line from $t(d)$ through $t(b)$. ($(t(b)-t(d))$ is multiplied by a scalar). When $\sharp B \geq 3$, this action is non trivial on $\breve{S}_B(T_{c(b)}M)$, $P(\Gamma)$
factors through the quotient of $F(B)$ by this action that has one less dimension.
\eop
\begin{lemma}
\label{lemsym}
Let $\Gamma \in {\cal E}_n$. Let $B$ be a subset of $V$ such that at least one element of $B$ belongs to exactly two edges of $\Gamma_B$. Let ${\cal E}(\Gamma)$ denote the set of labelled edge-oriented graph that are isomorphic to $\Gamma$ by an isomorphism that preserves the labels of the vertices, but that may change the labels and the orientations of the edges. Then
$$\sum_{\tilde{\Gamma}; \tilde{\Gamma} \in {\cal E}(\Gamma)}I_{\tilde{\Gamma},F(B)}[\tilde{\Gamma}]=0.$$
\end{lemma}
\noindent {\sc Proof: } Let $v_m$ be the vertex of $B$ with smallest label $m \in \{1,2,\dots, 2n\}$ that belongs to exactly two edges of $\Gamma_B$.
We first describe an orientation-reversing diffeomorphism of the complement of a codimension $3$ submanifold of $F(B)$. Let $v_j$ and $v_k$ denote the (possibly
equal) two other vertices of the two edges of $\Gamma_B$ that contain $v_m$.
Consider the linear transformation $S$ of the space $S(T_{c(b)}M^B/\mbox{diag}(T_{c(b)}M^B))$ of non-constant maps $f$ from $B$ to $T_{c(b)}M$
up to translations and dilations, that maps $f$ to $S(f)$ where\\
$S(f(v_{\ell}))=f(v_{\ell})$ if $v_\ell \neq v_m$, and,\\
$S(f(v_m))=f(v_j)+f(v_k)-f(v_m)$.\\
This is an orientation-reversing involution of $S(T_{c(b)}M^B/\mbox{diag}(T_{c(b)}M^B))$.
The set of elements of $\breve{S}_B(T_{c(b)}M)$
whose image under $S$ is not in $\breve{S}_B(T_{c(b)}M)$ is a codimension $3$ submanifold of $\breve{S}_B(T_{c(b)}M)$.
The fibered product of $S$ by the identity of the base $\breve{C}_{\{b\} \cup(V \setminus B)}(M)$ is an orientation-reversing smooth involution outside a codimension $3$ submanifold $F_S$ of $F(B)$. It is still denoted by $S$.
Now, let $\sigma(B;\Gamma)(\tilde{\Gamma})$ be obtained from $(\tilde{\Gamma} \in {\cal E}(\Gamma))$ by reversing the orientations of the edges of $\Gamma_B$ that contain $v_m$ and by exchanging
their labels. Then, as the following picture shows,
$$P(\tilde{\Gamma}) \circ S=P(\sigma(B;\Gamma)(\tilde{\Gamma})).$$
$$\begin{pspicture}[.2](0,0)(8,2.5)
\psset{xunit=1.2cm,yunit=1.2cm}
\rput[b](5.5,2.1){$a$}
\rput[t](2.5,.4){$\sigma(B;\Gamma)(a)$}
\rput[l](5.6,1.1){$b$}
\rput[r](2.2,1.3){$\sigma(B;\Gamma)(b)$}
\rput[r](.9,.6){$f(v_m)$}
\rput[r](3.9,2.1){$f(v_k)$}
\rput[l](4.2,.4){$f(v_j)$}
\rput[l](7.2,1.9){$S(f(v_m))$}
\psline{->}(4,.5)(5.4,1.2)
\psline{-}(5.4,1.2)(7,2)
\psline{->}(1,.5)(2.4,1.2)
\psline{-}(2.4,1.2)(4,2)
\psline{*->}(4,2)(5.5,2)
\psline{-*}(5.5,2)(7,2)
\psline{*->}(1,.5)(2.5,.5)
\psline{-*}(2.5,.5)(4,.5)
\end{pspicture}$$
Therefore,
$$\begin{array}{ll}
I_{\tilde{\Gamma},F(B)} &=\int_{F(B) \setminus F_S}P(\tilde{\Gamma})^{\ast}(\tilde{\Omega})\\
&=-\int_{F(B) \setminus F_S}S^{\ast}\left(P(\tilde{\Gamma})^{\ast}(\tilde{\Omega})\right)\\
&=-\int_{F(B) \setminus F_S}(P(\tilde{\Gamma}) \circ S)^{\ast}(\tilde{\Omega})\\ &=-\int_{F(B) \setminus F_S}P(\sigma(B;\Gamma)(\tilde{\Gamma}))^{\ast}(\tilde{\Omega})\\
&=-I_{\sigma(B;\Gamma)(\tilde{\Gamma}),F(B)}\end{array}$$
while $[\tilde{\Gamma}]=[\sigma(B;\Gamma)(\tilde{\Gamma})]$.
Now, $\sigma(B;\Gamma)$ defines an involution of ${\cal E}(\Gamma)$, and it is easy to conclude:
$$\sum_{\tilde{\Gamma}; \tilde{\Gamma} \in {\cal E}(\Gamma)}I_{\tilde{\Gamma},F(B)}[\tilde{\Gamma}]=\sum_{\tilde{\Gamma}; \tilde{\Gamma} \in {\cal E}(\Gamma)}I_{\sigma(B;\Gamma)(\tilde{\Gamma}),F(B)}[\sigma(B;\Gamma)(\tilde{\Gamma})]$$
$$=-\sum_{\tilde{\Gamma}; \tilde{\Gamma} \in {\cal E}(\Gamma)}I_{\tilde{\Gamma},F(B)}[\tilde{\Gamma}]=0.$$
\eop
The symmetry used in the above proof was observed by Kontsevich in \cite{ko}.
The three previous lemmas allow us to get rid of the pairs $(B;\Gamma)$ with $\sharp B \geq 3$ such that at least one element of $B$ does not belong to three edges. Therefore, since the $\Gamma$ are connected, we are left
with the pairs $(B;\Gamma)$ with $B=V$, that are treated by Lemma~\ref{lemfacetot} below, and with the pairs $(B;\Gamma)$
where $B \neq V$, $\sharp B=2$, and at least one element belongs to
exactly one edge of $\Gamma_B$. The following lemma allows us to get rid of this latter case where $\Gamma_B$ must be an edge.
\begin{lemma}
\label{lemihx}
Let $\Gamma \in {\cal E}_n$. Let $B$ be a subset of $V$
such that $\Gamma_B$ is made of an edge $e(\ell)$ with label $\ell$ oriented from a vertex $v_j$ to a vertex $v_k$.
Let $\Gamma/\Gamma_B$ be the labelled edge-oriented graph obtained from $\Gamma$
by collapsing $\Gamma_B$ down to one point.
(The labels of the edges of $\;\Gamma/\Gamma_B$ belong to $\{1, 2, \dots, 3n\} \setminus \{\ell\}$, the labels of the vertices of $\;\Gamma/\Gamma_B$ belong to $\{1, 2, \dots, 2n\} \setminus \{k\}$, $\Gamma/\Gamma_B$ has one four-valent vertex $(v_j=v_k)$ and its other vertices are trivalent.)
Let ${\cal E}(\Gamma;B)$ be the subset of ${\cal E}_n$ that contains the graphs $\tilde{\Gamma}$ whose edge with label $\ell$ goes from $v_j$ to $v_k$ and such that $\Gamma/\Gamma_B$ is equal to $\tilde{\Gamma}/\tilde{\Gamma}_B$. Then
$$\sum_{\tilde{\Gamma}; \tilde{\Gamma} \in {\cal E}(\Gamma;B)}I_{\tilde{\Gamma},F(B)}[\tilde{\Gamma}]=0.$$
\end{lemma}
\noindent {\sc Proof: }
$F(B)$ is fibered over $\breve{C}_{V \setminus \{v_k\}}(M)$ with fiber $ST_{c(v_j)}M$ that contains the direction of the vector from $c(v_j)$ to $c(v_k)$. The oriented face $F(B)$ and the map
$$P(\tilde{\Gamma}): (F(B) \subset C_{2n}(M)) \longrightarrow C_2(M)^{3n}$$
are the same for all the elements $\tilde{\Gamma}$ of ${\cal E}(\Gamma;B)$.
Therefore $I_{\tilde{\Gamma},F(B)}$ is the same for all the elements $\tilde{\Gamma}$ of ${\cal E}(\Gamma;B)$, the sum of the statement is
$$\sum_{\tilde{\Gamma}; \tilde{\Gamma} \in {\cal E}(\Gamma;B)}I_{\tilde{\Gamma},F(B)}[\tilde{\Gamma}]
=I_{{\Gamma},F(B)}\sum_{\tilde{\Gamma}; \tilde{\Gamma} \in {\cal E}(\Gamma;B)}[\tilde{\Gamma}],$$
and we are left with the study of the set ${\cal E}(\Gamma;B)$.
Let $\tilde{\Gamma} \in {\cal E}(\Gamma;B)$.
Let $a,b,c,d$ be the four half-edges of
$\Gamma/\Gamma_B$ that contain $(v_j=v_k)$. Let $e_1$ be the first half-edge of $e(\ell)$ that contains
$v_j$, and let $e_2$ be the other half-edge of $e(\ell)$. Then in $\tilde{\Gamma}$, $v_j$ belongs to $e_1$ and to two half-edges of $\{a,b,c,d\}$, and the corresponding unordered pair
determines $\tilde{\Gamma}$ as an edge-oriented labelled graph.
Thus, there are 6 graphs in ${\cal E}(\Gamma;B)$ labelled by the pairs of elements
of $\{a,b,c,d\}$.
Equip $\Gamma=\Gamma_{ab}$ with a vertex-orientation that reads
$(a,b,e_1)$ at $v_j$ and $(c,d,e_2)$ at $v_k$ and that is consistent
with its given edge-orientation (i.e. such that the edge-orientation of $H(\Gamma)$
is equivalent to its vertex-orientation).
A representative of the orientation of $H(\Gamma)$ reads
$( \dots, a,b,e_1, \dots , c,d,e_2, \dots )$
and is equivalent to the edge-orientation of $H(\Gamma)$ that is the same for all the elements of ${\cal E}(\Gamma;B)$.
Thus, cyclically permuting the letters $b,c,d$ gives rise to two other graphs
in ${\cal E}(\Gamma;B)$ equipped with a suitable vertex-orientation, that respectively reads\\
$(a,c,e_1)$ at $v_j$ and $(d,b,e_2)$ at $v_k$, or\\
$(a,d,e_1)$ at $v_j$ and $(b,c,e_2)$ at $v_k$,\\
The three other elements of ${\cal E}(\Gamma;B)$ with their suitable vertex-orientation are obtained from the three previous ones by exchanging the ordered pair before $e_1$ with the ordered pair before $e_2$. This amounts to exchanging the vertices $v_j$ and $v_k$ in the picture, and does not change the unlabelled vertex-oriented graph. The first three graphs can be represented by three graphs identical outside the pictured disk:
$$\begin{pspicture}[.2](0,-.2)(1.6,1)
\psset{xunit=1.2cm,yunit=1.2cm}
\rput[r](.05,1){$a$}
\rput[r](.2,0){$b$}
\rput[l](.8,0){$c$}
\rput[l](.55,1){$d$}
\rput[l](.55,.55){$v_k$}
\psline{-*}(.1,1)(.35,.2)
\psline{*-}(.5,.5)(.5,1)
\psline{-}(.75,0)(.5,.5)
\psline{->}(.25,0)(.5,.5)
\end{pspicture}
\;\; \mbox{,} \;\;
\begin{pspicture}[.2](0,-.2)(1.6,1)
\psset{xunit=1.2cm,yunit=1.2cm}
\rput[r](.05,1){$a$}
\rput[r](.2,0){$b$}
\rput[l](.8,0){$c$}
\rput[l](.55,1){$d$}
\rput[l](.55,.65){$v_k$}
\psline{*-}(.5,.6)(.5,1)
\psline{->}(.8,0)(.5,.6)
\psline{-}(.2,0)(.5,.6)
\pscurve[border=2pt]{-*}(.1,1)(.3,.3)(.7,.2)
\end{pspicture}
\;\; \mbox{and} \;\;
\begin{pspicture}[.2](0,-.2)(1.6,1)
\psset{xunit=1.2cm,yunit=1.2cm}
\rput[r](.05,1){$a$}
\rput[r](.2,0){$b$}
\rput[l](.8,0){$c$}
\rput[l](.55,1.05){$d$}
\rput[l](.55,.4){$v_k$}
\psline{<-}(.5,.35)(.5,1)
\psline{-*}(.75,0)(.5,.35)
\psline{-}(.25,0)(.5,.35)
\pscurve[border=2pt]{-*}(.1,1)(.2,.75)(.7,.75)(.5,.85)
\end{pspicture}
$$
Then the sum $\sum_{\tilde{\Gamma}; \tilde{\Gamma} \in {\cal E}(\Gamma;B)}[\tilde{\Gamma}]$ is zero thanks to IHX. \index{N}{IHX}
\eop
\begin{lemma}
\label{lemfacetot}
For any $\Gamma \in {\cal E}_n$,
$I_{{\Gamma},F(V)}=0$ if $\hat{\omega}^{(i)}_M$ and $\omega^{(i)}_M$ are fundamental with respect to $\tau_M$, or if $\hat{\omega}^{(i)}_M$ and $\omega^{(i)}_M$ coincide on $ST(B_M)$.
\end{lemma}
In the first case, the face $F(V)$ is identified via $\tau_M$ to $\breve{S}_V(\mathbb{R}^3) \times (M \setminus \infty)$, and the form $\tilde{\Omega}$ to be integrated can be pulled-back through the projection
onto the fiber. In the second case, for any $j \in \{1,2,\dots, 3n\}$, $p_j$ maps $F(V)$ into
$\partial C_2(M)$, and therefore $P(\Gamma)^{\ast}(\tilde{\Omega})=0$ on $F(V)$
thanks to Lemma~\ref{lemnolosseta}.
\eop
This ends the proof of
Proposition~\ref{propunb}, and hence the proofs of Proposition~\ref{propun}
and \ref{propzad}.
Since for any admissible form $\omega_M$ on $C_2(M)$, $z_n(\omega_M)$ only depends on
the restriction of $\omega_M$ to $ST(B_M)$, $z_n(\omega_M)$ will also be denoted by $z_n(\omega_{M|{ST(B_M)}})$.
\begin{proposition}
\label{propunbb}
For any admissible form $\omega$ on $C_2(M)$ and for any one-form $\eta$ on $C_2(M)$ that
reads $p_M(\tau_M)^{\ast}(\eta_{S^2})$ on $\partial C_2(M) \setminus ST(B_M)$, for some one-form $\eta_{S^2}$ on $S^2$ and for some trivialisation $\tau_M$ that is standard near $\infty$,
$$z_n(\omega +d \eta)-z_n(\omega)$$
$$=\sum_{\Gamma \in {\cal E}_n} \int_{F(V)}\sum_{i=1}^{3n}\left(\bigwedge_{j=1}^{i-1}P_j({\Gamma})^{\ast}(\omega) \wedge P_i({\Gamma})^{\ast}(\eta) \wedge \bigwedge_{j=i+1}^{3n}P_j({\Gamma})^{\ast}(\omega+d\eta)\right)[{\Gamma}].$$
\end{proposition}
\noindent {\sc Proof: }
Indeed, according to Proposition~\ref{propunb}, $$z_n(\left(\bigwedge_{j=1}^{i-1}P_j^{\ast}(\omega) \wedge \bigwedge_{j=i}^{3n}P_j^{\ast}(\omega+d\eta)\right))-z_n(\left(\bigwedge_{j=1}^{i}P_j^{\ast}(\omega) \wedge \bigwedge_{j=i+1}^{3n}P_j^{\ast}(\omega+d\eta)\right))=$$
$$= \sum_{\Gamma \in {\cal E}_n}
\int_{F(V)}\left(\bigwedge_{j=1}^{i-1}P_j({\Gamma})^{\ast}(\omega) \wedge P_i({\Gamma})^{\ast}(\eta) \wedge \bigwedge_{j=i+1}^{3n}P_j({\Gamma})^{\ast}(\omega+d\eta)\right)[{\Gamma}],$$
and the above statement is nothing but the sum over the $i$ in $\{1,\dots, 3n\}$ of these equalities.
\eop
\subsection{Forms over $S^2$-bundles}
\label{subsproofdeux}
\noindent{\sc Proof of Proposition~\ref{propdeux}:}
According to Proposition~\ref{propconffacedeux},
the codimension one faces of $S_V(E_1)$ are the fibered spaces over $S^4$
with fibers $f(B)(p^{-1}(x))$, for all the strict subsets $B$ of $V$ with cardinality at least $2$.
Then the independence of $\omega_T$ is proved as in the previous subsection,
using lemmas similar to Lemmas~\ref{lemdiscon}, \ref{lemedge}, \ref{lemsym}, \ref{lemihx}
that treat all the possible faces, and Proposition~\ref{propdeux} is proved.
\eop
Note that the proofs of these lemmas in fact show that
the image of $S_{2n}(E_1)$ under $\sum_{\Gamma \in {\cal E}_n}P(\Gamma)[\Gamma]$
is a
cycle whose homology class is in $H_{6n}(S_2(E_1)^{3n};{\cal A}(\emptyset)) $ even if ${\cal A}(\emptyset)$ is defined with integral coefficients. (Its boundary $$\sum_{(\Gamma,F);\Gamma \in {\cal E}_n, F \in \partial_1(S_{V}(E_1))}[P(\Gamma)(F)][\Gamma]$$ vanishes algebraically. ) Then $2^{3n}(2n)!(3n)!\xi_n$ \index{N}{ksin@$\xi_n$} is just
the evaluation of $\bigwedge_{i=1}^{3n}p_i^{\ast}[\omega_T]$ at the class of this cycle.
More generally, we have the following proposition:
\begin{proposition}
\label{propbunind}
Let $E$ be an $\mathbb{R}^3$-bundle over a base $W$ that is an oriented four-dimensional manifold.
Let $\eta$ denote a one-form on $S_2(E)$ and let $\omega$ denote a closed
two-form on $S_2(E)$.
Let \index{N}{znEomega@$z_n(E;\omega)$}
$$z_n(E;\omega)=\sum_{\Gamma \in {\cal E}_n} \int_{\breve{S}_{2n}(E)}\bigwedge_{i=1}^{3n}P_i({\Gamma})^{\ast}(\omega)[{\Gamma}].$$
Then $z_n(E;\omega+d\eta)-z_n(E;\omega)=\delta_n(E;\omega,\eta)$
with $\delta_n(E;\omega,\eta)=$
$$\sum_{\Gamma \in {\cal E}_n} \int_{\breve{S}_{2n}(E_{|\partial W})}\sum_{i=1}^{3n}\left(\bigwedge_{j=1}^{i-1}P_j({\Gamma})^{\ast}(\omega) \wedge P_i({\Gamma})^{\ast}(\eta) \wedge \bigwedge_{j=i+1}^{3n}P_j({\Gamma})^{\ast}(\omega+d\eta)\right)[{\Gamma}].$$
\end{proposition}
\noindent {\sc Proof: }
The contributions of the faces coming from the boundary
of $S_{2n}(\mathbb{R}^3)$ cancel as in the above case and we are left with the contributions
coming from the boundary of $W$. \eop
\begin{lemma}
\label{lemdepboun}
Let $W$ be a connected oriented compact four-dimensional manifold, let $E$ be the trivial $\mathbb{R}^3$-bundle $E=W \times \mathbb{R}^3$, and let $\omega$ denote a closed
two-form on $S_2(E)$. If the inclusion induces an injection from $H^2(W)$ to $H^2(\partial W)$ and a surjection from $H^1(W)$ to $H^1(\partial W)$, then
$z_n(E;\omega)$ \index{N}{znEomega@$z_n(E;\omega)$}
only depends on the restriction of $\omega$ to $\partial W \times S^2$.
\end{lemma}
\noindent {\sc Proof: } Indeed, a closed form $\omega^{\prime}$ that coincides with $\omega$ on $\partial W \times S^2$ would read $(\omega+d\eta)$ for some one-form $\eta$ whose restriction
to the boundary $\partial W \times S^2$ is closed and may be extended to $W \times S^2$ as a closed form $\eta^{\prime}$. Thus, $\omega^{\prime}=\omega +d(\eta -\eta^{\prime})$ and since $(\eta -\eta^{\prime})$ vanishes on $\partial W \times S^2$, Proposition~\ref{propbunind} guarantees that $z_n(E;\omega)=$
$z_n(E;\omega^{\prime})$.
\eop
Here, a {\em bundle morphism\/} $\psi$ from an $\mathbb{R}^3$-bundle $E$ to another one $E^{\prime}$ will always restrict to an isomorphism from a fiber of $E$
to a fiber of $E^{\prime}$. Such a bundle morphism induces bundle morphisms
that are still denoted by $\psi$ from $S_n(E)$ to $S_n(E^{\prime})$ for
every $n$.
Note that such a bundle morphism of $\mathbb{R}^3$-bundles is determined by
$\psi: S_2(E) \longrightarrow S_2(E^{\prime})$
up to a multiplication by a function from the base of $E$ to $\mathbb{R}$, that preserves all the maps $\psi: S_n(E) \longrightarrow S_n(E^{\prime})$.
\begin{lemma}
\label{lemtranspform}
Let $E$ be an $\mathbb{R}^3$-bundle over a base $W$ that is an oriented four-dimensional manifold and let $\omega$ denote a closed
two-form on $S_2(E)$.
Assume that there exist a bundle morphism $\psi$ from $E$ to an $\mathbb{R}^3$-bundle $E(X)$ over a base $X$, and a closed
two-form $\omega(X)$ on $S_2(E(X))$ such that $\omega=\psi^{\ast}(\omega(X))$.
If $X$ is a manifold of dimension $< 4$,
then $z_n(E;\omega)=0$, \index{N}{znEomega@$z_n(E;\omega)$}
and if
$\psi$ is an orientation-preserving diffeomorphism, then $z_n(E;\omega)=z_n(E(X);\omega(X))$.
\end{lemma}
\noindent {\sc Proof: }
Indeed, since the maps $\psi$ commute with the $P_i(\Gamma)$,
$$\int_{\breve{S}_{2n}(E)}\bigwedge_{i=1}^{3n}P_i({\Gamma})^{\ast}(\psi^{\ast}(\omega(X)))
=\int_{\breve{S}_{2n}(E)}\psi^{\ast}\left(\bigwedge_{i=1}^{3n}P_i({\Gamma})^{\ast}(\omega(X))\right).$$
Therefore, if $X$ is of dimension $< 4$, the dimension of $\breve{S}_{2n}(E(X))$
is less than the dimension of $\breve{S}_{2n}(E)$ and the integral vanishes.
If $\psi$ is an orientation-preserving diffeomorphism, then it induces an
orientation-preserving diffeomorphism from $\breve{S}_{2n}(E)$ to $\breve{S}_{2n}(E(X))$.
\eop
\subsection{The dependence on the trivializations}
\label{subdeptriv}
The closure of the face $F(V)$ \index{N}{FV@$F(V)$} in $C_V(M)$ is diffeomorphic via $\tau_M$ to $C_1(M) \times S_V(\mathbb{R}^3)$. When $(S_V=S_V(\mathbb{R}^3))$ is oriented as in Subsection~\ref{subfra}, the involved diffeomorphism preserves the orientation.
Since
any ordered pair $P$ included into $V=\{1,2,\dots, 2n\}$ gives rise to a restriction map
from $S_{2n}=S_V$ to $S_P=S_2=S^2$,
any edge-oriented labelled graph $\Gamma$ again induces a smooth map
$$P(\Gamma):S_{2n}(\mathbb{R}^3) \longrightarrow (S^2)^{3n}$$ whose $i^{th}$ projection $P_i(\Gamma)$ is the map associated to the edge labelled by $i$.
The key-proposition to study how $z_n(\omega_M)$ depends on the restriction of $\omega_M$ to $ST(B_M)$
is the following one.
\begin{proposition}
\label{propkeytriv}
Let $\omega_0$ and $\omega_1$ be two admissible two-forms on $C_2(M)$
that coincide on $\partial C_2(M) \setminus ST(B_M)$.
Let $\tau_M$ be a trivialisation of
$(M \setminus \infty)$ that is standard near $\infty$.
Identify $ST(B_M)$ to $B_M \times S^2$ with respect to $\tau_M$.
Then there exists a closed two-form $\omega$ on $ [0,1] \times B_M \times S^2 $
such that
\begin{itemize}
\item
$\omega$ coincides with
$\pi_{B_M \times S^2}^{\ast}\omega_1$ on $(\{1\} \times B_M \cup [0,1] \times \partial B_M) \times S^2$ and,
\item
$\omega$ coincides with
$\omega_0$ on $\{0\} \times B_M \times S^2 $,
\end{itemize}
and, for any such two-form $\omega$,
$$z_n(\omega_1)-z_n(\omega_0)=z_n([0,1] \times B_M \times \mathbb{R}^3;\omega)$$
where $$z_n([0,1] \times B_M \times \mathbb{R}^3;\omega)
=\sum_{\Gamma \in {\cal E}_n} \int_{[0,1] \times B_M \times S_V(\mathbb{R}^3)} \bigwedge_{i=1}^{3n}P_i(\Gamma)^{\ast}(\omega)[\Gamma].$$
\end{proposition}
\noindent {\sc Proof: }
First, the two-form $\omega$ exists because
the restriction induces an isomorphism from $H^2([0,1] \times B_M \times S^2;\mathbb{R})$ to $H^2(\partial([0,1] \times B_M \times S^2);\mathbb{R})$. See Lemma~\ref{lemdr}.
Next, $z_n([0,1] \times B_M \times \mathbb{R}^3;\omega)$ is independent of
the chosen closed extension $\omega$ by Lemma~\ref{lemdepboun}.
Now, $\left(z_n(\omega_0)+z_n([0,1] \times B_M \times \mathbb{R}^3;\omega)\right)$
is independent of $\omega_0$ because $[0,1] \times B_M \times S_V(\mathbb{R}^3)$
can be glued to $C_V(M)$ along the closure of $F(V)_{|B_M}$ that is identified to $\{0\} \times B_M \times S_V(\mathbb{R}^3)$ via $\tau_M$. The details of this argument can be written as follows.
Let $\tilde{\omega}_0$ be another admissible form on $\partial C_2(M)$ that coincides with $\omega_1$ outside $ST(B_M)$, and let
$\tilde{\omega}=\omega +d \eta$ be a closed two-form on
$[0,1] \times B_M \times S^2$ that
coincides\\
with $\tilde{\omega}_0$ on $\{0\} \times B_M \times S^2 $, and \\
with $\pi_{B_M \times S^2}^{\ast}\omega_1$ on $(\{1\} \times B_M \cup [0,1] \times \partial B_M) \times S^2 $. \\
We assume that $\eta$ vanishes
on $(\{1\} \times B_M \cup [0,1] \times \partial B_M) \times S^2$ without loss because the $H^1$ of this space is trivial.
In particular, according to Proposition~\ref{propbunind},
$$z_n([0,1] \times B_M \times \mathbb{R}^3;\tilde{\omega}) - z_n([0,1] \times B_M \times \mathbb{R}^3;\omega)=$$
$$-\sum_{\Gamma \in {\cal E}_n} \int_{0 \times B_M \times S_V(\mathbb{R}^3)}\sum_{i=1}^{3n}\left(\bigwedge_{j=1}^{i-1}P_j({\Gamma})^{\ast}(\omega_0) \wedge P_i({\Gamma})^{\ast}(\eta) \wedge \bigwedge_{j=i+1}^{3n}P_j({\Gamma})^{\ast}(\tilde{\omega}_0)\right)[{\Gamma}].$$
Similarly, $\tilde{\omega}_0$ and $\omega_0$ extend to $C_2(M)$ as $\tilde{\omega}_M$ and ${\omega}_M$, respectively, and there exists a one-form
$\eta^{\prime}$ on $C_2(M)$ such that $\tilde{\omega}_M=\omega_M +d \eta^{\prime}$
where $\eta^{\prime}$ vanishes on $\partial C_2(M) \setminus ST(B_M)$, and coincides with $\eta$ on $ST(B_M)$. Then
according to Proposition~\ref{propunbb},
$$z_n(\tilde{\omega}_M) - z_n(\omega_M)= - (z_n([0,1] \times B_M \times \mathbb{R}^3;\tilde{\omega}) - z_n([0,1] \times B_M \times \mathbb{R}^3;\omega)).$$
In particular, when $\tilde{\omega}_0=\omega_1$, we can choose the extension $\tilde{\omega}=\pi_{B_M \times S^2}^{\ast}(\omega_1)$ where
$$\pi_{B_M \times S^2}: [0,1] \times B_M \times \mathbb{R}^3 \longrightarrow \{1\} \times B_M \times \mathbb{R}^3$$
is the natural bundle morphism
and we
have $$z_n(\omega_0)+z_n([0,1] \times B_M \times \mathbb{R}^3;\omega)=$$
$$=z_n(\omega_1)+z_n([0,1] \times B_M \times \mathbb{R}^3;\pi_{B_M \times S^2}^{\ast}(\omega_1))$$
where, according to Lemma~\ref{lemtranspform}, $$z_n([0,1] \times B_M \times \mathbb{R}^3;\pi_{B_M \times S^2}^{\ast}(\omega_1))=0.$$
\eop
\begin{lemma}
If $\tilde{\tau}_M$ is a trivialization homotopic to $\tau_M$, then
$z_n(\tau_M)=z_n(\tilde{\tau}_M)$.
\end{lemma}
\noindent {\sc Proof: } When $\tilde{\tau}_M$ is homotopic to $\tau_M$,
there exists $g: [0,1] \times B_M \longrightarrow GL^+(\mathbb{R}^3)$ such that
$g$ maps a neighborhood of $([0,1] \times(B_M \setminus B_M(1)) \cup \{1\} \times B_M )$ to $1$, and, if $\tau_M(v \in T_m M)=(m,u \in \mathbb{R}^3)$, then
$\tilde{\tau}_M(v \in T_m M)=(m,g(0,m)(u))$. The map $g$ induces the bundle-morphism
$$\begin{array}{llll}\phi(g):&[0,1] \times B_M \times \mathbb{R}^3 &\longrightarrow &\mathbb{R}^3\\
&(t,m,v)& \mapsto & g(t,m)(v).
\end{array}$$
such that $\phi(g)^{\ast}(\omega_{S^2})$ satisfies the hypotheses of Proposition~\ref{propkeytriv}, and
$$z_n(\tau_M)-z_n(\tilde{\tau}_M)=z_n([0,1] \times B_M \times S^2;\phi(g)^{\ast}(\omega_{S^2})).$$
Then thanks to Lemma~\ref{lemtranspform}, the right-hand side vanishes.
\eop
This lemma concludes the proof of the first part of Proposition~\ref{proptrois}.
\begin{lemma}
\label{lemtrivind}
Let $G: M \setminus \infty \longrightarrow GL^+(\mathbb{R}^3)$ map $(M \setminus \infty) \setminus B_M(1)$ to $1$. Then
$z_n(\psi(G) \circ \tau_M)-z_n(\tau_M)$ is independent of $\tau_M$.
\end{lemma}
\noindent {\sc Proof: }
Let $\tilde{\tau}_M$ be another trivialisation. Then, there exists
$g:(M \setminus \infty) \longrightarrow GL^+(\mathbb{R}^3)$
such that $\tilde{\tau}_M=\psi(g) \circ \tau_M $.
The map $g$ induces automorphisms $\psi(g)$ on all $((M \setminus \infty) \times S_V)$.
Furthermore on $B_M \times S^2$,
$p_M(\tau_M) =p_{S^2}$,
$p_M(\tilde{\tau}_M)= p_{S^2} \circ \psi(g) $,
$p_M( \psi(G) \circ \tau_M)= p_{S^2} \circ \psi(G)$, and
$p_M(\psi(G) \circ \tilde{\tau}_M)=p_{S^2} \circ \psi(G) \circ \psi(g)$.
Thus, when $\omega$ is suitable to compute $\left(z_n(\tau_M)-z_n(\psi(G) \circ \tau_M)\right)$, according to Proposition~\ref{propkeytriv}, $\psi(g)^{\ast}(\omega)$ is suitable to compute $\left(z_n(\tilde{\tau}_M)-z_n( \psi(G) \circ \tilde{\tau}_M)\right)$, and since
this amounts to pull-back $\bigwedge_{i=1}^{3n}P_i(\Gamma)^{\ast}(\omega)$ by the orientation-preserving diffeomorphism $\psi(g)$ acting on $ I \times B_M \times S_V$, it does
not change the integrals. Therefore,
$$z_n(\tau_M)-z_n(\psi(G) \circ \tau_M)=z_n(\tilde{\tau}_M)-z_n( \psi(G) \circ \tilde{\tau}_M)
$$
and we are done.
\eop
The above lemma allows us to define
$$z_n^{\prime}(G)=z_n(\psi(G) \circ \tau_M)-z_n(\tau_M)$$
for any $G: M \setminus \infty \longrightarrow GL^+(\mathbb{R}^3)$ that maps $(M \setminus \infty) \setminus B_M(1)$ to $1$.
\begin{lemma}
\label{lemvalrho}
If $G$ maps the complement of a ball $B^3$ to the identity, and if $G$ is homotopic to $\rho$ on the quotient of this 3-ball by its boundary, then
$$z_n^{\prime}(G)=\delta_n.$$
\end{lemma}
\noindent {\sc Proof: }
Indeed, in this case, there exists a two-form $\omega$ on $ [0,1] \times B_M\times S^2 $
that coincides with $p_{S^2}^{\ast}(\omega_{S^2})$ near $\{1\} \times B_M \times S^2 $ and $[0,1] \times (B_M \setminus B^3) \times S^2$ and that coincides with
$(p_{S^2} \circ \psi(\tilde{\rho})^{-1})^{\ast}(\omega_{S^2})$ near $\{0\} \times B^3 \times S^2 $ where $\tilde{\rho}$ denotes the restriction of $G$ to $B^3$ that is homotopic to $\rho$. Then
$$z_n(\tau_M)-z_n(\psi(G)^{-1} \circ \tau_M)=\sum_{\Gamma \in {\cal E}_n} \int_{[0,1] \times B^3 \times S_V(\mathbb{R}^3)}\bigwedge_{i=1}^{3n}P_i(\Gamma)^{\ast}(\omega)[\Gamma]$$
since the forms vanish on $[0,1] \times (B_M \setminus B^3) \times S_V$.
Now, view the bundle $E_1$ of Subsection~\ref{subfra} as \index{N}{Eone@$E_1$}
$$E_1 \cong B^4 \times \mathbb{R}^3 \cup_{(\partial B^4=S^3=B^3 \cup_{S^2} -B^3) \times S^2} -B^4 \times \mathbb{R}^3$$
where
$(x,y)$ of the first copy $\partial B^4 \times \mathbb{R}^3$ is identified with
$(x,y)$ of the second copy if $x$ is in $(-B^3)$,
and with $\psi(\tilde{\rho})(x,y)$ otherwise. Then $\omega$ can be extended by
$p_{S^2}^{\ast}(\omega_{S^2})$ outside $ [0,1] \times (B^3) \times S^2 \subset -B^4 \times S^2$ on $S_2(E_1)$. The integrals over
$S_V\left(E_1 \setminus p^{-1}\left(([0,1] \times B^3) \subset -B^4\right)\right)$ are zero.
Therefore, $$z_n(\tau_M)-z_n(\psi(G)^{-1} \circ \tau_M)=\sum_{\Gamma \in {\cal E}_n} \int_{S_V(E_1)} \bigwedge_{i=1}^{3n}P_i(\Gamma)^{\ast}(\omega)[\Gamma]=\delta_n.$$
Now, $\omega$ represents
the Thom class of $E_1$, and we conclude with the help of Lemma~\ref{lemtrivind}.
\eop
\subsection{More on trivialisations of $3$-manifolds}
\label{submoretriv}
Let us now recall some more standard facts about homotopy classes of orientation-respecting trivialisations of $3$-manifolds.
Fix a trivialisation $\tau_M$ of $T(M \setminus \infty)$ that is standard near $\infty$.
Any other such will read $\psi(G) \circ \tau_M$ for a unique
$G: ((M \setminus \infty), M \setminus (\infty \cup B_M(1))) \longrightarrow (GL^+(\mathbb{R}^3),1)$, with the notation of Proposition~\ref{proppont}.
Then $(G \mapsto \psi(G) \circ \tau_M)$ induces a (non-canonical) bijection between the homotopy classes of trivialisations of $T(M \setminus \infty)$ that are standard near $\infty$, and the homotopy classes of maps from $(M,M \setminus B_M(1))$ to $(GL^+(\mathbb{R}^3),1)$.
This latter set is denoted by $[(M,M \setminus B_M(1)),(GL^+(\mathbb{R}^3),1)]$.
It canonically coincides with $[(M,M \setminus B_M(1)),(SO(3),1)]$.
Let $\Gamma$ be a topological group, and let $X$ be a topological space.
Define the product of two maps $f$ and $g$ from $X$ to $\Gamma$ as
$$\begin{array}{llll}fg: &X & \longrightarrow &\Gamma\\
&x &\mapsto & f(x)g(x).\end{array}$$
This product induces a group structure on the set $[X,\Gamma]$ of homotopy classes
of maps from $X$ to $\Gamma$. When $X=M$, this product induces a group structure on $[(M,M \setminus B_M(1)),(GL^+(\mathbb{R}^3),1)]$. Recall the easy lemma.
\begin{lemma}
\label{prodpi}
The usual product of $\pi_n(\Gamma)$ coincides with the product induced by the multiplication in $\Gamma$ (defined above with $X=S^n$).
\end{lemma}
\eop
Let $G_M(\rho):M \longrightarrow SO(3)$ \index{N}{GM@$G_M(\rho)$}
\index{N}{rho@$\rho$} be a map that sends the complement of a ball $B^3 \subset B_M(1)$ to the identity, and that is homotopic to $\rho$ on the quotient of this 3-ball by its boundary. Note that all such maps induce the same element
$[G_M(\rho)]$ in $[(M,M \setminus B_M(1)),(GL^+(\mathbb{R}^3),1)]$.
The elements of $[(M,M \setminus B_M(1)),(SO(3),1)]$ have a well-defined degree
that is the degree of one of their representative from $M$ to $SO(3)$.
\begin{lemma}
\label{lempreptrivun}
Let $M$ be a closed oriented $3$-manifold.
\begin{enumerate}
\item Any map $G$ from $(M,M \setminus B_M(1))$ to $(SO(3),1)$, such that
$$\pi_1(G): \pi_1(M) \longrightarrow \pi_1(SO(3))\cong \mathbb{Z}/2\mathbb{Z}$$
is trivial, belongs to the subgroup $<[G_M(\rho)]>$ of
$[(M,M \setminus B_M(1)),(SO(3),1)]$
generated by $[G_M(\rho)]$. \index{N}{GM@$G_M(\rho)$}
\item For any $[G] \in [(M,M \setminus B_M(1)),(SO(3),1)]$,
$$[G]^2 \in <[G_M(\rho)]>.$$
\item The group $[(M,M \setminus B_M(1)),(SO(3),1)]$ is abelian.
\item The degree is a group homomorphism from $[(M,M \setminus B_M(1)),(SO(3),1)]$ to $\mathbb{Z}$.
\item The morphism $$\begin{array}{llll}\frac{\mbox{deg}}{2}:&[(M,M \setminus B_M(1)),(SO(3),1)]\otimes_{\mathbb{Z}} \mathbb{Q} &\longrightarrow
&\mathbb{Q}[G_M(\rho)]\\
&[g] \otimes 1 &\mapsto &\frac{\mbox{deg}(g)}{2}[G_M(\rho)]\end{array}$$
is an isomorphism.
\end{enumerate}
\end{lemma}
\noindent {\sc Proof: } Assume that $\pi_1(G)$ is trivial. Choose a cell decomposition of $B_M$ with respect to its boundary
with no zero-cell, only one three-cell, one-cells and two-cells. Then after a homotopy, we may assume that $G$ maps the one-skeleton of $B_M$ to $1$. Next, since $\pi_2(SO(3)) = 0$, we may assume that $G$ maps the two-skeleton of $B_M$ to $1$, and therefore that $G$ maps the exterior of some $3$-ball to $1$. Now $G$ becomes a map from $B^3/\partial B^3=S^3$ to $SO(3)$, and its homotopy class is $k[\rho]$ in $\pi_3(SO(3))=\mathbb{Z}[\rho]$, where $(2k)$ is the degree of the map $G$ from $S^3$ to $SO(3)$. Therefore $G$ is homotopic to $G_M(\rho)^k$, and this proves the first assertion.
Since $\pi_1(G^2)=2\pi_1(G)$ is trivial, the second assertion follows.
For the third assertion, first note that $[G_M(\rho)]$ belongs to the center of
$[(M,M \setminus B_M(1)),(SO(3),1)]$ because it can be supported in a small ball disjoint
from the support (preimage of $SO(3) \setminus \{1\}$) of a representative of any other element. Therefore, according to the second assertion any square will be in the center. Furthermore, since any commutator induces the trivial map on $\pi_1(M)$, any commutator is in $<[G_M(\rho)]>$.
In particular, if $f$ and $g$ are elements of $[(M,M \setminus B_M(1)),(SO(3),1)]$,
$$(gf)^2=(fg)^2=(f^{-1}f^2g^2f)(f^{-1}g^{-1}fg)$$
where the first factor equals $f^2g^2=g^2f^2$. Exchanging $f$ and $g$ yields
$f^{-1}g^{-1}fg=g^{-1}f^{-1}gf$. Then the commutator that is a power of $[G_M(\rho)]$ has a vanishing square, and thus a vanishing degree. Then it must be trivial.
For the fourth assertion, it is easy to see that $\mbox{deg}(fg)=\mbox{deg}(f)+\mbox{deg}(g)$ when $f$ or $g$
is a power of $[G_M(\rho)]$, and that $\mbox{deg}(f^k)=k\mbox{deg}(f)$ for any $f$. In general,
$\mbox{deg}(fg)=\frac{1}2\mbox{deg}((fg)^2)=\frac{1}2\mbox{deg}(f^2g^2)=\frac{1}2\left( \mbox{deg}(f^2)+\mbox{deg}(g^2)\right)$, and the fourth assertion is proved.
In particular,
$$\begin{array}{llll}\frac{\mbox{deg}}{2}:&[(M,M \setminus B_M(1)),(SO(3),1)]\otimes_{\mathbb{Z}} \mathbb{Q} &\longrightarrow
&\mathbb{Q}[G_M(\rho)]\\
&[g] \otimes 1 &\mapsto &\frac{\mbox{deg}(g)}{2}[G_M(\rho)]\end{array}$$
is an isomorphism, and the last assertion follows, too.
\eop
\begin{lemma}
The map $z_n^{\prime}$ from $[(M,M \setminus B_M(1)),(SO(3),1)]$ to ${\cal A}_n(\emptyset)$ is a group homomorphism.
\end{lemma}
\noindent {\sc Proof: } According to Lemma~\ref{lemtrivind},
$z_n^{\prime}(fg)=z_n(\psi(f) \psi(g) \tau_M) -z_n( \psi(g) \tau_M)+z_n( \psi(g) \tau_M) -z_n(\tau_M)=z_n^{\prime}(f)+z_n^{\prime}(g)$. \eop
This lemma, together with Lemma~\ref{lemvalrho} that asserts that $z_n^{\prime}(G_M(\rho))=\delta_n$ and Lemma~\ref{lempreptrivun},
concludes the proof of Proposition~\ref{proptrois}. \eop
\subsection{Proof of Proposition~\ref{proppont}}
\label{subproofpont}
Recall that the first \indexT{Pontryagin class} $p_1(W)$ of a closed oriented 4-manifold $W$ is the obstruction to trivialise the complexification of its tangent bundle. It is defined like in Subsection~\ref{subpont}. See also \cite{milnorsta}.
According to \cite[Example 15.6]{milnorsta}, $p_1(\mathbb{C} P^2)=3$. \index{N}{pone@$p_1$}
We shall use the following Rohlin theorem that compares the two cobordism invariants of closed 4-manifolds.
\begin{theorem}[Rohlin]
When $W$ is a closed oriented 4-manifold,
$$p_1(W)=3\mbox{signature(W)}.$$
\end{theorem}
\begin{lemma}
\label{lempunrel}
Let $M$ be a $\mathbb{Q}$-sphere. Let $\tau_M$ be a trivialisation of $T(M \setminus \infty)$ that is trivial near $\infty$. Let $W$ and $W^{\prime}$ be two cobordisms between $B^3(3)$ and $B_M$. Then \index{N}{pone@$p_1$}
$$p_1(W; \tau(\tau_M))-p_1(W^{\prime}; \tau(\tau_M))= 3\left(\mbox{signature}(W) -\mbox{signature}(W^{\prime})\right).$$
\end{lemma}
\noindent {\sc Proof: }
Let $N(\partial W)$ be a regular neighborhood of $\partial W$ in $W$, or in $W^{\prime}$.
Let $\tau$ be a trivialisation of $TW \otimes \mathbb{C}$ defined in $N(\partial W)$.
Set $\tilde{W}=W \setminus \mbox{Int}(N(\partial W))$, and $\tilde{W}^{\prime}=W^{\prime} \setminus \mbox{Int}(N(\partial W))$.
Then
$$p_1(W;\tau)-p_1(W^{\prime};\tau)=p_1(\tilde{W};\tau)-p_1(\tilde{W}^{\prime};\tau)$$
does not depend on the trivialisation $\tau$ and equals
$p_1(\tilde{W}\cup_{\partial \tilde{W}} - \tilde{W}^{\prime})$.
According to Rohlins's theorem, this is $3\; \mbox{signature}(\tilde{W}\cup_{\partial \tilde{W}} - \tilde{W}^{\prime})$,
where $\tilde{W}\cup_{\partial \tilde{W}} - \tilde{W}^{\prime}$ is homeomorphic to $W \cup_{\partial W}(- W^{\prime})$ and $\partial W$ is homeomorphic to $M$.
Since $M$ is a $\mathbb{Q}$-sphere, the Mayer-Vietoris sequence makes clear that $$H_2(W \cup_M(- W^{\prime});\mathbb{R})=H_2(W;\mathbb{R}) \oplus H_2(W^{\prime};\mathbb{R}),$$
and it is easy to see that
$$\mbox{signature}(W \cup_M (-W^{\prime}))= \mbox{signature}(W) -\mbox{signature}(W^{\prime}),$$
and to conclude.
\eop
In particular, the definition of $p_1$ does not depend
on the chosen $4$-cobordism $W$ with signature $0$. It is clear that $p_1(\tau_M)$ only depends on the homotopy class of $\tau_M$. Proposition~\ref{proppont} is now the direct consequence of Lemmas~\ref{lempunind},
\ref{lemvarpun} and \ref{lembijpun} below.
Let ${\bf K}= \mathbb{R}$ or $\mathbb{C}$. Let $n \in \mathbb{N}$. The stabilisation maps induced
by the inclusions
$$\begin{array}{llll}i: & GL({\bf K}^n) & \longrightarrow & GL({\bf K} \oplus {\bf K}^n)\\
& g & \mapsto & (i(g): (x,y) \mapsto (x,g(y))\end{array}$$
will be denoted by $i$. The ${\bf K}$ (euclidean or hermitian) oriented vector space with the direct orthonormal basis $(v_1, \dots, v_n)$ will be denoted by ${\bf K}<v_1, \dots, v_n>$.
The inclusions $SO(n) \subset SU(n)$ will be denoted by $c$.
The projection from $SO(\mathbb{R}^4=\mathbb{R}<1,i,j,k>)$ to $S^3$ that maps
$g$ to $g(1)$ is denoted by $p$.
In particular, the long exact sequence associated to the fibration
$SO(3) \hookrightarrow SO(4) \hfl{p} S^3$ gives rise
to the exact sequence \index{N}{rho@$\rho$}
$$\pi_3(SO(3))=\mathbb{Z}[\rho] \hfl{i_{\ast}} \pi_3(SO(4)) \hfl{p_{\ast}} \pi_3(S^3)=\mathbb{Z}[\mbox{Id}] \longrightarrow \{0\}$$
Let $m_r$ \index{N}{mr@$m_r$}
denote the map from $S^3=S({\mathbb{H}})$ to $SO(\mathbb{R}^4={\mathbb{H}})$ be induced by the
right-multiplication. When $v \in S^3$ and $x \in {\mathbb{H}}$, $m_r(v)(x)=x.v$.
Define a section $\sigma$ of $p_{\ast}$, by setting
$$\sigma([\mbox{Id}])=[m_r].$$
In particular, $\pi_3(SO(4))$ is generated by $i_{\ast}([\rho])$ and $[m_r]$.
Let $m^{\mathbb{C}}_r$ \index{N}{mrC@$m^{\mathbb{C}}_r$}
denote the homeomorphism from $S^3=S({\mathbb{H}})$ to $SU(\mathbb{C}^2=\mathbb{C}<1,j>= {\mathbb{H}})$ be induced by the
right-multiplication. When $v \in S^3$ and $x \in {\mathbb{H}}$, $m^{\mathbb{C}}_r(v)(x)=x.v$.
$$m^{\mathbb{C}}_r(z_1 + z_2 j)=\left[ \begin{array}{cc} z_1 &-\overline{z}_2\\ z_2 &\overline{z}_1\end{array}\right].$$
$$\pi_3(SU(2))=\mathbb{Z}[m^{\mathbb{C}}_r]$$
Finally recall that $i^n_{\ast}: \pi_3(SU(2)) \longrightarrow \pi_3(SU(n+2))$
is an isomorphism for any natural number $n$,
and in particular, that \index{N}{itwo@$i^2(m^{\mathbb{C}}_r)$}
$$\pi_3(SU(4))=\mathbb{Z}[i^2(m^{\mathbb{C}}_r)].$$
The following lemma determines the map $$c_{\ast}: \pi_3(SO(4)) \longrightarrow \pi_3(SU(4)).$$
\begin{lemma}
\label{lempitroissoquatre}
$$c_{\ast}([m_r])=2[i^2(m^{\mathbb{C}}_r)].$$
$$c_{\ast}(i_{\ast}([\rho]))=-4[i^2(m^{\mathbb{C}}_r)].$$ \index{N}{rho@$\rho$}
$$\pi_3(SO(4))=\mathbb{Z}[m_r] \oplus \mathbb{Z} i_{\ast}([\rho]).$$
\end{lemma}
\noindent {\sc Proof: }
Let $m_{\ell}$ denote the map from $S^3=S({\mathbb{H}})$ to $SO(\mathbb{R}^4={\mathbb{H}})$ induced by the
left-multiplication. When $v \in S^3$ and $x \in {\mathbb{H}}$, $m_{\ell}(v)(x)=v.x$.
Let $\overline{m}_r=m_r^{-1}$.
When $v \in S^3$ and $x \in {\mathbb{H}}$, $\overline{m}_r(v)(x)=x.\overline{v}$.
Then in $\pi_3(SO(4))$, $$i_{\ast}([\rho])=[m_{\ell}] + [\overline{m}_r]=
[m_{\ell}] - [{m}_r],$$
thanks to Lemma~\ref{prodpi}.
Now, using the conjugacy of quaternions, $m_{\ell}(v)(x)=v.x=\overline{\overline{x}.\overline{v}}=\overline{\overline{m}_{r}(v)(\overline{x})}$.
Therefore $m_{\ell}$ is conjugated to $\overline{m}_{r}$ via
the conjugacy of quaternions that acts on $\mathbb{R}^4$ as a hyperplan symmetry.
Now, observe that since $U(4)$ is connected, the conjugacy by an element of $U(4)$ induces the identity on $\pi_3(SU(4))$.
Thus, $$c_{\ast}([m_{\ell}])=c_{\ast}([\overline{m}_{r}])=-c_{\ast}([{m}_{r}]),$$
and
$$c_{\ast}(i_{\ast}([\rho]))=-2 c_{\ast}([{m}_{r}]).$$
Therefore, we are left with the proof of the following sublemma
that implies that $i_{\ast}: \pi_3(SO(3)) \longrightarrow \pi_3(SO(4))$
is injective and thus, that
$$\pi_3(SO(4))=\mathbb{Z}[m_r] \oplus \mathbb{Z} i_{\ast}([\rho]).$$
\begin{sublemma} \index{N}{itwo@$i^2(m^{\mathbb{C}}_r)$}
$$c_{\ast}([m_r])=2[i^2(m^{\mathbb{C}}_r)].$$
\end{sublemma}
\noindent {\sc Proof: }
Let ${\mathbb{H}} + I {\mathbb{H}}$ denote the complexification of $\mathbb{R}^4= {\mathbb{H}}= \mathbb{R}<1,i,j,k>$.
Here, $\mathbb{C}=\mathbb{R} \oplus I\mathbb{R}$.
When $x \in {\mathbb{H}}$ and $v \in S^3$, $c(m_r)(v)(Ix)=Ix.v$, and $I^2=-1$.
Let $\varepsilon=\pm 1$, define
$$ \mathbb{C}^2(\varepsilon)=\mathbb{C}<\frac{\sqrt{2}}{2}(1+ \varepsilon Ii),\frac{\sqrt{2}}{2}(j+ \varepsilon Ik)>.$$
Consider the quotient $\mathbb{C}^4/\mathbb{C}^2(\varepsilon)$.
In this quotient, $Ii=-\varepsilon 1$, $Ik =-\varepsilon j$, and since $I^2=-1$,
$I1=\varepsilon i$ and $Ij=\varepsilon k$. Therefore this quotient is isomorphic to ${\mathbb{H}}$ as a real vector space with its complex structure $I= \varepsilon i$.
Then it is easy to see that $c(m_r)$ maps $\mathbb{C}^2(\varepsilon)$ to $0$ in this quotient. Thus $c(m_r)(\mathbb{C}^2(\varepsilon)) = \mathbb{C}^2(\varepsilon)$.
Now, observe that ${\mathbb{H}} + I {\mathbb{H}}$ is the orthogonal sum of $\mathbb{C}^2(1)$ and $\mathbb{C}^2(-1)$.
In particular, $\mathbb{C}^2(\varepsilon)$ is isomorphic to the quotient $\mathbb{C}^4/\mathbb{C}^2(-\varepsilon)$ that is isomorphic to $({\mathbb{H}};I= -\varepsilon i)$ and
$c(m_r)$ acts on it by the right multiplication. Therefore,
with respect to the orthonormal basis $\frac{\sqrt{2}}{2}(1-Ii, j-Ik, 1+Ii, j+Ik )$, $c(m_r)$ reads
$$c(m_r)(z_1+z_2j) =\left[\begin{array}{cccc}
z_1 & -\overline{z}_2 & 0 & 0\\
z_2 & \overline{z}_1 & 0 & 0\\
0 & 0 & \overline{z}_1=x_1-Iy_1 & -z_2\\
0 & 0 & \overline{z}_2 & z_1=x_1+Iy_1\\
\end{array} \right]$$
Therefore, the homotopy class of $c(m_r)$ (invariant under conjugacy by an element of $U(4)$) is the sum of the homotopy classes of
$$(z_1+z_2j) \mapsto \left[\begin{array}{cc}
m^{\mathbb{C}}_r & 0 \\
0 & 1
\end{array} \right] \;\; \mbox{and}\;\; (z_1+z_2j) \mapsto \left[\begin{array}{cc}
1 & 0 \\
0 & m^{\mathbb{C}}_r \circ \iota
\end{array} \right] $$
where $\iota (z_1 +z_2 j)= \overline{z}_1 +\overline{z}_2 j$.
Since the first map is conjugate by a fixed element of $SU(4)$ to $i^2_{\ast}(m^{\mathbb{C}}_r)$, it is homotopic to $i^2_{\ast}(m^{\mathbb{C}}_r)$, and since $\iota$ induces the
identity on $\pi_3(S^3)$, the second map is homotopic to $i^2_{\ast}(m^{\mathbb{C}}_r)$, too.
\eop
\begin{lemma}
\label{lempunind}
Consider $g: (B_M, ]1,3] \times S^2) \longrightarrow (SO(3),1)$ and
$$\begin{array}{llll}
\psi(g): &B_M \times \mathbb{R}^3 &\longrightarrow &B_M \times \mathbb{R}^3\\
&(x,y) & \mapsto &(x,g(x)(y))\end{array}$$ then $\left(p_1(\psi(g) \circ \tau_M)-p_1(\tau_M)\right)$ is independent of $\tau_M$.
\end{lemma}
\noindent {\sc Proof: } Indeed, $\left(p_1(\psi(g) \circ \tau_M)-p_1(\tau_M)\right)$ can be defined as the obstruction to extend the following trivialisation of the tangent bundle of
$[0,1] \times B_M$ restricted to the boundary. This trivialisation is
$T[0,1] \oplus \tau_M$ on $(\{0\} \times B_M) \cup ([0,1] \times \partial B_M)$ and $T[0,1] \oplus \psi(g) \circ \tau_M$ on $\{1\} \times B_M$. But this obstruction is the obstruction
to extend the map $\tilde{g}$ from $\partial([0,1] \times B_M)$ to $SO(4)$ that maps $(\{0\} \times B_M) \cup ([0,1] \times \partial B_M)$ to $1$ and that coincides with $i(g)$ on $\{1\} \times B_M$, viewed as a map from $\partial([0,1] \times B_M)$ to $SU(4)$, on $([0,1] \times B_M)$. This obstruction that lies in
$\pi_3(SU(4))$ since $\pi_i(SU(4))=0$, for $i<3$, is independent of $\tau_M$.
\eop
Define $p^{\prime}_1:[(M, M \setminus B_M(1)),(SO(3),1)] \longrightarrow \mathbb{Z}$
by $$p^{\prime}_1(g)=p_1(\psi(g) \circ \tau_M)-p_1(\tau_M).$$
\begin{lemma}
\label{lemvarpun}
$$p^{\prime}_1(g)=p_1(\psi(g) \circ \tau_M)-p_1(\tau_M)=-2\mbox{deg}(g).$$
\end{lemma}
\noindent {\sc Proof: } Lemma~\ref{lempunind} guarantees that $p^{\prime}_1$ is a group homomorphism. According to Lemma~\ref{lempreptrivun}, $p^{\prime}_1$ must read
$p^{\prime}_1(G_M(\rho)) \frac{\mbox{deg}}{2}$.
Thus, we are left with the proof that \index{N}{GM@$G_M(\rho)$} $$p^{\prime}_1(G_M(\rho))=-4.$$
Let $g=G_M(\rho)$, we can extend $\tilde{g}$ (defined in the proof of Lemma~\ref{lempunind}) by the constant map with value 1 outside
$[\varepsilon, 1] \times B^3 \cong B^4$ and, in $\pi_3(SU(4))$
$$[c(\tilde{g}^{-1}_{|\partial B^4})]=-(p_1(\psi(g) \circ \tau_M)-p_1(\tau_M))[i^2(m^{\mathbb{C}}_r)].$$
Since $\tilde{g}^{-1}_{|\partial B^4}$ is homotopic to $i(\rho)^{-1}$, Lemma~\ref{lempitroissoquatre} allows us to conclude.
\eop
\begin{lemma}
\label{lembijpun}
\begin{itemize}
\item If $M$ is a given $\mathbb{Z}$-sphere, then $p_1$ \index{N}{pone@$p_1$} defines a bijection from the set of homotopy classes of trivialisations of $M$ that are standard near $\infty$ to $4 \mathbb{Z}$.
\item For any $\mathbb{Z}$-sphere $M$, for any trivialisation $\tau_M$ of $M$ that is standard near $\infty$,
$$\left(p_1(\tau_M)-\mbox{dimension}(H_1(M;\mathbb{Z}/2\mathbb{Z}))\right) \in 2 \mathbb{Z}.$$
\end{itemize}
\end{lemma}
\noindent {\sc Proof: } Any closed oriented $3$-manifold $M$ bounds a $4$-dimensional manifold $W$
obtained from $B^4=[0, \varepsilon] \times B^3$ by attaching $b_2(W)$ two-handles
with even self-intersection \cite{kaplan}. We are going to prove the following sublemma.
\begin{sublemma}
There exists a trivialisation $\tau_M$ of $T(M \setminus \infty)$ that is standard near $\infty$ such that
$$p_1(W;\tau(\tau_M)) \equiv 2 b_2(W)\; \mbox{\rm mod}\; 4.$$
\end{sublemma}
\noindent {\sc Proof: } For our $W$, there exists a Morse function that coincides with the projection
onto $[0,1]$ near the boundary where $W$ looks like $[0,1] \times B^3$ or
$[0,1] \times B_M$ and whose only critical points are index two critical points
that correspond to the $b_2(W)$ two-handles. Let $X$ be the gradient field of this
function that is defined outside the critical points.
Let $B^4$ be a $4$-ball of $W$ that intersects $\partial W$ along a $3$-ball $B^3 \subset M$ and that contains all the critical points.
$W \setminus B^4$ is homotopy equivalent
to $W$ and is obtained from a regular neighborhood of $(\{0\} \times B^3) \cup
(-[0,1] \times S^2)$ by attaching two-handles.
The obstruction to extend the trivialisation $X \oplus \tau_S^3$ of $TW$ defined near $(\{0\} \times B^3) \cup
(-[0,1] \times S^2)$
to these handles is in $\pi_1(SO(4))=i_{\ast}(\pi_1(SO(3)))=\mathbb{Z}/2\mathbb{Z}$, it
is the self-intersection of the handles mod 2, and it vanishes.
Therefore, the trivialisation $X \oplus \tau_S^3$ of $TW$ defined near $(\{0\} \times B^3) \cup
(-[0,1] \times S^2)$
extends to $(W \setminus B^4)$ as a trivialisation of the form $X \oplus \tau$.
In particular, $\tau$ provides a trivialisation $\tau_M$ on $B_M \setminus B^3$ that is standard near $\infty$, and that can be extended to $B^3$ since $\pi_2(SO(3))=\{0\}$. Now, $X \oplus \tau$ is a frame on $\partial B^4$ that is viewed as a map from $\partial B^4$ to $SO(4)$, and, in $\pi_3(SU(4))$
$$[c_{\ast}(X \oplus \tau)]=-p_1(W;\tau(\tau_M))[i^2(m^{\mathbb{C}}_r)].$$
Note that $p(X \oplus \tau)=X$ and that $[X]=b_2(W)[\mbox{Id}]$ in $\pi_3(S^3)=H_3(S^3)$.
Indeed, $X$ defines a map from the complement $C$ in $B^4$ of small balls centered at
the critical points to $S^3$. In $C$, $\partial B^4$ is homologous to the sum of the boundaries of these small balls. Therefore, when $X_{\ast}$ denotes the map
from $H_3(C)$ to $H_3(S^3)$ induced by $X$, $[X]=X_{\ast}[\partial B^4]$
is the sum of the degrees of $X$ on the boundaries of the small balls.
Since $X$ is obtained from the outward normal field by a multiplication by
a matrix with two negative eigenvalues on the boundaries of these small balls, the degree is one for all these critical points, and we have proved that $[X]=b_2(W)[\mbox{Id}]$.
Therefore,
$$(X \oplus \tau) \in \left(b_2(W)m_r \oplus i_{\ast}(\pi_3(SO(3)))\right) \subset \pi_3(SO(4)),$$
and, according to Lemma~\ref{lempitroissoquatre},
$$[c_{\ast}(X \oplus \tau)] \in 2 b_2(W)\mathbb{Z} [i^2(m^{\mathbb{C}}_r)] + 4 \mathbb{Z} [i^2(m^{\mathbb{C}}_r)],$$
This concludes the proof of the sublemma.
\eop
Now, it follows from Lemma~\ref{lempunrel} that
$$\begin{array}{ll}p_1(\tau_M) &=p_1(W; \tau(\tau_M))-3\,\mbox{signature}(W)\\
&\equiv 2 b_2(W) -3\,\mbox{signature}(W)\; \mbox{\rm mod} \;4.\end{array}$$
Since $M$ is a $\mathbb{Q}$-sphere, $(\mbox{signature}(W)-b_2(W)) \in 2 \mathbb{Z}$, and therefore $$p_1(\tau_M)\equiv \mbox{signature}(W)\; \mbox{\rm mod} \;4.$$
When $M$ is a $\mathbb{Z}$-sphere the intersection form of $W$ is unimodular, therefore since the form is even the signature of $W$ is divisible by $8$ (see~\cite[Chap. V]{serre}),
and $p_1(\tau_M) \in 4 \mathbb{Z}$. Thus, by Lemmas~\ref{lempreptrivun} and \ref{lemvarpun}, $p_1$ maps the homotopy classes of trivialisations of $M$ that are standard near $\infty$ onto
$4\mathbb{Z}$. These lemmas also show that $p_1$ is bijective from the set of homotopy classes of trivialisations of $M$ that are standard near $\infty$ to
$4\mathbb{Z}$.
Lemma~\ref{lemvarpun} implies that for any pair $(\tau_M, \tau_M^{\prime})$ of trivialisations of $M$ that are standard near $\infty$,
$(p_1(\tau_M)-p_1(\tau_M^{\prime}))$ is even.
Now, since the intersection matrix of $W$ mod $2$ is a presentation matrix for
$H_1(M;\mathbb{Z}/2 \mathbb{Z})$ and since it can be written as the orthogonal sum of matrices
$\left[\begin{array}{cc}0&1\\1&0\end{array}\right]$ and a null matrix of dimension $\mbox{rank}(H_1(M;\mathbb{Z}/2 \mathbb{Z}))$, $$\mbox{signature}(W) \equiv \mbox{rank}(H_1(M;\mathbb{Z}/2 \mathbb{Z})) \;\mbox{\rm mod}\; 2$$
and we are done. This concludes the proof of Lemma~\ref{lembijpun} and the proof of Proposition~\ref{proppont}.
\eop
\subsection{Computation of $\xi_1$} \index{N}{ksione@$\xi_1$}
We use notation introduced in Subsection~\ref{subfra}.
\begin{proposition}
\label{propcptrois}
The projective space $\mathbb{C} P^3$ is homeomorphic to $-S_2(E_1)$. \index{N}{Eone@$E_1$} \index{N}{StwoEone@$S_2(E_1)$}
\end{proposition}
\begin{lemma}
The projective space $\mathbb{C} P^3$ is an $S^2$-bundle over $S^4$.
\end{lemma}
\noindent {\sc Proof: }
Let ${\mathbb{H}}= \mathbb{C} \oplus \mathbb{C} j$ be the quaternionic field, and let ${\mathbb{H}} P^1$ be the quotient of ${\mathbb{H}}^2 \setminus 0$ by the left multiplication by $({\mathbb{H}}^{\ast}={\mathbb{H}} \setminus \{0\})$. $${\mathbb{H}} P^1=S^4=\{(h_1:1); h_1 \in {\mathbb{H}}\} \cup_{{\mathbb{H}}^{\ast}} \{(1:h_2); h_2 \in {\mathbb{H}}\}.$$
where $(h_1:1)=(1:h_1^{-1})$ when $h_1 \neq 0$. The complex projective space
$\mathbb{C} P^3$ is the quotient of $(\mathbb{C}^4 \setminus \{0\}={\mathbb{H}}^2 \setminus 0)$ by the left multiplication by $\mathbb{C}^{\ast} \subset {\mathbb{H}}^{\ast}$. The projection from ${\mathbb{H}}^2 \setminus \{0\}$ to $S^4$ factors through $\mathbb{C} P^3$
that becomes a bundle over $S^4$ with fiber $_{\mathbb{C}^{\ast}}\setminus ({\mathbb{H}} \setminus \{0\})=\mathbb{C} P^1=S^2$.
\eop
\begin{lemma}
\label{lemcompglu}
Let $P_{13}=\left[\begin{array}{ccc}0&0&1\\0&1&0\\-1&0&0\end{array}\right] \in SO(3)$. Let
$$\begin{array}{llll}g_3: &S^3 & \longrightarrow &SO(3)\\
& h_1 & \mapsto & P_{13} \rho(h_1)^{-1} P_{13}^{-1}\end{array}$$
$$\mathbb{C} P^3=B^4 \times S^2 \cup_{\partial B^4 \times S^2 \stackrel{\psi(g_3)}{\rightarrow} \partial (-B^4) \times S^2} (-B^4 \times S^2)$$
\end{lemma}
\noindent {\sc Proof: }
Let $h_1 \in {\mathbb{H}}^{\ast}$. The fiber of $\mathbb{C}^4 \setminus \{0\}$ over $(h_1:1)$ is $\{(kh_1,k); k \in {\mathbb{H}}^{\ast}\}$.
The fiber of $\mathbb{C}^4 \setminus \{0\}$ over $(1:h_1^{-1})$ is $\{(\ell,\ell h_1^{-1}); \ell \in {\mathbb{H}}^{\ast}\}$ with $\ell=kh_1$.
Therefore,
$$\mathbb{C} P^3=B^4 \times \mathbb{C} P^1 \cup_{\psi(\gamma_3)} (-B^4 \times \mathbb{C} P^1)$$
where $\psi(\gamma_3)((h_1;[k]) \in \partial B^4 \times
\mathbb{C} P^1)=(\overline{h}_1;\gamma_3(h_1)([k]))$
and $\gamma_3(h_1)([k])=[k.h_1]$ in $_{\mathbb{C}^{\ast}} \setminus {\mathbb{H}}^{\ast}=\mathbb{C} P^1$, with $[k=z_1+z_2 j]=(z_1:z_2)$.
To express the action $g_3(h_1)$ of $\gamma_3(h_1)$ on $$S^2=\{(z \in \mathbb{C};h \in \mathbb{R});|z|^2 +h^2=1\},$$ we will use the inverse diffeomorphisms
$$\begin{array}{llll} \xi:& \mathbb{C} P^1 & \longrightarrow & S^2\\
&(z_1:z_2)& \mapsto & (\frac{2 z_1 \overline{z}_2}{|z_1|^2 +|z_2|^2},
h=\frac{|z_2|^2-|z_1|^2}{|z_1|^2 +|z_2|^2})\\
\xi^{-1}:& S^2& \longrightarrow &\mathbb{C} P^1\\
&(z;h) & \mapsto & \begin{array}{ll}(z:1+h)\;&\mbox{if}\; h \neq -1\\
(1-h:\overline{z})\;&\mbox{if}\; h \neq 1 \end{array}\;\;\end{array}$$
and write
$$g_3(h_1)=\xi \circ \gamma_3(h_1) \circ \xi^{-1}.$$
Let $(z;h) \in S^2$, $h \neq -1$. Let $h_1=z_3+z_4j \in S^3 \subset {\mathbb{H}}$.
$$ (z+(1+h)j)(z_3+z_4j)=z^{\prime}_1 +z^{\prime}_2 j$$
with $z^{\prime}_1=zz_3-(1+h)\overline{z}_4$, $z^{\prime}_2= zz_4 +(1+h)\overline{z}_3$, and
$$|z^{\prime}_1|^2 + |z^{\prime}_2|^2=|z|^2 +(1+h)^2=2+2h.$$
Then $$g_3(z_3+z_4j)(z;h)= \xi(\gamma_3(z_3+z_4j)(z:1+h))=\xi((z^{\prime}_1:z^{\prime}_2))=
(z^{\prime};h^{\prime}).$$
$$|z^{\prime}_2|^2=|z|^2|z_4|^2 +(1+h)^2|z_3|^2 +(1+h)(z z_3 z_4 + \overline{z z_3 z_4}).$$
$$\frac{|z^{\prime}_2|^2}{1+h}=1 +h(|z_3|^2-|z_4|^2) +(z z_3 z_4 + \overline{z z_3 z_4}).$$
$$h^{\prime}=\frac{2|z^{\prime}_2|^2-(|z^{\prime}_1|^2 + |z^{\prime}_2|^2)}{|z^{\prime}_1|^2 + |z^{\prime}_2|^2}
=\frac{|z^{\prime}_2|^2}{1+h}-1= h(|z_3|^2-|z_4|^2) +(z z_3 z_4 + \overline{z z_3 z_4}).$$
$$z^{\prime}_1\overline{z}^{\prime}_2=|z|^2 z_3 \overline{z}_4 -(1+h)^2 z_3 \overline{z}_4 + (1+h)z_3^2 z -(1+h) \overline{z}_4^2 \overline{z} $$
$$z^{\prime}=\frac{2 z^{\prime}_1\overline{z}^{\prime}_2}{|z^{\prime}_1|^2 + |z^{\prime}_2|^2}=\frac{z^{\prime}_1\overline{z}^{\prime}_2}{1+h}
=- 2h z_3 \overline{z}_4 + z_3^2 z - \overline{z}_4^2\overline{z}.$$
In particular, the map $g_3(z_3+z_4j)$ from $S^2$ to $S^2$ extends as an element of $GL(\mathbb{R}^3)$ still denoted by $g_3(z_3+z_4j)$ with the matrix
$$g_3(z_3+z_4j)=\left[\begin{array}{ccc}
\mbox{Re}(z_3^2-z_4^2) & \mbox{Im}(z_4^2-z_3^2) &
-2\mbox{Re}({z}_3\overline{z}_4) \\
\;\;\mbox{Im}(z_3^2+z_4^2) & \mbox{Re}(z_3^2+z_4^2) &
-2\mbox{Im}({z}_3\overline{z}_4)\\
2\mbox{Re}({z}_3{z}_4) & -2\mbox{Im}({z}_3{z}_4) & |z_3|^2-|z_4|^2
\end{array}\right].$$
Let us now compute the matrix of the conjugacy
$$\rho(z_3 + z_4 j) : v \mapsto (z_3 + z_4 j)v(\overline{z}_3- z_4 j).$$
$$(z_3 + z_4 j)i(\overline{z}_3- z_4 j)=i(|z_3|^2 - |z_4|^2) - 2 z_3z_4 k$$
$$(z_3 + z_4 j)j(\overline{z}_3- z_4 j)= (z_3^2+z_4^2) j + {z}_3\overline{z}_4 - z_4 \overline{z}_3$$
$$(z_3 + z_4 j)k(\overline{z}_3- z_4 j)=i (z_4 \overline{z}_3 + z_3\overline{z}_4) -z_4^2 k +z_3^2 k $$
$$\rho(z_3 + z_4 j)=\left[\begin{array}{ccc}
|z_3|^2 - |z_4|^2 & 2\mbox{Im}({z}_3\overline{z}_4) &
2\mbox{Re}({z}_3\overline{z}_4) \\
\;\;2\mbox{Im}({z}_3{z}_4) & \mbox{Re}(z_3^2+z_4^2) &
\mbox{Im}(z_4^2-z_3^2)\\
-2\mbox{Re}({z}_3{z}_4) & \mbox{Im}(z_3^2+z_4^2) &
\mbox{Re}(z_3^2-z_4^2)
\end{array}\right].$$
Therefore, $g_3(z_3 + z_4 j)=P_{13}\rho(z_3 + z_4 j)^{-1}P_{13}$, and we are done.
\eop
It is now easy to conclude the proof of Proposition~\ref{propcptrois}. Since $SO(3)$ is connected, the gluing map of Lemma~\ref{lemcompglu} is homotopic to $(v \mapsto \rho^{-1}(v))$. Now,
to conclude define the orientation-reversing diffeomorphism
$S$ from
$$ \mathbb{C} P^3\cong B^4 \times S^2 \cup_{\partial B^4 \times S^2 \stackrel{\psi(\rho^{-1})}{\rightarrow} \partial (-B^4) \times S^2} (-B^4 \times S^2)$$
to $$S_2(E_1)=B^4 \times S^2 \cup_{\partial B^4 \times S^2 \stackrel{\psi(\rho)}{\rightarrow} \partial (-B^4) \times S^2} (-B^4 \times S^2)$$
by
$$S((x,v) \in B^4 \times S^2 \subset \mathbb{C} P^3)=(x,v) \in -B^4 \times S^2 \subset S_2(E_1)$$
and
$$S((x,v) \in -B^4 \times S^2 \subset \mathbb{C} P^3)=(x,v) \in B^4 \times S^2 \subset S_2(E_1).$$
\eop
\begin{proposition}
\label{propxiun} \index{N}{ksione@$\xi_1$}
$$\xi_1= -\frac{1}{12}[\theta].$$
\end{proposition}
\noindent {\sc Proof: } The only degree one Jacobi diagram is
$$\theta=\begin{pspicture}[.4](-1,-.5)(2,1.5)
\pscircle(0.5,0.5){.5}
\psline{*-*}(0,.5)(1,.5)
\rput[r](-.1,.5){$1$}
\rput[l](1.1,.5){$2$}
\rput[b](.25,.6){\small b}
\rput[b](.75,.6){\small B}
\rput[r](.05,.8){\small c}
\rput[r](0.05,.2){\small a}
\rput[l](.95,.8){\small C}
\rput[l](.95,.2){\small A}
\end{pspicture}.$$
Orient its edges from $1$ to $2$, and orient $V(\theta)=\{1,2\}$ with its natural order.
Then the edge-orientation of $\theta$ is given by the order $(a,A,b,B,c,C)$ that is equivalent to the order
$(a,b,c,B,A,C)$ of the vertex-orientation where the vertices of $\theta$ are oriented by the picture.
Therefore,
$$\xi_1=\frac{1}{12}\int_{S_2(E_1)}\omega_T^3[\theta].$$
Recall that $H^2(S_2(E_1))\cong H^2(\mathbb{C} P^3)= \mathbb{Z}[\omega_{\mathbb{C} P^3}]$
where $\omega_{\mathbb{C} P^3}$ is Poincar\'e dual to $\mathbb{C} P^2$ and $\int_{\mathbb{C} P^1}\omega_{\mathbb{C} P^3}=1$.
Since the orientation-reversing diffeomorphism $S$ from $\mathbb{C} P^3$ to $S_2(E_1)$ restricts to an orientation-preserving diffeomorphism from a fiber
$\mathbb{C} P^1$ of $\mathbb{C} P^3$ to a fiber $S^2$ of $S_2(E_1)$,
$$\int_{S(\mathbb{C} P^1)}(S^{-1})^{\ast}(\omega_{\mathbb{C} P^3})=1=\int_{S(\mathbb{C} P^1)}\omega_T.$$
Since $H^2(S_2(E_1))\cong \mathbb{Z}$, this shows that $\omega_T=(S^{-1})^{\ast}(\omega_{\mathbb{C} P^3})$. \index{N}{omegaT@$\omega_T$} Then
$$12\xi_1=\int_{S_2(E_1)}(S^{-1})^{\ast}(\omega_{\mathbb{C} P^3})^3[\theta]=-\int_{\mathbb{C} P^3}\omega_{\mathbb{C} P^3}^3[\theta]=-[\theta].$$
\eop
\newpage
\section{Compactifications of configuration spaces}
\label{seccomp}
\setcounter{equation}{0}
In this section, we give a detailed description of the compactifications of the configuration spaces mentioned in Subsection~\ref{substaconf} and we prove all the statements of this subsection that is the introduction to this section.
These compactifications are similar to the Fulton and MacPherson compactifications \cite{fmcp} first used by Bott and Taubes in \cite{bt}. Here, we use the Poirier approach \cite{Po} to present them.
The used definitions and the used properties of blow-ups will be given in Subsection~\ref{subsdifblowup}.
\subsection{Topological definition of the compactifications}
For any subset $A$ of
$V$, recall the restriction map \index{N}{pA@$p_A$}
$$p_A: \breve{C}_V(M) \longrightarrow \breve{C}_A(M).$$
Let $M^A(\infty^A)$ \index{N}{MAinftyA@$M^A(\infty^A)$}
be the manifold obtained from $M^A$ by blowing-up $\infty^A=(\infty, \infty, \dots, \infty)$. When $\sharp A=1$, set $C(A;M)=M^A(\infty)$. When $\sharp A > 1$, define $C(A;M)$ \index{N}{CAM@$C(A;M)$}
from $M^A(\infty^A)$ by blowing-up the closure of
the {\em strict diagonal\/} of $(M \setminus \infty)^A$ made of the constant maps
from $A$ to $(M \setminus \infty)$. Proposition~\ref{propblodifdeux} asserts that $C(A;M)$ inherits
a canonical differentiable structure from the differentiable structure of $M^A$.
Let $\Pi_A: C(A;M) \longrightarrow M^A$ \index{N}{PiA@$\Pi_A$} be the canonical projection.
Consider the embedding
$$\iota =\prod_{A \subseteq V, A \neq \emptyset}p_A: \breve{C}_V(M) \longrightarrow
\prod_{A \subseteq V, A \neq \emptyset}C(A;M)$$
and identify $\breve{C}_V(M)$ with its image under $\iota$.
Define $C_V(M)$ \index{N}{CV@$C_V(M)$}
as a topological space as the closure of $\iota(\breve{C}_V(M))$
in the compact space $\prod_{A \subseteq V, A \neq \emptyset}C(A;M)$.
Note that when $\sharp V=1$, $C_V(M)$ is
homeomorphic to $C_1(M)$. \index{N}{Cone@$C_1(M)$}
We have the following lemma.
\begin{lemma}
\label{lemcond1}
Any $c=(c_A)_{A \subseteq V, A \neq \emptyset} \in C_V(M)$, satisfies the following property $(C1)$:
The restriction of $\Pi_V(c_V)$ to $A$ is equal to $\Pi_A(c_A)$.
\end{lemma}
\noindent {\sc Proof: } Indeed, the set made of the configurations that satisfy $(C1)$ for a given $A$ is closed since it is the preimage
of the diagonal of $(M^A)^2$ under a continuous map. Furthermore, this set contains $\breve{C}_V(M)$. Therefore, it contains $C_V(M)$.\eop
Since we shall use the differentiable structure of the $C(A;M)$ to define the structure of $C_V(M)$. We first study the former one in detail.
\subsection{Differentiable structure on a blow-up}
\label{subsdifblowup}
\begin{definition}
A {\em \indexT{dilation}} is a homothety with ratio in $]0,\infty[$.
\end{definition}
In general, when $V$ is a vector space $SV=S(V)=\frac{V \setminus \{0\}}{]0,\infty[}$
denotes the quotient of $(V \setminus \{0\})$ by the action of $]0,\infty[$ that always operates by scalar multiplication.
Recall that the {\em unit normal bundle\/} $SN_X(Z)$ of a submanifold $Z$
in a smooth manifold $X$ is a bundle over $Z$ whose fiber over $(z \in Z)$ is $S(\frac{T_zX}{T_zZ})$.
\begin{definition}
\label{defblodif}
As a set, the {\em \indexT{blow-up}\/} of $X$ along $Z$ is \index{N}{XZ@$X(Z)$}
$$X(Z)=(X \setminus Z) \cup SN_X(Z).$$
It is equipped with a canonical projection from $X(Z)$ to $X$ that is the identity
outside $SN_X(Z)$ and that is the bundle projection from $SN_X(Z)$ to $Z$ on $SN_X(Z)$. The following proposition defines the canonical smooth structure
of a blow-up.
\end{definition}
\begin{proposition}
\label{propblodifun}
Let $Z$ be a $C^{\infty}$ submanifold of a $C^{\infty}$ manifold $X$ that is transverse to the possible boundary $\partial X$ of $X$.
The \indexT{blow-up} $X(Z)$ \index{N}{XZ@$X(Z)$}
has a unique smooth structure of a manifold with corners such that
\begin{enumerate}
\item the canonical projection from $X(Z)$ to $X$ is smooth and restricts to a diffeomorphism from $X \setminus Z$ to its
image in $X$,
\item any
smooth diffeomorphism $\phi: [0,\infty[^c \times \mathbb{R}^n \longrightarrow X$ from $[0,\infty[^c \times \mathbb{R}^n$ to an open subset $\phi([0,\infty[^c \times \mathbb{R}^n)$ in $X$ whose image intersects $Z$ exactly along
$\phi([0,\infty[^c \times \mathbb{R}^{d-c} \times 0)$, for natural integers $c,d,k$ with $c \leq d$, provides a smooth embedding
$$\begin{array}{lll}([0,\infty[^c \times \mathbb{R}^{d-c}) \times [0, \infty[ \times S^{n+c-d-1} &\hfl{\tilde{\phi}} &X(Z) \\ (x,\lambda \in ]0, \infty[,v) &\mapsto & \phi(x,\lambda v) \\ (x,0,v) &\mapsto & D\phi(x,0)(v) \in SN_X(Z) \end{array}$$
with open image in $X(Z)$.
\end{enumerate}
\end{proposition}
\noindent {\sc Proof: }
We use local diffeomorphisms of the form $\tilde{\phi}$ and charts
on $X \setminus Z$ to build an atlas for $X(Z)$.
These charts are obviously compatible over $X \setminus Z$, and we need to
check compatibility for charts
$\tilde{\phi}$ and $\tilde{\psi}$ induced by embeddings $\phi$ and $\psi$ as in the statement.
For these, transition maps
read:
$$(x,\lambda,u) \mapsto (\tilde{x},\tilde{\lambda}, \tilde{u})$$
where
$$\tilde{x}=p_1 \circ \psi^{-1} \circ \phi(x,\lambda u)$$
$$ \tilde{\lambda} = \parallel p_2 \circ \psi^{-1} \circ \phi(x,\lambda u )\parallel$$
$$ \tilde{u} = \left\{ \begin{array}{ll}\frac{p_2 \circ \psi^{-1} \circ \phi(x,\lambda u )}{\tilde{\lambda}} &\mbox{if} \; \lambda \neq 0 \\
\frac{D\left(p_2 \circ \psi^{-1} \circ \phi(x,0 )\right)(u)dt}{\parallel D\left(p_2 \circ \psi^{-1} \circ \phi(x,0 )\right)(u)dt\parallel} &\mbox{if} \; \lambda = 0\end{array} \right. $$
In order to check that this is smooth, write
$$p_2 \circ \psi^{-1} \circ \phi(x,\lambda u ) = \lambda \int_0^1D\left(p_2 \circ \psi^{-1} \circ \phi(x,t \lambda u )\right)(u)dt $$
where the integral does not vanish when $\lambda$ is small enough.
More precisely, assuming $c=0$ for simplicity in the notation, since the restriction to $S^{n-d-1}$ of $D\left(p_2 \circ \psi^{-1} \circ \phi(x,0 )\right)$ is an injection, for any $u_0 \in S^{n-d-1}$, there exists a neighborhood of $(0,u_0)$ in $[0,\infty[ \times S^{n-d-1}$ such that for any $(\lambda,u)$ in this neighborhood, we have the following condition about the scalar product $$\langle D\left(p_2 \circ \psi^{-1} \circ \phi(x,\lambda u )\right)(u),D\left(p_2 \circ \psi^{-1} \circ \phi(x,0)\right)(u) \rangle \; > 0.$$
Therefore, there exists $\varepsilon > 0$ such that for any $\lambda \in [0,\varepsilon[$, and for any $u \in S^{n-d-1}$,
$$\langle D\left(p_2 \circ \psi^{-1} \circ \phi(x,\lambda u )\right)(u),D\left(p_2 \circ \psi^{-1} \circ \phi(x,0)\right)(u) \rangle \; > 0.$$
Then
$$ \tilde{\lambda} = \lambda \norm{\int_0^1D\left(p_2 \circ \psi^{-1} \circ \phi(x,t \lambda u )\right)(u)dt}$$
is a smooth function (defined even when $\lambda \leq 0$)
and
$$ \tilde{u} =
\frac{\int_0^1D\left(p_2 \circ \psi^{-1} \circ \phi(x,t \lambda u )\right)(u)dt}{\parallel\int_0^1D\left(p_2 \circ \psi^{-1} \circ \phi(x,t \lambda u )\right)(u)dt\parallel}$$ is smooth, too.
\eop
\begin{proposition}
\label{propblodifdeux}
Let $Y$ be a $C^{\infty}$ submanifold of a $C^{\infty}$ manifold $X$ without boundary, and let $Z$ be a $C^{\infty}$ submanifold of $Y$.
\begin{enumerate}
\item The boundary $\partial X(Z)$ of $X(Z)$ is canonically diffeomorphic to $SN_X(Z)$.
\item The closure $\overline{Y}$ of $(Y \setminus Z)$ in $X(Z)$ is a submanifold of $X(Z)$ that intersects $\partial X(Z)$ as the unit normal bundle $SN_Y(Z)$ of $Z$ in $Y$.
\item The blow-up $X(Z)(\overline{Y})$ of $X(Z)$ along $\overline{Y}$ has a canonical differential structure of a manifold with corners, and the preimage of \/ $\overline{Y} \subset X(Z)$ in $X(Z)(\overline{Y})$ under the canonical projection
$$X(Z)(\overline{Y}) \longrightarrow X(Z) $$
is a fibered space over $\overline{Y}$ with fiber the spherical normal bundle
of $Y$ in $X$ pulled back by $(\overline{Y} \longrightarrow Y)$.
\end{enumerate}
\end{proposition}
\noindent {\sc Proof: }
\begin{enumerate}
\item The first assertion is easy to observe from the charts in Proposition~\ref{propblodifun}.
\item Now, it is always possible to choose a chart $\phi$ as above such that
furthermore the image of $\phi$ intersects $Y$ exactly along
$\phi(\mathbb{R}^k \times 0)$, $k > d$.
Then, let us look at the induced chart $\tilde{\phi}$ of $X(Z)$ near a point of $\partial X(Z)$. \\
The intersection of $(Y \setminus Z)$ with the image of $\tilde{\phi}$ is $\tilde{\phi}\left(\mathbb{R}^d \times ]0,\infty[ \times (S^{k-d-1}\subset S^{n-d-1})\right)$. Thus, the closure of $(Y \setminus Z)$ intersects the image of $\tilde{\phi}$ as $$\tilde{\phi}\left(\mathbb{R}^d \times [0,\infty[ \times (S^{k-d-1}\subset S^{n-d-1})\right).$$
\item
Together with the above mentioned charts of $\overline{Y}$, the smooth injective map $$S^{k-d-1} \times \mathbb{R}^{n-k} \longrightarrow S^{n-d-1}$$ $$(u,y) \mapsto \frac{(u,y)}{\parallel (u,y) \parallel}$$
identifies $\mathbb{R}^{n-k}$
with the fibers of the normal bundle of $\overline{Y}$ in $X(Z)$. The blow-up process
will therefore replace $\overline{Y}$ by the quotient of the corresponding $(\mathbb{R}^{n-k} \setminus \{0\})$-bundle
by $]0,\infty[$ which is of course the pull-back under the natural projection $(\overline{Y} \longrightarrow Y)$ of the spherical normal bundle of $Y$ in $X$.
\end{enumerate}
\eop
\noindent{\sc Proof of Lemma~\ref{lemextproj}:}
According to Proposition~\ref{propblodifun}, near the diagonal of $\mathbb{R}^3$, we have a chart of $C_2(S^3)$
$$\psi: \mathbb{R}^3 \times [0,\infty[ \times S^2 \longrightarrow C_2(S^3)$$
that maps $( x \in \mathbb{R}^3, \lambda \in ]0,\infty[ ,y \in S^2)$ to $(x, x + \lambda y) \in (\mathbb{R}^3)^2$. Here, $p_{S^3}$ extends as the projection onto the $S^2$ factor.\\
Consider the embedding \index{N}{phiinfty@$\phi_{\infty}$}
$$\begin{array}{llll}\phi_{\infty}: &\mathbb{R}^3 &\longrightarrow &S^3\\
& \mu (x \in S^2) & \mapsto & \left\{\begin{array}{ll} \infty \;&\;\mbox{if}\; \mu=0\\
\frac{1}{\mu}x \;&\;\mbox{otherwise.} \end{array}\right.\end{array}$$
This chart identifies $S(T_{\infty}S^3)$ to $S(\mathbb{R}^3)$. When $\mu \neq 0$,
$$p_{S^3}(\phi_{\infty}(\mu x), y \in \mathbb{R}^3)= \frac{\mu y-x}{\norm{\mu y- x}}.$$
Then $p_{S^3}$ can be smoothly extended on $S(T_{\infty}S^3) \times \mathbb{R}^3$ by
$$p_{S^3}(D\phi_{\infty}(x) \in S(T_{\infty}S^3), y \in \mathbb{R}^3) = -x.$$
Similarly, set
$$p_{S^3}( x \in \mathbb{R}^3, D\phi_{\infty}(y \in S(\mathbb{R}^3)) \in S(T_{\infty}S^3)) = y.$$
Now, when $$(x,y) \in \left(S((\mathbb{R}^3)^2) \setminus S(\mbox{diag}((\mathbb{R}^3)^2)) \stackrel{(D\phi_{\infty})^2}{\cong} S((T_{\infty}S^3)^2) \setminus S(\mbox{diag}((T_{\infty}S^3)^2))\right),$$
and when $x$ and $y$ are not equal to zero,
$$p_{S^3}(\phi_{\infty}(\lambda x),\phi_{\infty}(\lambda y))=\frac{\frac{y}{\norm{y}^2}-\frac{x}{\norm{x}^2}}
{\norm{\frac{y}{\norm{y}^2}-\frac{x}{\norm{x}^2}}}
=\frac{\norm{x}^2y-\norm{y}^2x}
{\norm{\norm{x}^2y-\norm{y}^2x}}.$$
Therefore, $p_{S^3}$ smoothly extends on $M^2(\infty,\infty)$ outside the boundaries of $\infty \times C_1(M)$,
$C_1(M) \times \infty$ and $\mbox{diag}(C_1(M))$ as
$$p_{S^3}( (D\phi_{\infty})^2((x,y) \in S^5)) =\frac{\norm{x}^2y-\norm{y}^2x}
{\norm{\norm{x}^2y-\norm{y}^2x}}.$$
Let us check that $p_{S^3}$ smoothly extends over the boundary of the diagonal of $C_1(M)$. There is a chart of $C_2(M)$ near the preimage of this boundary
in $C_2(M)$
$$\psi_2: [0,\infty[ \times [0,\infty[ \times S^2 \times S^2 \longrightarrow C_2(S^3)$$
that maps $(\lambda \in ]0,\infty[ , \mu \in ]0,\infty[, x \in S^2, y \in S^2)$
to $(\phi_{\infty}(\lambda x), \phi_{\infty}(\lambda(x + \mu y)))$ where $p_{S^3}$ reads
$$(\lambda,\mu,x,y) \mapsto \frac{y- 2\langle x,y \rangle x -\mu x}
{\norm{y- 2\langle x,y \rangle x -\mu x}},$$ and therefore smoothly extends when $\mu=0$. We similarly check that $p_{S^3}$ smoothly extends over the boundaries of $(\infty \times C_1(M))$ and
$(C_1(M) \times \infty)$.
\eop
\subsection{The differentiable structure of C(A;M)} \index{N}{CAM@$C(A;M)$}
Recall that $M^A(\infty^A)$ is the manifold obtained from $M^A$ by blowing-up $\infty^A=(\infty, \infty, \dots, \infty)$. As a set, $M^A(\infty^A)$ is the union of $M^A \setminus \infty^A$ with the spherical tangent bundle $S\left((T_{\infty}M)^A \right)$ of $M^A$ at $\infty^A$.
Let $\overline{\mbox{diag}((M\setminus \infty)^A)}$ denote the closure in $M^A(\infty^A)$ of
the {\em strict diagonal \/} of $(M \setminus \infty)^A$ made of the constant maps. The boundary of $\overline{\mbox{diag}((M\setminus \infty)^A)}$
is the strict diagonal of $(T_{\infty}M \setminus 0 )^A $ up to dilation.
This allows us to see all the elements of $\overline{\mbox{diag}((M\setminus \infty)^A)}$ as constant maps from $A$ to $C_1(M)$, and provides a canonical diffeomorphism $p_1:\overline{\mbox{diag}((M\setminus \infty)^A)} \longrightarrow C_1(M)$.
Now, $C(A;M)$ is obtained from $M^A(\infty^A)$ by blowing-up $\overline{\mbox{diag}((M\setminus \infty)^A)}$.
Thus, as a set, $C(A;M)$ is the union of
\begin{itemize}
\item the set of non constant maps from $A$ to $M$,
\item the space $\frac{(T_{\infty}M)^A \setminus \mbox{diag}((T_{\infty}M)^A)}{]0,\infty[}$,
and,
\item the bundle over $\overline{\mbox{diag}((M\setminus \infty)^A)}=C_1(M)$ whose fiber at a constant map with value $x \in C_1(M)$ is
$$S\left(\frac{T_{\Pi_1(x)}M^A}{\mbox{diag}((T_{\Pi_1(x)}M)^A)} \right)$$
\end{itemize}
Note that $(\frac{T_{\Pi_1(x)}M^A}{\mbox{diag}((T_{\Pi_1(x)}M)^A)} \setminus 0)$ may be identified with
$((T_{\Pi_1(x)}M)^{A \setminus b} \setminus 0)$ for any $b \in A$ through $$[(v_a)_{a \in A}] \mapsto
(v_a - v_b)_{a \in (A \setminus b)}.$$
Recall that for $A \subset V$, $\Pi_A: C(A;M) \longrightarrow M^A$
denotes the canonical projection.
When $V$ is a euclidean vector space, $S(V)$ is simply the unit sphere of $V$.
\begin{example}
\label{exa1}{\bf Charts near $\Pi_A^{-1}(\mbox{diag}((M \setminus \infty)^A))$.}\\
Let $$\phi: \mathbb{R}^3 \longrightarrow M \setminus \infty$$ be a smooth embedding
that is a chart of $M$ near $\phi(0)=x$.
Let $A$ be a finite set of cardinality $\sharp A \geq 2$. Let $b \in A$.
Let us construct an explicit chart $\psi(A;\phi;b)$ of $C(A;M)$ near a point of
$\Pi_A^{-1}(x^A)$ where $x^A$ denotes the constant map of $M^A$ with value $x$.
We have the chart
$$\begin{array}{llll}\tilde{\psi}(A;\phi;b):& \mathbb{R}^3 \times (\mathbb{R}^3)^{(A \setminus b)} &\longrightarrow &M^A\\
&(y,(y_c)_{c \in A \setminus b}) & \mapsto & \left(c \mapsto \left\{ \begin{array}{ll}\phi(y)& \mbox{if}\; c=b\\\phi(y+y_c)& \mbox{if}\; (c \in A \setminus b)\end{array}\right.\right)
\end{array}$$
of submanifold for the strict diagonal,
and this induces the chart \index{N}{psiAphiB@$\psi(A;\phi;b)$}
$$\begin{array}{lll} \mathbb{R}^3 \times [0,\infty[ \times S((\mathbb{R}^3)^{(A \setminus b)}) &\hfl{\psi(A;\phi;b)} &C(A;M)\\
(y, \lambda \in ]0,\infty[, (y_c)_{c \in (A \setminus b)}) & \mapsto &
\left(c \mapsto \left\{ \begin{array}{ll}\phi(y)& \mbox{if}\; c=b
\\\phi(y+ \lambda y_c)& \mbox{if}\; (c \in A \setminus b)\end{array}\right.\right)\\
(y, 0, (y_c)_{c \in (A \setminus b)}) & \mapsto & \left(D\phi(y)(y_c)\right)_{c \in (A \setminus b)} \in S\left(\frac{(T_{\phi(y)}M)^A}{\mbox{\small diag}((T_{\phi(y)}M)^A)}\right)
\end{array}$$
for $C(A;M)$ in $\Pi_A^{-1}(\phi(\mathbb{R}^3)^A)$. \end{example}
Let $S\left(\frac{T(M\setminus \infty)^A}{\mbox{\small diag}(T(M\setminus \infty)^A)}\right)$ denote the total space of the fibration over
$(M\setminus \infty)$ whose fiber over $x \in (M\setminus \infty)$ is $S\left(\frac{T_xM^A}{\mbox{\small diag}(T_xM^A)}\right)$.
Let
$$\Pi_d: \Pi_A^{-1}(\mbox{diag}(M \setminus \infty)^A) \longrightarrow S\left(\frac{T(M\setminus \infty)^A}{\mbox{\small diag}(T(M\setminus \infty)^A)} \right),$$
denote the canonical projection.
An element in the target of $\Pi_d(\Pi_A^{-1}(x^A))$ will be seen as a non-constant map from $A$ to $T_xM$ up to translation and up to dilation.
\begin{lemma}
\label{lemcond2}
Any $c=(c_A)_{A \subseteq V, A \neq \emptyset} \in C_V(M)$, satisfies the following property $(C2)$:
For any two subsets $A$ and $B$ of $V$ such that the cardinality of $A$ is greater than $1$ and $A \subset B$,
if $c_B \in \Pi_B^{-1}(\mbox{diag}(M \setminus \infty)^B)$, then the restriction
to $A$ of
$\Pi_d(c_B)$ is a (possibly null) positive multiple of $\Pi_d(c_A)$.
\end{lemma}
\noindent {\sc Proof: }
Choose a basepoint $b \in A$ for $A$ and $B$.
Consider the projection $\Pi_{AB}$ of $\prod_{C \subseteq V, C \neq \emptyset}C(C;M) $ onto $C(A;M) \times C(B;M)$
in a neighborhood of some $c$ such that $x^B=\Pi_B(c_B)$ and $x^A=\Pi_A(c_A)$,
with $x \in M\setminus \infty$.
Then $$(\psi(A;\phi;b)^{-1} \times \psi(B;\phi;b)^{-1}) \circ \Pi_{AB}$$ map the elements of $\breve{C}_V(M)$ to elements of the form
$(y, \lambda_A, u_A, y, \lambda_B, u_B)$ where $y \in \mathbb{R}^3$, $\lambda_A,\lambda_B \in ]0,+\infty[$, $u_A \in (\mathbb{R}^3)^{A \setminus b}$,
$u_B \in (\mathbb{R}^3)^{B \setminus b}$, $\parallel u_A \parallel=\parallel u_B \parallel=1$ and,
$$\lambda_B p(u_B) =\lambda_A u_A$$
where $p$ is the natural projection (or restriction) from $(\mathbb{R}^3)^{B \setminus b}$ to $(\mathbb{R}^3)^{A \setminus b}$.
In particular, $p(u_B)$ and $u_A$ are colinear in $(\mathbb{R}^3)^{A \setminus b}$,
and their scalar product is $\geq 0$. These two conditions define a closed
subset of $\left((\mathbb{R}^3)^{A \setminus b}\right)^2$. Therefore, they must be satisfied in the image of the closure $C_V(M)$. Since they read as stated when $c_B \in \Pi_B^{-1}(\mbox{diag}(M \setminus \infty)^B)$ , that is when $\lambda_B=0$, (and hence $\lambda_A=0$, too) we are done. \eop
Let
$$\Pi_{\infty}: \Pi_A^{-1}(\infty^A) \longrightarrow S\left((T_{\infty}M)^A \right) \subset M^A(\infty^A)$$ denote the canonical projection.
An element in the target of $\Pi_{\infty}$ will be seen as a non-zero map from $A$ to $T_{\infty}M$ up to dilation.
\begin{example}
\label{exa2}{\bf Charts of $M^A(\infty^A)$ near $\Pi_A^{-1}(\infty^A)$.}\\
Let $$\phi_{\infty}: \mathbb{R}^3 \longrightarrow M $$ be a smooth embedding
such that $\phi_{\infty}(0)=\infty$. Then the composition
$$]0,\infty[ \times S((\mathbb{R}^3)^A) \hfl{\mbox{multiplication}} (\mathbb{R}^3)^A
\hfl{(\phi_{\infty})^A} M^A$$
induces the chart \index{N}{psiAphiinfty@$\psi(A;\phi_{\infty})$}
$$\begin{array}{llll}\psi(A;\phi_{\infty}):& [0,\infty[ \times S((\mathbb{R}^3)^A) &\longrightarrow & M^A(\infty^A)\\
&(\lambda,u) & \mapsto &\phi_{\infty} \circ \lambda u \;\;\;\;\;\;\; \mbox{when} \; \lambda \neq 0.
\end{array}$$
Here, $u$ is seen as a map from $A$ to $\mathbb{R}^3$.\\
Note that $\Pi_{\infty}(\psi(A;\phi_{\infty})(0,u))=D_0 \phi_{\infty} \circ u$.
\end{example}
\begin{lemma}
\label{lemcond3}
Any $c=(c_A)_{A \subseteq V, A \neq \emptyset} \in C_V(M)$, satisfies the following property $(C3)$:
For any two non-empty subsets $A$ and $B$ of $V$ such that $A \subset B$,
if $c_B \in \Pi_B^{-1}(\infty ^B)$, then the restriction
to $A$ of
$\Pi_{\infty}(c_B)$ is a (possibly null) positive multiple of $\Pi_{\infty}(c_A)$.
\end{lemma}
\noindent {\sc Proof: } This can be proved along the same lines as Lemma~\ref{lemcond2} using the chart of Example~\ref{exa2}, and this is left to the reader.
\eop
\begin{example}
\label{exa3}{ \bf Charts of $C(A;M)$ near the intersection of $\Pi_A^{-1}(\infty^A)$ and the closure of the strict diagonal of $(M \setminus \infty)^A$.}\\
Use the notation of the previous example~\ref{exa2}.
Let $b \in A$. Assume $\sharp A > 1$.
From $\tilde{\psi}(A;\phi_{\infty};b)$
$$\begin{array}{lll} ]0,\infty[ \times S\left(\mathbb{R}^3 \times (\mathbb{R}^3)^{(A \setminus b)}\right) &\longrightarrow &M^A\\
(\lambda;y,(y_c)_{c \in (A \setminus b)}) & \mapsto & \left(c \mapsto \left\{ \begin{array}{ll}\phi_{\infty}( \frac{\lambda}{\sqrt{\sharp A}}y)& \mbox{if}\; c=b\\\phi_{\infty}(\lambda(\frac{1}{\sqrt{\sharp A}}y+y_c))& \mbox{if}\; c\neq b\end{array}\right.\right)
\end{array}$$
we get a chart \index{N}{psiAphiinftyb@$\psi(A;\phi_{\infty};b)$}
$${\psi}(A;\phi_{\infty};b):[0,\infty[ \times S^2 \times [0,\infty[
\times S\left((\mathbb{R}^3)^{(A \setminus b)}\right) \longrightarrow C(A;M)$$
with the property that
$$\Pi_A({\psi}(A;\phi_{\infty};b)(\lambda,u,\mu,v))=
\phi_{\infty} \circ \lambda\left((\frac{1}{\sqrt{\sharp A}}u)^A + \mu v \right)$$
as a map from $A$ to $M$, where $v(b)=0$.
In particular $$\Pi_A^{-1}(\infty^A) \cap \mbox{Im}\left(\psi(A;\phi_{\infty};b)\right)=\psi(A;\phi_{\infty};b)\left(\{0\} \times S^2 \times [0,\infty[
\times S\left((\mathbb{R}^3)^{(A \setminus b)}\right)\right),$$
and
$$\begin{array}{llll} \Pi_{\infty}:& \Pi_A^{-1}(\infty^A) \cap \mbox{Im}\left(\psi(A;\phi_{\infty};b)\right) &\longrightarrow &S\left((T_{\infty}M)^A\right) \subset M^A(\infty^A)
\\ &
\psi(A;\phi_{\infty};b)(0,u,\mu,v) &\mapsto & D_0\phi_{\infty} \circ (\frac{1}{\sqrt{\sharp A}}u^A + \mu v)\end{array}$$
where $u^A$ stands for the constant map with value $u$.\\
The boundary $\Pi_{\infty}^{-1}(\mbox{diag}(T_{\infty}M)^A)$ of
$\overline{\mbox{diag}((M\setminus \infty)^A)}$
is $\psi(A;\phi_{\infty};b)(\{0\} \times S^2 \times \{0\}
\times S\left((\mathbb{R}^3)^{(A \setminus b)}\right))$. The projection $p_1$ of $\psi(A;\phi_{\infty};b)(0,u,0,v)$ onto the boundary of $C_1(M)$ is $D_0\phi_{\infty}(u) \in S(T_{\infty}(M))$.
\end{example}
Let
$$\Pi_{\infty,d}:\Pi_{\infty}^{-1}(\mbox{diag}(T_{\infty}M)^A)) \longrightarrow S\left(\frac{T_{\infty}M^A}{\mbox{diag}(T_{\infty}M)^A}\right)$$
denote the canonical map.
Note that it reads
$$\psi(A;\phi_{\infty};b)(0,u,0,v) \mapsto D_0\phi_{\infty}\circ v$$
in the above charts.
\begin{lemma}
\label{lemcond4}
Any $c=(c_A)_{A \subseteq V, A \neq \emptyset} \in C_V(M)$, satisfies the following property $(C4)$:
For any two subsets $A$ and $B$ of $V$ such that the cardinality of $A$ is greater than $1$ and $A \subset B$,
if $c_B \in \Pi_B^{-1}(\infty ^B)$, and if $\Pi_{\infty}(c_B)$ is a constant map (or is diagonal) then the restriction
to $A$ of
$\Pi_{\infty,d}(c_B)$ is a (possibly null) positive multiple of $\Pi_{\infty,d}(c_A)$.
\end{lemma}
\noindent {\sc Proof: } Again, this can be seen on the charts given in the previous example.
Consider the projection $\Pi_{AB}$ of $\prod_{C \subseteq V, C \neq \emptyset}C(C;M) $ onto $C(A;M) \times C(B;M)$
in a neighborhood of some $c$ such that $\infty^B=\Pi_B(c_B)$, $\Pi_{\infty}(c_B)$ is constant, $\infty^A=\Pi_B(c_A)$ and $\Pi_{\infty}(c_A)$ is constant.
Then $$(\psi(A;\phi_{\infty};b)^{-1} \times \psi(B;\phi_{\infty};b)^{-1}) \circ \Pi_{AB}$$
maps the elements of $\breve{C}_V(M)$ to elements of the form
$$(\lambda_A, u_A, \mu_A, v_A, \lambda_B, u_B,\mu_B, v_B)$$
where $\lambda_A,\lambda_B, \mu_A, \mu_B \in ]0,+\infty[$, $u_A,u_B \in S^2$,
$v_A \in S((\mathbb{R}^3)^{A \setminus b})$, $v_B \in S((\mathbb{R}^3)^{B \setminus b})$, and,
$$ \frac{\lambda_B}{\sqrt{\sharp B}}u_B =\frac{\lambda_A}{\sqrt{\sharp A}}u_A$$
$$ {\lambda_B\mu_B}p(v_B) ={\lambda_A\mu_A}v_A$$
where $p$ is the natural projection (or restriction) from $(\mathbb{R}^3)^{B \setminus b}$ to $(\mathbb{R}^3)^{A \setminus b}$.
Now, it is easy to conclude as before.\eop
\subsection{Sketch of construction of the differentiable structure of $C_V(M)$}
\label{subsketchdifcv} \index{N}{CV@$C_V(M)$}
In this subsection, we sketch the construction of the differentiable structure of $C_V(M)$ and we reduce the proofs of Propositions~\ref{propconfunc}, \ref{propconfuns}, \ref{propconffaceun}, \ref{propconffacedeux} to the proofs of Lemmas~\ref{lempropcons},~\ref{lempropconsadface} and Proposition~\ref{propcdeuxcoinc} stated
below.
We shall use the notation $A \subseteq B$ (resp. $A \subset B$) to say that $A$ is a subset (resp. strict subset) of $B$.
Define $\tilde{C}_V(M)$ \index{N}{CtildeV@$\tilde{C}_V(M)$}
to be the set of the elements $c=(c_A)_{A \subseteq V, A \neq \emptyset}$ of $$\prod_{A \subseteq V, A \neq \emptyset}C(A;M)$$
that satisfy the properties $(C1)$, $(C2)$, $(C3)$ $(C4)$, of Lemmas~\ref{lemcond1},
\ref{lemcond2}, \ref{lemcond3} and \ref{lemcond4}. These lemmas ensure that $C_V(M)$ is a subset of $\tilde{C}_V(M)$.
An element of $\tilde{C}_V(M)$ is a map $(\Pi_V(c_V)\in M^V)$ from $V$ to $M$ with additional
data that allow us to see
\begin{itemize}
\item the restricted configurations corresponding to a multiple point $x \neq \infty$ at any scale $A \subseteq \Pi_V(c_V)^{-1}(x)$ as a non-constant map $\Pi_d(c_A)$ from $A$ to
$T_xM$ up to dilation and translation,
\item the restricted configuration at a scale $A \subseteq \Pi_V(c_V)^{-1}(\infty)$ first as a non-zero map $\Pi_{\infty}(c_A)$ from $A$
to $T_{\infty}M$ up to dilation, and, if this latter map is constant,
\item with an additional zoom, the restricted configuration at a smaller scale as another independent non-constant map $\Pi_{\infty,d}(c_A)$ from $A$
to $T_{\infty}M$ up to dilation and translation.
\end{itemize}
with respective compatibity conditions $(C2)$, $(C3)$, $(C4)$.
Therefore, elements of $\tilde{C}_V(M)$ will be called {\em limit configurations.\/}
We are going to prove that $C_V(M)$ \index{N}{CV@$C_V(M)$}
is equal to $\tilde{C}_V(M)$ \index{N}{CtildeV@$\tilde{C}_V(M)$}
and to construct a differentiable structure for $C_V(M)$ by proving the following proposition.
\begin{proposition}
\label{propdifconf}
For any $c^0 \in \tilde{C}_V(M)$, \index{N}{CtildeV@$\tilde{C}_V(M)$}
there exist
\begin{enumerate}
\item $k \in \mathbb{N}$, and an open neighborhood $O$ of $0$ in $ [0,\infty[^k$,
(set $[0,\infty[^0=]0,\infty[^0=\{0\}$ if $k=0$)
\item an open neighborhood $W$ of a point $w^0$ in a smooth manifold $\tilde{W}$ without boundary,
\item an open neighborhood $U$ of $c^0$ in $\prod_{A \subseteq V, A \neq \emptyset}C(A;M)$,
\item a smooth map $\xi: O \times W \longrightarrow U$ such that $\xi(0;w^0)=c^0$,
$\xi(O \times W) \subset \tilde{C}_V(M)$, and $\xi((O \cap ]0,\infty[^k)\times W) = \breve{C}_V(M) \cap \xi(O \times W)$,
\item a smooth map $r: U \longrightarrow \mathbb{R}^k \times \tilde{W}$ such that
\begin{itemize}
\item $r \circ \xi$ is the identity of $O\times W$,
\item $r(U \cap \tilde{C}_V(M)) \subseteq O \times W$, and
\item the restriction of $\xi \circ r$ to $U \cap \tilde{C}_V(M)$ is the identity of $U \cap \tilde{C}_V(M)$.
\end{itemize}
(This implies that $\xi(O \times W)=U \cap \tilde{C}_V(M)$.)
\end{enumerate}
\end{proposition}
Proposition~\ref{propdifconf} easily implies that our $\xi$ form an atlas for
$\tilde{C}_V(M)$ that becomes a smooth manifold with corners and that
$C_V(M)=\tilde{C}_V(M)$. Furthermore, with such an atlas, the inclusion $\iota$ from $C_V(M)$ to $\prod_{A \subseteq V, A \neq \emptyset}C(A;M)$ will be smooth, and a map $f$ from a smooth manifold $X$ to $C_V(M)$ will be smooth if and only if $\iota \circ f$ is smooth.
When $\sharp V=1$, we observe at once that $\tilde{C}_V(M)$ is diffeomorphic to $C(V;M)$. Therefore, our two definitions of $C_1(M)$ coincide.
We shall prove the following proposition in Subsection~\ref{subpropcdeuxcoinc}.
\begin{proposition}
\label{propcdeuxcoinc}
Let $V=\{1,2\}$. Let $C_2(M)$ \index{N}{Ctwo@$C_2(M)$}
denote the manifold obtained from $M^2(\infty,\infty)$ by blowing up $\infty \times C_1(M)$,
$C_1(M) \times \infty$ and $\mbox{diag}(C_1(M))$
as in Subsection~\ref{subconfap}. Let $C_V(M)$ be the compactification of
$\breve{C}_2(M)$ defined in this subsection.
Then $C_2(M)$ is canonically diffeomorphic to $C_V(M)$.
\end{proposition}
\begin{lemma}
\label{lemdifconfimpconfunc}
Propositions~\ref{propdifconf} and \ref{propcdeuxcoinc} imply Proposition~\ref{propconfunc}.
\end{lemma}
\noindent {\sc Proof: }
It is obvious from Proposition~\ref{propdifconf} that $(\breve{C}_V(M) = \iota(\breve{C}_V(M)))$ is the interior of $C_V(M)$. On $\breve{C}_{V}(M)$, for $e=(a,b)$, $p_e$ is given by the projection on
$C(e;M)$ that determines the projections on $C(\{a\};M)$ and $C(\{b\};M)$.
These projections naturally extend from $\tilde{C}_V(M)$ to the closure of the image of $\breve{C}_{V}(M)$ in $C_e(M)$, and they will be smooth because they come from the smooth projections and because of the forms of our charts.
\eop
Proposition~\ref{propconfuns} is easier to prove than Proposition~\ref{propconfunc} and could be proved before.
Nevertheless, we shall focus on the proof of Proposition~\ref{propconfunc} and see
Proposition~\ref{propconfuns} as a particular case of Proposition~\ref{propconfunc} with the help of the following proposition
\ref{propdifconfad}.
Let $0^V$ denote the constant map with value $0$ in $(\mathbb{R}^3)^V$, where $\mathbb{R}^3=S^3 \setminus \infty$.
The preimage of $0^V$ under the canonical projection $\Pi_V:C(S^3;V) \longrightarrow (S^3)^V$ is the set of non-constant maps from $V$ to $T_0(\mathbb{R}^3)$ up to dilation and translation. This allows us to see
$\breve{S}_V(\mathbb{R}^3)$ as an open submanifold of $\Pi_V^{-1}(0^V)$.
Furthermore, for a given element $s_V$ of $\breve{S}_V(\mathbb{R}^3)$, there is a unique element of $\tilde{C}_V(S^3)$ whose projection on $C(V;M)$ is $s_V$ (by (C2) that determines its other projections). This allows us to see $\breve{S}_V(\mathbb{R}^3)$ as a subset of $\tilde{C}_V(S^3)$.
Set \index{N}{SVR@$S_V(\mathbb{R}^3)$}
$$S_V(\mathbb{R}^3)=(\Pi_V \circ p_V)^{-1}(0^V) \cap \tilde{C}_V(S^3).$$
$S_V(\mathbb{R}^3)$ is a compact set that contains $\breve{S}_V(\mathbb{R}^3)$.
Proposition~\ref{propconfuns} now becomes the consequence of the following proposition (together with Proposition~\ref{propdifconf}) by a proof similar to the proof of Lemma~\ref{lemdifconfimpconfunc}
above.
\begin{proposition}
\label{propdifconfad}
For any $c^0 \in \tilde{C}_V(S^3)$ such that $\Pi_V \circ p_V(c^0)=0^V$, in Proposition~\ref{propdifconf}, we have
\begin{enumerate}
\item $k \geq 1$, $\tilde{W}=\mathbb{R}^3 \times \tilde{\tilde{W}}$,
\item $S_V(\mathbb{R}^3) \cap U=\xi\left(O \times W \cap \left(\{0\} \times [0,\infty[^{k-1} \times \{0\} \times \tilde{\tilde{W}}\right)\right)$,
\item $\xi\left(O \times W \cap \left(\{0\} \times ]0,\infty[^{k-1} \times \{0\} \times \tilde{\tilde{W}}\right)\right) = \breve{S}_V(\mathbb{R}^3) \cap U$.
\end{enumerate}
\end{proposition}
Proposition~\ref{propdifconf} and Proposition~\ref{propdifconfad} are a consequence of the two following lemmas.
\begin{lemma}
\label{lempropcons}
Proposition~\ref{propdifconf} and Proposition~\ref{propdifconfad} are true when $\Pi_V(c^0_V)$ is a constant map of $M^V$.
\end{lemma}
\begin{lemma}
\label{lemimp}
(1) Lemma~\ref{lempropcons} implies Proposition~\ref{propdifconf}.\\
(2) Assume Lemma~\ref{lempropcons} is true.
Let $(A_i)_{i=1,2,\dots, s}$ be a partition of $V$ into nonempty subsets
$$V = \coprod_{i=1}^s A_i.$$
Let $\phi_i:\mathbb{R}^3 \longrightarrow M$, for $i=1, \dots, s$, be embeddings with disjoint images in $M$.
Let $U_A$ be the following open subset of $C(A;M)$.
$$U_A=\{c_A \in C(A;M); \Pi_A(c_A)( A \cap A_i ) \subseteq \phi_i(\mathbb{R}^3)\}$$
and define $$Q_V=\tilde{C}_V(M) \cap \prod_{\emptyset \subset A \subseteq V}U_A\;\; \mbox{and}
\;\; Q_{A_i}=\tilde{C}_{A_i}(M) \cap \prod_{\emptyset \subset A \subseteq {A_i}}U_A$$
Then the map $(Q_V \longrightarrow \prod_{i=1}^sQ_{A_i})$ induced by the restrictions is a diffeomorphism.
\end{lemma}
\noindent{\sc Proof of Lemma~\ref{lemimp}:}
(1) Let $c^0=(c^0_A)_{A \subseteq V, A \neq \emptyset} \in \tilde{C}_V(M)$.
Consider the map $\Pi_V(c_V)$ from $V$ to $M$ and set
$$\Pi_V(c_V)(V)=\{m_1, m_2, \dots, m_s\}$$
and $$A_i=\Pi_V(c_V)^{-1}(m_i)$$
Choose embeddings $\phi_i:\mathbb{R}^3 \longrightarrow M$, for $i=1, \dots, s,$ with disjoint images in $M$ such that $\phi_i(0)=m_i$.
Let $c^i=c^0_{|A_i}=(c^0_A)_{A \subseteq A_i, A \neq \emptyset} \in \tilde{C}_{A_i}(M)$ denote the restriction of $c^0$ to $A_i$.
According to Lemma~\ref{lempropcons}, we may find $k_i,U_i,O_i,W_i,w^0_i,\xi_i,r_i$ satisfying the conclusions
of Proposition~\ref{propdifconf} with $(c^i,A_i)$ instead of $(c^0,V)$, and after
a possible reduction of $U_i,O_i,W_i$, we may assume that $U_i \subseteq \prod_{A \subseteq A_i}U_A$.
Then, set $k=\sum_{i=1}^s k_i$, $O=\prod_{i=1}^s O_i$, $W=\prod_{i=1}^s W_i$,
$w^0=(w^0_i)_{i \in \{1, \dots, s\}}$.
$$U=\prod_{i=1}^s U_i \times \prod_{A \subseteq V; \forall i, A \cap (A \setminus A_i) \neq \emptyset}C(A;M)$$
Define $\xi((v,w)=(v_1, \dots, v_s,w_1, \dots, w_s))=(\xi(v,w)_A)_{A \subseteq V; A \neq \emptyset}$
by $\xi(v,w)_A=\xi_i(v_i,w_i)_A$ if $A \subseteq A_i$
and $\Pi_A(\xi(v,w)_A)(a\in A_i)=\Pi_{A_i}(\xi(v,w)_{A_i})(a)$.
When $A$ intersects all the $(A \setminus A_i)$, $\Pi_A(\xi(v,w)_A)$ is not constant. Since the restriction of $\Pi_A$ to the preimage of the set of non-constant maps is a diffeomorphism onto its image, $\xi(v,w)_A$ is smoothly well-determined for these $A$. Therefore $\xi$ is well-determined and smooth. Furthermore, $\xi(v,w)$ satisfies
$(C1)$ by construction and $\xi(v,w)$ satisfies the other conditions $(C2)$, $(C3)$ and $(C4)$ that are (thanks to $(C1)$ and to the choice of the $U_i$) conditions on some $\xi(v,w)_A$ and $\xi(v,w)_B$
for $A \subset B \subseteq A_i$. It is easy to see that $\xi(0,w^0)=c^0$,
and $\xi((O \cap ]0,\infty[^k)\times W) \subset \breve{C}_V(M)$ since the elements of $\breve{C}_V(M)$ are the elements $c$ of $\tilde{C}_V(M)$ such that $\Pi_V(c_V) \in M^V$ is an injective map from $V$ to $(M \setminus \infty)$.
We also easily see that $$\breve{C}_V(M) \cap \xi( O \times W) \subseteq \xi((O \cap ]0,\infty[^k)\times W).$$
When $r_i(u_i \in U_i)=(r^1_i(u_i) \in \mathbb{R}^{k_i}; r_i^2(u_i) \in W_i)$, define $$r((u_i)_{i \in \{1, \dots, s\}};(c_A)_{A \subseteq V; \forall i, A \cap (A \setminus A_i) \neq \emptyset})$$ $$=((r^1_i(u_i))_{i \in \{1, \dots, s\}};(r^2_i(u_i))_{i \in \{1, \dots, s\}}).$$
Now, it is easy to see that Lemma~\ref{lempropcons} implies Proposition~\ref{propdifconf}.
The second part (2) of the lemma follows from the above proof.
\eop
Assume that Proposition~\ref{propdifconf} is true and come back to the faces defined in Subsection~\ref{substaconf}.
First recall that
$F(\infty;V)=S_i(T_{\infty}M^V) \subseteq S((T_{\infty}M)^V)$
embeds in $C(V;M)$. This embedding is smooth and canonical.
Furthermore, by $(C1)$ and $(C3)$, there is a unique map of $F(\infty;V)$
into
$C_V(M)$ whose composition with the projection on $C(V;M)$ is the above embedding. Since the restrictions are smooth from $C(V;M) \cap S_i(T_{\infty}M^V)$ to the $C(A;M)$ for $A \subset V$, the charts of Proposition~\ref{propdifconf} for $C_V(M)=\tilde{C}_V(M)$ make clear that $F(\infty;V)$ smoothly injects into $C_V(M)$. Lemma~\ref{lemimp} allows us
to conclude that for any non-empty subset $B$ of $V$, $F(\infty;B)$ injects
into $C_V(M)$, smoothly and canonically. It is easy to see that the projections
$p_e$ associated to pairs of elements of $V$ restrict to the image of $F(\infty;B)$
as described in Subsection~\ref{substaconf}. The reader can similarly check
that, for any subset $B$ of $V$ with $(\sharp B \geq 2)$, $F(B)$ smoothly and canonically injects into $C_V(M)$ and that the restrictions of the $p_e$ to the images of the $F(B)$ are described in Subsection~\ref{substaconf}.
Obviously, the images of the $F \in \partial_1(C_V(M))$ are disjoint.
Let us inject $(f(B)=f(B)(\mathbb{R}^3))$ into $S_V(\mathbb{R}^3)$ where $B$ is a strict subset of $V$, $\sharp B \geq 2$.
Identify $\breve{S}_{\{b\} \cup(V \setminus B)}(\mathbb{R}^3)$ with a subspace of $S\left(\frac{(\mathbb{R}^3)^V}{\mbox{\small diag}((\mathbb{R}^3)^V)}\right)$ made of maps that are constant on $B$, by setting $c(B)=c(b)$. In particular, $\breve{S}_{\{b\} \cup(V \setminus B)}(\mathbb{R}^3)$ smoothly embeds into $C(V;S^3) \cap \Pi_V^{-1}(0^V)$. When $A$ is a non-empty subset of $V$ that is not a subset of $B$, $\breve{S}_{\{b\} \cup(V \setminus B)}(\mathbb{R}^3)$ smoothly projects to
$C(A;S^3) \cap \Pi_A^{-1}(0^A)$ by the restrictions imposed by
$(C1)$ and $(C2)$ (that do not determine anything for the subsets of $B$ where $c$
is constant).
Now, $\breve{S}_{B}(\mathbb{R}^3)$ smoothly embeds into $C(B;S^3) \cap \Pi_B^{-1}(0^B)$ and smoothly projects to
$C(A;S^3) \cap \Pi_A^{-1}(0^A)$, when $A$ is a non-empty subset of $B$, by the restrictions imposed by
$(C1)$ and $(C2)$. This allows us to define a canonical smooth injection of $f(B)(\mathbb{R}^3)$ into
$S_V(\mathbb{R}^3)$, and the $p_e$ have the desired form on the image. When $B$ and $B^{\prime}$ are two disjoint subsets of $V$, $f(B)$ and $f(B^{\prime})$
are disjoint.
The $F(\infty;B)$, $F(B)$ and $f(B)(\mathbb{R}^3)$ will be identified with their
images.
\begin{lemma}
\label{lempropconsadface}
Assume that Proposition~\ref{propdifconf} is true. In Proposition~\ref{propdifconf},
\begin{itemize}
\item when $\Pi_V(c^0_V)$ is a constant map with value $m \in (M \setminus \infty)$,\\ $k=1$ if and only if $c^0 \in F(V)$,
\item when $\Pi_V(c^0_V)$ is the constant map with value $\infty$,\\ $k=1$ if and only if $c^0 \in F(\infty;V)$,
\item when $\Pi_V(c^0_V)$ is the constant map $0^V$ of $(S^3)^V$,\\ $k=2$ if and only if $c^0 \in f(B)(\mathbb{R}^3)$ for some strict subset $B$ of $V$ with $\sharp B \geq 2$.
\end{itemize}
\end{lemma}
\noindent{\sc Proof of Proposition~\ref{propconffaceun} assuming Proposition~\ref{propdifconf} and Lemma~\ref{lempropconsadface}:}
Let $c$ belong to a codimension one face of $\tilde{C}_V(M)$.
As in the proof of Lemma~\ref{lemimp}, set
$\Pi_V(c_V)(V)=\{m_1, m_2, \dots, m_s\}$,
and $A_i=\Pi_V(c_V)^{-1}(m_i)$.
Choose embeddings $\phi_i:\mathbb{R}^3 \longrightarrow M$, for $i=1, \dots, s$ with disjoint images in $M$ such that $\phi_i(0)=m_i$. Then by Lemma~\ref{lemimp},
if $c_{|A_i}$ belongs to a codimension $d(i)$ face, then $c$ belongs to
a codimension $\left(\sum_{i=1}^sd(i) \right)$-face. Therefore, there exists
a unique $j$ such that $c_{|A_j}$ belongs to a codimension one face. Set $B=A_j$.
When $i \neq j$, $c_{|A_i}$ belongs to the interior $\breve{C}_{A_i}(M)$ of $C_{A_i}(M)$, and since $c_{|A_i}$ is constant, $A_i$ contains a unique element
and $c_{|A_i}$ does not map it to $\infty$.
Two cases occur. Either $c_B(B)=\{\infty\}$ and $c \in F(\infty;B)$, or $c_B(B)=\{c_B(b)\} \subset (M \setminus \infty)$ and $c \in F(B)$.
Therefore the union of the codimension one faces is a subset
of $\coprod_{F \in \partial_1(C_V(M))}F$.
Conversely, Lemma~\ref{lempropconsadface} and the local product structure of
Lemma~\ref{lemimp} make clear that $\coprod_{F \in \partial_1(C_V(M))}F$ is a subset of the union of codimension one faces.
Now, it is clear that every $F \in \partial_1(C_V(M))$ is connected.
Furthermore, the closure of any such $F$ does not meet any other
$F^{\prime} \in \partial_1(C_V(M))$.\\
Let us prove this for $F=F(\infty;B)$.
In the closure of $F(\infty;B)$ all the configurations map $B$
to $\infty$ therefore $\overline{F(\infty;B)}$ may only meet the
$F(\infty;A)$ such that $B \subset A$. Consider a configuration $c$ in $(\overline{F(\infty;B)} \cap F(\infty;A))$. With the notation of Example~\ref{exa2},
since $c \in \overline{F(\infty;B)}$, $p_A(c)=c_A=\psi(A;\phi_{\infty})(\lambda, u \in S((\mathbb{R}^3)^A))$, where $u$ maps $B$ to $0$;
then $\Pi_{\infty}(c_A)$
maps $B$ to $0$, but in this case $c \notin F(\infty;A)$.
A similar proof left to the reader leads to the same conclusion for $F=F(B)$.
Therefore, the $F$ are closed in the finite disjoint union $\coprod_{F \in \partial_1(C_V(M))}F$. Thus, they are the codimension one faces of $C_V(M)$,
and consequently, they smoothly embed in $C_V(M)$.
\eop
\noindent{\sc Proof of Proposition~\ref{propconffacedeux} assuming Lemma~\ref{lempropcons} and Lemma~\ref{lempropconsadface}:}
It is immediate from Lemma~\ref{lempropconsadface} and Proposition~\ref{propdifconfad}
that the disjoint union of the elements $f(B)(\mathbb{R}^3)$ of $\partial_1(S_V(\mathbb{R}^3))$
coincides with the union of the codimension one faces. A proof similar to the above one shows that the $f(B)$ are the connected components of this union.
Therefore, they are the codimension one faces of $S_V(\mathbb{R}^3)$ and they smoothly embed there.
\eop
Proposition~\ref{propcdeuxcoinc} will be proved in Subsection~\ref{subpropcdeuxcoinc}.
Apart from Proposition~\ref{propcdeuxcoinc}, we are left with the proofs of Lemmas~\ref{lempropcons} and \ref{lempropconsadface} about the structure of $\tilde{C}_V(M)$ near a configuration $c^0$ such that
$\Pi_V(c^0_V)$ is the constant map $m^V$
with value $m$. The case where $(m \neq \infty)$ will be treated in the next subsection. The case $(m = \infty)$ is similar though more complicated, it will be treated in Subsection~\ref{subsecinf}, but some arguments will not be repeated.
\subsection{Proof of Proposition~\ref{propdifconf} when $\Pi_V(c^0_V)=m^V$, $m \in M \setminus \infty$.}
\label{subsecnoninf}
Let $\phi: \mathbb{R}^3 \longrightarrow (M \setminus \infty)$ be a smooth embedding, $\phi(0)=m$.
If $\sharp V=1$, set $k=0$, $W=\mathbb{R}^3$, $w^0=0$, $U=\phi(\mathbb{R}^3) \subset C_1(M)$, $\xi=\phi$ and $r=\phi^{-1}$, and we are done. Assume $\sharp V \geq 2$.
Let $c^0=(c^0_A)_{A \subseteq V; A \neq \emptyset} \in \tilde{C}_V(M)$ be such that $\Pi_V(c^0_V)=m^V$.
\medskip
\noindent{\bf The tree $\tau(c^0)$ associated to the limit configuration $c^0$.\/} \index{N}{tauc@$\tau(c^0)$}
\medskip
We shall define a set $\tau(c^0)$ of subsets of $V$ with cardinality $\geq 2$ as follows.
The set is organized as a tree with $(V \in \tau(c^0))$ as a root.
The other elements of $\tau(c^0)$ are constructed inductively as follows.
Every element $A$ of $\tau(c^0)$ is the {\em daughter\/} of its unique {\em mother\/} $\hat{A}$ in $\tau(c^0)$, except for $V$ that
has no mother, and some elements have {\em daughters\/} (i.e. are the mother of these).
A daughter is strictly included into its mother, and any two daughters are disjoint.
Therefore, it is enough to construct the daughters of an element $A$.
By assumption,
$c^0_{A} \in \Pi_{A}^{-1}(\mbox{diag}(M \setminus \infty)^{A}) \subset
C(A;M)$.
Thus, $$\Pi_d(c^0_{A}) \in S\left(\frac{T_mM^{A}}{\mbox{diag}
(T_mM^{A})} \right)$$
defines a map from $A$ to $T_mM$ up to translation
and dilation.
The daughters of $A$ will be the preimages of multiple points.
The preimages of non-multiple points will be the {\em sons\/} of $A$.
$\tau(c^0)$ has the property that whenever $\{A,B\} \subseteq \tau(c^0)$,
either $A \subset B$, or $B \subset A$, or $A \cap B = \emptyset$.
Fix $c^0$, and $\tau=\tau(c^0)$.
For any $A \in \tau$ choose a basepoint $b(A)=b(A;\tau)$, \index{N}{bA@$b(A)$}
such that
if $A \subset B$, if $B \in \tau$, and if $b(B) \in A$, then $b(A)=b(B)$.
When $A \in \tau$, $D(A)$ denotes the set of daughters of $A$.
\medskip
\noindent{\bf Configuration spaces associated to $\tau$.\/}
\medskip
For any $A \in \tau$,
consider the following subsets of the unit sphere $S((\mathbb{R}^3)^V)$ of $(\mathbb{R}^3)^V$ equipped with its usual scalar product.
Define the set $C(A;b(A);\tau)$ \index{N}{CAbAtau@$C(A;b(A);\tau)$}
of maps $w:V \longrightarrow \mathbb{R}^3$ such that
\begin{itemize}
\item $\parallel w \parallel=1$
\item $w(b(A))=0$, $w(V \setminus A)=\{0\}$, and
\item $w$ is constant on any daughter of $A$.
\end{itemize}
It is easy to see that $C(A;b(A);\tau)$ \index{N}{CAbAtau@$C(A;b(A);\tau)$}
has a canonical differentiable structure
(and is diffeomorphic to a sphere of dimension $\left(3 (\sharp A -\sum_{i=1}^n\sharp A_i +n-1)-1\right)$ where $A_1, \dots, A_n$ are the daughters of $A$.
Note that $c^0_A=\psi(A;\phi;b(A))(0;0;w^0_A)$ with the notation of Example~\ref{exa1}
where the natural extension $w^0_A$ (by some zeros) of
$w^0_A \in (\mathbb{R}^3)^{A \setminus b(A)} \subset (\mathbb{R}^3)^V$ is in $C(A;b(A);\tau)$.
Define the set $O(A;b(A);\tau)$ \index{N}{OAbAtau@$O(A;b(A);\tau)$}
of maps $w:V \longrightarrow \mathbb{R}^3$ such that
\begin{itemize}
\item $\parallel w \parallel=1$
\item $w(b(A))=0$, $w(V \setminus A)=\{0\}$, and
\item Two elements of $A$ that belong to different children (daughters and sons)
of $A$ are mapped to different points of $\mathbb{R}^3$.
\end{itemize}
It is clear that $O(A;b(A);\tau)$ is an open subset of $S((\mathbb{R}^3)^{A \setminus b(A)})$ that contains $w^0_A$.
Set
$$W_A=O(A;b(A);\tau) \cap C(A;b(A);\tau)$$
$W_A$ is an open subset of the sphere $C(A;b(A);\tau)$.
\medskip
\noindent{\bf The data $U$, $W$, $w^0$ and $k$.\/}
\medskip
\begin{itemize}
\item $k=\sharp \tau$.
\item $\tilde{W}=\mathbb{R}^3 \times \prod_{A \in \tau} W_A$
\item $W$ will be an open neighborhood of $w^0=(0;(w_A^0)_{A \in \tau})$
in $\tilde{W}$.
\item $\tilde{U}=\prod_{A \in \tau}\psi(A;\phi;b(A))\left(\mathbb{R}^3 \times [0,\infty[ \times O(A;b(A);\tau)\right) \times \prod_{A \notin \tau}C(A;M)$
\item $U$ will be an open neighborhood of $c^0$ in $\tilde{U}$.
\end{itemize}
\medskip
\noindent{\bf Construction of $\xi$.\/}
\medskip
Let $$P=((\mu_A)_{A \in \tau};u; (w_A)_{A \in \tau}) \in \mathbb{R}^{\tau} \times W$$
and $$P^0=((0)_{A \in \tau};0; (w^0_A)_{A \in \tau}) =(0;w^0).$$
When $A \in \tau$, define
$$v_A = v_A(P)=\sum_{C \in \tau; C \subseteq A}\left(\prod_{D \in \tau; C \subseteq D \subset A} \mu_D \right) w_C \in S((\mathbb{R}^3)^{A \setminus b(A)}) \subset S((\mathbb{R}^3)^V)$$
Note that $v_A$ is a smooth function defined on $\mathbb{R}^k \times
\tilde{W}$, and that $v_A(P^0)=w_A^0$.
In particular, $\parallel v_A(P^0) \parallel=1$ and $\frac{v_A(P^0)}{\parallel v_A(P^0) \parallel}$ is in $O(A;b(A);\tau)$.
Therefore, we can choose neighborhoods $O$ of $0$ in $[0,\infty[^k$
and $W$ of $w^0$ in $\tilde{W}$, so that for any $P$ in $O \times W$, $\parallel v_A(P) \parallel \neq 0$ and
$\frac{v_A(P)}{\parallel v_A(P) \parallel}$ is in $O(A;b(A);\tau)$.
We choose $O$ and $W$ so that these properties are satisfied for any $A \in \tau$.
In order to define $\xi$, we define its projections $\xi_A(P)$ onto the factors
$C(A;M)$. First set
$$\xi_V(P)=\psi(V;\phi;b(V))(u;\mu_V;\frac{v_V}{\parallel v_V \parallel})$$
Then $$\Pi_V(\xi_V(P))(a)= \phi(u + \frac{\mu_V}{\parallel v_V \parallel}v_V(a))$$
When $A \in \tau$, set
$$\xi_A(P)=\psi(A;\phi;b(A))\left(u + \frac{\mu_V}{\parallel v_V \parallel}v_V(b(A));
\frac{{\parallel v_A \parallel}\prod_{D \in \tau; A \subseteq D \subseteq V} \mu_D}{\parallel v_V \parallel};\frac{v_A}{\parallel v_A \parallel}\right)$$
The latter definition makes sense because
$v_{\hat{A}}$ is not constant on $A$ since $\frac{v_{\hat{A}}(P)}{\parallel v_{\hat{A}}(P) \parallel}$ belongs to $O(A;b(A);\tau)$.
Indeed, either $\Pi_{\hat{A}}(\xi_{\hat{A}}(P))$ is non constant and then its restriction to $A$ is non constant, and we take the usual smooth
restriction, or $\Pi_{\hat{A}}(\xi_{\hat{A}}(P))$ is constant with value $\phi(v)$, and we take the restriction of the map $D_v \phi \circ {v}_{\hat{A}}$ from $\hat{A}$ to $T_{\phi(v)}M$ up to translation and dilation.
It is easy to check that this restriction is smooth from this open subset of $C({\hat{A}};M)$ to $C(A;M)$, by using appropriate charts of $C(A;M)$ and $C(\hat{A};M)$ as in Example~\ref{exa1} with the same basepoint
for $A$ and $\hat{A}$.
Thus, we defined a smooth map $\xi$ from $O \times W$ to $\tilde{U}$ such that $\xi(0;w^0)=c^0$.
\medskip
\noindent{\bf Checking that $\xi$ satisfies $(C1)$. \/}
\medskip
It is enough to check that $\Pi_A(\xi(P)_A)= \Pi_V(\xi_V(P))_{|A}$ for any $A \in \tau$.
Let $A \in \tau$, $a \in A$.
$$\Pi_V(\xi_V(P))(a)=
\phi\left(u + \frac{\mu_V}{\parallel v_V \parallel}\sum_{C \in \tau; a \in C}\left(\prod_{D \in \tau; C \subseteq D \subset V} \mu_D \right) w_C(a)\right)$$
where the elements $C$ of $\tau$ that contain $a$, are
\begin{enumerate}
\item the $C$ of $\tau$ such that $A \subset C$ that satisfy $w_C(a)=w_C(b(A))$, and
\item the $C$ of $\tau$ such that $a \in C \subseteq A$ that satisfy $w_C(b(A))=0$.
\end{enumerate}
In particular,$$u + \frac{\mu_V}{\parallel v_V \parallel}v_V(b(A))=
u + \frac{\mu_V}{\parallel v_V \parallel}\sum_{C \in \tau; A \subset C}\left(\prod_{D \in \tau; C \subseteq D \subset V} \mu_D \right) w_C(a).$$
Therefore,
$$\phi^{-1}(\Pi_V(\xi_V(P))(a))-\left(u + \frac{\mu_V}{\parallel v_V \parallel}v_V(b(A))\right) $$
$$=\frac{1}{\parallel v_V \parallel}\sum_{C \in \tau; a\in C; C \subseteq A}\left(\prod_{D \in \tau; C \subseteq D \subseteq V} \mu_D \right) w_C(a)$$
$$=\frac{\prod_{D \in \tau; A \subseteq D \subseteq V} \mu_D}{\parallel v_V \parallel}v_A(a).$$
\medskip
\noindent{\bf Checking that $\xi(O \times W) \subset \tilde{C}_V(M)$. \/}
\medskip
It is enough to check that $\xi(P)$ satisfies $(C2)$ since $\Pi_V(\xi_V(P))(V) \subset (M \setminus \infty)$.
Let $A \subset B \subseteq \Pi_V(\xi_V(P))^{-1}(x)$.
If $B$ is not in $\tau$, then $\hat{B} \subseteq \Pi_V(\xi_V(P))^{-1}(x)$
(see the construction of $\xi_B(P)$), and $\xi_B(P)$ is the non-trivial restriction of $\xi_{\hat{B}}(P)$. Therefore, for this proof, we may assume that $B \in \tau$. Similarly,
we may assume that $A \in \tau$. Then it is enough to check that the restriction of $v_B$ to $A$ up to translation is a $(\geq 0)$ multiple of $v_A$, and this is easy to observe in the defining formula for $v_A$.
\medskip
\noindent{\bf Checking that $\xi((O \cap ]0,\infty[^k)\times W) = \breve{C}_V(M) \cap \xi(O \times W)$. \/}
\medskip
Let us first prove that $\xi((O \cap ]0,\infty[^k)\times W) \subset \breve{C}_V(M)$. Since $(C1)$ is fulfilled in the image of $\xi$, it is enough to prove that $\Pi_V(\xi_V(P))$
is injective when the $\mu_A$ are non zero.
Let $a$ and $b$ be in $V$, and let $A$ be the smallest element of $\tau$ that contains both of them. Then $v_A(a) \neq v_A(b)$
since $\frac{v_A}{\parallel v_A \parallel}$ is in $O(A;b(A);\tau)$, thus $\Pi_A(\xi_A(P))$ separates $a$ and $b$, and we are done thanks to $(C1)$. Conversely, since as soon as a $\mu_A$ vanishes, the corresponding $\Pi_A(\xi_A(P))$ is constant, $$\breve{C}_V(M) \cap \xi(O \times W) \subseteq \xi((O \cap ]0,\infty[^k)\times W).$$
\medskip
\noindent{\bf Construction of $r$.\/}
\medskip
For any $A \in \tau$, choose
$b^{\prime}(A) \neq b(A) \in A$ \index{N}{bprimeA@$b^{\prime}(A)$}
to be either the element of a son of $A$ or a basepoint of a daughter of $A$ that does not contain $b(A)$. Note that $w^0_A(b^{\prime}(A)) \neq 0$.
The map $r$ will factor through the projection onto $$\prod_{A \in \tau}\psi(A;\phi;b(A))\left(\mathbb{R}^3 \times [0,\infty[ \times O(A;b(A);\tau)\right).$$
Let $$Q=\left(\psi(A;\phi;b(A))(u_A; \lambda_A; y_A) \right)_{A \in \tau}$$ be a point of this space and let
$$r(Q)=((\mu_A)_{A \in \tau}; u_V; (w_A)_{A \in \tau})$$ denote its image in $\mathbb{R}^k \times \tilde{W}$.
The map $u_V$ is already defined and smooth, and we need to define the $\mu_A$ and the $w_A$ as smooth functions of $(u_A; \lambda_A; y_A)_{A \in \tau}$.
Define
$w^1_A \in (\mathbb{R}^3)^A$ by
$$w^1_A(a)=\left\{\begin{array}{ll} y_A(a) & \mbox{if}\; a \in \left(A \setminus
(\cup_{B \in D(A)}B) \right)\\
y_A(b(B)) & \mbox{if}\; a \in B\;\mbox{and if} \;B \in D(A) \end{array}\right..$$
Then set
$$w_A=\frac{w^1_A}{\parallel w^1_A \parallel}$$
Since $y_A \in O(A;b(A);\tau)$, $\parallel w^1_A \parallel \neq 0$, and $w_A$ is smooth.
Then define $\mu_V=\lambda_V$, and for $A \in \tau$, $A \neq V$,
$$\mu_A=\frac{\parallel w_{\hat{A}}(b^{\prime}({\hat{A}})) \parallel \langle y_{\hat{A}}(b^{\prime}(A))-y_{\hat{A}}(b(A)),w_A(b^{\prime}(A)) \rangle }{\parallel y_{\hat{A}}(b^{\prime}({\hat{A}})) \parallel \; \parallel w_A(b^{\prime}(A)) \parallel^2}$$
Then it is clear that $r$ is smooth from $\tilde{U}$ to $\mathbb{R}^k \times \tilde{W}$.
\medskip
\noindent{\bf Checking that $r \circ \xi$ is the identity of $O \times W$.\/}
\medskip
We compute $$r \circ \xi\left(P=((\mu_A)_{A \in \tau};u; (w_A)_{A \in \tau})\right)=r((\xi_A(P))_{A \in \tau})$$
$$=((\hat{\mu}_A)_{A \in \tau}; u_V; (\hat{w}_A)_{A \in \tau}).$$
where
$$\xi_V(P)=\psi(V;\phi;b(V))(u;\mu_V;\frac{v_V}{\parallel v_V \parallel}),$$
$$\xi_A(P)=\psi(A;\phi;b(A))\left(u_A; \lambda_A;y_A=\frac{1}{\parallel v_A \parallel}v_A\right),$$
and $v_A$ has been defined in the construction of $\xi$.\\
We easily find $u=u_V$ and $\hat{\mu}_V=\mu_V$, and $\hat{w}_A=w_A$.
Since $$\left(v_{\hat{A}}(b^{\prime}(A)) - v_{\hat{A}}(b(A))\right)=\mu_A w_A(b^{\prime}(A))\;\;\;\mbox{and}\;\;\; v_{\hat{A}}=\parallel v_{\hat{A}} \parallel y_{\hat{A}}$$
$$\parallel v_{\hat{A}} \parallel\left(y_{\hat{A}}(b^{\prime}(A)) - y_{\hat{A}}(b(A))\right)=\mu_A w_A(b^{\prime}(A)).$$
Furthermore,
$$y_{\hat{A}}(b^{\prime}({\hat{A}})) =\frac{1}{\parallel v_{\hat{A}} \parallel}v_{\hat{A}}(b^{\prime}({\hat{A}})) =\frac{1}{\parallel v_{\hat{A}} \parallel}w_{\hat{A}}(b^{\prime}({\hat{A}})).$$
Therefore
$\hat{\mu}_A=\mu_A$.
Thus, $r \circ \xi$ is the identity of $O \times W$.
\medskip
\noindent{\bf Checking that $r(\tilde{U} \cap \tilde{C}_V(M)) \subseteq [0,\infty[^k \times \tilde{W}$.\/}
\medskip
When $Q=\left(\psi(A;\phi;b(A))(u_A; \lambda_A; y_A) \right)_{A \in \tau}$
comes from an element of $\tilde{C}_V(M)$, if $A$ and $B$ are two elements of $\tau$ such that $A \subset B$,
then for any $a \in A$,
$$u_B + \lambda_B y_B(a)=u_A + \lambda_A y_A(a),$$ and the map from $A$ to $\mathbb{R}^3$ that maps $a$ to
$(y_B(a)-y_B(b(A)))$ is a $(\geq 0)$ multiple of $y_A$.
$$r(Q)=((\mu_A)_{A \in \tau};u_V; (w_A)_{A \in \tau})$$
where
$$\mu_A=\frac{\parallel w_{\hat{A}}(b^{\prime}({\hat{A}})) \parallel \; \parallel y_{\hat{A}}(b^{\prime}(A))-y_{\hat{A}}(b(A)) \parallel}{\parallel y_{\hat{A}}(b^{\prime}({\hat{A}})) \parallel \; \parallel w_A(b^{\prime}(A)) \parallel}$$
when $A \in \tau$, $A \neq V$.
Indeed, $\left((y_{\hat{A}})_{|A} -(y_{\hat{A}}(b(A)))^A\right)$ is a $(\geq 0)$ multiple of $y_A$, and $y_A(b^{\prime}(A))$ is a $(\geq 0)$ multiple of $w_A(b^{\prime}(A))$. In particular, $\mu_A \geq 0$, $\mu_V=\lambda_V$ is also positive.
Also, note that $r(c^0)=P^0$.
Now, choose
$(\varepsilon > 0)$ such that $[0,\varepsilon[^k \subset O$,
reduce $O$ into
$[0,\varepsilon[^k$ and set
$$U=r^{-1}(]-\varepsilon,\varepsilon[^k \times W).$$
Then $r(U \cap \tilde{C}_V(M)) \subseteq O \times W$.
\medskip
\noindent{\bf Checking that $\xi \circ r_{|U \cap \tilde{C}_V(M)}$ is the identity of $U \cap \tilde{C}_V(M)$.\/}
Keep the above notation for $Q$ and $r(Q)$. Assume $Q \in U \cap \tilde{C}_V(M)$.
$$(\xi \circ r(Q))_A=\psi(A;\phi;b(A))(\tilde{u}_A;\tilde{\lambda}_A;\frac{v_A}{\norm{v_A}})$$
where $v_A$ is the vector associated to $r(Q)$ in the construction of $\xi$.
\medskip
\noindent{\em Proof that $y_A=\frac{v_A}{\norm{v_A}}$.}\\
\medskip
Let $b_0$ be an element of $A$. Inductively define
$$B^1=\hat{\{b_0\}} \subset B^2=\hat{B^{1}} \subset \dots \subset B^{i+1}=\hat{B^{i}} \subset \dots \subset B^k=A
.$$
Set $y_i= y_{B^i}$, $b_i=b(B^i)$, $b^{\prime}_i=b^{\prime}(B^{i})$ and $w_i= w_{B^i}$.
Then
$$v_A(b_0)=\sum_{i=1}^k \left( \prod_{j=i}^{k-1}
\frac{\parallel w_{j+1}(b^{\prime}_{j+1}) \parallel \; \parallel y_{j+1}(b^{\prime}_j)-y_{j+1}(b_j) \parallel}{\parallel y_{j+1}(b^{\prime}_{j+1}) \parallel \; \parallel w_j(b^{\prime}_j) \parallel}
\right)w_i(b_0)$$
where
$$w_i(b_0)=
\frac{\parallel w_i(b^{\prime}_i) \parallel}{\parallel y_{i}(b^{\prime}_i) \parallel }
y_i(b_{i-1}).$$
Therefore
$$v_A(b_0)=\sum_{i=1}^k \parallel w_{A}(b^{\prime}(A)) \parallel \left( \frac{ \prod_{j=i}^{k-1} \parallel y_{j+1}(b^{\prime}_j)-y_{j+1}(b_j) \parallel}{\prod_{j=i}^{k}\parallel y_{j}(b^{\prime}_j) \parallel }
y_i(b_{i-1})\right)$$
while
$$y_A(b_0)= \sum_{i=1}^k(y_A(b_{i-1})-y_A(b_i)),$$
and since, for $i \leq j$,
$$y_{j+1}(b_{i-1})-y_{j+1}(b_i)=\frac{\parallel y_{j+1}(b^{\prime}_j)-y_{j+1}(b_j)\parallel}{\parallel y_{j}(b^{\prime}_j) \parallel}\left(y_{j}(b_{i-1})-y_{j}(b_i)\right)
$$
$$y_A(b_{i-1})-y_A(b_i)=\prod_{j=i}^{k-1}\frac{\parallel y_{j+1}(b^{\prime}_j)-y_{j+1}(b_j)\parallel}{\parallel y_{j}(b^{\prime}_j) \parallel}
y_i(b_{i-1})$$
Thus,
$$v_A(b_0)= \frac{\parallel w_{A}(b^{\prime}(A)) \parallel}{\parallel y_A(b^{\prime}(A)) \parallel}y_A(b_0)$$
and $y_A=\frac{v_A}{\norm{v_A}}$.
\eop
Note that $\tilde{\lambda}_V=\mu_V=\lambda_V$, $\tilde{u}_V=u_V$, and therefore
$(\xi \circ r(Q))_V$ is the restriction of $Q$ to $V$ whose value in $M$ at $b(A)$ determines $\tilde{u}_A$ by $(C1)$ that is fulfilled in the image of $\xi$. Therefore $\tilde{u}_A=u_A$.
\medskip
\noindent{\em Proof that $\tilde{\lambda}_A ={\lambda}_A$ for $A \neq V$.}\\
\medskip
Now, let us compute $\tilde{\lambda}_A$ for $A \in \tau$, $A \neq V$.
Define
$$B^1=A \subset B^2=\hat{B^{1}} \subset \dots \subset B^{i+1}=\hat{B^{i}} \subset \dots \subset B^k=V
.$$
Again, set $y_i= y_{B^i}$, $b_i=b(B^i)$, $b^{\prime}_i=b^{\prime}(B^{i})$ and $w_i= w_{B^i}$.
$$\tilde{\lambda}_A=\frac{\lambda_V\parallel v_A \parallel}{\parallel v_V \parallel}
\prod_{i=1}^{k-1}
\left(
\frac{\parallel w_{i+1}(b^{\prime}_{i+1}) \parallel \; \parallel y_{i+1}(b^{\prime}_i)-y_{i+1}(b_i) \parallel}{\parallel w_i(b^{\prime}_i) \parallel \; \parallel y_{i+1}(b^{\prime}_{i+1}) \parallel }
\right)$$
where
$$\parallel v_A \parallel= \frac{\parallel w_A(b^{\prime}(A)) \parallel}{\parallel y_A(b^{\prime}(A)) \parallel}.$$
Therefore
$$\tilde{\lambda}_A=\frac{\lambda_V\parallel y_V(b^{\prime}(V)) \parallel}{\parallel y_A(b^{\prime}(A)) \parallel}
\prod_{i=1}^{k-1}
\left(
\frac{ \parallel y_{i+1}(b^{\prime}_i)-y_{i+1}(b_i) \parallel}{\parallel y_{i+1}(b^{\prime}_{i+1}) \parallel }
\right)$$
where
$$
\lambda_{B^i}=
\frac{\parallel y_{i+1}(b^{\prime}_i)-y_{i+1}(b_i) \parallel}
{ \parallel y_{i}(b^{\prime}_{i}) \parallel}
\lambda_{B^{i+1}}$$
$$\lambda_A= \lambda_V \prod_{i=1}^{k-1}
\left( \frac{\parallel y_{i+1}(b^{\prime}_i)-y_{i+1}(b_i) \parallel}
{ \parallel y_{i}(b^{\prime}_{i}) \parallel} \right)$$
Thus $\tilde{\lambda}_A=\lambda_A$.
When $A \notin \tau$, $(\xi \circ r(Q))_A$ is the restriction of $Q_{\hat{A}}$ to $A$, and we can conclude that the restriction of $\xi \circ r$ to ${U} \cap \tilde{C}_V(M)$ is the identity.
\eop
This concludes the proof of Proposition~\ref{propdifconf} in this case.
\eop
Proposition~\ref{propdifconfad} follows from a careful reading of the previous proof. Since $V \in \tau$, $k=\sharp V \geq 1$.
Choose the natural embedding $$\phi: \mathbb{R}^3 \longrightarrow S^3 = \mathbb{R}^3 \cup \{\infty\}.$$
The elements of ${S}_V(\mathbb{R}^3) \cap U$ (resp. $\breve{S}_V(\mathbb{R}^3) \cap U$) are the elements whose projection onto $C(V;M)$ is of the form
$\psi(V;\phi;b(V))(0 \in \mathbb{R}^3;\mu_V=0;y_V)$ for some $y_V$ (resp. for some injective $y_V$). In particular, the second item is true where $\mu_V$ is the distinguished real parameter that vanishes if and only if $\Pi_V(c_V)$ is constant.
Now, since
$v_V$ is injective if and only if all the $\mu_{D}$ are non-zero for $D \in \tau \setminus V$, the third item is true.
Proposition~\ref{propdifconfad} is proved. \eop
In this case, Lemma~\ref{lempropconsadface} also follows from a careful reading of the previous proof. Indeed, $k=1$ if and only if $\tau=\{V\}$, that is if and only
if $c_V^0 \in \breve{S}_V(T_{c(b)}M)$, that is if and only if $c^0 \in F(V)$.
Now, $k=2$ if and only if $\tau=\{V,B\}$ for some strict subset $B$ of $V$
with $\sharp B \geq 2$, that is if and only if :\\
$\Pi_d(c^0_V)$ is constant on $B$
and injective on $\{b\} \cup (V \setminus B)$, and $\Pi_d(c^0_B)$ is injective.\\
This is equivalent to say that under the assumptions of Lemma~\ref{lempropconsadface}, $c^0 \in f(B)(\mathbb{R}^3)$.
Lemma~\ref{lempropconsadface} is proved in this case.\eop
\subsection{Proof of Proposition~\ref{propdifconf} when $\Pi_V(c^0_V)=\infty^V$.}
\label{subsecinf}
Let $\phi_{\infty}: \mathbb{R}^3 \longrightarrow M$ be a smooth embedding, $\phi_{\infty}(0)=\infty$.
Let $c^0=(c^0_A)_{A \subseteq V; A \neq \emptyset} \in \tilde{C}_V(M)$ be such that $\Pi_V(c^0_V)$ maps every point of $V$ to $\infty$.
\medskip
\noindent{\bf The tree $\tau(c^0)$ associated to $c^0$.\/} \index{N}{tauc@$\tau(c^0)$}
\medskip
We shall define a set $\tau=\tau(c^0)$ of non-empty subsets of $V$ as follows.
The set is organized as a tree with $V$ as a root.
The other elements of $\tau$ are constructed inductively as follows.
Again, every element $A$ of $\tau$ is the {\em daughter\/} of its unique {\em mother\/} $\hat{A}$ in $\tau$, except for $V$ that
has no mother, and some elements have {\em daughters\/} (i.e. are the mother of these).
In order to define the daughters of $A \in \tau$, consider the map defined up to dilation \index{N}{Piinfty@$\Pi_{\infty}(c^0_{A})$}
$$\Pi_{\infty}(c^0_{A}):A \longrightarrow T_{\infty}(M).$$
\begin{itemize}
\item If this map $\Pi_{\infty}(c^0_{A})$
is non-constant, or if $A$ has only one element, then $A$ is {\em non-degenerate\/}.
In this case, let $A_0$ denote the preimage of $\{0\}$ under $\Pi_{\infty}(c^0_{A})$. If $A_0$ is non-empty,
$A_0$ is a daughter of $A$ and this daughter is said to be {\em special;\/}
the other daughters of $A$ will be the preimages of multiple points different from $0$ under $\Pi_{\infty}(c^0_{A})$.
The preimages of non-multiple points different from zero will be the {\em sons\/} of $A$.
\item If the map $\Pi_{\infty}(c^0_{A})$ is constant, and if $\sharp A \geq 2$, then $A$ is {\em degenerate\/}, and we consider the non-constant map defined up to translation and dilation \index{N}{Piinftyd@$\Pi_{\infty,d}(c^0_{A})$}
$$\Pi_{\infty,d}(c^0_{A}):A \longrightarrow T_{\infty}(M).$$
The daughters of $A$ are the preimages of multiple points under this map, and its sons are the preimages of the other points.
\end{itemize}
By definition $V$ is special, and an element $A \neq V$ of $\tau$ is special
if and only if $\Pi_{\infty}(c^0_{\hat{A}})(A)=\{0\}$.
Let $\tau_d$ \index{N}{taud@$\tau_d$}
be the set of the degenerate elements of $\tau$,
and let $\tau_s$ \index{N}{taus@$\tau_s$}
be the set of the special elements of $\tau$. When $A \in \tau$, $D(A)$ denotes the set of daughters of $A$. Note that
\begin{itemize}
\item $V \in \tau_s$,
\item $D(A \in \tau_d) \subset \tau_d \cap (\tau \setminus \tau_s)$
\item $D(A \notin \tau_d) \subseteq (\tau_d \cup \{A_0\})$,
\item $\tau =\tau_s \cup \tau_d$
\item If $A \neq V$, $A \in \tau_s$, then $\hat{A} \notin \tau_d$, therefore $\hat{A} \in \tau_s$.
\end{itemize}
In particular,
$$\tau_s=\{V=V(1),V(2), \dots, V(\sigma)\}$$
where $V(i)_0=V(i+1) \neq \emptyset$ if $i < \sigma$, and $V(\sigma)_0= \emptyset$.
Also note that $\tau_s \cap \tau_d \subseteq \{V(\sigma)\}$.
\index{N}{taud@$\tau_d$} \index{N}{taus@$\tau_s$}
Fix $c^0$, and $\tau=\tau(c^0)$.
For any $A \in \tau$ choose a basepoint
$b(A)=b(A;\tau)$, \index{N}{bA@$b(A)$}
such that
\begin{itemize}
\item $b_i=b(V(i))=b(V(\sigma))$ for any $i=1, \dots, \sigma$, and,
\item if $A \subset B$, if $B \in \tau$, and if $b(B) \in A$, then $b(A)=b(B)$.
\end{itemize}
\medskip
\noindent{\bf Configuration spaces associated to $\tau$.\/}
\medskip
Let $i \in \{1, \dots, \sigma\}$.
Define the smooth manifold $C(V(i);\tau)$ \index{N}{CVitau@$C(V(i);\tau)$}
as the following submanifold of the unit sphere $S((\mathbb{R}^3)^V)$ of $(\mathbb{R}^3)^V$ equipped with its usual scalar product.
The set $C(V(i);\tau)$ is the set of maps $w:V \longrightarrow \mathbb{R}^3$ such that
\begin{itemize}
\item $\parallel w \parallel=1$
\item $w(V(i)_0)=\{0\}$, $w(V \setminus V(i))=\{0\}$, and
\item \begin{itemize}
\item if $V(i) \notin \tau_d$, $w$ is constant on any daughter of $V(i)$, and,
\item if $V(i) \in \tau_d$, $w$ is constant.
\end{itemize}
\end{itemize}
Define the open subset $O(V(i);\tau)$ \index{N}{OVitau@$O(V(i);\tau)$}
of $S((\mathbb{R}^3)^{V(i)})$ as
the set of maps $w:V \longrightarrow \mathbb{R}^3$ such that
\begin{itemize}
\item $\parallel w \parallel=1$
\item $w(V \setminus V(i))=\{0\}$,
\item If $V(i) \notin \tau_d$, two elements of $V(i)$ that belong to different children (daughters and sons)
of $V(i)$ are mapped to different points of $\mathbb{R}^3$, and
\item $0 \notin w(V(i) \setminus V(i)_0)$.
\end{itemize}
Set $W^s_i=O(V(i);\tau) \cap C(V(i);\tau)$. $W^s_i$ is an open submanifold of the sphere $C(V(i);\tau)$.
Then after a proper scalar multiplication, the natural extension $s^0_i$ (by some zeros) of $(D_0\phi_{\infty})^{-1} \circ \Pi_{\infty}(c^0_{V(i)})$ is in $W^s_i$.
For any $A \in \tau_d$, consider the smooth manifold $C(A;b(A);\tau)$ and the open subset $O(A;b(A);\tau)$ of $S((\mathbb{R}^3)^{A \setminus b(A)})$ defined as in Subsection~\ref{subsecnoninf}. Set
$$W_A=O(A;b(A);\tau) \cap C(A;b(A);\tau).$$
Then after a proper normalization, the natural extension $w^0_A$ (by some zeros) of $(D_0\phi_{\infty})^{-1} \circ \Pi_{\infty,d}(c^0_A)$ is in $W_A$.
\medskip
\noindent{\bf The data $U$, $W$, $w^0$ and $k$.\/}
\medskip
\begin{itemize}
\item $k= \sharp \tau_s + \sharp \tau_d= \sigma + \sharp \tau_d$.
\item $\tilde{W}=\prod_{i=1}^{\sigma} W^s_i \times \prod_{A \in \tau_d} W_A$
\item $W$ will be an open neighborhood of $w^0=((s_i^0)_{i \in \{1, \dots, \sigma\}};(w_A^0)_{A \in \tau_d})$
in $\tilde{W}$.
\item $\tilde{U}=\prod_{A \in \tau_d}\psi(A;\phi_{\infty};b(A))\left([0,\infty[
\times S^2 \times [0,\infty[ \times O(A;b(A);\tau)\right) $\\
$\times
\prod_{A \in (\tau \setminus \tau_d)}\psi(A;\phi_{\infty})\left([0,\infty[
\times O(A;\tau)\right) \times \prod_{A \notin \tau}C(A;M)$\\
(with the charts of Examples~\ref{exa2} and \ref{exa3}).
\item $U$ will be an open neighborhood of $c^0$ in $\tilde{U}$.
\end{itemize}
\noindent{\bf Forgetting $\prod_{A \notin \tau}C(A;M)$.\/}
When $A \notin \tau$, let $\hat{A}$ be the smallest element in $\tau$ that contains
$A$. Let $C=\prod_{A \subseteq V, A \neq \emptyset}C(A;M)$, $C^{\tau}=\prod_{A \in \tau}C(A;M)$ and let $C^{\tau}_V(M)$ be the subspace of $C^{\tau}$ made of the elements that satisfy the restriction conditions $(C1)$, $(C2)$, $(C3)$, $(C4)$ of Lemmas~\ref{lemcond1},
\ref{lemcond2}, \ref{lemcond3}, \ref{lemcond4} that involve elements of $\tau$.
Let $p^{\tau}: C \longrightarrow C^{\tau}$ be the natural projection.
Define the following smooth map $$\iota^{\tau}: p^{\tau}(\tilde{U}) \longrightarrow C$$
by
$\iota^{\tau}(c=(c_A)_{A \in \tau})=(d_A)_{A \subseteq V, A \neq \emptyset}$, where $d_A=c_A$ when $A \in \tau$, and $d_A$ is the restriction of $c_{\hat{A}}$ to $A$
otherwise (so that the restriction conditions $(C1)$, $(C2)$, $(C3)$, $(C4)$
are satisfied for $(A,\hat{A})$). Note that such a restriction is well-defined and smooth
from $p_{\hat{A}}(\tilde{U})$ to $C(A;M)$ since $A$ is not contained
a daughter of $\hat{A}$. See the charts of Examples~\ref{exa2} and \ref{exa3}. In particular, $\iota^{\tau}$ is smooth. The proofs of the following assertions are left to the reader.
\begin{itemize}
\item $p^{\tau}(\tilde{C}_V(M)) \subseteq C^{\tau}_V(M)$.
\item $p^{\tau} \circ \iota^{\tau}_{|p^{\tau}(\tilde{U})}=\mbox{Identity}(p^{\tau}(\tilde{U}))$
\item $\iota^{\tau} \circ p^{\tau}_{|\tilde{U} \cap \tilde{C}_V(M)}=\mbox{Identity}(\tilde{U} \cap \tilde{C}_V(M))$.
\item $\iota^{\tau}(p^{\tau}(\tilde{U}) \cap C^{\tau}_V(M)) \subset \tilde{C}_V(M)$.
\end{itemize}
We shall prove the following lemma.
\begin{lemma}
\label{lemdifconfinfini}
There exist
\begin{enumerate}
\item $\varepsilon>0$,
$O= [0,\varepsilon[^k$,
\item an open neighborhood $W$ of $w^0$ in $\tilde{W}$,
\item an open neighborhood $U^{\tau}$ of $p^{\tau}(c^0)$ in $p^{\tau}(\tilde{U})$,
\item a smooth map $(p^{\tau} \circ \xi): O \times W \longrightarrow U^{\tau}$ such that
\begin{itemize}
\item $(p^{\tau} \circ \xi)(0;w^0)=p^{\tau}(c^0)$,
\item $(p^{\tau} \circ \xi)(O \times W) \subset {C}^{\tau}_V(M)$, and
\item $p_V \circ \xi(\omega \in O,w)$ is an injective map from $V$ to $(M \setminus \infty)$
if and only if $\omega \in ]0,\infty[^k$,
\end{itemize}
\item a smooth map $r^{\tau}: p^{\tau}(\tilde{U}) \longrightarrow \mathbb{R}^k \times \tilde{W}$ such that
\begin{itemize}
\item $r^{\tau} \circ p^{\tau} \circ \xi$ is the identity of $O\times W$,
\item $r^{\tau}(U^{\tau} \cap {C}^{\tau}_V(M)) \subseteq O \times W$, and
\item the restriction of $p^{\tau} \circ \xi \circ r^{\tau}$ to $U^{\tau} \cap C^{\tau}_V(M)$ is the identity of $U^{\tau} \cap C^{\tau}_V(M)$.
\end{itemize}
\end{enumerate}
\end{lemma}
This lemma implies Proposition~\ref{propdifconf} in this case because $\xi=\iota^{\tau} \circ (p^{\tau} \circ \xi)$, $U=(p^{\tau})^{-1}(U^{\tau})$ and
$r=r^{\tau} \circ p^{\tau}$ have the desired properties under its conclusions.
\medskip
\noindent{\bf Construction of $p^{\tau} \circ \xi$, $O$ and $W$.\/}
\medskip
\noindent Set $$P=((\nu_i)_{i \in \{1, \dots, \sigma\}};(\mu_A)_{A \in \tau_d};(s_i)_{i \in \{1, \dots, \sigma\}};(w_A)_{A \in \tau_d}) \in \mathbb{R}^{\tau_s} \times \mathbb{R}^{\tau_d} \times \tilde{W},$$
$$P^0=((0)_{i \in \{1, \dots, \sigma\}};(0)_{A \in \tau_d};(s^0_i)_{i \in \{1, \dots, \sigma\}}; (w^0_A)_{A \in \tau_d}) )=(0;w^0).$$
When $A \in \tau_d$, define
$$\tilde{w}_A =\tilde{w}_A(P)=\sum_{C \in \tau; C \subseteq A}\left(\prod_{D \in \tau; C \subseteq D \subset A} \mu_D \right) w_C \in S((\mathbb{R}^3)^{A \setminus b(A)}) \subset S((\mathbb{R}^3)^V).$$
$$\tilde{w}_A=w_A + \sum_{C \in D(A)}\mu_C \tilde{w}_C.$$
Set
$$\begin{array}{ll}
\tilde{s}_{\sigma} =s_{\sigma} + \mu_{V(\sigma)} \tilde{w}_{V(\sigma)} & \mbox{if}\; V(\sigma) \in \tau_d\\
\tilde{s}_{\sigma} =s_{\sigma} + \sum_{C \in D(V(\sigma))}\mu_C \tilde{w}_C & \mbox{otherwise.}
\end{array}$$
For $i= \sigma-1, \sigma-2, \dots, 1$, inductively define
$$\tilde{s}_i =s_i + \nu_{i+1} \tilde{s}_{i+1} +\sum_{C \in D(V(i)); C \neq V(i+1)}\mu_C \tilde{w}_C.$$
Define
$$\lambda_{V(r)}=\lambda_r=\prod_{i=1}^r \nu_i$$
so that $(\lambda_i \tilde{s}_i)_{|V(i+1)}=\lambda_{i+1} \tilde{s}_{i+1}$.
The $\tilde{w}_A$, $\lambda_r$, and $\tilde{s}_i$ are smooth functions defined on $\mathbb{R}^k \times \tilde{W}$, such that $\tilde{w}_A(P^0)=w_A^0$ and $\tilde{s}_i(P^0)=s_i^0$.
In particular, since the norms of these vectors are $1$ for $P^0$, we can choose neighborhoods $O$ of $0$ in $[0,\infty[^k$
and $W$ of $w^0$ in $\tilde{W}$, so that for any $P$ in $O \times W$
\begin{itemize}
\item the norms of the $\tilde{w}_A(P)$ and $\tilde{s}_i(P)$ do not vanish,
\item $\frac{\tilde{w}_A(P)}{\parallel \tilde{w}_A(P) \parallel} \in O(A;b(A);\tau)$, and,
\item $\frac{\tilde{s}_i(P)}{\parallel \tilde{s}_i(P) \parallel} \in O(V(i);\tau)$.
\end{itemize}
We choose $O$ and $W$ so that these properties are satisfied for any $A \in \tau_d$, and for any $i=1,2,\dots,\sigma$.
When $A \in \tau$, let ${V(i(A))}$ be the smallest element of $\tau_s$
such that $A \subseteq V(i(A))$.
Define $\tilde{s}_A \in (\mathbb{R}^3)^A$ as the restriction of $\tilde{s}_{i(A)}$ to $A$.
In order to define $p^{\tau} \circ \xi$, we define its projections $\xi_A(P)$ onto the factors
$C(A;M)$ for $A \in \tau$.\\
When ${A} \in \tau \setminus \tau_d$,
$\xi_A(P)=\psi(A;\phi_{\infty})(\lambda_{i(A)} \parallel \tilde{s}_A \parallel;
\frac{\tilde{s}_A}{\parallel \tilde{s}_A \parallel}).$\\
When ${A} \in \tau_d$,
$\xi_A(P)=\psi(A;\phi_{\infty};b(A))(\ell_A;u_A;m_A;v_A)$
with
$${\ell}_A=
\lambda_{i(A)} \sqrt{\sharp A}\parallel \tilde{s}_{A}(b(A)) \parallel$$
$${u}_A=\frac{\tilde{s}_{A}(b(A))}{\parallel \tilde{s}_{A}(b(A)) \parallel}$$
$${m}_A=\left(\prod_{D \in \tau_d; A \subseteq D \subseteq V(i(A))} \mu_D \right)
\frac{\parallel \tilde{w}_A \parallel}{\sqrt{\sharp A}\parallel \tilde{s}_{A}(b(A)) \parallel}$$
$${v}_A=\frac{\tilde{w}_A}{\parallel \tilde{w}_A \parallel}.$$
Thus, we defined a smooth map $p^{\tau} \circ \xi$ from $O \times W$ to $p^{\tau} (\tilde{U})$.
\medskip
\noindent{\bf Checking that $p^{\tau} \circ \xi$ satisfies $(C1)$. \/}
\medskip
Since the restriction of $\lambda_{i} \tilde{s}_i$ to $V(i+1)$ is
$\lambda_{i+1} \tilde{s}_{i+1}$, when $i \leq \sigma -1$, it is enough to check that for any $A$, $\Pi_A(\xi_A(P))$ is equal to
$\phi_{\infty} \circ \left(\lambda_{i(A)} (\tilde{s}_{i(A)})_{|A}\right)$.
When ${A} \notin \tau_d$
it is obvious.
Let us now consider the case when ${A} \in \tau_d$. \\
$$\tilde{s}_{A}=\mbox{constant map} + \left(\prod_{D \in \tau_d; A \subseteq D \subseteq V(i(A))} \mu_D \right) \tilde{w}_{{A}}.$$
Therefore
$$\phi_{\infty}^{-1} \circ \left(\Pi_A(\xi_A(P))\right)$$
$$=\lambda_{i(A)}\left( \left(\tilde{s}_{A}(b(A))\right)^A +
\left(\prod_{D \in \tau_d; A \subseteq D \subseteq V(i(A))} \mu_D \right)
\tilde{w}_A
\right)$$
$$=\lambda_{i(A)}\tilde{s}_{A}.$$
\medskip
\noindent{\bf Checking that $p^{\tau} \circ \xi$ satisfies $(C3)$. \/}
\medskip
Here, we need to check that when $A \subset B$, (and when $\lambda_{i(B)}=0$)
$(\tilde{s}_{i(B)})_{|A}$ is a $(\geq 0)$ multiple of $(\tilde{s}_{i(A)})_{|A}$.
Since $(\tilde{s}_{i(B)})_{|V(i(A))}=\left(\prod_{j=i(B)+1}^{i(A)}\nu_j \right) \tilde{s}_{i(A)}$, we are done.
\medskip
\noindent{\bf Checking that $p^{\tau} \circ \xi$ satisfies $(C2)$ and $(C4)$. \/}
\medskip
These conditions must be checked for some $A \subset B$, when the restriction of $\tilde{s}_{i(B)}$ to $B$ is constant.
In this case, since $\frac{\tilde{s}_{i(B)}}{\norm{\tilde{s}_{i(B)}}} \in O(V(i(B));\tau)$, ${B} \in \tau_d$, and
therefore
${A} \in \tau_d$.
These conditions say that, up to translation, $(\tilde{w}_{{B}})_{|A}$ is a $(\geq 0)$ multiple of $(\tilde{w}_{{A}})$
(when $\left(\prod_{D \in \tau_d; B \subseteq D \subseteq V(i(B))} \mu_D \right)=0$). They are realised with $\left(\prod_{D \in \tau_d; A \subseteq D \subset {B}} \mu_D \right)=0$ as a factor.
\medskip
\noindent {\bf We have proved that $p^{\tau} \circ \xi(O \times W) \subset {C}^{\tau}_V(M)$}
and it is easy to see that $p^{\tau} \circ \xi(P^0=(0;w^0))=p^{\tau}(c^0)$.
\medskip
\noindent{\bf Checking that $p_V \circ \xi((\mu_A,\nu_i) \in O,w \in W)$ is injective and does not reach $\infty$
if and only if all the $\mu_A$ and the $\nu_i$ are non zero. \/}
\medskip
Remember from the proof that $p^{\tau} \circ \xi$ satisfies $(C1)$, that the
restriction $\Pi_A(\xi_A(P))$ of $\Pi_V(\xi_V(P))$ is
$\phi_{\infty} \circ \left(\lambda_{i(A)} (\tilde{s}_{i(A)})_{|A}\right)$ for $A \in \tau$.
Now, $\Pi_V(\xi_V(P))$ is injective and does not reach $\infty$
if and only if $\lambda_{1}\tilde{s}_{1}$ is injective and does not reach $0$.
In particular, if $\Pi_V(\xi_V(P))$ is injective and does not reach $\infty$,
all the restrictions $\lambda_{i(A)} (\tilde{s}_{i(A)})_{|A}$ are injective and do not reach $0$ and this easily implies that the $\mu_A$ and the $\nu_i$ are non zero.
Conversely, assume that the $\mu_A$ and the $\nu_i$ are non zero, and let us prove that $\lambda_{1}\tilde{s}_{1}$ is injective and does not reach $0$. Since $\nu_1=\lambda_1$, it is enough to prove that $\tilde{s}_{1}$ is injective and does not reach $0$.
Let $a$ and $b$ be in $V$, and let $A$ be the smallest element of $\tau$ that contains both of them. If $A \in \tau_d$, $\tilde{w}_A(a) \neq \tilde{w}_A(b)$ and $\tilde{s}_{i(A)}(a) \neq \tilde{s}_{i(A)}(b)$. If $A \notin \tau_d$, $A=V(i(A))$, and $\tilde{s}_{i(A)}(a) \neq \tilde{s}_{i(A)}(b)$. Since $\tilde{s}_{1\,|V(i(A))}$ is a non zero multiple
of $\tilde{s}_{i(A)}$, it separates $a$ and $b$, and $\tilde{s}_{1}$ is injective. If $\tilde{s}_{1}(a)=0$, then $\tilde{s}_{i(\{a\})}(a)=0$, and this
is impossible, therefore $\tilde{s}_{1}$ does not reach $0$.
\medskip
\noindent{\bf Construction of $r^{\tau}$.\/}
\medskip
Let $c \in p^{\tau}(\tilde{U})$.
$$c=\left(\left(\psi(A;\phi_{\infty};b(A))(\ell_A;u_A;m_A;v_A)\right)_{A \in \tau_d};\left(\psi(A;\phi_{\infty})(\ell_A;S_A)\right)_{A \in (\tau \setminus \tau_d)}
\right).$$
We shall define
$$r^{\tau}(c)=((\nu_i)_{i \in \{1, \dots, \sigma\}};(\mu_A)_{A \in \tau_d};(s_i)_{i \in \{1, \dots, \sigma\}};(w_A)_{A \in \tau_d}).$$
\medskip
\noindent{\em Definition of $w_A$, for $A \in \tau_d$.}\\
Let $A \in \tau_d$. Define $w^1_A \in (\mathbb{R}^3)^A$ by
$$w^1_A(a)=\left\{\begin{array}{ll} v_A(a) & \mbox{if}\; a \in \left(A \setminus
(\cup_{B \in D(A)}B) \right)\\
v_A(b(B)) & \mbox{if}\; a \in B\;\;\;\;\mbox{with} \;\;\;\;B \in D(A) \end{array}\right.$$
and set
$$w_A=\frac{w^1_A}{\parallel w^1_A \parallel}.$$
\medskip
\noindent{\em Definition of $s_i$, for $i \in \{1, \dots, \sigma\}$.}\\
Let $V(i) \notin \tau_d$. Define $s^1_i \in (\mathbb{R}^3)^{V(i)}$ by
$$s^1_i(a)=\left\{\begin{array}{ll} 0 & \mbox{if}\; a \in V(i)_0 \\
S_{V(i)}(a) & \mbox{if}\; a \in \left(V(i) \setminus
(\cup_{B \in D(V(i))}B) \right)\\
S_{V(i)}(b(B)) & \mbox{if}\; a \in B\;\mbox{where} \;B \in D(V(i))\;\mbox{and} \;B \neq V(i)_0 \end{array}\right.$$
and set
$$s_{V(i)}=s_i=\frac{s^1_i}{\parallel s^1_i \parallel}.$$
If $V(\sigma) \in \tau_d$, then
$$s_{\sigma}=\left(\frac{u_{V(\sigma)}}{\sqrt{\sharp V(\sigma)}}\right)^{V(\sigma)}.$$
\medskip
\noindent{\em Definition of $\mu_A$, for $A \in \tau_d$.}\\
For any $A \in \tau$ such that $(\sharp A \geq 2)$, choose
$b^{\prime}(A) \neq b(A) \in A$ \index{N}{bprimeA@$b^{\prime}(A)$}
to be either the element of a son of $A$ or a basepoint of a daughter of $A$ that does not contain $b(A)$. \\
\begin{itemize}
\item If $A \in \tau_d$, (if $A \notin \tau_s$,) and if $\hat{A} \in \tau_d$,
$$\mu_A=\frac{ \langle v_{\hat{A}}(b^{\prime}(A))-v_{\hat{A}}(b(A)), w_{A}(b^{\prime}(A)) \rangle }
{ \langle w_{A}(b^{\prime}(A)), w_{A}(b^{\prime}(A)) \rangle }
\frac{\parallel w_{\hat{A}}(b^{\prime}(\hat{A})) \parallel}{\parallel v_{\hat{A}}(b^{\prime}(\hat{A})) \parallel}.$$
\item If $A \in \tau_d$, if $A \notin \tau_s$, and if $\hat{A} \notin \tau_d$,
$$\mu_A=\frac{ \langle S_{\hat{A}}(b^{\prime}(A))-S_{\hat{A}}(b(A)), w_{A}(b^{\prime}(A)) \rangle }
{ \langle w_{A}(b^{\prime}(A)), w_{A}(b^{\prime}(A)) \rangle }
\frac{\parallel s_{\hat{A}}(b^{\prime}(\hat{A})) \parallel}{\parallel S_{\hat{A}}(b^{\prime}(\hat{A})) \parallel}.$$
\item If $V(\sigma) \in \tau_d$, then
$\mu_{V(\sigma)}=\frac{m_{V(\sigma)}}{\parallel \tilde{w}_{V(\sigma)}\parallel}$
with
$$\tilde{w}_{V(\sigma)}=\sum_{C \in \tau; C \subseteq V(\sigma)}\left(\prod_{D \in \tau; C \subseteq D \subset V(\sigma)} \mu_D \right) w_C.$$
\end{itemize}
\medskip
\noindent{\em Definition of $\nu_i$, for $i \in \{1, \dots, \sigma\}$.}\\
Set $S_i=S_{V(i)}$ when $V(i) \notin \tau_d$, and set $b^{\prime}_i=b^{\prime}(V(i))$ when $\sharp V(i) > 1$.
\begin{itemize}
\item
When $i \geq 2$,
\begin{itemize}
\item If $i=\sigma$, and, if $V(\sigma) \in \tau_d$ or if $\sharp V(\sigma)=1$, then
$$\nu_{i}=\frac{ \langle S_{i-1}(b_{\sigma}),s_{i}(b_{\sigma}) \rangle }{ \langle s_{i}(b_{\sigma}),s_{i}(b_{\sigma}) \rangle }
\frac{\parallel s_{i-1}(b^{\prime}_{i-1}) \parallel}{\parallel S_{i-1}(b^{\prime}_{i-1})\parallel}.$$
\item Otherwise,
$$\nu_{i}=\frac{ \langle S_{i-1}(b^{\prime}_{i}),s_{i}(b^{\prime}_{i}) \rangle }{ \langle s_{i}(b^{\prime}_{i}),s_{i}(b^{\prime}_{i}) \rangle }
\frac{\parallel s_{i-1}(b^{\prime}_{i-1}) \parallel}{\parallel S_{i-1}(b^{\prime}_{i-1})\parallel}$$
\end{itemize}
\item \begin{itemize}
\item If $V \in \tau_d$, or if $\sharp V=1$, $\nu_1=\ell_V$.
\item If $V \notin \tau_d$ and if $\sharp V > 1$,
$$\nu_1=\ell_V\frac{ \langle S_V(b^{\prime}_{1}),s_1(b^{\prime}_{1}) \rangle }{ \langle s_1(b^{\prime}_{1}),s_1(b^{\prime}_{1}) \rangle }.$$
\end{itemize}
\end{itemize}
Then it is clear that $r^{\tau}$ is smooth from $p^{\tau}(\tilde{U})$ to $\mathbb{R}^k \times \tilde{W}$.
\medskip
\noindent{\bf Checking that $r^{\tau} \circ p^{\tau} \circ \xi$ is the identity of $O \times W$.\/}
\medskip
We compute $$r^{\tau} \circ p^{\tau} \circ \xi(P=((\nu_i)_{i \in \{1, \dots, \sigma\}};(\mu_A)_{A \in \tau_d};(s_i)_{i \in \{1, \dots, \sigma\}};(w_A)_{A \in \tau_d}))$$
$$=((\nu^{\prime}_i)_{i \in \{1, \dots, \sigma\}};(\mu^{\prime}_A)_{A \in \tau_d};(s^{\prime}_i)_{i \in \{1, \dots, \sigma\}};(w^{\prime}_A)_{A \in \tau_d}).$$
It is clear that $w^{\prime}_A=w_A$ for any $A \in \tau_d$ and that $s^{\prime}_i=s_i$ if $V(i) \notin \tau_d$.\\
If $V(\sigma) \in \tau_d$, then $s_{\sigma}$ is constant and $\frac{\tilde{s}_{\sigma}(b_{\sigma})}{\parallel \tilde{s}_{\sigma}(b_{\sigma})\parallel}= \frac{s_{\sigma}(b_{\sigma})}{\parallel s_{\sigma}(b_{\sigma})\parallel}$.\\ Thus
$u_{V(\sigma)}= \frac{s_{\sigma}(b_{\sigma})}{\parallel s_{\sigma}(b_{\sigma})\parallel}$ and $s^{\prime}_{\sigma}=\left(\frac{u_{V(\sigma)}}{\sqrt{\sharp V(\sigma)}}\right)^{V(\sigma)}
=s_{\sigma}$.\\
\noindent{\em Checking that $\mu^{\prime}_A=\mu_A$, for $A \in \tau_d$.}\\
\begin{itemize}
\item If $A \in \tau_d$, if $A \notin \tau_s$, and if $\hat{A} \in \tau_d$,
then $$v_{\hat{A}}
=\frac{\tilde{w}_{\hat{A}}}{\parallel \tilde{w}_{\hat{A}} \parallel}
=\frac{\parallel {v}_{\hat{A}}(b^{\prime}(\hat{A})) \parallel}{\parallel \tilde{w}_{\hat{A}}(b^{\prime}(\hat{A})) \parallel}\tilde{w}_{\hat{A}}
=\frac{\parallel {v}_{\hat{A}}(b^{\prime}(\hat{A})) \parallel}{\parallel {w}_{\hat{A}}(b^{\prime}(\hat{A})) \parallel}\tilde{w}_{\hat{A}}$$
and
$$\tilde{w}_{\hat{A}}(b^{\prime}(A))-\tilde{w}_{\hat{A}}(b(A))
=\mu_A \left( w_{A}(b^{\prime}(A))-w_A(b(A)) \right)=\mu_A w_{A}(b^{\prime}(A)).$$
Therefore,
$\mu^{\prime}_A=\mu_A.$
\item If $A \in \tau_d$, if $A \notin \tau_s$, and if $\hat{A} \notin \tau_d$,
$$S_{\hat{A}}=\frac{\tilde{s}_{\hat{A}}}{\parallel \tilde{s}_{\hat{A}} \parallel}=\frac{\parallel {S}_{\hat{A}}(b^{\prime}(\hat{A})) \parallel}{\parallel \tilde{s}_{\hat{A}}(b^{\prime}(\hat{A})) \parallel}\tilde{s}_{\hat{A}}
=\frac{\parallel {S}_{\hat{A}}(b^{\prime}(\hat{A})) \parallel}{\parallel {s}_{\hat{A}}(b^{\prime}(\hat{A})) \parallel}\tilde{s}_{\hat{A}}$$
and $$\tilde{s}_{\hat{A}}(b^{\prime}(A))-\tilde{s}_{\hat{A}}(b(A))
=\mu_A \left( w_{A}(b^{\prime}(A))-w_A(b(A)) \right).$$
Therefore,
$\mu^{\prime}_A=\mu_A.$
\item If $V(\sigma) \in \tau_d$, then
$m_{V(\sigma)}=\mu_{V(\sigma)}
\frac{\parallel \tilde{w}_{V(\sigma)} \parallel}
{\sqrt{{\sharp V(\sigma)}} \norm{\tilde{s}_{\sigma}(b_{\sigma})}}$,\\
where $\tilde{s}_{\sigma}(b_{\sigma})={s}_{\sigma}(b_{\sigma})
=\frac{{s}_{\sigma}(b_{\sigma})}{\sqrt{{\sharp V(\sigma)}} \norm{{s}_{\sigma}(b_{\sigma})}}$.\\
Thus, $m_{V(\sigma)}=\mu_{V(\sigma)}
\parallel \tilde{w}_{V(\sigma)} \parallel$, and $\mu^{\prime}_{V(\sigma)}=\mu_{V(\sigma)}$.
\end{itemize}
\medskip
\noindent{\em Proving that $\nu^{\prime}_i=\nu_i$, for $i \in \{1, \dots, \sigma\}$.}\\
\begin{itemize}
\item
When $i \geq 2$,\\
$S_{V({i-1})}=S_{i-1}=\frac{\tilde{s}_{i-1}}{\parallel \tilde{s}_{i-1} \parallel}= \frac{\parallel S_{i-1}(b^{\prime}_{i-1}) \parallel}{\parallel s_{i-1}(b^{\prime}_{i-1})\parallel} \tilde{s}_{i-1} $, $\tilde{s}_{i-1}(b_{\sigma})=\nu_{i}s_{i}(b_{\sigma})$, and\\
$\tilde{s}_{i-1}(b^{\prime}_{i})=\nu_{i}s_{i}(b^{\prime}_{i})$ if $V(i) \notin \tau_d$ and $\sharp V(i) > 1$.
Therefore, $\nu^{\prime}_{i}=\nu_{i}$.
\item
If $V \in \tau_d$, or if $\sharp V=1$,\\
$\nu^{\prime}_1=\ell_V$
where $\ell_V= \nu_1 \sqrt{\sharp V}\parallel \tilde{s}_V(b(V)) \parallel$, and $\tilde{s}_V(b(V))=s_V(b(V))=\frac{1}{\sqrt{\sharp V}}u_V$.
Therefore, $\nu_1=\ell_V$, and we are done.
\item If $V \notin \tau_d$ and if $\sharp V > 1$,\\
$\nu^{\prime}_1=\ell_V\frac{ \langle S_V(b^{\prime}_{1}),s_1(b^{\prime}_{1}) \rangle }{ \langle s_1(b^{\prime}_{1}),s_1(b^{\prime}_{1}) \rangle }$, $\ell_V=\nu_1 \parallel \tilde{s}_1 \parallel$,
where $S_V=\frac{\tilde{s}_1}{\parallel \tilde{s}_1 \parallel}=
\frac{ \langle S_V(b^{\prime}_{1}),\tilde{s}_1(b^{\prime}_{1}) \rangle }{ \langle \tilde{s}_1(b^{\prime}_{1}),\tilde{s}_1(b^{\prime}_{1}) \rangle } \tilde{s}_1$, and $\tilde{s}_1(b^{\prime}_{1})=s_1(b^{\prime}_{1})$.
Thus, $\nu^{\prime}_1=\nu_1.$\\
\end{itemize}
\medskip
Thus, $r^{\tau} \circ p^{\tau} \circ \xi$ is the identity of $O \times W$.
\medskip
\noindent{\bf Checking that $r^{\tau}(p^{\tau}(\tilde{U}) \cap {C}^{\tau}_V(M)) \subseteq [0,\infty[^k \times \tilde{W}$.\/}
\medskip
Let $c=(c_A)_{A; A \in \tau} \in p^{\tau}(\tilde{U}) \cap {C}^{\tau}_V(M)$
with
$$c_A=\left\{\begin{array}{ll}\psi(A;\phi_{\infty};b(A))(\ell_A;u_A;m_A;v_A) & \mbox{if}
\;A \in \tau_d\\
\psi(A;\phi_{\infty})(\ell_A;S_A) & \mbox{if}
\; A \in (\tau \setminus \tau_d)\end{array}\right.$$
\begin{itemize}
\item If $A \in \tau_d$, (if $A \notin \tau_s$,) and if $\hat{A} \in \tau_d$,
then $\left(v_{\hat{A}}(b^{\prime}(A))-v_{\hat{A}}(b(A))\right)$ is a $(\geq 0)$
multiple of $\left(v_{A}(b^{\prime}(A))-v_{A}(b(A))\right)$ that is a positive multiple of $w_{A}(b^{\prime}(A))$, therefore,
$$\mu_A=\frac{ \parallel v_{\hat{A}}(b^{\prime}(A))-v_{\hat{A}}(b(A)) \parallel }
{\parallel w_{A}(b^{\prime}(A))\parallel }
\frac{\parallel w_{\hat{A}}(b^{\prime}(\hat{A})) \parallel}{\parallel v_{\hat{A}}(b^{\prime}(\hat{A})) \parallel} \geq 0.$$
\item If $A \in \tau_d$, if $A \notin \tau_s$, and if $\hat{A} \notin \tau_d$, similarly,
$$\mu_A=\frac{ \parallel S_{\hat{A}}(b^{\prime}(A))-S_{\hat{A}}(b(A)) \parallel }
{ \parallel w_{A}(b^{\prime}(A)) \parallel }
\frac{\parallel s_{\hat{A}}(b^{\prime}(\hat{A})) \parallel}{\parallel S_{\hat{A}}(b^{\prime}(\hat{A})) \parallel} \geq 0.$$
\item If $V(\sigma) \in \tau_d$, then
$\mu_{V(\sigma)}=\frac{m_{V(\sigma)}}{\parallel \tilde{w}_{V(\sigma)}\parallel} \geq 0.$
\item
When $i \geq 2$,
\begin{itemize}
\item If $i=\sigma$, and, if $V(\sigma) \in \tau_d$ or if $\sharp V(\sigma)=1$, then
$$\nu_{i}=\frac{\parallel S_{i-1}(b_{\sigma})\parallel }{ \parallel s_{i}(b_{\sigma})\parallel }
\frac{\parallel s_{i-1}(b^{\prime}_{i-1}) \parallel}{\parallel S_{i-1}(b^{\prime}_{i-1})\parallel} \geq 0.$$
\item Otherwise,
$$\nu_{i}=\frac{ \parallel S_{i-1}(b^{\prime}_{i})\parallel }{ \parallel s_{i}(b^{\prime}_{i})\parallel }
\frac{\parallel s_{i-1}(b^{\prime}_{i-1}) \parallel}{\parallel S_{i-1}(b^{\prime}_{i-1})\parallel} \geq 0$$
\end{itemize}
\item \begin{itemize}
\item If $V \in \tau_d$, or if $\sharp V=1$, $\nu_1=\ell_V \geq 0$.
\item If $V \notin \tau_d$ and if $\sharp V > 1$,
$$\nu_1=\ell_V\frac{ \parallel S_V(b^{\prime}_{1})\parallel }{ \parallel s_1(b^{\prime}_{1})\parallel} \geq 0.$$
\end{itemize}
\end{itemize}
Therefore all the $\mu_A$ and the $\nu_i$ are positive.
Also, note that $r^{\tau}(p^{\tau}(c^0))=P^0$.
Now, choose
$(\varepsilon > 0)$ such that $[0,\varepsilon[^k \subset O$,
reduce $O$ into
$[0,\varepsilon[^k$ and set
$$U^{\tau}=(r^{\tau})^{-1}(]-\varepsilon,\varepsilon[^k \times W).$$
Then $p^{\tau} \circ \xi(O \times W) \subseteq U^{\tau}$,
$r^{\tau}(U^{\tau} \cap {C}^{\tau}_V(M)) \subseteq O \times W$.
\medskip
\noindent{\bf Checking that $p^{\tau} \circ \xi \circ r^{\tau}_{|U^{\tau} \cap {C}^{\tau}_V(M)}$ is the identity.\/}
Keep the above notation introduced to check that $r^{\tau}(p^{\tau}(\tilde{U}) \cap {C}^{\tau}_V(M)) \subseteq [0,\infty[^k \times \tilde{W}$
and assume that $c \in U^{\tau} \cap {C}^{\tau}_V(M)$.
\medskip
\noindent{\em Introducing more notation to prove that
$\xi_A \circ r^{\tau}(c)=c_A$, for any $A \in \tau$.} \\
$$r^{\tau}(c)=((\nu_i)_{i \in \{1, \dots, \sigma\}};(\mu_A)_{A \in \tau_d};(s_i)_{i \in \{1, \dots, \sigma\}};(w_A)_{A \in \tau_d}).$$
Define $\tilde{w}_A$, for $A \in \tau_d$, $\tilde{s}_i$ and $\lambda_i$, for
$i \in \{1, \dots, \sigma\}$ as in the construction of $p^{\tau} \circ \xi$. Then
$$\xi_A(r^{\tau}(c))=\left\{\begin{array}{ll}\psi(A;\phi_{\infty};b(A))(\tilde{\ell}_A;\tilde{u}_A;\tilde{m}_A;\frac{\tilde{w}_{A}}{\parallel \tilde{w}_{A} \parallel}) & \mbox{if}
\;A \in \tau_d\\
\psi(A;\phi_{\infty})(\lambda_{i(A)} \parallel \tilde{s}_{i(A)} \parallel;
\frac{\tilde{s}_{i(A)}}{\parallel \tilde{s}_{i(A)} \parallel}) & \mbox{if}
\; A \in (\tau \setminus \tau_d)\end{array}\right.$$
where, for $A \in \tau_d$,
$$\tilde{\ell}_A=
\lambda_{i(A)} \sqrt{\sharp A}\parallel \tilde{s}_{i(A)}(b(A)) \parallel,$$
$$\tilde{u}_A=\frac{\tilde{s}_{i(A)}(b(A))}{\parallel \tilde{s}_{i(A)}(b(A)) \parallel},$$
and
$$\tilde{m}_A=\left(\prod_{D \in \tau_d; A \subseteq D \subseteq V(i(A))} \mu_D \right)
\frac{\parallel \tilde{w}_{A} \parallel}{\sqrt{\sharp A}\parallel \tilde{s}_{i(A)}(b(A)) \parallel}.$$
\medskip
\noindent{\em Proving that $v_A=\frac{\tilde{w}_{A}}{\parallel \tilde{w}_{A} \parallel}$ when $A \in \tau_d$.}\\
Since $\norm{v_A}=1$, it suffices to prove that
$$\tilde{w}_A=\frac{1}{\norm{w^1_A}}v_A$$
where because $w_A=\frac{1}{\norm{w^1_A}}w^1_A$
$$\norm{w^1_A}=\frac{\norm{w^1_A(b^{\prime}(A))}}{\norm{w_A(b^{\prime}(A))}}=\frac{\norm{v_A(b^{\prime}(A))}}{\norm{w_A(b^{\prime}(A))}}.$$
\begin{itemize}
\item When $A$ has no daughters, since $\tilde{w}_A=w_A=w^1_A=v_A$ and $\norm{w^1_A}=1$, we are done.
\item Assume that $\tilde{w}_C=\frac{1}{\norm{w^1_C}}v_C$ for any $C \in D(A)$.
Let $C \in D(A)$. \\
Since $c \in {C}^{\tau}_V(M)$, $\left((v_A)_{|C}-(v_A(b(C)))^C\right)$ is a positive multiple of $v_C$, while
$(v_C)_{|\{b(C),b^{\prime}(C)\}}$ is a positive multiple of $w_C$ that vanishes at $b(C)$. Therefore,
$$\mu_C=\frac{1}{\norm{w^1_A}}\frac{\norm{v_A(b^{\prime}(C))-v_A(b(C))}}{\norm{w_C(b^{\prime}(C))}},$$
$$\frac{\mu_C}{\norm{w^1_C}}=\frac{1}{\norm{w^1_A}}\frac{\norm{v_A(b^{\prime}(C))-v_A(b(C))}}{\norm{v_C(b^{\prime}(C))}},$$
and
$$\norm{w^1_A}\tilde{w}_A=w^1_A + \sum_{C \in D(A)}\frac{\norm{v_A(b^{\prime}(C))-v_A(b(C))}}{\norm{v_C(b^{\prime}(C))}} v_C=v_A.$$
\end{itemize}
This proves that $\tilde{w}_A=\frac{1}{\norm{w^1_A}}v_A$ by induction.
\medskip
\noindent{\em Proving that $S_A=\frac{\tilde{s}_{i(A)}}{\parallel \tilde{s}_{i(A)} \parallel}$ when $A \in (\tau \setminus \tau_d)$.}\\
If $\sharp A =1$, then $A=V(\sigma)$, $\tilde{s}_{i(A)}=s_{i(A)}=S_A$, and we are done.\\
Assume $A=V(i) \notin \tau_d$ and $\sharp A > 1$.\\
We need to prove that, $S_i=S_{V(i)}=\frac{\tilde{s}_{i}}{\parallel \tilde{s}_{i} \parallel}$.
Again, it is enough to prove that $\tilde{s}_{i}=\frac{1}{\norm{s^1_i}}S_i$
where, since $s_i=\frac{1}{\norm{s^1_i}}s^1_i$,
$$\norm{s^1_i}=\frac{\norm{s^1_i(b^{\prime}_i)}}{\norm{s_i(b^{\prime}_i)}}
=\frac{\norm{S_i(b^{\prime}_i)}}{\norm{s_i(b^{\prime}_i)}}$$
Let $C \in D(V(i))$, $C \neq V(i)_0$.\\
Then $\left(S_i(b^{\prime}(C))-S_i(b(C))\right)$ is a $(\geq 0)$ multiple of
$\left(v_C(b^{\prime}(C))=\norm{w^1_C}w_C(b^{\prime}(C))\right)$ (see the previous proof),
$$\mu_C=\frac{1}{\norm{s^1_i}}\frac{\norm{S_i(b^{\prime}(C))-S_i(b(C))}}{\norm{w_C(b^{\prime}(C))}},$$
and,
$$\frac{\mu_C}{\norm{w^1_C}}=\frac{1}{\norm{s^1_i}}\frac{\norm{S_i(b^{\prime}(C))-S_i(b(C))}}{\norm{v_C(b^{\prime}(C))}}.$$
Therefore
$$\begin{array}{ll}\norm{s^1_i}(\tilde{s}_{i})_{|(V(i) \setminus V(i)_0)} &
=s^1_i + \sum_{C \in D(V(i)); C \neq V(i)_0}\frac{\mu_C}{\norm{w^1_C}}\norm{s^1_i}
v_C\\
&
=s^1_i + \sum_{C \in D(V(i)); C \neq V(i)_0}\frac{\norm{S_i(b^{\prime}(C))-S_i(b(C))}}{\norm{v_C(b^{\prime}(C))}}
v_C\\
& =(S_i)_{|(V(i) \setminus V(i)_0)}\end{array}$$
\begin{itemize}
\item When $V(\sigma) \notin \tau_d$, and when $i=\sigma$,\\
$V(i)_0=\emptyset$ and we can conclude.
\item When $V(\sigma) \in \tau_d$, and when $i=\sigma-1 \geq 1$, \\
By definition of $\nu_{\sigma}$, $$\nu_{\sigma}=\frac{\norm{S_i(b_{\sigma})}}{\norm{s_{\sigma}(b_{\sigma})}}\frac{\norm{s_i(b^{\prime}_i)}}{\norm{S_i(b^{\prime}_i)}}=\sqrt{\sharp V(\sigma)}\frac{\norm{S_i(b_{\sigma})}}{\norm{s^1_i}}.$$
Therefore
$$\begin{array}{ll}\norm{s^1_i}(\tilde{s}_{i})_{|V(i)_0} &
=\norm{s^1_i}\nu_{\sigma} \tilde{s}_{\sigma} =\norm{s^1_i}\nu_{\sigma} \left({s}_{\sigma} + \mu_{V(\sigma)} \tilde{w}_{V(\sigma)} \right)\\
&
=\sqrt{\sharp V(\sigma)}\norm{S_i(b_{\sigma})}\left({s}_{\sigma} + \mu_{V(\sigma)} \norm{\tilde{w}_{V(\sigma)}}v_{V(\sigma)} \right)\\
&
=\sqrt{\sharp V(\sigma)}\norm{S_i(b_{\sigma})}\left(\left(\frac{{u}_{V(\sigma)}}{\sqrt{\sharp V(\sigma)}}\right)^{V(\sigma)} + m_{V(\sigma)} v_{V(\sigma)} \right)\\
& =(S_i)_{|V(\sigma)}\end{array}$$
since the latter right-hand side is a $(\geq 0)$ mutiple of the previous right-hand side, and since the norms of their values at $b_{\sigma}$ are the same.
\item When $V(i+1) \notin \tau_d$, and when $\tilde{s}_{i+1}=\frac{1}{\norm{s^1_{i+1}}}S_{i+1}$,
$$(\tilde{s}_{i})_{|V(i+1)}= \nu_{i+1} \tilde{s}_{i+1}= \frac{\nu_{i+1}}{\norm{s^1_{i+1}}} S_{i+1}$$
\begin{equation}
\label{eq317}
\nu_{i+1}=\left\{\begin{array}{ll} \frac{\norm{s^1_{i+1}}}{\norm{s^1_{i}}}
\frac{\norm{S_i(b_{\sigma})}}{\norm{S_{i+1}(b_{\sigma})}} &
\mbox{if} \; i+1=\sigma \mbox{, and if}\; \sharp V(\sigma)=1,\\
\frac{\norm{s^1_{i+1}}}{\norm{s^1_{i}}}
\frac{\norm{S_i(b^{\prime}_{i+1})}}{\norm{S_{i+1}(b^{\prime}_{i+1})}} & \mbox{otherwise.}
\end{array}\right.\end{equation}
Thus, in the second case,
$$\norm{s^1_{i}}(\tilde{s}_{i})_{|V(i+1)}=\frac{\norm{S_i(b^{\prime}_{i+1})}}{\norm{S_{i+1}(b^{\prime}_{i+1})}}S_{i+1}=(S_i)_{|V(i+1)}.$$
In any case, $\norm{s^1_{i}}(\tilde{s}_{i})_{|V(i+1)}=(S_i)_{|V(i+1)}$.
\end{itemize}
We conclude that $\tilde{s}_{i}=\frac{1}{\norm{s^1_{i}}}S_{i}$ for any $i$ such that $V(i) \notin \tau_d$, with a decreasing induction on $i$.
\medskip
\noindent{\em Proving that $\ell_A=\lambda_A \norm{\tilde{s}_A}$ when $A \in (\tau \setminus \tau_d)$.}\\
If such an $A$ exists, $V \notin \tau_d$. Let $A=V(i) \notin \tau_d$, and let us prove that $\ell_{V(i)}= \lambda_i \norm{\tilde{s}_i}$\\ where
$S_i=\norm{{s}^1_i}\tilde{s}_i$, $\norm{\tilde{s}_i}=\frac{1}{\norm{{s}^1_i}}$, $\lambda_i= \prod_{j=1}^{i}\nu_j$ and $\nu_1=\ell_V \norm{{s}^1_1}$. (The latter equality is obvious if $\sharp V=1$, otherwise $V \notin \tau_d$ and $\nu_1=\ell_V \frac{\norm{S_V(b^{\prime}_1)}}{\norm{\tilde{s}_V(b^{\prime}_1)}}$.)\\
If $i=1$, we are done.\\
Otherwise, for any $j < i$, according to \ref{eq317},
$$\nu_{j+1}=\frac{\norm{s^1_{j+1}}}{\norm{s^1_{j}}}
\frac{\norm{(S_j)_{|V(j+1)}}}{\norm{S_{j+1}}}$$
Therefore,
$$\begin{array}{ll} \lambda_i \norm{\tilde{s}_i}&=\norm{\tilde{s}_i}\prod_{j=1}^{i}\nu_j\\
&=\ell_V\frac{\norm{{s}^1_1}}{\norm{{s}^1_i}}\prod_{j=1}^{i-1}\nu_{j+1}\\
&=\ell_V\prod_{j=1}^{i-1}\frac{\norm{(S_j)_{|V(j+1)}}}{\norm{S_{j+1}}}
\end{array}$$
where $\ell_{V(j)}(S_j)_{|V(j+1)}=\ell_{V(j+1)}S_{j+1}$.
It follows that $\ell_{V(i)}= \lambda_i \norm{\tilde{s}_i}$ by induction on $i$.
\medskip
\noindent{\em Proving that $\tilde{u}_{V(\sigma)}=u_{V(\sigma)}$, $\tilde{m}_{V(\sigma)}=m_{V(\sigma)}$ and $\tilde{\ell}_{V(\sigma)}=\ell_{V(\sigma)}$ when ${V(\sigma)} \in \tau_d$.}\\
Let ${V(\sigma)} \in \tau_d$. We already know that $v_{V(\sigma)}=\frac{\tilde{w}_{V(\sigma)}}{\norm{\tilde{w}_{V(\sigma)}}}$.
In particular,
$$\begin{array}{ll}\tilde{s}_{\sigma}&=s_{\sigma} + \mu_{V(\sigma)} \tilde{w}_{V(\sigma)}\\
&=s_{\sigma} + \mu_{V(\sigma)} \norm{\tilde{w}_{V(\sigma)}}{v}_{V(\sigma)}\\
&=\left(\frac{u_{V(\sigma)}}{\sqrt{\sharp V(\sigma)}} \right)^{V(\sigma)} + m_{V(\sigma)}{v}_{V(\sigma)},\\ \end{array}$$
$$\tilde{u}_{V(\sigma)}=\frac{\tilde{s}_{\sigma}(b({V(\sigma)}))}{\parallel \tilde{s}_{\sigma}(b({V(\sigma)})) \parallel}=u_{V(\sigma)},$$
$$\sqrt{\sharp V(\sigma)}\parallel \tilde{s}_{\sigma}(b({V(\sigma)})) \parallel=1,$$
$$\tilde{m}_{V(\sigma)}= \mu_{V(\sigma)}\frac{\norm{\tilde{w}_{V(\sigma)}}}{\sqrt{\sharp V(\sigma)}\parallel \tilde{s}_{\sigma}(b({V(\sigma)})) \parallel}=m_{V(\sigma)}.$$
We are left with the proof that $\tilde{\ell}_{V(\sigma)}=\ell_{V(\sigma)}$.
\begin{itemize}
\item
When $\sigma=1$,\\
$V(\sigma)=V$, $\tilde{\ell}_{V}=\lambda_1=\nu_1=\ell_V$, and we are done.
\item
When $\sigma > 1$,\\
Since $p^{\tau} \circ \xi \circ r^{\tau}(c) \in {C}^{\tau}_V(M)$,\\
$$\tilde{\ell}_{V(\sigma)} \left(\left(\frac{\tilde{u}_{V(\sigma)}}{\sqrt{\sharp V(\sigma)}} \right)^{V(\sigma)} + \tilde{m}_{V(\sigma)}
\frac{\tilde{w}_{V(\sigma)}}{\parallel \tilde{w}_{V(\sigma)}\parallel} \right)
=\lambda_{\sigma -1} (\tilde{s}_{\sigma -1})_{|V(\sigma)},$$
Thus, $\tilde{\ell}_{V(\sigma)}\tilde{s}_{\sigma}=\ell_{V(\sigma -1)}(S_{V(\sigma -1)})_{|V(\sigma)}.$\\
Since $c \in {C}^{\tau}_V(M)$,\\
$${\ell}_{V(\sigma)}\tilde{s}_{\sigma}=\ell_{V(\sigma -1)}(S_{V(\sigma -1)})_{|V(\sigma)}.$$
This implies that $\tilde{\ell}_{V(\sigma)}=\ell_{V(\sigma)}$.
\end{itemize}
\medskip
\noindent{\em Proving that $\ell_A=\tilde{\ell}_A$, $\tilde{m}_A=m_A$
and $\tilde{u}_A=u_A$ when $A \in \tau_d$, $A \neq V(\sigma)$.}\\
We already know that $\Pi_V(c_V)=\Pi_V(\xi_V \circ r^{\tau}(c))$ in $M^V$.
This map from $V$ to $M$ may be written as $\phi_{\infty} \circ f$
where $f \in (\mathbb{R}^3)^V = \ell_V S_V$ if $V \notin \tau_d$, and
$f=\ell_V \tilde{s}_{1}$ if $V \in \tau_d$.
Since both $c$ and $p^{\tau} \circ \xi \circ r^{\tau}(c)$ satisfy $(C1)$, then
$\ell_A=\norm{f(b(A))}\sqrt{\sharp A}$ and $\tilde{\ell}_A=\norm{f(b(A))}\sqrt{\sharp A}$.
Therefore, $\ell_A=\tilde{\ell}_A$.\\
Now, $f_{|A}$ is a $(\geq 0)$ multiple of
$\left( (u_A/ \sqrt{\sharp A})^A +m_A v_A) \right)$ and
$\left( (\tilde{u}_A/ \sqrt{\sharp A})^A +\tilde{m}_A v_A \right)$.
Thus, if $f_{|A}\neq 0$, using $$\left( (u_A/ \sqrt{\sharp A})^A +m_A v_A \right)(b(A))=(u_A/ \sqrt{\sharp A})\;\;\;\mbox{and}\;\;\;\norm{u_A}=1,$$ we easily
conclude that
$$(u_A/ \sqrt{\sharp A})^A +m_A v_A =(\tilde{u}_A/ \sqrt{\sharp A})^A +\tilde{m}_A v_A.$$
Now, if $f_{|A} = 0$, since $f_{|V(i(A))}$ is a $(\geq 0)$ multiple of
$\tilde{s}_{i(A)}$, and since $\tilde{s}_{i(A)}$ does not vanish on $A$, we deduce that $f_{|V(i(A))}=0$.
Then $(C3)$ implies that $(\tilde{s}_{i(A)})_{|A}$ is a $(\geq 0)$ multiple of
$\left( (u_A/ \sqrt{\sharp A})^A +m_A v_A) \right)$ and
$\left( (\tilde{u}_A/ \sqrt{\sharp A})^A +\tilde{m}_A v_A) \right)$, and we conclude as before that
$$(u_A/ \sqrt{\sharp A})^A +m_A v_A =(\tilde{u}_A/ \sqrt{\sharp A})^A +\tilde{m}_A v_A.$$
This implies of course that $\tilde{m}_A=m_A$
and $\tilde{u}_A=u_A$. \eop
This finishes the proof of Lemma~\ref{lemdifconfinfini} and thus Proposition~\ref{propdifconf} is proved.
\eop
To prove Lemma~\ref{lempropconsadface} in this case, we again look at the above proof. Here, $k=1$ if and only if $\tau_s=\{V\}$ and $\tau_d=\emptyset$, that is if and only if $\Pi_{\infty}(c^0_V)$ is an injective map from $V$ to $(T_{\infty}M \setminus 0)$ (up to dilation),
that is if and only if $c^0 \in F(\infty;V)$.
Lemma~\ref{lempropconsadface} is now proved.\eop
\subsection{Proof of Proposition~\ref{propcdeuxcoinc}}
\label{subpropcdeuxcoinc}
First define a smooth map of the following form
$$\begin{array}{llll}H:&C_2(M) &\longrightarrow &C(V;M) \times C(\{1\};M) \times C(\{2\};M)\\
&c &\mapsto &(p_2(c),r_1(c),r_2(c))\end{array}$$
whose image will be in $C_V(M)$.
Since $C_2(M)$ is a blow-up of $C(V;M)$ along $\infty \times C_1(M)$ and
$C_1(M) \times \infty$, we get the canonical smooth projection
$$p_2:C_2(M) \longrightarrow C(V;M) (\hfl{\Pi_V} M^V).$$
Let $i \in \{1,2\}$.
Let $p_{\{i\}}: M^V \longrightarrow M^{\{i\}}$ be the canonical restriction, and let $$\tilde{r}_i=p_{\{i\}} \circ \Pi_V \circ p_2: C_2(M) \longrightarrow M.$$
Let $r_i$ denote the restriction
of $\tilde{r}_i$ from $ (\Pi_V \circ p_2)^{-1}\left((M \setminus \infty)^2\right)$ to $(M \setminus \infty)$.
We are now going to define a smooth extension of the $r_i$ to $C_2(M)$ so that $H(C_2(M)) \subseteq C_V(M)$.
Let $\Pi_1:C_1(M) \longrightarrow M$ be the canonical projection.
Since the blow-up of $M \times (M\setminus \infty)$ along $\infty \times (M \setminus \infty)$ is canonically diffeomorphic to the product of the blow-up
of $M$ at $\infty$ by $(M \setminus \infty)$, it is easy to observe the following lemma.
\begin{lemma}
For any two disjoint open subsets $V_1$ and $V_2$ of $M$, there is a canonical
diffeomorphism
$$(\Pi_V \circ p_2)^{-1}(V_1 \times V_2) \hfl{(r_1,r_2)} \Pi_1^{-1}(V_1) \times \Pi_1^{-1} (V_2)$$
where $r_1$ and $r_2$ coincide with the previous maps $r_1$ and $r_2$ wherever
it makes sense.
\end{lemma}
\eop
Let us now prove that $r_i$ extends to a smooth projection from $C_2(M)$ onto
$C_1(M)$.
This extension will be necessarily unique and canonical, it will be denoted by $r_i$.
By symmetry, we only consider the case $i=1$.
It remains to define $r_1$ on $(\Pi_V \circ p_2)^{-1}(\infty,\infty)$
and to prove that it is smooth there.
The canonical smooth projection $\Pi_{M^2(\infty,\infty)}$ from
$C_2(M)$ to $M^2(\infty,\infty)$ maps $(\Pi_V \circ p_2)^{-1}(\infty,\infty)$
to $\left(ST_{(\infty,\infty)}M^2\right)$.
In turn, $ST_{(\infty,\infty)}M^2 \setminus S(0 \times T_{\infty}M)$
projects onto $ST_{\infty}M$ as the first coordinate.
It is easy to see that this smoothly extends the definition of $r_1$ outside $\Pi_{M^2(\infty,\infty)}^{-1}\left(S(0 \times T_{\infty}M) \right)$. To conclude, we recall the structure of $C_2(M)$ near $\Pi_{M^2(\infty,\infty)}^{-1}\left(S(0 \times T_{\infty}M) \right)$.
According to Proposition~\ref{propblodifdeux}, since the normal bundle of $\infty \times M$ at $(\infty,\infty)$ in $M^2$ is $(T_{\infty}M \times \{0\})$, $\Pi_{M^2(\infty,\infty)}^{-1}\left(S(0 \times T_{\infty}M) \right)$ is the product
$ST_{\infty}M \times S(0 \times T_{\infty}M)$. Define $r_1$ as the projection
on the fiber $ST_{\infty}M \subset C_1(M)$ in this product.
From the chart $\phi_{\infty}^2: (\mathbb{R}^3)^2 \longrightarrow M^2$, that induces the chart
$$\psi_1:[0,\infty[ \times S((\mathbb{R}^3)^2)\cong S^5 \longrightarrow M^2(\infty,\infty)$$
such that $\Pi_V \circ \psi_1(\lambda;(x,y))= (\phi_{\infty}( \lambda x),\phi_{\infty}( \lambda y))$
that in turn, induces the chart near
$S( 0 \times T_{\infty}M )$
$$\psi_2:[0,\infty[ \times \mathbb{R}^3 \times S^2 \longrightarrow M^2(\infty,\infty)$$
such that $\Pi_V \circ \psi_2(\lambda;(x,y))= (\phi_{\infty}( \lambda x),\phi_{\infty}( \lambda y))$, we get a chart
$$\psi_3:[0,\infty[ \times ([0,\infty[ \times S^2) \times S^2 \longrightarrow C_2(M)$$
such that $\Pi_V \circ p_2 \circ \psi_3(\lambda;\mu;x;y))= (\phi_{\infty}( \lambda \mu x),\phi_{\infty}( \lambda y))$. Using a similar chart for $C_1(M)$, $r_1$ will read
$$(\lambda;\mu;x;y) \mapsto (\lambda\mu;x)$$
and is smooth.
Now, our map $H$ is well-defined and smooth.
The elements of $H(C_2(M))$ satisfy $(C1)$ and $(C3)$ (of Lemmas~\ref{lemcond1} and \ref{lemcond3}). Thus, since $V$ has two elements, $H(C_2(M)) \subseteq C_V(M)$ and we have defined a smooth map $H$ from
$C_2(M)$ to $C_V(M)$, that extends the identity of $\breve{C}_2(M)$.
Let $p_V: C_V(M) \longrightarrow C(V;M)$ be the canonical projection.
Define
$$K: C_V(M) \longrightarrow C_2(M)$$
so that
$$K(c_V,c_1,c_2) = \left\{\begin{array}{ll}
p_2^{-1}(c_V) &\;\mbox{if}\; c_V \notin (\infty \times C_1(M)) \cup (C_1(M) \times \infty)\\
(r_1,r_2)^{-1}(c_1,c_2)&\;\mbox{if}\;\Pi_V(c_V) \;\mbox{is not constant.}
\end{array} \right.
$$
The map $K$ is consistently defined outside $p_V^{-1}((\infty \times \partial C_1(M)) \cup (\partial C_1(M) \times \infty))$, and it is smooth there.
We shall extend $K$ by the canonical identifications on $p_V^{-1}((\infty \times \partial C_1(M)) \cup (\partial C_1(M) \times \infty))$ and
use the charts of $C_V(M)$, near $p_V^{-1}((\infty \times \partial C_1(M)) \cup (\partial C_1(M) \times \infty))$ to prove the smoothness.
For example, $p_V^{-1}((\infty \times \partial C_1(M))$ is the subset of
$S(\{0\} \times T_{\infty}M) \times S(T_{\infty}M \times \{0\}) \times S(\{0\} \times T_{\infty}M)$ where the first and third coordinate coincide. There,
$\tau_d=\emptyset$, $\tau_s=\{V,\{1\}\}$, and the chart
$\xi$ of Subsection~\ref{subsecinf} reads:\\
$(\nu_1,\nu_2,s_1=s_V,s_2=s_{\{1\}}) \mapsto ( \psi(V;\phi_{\infty})(\nu_1\norm{s_V + \nu_2 s_{\{1\}}};\frac{s_V + \nu_2 s_{\{1\}}}{\norm{s_V + \nu_2 s_{\{1\}}}}),$\\
$\psi(\{1\};\phi_{\infty})(\nu_1\nu_2;s_{\{1\}}),
\psi(\{2\};\phi_{\infty})(\nu_1;{s_V(2)}))$.\\
Mapping $\xi(\nu_1,\nu_2,s_1=s_V,s_2=s_{\{1\}})$ to $\psi_3(\nu_1,\nu_2,s_{\{1\}},s_V(2))$ (where $\psi_3$ is defined above in this subsection)
smoothly extends $K$ to $p_V^{-1}((\infty \times \partial C_1(M))$. Similarly, $K$ smoothly extends to $p_V^{-1}(\partial C_1(M) \times \infty)$.
Then $K$ and $H$ are smooth maps that extend the identity of $\breve{C}_2(M)$ that is dense in both spaces. Therefore $K$ and $H$
are inverse of each other and they are diffeomorphisms.
\eop
\newpage
\addcontentsline{toc}{section}{References}
|
{
"timestamp": "2004-11-04T14:48:56",
"yymm": "0411",
"arxiv_id": "math/0411088",
"language": "en",
"url": "https://arxiv.org/abs/math/0411088"
}
|
\section{Introduction}
An intriguing aspect of quantum mechanics is that even at absolute
zero temperature quantum fluctuations prevail in a system, whereas
all thermal fluctuations are frozen out. These quantum
fluctuations are able to induce a macroscopic phase transition in
the ground state of a many-body system, when the ratio of two
competing terms in the underlying Hamiltonian is varied across a
critical value [1-4]. The instability of the quantum critical
behavior with respect to (w.r.t.) the disorder can be interpreted
as a signal for phenomena of localization. Recently some of the
new developments in the localization problem \cite{QCP:2000} have
become a major theme in the condense matter research. One example
is the strongly-interacting electron (non-Fermi) liquid with
different strengths of disorder \cite{RMP:2001A}. Interesting
issues are the quantum phase transition (QPT) and quantum critical
point (QCP). Both effects of disorder and interaction are closely
relevant to the weak and strong localization
\cite{Lat:Disord,MIT:JPC}. They are then related to the
metal-insulator transition in two-dimensions (2D). Although most
of theories proposed before are based on the Fermi liquid
behavior, new insights could be obtained considering the possible
analogy with superconducting transition which might be related to
the bosonic system [2-3]. \newline
In the last two decades a variety of metals have been discovered
which display thermodynamic and transport properties at low
temperatures which are fundamentally different from those of the
usual metallic systems which are well described by the Landau
Fermi-liquid theory. The resistivity in a variety of high mobility
2D electron/hole systems is seen experimentally to exhibit a
number of interesting anomalies that don't as yet have an adequate
theoretical understanding.
Note that, at sufficiently low electron densities, an ideal
two-dimensional electron systems becomes strongly correlated,
because the kinetic energy is overpowered by energy of
electron-electron interactions (exchange and correlation energy).
The interaction strength is normally described by the Wigner-Seitz
radius, $r_s =1/(\pi n_s)^{1/2} a_B$ (where $n_s$ is the electron
density and $a_B$ is the effective Bohr radius in semiconductor).
Till now, for rather large $r_s$ ($\gg 1$) and together with
disorder, the nature of this metal-insulator transition (MIT)
remains the subject of ongoing debate [6,8-9].
\newline Motivated by the analogy
between electrons in periodic or disordered metals and waves in
classical acoustical systems [10-12] an investigation for
observing possible QCP in MIT or relevant localization
\cite{Lat:Disord} using the quantum discrete kinetic model [13-14]
was performed and will be presented here. In present approach the
\"{U}hling-Uhlenbeck collision term [13] which could describe the
collision of a gas of dilute hard-sphere Fermi- or Bose-particles
by tuning a parameter $\gamma$ : a Pauli-blocking factor (or
$\gamma f$ with $f$ being a normalized (continuous) distribution
function giving the number of particles per cell) is adopted
together with a disorder or free-orientation ($\theta$ which is
related to the relative direction of scattering of particles
w.r.t. to the normal of the propagating plane-wave front) into the
quantum discrete kinetic model which can be used to obtain
dispersion relations of (plane) sound waves propagating in quantum
gases. We then study the quantum critical behavior based on the
acoustical analog [15-17] which has been verified before. The
possible phase diagram for MIT and/or resistance(or
disorder)-scattering amplitude curves (as the temperature is
decreased ) we obtained resemble qualitatively those proposed in
[5] (cf Fig. 4 therein). The non-Fermi liquid behavior was also
clearly illustrated here.
\newline We firstly introduce the concept of acoustical
analog [15] in brief.
In a mesoscopic system, where the sample size is smaller than the
mean free path for an elastic scattering, it is satisfactory for a
one-electron model to solve the time-independent Schr\"{o}dinger
equation :
$-({\hbar^2}/{2m}) \nabla^2 \psi + V' (\vec{r}) \psi = E \psi$
or (after dividing by $-\hbar^2/2m$)
$\nabla^2 \psi + [q^2 - V (\vec{r})] \psi = 0$,
where $q$ is an (energy) eigenvalue parameter, which for the
quantum-mechanic system is $\sqrt{2mE/\hbar^2}$. Meanwhile, the
equation for classical (scalar) waves is
$\nabla^2 \psi - ({\partial^2 \psi}/{c^2 \,\partial t^2})
=0$
or (after applying a Fourier transform in time and contriving a
system where $c$ (the wave speed) varies with position $\vec{r}$)
$\nabla^2 \psi + [q^2 - V (\vec{r})] \psi = 0$,
here, the eigenvalue parameter $q$ is $\omega/c_0$, where $\omega$
is a natural frequency and $c_0$ is a reference wave speed.
Comparing the time dependencies one gets the quantum and classical
relation $E= \hbar \omega$ [15-17].\newline
We assume that the gas is composed of identical hard-sphere
particles of the same mass \cite{U:U,Chu:PhD,Platkowski:1988}. The
velocities of these particles are restricted to, e.g., : ${\bf
u}_1, {\bf u}_2, \cdots, {\bf u}_p$, $p$ is a finite positive
integer. The discrete number densities of particles are denoted by
$N_i ({\bf x},t)$ associated with the velocities ${\bf u}_i$ at
point ${\bf x}$ and time $t$. If only nonlinear binary collisions
and the evolution of $N_i$ are considered, we have
\begin{equation}
\frac{\partial N_i}{\partial t}+ {\bf u}_i \cdot \nabla N_i
= F_i \equiv \frac{1}{2}\sum_{j,k,l} (A^{ij}_{kl} N_k N_l - A_{ij}^{kl}
N_i N_j),
\hspace*{3mm} i \in\Lambda =\{1,\cdots,p\},
\end{equation}
where $(i,j)$ and $(k,l)$ ($i\not=j$ or $k\not=l$) are admissible
sets of collisions [13-14,16-18]
Here, the summation is taken over all $j,k,l \in \Lambda$, where
$A_{kl}^{ij}$ are nonnegative constants satisfying [13-14,18]
$ A_{kl}^{ji}=A_{kl}^{ij}=A_{lk}^{ij}$,
$ A_{kl}^{ij} ({\bf u}_i +{\bf u}_j -{\bf u}_k -{\bf u}_l )=0$,
and $A_{kl}^{ij}=A_{ij}^{kl}$.
The conditions defined for the discrete velocities above require
that there are elastic, binary collisions, such that momentum and
energy are preserved, i.e.,
${\bf u}_i +{\bf u}_j = {\bf u}_k +{\bf u}_l$,
$|{\bf u}_i|^2 +|{\bf u}_j|^2 = |{\bf u}_k|^2 +|{\bf u}_l|^2$,
are possible for $1\le i,j,k,l\le p$.
We note that, the summation of $N_i$ ($\sum_i N_i$) : the total
discrete number density here is related to the macroscopic density
: $\rho \,(= m_p \sum_i N_i)$, where $m_p$ is the mass of the
particle \cite{Platkowski:1988}.
\newline
Together with the introducing of the \"{U}hling-Uhlenbeck
collision term \cite{U:U} :
$F_i$ $=\sum_{j,k,l} A^{ij}_{kl} \,[ N_k N_l$ $(1+\gamma N_i)(1+\gamma N_j)$ $-
N_i N_j (1+\gamma N_k)(1+\gamma N_l)]$,
into equation (1), for $\gamma <0$ (normally, $\gamma=-1$), we can
then obtain a quantum discrete kinetic equation for a gas of
Fermi-particles; while for $\gamma
> 0$ (normally, $\gamma=1$) we obtain one for a gas of Bose-particles,
and for $\gamma =0$ we recover the equation (1). \newline
Considering binary collisions only, from equation above, the
model of quantum discrete kinetic equation for Fermi or Bose gases
proposed in [13-14] is then a system of $2n(=p)$ semilinear
partial differential equations of the hyperbolic type :
\begin{displaymath}
\frac{\partial}{\partial t}N_i +{\bf v}_i \cdot\frac{\partial}{\partial
{\bf x}} N_i =\frac{c S}{n} \sum_{j=1}^{2n} N_j N_{j+n}(1+\gamma N_{j+1})
(1+\gamma N_{j+n+1})-
\end{displaymath}
\begin{equation}
\hspace*{18mm} 2 c S N_i N_{i+n} (1+\gamma N_{i+1})(1+\gamma
N_{i+n+1}),\hspace*{24mm} i=1,\cdots, 2 n,
\end{equation}
where $N_i=N_{i+2n}$ are unknown functions, and ${\bf v}_i$ =$ c
(\cos[\theta+(i-1) \pi/n], \sin[\theta+(i-1)\pi/n])$; $c$ is a
reference velocity modulus and the same order of magnitude as that
($c$, the sound speed in the absence of scatters) used in Ref. 9,
$\theta$ is the orientation starting from the positive $x-$axis to
the $u_1$ direction and could be thought of as a parameter for
introducing a {\it disorder}
\cite{RMP:Loc2001,Local:1996,Chu:2001,ChuA:2001,ChuA:2002A}, $S$
is an effective collision cross-section for the collision system.
\newline Since passage of the sound wave will cause a small departure
from an equilibrium state and result in energy loss owing to
internal friction and heat conduction, we linearize above
equations around a uniform equilibrium state (particles' number
density : $N_0$) by setting $N_i (t,x)$ =$N_0$ $(1+P_i (t,x))$,
where $P_i$ is a small perturbation.
After some similar manipulations as mentioned in [14-17], with
$B=\gamma N_0 <0$ \cite{U:U,Chu:PhD}, which gives or defines the
(proportional) contribution from the Fermi gases (if $\gamma < 0$,
e.g., $\gamma=-1$), we then have
\begin{equation}
[\frac{\partial^2 }{\partial t^2} +c^2
\cos^2[\theta+\frac{(m-1)\pi}{n}]
\frac{\partial^2 }{\partial x^2} +4 c S N_0 (1+B) \frac{\partial
}{\partial t}] D_m= \frac{4 c S N_0 (1+B)}{n} \sum_{k=1}^{n} \frac{\partial
}{\partial t} D_k ,
\end{equation}
where $D_m =(P_m +P_{m+n})/2$, $m=1,\cdots,n$, since $D_1 =D_m$
for $1=m$ (mod $2 n)$. \newline
We are ready to look for the solutions in the form of plane wave
$D_m$= $a_m$ exp $i (k x- \omega t)$, $(m=1,\cdots,n)$, with
$\omega$=$\omega(k)$. This is related to the dispersion relations
of 1D (forced) plane wave propagation in Fermi gases. So we have
\begin{equation}
(1+i h (1+B)-2 \lambda^2 cos^2 [\theta+\frac{(m-1)\pi}{n}]) a_m -\frac{i h (1+B)}{n}
\sum_{k=1}^n a_k =0 , \hspace*{6mm} m=1,\cdots,n,
\end{equation}
where
\begin{displaymath}
\lambda=k c/(\sqrt{2}\omega), \hspace*{18mm} h=4 c S N_0 /\omega
\hspace*{6mm} \propto \hspace*{2mm} 1/K_n,
\end{displaymath}
where $h$ is the rarefaction parameter of the gas; $K_n$ is the
Knudsen number which is defined as the ratio of the mean free path
of gases to the wave length of the plane (sound) wave.
\newline
We can obtain the complex spectra ($\lambda=\lambda_r +$ i
$\lambda_i$; $\lambda_r = k_r c/(\sqrt{2}\omega)$: sound
dispersion, a relative measure of the sound or phase speed;
$\lambda_i = k_i c/(\sqrt{2}\omega)$ : sound attenuation or
absorption) from the complex polynomial equation above. Here, $B$
could be related to the occupation number of different-statistic
particles of gases
To examine the critical region possibly tuned by the
Pauli-blocking measure $B=\gamma N_0$ and the disorder $\theta$,
as evidenced from our preliminary results : $\lambda_i =0$ for
cases of $B=-1$ or $\theta=\pi/4$ [17], we firstly check those
spectra near $\theta=0$, say, $\theta=0.005$ and $\theta=\pi/4
\approx 0.7854$, say, $\theta=0.78535$ for a $B$-sweep ($B$
decreases from 1 to -1). We plot them into figures 1, 2,
respectively. Note that, as the disorder or free-orientation
$\theta$ is not zero, there will be two kinds of propagation of
the disturbance wave : sound and diffusion modes [16-17,19-20].
The latter (anomalous) mode has been reported in Boltzmann gases
[19,21] and is related to the propagation of entropy wave which is
not used in the acoustical analog here. The absence of (further)
diffusion (or maximum absorption) for the sound mode at certain
state ($h$, corresponding to the inverse of energy $E$; cf. Refs.
11 or 15) is classified as a localized state [11,15,17]. The state
of decreasing $h$ corresponds to that of $T$ (absolute
temperature) decreasing as the mean free path is increasing
(density or pressure decreasing).
\newline We can observe the max. $\lambda_i$ (absorption of sound
mode, relevant to the localization length according to the
acoustical analog [15,17]) drop to around four orders of magnitude
from $\theta=0.005$ to $0.78535$! This is a clear demonstration of
the effect of disorder. Meanwhile, once the Pauli-blocking measure
($B$) increases or decreases from zero (Boltzmann gases), the
latter (Fermi gases : $B <0$) shows opposite trend compared to
that of the former (Bose gases : $B>0$) considering the shift of
the max. $\lambda_i$ state ($\delta h$). $\delta h >0$ is for
Fermi gases ($|B|$ increasing), and the reverse ($\delta h <0$)
is for Bose gases ($B$ increasing)! This illustrates partly the
electron-electron interaction effect (through the Pauli exclusion
principle). These results will be crucial for further obtaining
the phase diagram (flow to metallic or insulating state as the
density or temperature is decreased) tuned by both disorder and
the (electron-electron) interaction below. Here, $B=-1$ or
$\theta=0, \pi/4$ might be fixed points commented in [22].
\newline
To check what happens
when the temperature is decreased (or the density or $h$ is
decreased) to near $T=0$, we collect all the data based on the
acoustical analog from the dispersion relations (especially the
absorption of sound mode) we calculated for ranges in different
degrees of disorder (here, $\theta$ is up to $\pi/4$ considering
single-particle scattering and binary collisions; in fact, effects
of $\theta$ are symmetric w.r.t. $\theta=\pi/4$ for
$0\le\theta\le\pi/2$ [17]) and Pauli-blocking measure. After that,
we plot the possible phase diagram for the (dimensionless)
conductivity vs. the absolute (dimensionless) temperature into
Fig. 3 (for different $B$s : $B=-0.9, -0.7, -0.5, -0.3, 0.1$).
Here, MFP is
the mean free path and the temperature vs. MFP relations could be
traced from [23] (cf. Fig. 3 therein). Note that, the resistivity
or resistance is proportional to the strength of disorder in 2D
(in the sense that for weak disorder it is given by $1/(k_F l)$,
in units of $\hbar/e^2$; $k_F$ is the Fermi wave number, $l$ is
the mean free path associated with the usual Drude conductivity)
[5]. This figure shows that as the temperature decreases to a
rather low value, the resistivity (or strength of disorder) will
decrease sharply (at least for Bose or Fermi gases). There is no
doubt that this result resembles qualitatively that proposed
before [5,24-25].
\newline
To know the detailed effects of electron-electron interactions
(tuned by $B$s here), we plot the corresponding figure as shown in
Fig. 4. Each contour line (flow path relevant to a specific $B$
tuned by the disorder) represents the behavior when $T$ is
decreased, and different (flow) lines represent different values
of scattering amplitude ($S_b$ $\propto K_n$ or $r_s$).
Interesting results are (i) there seems to be a quantum phase
transition boundary (interface or regime) for bosonic-like
particles ($B>0$) which resembles that of high-temperature
superconducting phase transition due to doping if we treat the
strength of disorder to be equivalent to the doping amount! (ii)
as evidenced in the top part of this plot for the Bose gases (near
QCPs for smaller resistivity together with rather large arrows),
it confirms Larkin's comment (cf page 793 in [2] by Larkin) : both
Bose and Fermi approached are important for the (QP) transition;
Bose approach is more useful in a small region close to the
transition!
\newline To make sure we already recover previous proposed results
(possible renormalization group (RG) flows for disorder plus
interactions or scattering amplitude-resistance curves or
suggested phase diagram tuned by $r_s$ and disorder, cf. [5] or
[24-25]), we summarize our results by illustrating them into a 3D
plot as demonstrated in Fig. 5. Crossover lines separate those
flows which begin at $T>0$ at small resistance (disorder or the
inverse of conductance : $1/G$) and flow initially, as the
temperature is lowered, toward larger resistance but then are
repelled by the fixed point and flow to large $S_b$ (or $r_s$) and
large G (metallic behavior) from those which begin at larger
disorder (or smaller G) and flow toward very-large disorder
(insulating behavior). Again, this result resembles that proposed
before (cf. [5] or [24-25] therein). Note that, this flow is also
similar to those in [24-25] (e.g., cf. Fig. 41 by Aoki [24]).
Meanwhile, as expected before, at $T=0$, a 2D system would become
a Wigner crystal phase at $r_s \ge 37$ [26]. This lies possibly
near or above the rather large $S_b$ position with zero strength
of disorder in our illustrations. Possible QCPs happen around $S_b
\sim 80$ for $B=0.85$ but $S_b \sim 10$ for $B=0.1$.
\newline To conclude in brief, our illustrations here, although
are based on the acoustical analog of our quantum discrete
kinetic calculations, can indeed show the non-Fermi liquid and
quantum critical behavior for the metal-insulator transition in 2D
[27-31] as the temperature is decreased to rather low values.
|
{
"timestamp": "2004-11-25T07:22:18",
"yymm": "0411",
"arxiv_id": "cond-mat/0411627",
"language": "en",
"url": "https://arxiv.org/abs/cond-mat/0411627"
}
|
\section{Introduction}
Recent experimental breakthrough works \cite{lib}, \cite{wan},
\cite{pog}, \cite{ash}, \cite{han}, \cite{koc} on DNA strand
separation in transcription, denaturation and other processes, have
made it possible to develop a detailed understanding of such
fundamental steps of life (e.g. protein production from the DNA
template, see below). There has been a parallel extensive work on
theoretical models to describe the corresponding ``bubble'' dynamics.
One line of research concentrated on the nonlinear nature of the
denaturation opening, see e.g. \cite{eng}, \cite{fed}, \cite{yak1},
\cite{mut}, \cite{pey}, \cite{dau}, \cite{bog}, \cite{bar},
\cite{bha}, \cite{cam}. In these models, the denaturation bubble is
described either as a breather or a kink of nonlinear Klein-Gordon
(NLKG) or of sine-Gordon type equations. Statistical models to include
temperature and noise were also developed \cite{kaf}, \cite{the}. A
number of detailed numerical studies of local DNA opening have been
carried out, which are detailed enough to be compared with
experimental data on long chains of DNA strands. These include, among
others, studies of Lavery and colleagues, by methods of molecular
mechanics \cite{ram}, \cite{ber}, and molecular/Brownian dynamics
\cite{bri}, \cite{gui}; molecular dynamics simulations of DNA by the
group of Beveridge \cite{bev}, \cite{mcc}, \cite{mcc1} and Langowski
\cite{bus}, \cite{lan}. In a series of works, Schlick and her group
developed comprehensive models of DNA dynamics. This approach uses a
bead model of DNA chains, where the energy of the chain depends on
twist, stretch, bend and hydrodynamic-mediated inter-base
interactions. This system is then assumed to obey the Langevin
equation, and both inertial and overdamped cases are studied in
detail, \cite{bea1}, \cite{yan}, \cite{rama}, \cite{sch}.
The simulation of dynamics of DNA on relevant time scales of
transcription and folding is extremely computationally extensive. One
of the important features is that the relaxation times are at the
picosecond level, while we need to follow the dynamics in the
millisecond and second level, \cite{sch1}. Therefore it is desirable to
find a nonlinear equation that governs the bubble dynamics, while
taking into account all the physics, including the double stand
nature of DNA and sequence dependence, curvature, solvent effects etc.
Our model attempts to understand the bubble in the double stranded DNA
using a classical mechanical model, which is in many ways similar to
models discussed above. We start by looking at a chain of connected
masses, and take the by-layer structure into account. Dissipation of
the molecular dynamics in fluid is usually derived from the
interactions between the DNA and the polarization and kinematic
properties of the water molecules surrounding it. The result is a
friction force proportional to the velocity and stochastic term to
describe the effect of thermal motion. However, this assumes that the
DNA is a point particle, with no shape and internal degrees of
freedom. The effect of internal degrees of freedom may also be
relevant. It can be shown that coupling of a system to a {\it
conservative} big system leads to a form of dissipation for the small
system. In fact, a simple motivating example was solved exactly by
Lamb in 1900, where he showed that coupling an oscillator to an
infinite string leads to dissipation. For a review of recent works in
this direction see \cite{sof}. We then ask what will be the effect of
the internal vibrational and other modes of each base in the DNA. By
modeling each such base as a string we derive the leading order
effect, and show it is dissipative, yet can not be incorporated in the
usual friction terms as it is curvature dependent. Hence we refer to
this contribution as {\it curvature dissipation}. Just like the usual
friction it depends only on the first derivative in time. On the
other hand it also contains the derivatives of the amplitude with respect to $x$,
the position along the chain.
It is interesting to note that curvature effects are sometimes
relevant to friction, see e.g. \cite{lig}, \cite{wig},
\cite{gol}. However, the kind of dissipation considered in these and
other works comes from the interaction of a curved object with the
surrounding fluid. The curvature dissipation introduced in this paper
comes from the {\it internal} motions of the molecule and would enter the
equation of motion even if the motion took place in vacuum.
Our main result is an equation of motion for a double-stranded DNA
which allows for stable, pinned, localized solutions. These solutions
(a kink and an anti-kink) can be used to model denaturation bubbles in
many biological systems. At this first stage of developing a new
theory, we addressed the following questions: What are the static
parameters of denaturation bubbles (their steepness and their
longitudinal size)? How much energy does a bubble require to be
moved along the DNA chain (or, alternatively, how difficult is it to
keep it in place?) What defines the direction of motion of a bubble?
How can a bubble collapse?
This paper is organized as follows. We start by presenting several
biological systems where denaturation bubbles play an important role
(Section~\ref{bub}). In Section~\ref{der} we outline the derivation of
the new equation of motion for a double-stranded DNA. In Section
\ref{solutions} we find relevant solutions of this equation and study
their stability; we prove that our model supports {\it stable, pinned}
localized solutions. We also present numerical stability results for a
more complicated, spatially inhomogeneous system. In Section~\ref{bio}
we discuss properties of our model in the context of several
biological scenarios. We present model predictions on the size and
shape of the bubble, energetic requirements for bubble motion,
directionality of bubble motion, and bubble collapse. We identify the
parameters that have to be measured to validate the model. We also
define the relative importance of curvature dissipation. Section
\ref{concl} is reserved for conclusions.
\section{\label{bub} Denaturation bubbles in biological systems}
Here we list several examples of biological phenomena where
denaturation bubbles are essential. It is remarkable that bubbles are
found at the very basis of life: reproduction (both mitosis and
meiosis) and protein synthesis.
\paragraph{RNA transcription.} A denaturation bubble plays
the central role in RNA transcription, the first step in protein
synthesis \cite{cell}, see figure \ref{fig:tran}. The process begins
when an RNA polymerase (RNAP) enzyme molecule binds to the promoter
sequence of the DNA. It starts the transcription by opening up a local
region of about 20 base-pairs on the double helix to expose the
nucleotides. One of the two strands serves as a template for
complimentary base-pairing with incoming monomers, which begin an RNA
chain. The RNAP molecule then moves stepwise along the DNA, unwinding
the DNA helix just ahead to expose a new region for base-pairing, and
rewinding the region just behind. In this process, a short region
(about 8-9 base-pairs) of DNA-RNA helix is formed briefly, after which
the newly-built region of the single-stranded DNA molecule is released
to allow the rewinding of the DNA-DNA helix. The rate of transcription
at $37$ C is about 30 nucleotides per second. A typical size of a
completed RNA chain is between $70$ and $10,000$ nucleotides.
\paragraph{Nucleotide excision repair.} A denaturation bubble plays central
role in a repair process called {\it nucleotide excision
repair}. There, a damaged site of the DNA is recognized, and then a
bubble is formed around it, which is about $25$ \cite{eva} or $20$
\cite{mu} base pairs long for humans, and is shorter ($\sim 6$ base
pairs) in E. coli \cite{zou}. The bubble is created by a helicase
which plays a similar part at the initiation of RNA transcription. The
repair then proceeds by single strand incision at both sides of the
lesion, a removal of the damaged part from the bubble area, DNA repair
synthesis to replace the gap and ligation of the remaining single
stranded nick.
\paragraph{Other biological systems.} A DNA bubble occurs in a variety of situations
besides transcription and nucleotide excision repair. An expanding
bubble is formed at DNA replication. Also, we will mention the
process of meiotic recombination, where a type of a helicase (RecBCD)
propels itself along the DNA \cite{bia} creating a bubble, until a
recognition site is encountered, where the traveling loop of DNA is
cut, which initiates the genetic recombination event.
\section{\label{der}The new equation:\protect\\
motivation and derivation outline}
The motion of the double strand is usually modeled by means of some
nonlinear equation of the form,
$$\dot z=bz_{xx}-\partial V/\partial z+\mbox{random forcing}+\mbox{higher order friction terms},$$
where $z$, the transversal displacement of the nucleotides, is a
function of space, $x$, and time, $t$, and the dot stand for its
time-derivative. The coefficient $b$ is the ``spring constant'' of the
longitudinal interactions modeled as (non)-linear springs, and $V$ is
the potential. Different authors proposed various shapes of the
nonlinearity corresponding to the hydrogen-bond potential, $V$,
introduced nonlinearity in the ``elasticity properties'' of the
sugar-phosphate backbone, and included extra degrees of freedom coming
from the secondary structure of the DNA, as well as chiral forces and
torques. The energy terms could be very sophisticated, and often
include twist, stretch and bend of the molecule.
One common feature of nonlinear models of DNA dynamics can be
identified as follows: they rely on a somewhat {\it ad hoc} assumption that the
coupling between neighboring nodes of the lattice occurs by means of
non-material springs. The main point of this paper is to argue that
the interaction between oscillators has a different form and is better
described as coupling by ``strings'', rather than ``springs''. Roughly
speaking, we can say that some of the energy of transversal
oscillations of the double strand gets absorbed in the motion of the
material connecting the neighboring nodes. The relevant forces are
proportional to the momentum, that is, to the time-derivative of the
displacement, $z$. To leading order, this leads to a mixed-derivative
term in the master equation,
$$a(\dot{z}_{m-1}-2\dot{z}_m+\dot{z}_{m+1}).$$
The continuous version of the equation, in its simplest form, will read,
\begin{equation}
\label{newsg}
\dot z= bz_{xx}+a\dot{z}_{xx}-\partial V/\partial z+\mbox{random
forcing}.
\end{equation}
In equation (\ref{newsg}) we assume that the
constants, $a,b>0$. Here we outline the main ideas behind the
derivation of equation (\ref{newsg}). This is an equation of motion for
the node $z_m$ in the direction perpendicular to the molecule, in the
overdamped limit, where the influence of the second time-derivative
term can be neglected (see the end of Section \ref{bio} for the inertial
limit). The expression that multiplies the constant $b$ comes from the
forces acting on each node from its neighbors due to stretching (the
vibrations are not taken into account). The term $-\partial V/\partial
z$ comes from the potential forces of interaction of the two units
across the double strand. The term $\dot z$ represents the usual
friction, since the motion takes place in a viscous medium. The third
derivative term multiplying the constant $a$, the curvature
dissipation term, reflects the loss of energy due to vibrational modes
of the longitudinal connections among the nodes. In what follows, we
will outline the derivation of this new term.
In modeling double stranded molecules, one should consider the fact
that each longitudinal link is in fact a many-particle molecule, and
therefore has a large number of degrees of freedom. Such a molecule
should then be described by a dispersive system with many degrees of
freedom. In the simplest classical approximation we treat it as a
string (rather than a massless spring) of some fixed length. Note that
a more general dispersion relation than the usual string will not
change qualitatively our analysis.
The critical difference between a ``spring'' and a ``string'' is that
a string will effectively act as a reservoir which absorbs some of the
oscillatory energy. Therefore, we expect the motion of the ends of the
links to obey an equation that contains a dissipative correction. Such
a correction can, to the leading order, be approximated by the terms
with $\dot{z}$. To see the origin of the $\dot{z}$ terms, let us
consider the node $z_m$ and solve the wave equation to the left and to
the right from it, see Fig. \ref{fig:vars}. Let $u(x,t)$ denote the
position of the string at point $x$ at time $t$. We have, to the left
of the node $z_m$,
$$u_{tt}-g^2u_{xx}=0,\quad u(0,t)=z_{m-1}(t),\quad u(L,t)=z_m(t),$$
where $L$ is the length of the connection and $g$ is the speed of
sound in the string. Similarly, to the right from the node $z_m$ we have,
$$\tilde u_{tt}-\tilde g^2\tilde u_{xx}=0,\quad \tilde u(0,t)=z_{m}(t),\quad \tilde u(\tilde L,t)=z_{m+1}(t),$$
where the constants do not have to be the same.
The force exerted on the node from the right in the direction
perpendicular to the string is proportional to
$\tilde{u}_x\vert_{x=0}$, and the force from the left is proportional
to $u_x\vert_{x=L}$. By examining the solution of the wave
equation, we can show that the spatial and temporal derivatives are
linearly dependent; therefore, the force can be defined in terms of
$\tilde{u}_t\vert_{x=0}$ and $u_t\vert_{x=L}$. In turn, the solution
$\tilde{u}(x,t)$ is a linear functional of the boundary conditions,
$z_{m}(t)$ and $z_{m+1}(t)$, thus $\tilde{u}_t\vert_{x=0}$ is a linear
functional of $\dot{z}_{m}(t)$ and $\dot z_{m+1}(t)$, which we denote
$G(\dot{z}_{m}(t),\dot z_{m+1}(t))$. Similarly, $u_t\vert_{x=L}$ is a
linear functional of $\dot z_{m-1}(t)$ and $\dot z_{m}(t)$. Therefore,
the force from the moving springs can be expressed as
$$\tilde G(\dot{z}_{m}(t),\dot z_{m+1}(t))-G(\dot{z}_{m_1}(t),\dot z_{m}(t)).$$
In equation (\ref{newsg}) we used a very simple model for $G$ and
$\tilde G$, where they were just linear functions of their
variables. This gave rise to the term
$a(\dot{z}_{m-1}+\dot{z}_{m+1})-2a'\dot{z}_m$. Setting $a'=a$ and
taking the continuous limit, leads to equation (\ref{newsg}), which
corresponds simply to $G(y_1,y_2)=\tilde G(y_1,y_2)=a(y_2-y_1)$. A
complete derivation of the functionals $G$, $\tilde G$ will be
presented elsewhere.
\section{\label{solutions}Localized solutions and their stability}
In the literature, the DNA denaturation bubble is often modeled in
terms of kinks or breathers. However, both types of localized
solutions have several problems \cite{cam}. Breathers generically lose
stability as the level of discretization of the lattice becomes lower
\cite{aub}. Kinks, on the other hand, are very difficult to pin, even
on a lattice. As the degree of discretization decreases, the
Peierls-Nabarro barrier that keeps a kink from moving decreases
exponentially \cite{wil}, \cite{joo}. Therefore, a very small amount
of energy can set a kink in motion.
In this section, we will describe solutions of equations of type
(\ref{newsg}) which have properties relevant to many biological
systems. Namely, we will study equation
\begin{equation}
\label{nof}
\dot z= bz_{xx}+a\dot{z}_{xx}-\partial V/\partial z,
\end{equation}
and prove that it supports {\it stable pinned localized solutions}.
The effects of random forcing (equation (\ref{newsg})), which is an
integral feature of dynamics on the relevant scales, has to be
analyzed separately. Stability of localized solutions in the system
without random forcing is a necessary condition for their stability
once the temperature effects have been added.
\paragraph{Kinks, solitons and the energy functional.} The exact shape of
a stationary localized solution, $\bar z(x)$, is found from the
equation
\begin{equation}
\label{solit}
b\bar{z}_{xx}-\partial V(z)/\partial z\vert_{z=\bar z}=0.
\end{equation}
Note that the nature of the solution $\bar z(x)$, will depend on the
form of the potential, $V,$ as a function of $z$. Let us suppose that
$V(z)$ is a smooth function. Integrating equation (\ref{solit}) in
$x$, we can see that the quantity $C=b\bar z_x^2/2-V(\bar z)$ is a
constant along the solution for $-\infty < x<\infty$. Using this
property, we can see that a {\it topological kink} (or antikink)
solution exists only if the potential, $V(z)$, as a function of $z$,
has at least two minima, say, at points $z_1$ and $z_2$, such that
$V(z_1)=V(z_2)$, see figure \ref{fig:pot}; here $z_1<z_2$ are some
real numbers. The kink will satisfy the conditions at infinity,
$$\lim _{x\to -\infty}\bar z(x)=z_1,\quad \lim _{x\to
+\infty}\bar z(x)=z_2.$$
For the antikink, we have
$$\lim _{x\to -\infty}\bar z(x)=z_2,\quad \lim _{x\to
+\infty}\bar z(x)=z_1.$$
A different type of localized solutions is a soliton. A soliton
solution is possible whenever the function $V(z)$ has a local minimum,
say, at a point $z=z_0$. There will be another point, $z'_0$, with
$V(z_0)=V(z'_0)$. The soliton solution will have a maximum (or
minimum) value of $z_0'$, and the following condition at infinity:
$$\lim _{x\to -\infty}\bar z(x)=\lim _{x\to
+\infty}\bar z(x)=z_0.$$
Let us define distance between functions, $z(x)$ and $v(x)$, as
$$d(z,v)=\int_{-\infty}^\infty [(z-v)^2+(z_x-v_x)^2]\,dx.$$
The energy functional of equation (\ref{nof}), is given by
\begin{equation}
\label{energy}
E\{z\}=\int_{-\infty}^\infty \left[bz_x^2/2+V(z)-V_\infty\right]\,dx.
\end{equation}
Here, $V_\infty$ is some constant; we subtract this constant in order
to make sure that a localized solution has finite energy. For solitons
we take $V_\infty=V(z_0)$. For kinks, we have $V_\infty=V(z_1)$, see
figure \ref{fig:pot}. With this choice of the constant, in each case,
the function $E\{z\}$ is defined for solutions $z(x,t)$, such that
$d(z,\bar z)<\infty$, where $\bar z(x)$ is the localized
solution. Using equation (\ref{nof}), it is easy to show that
\begin{equation}
\label{decay}
\frac{dE\{z\}}{dt}=-a\int_{-\infty}^\infty z_{xt}^2\,dx-\int_{-\infty}^\infty z_{t}^2\,dx\le0,
\end{equation}
which means that starting from any initial conditions (for which
$E\{z\}$ is defined), the solution will always decrease the energy
functional.
\paragraph{Stability of kinks.} Here we will show that kinks are stable. The analysis
for antikinks is similar. Let us prove that the energy functional,
$E\{z\}$, has a local minimum at the point $z=\bar z$, where $\bar z$
is a kink satisfying stationary equation (\ref{solit}). Let us
calculate the gradient and the curvature of $ E\{z\}$ at $\bar z$. We
have,
$$\left(\frac{\delta E\{z\}}{\delta z},\psi\right)_{z=\bar z}=\int_{-\infty}^{\infty}\left(b\bar z_x\psi_x+\frac{\partial V(\bar z)}{\partial \bar z}\psi\right)\,dx=0$$
for all test functions, $\psi(x)$, in the appropriate space. Here and
below we use the short-hand notation, $\partial Q(\bar z)/\partial
\bar z\equiv \partial Q(z)/\partial z\vert_{z=\bar z}$, where $Q(z)$
is a function of $z$. Next, we evaluate
$$\left(\psi, \frac{\delta^2 E\{z\}}{\delta z^2}\psi\right)_{z=\bar z}=\int_{-\infty}^{\infty}\left(b\psi_x^2+\frac{\partial^2 v(\bar z)}{\partial \bar z^2}\psi^2\right)\,dx=\int_{-\infty}^{\infty}\left(\psi,H\psi\right)\,dx,$$
where the self-adjoint operator $H$ is given by
$$H=-b\frac{\partial^2}{\partial x^2}+\frac{\partial^2V(\bar z)}{\partial \bar z^2}.$$
Using Weyl's theorem, it is easy to show that this operator has a
positive continuous spectrum. Indeed, we have
$$\lim_{x\to \infty}\frac{\partial^2 V(\bar z)}{\partial \bar z^2}=V''(z_1)>0,\quad \lim_{x\to -\infty}\frac{\partial^2 V(\bar z)}{\partial \bar z^2}=V''(z_2)>0, $$
that is, for large values of $x$, the potential $V''$ approaches the
value of the curvature at its minima, see figure \ref{fig:pot}. The
continuous spectrum must therefore be positive. The only negative
contribution could come from the discrete spectrum. In order to
exclude this possibility, let us consider the eigenfunction $\bar
z_x$, corresponding to the horizontal translation of the kink. This
eigenvector corresponds to the eigenvalue zero. Indeed,
differentiating equation (\ref{solit}) in $x$, we obtain $H\bar
z_x=0$. On the other hand, this eigenfunction is positive for
monotonically increasing kinks. Using Sturm's oscillation theorem we
conclude that this eigenfunction is the ground state of the operator,
which means that all other localized solutions, if they exist, have
nonnegative eigenvalues. Therefore, $(\psi,H\psi)\ge 0$.
We have proved that the function $\bar z$ is a local minimum of the
energy functional $E\{z\}$. Therefore, starting from any solution in a
vicinity of the kink, the system will return to the kink. This
concludes the stability analysis.
\paragraph{Instability of solitons.} The above argument breaks down in the case
of solitons. It will remain the same up to the point where we look at
the second derivative of $ E\{z\}$ at the point $z=\bar z(x)$,
the soliton solution. The operator $H$ satisfies,
$$\lim_{|x|\to \infty}\frac{\partial^2 V(\bar z)}{\partial \bar z^2}=V''(z_0)>0,$$
so the continuous spectrum is positive. The minimum of $H(x)$ is
negative, because between the points $z_0$ and $z'_0$ there must be a
point, $z_*$ such that $V''(z_*)<0$, see figure \ref{fig:pot}. The
discrete spectrum must have a negative eigenvalue, because the
translational mode with a zero eigenvalue, $\bar z_x$, is not a
positive function in the case of a soliton. Therefore, by Sturm's
oscillation theorem, there will be another eigenfunction (the ground
state) with an eigenvalue between $V''(z_*)<0$ and zero. This suggests
that the soliton solution is a saddle point for the energy
functional. An infinitesimal perturbation in the ``right'' direction
will destabilize the solution and bring the system to a different
stationary state, with a lower energy, e.g. the solution
$z(x)=z_0=const$.
\paragraph{Nonhomogeneous chains: numerical stability results.} In the model discussed so far, we treated all base-pairs as if they were identical. A more accurate
model for a double-stranded DNA will distinguish between two types of
hydrogen bonds, A--T and G--C. The two bonds are characterized by
different potentials, namely, $D_{A-T}=0.05\, eV$ and $D_{G-C}=0.075\,
eV$. In order to model a non-homogeneous DNA sequence, we can use a version of a
discretized equation,
\begin{equation}
\label{discrete}
\dot z_m=b(z_{m-1}-2z_m+z_{m+1})+a(\dot z_{m-1}-2\dot z_m+\dot z_{m+1})-\frac{\partial V_m(z)}{\partial z}\bigg\vert_{z=z_m},
\end{equation}
where $z_m(t)$ is the vertical displacement of each nucleotide, and
instead of one potential $V(z)$, we have functions $V_m(z)$ which
represent interaction between two nucleotides in each base pair. The
interaction potentials are allowed to differ from site to site.
We have performed numerical experiments where the functions $V_m\in
\{V_{A-T},V_{G-C}\}$ were taken from some distribution. It appears
that the stability properties of the kink are not affected by this
type of randomness as long as the values $V_{A-T}$ and $D_{G-C}$ are
not too far apart.
\paragraph{Inertial systems.} Finally, we consider systems where the motion
is not overdamped, such as DNA molecules in non-soluble media. In this
situation we need to include the kinematic terms corresponding to
elastic modes, $\ddot z$. Such cases arise in many applications
\cite{por}, including nanowires made of DNA strands, DNA on dry
surfaces, DNA held by electric fields and other nanodevices. In these
cases, the propagation of bubbles is governed by the following
equation:
\begin{equation}
\label{kap}
\kappa \ddot{z}=a\dot{z}_{xx}+bz_{xx}-\dot z-{\partial \tilde V}/{\partial z}+f(x,t),
\end{equation}
where $f(t)$ is the external force corresponding to the tweezers,
electric/magnetic fields etc. Our stability
analysis holds almost exactly as before. In the absence of $f(x,t)$, we need to introduce the energy functional,
$$\tilde E\{z\}=\int_{-\infty}^\infty \left[\frac{1}{2}\left(\kappa z_t^2+bz_x^2\right)+V(z)-V_\infty\right]\,dx.$$
Let $\bar z$ be the stationary localized solution, as before. It is
clear that if $\bar z$ is a local minimum of $E\{z\}$, then it is also
a local minimum of $\tilde E\{z\}$, since $z_t^2\ge 0$ and $\bar
z_t=0$. Therefore, the stability results for the kink hold in this
case.
The potential term in the absence of RNA polymerase is different, and
a bubble life-time analysis can be performed using equation
(\ref{kap}). In this case, the lifetime may be nonsmall, due to the
absence of stochastic noise.
\section{\label{bio}Biological applications}
The equation of motion derived here, with a curvature dissipation
term, can serve as a starting point to design detailed models of many
biological systems where a denaturation bubble plays a role, see
Section \ref{bub}. However, this is not the goal of the present
paper. In fact, at this stage we are still quite far from grasping all
the features of such complex biological phenomena as RNA
transcription, or nucleotide excision repair. For example, in order to
describe RNA transcription, a model must contain information on the
RNA polymerase molecule. In the present work we are mostly concerned
with properties of the double-stranded DNA molecule. A natural
question is, what is the value of this modeling for studies of real
biological systems?
We will answer this question by using the following analogy. Let us
suppose that we need to model a cruise ship. In order to accomplish
this task, we need to include all the details of the ship's
design. However, no such model would be any good unless we understand
the basic properties of water! So a reasonable start for modeling a cruise
ship is good old fluid mechanics. Would water be able to hold a
massive object without sinking it? Can a (generic heavy) object move
along in water? How much energy does such motion take? And so on. In
the case of modeling RNA transcription, we first need to understand
how double-stranded DNA moves, and how a denaturation bubble forms,
before we can begin talking about details of the transcription process
itself. The model developed here addresses the following questions:
What is the generic shape of a denaturation bubble? What is its size?
Can a denaturation bubble be stable? How much energy does it take to
move it along the DNA molecule? How can a bubble collapse?
\paragraph{Shape and size of the bubble.} A denaturation bubble can be modeled as a
solution of (\ref{newsg}) which consists of a kink and an antikink,
see Fig. \ref{fig:kink}. If the kink and the antikink are sufficiently far
apart, we can say that they do not interact and can coexist for a long
time. The width of a kink is roughly given by
\begin{equation}
\label{ww}
w=\sqrt{b/\Delta V},
\end{equation}
where $\Delta V$ is the potential barrier of the interaction energy of
nucleotides across the double strand, given by the difference between
$V(z)$ at its maximum, and at its minimum.
The longitudinal size of the bubble, $n$, is given by the distance between the
kink and the antikink. We must require that
\begin{equation}
\label{size}
n\gg \sqrt{b/\Delta V}
\end{equation}
in order for the bubble to be stable, see Fig. \ref{fig:kink}. Note
that the size of the bubble in this model is not defined by intrinsic
properties of the DNA molecule (except for the constraint that a
bubble cannot be too small, to satisfy condition (\ref{size})
above). This means that the bubble size can be different under
different circumstances. For instance, in RNA transcription process it
is defined by the RNA polymerase molecule. The size of the bubble
created in the process of nucleotide excision repair is defined by the
appropriate helicase. Finally, the denaturation region formed during
DNA replication or meiotic recombination does not have a fixed size,
as it is created by a moving helicase which opens up the DNA double
helix on one side of the bubble. This suggests that modeling
denaturation bubbles as a pair of two independent localized solutions
(the kink and the antikink of Fig. \ref{fig:kink}) is consistent with
biological reality, more so than using one localized solution like a
breather or a soliton.
In order to relate the model's prediction, equation (\ref{ww}), to
biological systems, we need to know numerical values for $b$ and
$\Delta V$.
\paragraph{Measurements of ``static'' parameters of the bubble.} The quantities
relevant for the shape of the bubble (formula (\ref{ww})) are given by
$$b=\frac{Kh^2}{D},\quad \Delta V=\alpha^2h^2,$$
where $D$ is the depth of the hydrogen bond potential, $h$ is the
longitudinal distance between nucleotide pairs, $\alpha$ is the width
of the potential well and $K$ is the ``spring constant'' of the DNA
sugar-phosphate backbone. The first three parameters can be measured
relatively accurately, whereas $K$ presents a problem.
The depth of the hydrogen bond potential, $D$, has
been estimated to be $D_{A-T}=0.05\, eV$ and $D_{G-C}=0.075\, eV$ for
the two types of pairing. The parameter $\alpha$ that defines the
width of the potential well is taken to be $\alpha=2.55\,A$ in
\cite{pey}, $\alpha=4.45\,A^{-1}$ in \cite{bar}, $\alpha=4\,A^{-1}$ in
\cite{cam}. The distance between pairs is $h=3.4\,A$ (\cite{cam} and
\cite{bar}).
A more difficult quantity to measure is the ``spring constant'' $K$,
of the DNA sugar-phosphate backbone.\footnote{Large discrepancies in
the values of the spring constant are not surprising. Our models
suggests that the ``spring'' properties of the DNA, that is, the
coefficient $b$ in equation (\ref{newsg}), is not the entire
story. Energy losses due to vibrational modes of the nucleotides have
to be taken into account, which can in principle be done by measuring
the spectrum of vibrational modes. } In \cite{pey} it was merely
estimated from the model to give a realistic denaturation temperature;
the corresponding value is $3.0\times 10^{-3}\,eV/A^2$. However, the
paper by \cite{kam} suggests that this value is much larger, the
measured parameter is $K=0.22 \,eV/A^2$. An even larger value,
$K=1.0eV/A^2$, is quoted in \cite{bar}. The paper by \cite{ger} uses
the value $K=0.026\, eV/A^2$ (however, this value has been estimated
for RNA and includes effects of the secondary structure). Note that
other experimental measurements give very different values, see
\cite{smi}, \cite{ben}, where the spring constant is found to be very
small, of the order of $10^{-6}\, eV/A^2$. However, it must be noted
that in those experiments the spring constant of the DNA molecule as a
whole was measured as opposed to local elasticity properties of the
sugar-phosphate backbone, and it is the latter quantity which is of
interest to us.
With the information that we have so far, we can obtain the value of
$w$ between $0.18$ (for $K=0.2\, eV$, $\alpha=4.45\, A$ and $D=0.33\,
eV$) and $1.96$ (for $K=1\,eV$, $\alpha=2.55$ and $D=0.4\,eV$). This
means that the number of nodes in the ``knee'' of the kink is of the
order one. This estimate is consistent with the picture of RNA
transcription (Section \ref{bub}) where an RNA polymerase enzyme
molecule opens up only a few base-pairs to complete the transcription
of a small portion of the DNA template, with 3 or fewer nucleotides
forming the ``sides'' of the bubble.
\paragraph{Energy needed to move the bubble.} In potential systems,
such as nonlinear Klein-Gordon equation, a whole family of moving
kinks, $\bar{z}(x-vt)$, exists for any velocity $v$. Therefore, moving
a kink along a lattice does not take any energy. In the new equation,
this is not the case. In order to move the bubble along the DNA
molecule, an external force must be applied.
This has relevance for many biological systems involving denaturation
bubbles. In the context of RNA polymerase, we can ask: how strong a
push does a transcription bubble need to travel along the DNA? The
RNAP molecule is thought to be a molecular motor, which uses the
energy of ribonucleoside triphosphades to propel itself in the 3'-5'
direction along the coding strand of the DNA molecule \cite{gel}. Our
model implies that the RNAP ``drags'' the transcription bubble
(consisting of a kink and and antikink) along, using the appropriate
fraction of its total energy. The same holds for expanding
denaturation regions during the process of DNA replication and meiotic
recombination. How much energy does a helicase need to propel a
traveling loop of DNA?
In the context of nucleotide excision repair, one can ask the
opposite question: how stable is the bubble? How easy is it to keep it
in place for as long as it takes to perform the repair?
Theoretically we can address these questions in the framework of our
model. Using equation (\ref{decay}), we can calculate how much energy
it takes to move a kink with velocity $v$ for time $\Delta t$:
\begin{equation}
\label{ener}
\Delta E=v^2\Delta t\int_{-\infty}^\infty \left(\bar z_x^2+a\bar z_{xx}^2\right)\,dx.
\end{equation}
We can see that energy losses come from two sources: the first term
under the integral is the usual dissipation. The second term is the
curvature dissipation, that is, the loss due to internal vibrational
modes of the DNA molecule. This is the novel contribution of the
present model.
In order to obtain a quantitative prediction, several detailed
measurements must be performed. First of all, the shape of the bubble
has to be identified, to find the slope, $z_x$ and the curvature,
$z_{xx}$ along the bubble. Then, the velocity of motion, $v$, has to
be estimated during a time-interval, $\Delta t$. Finally, the
contribution of dissipation and curvature dissipation must be
identified. This is the most difficult task. Measuring the spectrum
of vibrational modes will eventually lead to the information necessary
to estimate the coefficient $a$ in equations (\ref{newsg}) and
(\ref{ener}). ?? AVY ADD SOMETHING ??
\paragraph{Direction of the bubble motion.} According to our
model, the bubble motion direction is defined externally, by the
``motor'' which propels the kink along the DNA chain. In biological
systems, bubble motion happens in a fixed direction. For example, in
RNA transcription, the process of elongation always proceeds in the
5'--3' direction (i.e. the RNA polymerase moves along the template
strand of DNA in the 3'--5' direction). Therefore, the ``polarity'' of
the coding strand defines the arrow of motion. Our suggestion is that
it is the molecular motor (RNAP) that recognizes the directionality of
the DNA strand, and the bubble itself can be moved in either
direction.
\paragraph{Relative importance of curvature dissipation.} Recent measurements of
both intermolecular and intramolecular vibrational modes of
nucleotides show their significance for DNA dynamics. In the works of
\cite{lee}, \cite{lee2}, the Raman spectrum of nucleotides is measured
in the range from $200$ to $4000$ cm$^{-1}$. These modes correspond to
the internal vibrations within the molecule, and they are in the same
energy range as the hydrogen bonds between the strands. Moreover, when
the measurements are done at low temperature ($10$--$20$K) one
observes that broad absorption lines are in fact many resonances,
fused together due to thermal fluctuations. Other modes correspond to
vibrations of two coupled nucleotides; they have a lower energy and
therefore are easier to excite. These modes have also been measured
and are typically in the range from $30$ to $150$cm$^{-1}$. Some of
them are measured by \cite{fis} in experiments on crystals, and by
\cite{bol} on the molecular level; see also \cite{wilma}. It is clear
from this abundance of the modes at the relevant energy scales, that a
realistic temporal description, as required for example for bubble
motion in transcription, must adequately incorporate the corresponding
contributions.
We can use simple energetic considerations to estimate the effect of
curvature dissipation on the DNA dynamics. As the bubble propagates
through the DNA, it excites many internal modes. Therefore, we need to
compare the energies of the motion of the bubble with the vibrational
modes. The number of relevant modes of one nucleotide is multiplied
by the number of points where the curvature is not zero, about 6 (that
is, $12$ nucleotides). Using the fact that each vibrational mode of
nucleotides, as well as nucleotide-nucleotide couplings, is of the
order of $10^{-3}$ $eV$ \cite{wilma}, we can see that each vibrational
mode of the 6 involved base-pairs contributes an amount which is about
10-20\% of the difference of the base-coupling energies ($0.25\,eV$).
It is now possible to estimate the value of the coefficient $a$ in
equation (\ref{newsg}). Let us suppose that the bubble moves with a constant
speed, $v$. Then, we assume that it excites $12$ nucleotides, with the total
energy $e$. Then the energy change per unit time is given by $ev/d$, where
$d$ is the size of the kink (say, $d=3$ base pairs). Our formula for the rate of energy change related to internal modes is given by $av^2\int\bar z_{xx}^2\,dx$.
Therefore
$$a\sim \frac{e}{vd\int \bar z_{xx}^2\,dx},$$
where the integral is completely determined by the shape of the kink
and has support ($\sim d$) of a few base points. Now, notice that we
can eliminate the dependence on $v$ using the fact that E, the total
energy is proportional to $ v^2$. Therefore $\Delta E /E =ev/dE \sim a
$ where $ \Delta E$ stands for the energy loss per unit time.
?? AVY PLEASE CHECK ??
\paragraph{Bubble collapse.} Our approach can find applications in modeling transcription
termination. There are two ways in which transcription is
terminated. $\rho$-independent termination involves a specific
sequence prone to forming a hairpin. $\rho$-dependent termination
requires a subunit of RNAP which utilizes the energy of ATP to stop
the transcription. In order to model this, we can use equation
(\ref{discrete}), a discrete version of the equation of motion where
the two different types of hydrogen bonds, A-T and G-C, are taken into
account.
The process of $\rho$-independent termination can be modeled by introducing a
large perturbation in the sequence of $V_m$ (hydrogen
bonds). Simulations show that a particularly {\it small} value of
$V_m$ at one site can lead to a collapse of the kink and the antikink
on each other. The $\rho$-dependent termination can be modeled by
adding a large perturbation somewhere between the kink and
antikink. Say, if the value of $z$ outside the bubble is $z_1$, and it
is $z_2$ inside, setting several (strategically chosen) nodes inside
the kink back to the value $z_1$ may cause a collapse of the
kink-antikink pair.
The behavior of our model is in qualitative agreement with reality. At
this stage, we can only suggest that the equations of motion that we
derived for the dynamics of a double-stranded DNA molecule allow for a
bubble collapse if appropriate forcing is applied. A more detailed
model, based on particular sequences, must be devised to give
quantitative predictions.
\section{\label{concl} Conclusions}
We have introduced a nonlinear equation of motion describing the
dynamics of double-stranded DNA. Along with the usual dissipation
term, it contains a curvature dissipation term, corresponding to the
loss of energy to the many vibrational modes of the DNA molecule. This
equation allows for a localized, pinned solution which can
be relevant for modeling DNA denaturation bubble because of the
following useful properties:
\begin{itemize}
\item[(i)] It is not a breather, that is, its existence does not depend on the fast transversal vibrations;
\item[(ii)] It is pinned, that is, it will not travel along the DNA when perturbed in the longitudinal direction; in fact, it requires finite energy to move;
\item[(iii)] It is stable, and its stability can be proved rigorously.
\end{itemize}
There are many biological processes involving DNA denaturation
bubbles, such as RNA transcription, nucleotide excision repair, DNA
replication and meiotic recombination. When modeling these and other
processes, the basic equation of motion for the double-stranded
molecule must allow for stable solutions corresponding to local
opening of the DNA. In this first paper we have suggested a framework
for such modeling.
The dynamical formulation we use, allows the incorporation of other
important structural factors. The first thing we need to include is
the effect of chain content. This is easily done by making the strength
of the coupling between the strands change value according to whether
it is GT or TC. We can also include the effect of content by changing
the "string" constant as we move from type base to base along the
chain; for this we can use the information on relevant excited modes
of each such molecule. We note also that the effect of stacking, which
was considered by many authors before (see e.g. \cite{bar2},
\cite{bar3}) can be included in a similar way by adding "spin" degrees
of freedom. There is no reason to believe that existence and stability
results would change in the modified system.
Most importantly, the effect of curvature of the DNA molecule, can
also be implemented by making the coefficients of the discrete
Laplacian position-dependent. It is not easy to see how this can be
done in a nonhamiltonian, energy landscape type models. The
implications of these modifications may be very important, and will be
studied in a forthcoming work. Here we only mention that the {\it
curvature effects} play a central role in the DNA dynamics, and can
have important consequences for the regulation and the dynamics of the
transcription process. For example, when the DNA region is tightly
bound, curled around a chromatin, transcription initiation is
impossible. But when the curvature is lowered, by the action of
appropriate enzymes, the process can begin. Once the process has
started, the curvature will affect the velocity of propagation of the
bubble. Most importantly, we conjecture that, in some places, it will
also change the effective energy landscape, to the point of creating,
or moving of {\it arrest points}. Such points are critical to
understanding transcription regulation.
Finally, we would like to describe details of biochemical reaction
which include more players. In upcoming papers we will concentrate on
the process of RNA transcription and show how the equation of motion
for the double-stranded DNA can be coupled with an explicit equation
for the RNA polymerase molecule. Unfortunately, complicated models
like this do not often allow for a clear and rigorous mathematical
analysis. The advantage of the present model is its transparent
behavior. It will serve an a building block for more complicated
systems.
\newpage
|
{
"timestamp": "2004-12-21T23:44:54",
"yymm": "0411",
"arxiv_id": "cond-mat/0411621",
"language": "en",
"url": "https://arxiv.org/abs/cond-mat/0411621"
}
|
\section{Introduction}
We shall be concerned with discrete signals $S=(S(0),\ldots, S(N-1)) \in \mathbb{C}^N$ and their Fourier
transforms $\hat{S}=(\hat{S}(0),\ldots,\hat{S}(N-1))$, defined by
$\hat{S}(\omega)=\frac{1}{\sqrt{N}}\sum_{t=0} ^{N-1} S(t)e^{-2 \pi i \omega t/N}$. In terms of the Fourier basis
functions $\phi_{\omega}(t) = \frac{1}{\sqrt{N}} e^{2 \pi i \omega t/N}$, $S$ can be written as $S
= \sum_{\omega=0}^{N-1} \hat{S}(\omega)\phi_{\omega}(t)$; this is the (discrete) Fourier representation of $S$.
In many situations, a few large Fourier coefficients already capture the major time-invariant wave-like information
of the signal and very small Fourier coefficients can thus be discarded. The problem of finding the (hopefully few)
largest Fourier coefficients of a signal that describe most of the signal trends, is a fundamental task in
Fourier Analysis. Techniques to solve this problem are very useful in data compression, feature extraction, finding
approximating periods and other data mining tasks \cite{GGIMS}, as well as in situations where multiple scales
exist in the domain (as in e.g. materials science), and the solutions have sparse modes in the frequency domain.
Let $S$ be a signal that is known to have a sparse $B$-term Fourier
representation with $B\ll N$, i.e.,
\begin{equation}\label{Brepn}
S(t)=\frac{1}{\sqrt{N}}(a_{1}e^{i2\pi\omega_{1}t/N}+\ldots+a_{B}e^{i2\pi\omega_{B}t/N}),
\end{equation}
and let us assume that it is possible to evaluate $S$, at arbitrary $t$, at cost $O(1)$ for every evaluation.
To identify the parameters $a_{1},\ldots,a_{B},\omega_{1},\ldots,\omega_{B}$, one can use the Fast Fourier
Transform (FFT). Starting from the $N$ point-evaluations $S(0), \ldots, S(N-1)$, the FFT computes all the
Fourier coefficients; one can then take the largest $B$ coefficients and the corresponding modes. The time cost
for this procedure is $\Omega (N \log N)$; this can become very expensive if $N$ is huge. (Note that
all logarithms in this paper are with base 2, unless stated otherwise.) The problem becomes worse in higher dimensions. If one uses grids of size $N$ in
each of $d$ dimensions, the total number of points is $N^{d}$ and the FFT procedure takes $\Omega(d N^d \log N)$ time.
It follows that
identifying a sparse number of modes and amplitudes is expensive for even fairly modest $N$.
Our goal in this paper is to discuss much faster algorithms that can identify the coefficients $a_{1},\ldots,a_{B}$ and
the modes $\omega_{1},\ldots,\omega_{B}$ in equation (\ref{Brepn}). These algorithms will not use all the samples
$S(0),\ldots, S(N-1)$, but only a very sparse subset of them.
In fact, we need not restrict ourselves to signals that are exactly equal to a $B$-term representation. Let us
denote the optimal $B$-term Fourier representation of a signal $S$ by $R_{opt}^B(S)$; it is simply a truncated version of the Fourier representation of $S$, retaining only the $B$ largest coefficients.
We are then interested in identifying (or finding a close approximation to) $R_{opt}^B(S)$ via a fast algorithm.
The papers \cite{GGIMS} \cite{Mansour} \cite{GMS} provide such algorithms; all compute a (near-)optimal $B$- term Fourier
representation $R$ in time and space $poly(B,\log(1/\delta),\ \log N,\log M,1/\epsilon)$, such that $\Vert
S-R\Vert_2^{2}\leq(1+\epsilon)\Vert S-R_{opt}^B(S)\Vert_2^{2}$, with success probability at least $1-\delta$,
where $M$ is an a priori given upper bound on $\|S\|_2$. The
algorithms in these papers share the property that they need only some random subsets of the input rather than all the data;
they differ in many details: the different papers assume different conditions on $N$, (for example, $N$ is assumed to be
a power of 2 or a small prime number in \cite{Mansour}; N may be arbitrary
but is preferably a prime in \cite{GGIMS}); the algorithms also use different
schemes to locate the significant modes. (Here we say a mode $\omega$ is significant if for
some pre-set $\eta$, $|\hat{S}(\omega)|^2 \geq \eta \|S\|^2$.) Mansour and Sahar \cite{Sahar} implemented a similar algorithm
for Fourier analysis on the set $\mathbb{Z}_2^n$, where our algorithm is for Fourier analysis on $\mathbb{Z}_N$.
The results of \cite{GGIMS} can be extended to more general representations, with respect to a particular basis or a family of bases; examples are wavelet
bases, wavelet packets or Fourier bases. We shall use the acronym RA$\ell$STA (Randomized Algorithm for Sparse
Transform Analysis) for this family of algorithms. We here restrict ourselves to the Fourier case and thus RA$\ell$SFA.
For a wide range of applications, the speed potential suggested by the sublinear cost of these
algorithms is of great importance. In this paper, we concentrate on the approach proposed in \cite{GGIMS}.
Note that \cite{GGIMS} gives a theoretical rather than a practical analysis in the sense that it does not discuss parameter settings;
it gives few hints about the order of the polynomial in $B$ and $\log N$; in fact, a straightforward implementation of RA$\ell$SFA following the set-up of \cite{GGIMS} turns out to be too
slow to be practical, so that none of the direct implementation work was published. In addition, \cite{GGIMS} did not discuss extensions
to higher dimensions, where the pay-off of RA$\ell$SFA versus the FFT is expected to be larger.
Our main result in this paper is a version of RA$\ell$SFA that addresses these problems. We give
theoretical and heuristic arguments for the setting of parameters; we introduce some new ideas that produce a practical RA$\ell$SFA implementation. Our new version can outperform the FFTW when $N$ is around $70,000$ and $B$ is small.
\textbf{ A Motivating Example.} RA$\ell$SFA is an exciting replacement for the FFT to solve multiscale models.
Typically, one wants to simulate a multiscale model in several dimensions with both a microscopic and a
macroscopic description. The solution to the model has rapidly oscillating coefficients with period proportional to a small parameter
$\epsilon$. For examples of multiscale problems of size $N$ that are dominated by the behavior of $B\ll N$ Fourier components, see e.g \cite{BLP}. In a traditional (pseudo-)spectral method, one computes the spatial derivatives by the FFT
and Inverse FFT at each time iteration; consequently the time to find the Fourier representation of a signal is
the determining factor in the overall time of simulation. In multiscale problems,
where only a small number of Fourier modes contribute to the energy of an initial condition and coefficient
functions, we expect that RA$\ell$SFA will
significantly speed up the calculation for large $N$. In fact, a preliminary study has shown \cite{OLOF}
that for some transport and diffusion equations with multiple scales, using only significant frequencies to approximate intermediate solutions does not substantially degrade the quality of the
approximate final solution to the multiscale problem. By using the most significant frequencies and RA$\ell$SFA instead
of all frequencies and the FFT, we could replace a superlinear algorithm by a poly-log (polynomial in the logarithm)
algorithm. The corresponding decrease of the running time would make it possible to handle a larger number of grid points in
high dimensions. We shall present detailed applications
of this algorithm in multiscale problems in \cite{ZDR}.
\textbf{Notation and Terminology.} For any two frequencies $\omega_1$, $\omega_2$,
where $\omega_1 \neq \omega_2$, we say that $\hat{S}(\omega_1)$ is bigger than
$\hat{S}(\omega_2)$ if $| \hat{S}(\omega_1)|>|\hat{S}(\omega_2)|$. The squared norm $\|S\|_2^2=\sum_{t=0}^{N-1}|S(t)|^2$
of $S$ is also called the energy of $S$; we shall refer to
$| \hat{S}(\omega) |^2$ as the energy of the Fourier coefficient $\hat{S}(\omega)$. Similarly, the energy of a
set of Fourier coefficients is the sum of the squares of their magnitudes. We shall use only the $\ell ^2$-norm in this
paper; for convenience, we therefore drop the subscript from now on, and denote $\|F\|_2^2$ by $\|F\|^2$
for any signal $F$.
We denote the convolution by $F*G$, $(F*G)(t)=\sum_s{F(s)G(t-s)}$. It follows that $\widehat{F*G}=\sqrt N \hat{F}\hat{G}$.
We denote by $\chi_T$ the
signal that equals 1 on a set $T$ and zero elsewhere. The index to $\chi_T$ may be either time or frequency; this is
made clear from context. For more background on Fourier analysis, see \cite{weaver}. The support $supp(F)$ of
a vector $F$ is the set of $t$ for which $F(t)\neq 0$. A signal is 98$\%$ pure if there exists a frequency $\omega$ and some signal $\rho$, such
that $S=a \phi_{\omega}+\rho$ and $|a|^2\geq 0.98\|S\|^2$.
RA$\ell$SFA is a randomized algorithm. By this, we do {\bf not} mean the signal is randomly chosen from some kind of
distribution, with our timing and memory requirement estimates holding
with respect to this distribution; on the contrary, the signal, once given to us, is {\bf fixed}.
The randomness lies in the algorithm. After random sampling, certain operations are
repeated many times, on different subsets of samples, and averages and medians of the results are computed.
We set in advance
a desired probability of success $1-\delta$, where $\delta>0$ can be arbitrarily small. Then the claim is that
for each arbitrary input $S$, the algorithm succeeds with probability $1-\delta$,
i.e., gives a $B$-term estimate $R$ such that
$\|S-R\|^2 \leq (1+\epsilon) \|S-R_{opt}^B\|^2$. For given $\epsilon$, $\delta$, numerical experiments show that the
algorithm may take $O(B^2 \log N)$ time and space.
\textbf{Organization.} The chapters are organized as follows. Section 2 shows the testbed and numerical experiments about the
comparison of our RA$\ell$SFA and the FFTW. In Section 3, we
introduce all the new techniques and ideas of RA$\ell$SFA (different from \cite{GGIMS}) and its extension to multi-dimensions.
\section{Testbed and Numerical Results of RA$\ell$SFA}
In this section, we present numerical results of RA$\ell$SFA. We begin in Section
\ref{subset:onedim} with comparing the running time of RA$\ell$SFA and the FFTW
for some one dimensional test examples. In Section \ref{sect:ntwodim},
the performances of two dimensional RA$\ell$SFA and the FFTW for some test signals
are shown.
The randomness of the algorithm implies that the performance differs each time for
the same group of parameters. Hence, we give the average data, bar and quartile
graph based on 100 runs as well as the fastest data among these experiments. The
popular software FFTW \cite{FJ} version 2.1.5 is used to determine the
timing of the Fast Fourier Transform for the same data.
The test signals are either superpositions of $B\ll N$ modes in the frequency domain, that is, $S=\sum_{j=1}^{B} c_j
\phi_{\omega_j}$, contaminated with Gaussian white noise, or signals for which the Fourier coefficients exhibit rapid decay, so that a $B$-mode approximation with $B\ll N$ will already be very accurate. Different choices of the $\omega_j$ were checked; these did not influence the whole execution time. These choices included cases where some frequencies were close; note that this is the ``hard'' case for most
estimation algorithms. For RA$\ell$SFA, which contains random scrambling
operations (that are later described), the distance between the modes does not
matter if $N$ is prime. If $N$ is not prime, then $gcd(\omega_1-\omega_2, N)$
cannot decrease by the scrambling operation, so that different $(\omega_1,
\omega_2)$ pairs may (in theory) lead to different performances; in practice,
this doesn't seem to matter. In all these situations,
RA$\ell$SFA reliably estimates the size and locations of the few largest coefficients. We also set other parameters as follows: accuracy factor
$\epsilon=10^{-2}\|S\|$, failure probability $\delta=0.05$.
The parameter choices in the algorithm are quite tricky. The
theoretical bounds given in \cite{GGIMS} do not work well in practice; instead
much smaller parameters and heuristic settings work more efficiently.
All the experiments were run on an AMD Athlon(TM) XP1900+ machine with Cache size 256KB, total memory 512 MB, Linux kernel version 2.4.20-20.9 and compiler gcc version 3.2.2.
\subsection{Numerical Results in one dimension}
\label{subset:onedim}
The first implementation results of RA$\ell$SFA were not published; the program was
basically a proof of concept, not optimized. With the choices and parameters
described in \cite{GGIMS}, it was extremely slow and thus not practical for
real-world applications. The implementation we present here runs several order
of magnitude faster; this involves introducing many adjustments and ideas to
the algorithm of \cite{GGIMS}. (See Section 3 for details.)
The goal of this paper is to check the possibility to replace the FFT with RA$\ell$SFA for sparse and long signals. Therefore, we focus on comparing the performance of RA$\ell$SFA and FFTW in the following subsections.
\subsubsection{Experiments for an Eight-mode Representation}
\label{subset:eightoned}
We begin with the experiments for recovering a signal consisting of eight modes (with and without noise).
In the noisy signal case, the noise is a Gaussian white noise with signal-to-noise ratio
($SNR$, defined as $10\log_{10}\frac{\|S\|^2}{N\sigma^2}$) approximately 5dB. The coefficients are randomly taken
from the interval $[1,10]$ and the significant modes from $[0, N-1]$.
Two kinds of running time for each algorithm are provided. One is the total running time and
another is the running time excluding the sampling time. As we know, the FFT
takes $\Omega(N)$ to compute all signal values. On the other hand, our algorithm
doesn't need all the sample values. All our conclusions are based on the time
{\it excluding} the sampling. However, we still list the running time including sampling time
as well because of the existence of various forms of data in practice. For example, in pseudospectral
applications, the data need to be computed from a
B-superposition, which may take $O(B)$ per sample. It is possible to sample more quickly, which
is addressed in \cite{GMS}. On the other hand, if the data is
already stored in a file or a disk, we simply get them without any computation.
In all these cases, we assume the data is either already in memory or
available through computation. Thus we don't need to go through every data, which would take time $O(N)$.
Table \ref{tab:B8onedim} provides a
comparison of the running times of the FFTW and RA$\ell$SFA for eight-mode clean and noisy signals.
In the beginning when $N$ is small, the FFTW is almost instantaneous. As
the signal length $N$ increases, its time grows superlinearly. On the contrary, RA$\ell$SFA takes longer
time in smaller
$N$ cases; however the time cost remains almost constant regardless of the
signal length. In addition, the benchmark FFTW
software fails to process more than $10^8$ data because it runs out of the memory space. In contrast,
RA$\ell$SFA has no difficulty at all since it does not need all the data. A simple interpolation from the entries in Table
\ref{tab:B8onedim} predicts that RA$\ell$SFA beats the FFTW when $N>15,200$ for eight-mode signals, all the more convincingly when $N$ is larger.
If we compare the time including sample computation, the cross-over point would be $N=70,000$. The table also
shows the linear relationship between the time cost and the logarithm of the length $N$.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
Length& \multicolumn{2}{l|}{Time of}& \multicolumn{1}{l|}{ Time of} & \multicolumn{2}{l|}{Time of RA$\ell$SFA}&\multicolumn{1}{l|}{Time of FFTW}\\
N & \multicolumn{2}{l|}{RA$\ell$SFA} & \multicolumn{1}{l|}{FFTW} &\multicolumn{2}{l|}{(excluding sampling) }&
\multicolumn{1}{l|}{(excluding sampling)}\\ \cline{2-6}
&clean& noisy& &clean& noisy& \\ \hline \hline
$10^{3}$&
0.22 & 0.25 & 0 &0.01 &0.02 &
0 \\ \hline $10^{4}$&
0.25&
0.29 & 0.04& 0.03 & 0.04&
0.01
\\ \hline $10^{5}$&
0.32 &
0.34& 0.46&0.05 & 0.05&
0.17 \\ \hline
$10^{6}$&
0.37 & 0.41&5.01 & 0.07&
0.08 &
2.23\\ \hline $10^{7}$&
0.44 & 0.48& 54.57 & 0.10&
0.11 &
26.24\\ \hline
\end{tabular}
\end{center}
\caption{Time Comparison between RA$\ell$SFA and FFTW (B=8) based on 100 runs. ``Clean'' means that the test signal is pure. ``Noisy'' means the signal is contaminated with noise of $SNR=5dB$. ``Excluding Sampling'' column lists the running time without precomputation of sample values.} \label{tab:B8onedim}
\end{table} \addtocounter{figure}{+1}
As can be expected from a randomized algorithm, RA$\ell$SFA has a different performance in each run. Figure \ref{fig:B8bar1}
illustrates the spread of the execution time (including sampling) for pure signals over 100 runs.
\begin{figure}[htbp]
\begin{center}
\includegraphics[%
width=6cm]{NBarB8pure.eps}
\includegraphics[%
width=6cm]{NquarB8pure.eps}
\caption{Comparing the total running times of 8-mode RA$\ell$SFA for 100 different runs of the randomized algorithm. Left: mean and
variance as a function of $N$; right: median, quartiles and total spread of the runs as a function of $N$, $B=8$}\label{fig:B8bar1}
\end{center}
\end{figure} \addtocounter{table}{+1}
\subsubsection{Experiments with Different Levels of Noise}
\label{subset:noise}
In the experiments above, we compared the performance of clean and slightly noisy signals. Here, we shall push
the noise level much higher, keeping $N$ and $B$ fixed to illustrate the effect of noise.
Also, instead of allowing the algorithm to run for $poly(B, \log N, 1/\epsilon, \log(1/\delta) )$ iterations,
we set a smaller fixed upper bound (so that the success probability is no longer $1-\delta$).
When noise is present,
it influences the success probability with which modes with small amplitude are detected. To explore this, we ran an experiment
with only a single mode; we kept the amplitude of the mode constant and increased the noise. Figure \ref{fig:noise} (left) shows the \textit{success probability} of the detection of the single mode by the algorithm (estimated
by running 100 trials each time and recording the number that were successful) for three different settings of the maximum number of iterations.
The dependence of the \textit{running time} on the $SNR$ in the case of detection of a single mode is illustrated in
Figure \ref{fig:noise} (right), where we show the results of the average over 100 runs for every data point, with only a very loose a priori
restriction on the running time ($\leq$1000 iterations); only parameter settings with over $50\%$ success probability were
taken into account.
\begin{figure}[htbp]
\begin{center}
\includegraphics[%
width=6cm]{succSNR.eps}
\includegraphics[%
width=6cm]{timeSNR.eps}
\caption{Experiments for signal $S=\phi_0+noise$ with length $N=10,009$. Compare the success rate
and running time of RA$\ell$SFA when the total number of iterations is bounded by 100, 500 and 1000 (respectively), based on 100 different runs of the randomized algorithm
in each case. Left: success probability of RA$\ell$SFA as $SNR$ decreases; right: running time of RA$\ell$SFA as $SNR$ decreases (we only show the running time when success probability is greater than $50\%$). Note that the abscissa show $-SNR$ each time, meaning that the $\ell^2$-norm of the noise is much larger than that of the signal in the regimes
illustrated here; for instance, $SNR=-60dB$ means that the $\ell^2$-norm of the noise equals $1,000 \times \ell^2 \,norm \,of \,the \,signal $.} \label{fig:noise}
\end{center}
\end{figure} \addtocounter{table}{+1}
This experiment indicates
that it is possible to detect modes that are significantly weaker than the noise, within limits, of course. If the amplitude
of the signal is too weak, then trying to detect it may waste many resources. In practice we
shall put our cut-off on the amplitude at about one sixth of the noise level, i.e., at $\sigma/6$; this can of course be adjusted
depending on whether one wishes fast speed or not.
Although $SNR$ is the standard characterization of noise intensity, it is not clear that it is the parameter that
matters most for our algorithm. We therefore also ran an experiment in which we compare the results for two
different values of $N$: 10,009 (as in the Figures above) and 100,003, respectively.
The second value of $N$ is about 10 times larger than the first; for the same choices of $\sigma$ and $c$ (the amplitude of the single mode), the $SNR$
for the second $N$ is smaller by $10dB$. Table \ref{tab:noisesnr}, comparing the performance for these two values of $N$ and several choices of
$\sigma$, shows that the value of $\sigma$ itself rather than $SNR$ governs the running time and success probability.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c||c|c|c||c|c|c|}
\hline
& \multicolumn{3}{c||}{$N_1=10,009$}& \multicolumn{3}{c|}{$N_2=100,003$}\\ \hline
$\sigma$& success probability & time & $SNR$ & success probability & time & $SNR$
\\ \hline 2& $100\%$ & 0.11& -46.02& $100\%$&0.19 & -56.02\\ \hline
2.5 & $93\%$ & 0.32& -47.96& $77\%$& 0.55& -57.96\\ \hline 3& $49\%$ & 0.38& -49.54& $27\%$&0.61 &-59.54 \\ \hline 3.5& $21\%$ & 0.45& -50.88& $10\%$& 0.38&-60.88
\\ \hline 4& $13\%$ & 0.38& -52.04& $1\%$& 0.37&-62.04 \\ \hline
\end{tabular}
\end{center}
\caption{Exploring the dependence on $\sigma$ versus $SNR$ of the influence of the noise on processing
the signal $S=\phi_0+noise$, where the noise is gaussian $N(0, \sigma)$. For two different values of $N$,
$N_1=10,009$ and $N_2=100,003 \approx 10N_1$, respectively, and a range of values for $\sigma$, we determined the success
probability within 100 runs, and the average running time for successful runs. In both cases we see a clear transition as
$\sigma$ increases; the location of the transition (between 2.5 and 4 for $N_1$, between 2 and 3.5 for $N_2$) shifts slightly
with $N$, but it is nevertheless clear that $\sigma$ is a better parameter to track than $SNR$: in fact, the largest choice for
$\sigma$, $\sigma=4$, still has lower $SNR$ in the case $N=N_1$ than the smallest choice, $\sigma=2$, for $N=N_2$, yet the success
probability and running time are much worse. } \label{tab:noisesnr}
\end{table} \addtocounter{figure}{+1}
\subsubsection{Experiments with Different Numbers of Modes}
\label{sect:diffB}
The crossover points for $N$ are different for signals with different $B$; the number of modes has an important
influence on the running time. To investigate this, we experimented with fixed $N$ (we took a prime number
$N=2,097,169$ (a prime number) for RA$\ell$SFA and $N=2^{21}=2,097,152$ for FFTW) but varying $B$. In all cases, we take $S$ to be
a superposition of exactly $B$ modes, i.e., $S(t)=\sum_{i=1}^{B}{c_{i}\phi_{\omega_i}}$ for some $B$. Table
\ref{tab:diffB} compares the running time for different $B$ using the FFTW and RA$\ell$SFA. For small $B$, RA$\ell$SFA
takes less time because $N$ is so large. The execution time for the FFT can be taken to include the time for
evaluation of all the samples (which increases linearly in $B$) or not (in which case the execution time is
constant to $B$). In both cases, the FFTW overtakes RA$\ell$SFA as $B$ increases; the execution time of the FFTW
is constant or linear in the number of modes $B$ (depending on whether the evaluation of samples is included),
while that of RA$\ell$SFA is polynomial of higher order. For $N=2,097,169$, the FFTW is faster than RA$\ell$SFA
when $B \geq 33$. By regression techniques on the experimental data, one empirically finds that the order of $B$
in RA$\ell$SFA is quadratic. This is the main disadvantage of RA$\ell$SFA. (Although this
nonlinearity in $B$ was expected by the authors of \cite{GGIMS}, the observation that it played such an
important role even for modest $B$ was the motivation for Gilbert, Muthukrishnan and Strauss to construct in
\cite{GMS} a different version of RA$\ell$SFA that is linear in $B$ for all $N$.) Hence, RA$\ell$SFA is most
useful for a long signal with a small number of modes.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
Number of modes & Time of & Time of & Time of RA$\ell$SFA & Time of FFTW\\
B & RA$\ell$SFA & FFTW & (exclude sampling )& (exclude sampling)\\ \hline
$2$&
0.05 &
7.49&
0.03 &
5.46\tabularnewline
$4$&
0.14 &
9.38&
0.05 &
5.46\tabularnewline
$8$&
0.35 &
13.22 &
0.07 &
5.46\tabularnewline
$16$&
2.48 &
20.92 &
0.83 &
5.46\tabularnewline
$32$&
15.53 &
36.28 &
4.13 &
5.46\tabularnewline
$64$&
107.55 &
67.16 &
39.55&
5.46 \tabularnewline
\hline
\end{tabular}
\end{center}
\caption{Time Comparison between RA$\ell$SFA and FFTW for Different $B$ when $N \approx 2,097,169$} \label{tab:diffB}
\end{table} \addtocounter{figure}{+1}
\subsubsection{Experiments with Signals that have infinitely many modes with rapid decay in frequency}
For our final batch of one-dimensional experiments, we ran the algorithm on the signal $S=1/(1.5+\cos 2\pi t)+noise$. In continuous time, the clean signal has infinitely many modes with amplitudes that decay
exponentially as the frequency of the mode increases. We ran the experiment with a white Gaussian noise once with $SNR$
$-20dB$ and a second time with $SNR=-8dB$, with $N=1000$. The threshold for the amplitudes of modes we wished to find was adjusted to the
noise level in both cases.
\begin{figure}[htbp]
\begin{center}
\includegraphics[%
width=7cm]{decayral6.eps}
\includegraphics[%
width=7cm]{decayfft6.eps}
\caption{For signal $S=1/(1.5+\cos 2\pi t)+noise$ with $SNR=-20dB$. Compare the approximation effect by RA$\ell$SFA and FFTW. Left:
approximation of the significant coefficients
by RA$\ell$SFA; the relative approximation error is $0.74\%$; right: approximation of the significant coefficients
by FFTW. } \label{fig:decay}
\end{center}
\end{figure} \addtocounter{table}{+1}
\begin{figure}[htbp]
\begin{center}
\includegraphics[%
width=7cm]{decayral7.eps}
\includegraphics[%
width=7cm]{decayfft7.eps}
\caption{For signal $S=1/(1.5+\cos 2\pi t)+noise$ with $SNR=-8dB$. Compare the approximation effect by RA$\ell$SFA and FFTW. Left:
approximation of the significant coefficients
by RA$\ell$SFA; the relative approximation error is $0.4\%$;right: approximation of the significant coefficients
by FFTW. (this is for the one run illustrated. In other runs, it makes similarly one or two mistakes, not necessarily at the same modes.)} \label{fig:decay1}
\end{center}
\end{figure} \addtocounter{table}{+1}
The results are shown in Figure \ref{fig:decay} ($SNR=-20dB$)and Figure \ref{fig:decay1} ($SNR=-8dB$), respectively.
For $SNR=-20dB$, the Fourier coefficients obtained by FFTW are all very close to the ``noise floor'', i.e., they
lie in a band of amplitude close to the value of $\sigma$. For $SNR=-8dB$, $\sigma$ is smaller ($\sigma=2.6$), and we find
the ``noise floor'' in the FFTW computation at this lower level. The three largest modes of the signal have amplitudes significantly higher
than this $\sigma$, and FFTW finds them with reasonable accuracy. In contrast, RA$\ell$SFA (shown on the left
in both figures; only 1 run is shown) hits all the coefficients exceeding $\sigma$ ``on the nose'', in both cases; it also finds all the central 15 modes exactly in the
$SNR=-8dB$ case, even if they have values significantly smaller than $\sigma$. This experiment illustrates the great robustness
of RA$\ell$SFA to noise and its ability to detect harmonic components with smaller energy than the white noise,
already seen in \ref{subset:noise}.
\subsection{Numerical Results in Two Dimensions}
\label{sect:ntwodim}
The number of grid points depends exponentially on the dimension. To achieve reasonable accuracy, a minimum $N$
is required in each dimension; however, when $d>1$, the FFTW has great difficulty in handling the corresponding $N^d$
points for even modest $N$. RA$\ell$SFA does not have this problem.
\subsubsection{Experiments for Eight-mode Signals in Two Dimensions}
We take the signal $S=\sum_{k=1}^{B}c_{k}\phi_{\omega_{x,k}}\phi_{\omega_{y,k}}$, where
$B=8,\epsilon=10^{-2}\|S\|,\delta=0.05$. The parameter $N$ is the number of grid points in
each dimension, random complex constants $c_k$ with real and imaginary parts in $[1, 10]$, and $\omega_{x,k}$ and $\omega_{y,k}$
are random integers from ${0, \ldots, N-1}$. As Table
\ref{tab:B2D2} shows, two dimensional RA$\ell$SFA surpasses two
dimensional FFTW when $N \geq 1500$. In particular, when $N=5000$ and the computation for samples is
not included, the FFTW takes 21 seconds and RA$\ell$SFA only less than 5 second. When we include the
sampling time, the crossover point becomes $N=900$.
The crossover point for $N$ is 70000 for $d=1$, and 900 for $d=2$; if we conjecture that the crossover $N$ for 2-mode in $d$ dimensions is given
by $c_2 n_2^{\frac{1}{d}}$, then this leads us to guess that the crossover $N$ for $d=3$ may be close to 210.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Length& \multicolumn{2}{c|}{Time of}& \multicolumn{1}{c|}{ Time of} & \multicolumn{2}{c|}{Time of RA$\ell$SFA}&\multicolumn{1}{c|}{Time of FFTW}\\
N & \multicolumn{2}{c|}{RA$\ell$SFA} & \multicolumn{1}{c|}{FFTW} &\multicolumn{2}{c|}{(excluding sampling) }&
\multicolumn{1}{c|}{(excluding sampling)}\\ \cline{2-6}
&clean& noisy& &clean& noisy& \\ \hline \hline
$100$&
3.41& 3.64 & 0.05 &0.88 &1.05 &
0.04 \\ \hline $1000$&
4.11 & 4.54 & 4.87 &1.04 &1.25 &
0.20 \\ \hline $2000$&
4.76&
4.91 & 20.86& 1.31 & 1.44&
2.12 \\ \hline $3000$&
4.55 &
5.37& 47.73& 1.33 & 1.70&
5.62\\ \hline $4000$&
5.41 & 5.59&85.89 & 1.41&
1.51 &
10.74\\ \hline $5000$&
6.03 & 6.20& 138.27 & 1.56&
1.66&
20.98\\ \hline
\end{tabular}
\end{center}
\caption{Time Comparison between RA$\ell$SFA and FFTW (B=8) based on 100 runs. ``Clean'' means that the test signal is pure. ``Noisy'' means the signal is contaminated with noise of $SNR=-4dB$. ``Excluding Sampling'' column lists the running time without including precomputation of sample values.} \label{tab:B2D2}
\end{table} \addtocounter{figure}{+1}
\subsubsection{Experiments for Signals with Different Number of Modes $B$}
As in one dimension, the number of modes $B$ is the bottleneck for applying RA$\ell$SFA freely to signals that
are not so sparse. Suppose the signal is of the form $S(t)=\sum_{k=1}^{B}{c_{k}\phi_{\omega_{x,k}} \phi_{\omega_{y,k}}}$, with $N=3001$
for RA$\ell$SFA and $3000$ for FFTW. Table \ref{tab:diffBD2} illustrates the relationship between running time
and the number of modes $B$. Time increases depends polynomially on the number of terms $B$. When $N=3001$, the
crossover points for the FFTW to surpass RA$\ell$SFA are at $B=10$ and $B=17$ respectively, for including and
excluding sample computation cases. This implies the influence of $B$ on the execution time is far from
negligible.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
Number of modes & Time of & Time of & Time of RA$\ell$SFA & Time of FFTW\\
B & RA$\ell$SFA & FFTW & (exclude sampling )& (exclude sampling)
\tabularnewline
\hline
$2$&
0.15 &
16.45 &
0.08& 5.64\tabularnewline $4$&
0.52 &
26.81 &
0.14& 5.64\tabularnewline $8$&
4.55 &
47.73 &
1.343& 5.64\tabularnewline $12$&
19.37 &
68.47&
8.82& 5.64\tabularnewline $16$&
48.69 &
89.13&
9.13 & 5.64\tabularnewline $20$&
114.80 &
109.88&
22.75& 5.64 \tabularnewline \hline
\end{tabular}
\end{center}
\caption{Time Comparison between RA$\ell$SFA and the FFTW for signals with different $B$ in 2 dimensions when
$N=3000$} \label{tab:diffBD2}
\end{table} \addtocounter{figure}{+1}
\section{Theoretical Analysis and Techniques of RA$\ell$SFA}
We hope the numerical results have whetted the reader's appetite for
a more detailed explanation of the algorithm. Before explaining the structure of RA$\ell$SFA as implemented by us, we review the basic idea of the algorithm.
Given a signal consisting of several frequency modes with different amplitudes, we could split it into several
pieces that have fewer modes. If one such piece had only a single mode, then it would be fairly easy to
identify this mode, and then to approximately find its amplitude. If the piece were not uni-modal, we could, by repeating
the splitting, eventually get uni-modular pieces. In order to compute the amplitudes, we need to ``estimate
coefficients.'' To verify the location of the modes in the frequency domain and concentrate on the most significant part of
the energy, we use ``group testing.'' An estimation that recurs over and over again
in this testing is the ``evaluation of norms.'' The first splitting of the signal is done in the ``isolation'' step.
The different steps are carried out on many different variants of the signals, each obtained by a random translation
in the frequency domain (corresponding to a modulation and the inverse dilation in the time
domain). Because the signal is sparse in the frequency domain, the different modes are highly likely to be well
separated after these random operations, facilitating isolation of individual modes.
The main skeleton of the algorithm was already given in \cite{GGIMS}; in our discussion here, we introduce new ideas and give the
corresponding theoretical analysis. We also explain how to set parameters that are either not mentioned or
loose in \cite{GGIMS}. In Section \ref{sect:num}, the total scheme of RA$\ell$SFA is given. In Section \ref{sect:coeff}, we show
the theoretical basis to choose parameters for estimating coefficients, and introduce some techniques to
speed up the algorithm. In Section \ref{sect:norm}, we set the parameters for norm estimation. Section \ref{sect:isolation} presents the
heuristic rules to pick the filter width for the isolation procedure. This is one of the key factors
determining the speed. A new filter is proposed for Group Testing in Section \ref{sect:group}, which works more efficiently.
Section \ref{sect:sample} discusses how to evaluate a random sample from a signal.
Finally, we discuss the extension to higher dimensions in Section \ref{sect:twodim}.
\subsection{Set-up of RA$\ell$SFA} \label{sect:num}
The following theorem is the main result of \cite{GGIMS}.
\begin{theorem}
Let an accuracy factor $\epsilon$, a failure probability $\delta$, and a sparsity target $B \in \mathbb{N}, B<N$
be given. Then for an arbitrary signal \(S\) of length $N$, RA$\ell$SFA will find, at a cost in time and space of
order \(poly(B,\log(N), \frac{1}{\epsilon}, \log(\frac{1}{\delta}))\) and with probability exceeding $1-\delta$, a
\(B-\)term approximation \(R\) to $S$, so that \(\|S-R\|^2 \leq (1+\epsilon)\|S-R_{opt}^B(S)\|^2_2\).
\end{theorem}
It is especially striking that the near-optimal representation $R$ can be built in sublinear time i.e.,
$poly(\log N)$ instead of the $O(N \log N)$ time requirement of the FFT. RA$\ell$SFA's speed will surpass the FFT as long as
the length of the signal is sufficiently large. In particular, if $S=R^B_{opt}(S)$ (that is, $\hat{S}(\omega)$
vanishes for all but $B$ values of $\omega$), then $\|S-R\|^2=0$, i.e., RA$\ell$SFA constructs $S$
without any error, at least in theory; in practice this means the error is limited by accuracy issues.
The main procedure is a Greedy Pursuit with the following steps:
\begin{algorithm} {\sc Total Scheme} \\
Input: signal $S$, the number of nonzero modes $B$ or its upper bound, accuracy factor $\epsilon$, success probability $1-\delta$, an upper bound of the signal energy $M$, the standard deviation of the white Gaussian noise $\sigma$, a ratio $\iota$ for relative precision.
\begin{enumerate}
\item Initialize the representation signal $R$ to 0, set the maximum number of iterations $T=B\log(N)\log(\delta)/\epsilon^{2}$,
\item Test whether $\Vert S-R \Vert^2 \leq \iota \|R\|^2$. If yes, return the representation signal $R$ and the whole algorithm ends; else go to step 3.
\item Locate Fourier Modes $\omega$ for the signal $S-R$ by the isolation and group test procedures below.
\item Estimate Fourier Coefficients at $\omega$: $\widehat{(S-R)}(\omega)$.
\item If the total number of iterations is less than $T$, go to 2; else return the representation $R$.
\end{enumerate}
\end{algorithm}
The test at stage 2, which is not in \cite{GGIMS}, can allow us to end early. The criterion $\|S-R\|^2 \leq \iota \|R\|^2$,
where $\iota$ is a small number chosen heuristically, is suitable when one expects that $S$ is sparse, up to a small energy contribution.
(Note that step 2 does not use the exact value of $\|S-R\|^2$, which is not known; we use
a procedure called norm estimation (see below) to give a rough estimate; this is good enough for the stop criterion. Other criteria could be substituted when appropriate.)
In practice, we would not know how many modes a signal has. In fact, the algorithm
doesn't really need to know $B$: it can just proceed until the residual energy
is estimated to be below threshold. (The value of $B$ is used only to set the
maximum number of iterations, and the width of a filter in the isolation
procedures below. For the maximum number $T$, a loose upper bound on $B$
suffices; the isolation filter width depends only very weakly on $B$.) If
either the residual energy or the threshold is large, the program would
continue. Note that for each iteration of the algorithm, we take new random
samples from the signal $S$.
\subsection{Estimate Individual Fourier Coefficients}
\label{sect:coeff}
The original RA$\ell$SFA only shows the validity of estimating coefficients without mentioning parameter
settings. Here we introduce a new technique to achieve better and faster estimation;
in the process, we give another proof
of Lemma 2 in \cite{GGIMS} that contains explicit parameter choices.
\begin{algorithm}\label{alg:coeff} {\sc Estimate Individual Fourier Coefficients} \\
Input: signal $S$, success probability $1-\delta$, and accuracy factor $\epsilon$.
\begin{enumerate}
\item Randomly sample from signal $S$ with indices
$t_{i,j}$: $S(t_{i,j})$, $i=1,\ldots, 2\log(1/\delta)$, $j=0,\ldots,8/\epsilon^2$ .
\item Take the empirical mean of the $\left \langle S(t_{i,j}), \phi_{\omega}(t_j) \right \rangle$, $j=0,\ldots, 8/\epsilon^2$, store as $mean(i)$.
\item Take the median $y=median (mean(i))$, $i=1,\ldots, 2\log(1/\delta)$.
\item Return $y$.
\end{enumerate}
\end{algorithm} \label{alg:estcoef}
\begin{lemma} \label{lem:coef1}
Every application of Algorithm {\em \ref{alg:coeff}} constructs a realization of a random variable $Z$, that
estimates the Fourier coefficient \(\hat{S}(\omega)\), good up to tolerance \(\epsilon^2 \|S\|^2\) with high
probability \(1-\delta\), i.e.,
\begin{equation}
Prob \left (|Z-\hat{S}(\omega)|^2 \geq \epsilon^2 \|S\|^2 \right ) \leq \delta.
\end{equation}
\end{lemma}
\begin{proof}
Define a random vector $V$ as follows:
\begin{equation}
V=\left (0,0,\cdots,NS(t),0,\cdots,0 \right )=N\delta_{t}S(t).
\end{equation}
where $t$ is chosen uniformly and randomly from $\{ l:\, l=1,\cdots,N\}$. Then the expectation of $V$ is
\begin{equation}
E(V)=\frac{1}{N}\sum_{t}{NS(t)\delta_{t}}.
\end{equation}
Let X be the random variable $X=\langle V,\phi_{\omega}\rangle$, where $\phi_{\omega}(t)=N^{-\frac{1}{2}}e^{-2\pi i\omega t/N}$. We have
\begin{equation}E[X]=\frac{1}{N}\sum_{t}{NS(t)\phi_{\omega}(t)}=\hat{S}(\omega),\end{equation} and
\begin{equation}E \left (|X-\hat{S}(\omega)|^{2}\right )\leq E(|X|^{2}) =\frac{1}{N}\sum_{t}{\left |\frac{N}{\sqrt{N}}S(t)e^{-2\pi i\omega t/N} \right |^{2}}=\Vert S\Vert_2^{2}.\end{equation}
Define another random vector $W$ as
the average of $L$ independent realization of $V$, with $L = 8 \epsilon^{-2}$. Let a random variable
\begin{equation}Y=\langle W,\phi_{\omega}\rangle. \end{equation}
Then $E[Y]=\hat{S}(\omega)$ and $var[Y]=var[X]/L=\epsilon^{2}\Vert S\Vert^{2}/8$,
so that $Prob \left (|Y-\hat{S}(\omega)|^{2}\geq\epsilon^{2}\Vert S\Vert^{2} \right)\leq 1/8$. \\
Set $Z=median_K Y$, where $K=2\log(1/\delta)$. If $|Z-\hat{S}(\omega)|^2 \geq \epsilon^2 \|S\|^2$, then
for at least half of the $Y$s, we have
\begin{equation}
|Y-\hat{S}(\omega)|^2 \geq \epsilon^2 \|S\|^2.
\end{equation}
Therefore
\begin{align}
P\left (|Z-\hat{S}(\omega)|^{2}\geq\epsilon^{2}\Vert S\Vert_2^{2} \right) &\leq \sum_{j=K/2} ^K {K \choose j} \left ( \frac{1}{8} \right )^j \nonumber \\
&\leq 8^{-K/2} 2^K = 2^{-K/2} \leq\delta.
\end{align}
So with probability $1-\delta$, $Z$ is a good estimate of the Fourier Coefficient $\hat{S}(\omega)$, good up to
tolerance $\epsilon^{2}\Vert S\Vert^{2}.$ \qquad\end{proof}
Several observations and new techniques can speed up the coefficient estimation even further.
One observation is that fewer samples are already able to give an estimation with desirable accuracy and probability.
Our arguments indicate that $16 \epsilon^{-2} |\log (\delta)|$ samples per coefficient suffice to obtain good approximations of the
coefficients. The estimates used to obtain this bound are rather coarse, however. In a practical implementation, if a multi-step evaluation is used (see below),
it turns out that three steps, in which every step uses 10 samples per mean, and 5 means per median, for a total of 150 samples (per coefficient)
already determine the coefficient with accuracy $\epsilon = 10^{-4}$. The major factor in this drastic reduction (from $16\cdot 10^8|\log \delta|$ to 150)
is the much smaller number of means used; in practice, the dependence on $\epsilon$ grows much slower than $\epsilon^{-2}$ as $\epsilon \rightarrow 0$
If the signal is contaminated by noise or has more than one significant mode, we need more samples for a good estimation of the same accuracy.
An additional difference with the sampling described in \cite{GGIMS} is that one can replace individual random
samples by samples on short arithmetic progressions with random initial points. This technique became one of
several components in the RA$\ell$SFA version of \cite{GMS} that adapted the original algorithm in order to obtain
linearity in $B$. For a description of the arithmetic progression sampling, we refer to \cite{GMS}.
Surprisingly, this change not only improves the speed, but also gives a closer approximation than simply random
sampling, using the same number of samples.
Another idea is a coarse-to-fine multi-step estimation of the coefficients. There are several reasons for not
estimating coefficients with high accuracy in only one step. One of them is that increasing the accuracy $\epsilon$
means a corresponding quadratic increase of the number of samples $O(|\log\delta| \epsilon^{-2})$.
A multi-step procedure, which produces only an approximate estimate of the coefficients in each step,
achieves better accuracy and speed. To explain how this works, we need the
following lemma.
\begin{lemma}
Given a signal $S$, let $\omega_1, \ldots, \omega_q$ be $q$ different frequencies, and define
$\beta \colon = \left [ \|S\|^2_2 - \sum_{i=1}^q |\hat{S}(\omega_i)|^2 \right ]/ \|S\|^2_2$.
Estimate the coefficients $\hat{S}(\omega_i)$ where
$i=1, \ldots, q$ by the following iterative algorithm: apply Algorithm {\em \ref{alg:coeff}} with precision $\hat{\epsilon}$ and probability of failure $\delta$;
keep the parameters fixed throughout the iterative procedure, and let $Z^n_i$, $i=1, \ldots, q$, be the estimate (at the $n$-th iteration)
of the $\omega_i$-th Fourier coefficient of $S-\sum_{k=1}^{n-1} \sum_{j=1}^q Z^k_j \phi_{\omega_j}$. The total estimate $R_n$ after the $n$-th
iteration is thus $R_n = \sum_{k=1}^n \sum_{j=1}^q Z^k_j \phi_{\omega_j}$. Then
\begin{equation}
\sum_{j=1}^q |\hat{S}(\omega_j) - \hat{R}_n(\omega_j)|^2 \leq \frac{q \hat{\epsilon}^2}{1-q \hat{\epsilon}^2} \beta \|S\|^2 + (q \hat{\epsilon}^2)^n \|S\|^2,
\end{equation}
with probability exceeding $(1-\delta)^{nq}$.
\end{lemma}
\begin{proof}
(This is essentially a simplified version of proof for Lemma 10 in \cite{GGIMS})\\
By Lemma \ref{lem:coef1},
\begin{equation}
|Z_i^n + \sum_{k=1}^{n-1} Z_i^k - \hat{S}(\omega_i)|^2 \leq \hat{\epsilon}^2 \|S-R_{n-1}\|^2,
\end{equation}
with probability exceeding $1-\delta$. It follows that
\begin{equation}
\sum_{i=1}^q |\hat{S}(\omega_i) - \sum_{k=1}^n Z^k_i|^2 \leq q \hat{\epsilon}^2 \|S-R_{n-1}\|^2,
\end{equation}
so that
\begin{align}
\|S-R_n\|^2 &\leq \sum_{\omega \notin \{ \omega_1, \ldots, \omega_q\}} |\hat{S}(\omega)|^2 + q \hat{\epsilon}^2 \|S-R_{n-1}\|^2 \nonumber \\
& = \|S\|^2 - \sum_{i=1}^q |\hat{S}(\omega_i)|^2 + q \hat{\epsilon}^2 \|S-R_{n-1}\|^2, \\ \nonumber
& = \beta \|S\|^2 + q \hat{\epsilon}^2 \|S-R_{n-1}\|^2\label{eq:cof}
\end{align}
with probability exceeding $(1-\hat{\delta})^q$. \\
Consider now the sequence $(a_n)$, defined by $a_n = \beta \|S\|^2 + q \hat{\epsilon}^2 a_{n-1}$, where $a_0=\|S\|^2$. It is easy to see that
\begin{align}
a_n & = \beta \|S\|^2 \sum_{k=0}^{n-1} (q \hat{\epsilon}^2)^k + (q \hat{\epsilon}^2 )^n \|S\|^2 \\ \nonumber
& = \beta \|S\|^2 \frac{1-(q \hat{\epsilon}^2)^n}{1-(q \hat{\epsilon}^2)} + (q \hat{\epsilon}^2 )^n \|S\|^2.
\end{align}
It then follows by induction that $\|S-R_n\|^2 \leq a_n$, with probability exceeding $(1-\hat{\delta})^{nq}$, for all $n$; we have thus
\begin{align}
\|S-R_n\|^2 & \leq \beta \|S\|^2 \frac{1-(q \hat{\epsilon}^2)^n}{1-(q \hat{\epsilon}^2)} + (q \hat{\epsilon}^2 )^n \|S\|^2 \\ \nonumber
& \leq \beta \|S\|^2 \frac{1}{1-(q \hat{\epsilon}^2)} + (q \hat{\epsilon}^2 )^n \|S\|^2,
\end{align}
or equivalently,
\begin{equation}
\sum_{j=1}^q |\hat{S}(\omega_j) - \hat{R}_n(\omega_j)|^2 = \|S-R_n\|^2 - \beta \|S\|^2 \leq \beta \|S\|^2 \frac{q \hat{\epsilon}^2}{1-q \hat{\epsilon}^2}
+ (q \hat{\epsilon}^2 )^n \|S\|^2,
\end{equation}
with probability exceeding $(1-\delta)^{qn}$. \qquad \end{proof}
The above lemma shows that repeated rough estimation can be more efficient than a single accurate estimation. To make this clear, if we set
\begin{equation}
q \epsilon^2 = \beta \frac{q \hat{\epsilon}^2 }{1-q \hat{\epsilon}^2 } + (q \hat{\epsilon}^2)^n, \, \, \, \,(1 - \delta)^q = (1-\hat{\delta})^{nq},
\label{eq:cofg}
\end{equation}
then a one-step procedure with parameters $\epsilon$, $\delta$ will achieve the same precision as an $n$-step iterative procedure with parameters
$\hat{\epsilon}$, $\hat{\delta}$. The one-step procedure will use $C q \epsilon^{-2} |\log(\delta)|$ sampling steps; the iterative procedure
will use $C nq \hat{\epsilon}^{-2} |\log(\hat{\delta})|$. It follows that the $n$-step iterative procedure will be more efficient, i.e., obtain the same
accuracy with the same probability while sampling {\it fewer} times, if
\begin{equation}
n \hat{\epsilon}^{-2} |\log (\hat{\delta})| \leq \epsilon^{-2} |\log (\delta)|,
\label{eq:cofgg}
\end{equation}
under the constraints (\ref{eq:cofg}). If $\beta=0$ (that is, if $S$ is a pure $q$-component signal), then this condition reduces
(under the assumption that $\hat{\delta}$, $\delta$ and $\hat{\epsilon}$,
$\epsilon$ are small, so that $\frac{q \hat{\epsilon}^2 }{1-q \hat{\epsilon}^2 } \simeq q \hat{\epsilon}^2$, $(1-\hat{\delta})^n \simeq 1-n \hat{\delta}$) to
\begin{equation}
n \left ( |\log \delta| + n \right ) (q \hat{\epsilon}^2)^{n-1} \leq |\log \delta|,
\label{eq:cofg1}
\end{equation}
which is certainly satisfied if $\hat{\epsilon}$ is sufficiently small and $n$ sufficiently large. If $\beta \neq 0$, matters are more complicated,
but by a simple continuity argument we expect the condition still to be satisfied if $\beta$ is sufficiently small. If $\beta$
is too large, (e.g. if $\beta > n_0^{-1}$, where $n_0$ is the minimum value of $n$ for which (\ref{eq:cofg1}) holds), then there are no choices of $n$,
$\hat{\epsilon}$, $\hat{\delta}$ that will satisfy (\ref{eq:cofg}) and (\ref{eq:cofgg}). On the other hand, $\beta$ can be large only if $S$ has important modes not included in ${\omega_1, \ldots, \omega_q}$. In practice, we use the multi-step procedure after the most important modes have been identified so that $\beta$ is small. For sufficiently small $\beta$, we do gain by taking the iterative
procedure. For example, assume that $\beta=10^{-2}$, for a signal of type $S = \phi_1+\phi_2$ with $N=1000$, $q=B=2$, $\delta=2^{-7}$,
$\epsilon=4 \cdot 10^{-4}$,
and with $n=3$, theoretically we would then use 450,000 samplings for the one-step
procedure, versus 150 samples for the iterative procedure. Note that we introduced the parameter $\beta$ only for expository
purposes. In practice, we simply continue with the process of identifying modes and roughly estimating their coefficients until our estimate of the residual signal is small; at that point, we switch to the above multi-step estimation procedure.
\subsection{Estimate Norms}
\label{sect:norm}
The basic principle to locate the label of the significant frequency is to estimate the energy of the new
signals obtained from isolation and group testing steps. The new signals are supported on only a small number of taps in
the time domain and have 98$\%$ of their energies concentrated on one mode. The original analysis in
\cite{GGIMS} only gave its loose theoretical bound. Here we find the empirical parameters, i.e., the number of
samples for norm estimation.
Here is a new scheme for estimating norms, which uses much fewer samples than the original one and still achieves good estimation.
It can ultimately be used to find the significant mode in conjunction with Group Testing and MSB, below.
\begin{algorithm} \label{alg:norm} {\sc Estimate Norms}
Input: signal $S$, failure probability $\delta$.
\begin{enumerate}
\item Initialize: the number of samples: $r=\lfloor 12.5 \ln (1/\delta) \rfloor $.
\item Take $r$ independent random samples from the signal $S$: $S(i_1), \ldots, S(i_r)$, where $r$ is a multiple of 5.
\item Return $ N \times$ ``60-th percentile of'' ${|S(i_1)|^2, \ldots, |S(i_r)|^2}$.
\end{enumerate}
\end{algorithm}
The following lemma presents the theoretical analysis of this algorithm.
\begin{lemma}\label{lem:norm}
If a signal $S$ is $93\%$ pure, the number of samples $r>12.5 \ln (1/\delta)$, the output of Algorithm {\em \ref{alg:norm}} gives an estimation $X$ of its energy
which exceeds $0.3 \|S\|^2$ with probability exceeding $1-\delta$.
\end{lemma}
\begin{proof}
Without loss of generality, suppose that $\|S\|=1$. Suppose the signal $S=a\phi_{\omega}+e$, where $|a|^2>0.93 \|S\|^2$, and $\phi_{omega}$ and $e$ are orthogonal.
We shall sample the signal $S$ independently for $r$ times, as stated in Algorithm \ref{alg:norm}. Note that we do not impose that
samples be taken at different time positions; with very small probability, the samples could coincide. Let $T=\{t:N|S(t)|^2< 0.3 \|S\|^2 \}$.
Hence, for any $t\in T$, we have $\sqrt{N} |S(t)|<\sqrt{0.3}=0.5477$. Also by the purity of $S$, we have $\|e\|^2 \leq 0.07$. Since $|S(t)|\geq |a \phi_{\omega}(t)|-|e(t)|$, we obtain
\begin{equation}
\sqrt{N}|e(t)|>|a|-\sqrt{N}|S(t)|.
\end{equation}
then for any $t\in T$,
\begin{equation}
\sqrt{N}|e(t)|>\sqrt{0.93}-\sqrt{0.3}.
\end{equation}
Therefore,
\begin{equation}
0.07N \geq N \|e\|^2 \geq N \sum_{t\in T}|e(t)|^2 \geq (\sqrt{0.93}-\sqrt{0.3})^2 |T|.
\end{equation}
It follows that
\begin{equation}
|T| \leq 0.403 N
\end{equation}
Let $\alpha = \frac{|T|}{N}$; the above inequality becomes
$0 \leq \alpha \leq 0.403$. \\
Consider now the characteristic function $\chi_T$ of the set $T$,
\begin{equation}
\chi_T(t) = \begin{cases}
1 & \text{if $t \in T$} \\
0 & \text{otherwise},
\end{cases}
\end{equation}
and define the random variable $X_T$ as $\chi_T(i)$, where $i$ is picked randomly. Then we have
\begin{equation}
E(X_T)=\frac{|T|}{N} \leq 0.403,
\end{equation}
and
\begin{equation}
E(e^{X_T z}) = e^0 Prob(\chi_T(i)=0) + e^z Prob(\chi_T(i)=1) = 1-\alpha + \alpha e^z.
\end{equation}
Suppose now we sample the signal $S$ $r$ times independently, and obtain $S(t_1), \ldots, S(t_r)$, where $t_1, \ldots, t_r \in [0,N]$. Take the \mbox{\it {60-th}} percentile of the numbers $N|S(t_1)|^2, \ldots, N|S(t_r)|^2$.
By Chernoff's standard argument, we have for $z>0$
\begin{align}
Prob \left ( \text{60-th percentile} < 0.3 \|S\|^2 \right ) & = Prob \left (0.6r\,\, \text{of the samples'} \,\,
t \,\, \text{belong to T} \right ) \nonumber \\
& = Prob(\chi_T(t_1)+ \ldots + \chi_T(t_r) > 0.6r) \nonumber \\
& \leq e^{-0.6rz} E(e^{z \sum_{j=1}^r \chi_T(t_j)} ) \nonumber \\
&= \left [ (1-\alpha) e^{-0.6z} + \alpha e^{0.4 z} \right ]^r.
\end{align}
Take $z=\ln (1.5(1-\alpha)/ \alpha)$, then
\begin{equation} \label{eq:norm}
(1-\alpha)e^{-0.6z} + \alpha e^{0.4z} = 1.96 \alpha^{0.6} (1-\alpha)^{0.4}.
\end{equation}
The right hand side of (\ref{eq:norm}) is increasing in $\alpha$ on the interval $[0, 0.403]$; since $\alpha \leq 0.403$,
we obtain an upper bound by substituting 0.403 for $\alpha$:\begin{eqnarray}
\left [ (1-\alpha) e^{-0.6z} + \alpha e^{0.4 z} \right ]^r = \left [ 1.96 \alpha^{0.6} (1-\alpha)^{0.4} \right ]^r \leq e^{-0.08r}.
\end{eqnarray}
So for $r \geq 12.5 \ln (1/\delta)$, we have
\begin{align}
Prob(\text{Output of Algorithm}\, 3.6 \geq 0.3 \|S\|^2)& =Prob( \text{60-th percentile of}\, N|S(t)|^2 \geq 0.3 \|S\|^2) \\ \nonumber
& \geq 1-\delta.
\end{align} \qquad\end{proof}
In practice, we often generate signals that are not so pure and thus need more samples for norm estimation.
Although the estimation is sometimes pretty far away from the true value, it gives a rough idea of where the significant mode might be. When we desire more accuracy,
a smaller constant $C$ in the number of samples $C\log(1/\delta)$ is chosen. In the statement of the algorithm, we choose $r$ to be a multiple of 5, so that
the \mbox{\it {60-th}} percentile would be well-defined. In practice, it works equally well to take $r$ that are not multiples
of 5 and to round down, taking the $\lfloor 3r/5 \rfloor$-th sample in an increasingly ordered set of samples.
We shall also need an upper bound on the outcome of Algorithm \ref{alg:norm}, which should hold regardless of whether
the signal $S$ is highly pure or not. This is provided by the next lemma, which proves that for general signals, Algorithm \ref{alg:norm} produces an estimation of the energy, that is less than $2\|S\|^2$ with high probability.
\begin{lemma} \label{lem:norm2}
Suppose Algorithm \ref{alg:norm} generates an estimation $X$ for $\|S\|^2$, then
\begin{equation}
Prob(X \geq 2\|S\|^2) \leq \left ( \frac{1 }{2} \right )^{0.144 \ln(1/\delta)} = \delta^{0.1}.
\end{equation}
\end{lemma}
\begin{proof}
Suppose $r$ independent random samples are $S(t_1), S(t_2), \ldots, S(t_r)$, then
\begin{equation}
Prob( N|S(t_i)|^2 \geq 2 \|S\|^2) \leq \frac{ N E(|S(t_i)|^2) }{ 2 \|S\|^2} = 1/2.
\end{equation}
Since $X$ is the 60-th percentile of the sequence $NS(t_1), \ldots, NS(t_r)$, with $r=0.36 \ln (1/\delta)$,
\begin{equation}
Prob( X \geq 2 \|S\|^2) \leq \left( Prob( N|S(t_i)|^2 \geq 2 \|S\|^2) \right)^{0.144 \ln(1/\delta)} \leq \left ( \frac{1 }{2} \right )^{0.144 \ln(1/\delta)} = \delta^{0.1}.
\end{equation}
\qquad\end{proof}
\subsection{Isolation}
\label{sect:isolation}
Isolation processes a signal $S$ and returns a new signal with significant frequency $\omega$, with 98$\%$ of
the energy concentrated on this mode. A frequency $\omega$ is called ``significant'' for $S$ , if
$|\hat{S}(\omega)|>\eta\|S\|^2$, where $\eta$ is a threshold, fixed by the implementation, which may be fairly
small. More precisely, the isolation step returns a series of signals $F_0, F_1, \ldots, F_{r}$, such that, with high probability,
$|\hat{F}_j(\omega)|^2 \geq 0.98 \|F_j\|^2$ for some $j$, that is, at least one of the $F_0, F_1, \ldots, F_{r}$ is $98\%$ pure.
Typically, not all of the $F_i$s are pure. We shall nevertheless apply the further steps of the algorithm to each of the
$F_i$s, since we don't know which one is pure. An impure $F_i$ may lead to a meaningless value for the putative mode
$\tilde{\omega}_i$ located in $F_i$. This is detected by the computation of the corresponding coefficients: only when
the coefficient corresponding to a mode is significant do we output the mode and its coefficient. Some impure signals might output an insignificant mode.
Hence, we estimate and compare their coefficients to check the significance of the modes.
Finally, we only output the modes with significant coefficients.
The discussion in \cite{GGIMS} proposes a B-tap box-car filter in the time domain, which corresponds to a
Dirichlet filter with width $\frac{N}{B}$ in the frequency domain. The whole frequency region would be covered
by random dilation and translations of this filter.
Notation: as in \cite{weaver}, we define a box-car filter $H_k$ as
$H_{k}(t)=\frac{\sqrt{N}}{2k+1}\chi_{[-k,k]}$, where $k \in \mathbb{N}$.
\begin{lemma}
\begin{enumerate}
\item For all \(k\),
\begin{equation}\label{ji}
\hat{H}_{k}(\omega) = \frac{1}{2k+1}\sum_{t=-k}^{k}e^{\frac{-2 \pi i \omega t}{N}}
= \frac{\sin(\pi(2k+1)\omega/N)}{(2k+1)\sin(\pi \omega/N)}.
\end{equation}
\item Notation: $H_{k,j}(t) = e^{2 \pi i j t /(2k+1)}H_k(t)$ in the time domain, which is equivalent to a shift of $\hat{H}_k(\omega)$ by $jN/(2k+1)$ in the
frequency domain.
\item Notation: Define $R_{\theta, \sigma}S(t)$ by $R_{\theta, \sigma}S(t)= e^{-2 \pi i \theta t/\sigma N} F(t/\sigma)$, so that
$\widehat{R_{\theta, \sigma} S}=\hat{S}(\sigma \omega + \theta)$., where $\widehat{R_{\theta,
\sigma}}$ is a dilation and shift operator in the frequency domain.
\end{enumerate}
\end{lemma}
More detailed description of the Box-car filter can be found in \cite{GGIMS}.
The isolation procedure in \cite{GGIMS} randomly permutes the signal $S$ and then convolves it with a shifted
version of $H_{k,j}$ to get a series of new signals $F_j = H_{k,j}* R_{\theta, \sigma} S$, where $j=0, \ldots, 2k$.
This scheme does not work well in practice. In the new version of the isolation steps, each $F_j = H_k
* R_{\theta_j, \sigma_j} S$ corresponds to different randomly generated dilation and modulation factors,
with $j=0, \ldots, \log(1/\delta)$, the parameters $\sigma_j$ and $N$ are relatively prime. These factors are taken at random between 0 and $N-1$. The following lemma is similar to Lemma 8 in \cite{GGIMS} for
the new isolation step, with more explicit values of the parameters.
\begin{lemma} {\em \cite{GGIMS}}
Let a signal $S$ and a number $\eta$ be given, and create $\log(1/\delta)$ new signals: $F_0, \ldots,
F_{\log(1/\delta)}$ with $F_j = H_k*R_{\theta_j, \sigma_j}S$, where $j=0, \ldots, \log (1/\delta)$. If $k \geq 12.25 (1-\eta)\pi^2/\eta$
, then for each $\omega$ such that $|\hat{S}(\omega)|^2 \geq \eta \|S\|^2$, there
exists some $j$ such that with high probability $1-\delta$, the new signal $F_j$ is $98\%$ pure.
\end{lemma}
\begin{proof}
Suppose $\sigma_j^{-1} (\omega - \theta_j)$ falls into the pass region of the $H_k$ filter, i.e., that $\left |\sigma_j^{-1} (\omega -
\theta_j \right | \leq \frac{N}{2(2k+1)}$.
We know that
\begin{equation}
\left | \hat{H}_k \left (\sigma_j^{-1} (\omega -
\theta_j) \right ) \right | \geq 2/\pi,
\end{equation}
so that
\begin{equation}
\left |\hat{F}_j \left (\sigma_j ^{-1} (\omega-\theta_j) \right ) \right |^2 \geq (2/\pi)^2 \left
|\hat{S}(\omega) \right |^2 \geq (2/\pi)^2 \eta \|S\|^2.
\end{equation}
greater than the average value, $1/(2k+1)$, of $|H^k|^2$.
Since $|\hat{H}_k(\sigma_j^{-1}(\omega-\theta_j))|^2$ is greater than the average value of $\hat{H}_k$, we have
\begin{equation}
\frac{\sum_{\omega' \neq \sigma_j^{-1}(\omega-\theta_j)} |\hat{H}_k(\omega')|^2}{N-1} \leq \frac{\|H_k\|^2}{N}=
\frac{1}{2k+1}.
\end{equation}
Moreover, $\sum_{\omega'' \neq \omega} |\hat{S}(\omega')|^2 \leq (1-\eta)\|S\|^2$. In
particular, $|\hat{S}(\omega')|^2 \leq (1-\eta)\|S\|^2$ if $\omega' \neq \omega$. We then have
\begin{equation}
E\left[\sum_{\omega'\neq\sigma_j^{-1}(\omega-\theta_j)} | \hat{F}_j(\omega')|^{2} \bigg |
-\frac{1}{2}N/(2k+1)\leq\sigma_j^{-1}(\omega-\theta_j)\leq \frac{1}{2}N/(2k+1)\right]\leq\frac{(1-\eta)\|
S\|^{2}}{2k+1}.
\end{equation}
Define $X$ to be the random variable
\begin{equation}
X= \left \{ \sum_{\omega' \neq \sigma_j^{-1}(\omega-\theta_j)} | \hat{F}_j(\omega')|^{2} \bigg |
-\frac{1}{2}N/(2k+1)\leq\sigma_j^{-1}(\omega-\theta_j)\leq \frac{1}{2}N/(2k+1) \right \}.
\end{equation}
For this random variable, we have
\begin{eqnarray}
Prob \left ( \frac{X}{|\hat{F}_{j}(\sigma_j^{-1}(\omega-\theta_j))|^{2}}\geq 1/49 \right ) &=Prob \left (X\geq|\hat{F}_{j}(\sigma_j^{-1}(\omega-\theta_j))|^{2}/49\right ) \nonumber \\
& \leq\frac{E(X)}{|\hat{F}_{j}(\sigma_j^{-1}(\omega-\theta_j))|^{2}/49}\leq\frac{49(1-\eta)\pi^{2}}{4 \eta (2k+1)}.
\end{eqnarray}
Since $k \geq 12.25 (1-\eta)\pi^2/\eta$, the right hand side of (4.37) is $\leq 1/2$, meaning that the signal $F_j$ is 98$\%$ pure with
probability $\geq 1/2$.
The success probability, i.e., the probability of obtaining at least one $F_j$ that is 98$\%$ pure, can be boosted from $\frac{1}{2}$
to probability $1-\delta$ by repeating $O(\log(1/\delta))$ times, i.e., generating $O(\log(1/\delta))$ signals.
\qquad\end{proof}
The above lemma gives a lower bound for the filter width. Obviously, the larger the width in the time domain,
the higher the probability that the frequency will be successfully isolated. However, a larger width leads to
more evaluations of the function and therefore more time for each isolation step. One needs to balance carefully
between the computational time for each iteration step and the total number of iterations.
Based on several numerical experiments, we found that a very narrow filter is preferable and gives good
performance; for instance, the filter with three-tap width, i.e., $k=1$
works best for a signal with 2 modes. For the choice $k=4$, the algorithm ends after fewer iterations; however,
each iteration takes much more time. The choice of a 9-tap width filter makes the code four times
slower in total.
The filter width is weakly determined by the number of modes in the signal, not by the length of the signal.
Through experimentation, we found that when the number of modes is less than 8, the 3-tap width filter works
very well; as the number of modes increases, larger width filters are better. Numerical experiments suggests
a sublinear relationship between the width of the filter and the number of modes; in our experiments a 5-tap filter still
sufficed for $B=64$.
\subsection{Group Testing}
\label{sect:group}
After the isolation returns several signals, at least one of which is 98$\%$ pure with high probability, group
testing aims at finding the most significant mode for each. We use a procedure called Most
Significant Bit (MSB) to approach the mode recursively.
In each MSB step, we use a Box-car filter $H_k$ to subdivide the whole region into $2k+1$ subregions. By
estimating the energies and comparing the estimates for all these new signals, we find the one with maximum
energy, and we exclude those that have estimated energies much smaller than this maximum energy. We then repeat
on the remaining region, a more precisely on the region obtained by removing the largest chain of excluded
intervals; we dilate so that this new region fills the whole original interval, and split again. The successive
outputs of the retained region gives an increasingly good approximation to the dominant frequencies. The following are the Group testing procedures:
\begin{algorithm} \label{alg:group}{\sc Group Testing} \\
Input: signal $F$, the length $N$ of the signal $F$. \\
Initialize: set the signal $F$ to $F_0$, iterative step $i=0$, the length $N$ of the signal, the accumulation factor $q=1$. \\
In the $i$th iteration,
\begin{enumerate}
\item If $q \geq N $, then return 0.
\item Find the most significant bit $v$ and the number of significant intervals $c$ by the procedure MSB.
\item Update $i=i+1$, modulate the signal $F_i$ by $\frac{(v+0.5)N}{4(2k+1)}$ and dilate it by a factor of $4(2k+1)/c$. Store it in $F_{i+1}$.
\item Call the Group testing again with the new signal $F_{i}$, store its result in $g$.
\item Update the accumulation factor $q = q * 4(2k+1)/c$.
\item If $g> N/2$, then $g = g -N$.
\item return $mod( \lfloor \frac{cg}{4(2k+1)}+ \frac{(v+1/2)N}{4(2k+1)}+0.5 \rfloor, N)$;
\end{enumerate}
\end{algorithm}
The MSB procedure is as follows.
\begin{algorithm} \label{alg:msb}{\sc MSB (Most Significant Bit)} \\
\text{}\hspace{10mm} Input: signal $F$ with length $N$, a threshold $0<\eta<1$.
\begin{enumerate}
\item Get a series of new signals $G_j(t) =F(t) \star (e^{2 \pi i j t/4(2k+1)} H_k )$, $j=0, \ldots, 8k+4$. That is, each signal $G_j$ concentrates on the
pass region $[ \frac{(j-1/2)N}{4(2k+1)}, \frac{(j+1/2)N}{4(2k+1)}]:=pass_j$.
\item Estimate the energies $e_j$ of $G_j$, $j=0, \ldots, 8k+4$.
\item Let $l$ be the index for the signal with the maximum energy.
\item Compare the energies of all other signals with the $l$th signal. If $e_i < \eta e_l$
, label it as an interval with small energy.
\item Take the center $v_s$ of the longest chain of consecutive small energy intervals, suppose there are $c_s$ intervals altogether in this chain.
\item The center of the large energy intervals is $v = 4(2k+1)-v_s$, the number of intervals with large energy is $c = 4(2k+1)-c_s$.
\item If $c>4(2k+1)/2$, then do the original MSB {\em \cite{GGIMS}} to get $v$ and set $c=2$, and $v=
center\, of\, the\, interval\, with\, maximal\, energy$.
\item Output the dilation factor $c$ and the most significant bit $v$.
\end{enumerate}
\end{algorithm}
\begin{lemma}
Given a signal \(F\) with \(98\)$\%$ purity, suppose \(G_j(t) =F * e^{2 \pi i j t /4(2k+1)} H_k(t)\). If \(k \geq 2
\), then Algorithm {\em \ref{alg:group}} can find the significant frequency $\omega$ of the signal $F$ with high
probability.
\end{lemma}
\begin{proof}
Suppose the filter width of $H_k$ is $2k+1$. Observe that, for some $j$, $0\leq j \leq 4(2k+1)$, $\omega \in pass_j$.
Without loss of generality, assume $j=0$. Now consider the signal $G_0$. Since $\omega \in pass_0$, the Fourier coefficient
$\hat{G}_0(\omega)$ satisfies
\begin{align}
|\hat{G_0}(\omega)|^2 & \geq \left( \frac{\sin(\pi/8)}{(2k+1)sin(\pi/8(2k+1))} \right ) ^2 |\hat{F}(\omega)|^2 \\ \nonumber
& \geq \left( \frac{\sin(\pi/8)}{(2k+1)\sin(\pi/8(2k+1))}\right)^2 (0.98) \|F\|^2 \\ \nonumber
& \geq 0.9744^2 \cdot 0.98 \|F\|^2 \approx 0.93 \|F\|^2.
\end{align}
for all $k>0$. It follows from Lemma \ref{lem:norm}, that the output of Algorithm \ref{alg:norm}, applies to $G_0$, estimate that is at least
\begin{equation}
0.3\|G_0\|^2 \geq 0.3 |\hat{G_0}(\omega)|^2 \geq 0.3 \cdot 0.98 \left( \frac{\sin(\pi/8)}{(2k+1)\sin(\pi/8(2k+1))}\right)^2 \|F\|^2 .
\end{equation}
On the other hand, now consider $G_5$. Note that
\begin{align}
|\hat{G_5}(\omega)| = |\hat{F}(\omega)||\widehat{H_{k}}(\omega)| & \leq \frac{1}{(2k+1)\sin(9 \pi/8(2k+1))} |\hat{F}(\omega)| \\ \nonumber
&\leq \frac{1}{(2k+1)\sin(9\pi/8(2k+1))} \|F\|.
\end{align}
Also,
$\| G_{5}\|^{2}-|\hat{G}_{5}(\omega)|^{2} \leq 0.02\| F\|^{2}$, because $F$ is $98\%$ pure. Thus
\begin{equation}
\| G_{5}\|^{2}\leq|\hat{G}_{5}(\omega)|^{2}+0.02\| F\|^{2}=\left( \left( \frac{1}{(2k+1)\sin(9 \pi/8(2k+1))}\right)^2 +0.02 \right) \| F\|^{2}.
\end{equation}
By Lemma \ref{lem:norm2}, if we use Algorithm \ref{alg:norm}, the estimation result for $G_5$ will be at most $2 \| G_{5}\|^{2}$ with high probability.
It is easy to show that the inequality
\begin{equation}
0.294 \left( \frac{\sin(\pi/8)}{(2k+1)\sin(\pi/8(2k+1))}\right)^2 \geq 2 \left( \frac{1}{(2k+1)\sin(9 \pi/8(2k+1))}\right)^2 +0.04
\end{equation}
holds for all $k>0$. The same argument applies to $G_j$ with $5\leq j\leq 4(2k+1)-5$. It follows that, with high probability, the result of applying Algorithm
\ref{alg:norm} to $G_0$ will give a result that exceeds the result obtained by applying Algorithm \ref{alg:norm} to $G_j$ with $5 \leq j \leq 4(2k+1)-5$.
In general, if the pass region is at some $j_{0}$, we can compare $\| G_{j_{0}}\|^{2}$with
$\| G_{j}\|^{2}$for all $|j-j_{0}|\geq5$. If there is some $j_{0}$ for which
the estimation of $\| G_{j_{0}}\|^{2}$ is apparently larger than
$\| G_{j}\|^{2}$, then we conclude $\omega\notin pass_{j}$; otherwise, possibly
$\omega\in pass_{j}$. By the above argument, we can eliminate
$4(2k+1)-9$ consecutive pass regions out of the $4(2k+1)$, leaving a cyclic interval
of length at most $\frac{9N}{4(2k+1)}$. In order for the residual region to be smaller or equal
to half of the whole region, we need $4(2k+1) \geq 18$, which is equivalent to the condition $k\geq 2$.
In the recursive steps, let $P$ denote a cyclic interval with size
at most $\frac{9N}{4(2k+1)}$ that includes all the possibilities for $\omega$.
Let $v$ denote its center. Then generate a new signal $F_{1}(t)=e^{-2\pi ivt/N}F(t)$; this
is a shift of the spectrum of $F$ by $-v$. Thus the frequency $\omega-v$
is the biggest frequency of $F_{1}(t)$., which is in the range of
$-\frac{4.5N}{4(2k+1))}$ to $+\frac{4.5N}{4(2k+1))}$. We will now seek $\omega-v$.
Since we rule out a fraction of $\frac{(8k-5)N}{4(2k+1)}$ length of the whole region, we may
dilate the remainder by $\lfloor 4(2k+1)/9 \rfloor$, which can be accomplished in
the time domain by dilating $F_{1}$ by $\frac{9}{4(2k+1)}$. Thus the interval of
length just less than $\frac{9N}{4(2k+1)}$ known to contain $\omega-v$ is dilated
to the alternate positions in an interval of length just less than
$N$. We then rule out again $\frac{8k-5}{4(2k+1)}$ of this dilated frequency
domain, leaving a remainder of length at most $\frac{9}{4(2k+1)}$ length. Then we undo the dilation,
getting an interval of length just less than $\frac{9N}{(4(2k+1))}$, centered
at some $v_{2}$, which is the second most significant bit of $\omega$ in
a number base $\lfloor \frac{4(2k+1)}{9} \rfloor$. We would repeat this process to get the other bits
of $\omega$. By getting a series of $v_{1},\ldots,v_{\lfloor\log_{4(2k+1)/9}N\rfloor+1}$,
we can recover the $\omega$. \qquad\end{proof}
In fact, a narrower filter with a larger shift width than $\frac{N}{4(2k+1)}$ works fine and makes the algorithm faster in practice.
Heuristically, we find that the optimal number of
taps for small $B$ cases is 3. Suppose the MSB filter width is 3 and each MSB rules out 2 intervals out of 3,
then the total number of recursive group test is $\log_3 N$. Then the computational cost is $3 \log_3 N$ norm estimations and $2 \log_3 N$ comparisons. Numerical experiments suggests that $k$ is probably linear in $\log B$. The shift width we use in practice is $\frac{N}{2k+1}$.
We find that the output of group testing in both the original and the present version of RA$\ell$SFA might differ from the true mode by one place.
We suspect that the reason is that all the float operations and the conversion to integers introduce and accumulate some error into the final
frequency. As a solution, the coefficients of nearby neighbors are also estimated roughly to determine the true significant modes.
\subsection{Sample from a transformed signal}
\label{sect:sample}
A key issue in the implementation consists of
obtaining information (by sampling) from a signal after it has been dilated,
modulated, or even convolved. We briefly discuss here how to carry out this
sampling in discrete signals.
First, we consider a dilated and modulated signal,
for example, in the isolation procedure which uses $R_{\theta,
\sigma}S(t)=e^{-2\pi i \theta t/\sigma N}S(t/\sigma)$, which is equivalent to
$\widehat{(R_{\theta, \sigma}S)}(\omega) = \hat{S}(\sigma \omega + \theta)$ in
the frequency domain. Here $\sigma$ and $\theta$ are chosen uniformly and
randomly, from $0$ to $N-1$ for $\theta$, and from $1$ to $N-1$ for $\sigma$.
The sample $R_{\theta, \sigma}F(t)$, where $t \in \lbrace 0, 1, \ldots,
N-1\rbrace$,is then $e^{-2 \pi i \theta t /\sigma N}(R_{\theta, \sigma})F(t)=e^{-2 \pi i
\theta t /\sigma N}F(\sigma^{*}t)$, where $\sigma^*$ is chosen so that $\sigma^{*}
\sigma =1(mod\, N)$. If $N$ is prime, then we can always find (a unique value
for) $\sigma^{*}$ for arbitrary $\sigma$; if $N$ is not prime, $\sigma^{*}$ may
fail to exist for some choices of $\sigma$. Our program uses the Euclidean
algorithm to determine $\sigma^{*}$; when $N$ is not prime and $\sigma$ and $N$
are not co-prime, the resulting candidates for $\sigma^{*}$ are not correct and
may lead to estimates for the modes that are incorrect; these mistakes are
detected automatically by the algorithm when it estimates the corresponding
coefficient and finds it to be below threshold.
We also need to sample from convolved signals, e.g. $S*H_k(t)$. Because $H_k$ has only $2k+1$ taps, only
$2k+1$ points contribute to the calculation of the convolution. Since
$S*H_k(t)=\sum_{i=-k}^k H_k(i) S(t-i)$, we need only the values $S(t-i)$,
$i=-k,\ldots,k$, all of which we sample.
\subsection{Extension to a Higher Dimensional Signal}
\label{sect:twodim}
The original RA$\ell$SFA discusses only the one dimensional case. As explained earlier, it is of particular interest
to extend RA$\ell$SFA to higher dimensional cases because there its advantage over the FFT is more pronounced.
In $d$ dimensions, the Fourier basis function is
\begin{equation}
\phi_{\vec \omega (\vec x) } = \phi_{\omega_1, \ldots, \omega_d}(x_1, \ldots, x_d) = N^{-\frac{d}{2}} e^{i 2 \pi
\omega_1 x_1 / N +\ldots +i 2 \pi \omega_d x_d / N } = N^{-\frac{d}{2}} e^{i 2 \pi \vec \omega_i \vec x_i /N};
\end{equation}
the representation of a signal is
\begin{equation}
S(x_1, \ldots, x_d) = \sum_{i=1} ^N {c_i \phi_{\omega_{i,1},\ldots,\omega_{i,d}}}.
\end{equation}
Suppose the dimension of the signal is $d$, denote $\vec x = (x_1, x_2, \ldots, x_d)$, $\vec \omega =
(\omega_1, \ldots, \omega_d)$.
The total scheme remains much the same as in one dimension:
\begin{algorithm} \label{alg:twodim}{\sc Total Scheme in $d$ dimensions} \\
Input: signal $S$, the number of nonzero modes $B$ or its upper bound, accuracy factor $\epsilon$, success probability $1-\delta$, an upper bound of the signal energy $M$, the standard deviation of the white Gaussian noise $\sigma$.
\begin{enumerate}
\item Initialize the representation signal $R$ to 0, set the maximum number of iterations $T=B\log(N)\log(\delta)/\epsilon^{2}$,.
\item Test whether $\Vert S-R \Vert^2 \leq \iota \|R\|^2$. If yes, return the representation signal $R$ and the whole algorithm ends; else go to step 3.
\item Locate Fourier Modes $\vec \omega$ for the signal $S-R$ by the new isolation and group test procedures.
\item Estimate Fourier Coefficients at $\vec \omega$: $\widehat{(S-R)}(\vec \omega)$.
\item If the total number of iterations is less than $T$, go to 2; else return the representation $R$.
\end{enumerate}
\end{algorithm}
The most important modification with respect to the one dimensional case lies in the procedure to carry out step 3 of Algorithm \ref{alg:twodim}. We adapt the technique for
frequency identification to fit the high dimensional case;
it is given by the following procedure.
\begin{algorithm} {\sc Locate the Fourier mode in $d$ dimensions}\label{locate2d}
Input: signal $S$, accuracy factor $\epsilon$, success probability $1-\delta$, an upper bound of the signal energy $M$.
\begin{enumerate}
\item Random permutations in d dimension.
\item Isolate in one (arbitrarily picked) dimension $i$ to get a new signal $F(t)=S*H_k(t)$.
\item For each dimension $i'$, find the $i'$th component $\vec \omega^*_{i'}$ of the significant frequency by Group Testing for the signal $F$ in the $i'$th dimension.
\item Finally, estimate the Fourier coefficients in the frequency $\vec \omega =
(\omega^*_0, \ldots, \omega^*_{d-1})$.
Keep the frequency d-tuple if its Fourier coefficient is large.
\end{enumerate}
\end{algorithm}
Note that the computational cost of the above algorithm is quadratic in the number of dimensions. The
permutation involves a $d \times d$ matrix\footnote{ Note that generalizing to $d$ dimensions our 1-dimensional
practice of checking not only the central frequency found, but also nearby neighbors, would make this algorithm
exponential in $d$, which is acceptable for small $d$. For large $d$, we expect it would suffice to check
a fixed number of randomly picked nearby neighbors, removing the exponential nature of this technical feature.}
The group test procedure in each dimension processes the {\it
same} isolation signal. If a filter with $B$ taps is used for the isolation, then it captures at least one
significant frequency in the pass region with probability $1/B$. The basic idea behind this procedure is that,
because of the sparseness of the Fourier representation, cutting the frequency domain into slices of width $1/B$
in 1 dimension, leaving the other dimensions untouched, leads to, with positive probability, a separation of the
important modes into different slices. After this essentially 1-dimensional isolation, we only need to identify
the coordinates of the isolated frequency mode. After isolation, we assume $F(\vec x)=A e^{2 \pi i \vec \omega
\cdot \vec x/N}$, where $A$ and $\vec \omega$ are unknown. To find $\omega_{j'}$, we sample in the $j'$-th
coordinate only, keeping $x_1, \ldots, x_{j'-1}, x_{j'+1},\ldots, x_d$ fixed, so that (for this step) $F(\vec
x)$ can be viewed as $A e^{2 \pi i \vec \omega \cdot \vec x/N}= \tilde{A}e^{2 \pi i \omega_{j'} x_{j'}/N} $,
where $\tilde{A}=Ae^{2 \pi i (x_1 \omega_1 + \ldots + x_{j'-1}\omega{j'-1}+x_{j'+1}\omega{j'+1}+ \ldots+x_d
\omega_d)}$, remains the same for different $x_{j'}$ and has the same absolute value as $A$, which we can do in
each dimension separately by the following argument.
If we just repeated the 1-dimensional technique in each dimension, that is, carried out isolation in each of the $d$ dimensions sequentially,
the time cost would be exponential in the dimension $d$. We discuss now in some detail the steps 1, 2, 3 of Algorithm \ref{locate2d}.
\subsubsection{Random Permutations}
\label{sect:permute}
In one dimensional RA$\ell$SFA, the isolation part includes random permutations and the construction of signals with one frequency dominant.
However, the situation is more complicated in higher dimensions, which is why we separated out the permutation step in the algorithm.
Recall that in one dimension, the signal is dilated and modulated randomly in order to
separate possibly neighboring frequencies. In higher dimensions, different modes can have identical coordinates in some of
the dimensions; they would continue to coincide in these dimensions if we just applied ``diagonal'' dilation, i.e., if we carried
out dilation and modulation sequentially in the different dimensions. To separate such modes, we need to use
random matrices. We transform any point
$(x_1,x_2,\ldots, x_d)$ into $(y_1,\ldots, y_d)$ given by
\begin{equation}
\left(\begin{array}{c}
y_1\\
\vdots \\
y_d \end{array}\right)=\left(\begin{array}{cccc}
a_{11} & a_{12}& \ldots & a_{1d}\\
\vdots & \vdots & \vdots& \vdots \\
a_{d1} & a_{d2}& \ldots & a_{dd}\end{array}\right)\left(\begin{array}{c}
x_1\\
\vdots \\
x_d \end{array}\right)+\left(\begin{array}{c}
b_1\\
\vdots \\
b_d \end{array}\right)
\end{equation}
where $A=\left (a_{ij}\right )$ is a random and invertible matrix, the $a_{ij}$ and the $b_i$ are chosen randomly, uniformly and independently,
and the arithmetic is modulo $N$.
For example, if $d=2, N=7, a_{11}=1, a_{12}=3, a_{21}=5, a_{22}=2, b_1=0, b_2=5 $, that is,
\begin{equation}
\left(\begin{array}{c}
y_1\\
y_2 \end{array}\right)=\left(\begin{array}{cc}
1 & 3\\
5 & 2 \end{array}\right)\left(\begin{array}{c}
x_1\\
x_2 \end{array}\right)+\left(\begin{array}{c}
0\\
5 \end{array}\right)
\end{equation}
the point $(1,2)$ gets mapped to
$(0,0)$, $(1,3)$ to $(3,2)$, and $(0,3)$ to $(2,4)$: even though points $(1,2)$ and $(1,3)$ have the same first coordinate, their
images don't share a coordinate; the same happens with points $(1,3)$ and $(0,3)$. For each dimension $i'$, the $i'$th components of frequencies are mapped by pairwise
independent permutations. Even adjacent points that differ in only one coordinate are destined to be separate with high
probability after these random permutations.
\subsubsection{Isolation}
\label{sect:twodimisolation}
After the random permutations, the high dimensional version of isolation can construct a sequence
$F_{0},F_{1},\ldots$ of signals, such that , for some j, $|\hat{F_{j}}(\omega^{'})|^{2} \geq 0.98\| F\|^{2}$.
\begin{algorithm} \label{alg:twodimiso}{\sc High Dimensional Isolation} \\
\text{}\hspace{10mm} Choose an arbitrary dimension $i$.
\begin{enumerate}
\item Filter on the dimension $i$ and leave all other dimensions alone, get the signal
\begin{equation}
F =S \star H_{k},
\end{equation}
where $H_{k}=\frac{\sqrt{N}}{2k+1}\chi_{[-k,k]}$ filters on the dimension $i$; the other dimensions are not affected.
\item Output new signals $F$ to be used in the Group Testing.
\end{enumerate}
\end{algorithm}
\subsubsection{Group Testing for Each Dimension}
\label{sect:twodimgroup}
After the random permutation and isolation, we expect a $d$-dimensional signal with most of its energy
concentrated on one mode. The isolation
step effectively separates the $d$-dimensional frequency domain in a number of $d$-dimensional slices. Group
testing has to subdivide these slices.
One naive method is to apply $d$ dimensional filters in group testing, concentrating on $d$-dimensional cubic subregions in group testing that cover the whole area.
However, this leads to more cost. If the number of taps of this filter in one dimension is $2k+1$, we
obtain $(2k+1)^{d}$ subregions. Estimating the energies of all subregions slows down the total running time.
Consequently we instead locate each component of the significant frequency label
separately. That is, we only use a filter to focus on one dimension and leave other dimensions alone.
The energy of $2k+1$ regions are computed in every dimension. Hence, we
need to estimate the norm of $d(2k+1)$ intervals in total. This makes Group Testing linear in the number of dimensions,
instead of exponential as in the naive method.
Here is the procedure in Group Test:
\begin{algorithm} \label{alg:twodimgroup}{\sc High Dimensional Group Test} \\
\text{}\hspace{10mm} For $i'=1,\ldots, d$
\begin{enumerate}
\item Construct signals $\tilde{G}_j^{(i')}=F(t) * ( e^{2 \pi i j t_{i'}/(2l+1)} H_{l})$, $j=1,\ldots, 2l+1$,
where $H_l$ filters on $i'$th dimension and leave all other dimensions alone;
\item Estimate and compare the energy of each $\tilde{G}_j^{(i')}$, $j=1,\ldots, 2l+1$, use the similar procedure in one
dimensional group testing procedure. Find the candidates $\omega^{*}_{i'}$.
\end{enumerate}
\end{algorithm}
The reader may wonder how sampling works out for this $d$-dimensional algorithm. In Algorithm \ref{alg:twodimgroup}, we will need to
sample $\tilde{G}_j^{(i')} $ (which is the convolution of the (permuted version of) signal $S$ with 2 filters) to estimate its
energy; because filtering is done only in the $i'$-th dimension, we shall sample
$\tilde{G}_j^{(i')}(x_1, \ldots, x_{i'-1}, x_{i'}, x_{i'+1}, \ldots, x_d)$ for different $x_{i'}$, but keeping the other $x_j$ fixed, where $j \neq i'$.
The signal $F$ itself comes from the Isolation step, in which we filter in direction $i$, for which $S$ needs to be sampled, in this dimension only.
Together, for each choices $i'$ in Algorithm \ref{alg:twodimiso} and \ref{alg:twodimgroup}, this implies we have $(2k+1)\times(2l+1)$ different samples
of (the permuted version of) $S$, in which all but the $i$th coordinates of the samples $\vec x$ are identical.
\section{Conclusion}
We provide both theoretical and experimental evidence to support the advantage of the implementation of RA$\ell$SFA proposed here over the original one sketched in \cite{GGIMS}. Moreover, we extend RA$\ell$SFA to high dimensional cases. For functions with few, dominant Fourier modes, RA$\ell$SFA outperforms the FFT as $N$ increases. We expect that RA$\ell$SFA will be useful as a substitute for the FFT in potential applications that require processing such sparse signals or computing $B$-term approximations. This paper is just the beginning of a series of our papers and researches, many of which are in preparation. For example, the strong dependence of running time on the number of modes $B$ will be further lessened, and thus the algorithm would work for more interesting signals \cite{GMS}. Also, the application of RA$\ell$SFA in multiscale problems will be discussed in \cite{ZDR}.
\section*{Acknowledgments}
For discussions that were a great help, we would like to thank Bjorn Engquist, Weinan E, Olof Runborg, and Josko Plazonic.
|
{
"timestamp": "2005-06-10T05:05:50",
"yymm": "0411",
"arxiv_id": "math/0411102",
"language": "en",
"url": "https://arxiv.org/abs/math/0411102"
}
|
\section{\label{sec:intro}Introduction}
Crystal-field (CF) theory\cite{newmanbook} is one of the most
powerful theoretical methods to deal with the magnetic properties
of rare-earth (RE) and actinide (An) ions, and Stevens' operator
equivalents formalism is still the most commonly used to analyze
experimental data due to its simplicity. Unfortunately, this
approach concentrates only on the CF splittings within the
lowest-lying $^{2S+1}L_{J}$ multiplet of the considered ion,
completely neglecting the contributions of excited multiplets
(``$J$ mixing''). Although the task of diagonalizing the large
matrices related to the full $f^{n}$ configuration, including
different $J$ multiplets, is relatively easy to perform
numerically by means of today's computers, Stevens' approach often
makes it possible to obtain analytical expressions for physical
quantities of interest for systems of sufficiently high symmetry,
thus leading to a deeper insight on the physics of several
compounds.
In the present paper, we discuss a perturbative approach which
retains the validity of Stevens' formalism while correctly taking
into account $J$-mixing effects. This method, which has led to
interesting results for transition-metal (TM) based molecular
clusters\cite{liviotti,prlfe8} and ferromagnetic exchange-driven
RE-TM intermetallic compounds\cite{magnani} is now applied to
evaluate the intramultiplet CF splittings in light An and RE ions.
We exploit the method to analyze the CF of actinide dioxides.
These large-gap semiconductors are among the most studied actinide
compounds. Although $f$-electrons are well localized, the
complexity of the magnetic Hamiltonian, which includes CF and
magnetoelastic single-ion interactions, phonon-transmitted
quadrupolar interactions, and multipolar superexchange couplings
between neighboring ions, leads to a number of interesting and
unusual physical phenomena. Among them, we mention the proposed
octupolar phase transition in NpO$_2$,\cite{oct1,oct2} the
observed CF-phonon bound states in NpO$_2$,\cite{bound} and the
peculiar static and dynamic phenomena produced by magnetoelastic
interactions in UO$_2$,\cite{sasaki,cowley,uo299} some of which
are not yet fully understood.
The CF potential is the fundamental building block of any
theoretical model of the properties of dioxides, since this
influences the single-ion behavior to a large extent. In
particular, it determines which degrees of freedom of the $f$
shell are left unquenched and the size of the corresponding
multipole moments, which account for the low-$T$ physical
properties. Most of the published theoretical approaches are based
on the above-mentioned Stevens' treatment of the CF, which
includes only the lowest Russell-Saunders or Intermediate-Coupling
multiplet of the ion. If one takes as starting point the CF of
UO$_2$, on which very detailed information is available by
inelastic neutron scattering (INS) experiments,\cite{amoretti}
then scaling the CF of UO$_2$ within the Stevens' framework (to
take into account the different ionic radii) provides a good CF
model for NpO$_2$. However, the same scheme applied to PuO$_2$ is
only qualitatively satisfactory, since it reproduces the correct
level sequence but it underestimates the observed energy
splitting.
Moreover, this approach is not internally consistent because the
so-obtained CF parameters yield different results when additional
ionic multiplets are included in the calculation. On the other
hand, the increased complexity of $J$ mixing calculations makes
particularly hard to find CF parameter sets working consistently
over the various compounds. Indeed, different sets have been
proposed so far for dioxides within $J$ mixing calculations. In
particular, in NpO$_2$ two distinct and equally good sets of
parameters had been obtained.\cite{amorettinpo2}
By our perturbative $J$ mixing approach, we have been able to
obtain a unique set which works well over all the considered
compounds.
\section{\label{sec:model}The perturbative $J$-mixing model}
Following Ref.~\onlinecite{slichter}, the total free-ion and
crystal-field Hamiltonian $H=H_{FI}+H_{CF}$, with
\begin{equation}
H_{CF}=\sum_{k,q}B_{k}^{q}C_{q}^{(k)} \label{hcfgeneral}
\end{equation}
can be rewritten in the form
\begin{equation}
H=H_{0}+H_{1}+H_{2},
\end{equation}
where in the present case $H_{0}$ coincides with $H_{FI}$ and
$H_{1}$ and $H_{2}$ are chosen so that the former has nonzero
matrix elements only between states belonging the the same
$^{2S+1}L_{J}$ multiplet. It is possible to define a Hermitian
operator $\Omega$ such that the matrix element of the transformed
Hamiltonian $H^{\prime}=e^{-i\Omega}He^{i\Omega}$ are very small
in the off-diagonal blocks, thus restoring the possibility to use
an isolated-multiplet approach. In this framework,\cite{slichter}
\begin{equation}
\left<\alpha J M\left|H^{\prime}\right|\alpha J M^{\prime}\right>=
E_{0 \alpha J}\delta_{MM^{\prime}}+
\left<\alpha J M\left|H_{1}\right|\alpha J M^{\prime}\right>
\label{daslichter}
\end{equation}
\[
-
\sum_{\alpha^{\prime \prime} J^{\prime \prime} M^{\prime \prime}}
\frac{\left<\alpha J M\left|H_{2}\right|\alpha^{\prime \prime} J^{\prime \prime} M^{\prime \prime}\right>
\left<\alpha^{\prime \prime} J^{\prime \prime} M^{\prime \prime}\left|H_{2}\right|\alpha J M^{\prime}\right>}
{E_{0 \alpha^{\prime \prime} J^{\prime \prime}}-E_{0 \alpha J}},
\]
where $\alpha, J$ label free ion manifolds and $E_{0 \alpha J}$
are the eigenvalues of $H_0$. For clarity, in the following we
will label the states as in the Russell-Saunders scheme, where
$\alpha$ coincides with $(L,S)$ and any additional quantum number
necessary to identify the terms; yet the actual calculation will
include Intermediate Coupling corrections to the eigenfunctions.
Once we limit our calculations to the ground $J$ multiplet only,
the first term on the right-hand side of the above equation
represents a uniform energy shift of the whole multiplet, while
the second term is the usual ground-multiplet CF Hamiltonian $H^{(J)}$.
The effect of $J$-mixing is accounted for by the third term, which
will be considered as an extra contribution to the ground-multiplet
Hamiltonian and labelled $H^{(J)}_{mix}$ (we maintain the
redundant superscript $(J)$ notation in order to emphasize that
the newly obtained $J$-mixing Hamiltonian {\it also} acts on the
ground multiplet {\it only}).
In the case of light actinides, the most important $J$ mixing contribution comes from
the two lowest $J$-multiplets, i.e. $^{2S+1}L_{J}$ and $^{2S+1}L_{J+1}$.
Therefore, for the sake of simplicity we restrict our analysis to the case of these two multiplets,
separated by an energy gap $\Delta$ by the spin-orbit
interaction. From Eq.~\ref{daslichter}, $H^{(J)}_{mix}$ can be written as
\[
\left< J M \left| H^{(J)}_{mix} \right| J M^{\prime} \right> = -
\sum_{M^{\prime \prime}} \sum_{k,q}\sum_{k^{\prime},q^{\prime}}
B_{k}^{q} B_{k{\prime}}^{q{\prime}}
\]
\begin{equation}
\times \frac{
\left< J M \left| C_{q}^{(k)} \right| J+1 M^{\prime \prime} \right>
\left< J+1 M^{\prime \prime} \left| C_{q^{\prime}}^{(k^{\prime})} \right| J M^{\prime} \right>}
{\Delta}.
\label{mix1}
\end{equation}
The Wigner-Eckart theorem, in the form
\begin{equation}
\left\langle J_{1}M_{1}\left| C_{q}^{(k)}\right| J_{2}M_{2}\right\rangle
=\left( -1\right) ^{J_{1}-M_{1}}\left\langle J_{1}\left\| C^{(k)}\right\| J_{2}
\right\rangle \left(
\begin{array}{ccc}
J_{1} & k & J_{2} \\
-M_{1} & q & M_{2}
\end{array}
\right)
\label{wigeck}
\end{equation}
allows us to get rid of the sum over $M^{\prime \prime}$ in
Eq.~\ref{mix1}, since the $3j$ symbol in Eq.~\ref{wigeck} equals
zero if $M_{2} \neq M_{1} - q$. The products of $3j$ symbols can
be rewritten as linear combinations of matrix elements of Stevens
operators $O_{k}^{q}$,\cite{liviotti,magnani} so that
\begin{equation}
H^{(J)}_{mix} = \sum_{k,q,k^{\prime},q^{\prime}}
\frac{B_{k}^{q} B_{k^{\prime}}^{q^{\prime}}}{\Delta}M_{k+k^{\prime}}^{q+q{\prime}}
\end{equation}
where the ``mixing operators'' $M_{n}^{m}$ are\cite{liviotti}
\begin{equation}
M_{n}^{m}=\sum_{p=0}^{n} c_{n,p}^{(m)} O_{p}^{m}
\end{equation}
with conveniently defined $c_{n,p}^{(m)}$ coefficients.
Let us consider the simple but important case of cubic symmetry,
for which the CF Hamiltonian (\ref{hcfgeneral}) has the form
\begin{equation}
H_{CF}=B_{4}^{0}\left[ C_{0}^{(4)}+\sqrt{\frac{5}{14}}\left(
C_{4}^{(4)}+C_{-4}^{(4)}\right) \right]
+B_{6}^{0}\left[ C_{0}^{(6)}-\sqrt{%
\frac{7}{2}}\left( C_{4}^{(6)}+C_{-4}^{(6)}\right) \right] .
\label{hcubic}
\end{equation}
Restricting the calculations within the ground multiplet only and using the
Stevens' operator equivalents formalism, Eq.~(\ref{hcubic}) becomes
\begin{equation}
H_{CF}^{(J)}=\frac{B_{4}^{0}}{8}\beta \left( O_{4}^{0}+5O_{4}^{4}\right) +%
\frac{B_{6}^{0}}{16}\gamma \left( O_{6}^{0}-21O_{6}^{4}\right) ,
\label{hcfj}
\end{equation}
$\beta$ and $\gamma$ being the fourth- and sixth-order Stevens factors.
To introduce another common notation,\cite{newmanbook} we define
\begin{equation}
V_{4}=\frac{B_{4}^{0}}{8}=A_{4}\left< r^{4} \right> ;
V_{6}=\frac{B_{6}^{0}}{16}=A_{6}\left< r^{6} \right> ,
\label{notation}
\end{equation}
where $\left< r^{n} \right>$ are the expectation values of the
$r^{n}$ operator over the appropriate $f$-electron wavefunction.
It is found that $H^{(J)}_{mix}$ maintains the cubic symmetry, and
has the form
\begin{equation}
H_{mix}^{(J)}=\nu _{4}\left( O_{4}^{0}+5O_{4}^{4}\right)
+\nu_{6}\left( O_{6}^{0}-21O_{6}^{4}\right)
+\nu _{8}\left(
O_{8}^{0}+28O_{8}^{4}+65O_{8}^{8}\right) + \ldots ,
\label{hmixcub}
\end{equation}
where we did not explicitly write the terms containing operators
of rank higher than 8 since $O_{k}^{q} = 0$ for $k > 2J$, and
$J=9/2$ is the maximum possible value for the ground state of
light lanthanides and actinides.\cite{notaopstevens} The
coefficients appearing in Eq.~(\ref{hmixcub}) are dependent on $J$
and can be written as
\[
\nu _{k}=\frac{1}{\Delta} \left[
\nu _{k}^{(4,4)}\left( V_{4}\left\langle J\left\| C^{(4)}\right\|
J+1\right\rangle \right) ^{2}
+\nu _{k}^{(6,6)}\left( V_{6}\left\langle
J\left\| C^{(6)}\right\| J+1\right\rangle \right) ^{2}
\right.
\]
\begin{equation}
\left. +\nu_{k}^{(4,6)} \left( V_{4}\left\langle J\left\|
C^{(4)}\right\| J+1\right\rangle V_{6}\left\langle J\left\|
C^{(6)}\right\| J+1\right\rangle \right) \right] ,
\label{coefficient}
\end{equation}
and a list of $\nu _{k}^{(m,n)}$ for the ground multiplets of
$f^{n}$ configurations with $1 \leq n \leq 5$ is given in
Table~\ref{tab:table1}. The reduced matrix elements $\left\langle
J\left\| C^{(k)}\right\| J+1\right\rangle$ are calculated by using
the Intermediate Coupling free-ion wavefunctions.
Ions with 6 $f$ electrons have a $J=0$ ground singlet, so that no
intramultiplet energy splitting can exist and $J$-mixing effects
are evident only in the wavefunction composition; this case is
then impossible to study by the present approach. As for ions with
half-filled $f$ shell such as Gd$^{3+}$, Cm$^{3+}$, and Bk$^{4+}$,
$J$ mixing is generally negligible and it hardly affects any
physical property. Finally, the perturbative $J$-mixing approach
is in principle suitable to study heavy $f$-electron ions ($8 \leq
n \leq 13$), taking into account that the ground $J$ multiplet is
mixed with $J-1$ states, instead of $J+1$. However, in this case
the advantage of using the perturbative model with respect to the
numerical diagonalization of $H$ over a complete $f^{n}$ basis
could be significantly reduced, because: {\it i}) the use of
Stevens operators of rank 10 (12) becomes necessary for $J \geq 5$
(6); {\it ii}) it is not always possible to diagonalize
$H^{(J)}_{CF}+H^{(J)}_{mix}$ analytically, even in cubic symmetry.
In any case, the expected $J$-mixing strength is much smaller for
heavy than for light elements.
\section{\label{sec:f1}A particular case: the $f^{1}$ configuration}
In order to show in detail how the present model can be applied to the
study of rare-earth and actinide compounds, let us start from the simplest
possible configuration: an ion with a single $f$ electron.
The only interaction present in the free-ion Hamiltonian is the spin-orbit
coupling,
\begin{equation}
H_{FI}=\Lambda {\bf L\cdot S}.
\end{equation}
The $f^{1}$
spectra is composed of two multiplets only, $^{2}F_{5/2}$ (ground state) and $^{2}F_{7/2}$;
adding a crystal field of cubic symmetry, the $14\times 14$ matrix
representing the $^{2}F$ term is made up of two $4\times 4$ and two $3 \times 3$
diagonal blocks. The complete Hamiltonian $H$ can then be
analytically diagonalized, and the resulting energy gap between the
$\Gamma_{7}$ doublet and a $\Gamma_{8}$ quartet composing the
ground multiplet is
\[
E_{\Gamma_{7}}-E_{\Gamma_{8}} = \frac{1}{1716} \bigg[
3744V_{4}-4480V_{6}
\]
\[
+\sqrt{25600\left( 13V_{4}-84V_{6}\right)^{2}
-137280\left( 13V_{4}-84V_{6}\right)\Lambda+
9018009\Lambda^{2}}
\]
\begin{equation}
-\sqrt{16384\left(13V_{4}+70V_{6}\right)^{2}-
219648\left(13V_{4}+70V_{6}\right)\Lambda+9018009
\Lambda^{2}}~\bigg] . \label{f1compl}
\end{equation}
Although the perturbative approach is not particularly useful here
since an exact analytical solution can be obtained, let us study
this simple case in order to clarify the details of the process
and to understand its limits of validity. No intermediate coupling
occurs, and $\beta=2/315$, $\gamma=0$, $\left< J \left\| C^{4}
\right\| J+1 \right> = 2 \sqrt{10/77}$, and $\left< J \left\|
C^{6} \right\| J+1 \right> = -10 \sqrt{2/143}$; moreover, the
spin-orbit gap can be expressed as $\Delta=\Lambda(J+1)=7\Lambda
/2$, with $\Lambda = 100.5~{\rm meV}$. Diagonalizing
$H_{CF}^{(J)}+H_{mix}^{(J)}$ (see Appendix for details), we find
\begin{equation}
E_{\Gamma_{7}}-E_{\Gamma_{8}} = \frac{16}{7}V_{4} +
\frac{1}{\Lambda}\left(
\frac{20480}{124509}V_{4}^{2}
-\frac{614400}{77077}V_{4}V_{6}+
\frac{204800}{20449}V_{6}^{2} \right) ,
\label{f1appr}
\end{equation}
which corresponds exactly to the series expansion of
Eq.~(\ref{f1compl}) up to the first order in $\Lambda^{-1}$.
The intramultiplet energy gap calculated above can be directly
measured by means of spectroscopic techniques; for example,
inelastic neutron scattering measurements for PrO$_{2}$ (where
praseodymium ions have valence 4+, therefore presenting a $4f^{1}$
electronic configuration) have shown that
$E_{\Gamma_{7}}-E_{\Gamma_{8}} = 131~\rm{meV}$.\cite{boothroyd}
Figure~\ref{fig:fig1} shows the possible solutions of this
equation in terms of the crystal-field parameters $V_{4}$ and
$V_{6}$, with three different expressions: the full black line
corresponds to the exact diagonalization of the complete $f^{1}$
Hamiltonian [Eq.~(\ref{f1compl})]; the dashed vertical line is
obtained by neglecting $J$ mixing and using Stevens' approximation
$E_{\Gamma_{7}}-E_{\Gamma_{8}} = (16/7)V_{4}$; finally, the
dashed-dotted line takes $J$ mixing into account perturbatively by
the present approach [Eq.~(\ref{f1appr})]. The results are
satisfactory: the agreement between the exact (full) and
approximate (dashed-dotted) curves is qualitatively much better if
compared with Stevens' approximation (dashed line), and the
quantitative contribution of the excited states is correctly
estimated in the small-CF range $\left| V_{k}/\Lambda\right|<1$.
It may be noticed that the three curves of Fig.~\ref{fig:fig1}
almost coincide when $V_{6} \simeq 0$, since in this case the
largest part of the $J$-mixing correction (which is linear in
$V_{6}$) vanishes [this can be verified by observing the relative
magnitude of the different coefficients in Eq.~(\ref{f1appr})].
For the same reason, for small $V_{6}$, the value of $V_{4}$ is
slightly underestimated for $V_{6}<0$ and overestimated for
$V_{6}>0$. As the spin-orbit interaction is stronger for heavier
ions and the gap between the two lowest multiplets grows with $J$,
we expect our model to have a reasonably good performance over the
whole lanthanide and actinide series.
\section{\label{sec:ano2}The cubic phase of actinide dioxides}
In this Section, the perturbative $J$-mixing model outlined so far
will be applied to interpret the intramultiplet crystal-field
splittings observed by inelastic neutron scattering (INS) for
actinide dioxides AnO$_{2}$ (An = U, Np, Pu). The crystal-field
analysis will be performed in terms of the parameters $A_{4}$ and
$A_{6}$ [Eq.~(\ref{notation})] instead of $V_{4}$ and $V_{6}$,
using the values of $\left< r^{n} \right>$ given in
Ref.~\onlinecite{raggimedi}. Although recent density functional
studies have pointed out a certain degree of covalency for the
$An$-O bond in these systems,\cite{wu} this does not prevent one
from using the crystal field theory to analyze experimental data;
it would be quite more difficult than in a ionic compound to
calculate the CF parameters from first principles, but this is not
our aim. We will demonstrate that the present method can be used
for an analytical study of the experimental results over a wide
range of parameters and compositions.
INS spectra for UO$_{2}$ in the paramagnetic phase (above $T_{N} =
30.8 K$) display peaks between 150 ad 185
meV,\cite{kernuo2,amoretti} which have been attributed to
transitions between the $\Gamma_{5}$ ground
state\cite{kernuo2,amoretti,rahman} and excited $\Gamma_3$ and
$\Gamma_4$ states, and no other magnetic transitions were reported
up to 700 meV\cite{amoretti} (it is worth to recall that the
$\Gamma_{5} \rightarrow \Gamma_{1}$ transition is not
dipole-allowed so, if present, it will display a very small
intensity with respect to the other two transitions).
Figure~\ref{fig:fig2} shows the values of $A_{4}$ and $A_{6}$ for
which the possible transitions lie within the experimentally
observed range according to the perturbative model. In the
previous Section, we have studied the quantitative discrepancy
between the model's predictions and the exact results, which was
found to be significant for large CF parameters. Following these
estimates, we have determined a ``safe zone'' (indicated in
Fig.~\ref{fig:fig2} by a dashed ellipse) within which the true set
of CF parameters for UO$_2$ is located with high degree of
confidence.
For the paramagnetic phase of NpO$_{2}$ we have followed Amoretti et
al.,\cite{amorettinpo2} who observed a broad magnetic signal centered at 55 meV in
the INS spectra and attributed this peak to a transition between the two $\Gamma_{8}$
quartets. They gave two possible solutions for the paramagnetic phase, labelled 3
and 4 in Fig.~\ref{fig:fig3}; we show that actually there are infinite possible solutions,
divided in two branches.
PuO$_2$ displays a temperature-independend magnetic susceptibility
below 1000 K,\cite{raphael} so that the CF ground state is
expected to be the $\Gamma_{1}$ singlet. Magnetic-dipole matrix
elements involving this state within the $^{5}I_{4}$ multiplet are
zero except with the $\Gamma_{4}$ triplet. Indeed, only one peak
centered at 123 meV is observed in PuO$_{2}$ INS
spectra.\cite{kernpuo2} According to our perturbative model there
are again infinite possible solutions, plotted in
Fig.~\ref{fig:fig3}, which cover a large area of the
$A_{4}$-vs-$A_{6}$ diagram.
It is clear that, in the case of AnO$_{2}$, the INS data analysis
cannot give unambiguous results if every compound is treated
separately; this can also be inferred from the widely scattered
sets of parameters which are found in the literature (some of
which are listed in Table~\ref{tab:table2} and displayed in
Fig.~\ref{fig:fig3}). On the other hand, if we assume that the
$A_{k}$ parameters are approximately the same for all
isostructural compounds,\cite{newmanbook} an inspection of
Fig.~\ref{fig:fig3} shows that the only area of the diagram where
common solutions for $An =$U, Np, Pu might exist is around the
point labelled 4, which corresponds to one of the two solutions
(the ``strong $J$-mixing'' one) proposed by Amoretti et
al.\cite{amorettinpo2} for NpO$_{2}$ ($A_{4}=-19.6~{\rm
meV}/a_{0}^{4}$ ; $A_{6}=0.666~{\rm meV}/a_{0}^{6}$). In order to
verify this result, we have calculated the INS transition energies
for UO$_{2}$, NpO$_{2}$, PuO$_{2}$, and PrO$_{2}$ with this set of
parameters by numerical diagonalization of the complete $f^{n}$
configuration Hamiltonian (Table~\ref{tab:table3}). Moreover, as a
further test, we have calculated the magnetic susceptibility for
AmO$_{2}$ and CfO$_{2}$ with the same parameters
(Fig.~\ref{fig:fig4}; the measurements can be found in
Ref.~\onlinecite{karraker} and Ref.~\onlinecite{moore}
respectively).
In all the cases examined so far, the comparison with experimental
results is quite good, considering that
a $100\%$ exact scaling of the CF potential is not expected to hold.
Therefore, our results point towards a coherent unified picture
for the CF potential in actinide dioxides.
The splitting predicted
for PrO$_2$ seems less satisfactory as the measured value of the
gap $E(\Gamma _{7}) -E(\Gamma _{8}) = 131$ meV is quite larger
than the value calculated with the solution we propose. However,
the PrO$_2$ case is complicated by magnetoelastic interactions
that are known to affect heavily the physics of this compound by
increasing the bare value of the CF gap.\cite{boothroyd} Hence, a
value of this bare gap of the order of 80 meV is fully realistic.
One feature which cannot be accounted for by the CF models
proposed so far is the temperature-independence of the magnetic
susceptibility of PuO$_{2}$ above 600 K. Indeed, it is obvious
that if there is a magnetic gap of 123 meV as observed by neutron
scattering, when this level becomes thermally populated it will
contribute to the susceptibility, no matter the mechanism which
generates the splitting (unless this contribution is accidentally
canceled by other contributions). Indeed, even a completely
different approach based on Density-Functional-Theory calculations
cannot remove the discrepancy between magnetic susceptibility and
neutron scattering experiments.\cite{mamo} Using the diagram in
Fig.~\ref{fig:fig3} as a guide we have been able to find a
solution for PuO$_{2}$ ($A_{4}=-26.7~{\rm meV}/a_{0}^{4}$,
$V_{6}=1.68~{\rm meV}/a_{0}^{6}$) which leads to
$E_{\Gamma_{4}}-E_{\Gamma_{1}}=134~{\rm meV}$ and a flat $\chi
(T)$ curve below 1000 K. For this set the Curie contribution of
the excited triplet and the off-diagonal Van Vleck contribution of
the $\Gamma_{1}-\Gamma_{4}$ pair accidentally combine into an
almost $T$-independent susceptibility. However, this solution is
quite unstable even for small variation of the CF parameters;
moreover, it does not give good results if applied to UO$_{2}$ and
NpO$_{2}$.
\section{Conclusions}
We have developed a perturbative approach to $J$ mixing which
allows the Stevens' formalism to be recovered by replacing the
original CF hamiltonian with an effective one operating within the
same $J$ multiplet. We have applied the method to the study of the
CF in actinide dioxides. These compounds have extremely
interesting physical properties, which are determined by the CF to
a large extent. It is therefore important to reach a precise
understanding of the CF, which is the basic building block of all
theoretical efforts devoted to understand these properties. Most
studies of the CF potential rely on the Stevens' approach, which
yields a solution working fairly consistently over several
compounds. Yet, this solution does not work satisfactorily anymore
when $J$ mixing is included, and the practical impossibility to
perform exact $J$ mixing calculations in wide ranges of the
parameter space has prevented so far the identification of a
better alternative. By hugely decreasing the numerical effort in
favor of analytical calculations, our method has allowed us to
find a $J$ mixing solution ($A_{4}=-19.6~{\rm meV}/a_{0}^{4}$ ,
$A_{6}=0.666~{\rm meV}/a_{0}^{6}$) which works consistently over
all the considered compounds.
Although in the present paper we have only studied cubic systems
and included the two lowest multiplets only (which in dioxides
account for almost 100\% of $J$ mixing), the method can be used
for any point symmetry, at the price of additional terms in the
effective Hamiltonian; also, as many multiplets as necessary can
be included in the calculation, leading to additional
contributions to the $\nu_k$ coefficients of
Eq.~(\ref{coefficient}). Even in this case our method, by allowing
to perform a quantitative analysis of experimental data by means
of a Stevens-like Hamiltonian containing higher-rank operators, is
much more efficient than fully numerical diagonalization. In
particular, wide ranges of the parameter space can be easily
investigated, allowing to produce quite easily diagrams such as
Figs.~\ref{fig:fig2} and~\ref{fig:fig3} which would otherwise be
very hard (if not impossible, in some cases) to obtain.
|
{
"timestamp": "2004-11-25T17:25:25",
"yymm": "0411",
"arxiv_id": "cond-mat/0411649",
"language": "en",
"url": "https://arxiv.org/abs/cond-mat/0411649"
}
|
\section{Introduction}\label{sec:introduction}
We work over an algebraically closed field $k$.
An isolated singular point of an algebraic surface
is called a \emph{cusp} if it is a rational double point of type $A\sb 2$
(Artin~\cite{Artin62, Artin66, Artin77}).
In characteristic $0$,
the number of cusps on a normal $K3$ surface is at most nine.
Barth showed in~\cite{Barth98} that
a complex normal $K3$ surface $Y$ has nine cusps as its only singularities
if and only if
$Y$ is the quotient of an abelian surface by a cyclic group of order $3$.
This is a generalization of the result of~\cite{Nikulin75},
in which Nikulin showed that
a complex normal $K3$ surface $Y$ has sixteen nodes as its only singularities
if and only if
$Y$ is the quotient of an abelian surface by the involution.
In~\cite{Barth2000},
Barth classified normal $K3$ surfaces with nine cusps according to the degrees of the polarizations.
In positive characteristics, however,
there exist normal $K3$ surfaces $Y$ such that the singular locus $\operatorname{\rm Sing}\nolimits Y$
of $Y$ consists of \emph{ten} cusps.
The purpose of this paper is to investigate such $K3$ surfaces.
\par
\medskip
A smooth $K3$ surface $X$
is called \emph{supersingular} (in the sense of Shioda~\cite{Shioda}) if
the N\'eron-Severi lattice $\operatorname{{\it NS}}\nolimits (X)$ of $X$ is of rank $22$.
Supersingular $K3$ surfaces exist only in positive characteristics.
Let $X$ be a supersingular $K3$ surface in characteristic $p>0$.
Artin~\cite{Artin74} showed that
there exists a positive integer $\sigma (X)\le 10$
such that
$\operatorname{\rm disc}\nolimits \operatorname{{\it NS}}\nolimits (X)=-p\sp{2\sigma (X)}$ holds.
This integer $\sigma (X)$ is called the \emph{Artin invariant of $X$}.
We denote by
$U(m)$
the lattice of rank $2$
whose intersection matrix is equal to
$$
\begin{pmatrix}
0 & m \\
m & 0
\end{pmatrix}.
$$
Our main results are Theorems 1.1 and 1.4 - 1.6.
\begin{theorem}\label{thm:A}
Let $Y$ be a normal $K3$ surface such that $\operatorname{\rm Sing}\nolimits Y$ consists of ten cusps,
and $\rho: X\to Y$ the minimal resolution of $Y$.
Let $R\sb{\rho}$ be the sublattice of $\operatorname{{\it NS}}\nolimits (X)$ generated by the classes of
the $(-2)$-curves that are contracted by $\rho$.
Then the following hold:
{\rm (1)}
The characteristic of the ground field $k$ is $3$.
{\rm (2)}
The orthogonal complement $R\sb\rho\sp{\perp}$ of $R\sb{\rho}$ in $\operatorname{{\it NS}}\nolimits (X)$
is isomorphic to either $U(1)$ or $U(3)$.
{\rm (3)}
If $R\sb\rho\sp{\perp}\cong U(1)$, then $\sigma (X)\le 5$,
while
if $R\sb\rho\sp{\perp}\cong U(3)$, then $\sigma (X)\le 6$.
\end{theorem}
Before we state the other main results, we fix the terminology below.
\begin{definition}
Let $L$ be a line bundle on a smooth $K3$ surface $X$.
We say that $L$ is \emph{very ample modulo $(-2)$-curves}
if the following hold:
\begin{itemize}
\item[{\rm (i)}] The complete linear system $|L|$ has no fixed components,
and hence has no base points by~\cite[Corollary 3.2]{SD}.
In particular, $|L|$ defines a morphism $\Phi\sb{|L|} : X\to \P\sp{N}$, where $N=L^2/2+1$.
\item[{\rm (ii)}] The morphism $\Phi\sb{|L|}$ is birational onto the image
$Y\sb{(X, L)}:=\Phi\sb{|L|} (X)$.
\end{itemize}
A \emph{polarized $K3$ surface} is a pair $(X, L)$ of a $K3$ surface $X$ and a line bundle $L$
on $X$ that is very ample modulo $(-2)$-curves.
The \emph{degree} of a polarized $K3$ surface $(X, L)$ is defined to be $L^2$.
\end{definition}
\begin{definition}
Let $(X, L)$ be a polarized $K3$ surface.
We denote by
$$
\rho\sb L: X\to Y\sb{{(X, L)}}
$$
the birational morphism induced by $|L|$.
By~\cite[Theorem 6.1]{SD},
$\rho\sb L$ is a contraction of an $ADE$-configuration of $(-2)$-curves on $X$.
We denote by $R\sb{(X, L)}$ the sublattice of $\operatorname{{\it NS}}\nolimits (X)$ generated by the classes of
the $(-2)$-curves that are contracted by $\rho\sb L$.
We also denote by $\mathord{\mathcal R}\sb{(X, L)}$ the $ADE$-type of the configuration of
these $(-2)$-curves.
\end{definition}
Note that $\mathord{\mathcal R}\sb{(X, L)}=10A\sb 2$ is equivalent to saying that $\operatorname{\rm Sing}\nolimits Y\sb{{(X, L)}}$ consists of ten cusps. The degree of $(X, L)$ can be completely determined:
\begin{theorem}\label{thm:U1}
The following conditions on a positive integer $d$ are equivalent:
\begin{itemize}
\item[{\rm (i)}]
$d=2ab$, where $a$ and $b$ are integers $\ge 3$ such that $a\ne b$.
\item[{\rm (ii)}]
There exists a polarized supersingular $K3$ surface $(X, L)$
of degree $d$
such that $\mathord{\mathcal R}\sb{(X, L)}=10A\sb 2$ and
$R\sb{{(X, L)}}\sp{\perp}\cong U(1)$.
\item[{\rm (iii)}]
Every supersingular $K3$ surface $X$ in characteristic $3$ with $\sigma (X)\le 5$
admits a line bundle $L$ such that
$(X, L)$ is a polarized $K3$ surface of degree $d$ satisfying
$\mathord{\mathcal R}\sb{(X, L)}=10A\sb 2$ and
$R\sb{{(X, L)}}\sp{\perp}\cong U(1)$.
\end{itemize}
\end{theorem}
\begin{theorem}\label{thm:U3}
The following conditions on a positive integer $d$ are equivalent:
\begin{itemize}
\item[{\rm (i)}]
$d\equiv 0 \bmod 6$.
\item[{\rm (ii)}]
There exists a polarized supersingular $K3$ surface $(X, L)$
of degree $d$
such that $\mathord{\mathcal R}\sb{(X, L)}=10A\sb 2$ and
$R\sb{{(X, L)}}\sp{\perp}\cong U(3)$.
\item[{\rm (iii)}]
Every supersingular $K3$ surface $X$ in characteristic $3$ with $\sigma (X)\le 6$
admits a line bundle $L$ such that
$(X, L)$ is a polarized $K3$ surface of degree $d$ satisfying
$\mathord{\mathcal R}\sb{(X, L)}=10A\sb 2$ and
$R\sb{{(X, L)}}\sp{\perp}\cong U(3)$.
\end{itemize}
\end{theorem}
Supersingular $K3$ surfaces with ten cusps can be obtained as purely inseparable triple covers of
the smooth quadric surface $Q=\P\sp 1\times \P\sp 1$.
From now on to the end of this paragraph,
we assume that $k$ is of characteristic $3$.
For integers $a$ and $b$,
we denote by $\mathord{\mathcal O}\sb{Q} (a, b)$ the invertible sheaf
$\operatorname{\rm pr}\nolimits\sb 1 \sp * \mathord{\mathcal O}\sb{\P\sp 1} (a) \otimes \operatorname{\rm pr}\nolimits\sb 2 \sp * \mathord{\mathcal O}\sb{\P\sp 1} (b)$ of $Q=\P\sp 1\times \P\sp 1$,
and by $L\sb{Q} (a, b)\to Q=\P\sp 1\times \P\sp 1$ the corresponding line bundle.
Because we are in characteristic $3$,
the differential map
$$
d : H\sp 0 (Q, \mathord{\mathcal O}\sb{Q} (3, 3)) \to H\sp 0 ( Q, \Omega\sb{Q} \sp 1 (3, 3))
$$
is well-defined by the isomorphism $L\sb{Q} (3, 3)\cong L\sb{Q}(1, 1)\sp{\otimes 3}$.
For $G\in H\sp 0 (Q, \mathord{\mathcal O}\sb{Q} (3, 3))$, we denote by $Z(dG)$ the subscheme of $Q$ defined by $dG=0$.
If $\dim Z(dG)=0$, then
$$
\textrm{length}\; \mathord{\mathcal O}\sb{Z(dG)}=c\sb 2 ( \Omega\sb{Q} \sp 1 (3, 3)) =10
$$
holds, where $c\sb 2$ is the second Chern class.
We put
$$
\mathord{\mathcal U}\sb{3,3}:=\set{G\in H\sp 0 (Q, \mathord{\mathcal O}\sb{Q} (3, 3)) }{\textrm{$Z(dG)$ is reduced and of dimension $0$}},
$$
which is a Zariski open dense subset of $H\sp 0 (Q, \mathord{\mathcal O}\sb{Q} (3, 3))$.
For a non-zero $G\in H\sp 0 (Q, \mathord{\mathcal O}\sb{Q} (3, 3))$, we denote by
$$
\pi\sb G : Y\sb G \to Q=\P\sp 1\times \P\sp 1
$$
the purely inseparable triple cover of $Q$ defined by
$$
W^3=G,
$$
where $W$ is a fiber coordinate of the line bundle $L\sb{Q}(1, 1)$.
It is easy to see that $G$ is contained in $\mathord{\mathcal U}\sb{3,3}$ if and only if
$Y\sb G$ is a normal $K3$ surface
such that $\operatorname{\rm Sing}\nolimits Y\sb G =\pi\sb G\sp{-1} (Z(dG))$
consists of ten cusps.
In particular, if $G\in \mathord{\mathcal U}\sb{3,3}$, then
the minimal resolution $X\sb G$ of $Y\sb G$ is a supersingular $K3$ surface
with Artin invariant $\le 6$ by Theorem~\ref{thm:A}.
Conversely, we have the following:
\begin{theorem}\label{thm:insep}
Let $X$ be a supersingular $K3$ surface in characteristic $3$ with Artin invariant $\le 6$.
Then there exists $G\in \mathord{\mathcal U}\sb{3,3}$ such that $X$ is isomorphic to $X\sb G$.
\end{theorem}
\noindent
We put
$$
\mathord{\mathcal V}\sb{1,1}:=\set{H^3\in H^0 (Q, \mathord{\mathcal O}\sb{Q}(3,3))}{H\in H^0(Q, \mathord{\mathcal O}\sb{Q}(1,1))},
$$
which is an additive group
acting on $\mathord{\mathcal U}\sb{3,3}$ by $G\mapsto G+H^3$ $(G\in \mathord{\mathcal U}\sb{3,3}, H^3\in \mathord{\mathcal V}\sb{1,1})$.
For $G, G\sp{\prime}\in \mathord{\mathcal U}\sb{3, 3}$, the triple covers
$Y\sb G$ and $Y\sb{G\sp{\prime}}$ are isomorphic over $Q$ if and only if
$G=cG\sp{\prime} +H^3$ holds for some $c\in k\sp{\times}$ and $H^3\in \mathord{\mathcal V}\sb{1,1}$.
Hence the space
$$
\mathord{\hbox{\mathgot M}}:=(\operatorname{{\it PGL}}\nolimits (2, k)\times \operatorname{{\it PGL}}\nolimits(2, k))\backslash \P\sb * (\mathord{\mathcal U}\sb{3,3}/ \mathord{\mathcal V}\sb{1,1})
$$
is a moduli space of supersingular $K3$ surfaces in characteristic $3$
with Artin invariant $\le 6$.
We remark that,
since $\dim \mathord{\mathcal U}\sb{3,3}=16$ and $\dim \mathord{\mathcal V}\sb{1,1}=4$,
we have
$$
\dim \mathord{\hbox{\mathgot M}} = 16-4-1-(3+3)=5,
$$
as is predicted from the result of Artin~\cite{Artin74}.
In particular, the {\it unique} supersingular $K3$ surface of Artin
invariant $1$ has the following precise model:
\begin{example}\label{example:G0}
We put
$$
G\sb 0:=(x^3-x)(y^3-y),
$$
where $x$ and $y$ are affine coordinates of the two factors of $Q=\P\sp 1\times \P\sp 1$.
Then $Z(dG\sb 0)$ is equal to
$$
\set{(\alpha, \beta)}{\alpha, \beta\in \mathord{\mathbb F}\sb 3 }\cup \{(\infty, \infty)\}.
$$
Therefore $G\sb 0\in \mathord{\mathcal U}\sb{3,3}$.
It can be shown that
the Artin invariant of the supersingular $K3$ surface $X\sb{G\sb 0}$
is $1$.
See Example~\ref{example:sigmaG0}.
\end{example}
\par
\medskip
Supersingular $K3$ surfaces in characteristic $2$ with $21$ nodes are investigated in
\cite{Shimada2003, Shimada2004, Shimada2004moduli}.
In particular,
it was shown there that every supersingular $K3$ surface in characteristic $2$
is birational to a purely inseparable double cover of the projective plane
with $21$ nodes;
that is,
every supersingular $K3$ surface in characteristic $2$ is obtained as a generic Zariski surface~\cite{BL}.
Quasi-elliptic $K3$ surfaces in characteristic $3$ with a section and ten singular fibers of type
$A\sb 2^*$
are constructed explicitly in~\cite{Ito92}.
The Artin invariants of these supersingular $K3$ surfaces are $\le 5$.
A family of smooth quartic surfaces in characteristic $3$ containing an $ADE$-configuration
of lines of type $10A\sb 2$ is constructed in~\cite{Shimada92}.
A general member of the family is of Artin invariant $6$.
See Example~\ref{example:quartic} for details.
\par
\medskip
This paper is organized as follows.
In \S\ref{sec:disc},
we review the theory of discriminant forms of lattices due to Nikulin~\cite{Nikulin79}.
In \S\ref{sec:NS},
we quote from Artin~\cite{Artin74}, Rudakov-Shafarevich~\cite{RS},
Saint-Donat~\cite{SD} and Nikulin~\cite{Nikulin91}
some known facts about N\'eron-Severi lattices and
polarizations of supersingular $K3$ surfaces.
In \S\ref{sec:thmA},
we prove Theorem~\ref{thm:A} using the theory of discriminant forms.
In \S\ref{sec:U3} and \S\ref{sec:U1},
we prove Theorems~\ref{thm:U3}~and~\ref{thm:U1}.
We reduce the problem of existence of the polarizations on a supersingular $K3$ surface
to a problem of existence of ternary codes with certain properties,
and solve the latter by computer.
In \S\ref{sec:insep},
we prove Theorem~\ref{thm:insep}.
The proof presented here seems to be quite lattice-intensive.
We think there should be a more elementary proof.
See Question~\ref{question1}.
\par
\medskip
{\bf Acknowledgment.} This work was done during the second author's visit
to Hokkaido University who likes to express his thanks for the very warm hospitality.
\section{Discriminant forms of lattices}\label{sec:disc}
For a finite abelian group $A$ and a prime integer $p$,
we denote by
$$
A=A\sb{(p)}\times A\sb{(p\sprime)}
$$
the decomposition of $A$ into the $p$-part $A\sb{(p)}$ and the $p$-prime-part $A\sb{(p\sprime)}$ of $A$.
A lattice is, by definition,
a free $\mathord{\mathbb Z}$-module of finite rank with a non-degenerate symmetric
$\mathord{\mathbb Z}$-valued bilinear form.
A lattice $\Lambda$ is said to be \emph{even} if $v^2\in 2\mathord{\mathbb Z}$ holds for every $v\in \Lambda$.
Let $\Lambda$ be an even lattice.
We denote by $\Lambda\sp{\vee}$ the \emph{dual lattice} $\operatorname{\rm Hom}\nolimits (\Lambda, \mathord{\mathbb Z})$.
We have a natural embedding $\Lambda\hookrightarrow \Lambda\sp{\vee}$
of finite cokernel, and a symmetric bilinear form
$\Lambda\sp{\vee}\times\Lambda\sp{\vee}\to\mathord{\mathbb Q}$ that extends the $\mathord{\mathbb Z}$-valued symmetric bilinear form on $\Lambda$.
We put
$$
\Disc {\Lambda} :=\Lambda\sp{\vee}/\Lambda,
$$
and call it the \emph{discriminant group of $\Lambda$}.
We then define the \emph{discriminant form}
\begin{eqnarray*}
\Discform{\Lambda} &:& \Disc{\Lambda} \to \mathord{\mathbb Q}/2\mathord{\mathbb Z} \quad\textrm{and}\\
\Discformb{\Lambda} &:& \Disc{\Lambda} \times \Disc{\Lambda} \to \mathord{\mathbb Q}/\mathord{\mathbb Z}
\end{eqnarray*}
by
\begin{eqnarray*}
\Discform{\Lambda} (\bar v)&:=&v^2\bmod 2\mathord{\mathbb Z}\quad\textrm{and}\\
\Discformb{\Lambda} (\bar v, \bar w)&:=&vw\bmod \mathord{\mathbb Z}=
(\Discform{\Lambda}(\bar v +\bar w) -\Discform{\Lambda}(\bar v)-\Discform{\Lambda}(\bar w))/2,
\end{eqnarray*}
where $v, w\in\Lambda\sp{\vee}$, and $\bar v:=v \bmod \Lambda$, $\bar w :=w \bmod \Lambda$.
Let $p$ be a prime integer dividing $| \Disc{\Lambda}| =|\operatorname{\rm disc}\nolimits\Lambda|$.
Then $\Disc{\Lambda}\sb{(p)}$ and $\Disc{\Lambda}\sb{(p\sprime)}$ are
orthogonal with respect to $\Discformb{\Lambda}$.
We put
\begin{eqnarray*}
{\Discform{\Lambda}}\sb{(p)}:= \Discform{\Lambda}| \Disc{\Lambda}\sb{(p)}, &&
{\Discform{\Lambda}}\sb{(p\sprime)}:= \Discform{\Lambda}| \Disc{\Lambda}\sb{(p\sprime)}, \\
{\Discformb{\Lambda}}\sb{(p)}:= \Discformb{\Lambda}| \Disc{\Lambda}\sb{(p)}\times \Disc{\Lambda}\sb{(p)}, &&
{\Discformb{\Lambda}}\sb{(p\sprime)}:= \Discformb{\Lambda}| \Disc{\Lambda}\sb{(p\sprime)}\times \Disc{\Lambda}\sb{(p\sprime)}.
\end{eqnarray*}
For a subgroup $H$ of $\Disc{\Lambda}$,
we denote by $H\sp{\perp}$ the orthogonal complement of $H$ with respect to $\Discformb{\Lambda}$.
Note that $(H\sp{\perp})\sb{(p)}$ is canonically isomorphic to
$$
(H\sb{(p)})\sp{\perp}:=\set{x \in \Disc{\Lambda}\sb{(p)}}{\hbox{${\Discformb{\Lambda}}\sb{(p)} (x, y)=0$ for any $y\in H\sb{(p)}$ }}.
$$
We will use the notation $H\sp{\perp}\sb{(p)}$
to denote $(H\sp{\perp})\sb{(p)}=(H\sb{(p)})\sp{\perp}$.
A subgroup $H\subset \Disc{\Lambda}$ is called \emph{isotropic} if $\Discform{\Lambda}|H$ is constantly equal to $0$.
If $H$ is isotropic, then $H$ is contained in $H\sp{\perp}$.
Note that we have
$$
(H\sp{\perp}/H)\sb{(p)}=H\sp{\perp}\sb{(p)}/H\sp{\phantom{\perp}}\sb{(p)}.
$$
An \emph{overlattice} of $\Lambda$ is, by definition,
a submodule $\Lambda\sp{\prime}$ of $\Lambda\sp{\vee}$ containing $\Lambda$ such that the $\mathord{\mathbb Q}$-valued symmetric bilinear form on
$\Lambda\sp{\vee}$ takes values in $\mathord{\mathbb Z}$ on $\Lambda\sp{\prime}$.
\begin{proposition}[Nikulin \cite{Nikulin79}]\label{prop:nikulin}
Let
$\operatorname{\rm pr}\nolimits\sb{\Lambda} : \Lambda\sp{\vee} \to \Disc{\Lambda}$
be the natural projection.
The correspondence
$$
H\mapsto \Lambda\sb H :=\operatorname{\rm pr}\nolimits\sb{\Lambda}\sp{-1} (H)
$$
gives a bijection from
the set of isotropic subgroups of $\Disc{\Lambda}$
to the set of even overlattices of $\Lambda$.
For an isotropic subgroup $H$,
the discriminant group of $\Lambda\sb{H}$ is isomorphic to $H\sp{\perp}/H$.
\end{proposition}
\begin{remark}
If $\Lambda$ is of rank $r$, then $\Disc{\Lambda}$ is generated by $r$ elements.
\end{remark}
\par
\medskip
A vector $v$ in an even \emph{negative-definite} lattice $\Lambda$ is called a \emph{root}
if $v^2=-2$.
We denote by
$\operatorname{{\rm Roots}}\nolimits (\Lambda)$ the set of roots in $\Lambda$.
It is known that $\operatorname{{\rm Roots}}\nolimits (\Lambda)$
forms a root system of type $ADE$~(\cite{B, E}).
An even negative-definite lattice $\Lambda$ is called a \emph{root lattice} if it is generated by $\operatorname{{\rm Roots}}\nolimits (\Lambda)$.
\par
\medskip
Let $\mathord{\mathbb Z} [10 A\sb 2 ]$ denote the root lattice of type $10A\sb 2$.
Then $\mathord{\mathbb Z} [10 A\sb 2 ]$ is generated by roots
$c\sb i, d\sb i$ $(i=1, \dots, 10)$
satisfying
$$
c\sb i^2=d\sb i^2=-2, \quad c\sb i d\sb i=1, \quad\textrm{and}\quad
\langle c\sb i, d\sb i\rangle \perp \langle c\sb j, d\sb j\rangle \;\;\textrm{if $i\ne j$}.
$$
We have
$$
\operatorname{{\rm Roots}}\nolimits(\mathord{\mathbb Z}[10A\sb 2])=\{ \pm c\sb i, \pm d\sb i, \pm (c\sb i + d\sb i)\;\;\; (i=1, \dots, 10)\},
$$
and
$$
\mathord{\mathbb Z}[10A\sb 2]\sp{\vee}=\set{\sum\sb{i=1}\sp{10} (s\sb i c\sb i+t\sb i d\sb i)/3}{s\sb i, t\sb i\in \mathord{\mathbb Z}, \;
s\sb i+t\sb i\equiv 0 \bmod 3\;\; (i=1, \dots, 10)}.
$$
We put
$$
\gamma\sb i := (c\sb i + 2 d\sb i)/3\;\bmod\; \mathord{\mathbb Z} [10 A\sb 2 ]\;\;\in\;\; \Disc{\mathord{\mathbb Z}[10A\sb 2]}.
$$
Then we have
$$
\Disc{\mathord{\mathbb Z}[10A\sb 2]}=\mathord{\mathbb F}\sb 3 \gamma\sb 1 \oplus \cdots \oplus \mathord{\mathbb F}\sb 3 \gamma\sb{10},
$$
and
\begin{equation}\label{eq:discform10At}
\Discform{\mathord{\mathbb Z}[10A\sb 2]}(x\sb 1 \gamma\sb 1+\cdots+x\sb{10}\gamma\sb{10}) =
-2 (x\sb 1^2 + \cdots + x\sb{10}^2)/3 \;\;\in\;\;\mathord{\mathbb Q}/2\mathord{\mathbb Z}.
\end{equation}
For a vector
$$
\mathord{\bf x}=(x\sb 1, \dots, x\sb{10})=x\sb 1 \gamma\sb 1+\cdots+x\sb{10}\gamma\sb{10}\;\in\; \Disc{\mathord{\mathbb Z}[10A\sb 2]}\cong \mathord{\mathbb F}\sb 3\sp{10},
$$
we define the \emph{Hamming weight} $\operatorname{\rm wt}\nolimits (\mathord{\bf x})$ of $\mathord{\bf x}$ by
$$
\operatorname{\rm wt}\nolimits (\mathord{\bf x} ):=|\set{i}{x\sb i\ne 0}|\;\;\in\;\;\mathord{\mathbb Z}\sb{\ge 0}.
$$
Then, for a vector $r\in \mathord{\mathbb Z}[10A\sb 2]\sp{\vee}$, we have
\begin{equation}\label{eq:wtr}
r^2\le -(2/3) \operatorname{\rm wt}\nolimits(\bar r), \quad\textrm{where $\bar r :=r \bmod \mathord{\mathbb Z}[10A\sb 2] \;\in\; \Disc{\mathord{\mathbb Z}[10A\sb 2]}$}.
\end{equation}
Moreover,
\begin{equation}\label{eq:wtrinv}
\parbox{10cm}{
for a vector $\mathord{\bf x}\in \Disc{\mathord{\mathbb Z}[10A\sb 2]}$,
there exists a vector $r\in \mathord{\mathbb Z}[10A\sb 2]\sp{\vee}$ such that
$\bar r =\mathord{\bf x}$ and
$r^2= (-2/3) \operatorname{\rm wt}\nolimits(\mathord{\bf x})$ hold.
}
\end{equation}
\par
\bigskip
Let $e$ and $f$ be basis of the lattice $U(m)$ satisfying
$$
e^2=f^2=0, \qquad ef=m.
$$
We put $e\sp{\vee}:= f/m$ and $f\sp{\vee} :=e/m$.
Then $\Disc{U(m)}\cong (\mathord{\mathbb Z}/ m\mathord{\mathbb Z})\sp 2 $ is generated by
$$
\bar e\sp{\vee}:=e\sp{\vee} \bmod U(m) \quad\textrm{and}\quad
\bar f\sp{\vee}:=f\sp{\vee} \bmod U(m),
$$
and the discriminant form is given by
\begin{equation}\label{eq:discformUm}
\Discform{U(m)}(y\sb 1 \bar e\sp{\vee} + y\sb 2 \bar f\sp{\vee})= 2 y\sb 1 y\sb 2 /m \;\;\in\;\;\mathord{\mathbb Q}/2\mathord{\mathbb Z}.
\end{equation}
\section{N\'eron-Severi lattices of supersingular $K3$ surfaces}\label{sec:NS}
A lattice $\Lambda$ is called \emph{hyperbolic}
if the signature of $\Lambda$ is $(1, \operatorname{\rm rank}\nolimits \Lambda -1)$.
Let $p$ be a prime integer.
A lattice $\Lambda$ is called \emph{$p$-elementary}
if $\Disc{\Lambda}$ is a $p$-elementary abelian group;
that is, $p \Lambda\sp{\vee} \subseteq \Lambda$ holds.
An overlattice of a hyperbolic $p$-elementary lattice is
again hyperbolic and $p$-elementary.
\par
\medskip
The following is due to Artin~\cite{Artin74} and Rudakov-Shafarevich~\cite{RS}.
\begin{theorem}\label{thm:ARS}
Let $X$ be a supersingular $K3$ surface in characteristic $p>0$.
Then $\operatorname{{\it NS}}\nolimits (X)$ is an even hyperbolic $p$-elementary lattice.
\end{theorem}
The following is due to Rudakov-Shafarevich~\cite[Section 1]{RS}.
\begin{theorem}\label{thm:NRS}
Suppose that $p$ is odd.
Let $\sigma$ be a positive integer $\le 10$.
Then the lattice $N$ with the following properties is unique up to isomorphisms:
{\rm (i)} $N$ is even, hyperbolic of rank $22$, and
{\rm (ii)} $\Disc{N} \cong \mathord{\mathbb F}\sb p \sp{2\sigma}$.
\end{theorem}
From now on to the end of this section,
we assume that $p$ is \emph{odd}.
We denote the lattice $N$ in Theorem~\ref{thm:NRS} by $N\sb{p, \sigma}$.
Let $X$ be a supersingular $K3$ surface in characteristic $p$
with $\sigma (X)=\sigma$.
By Theorems~\ref{thm:ARS} and~\ref{thm:NRS},
there exists an isometry
$$
\phi : N\sb{p, \sigma} \smash{\mathop{\;\to\;}\limits\sp{\sim\;}} \operatorname{{\it NS}}\nolimits(X).
$$
More precisely, we have the following:
\begin{proposition}\label{prop:h}
Let $h$ be a vector of $N\sb{p, \sigma}$ such that $h^2\ge 4$,
and let $X$ be a supersingular $K3$ surface in characteristic $p$
with $\sigma (X)=\sigma$.
{\rm (1)}
The following conditions are equivalent:
\begin{itemize}
\item[{\rm (i)}]
There exist no vectors $u\in N\sb{p, \sigma}$ satisfying $hu=1$ or $2$ and $u^2=0$,
and there exist no vectors $b\in N\sb{p, \sigma}$ satisfying $h=2b$ and $b^2=2$.
\item[{\rm (ii)}]
There exists an isometry $\phi : N\sb{p, \sigma} \smash{\mathop{\;\to\;}\limits\sp{\sim\;}} \operatorname{{\it NS}}\nolimits(X)$
such that $\phi (h)$ is the class $[L]$ of a line bundle $L$ that is very ample
modulo $(-2)$-curves.
\end{itemize}
{\rm (2)}
Suppose that the conditions in {\rm (1)} are fulfilled,
and let $L$ be a line bundle very ample
modulo $(-2)$-curves such that $\phi (h)=[L]$ by some isometry $\phi$.
Then $Y\sb{(X, L)}$ has only rational double points as its singularities,
and the $ADE$-type $\mathord{\mathcal R}\sb{(X, L)}$ of
$\operatorname{\rm Sing}\nolimits Y\sb{{(X, L)}}$ is equal to that of the root system
$$
\operatorname{{\rm Roots}}\nolimits (h\sp{\perp}):=\set{r\in N\sb{p, \sigma}}{rh=0,\; r^2=-2}.
$$
\end{proposition}
For the proof, we use the following results due to
Nikulin~\cite[Proposition 0.1]{Nikulin91} and
Saint-Donat~\cite[Section 5]{SD}.
\begin{proposition}[Nikulin \cite{Nikulin91}]\label{prop:nikulin1}
Let $L$ be a nef line bundle on a $K3$ surface $X$
with $L^2>0$.
If $|L|$ has a fixed component,
then
$|L|$ is equal to $m|U|+\Gamma$,
where $\Gamma$ is the fixed part of $|L|$,
$|U|$ is an elliptic pencil,
and
$U^2=0$, $U\Gamma=1$, $\Gamma^2=-2$, $m=\dim |L|=L^2/2+1$ hold.
If $|L|$ has no fixed components,
then a general member of $|L|$ is irreducible and $\dim |L|=L^2/2+1$.
\end{proposition}
\begin{proposition}[Saint-Donat \cite{SD}]\label{prop:SD1}
Let $|L|$ be a complete linear system without fixed components on a $K3$ surface $X$
such that $L^2\ge 4$.
Then the morphism $\Phi\sb{|L|}$ fails to be birational onto its image
if and only if one of the following holds:
\begin{itemize}
\item[{\rm (i)}]
There exists an irreducible curve $U$ such that $U^2=0$ and $UL=2$.
\item[{\rm (ii)}]
There exists an irreducible curve $B$ such that $B^2=2$ and $L=\mathord{\mathcal O}\sb X (2B)$.
\end{itemize}
\end{proposition}
\begin{proof}[Proof of Proposition~\ref{prop:h}]
The assertion (2) follows from~\cite[Theorem 6.1]{SD}
and~\cite[Lemma 2.4]{Shimada2003}. We now prove (1).
\par
Suppose that the condition (i) in (1) holds.
By~\cite[Section 3, Proposition 3]{RS},
there exists an isometry $\phi : N\sb{p, \sigma} \smash{\mathop{\;\to\;}\limits\sp{\sim\,}} \operatorname{{\it NS}}\nolimits (X)$ such that $\phi (h)$ is
the class of a \emph{nef} line bundle $L$.
By Proposition~\ref{prop:nikulin1},
$|L|$ is fixed component free.
By Proposition~\ref{prop:SD1},
$\Phi \sb{|L|}$ is birational onto its image. So (ii) is true.
\par
Conversely, suppose that (ii) holds.
We assume that there exists a vector $u\in N\sb{p, \sigma}$ satisfying $hu=1$ or $2$ and $u^2=0$,
and derive a contradiction by the argument in~\cite[Proof of Proposition 1.7]{U}.
By the Riemann-Roch theorem,
$\phi (u)$ is the class $[U]$ of an effective divisor $U$
such that $\dim |U|\ge 1$.
Let $D+\Delta$ be a general member of $|U|$,
where $\Delta$ is the fixed part of $|U|$.
We have $D\ne 0$ and $D^2\ge 0$.
If $DL=0$, then $D^2<0$ would follow by Hodge index theorem,
a contradiction.
Since $L$ is nef, $\Delta L\ge 0$.
Therefore, we have $DL=1$ or $2$.
Then the image of $D$ by $\Phi\sb{|L|}$ is either a line or a plane conic.
In any case, we have $\dim |D|=0$,
which is a contradiction.
\par
Next we assume that there exists a vector $b\in N\sb{p, \sigma}$
such that $h=2b$ and $b^2=2$.
Let $B$ be an effective divisor such that $\phi (b)=[B]$.
Since $[B]=[L]/2$,
$B$ is nef.
If there exists an irreducible member in $|B|$,
then Proposition~\ref{prop:SD1} implies that $\Phi\sb{|L|}$ is not birational
onto its image.
If there exist no irreducible members in $|B|$,
then Proposition~\ref{prop:nikulin1} implies that
$|B|$ has a fixed component, and $|B|$ is written as $2 |U|+\Gamma$,
where $UB=1$ and $U^2=0$.
Then $UL=2$ follows. Hence $\Phi\sb{|L|}$ is not birational onto its image,
and we get a contradiction. So (i) is true.
Thus the assertion (1) is proved.
\end{proof}
\begin{remark}\label{rem:degree8}
If there exists a vector $b$ such that $h=2b$ and $b^2=2$,
then $h$ is of degree $8$.
\end{remark}
\section{Proof of Theorem~\ref{thm:A}}\label{sec:thmA}
Theorem~\ref{thm:A} follows from the structure theorem of N\'eron-Severi lattices of supersingular $K3$ surfaces
(Theorems~\ref{thm:ARS} and~\ref{thm:NRS}), and a purely lattice-theoretic Lemma~\ref{lemma:lattice} below.
A sublattice $\Lambda\sp{\prime}\subset \Lambda$ is called \emph{primitive in $\Lambda$} if
$(\Lambda\sp{\prime}\otimes\mathord{\mathbb Q}) \cap \Lambda=\Lambda\sp{\prime}$ holds.
\begin{lemma}\label{lemma:lattice}
Let $N$ be an even hyperbolic $p$-elementary lattice of rank $22$
such that $\Disc{N}$ is isomorphic to $\mathord{\mathbb F}\sb p \sp{ 2 \sigma}$,
where $\sigma$ is a positive integer.
Suppose that $N$ contains a sublattice $R$ isomorphic to $\mathord{\mathbb Z} [10A\sb 2]$.
Then $p=3$, and the orthogonal complement $R\sp{\perp}$ of $R$ in $N$ is isomorphic to $U(1)$ or $U(3)$.
If $S\cong U(1)$, then $\sigma\le 5$,
while if $S\cong U(3)$, then $\sigma\le 6$.
\end{lemma}
\begin{proof}
We put $S:=R\sp{\perp}$,
which is an even hyperbolic lattice of rank $2$ primitive in $N$.
Then $N$ is an overlattice of the orthogonal direct sum $R\oplus S$.
We put
$$
H:= N/(R\oplus S).
$$
Clearly, we may assume that $H \ne (0)$.
\par
Note that $H$ is an isotropic subgroup of $ \Disc{R\oplus S}=\Disc{R} \oplus \Disc{S}$ with respect to
$\Discform {R\oplus S}=\Discform{R}\oplus\Discform{S}$,
and
$\Disc{N}\cong H\sp{\perp}/H$ is a $p$-elementary abelian group.
Since $S$ is primitive in $N$,
we have
\begin{equation}\label{eq:HcapS}
H\cap (0 \oplus \Disc{S}) =0.
\end{equation}
Let $l$ be a prime integer different from $3$ and $p$.
Assume that $\Disc{S}\sb{(l)}$ is not $0$.
Since $\Disc{N}\sb{(l)}=0$,
we see that $H\sb{(l)}$
is not $0$.
Since $\Disc {\mathord{\mathbb Z}[10A\sb 2]}\sb{(l)}=0$,
we have $H\sb{(l)}\subset (0 \oplus \Disc{S}\sb{(l)})$,
which contradicts~\eqref{eq:HcapS}.
Hence we obtain
\begin{equation}\label{eq:DiscSlpart}
\Disc{S}\sb{(l)}=0 \quad\textrm{for any prime $l$ distinct from $ 3$ and $ p$}.
\end{equation}
Let $m\sb 3 : \Disc{S}\sb{(3)} \to \Disc{S}\sb{(3)}$
be the homomorphism given by $m\sb 3 (x):=3x$.
Since every element of $\Disc{R}$ is annihilated by multiplication by $3$,
the image $H\sb{(3)}\sp S\subset \Disc{S}\sb{(3)}$ of
$H\sb{(3)}\subset \Disc{R}\sb{(3)}\oplus \Disc{S}\sb{(3)}$
by the projection to the factor $\Disc {S}\sb{(3)}$
is contained in $\operatorname{\rm Ker}\nolimits m\sb 3$ by~\eqref{eq:HcapS}:
\begin{equation}\label{eq:inKer}
H\sb{(3)}\sp S \;\;\subseteq\;\; \operatorname{\rm Ker}\nolimits m\sb 3.
\end{equation}
Therefore, $\Im m\sb 3$ is contained in the orthogonal complement
of $H\sb{(3)}\sp S$
with respect to ${\Discform{S}}\sb{(3)}$.
Hence we obtain
\begin{equation}\label{eq:Imm}
0\oplus \Im m\sb 3 \subset H\sp{\perp}\sb{(3)}.
\end{equation}
\par
\medskip
We assume $p\ne 3$, and derive a contradiction.
By~\eqref{eq:DiscSlpart},
we have
\begin{equation}\label{eq:prod}
\Disc{S} = \Disc {S}\sb{(3)} \times \Disc{S}\sb{(p)}.
\end{equation}
Since $\Disc{R}\sb{(p)}=0$,
the property~\eqref{eq:HcapS} implies $H\sb{(p)}=0$.
Therefore $\Disc{N}=\Disc{N}\sb{(p)}$ is isomorphic to $\Disc{S}\sb{(p)}$.
Since $\dim \sb{\mathord{\mathbb F}\sb p} \Disc{N}=2\sigma$ is positive and even,
and $S$ is of rank $2$,
we obtain
\begin{equation}\label{eq:ASp}
\Disc {S}\sb{(p)} \cong \mathord{\mathbb F}\sb p\sp{ 2}.
\end{equation}
On the other hand,
from $\Disc{N}\sb{(3)}=0$, we obtain
\begin{equation}\label{eq:H3}
H\sb{(3)}=H\sp{\perp}\sb{(3)}.
\end{equation}
By~\eqref{eq:HcapS},~\eqref{eq:Imm} and~\eqref{eq:H3},
we obtain $\Im m\sb 3 =0$; that is, $\Disc{S}\sb{(3)}$ is $3$-elementary.
From~\eqref{eq:H3},
we have
$10 + \dim \sb{\mathord{\mathbb F}\sb 3} \Disc{S}\sb{(3)}=2 \dim \sb{\mathord{\mathbb F}\sb 3} H\sb{(3)}$,
and hence
$\dim\sb{\mathord{\mathbb F}\sb 3 } \Disc{S}\sb{(3)}$ is even.
Since $S$ is of rank $2$,
we obtain
\begin{equation}\label{eq:AS3}
\Disc {S}\sb{(3)} \cong 0 \;\;\textrm{or}\;\; \mathord{\mathbb F}\sb 3\sp{ 2}.
\end{equation}
Suppose that $\Disc {S}\sb{(3)} \cong 0$.
Then $H\sb{(3)}$ can be regarded as an isotropic subgroup
of $\Disc{R}$ with respect to $\Discform{R}$.
Because $H\sb{(3)}=H\sp{\perp}\sb{(3)}$,
the corresponding overlattice of $R$ would be an even unimodular negative-definite lattice
of rank $20$.
This contradicts the classification of unimodular lattices~(\cite[Chapter V]{Serre}).
Suppose that $\Disc {S}\sb{(3)} \cong \mathord{\mathbb F}\sb 3\sp{ 2}$.
By~\eqref{eq:prod} and~\eqref{eq:ASp},
$S$ is an even indefinite lattice of rank $2$
such that $\Disc{S}\cong (\mathord{\mathbb Z}/3p\mathord{\mathbb Z})\sp{ 2}$.
By the classification of indefinite lattices of rank $2$~(\cite[Chapter 15, Section 3]{CS}),
we see that the intersection matrix of $S$ with respect to an appropriate basis is
$$
\begin{pmatrix}
0 & 3p \\
3p & 0
\end{pmatrix},
\qquad
\textrm{or}
\qquad
p=2\;\;\textrm{and}\;\;
\begin{pmatrix}
6 & 6 \\
6 & 0
\end{pmatrix}.
$$
In any case, the quadratic form $(\Disc{S}\sb{(3)}, {\Discform{S}}\sb{(3)})$ is isomorphic to
$$
\left(\:\mathord{\mathbb F}\sb 3\sp{ 2}, \: \left[ \begin{matrix} 0 & 1/3 \\ 1/3 & 0 \end{matrix} \right] \:\right)
\;\cong\;
(\Disc{U(3)}, \Discform{U(3)}).
$$
Therefore the isotropic subgroup $H\sb{(3)}$ of $\Disc {R}\oplus \Disc{S}\sb{(3)}$
satisfying $H\sb{(3)}=H\sp{\perp}\sb{(3)}$ would yield an even hyperbolic unimodular lattice of rank $22$
as an overlattice of $R\oplus U(3)$,
which again contradicts the classification of unimodular lattices.
Therefore $p=3$ is proved.
\par
\medskip
By~\eqref{eq:DiscSlpart}, we have
$\Disc {S}=\Disc {S}\sb{(3)}$,
and hence $H=H\sb{(3)}$ holds.
Suppose that $(\xi, \eta)\in H\sp{\perp}$,
where $\xi\in \Disc{R}$ and $\eta\in \Disc{S}$.
Since $H\sp{\perp}/H$ is $3$-elementary,
we have $(3\xi, 3\eta)=(0, 3\eta)\in H$.
By~\eqref{eq:HcapS},
we have $3\eta=0$.
Therefore
the image $(H\sp{\perp} )\sp S\subset \Disc{S}$ of
$H\sp{\perp}\subset \Disc{R}\oplus \Disc{S}$
by the projection to the factor $\Disc {S}$
is contained in $\operatorname{\rm Ker}\nolimits m\sb 3$:
\begin{equation}\label{eq:inKer2}
(H\sp{\perp})\sp S \;\;\subset\;\; \operatorname{\rm Ker}\nolimits m\sb 3.
\end{equation}
Next we will show that $S$ is isomorphic to $U(1)$ or $U(3)$.
Since $H\sp{\perp}/H$ is $3$-elementary,
\eqref{eq:HcapS} and~\eqref{eq:Imm} implies that $m\sb 3 (\Im m\sb 3)=0$;
that is, $9x=0$ for any $x\in \Disc {S}$.
Since
$$
2\sigma =\dim\sb{\mathord{\mathbb F}\sb 3} (H\sp{\perp}/H) =10 + \log\sb 3 | \Disc{S}| - 2 \log \sb 3 |H|
$$
is even and $S$ is of rank $2$,
$\Disc {S}$ is isomorphic to $0$, $\mathord{\mathbb F}\sb 3\sp 2$, $\mathord{\mathbb Z}/9\mathord{\mathbb Z}$ or $(\mathord{\mathbb Z}/9\mathord{\mathbb Z})\sp 2$.
We first assume that $\Disc{S}$ is a cyclic group of order $9$, and derive a contradiction.
Let $\gamma$ be a generator of $\Disc{S}$.
We have $\Im m\sb 3 =\operatorname{\rm Ker}\nolimits m\sb 3 =\langle 3\gamma\rangle$.
Let $H\sp R\subset \Disc{R}$ and $H\sp S\subset \Disc{S}$ be the images of
$H\subset \Disc{R}\oplus \Disc{S}$ by the projections to
the factors $\Disc{R}$ and $\Disc{S}$, respectively.
\begin{claim}\label{claim:1}
We have
$$
H\sp{\perp}=(H\sp R)\sp{\perp} \oplus (H\sp S)\sp{\perp},
$$
where
$(H\sp R)\sp{\perp}\subset \Disc{R}$ and $(H\sp S)\sp{\perp}\subset \Disc{S}$ are
the orthogonal complements of $H\sp R$ and $H\sp S$
with respect to $\Discform{R}$ and $\Discform{S}$, respectively.
In particular, we have $ (H\sp S)\sp{\perp}=(H\sp{\perp})\sp S$.
\end{claim}
\begin{proof}
It is obvious that
$H\sp{\perp}$ contains $(H\sp R)\sp{\perp} \oplus (H\sp S)\sp{\perp}$.
Suppose that $(\xi, \eta)\in H\sp{\perp}$, where $\xi\in \Disc{R}$ and $\eta\in \Disc{S}$.
By~\eqref{eq:inKer2}, we have $\eta\in\operatorname{\rm Ker}\nolimits m\sb 3=\Im m\sb 3$.
By~\eqref{eq:Imm}, we have $(0, \eta)\in H\sp{\perp}$ and hence $(\xi, 0)\in H\sp{\perp}$
hold.
Because $(\xi, 0)\in (H\sp R)\sp{\perp}$ and $(0, \eta)\in (H\sp S)\sp{\perp}$,
Claim~\ref{claim:1} is proved.
\end{proof}
Because $H\sp{\perp}/H$ is $3$-elementary,
we have $(0, \gamma)\notin H\sp{\perp}$
by~\eqref{eq:HcapS}.
Hence we obtain
\begin{equation}\label{eq:ne}
(H\sp S)\sp{\perp}=(H\sp{\perp})\sp S \ne \Disc{S}.
\end{equation}
By~\eqref{eq:inKer}, $H\sp S$ is either $0$ or $\operatorname{\rm Ker}\nolimits m\sb 3$.
If $H\sp S=0$, then $(H\sp S)\sp{\perp}=\Disc{S}$ and we get a contradiction to~\eqref{eq:ne}.
Suppose that $H\sp S=\operatorname{\rm Ker}\nolimits m\sb 3$.
Then $(H\sp S)\sp{\perp}\supset \Im m\sb 3$, and hence
$(H\sp S)\sp{\perp}= \Im m\sb 3$ by~\eqref{eq:ne}.
In particular,
we have
\begin{equation}\label{eq:one}
\log \sb 3 |(H\sp S)\sp{\perp}|=1.
\end{equation}
Since $|(H\sp R)\sp{\perp}|=3^{10}/ |H\sp R|$ and $H\cong H\sp R$ by~\eqref{eq:HcapS},
we see that
$$
2\sigma = \log \sb 3 |H\sp{\perp}/H|=\log \sb 3 |(H\sp R)\sp{\perp}| + \log \sb 3 |(H\sp S)\sp{\perp}| -\log \sb 3 |H|
=10-2 \log \sb 3 |H| +1
$$
is odd by~\eqref{eq:one}, which is absurd.
Therefore $\Disc{S}\not\cong \mathord{\mathbb Z}/9\mathord{\mathbb Z}$.
Because $S$ is an even lattice,
the classification of indefinite lattices of rank $2$~(\cite[Chapter 15, Section 3]{CS})
implies the following:
\begin{eqnarray*}
\Disc{S}=0 & \;\;\Longrightarrow\;\; & S\cong U(1), \\
\Disc{S}\cong \mathord{\mathbb F}\sb 3\sp 2 & \;\;\Longrightarrow\;\; & S\cong U(3), \\
\Disc{S}\cong (\mathord{\mathbb Z}/9\mathord{\mathbb Z})\sp 2 & \;\;\Longrightarrow\;\; & S\cong U(9).
\end{eqnarray*}
Next we assume $S\cong U(9)$, and derive a contradiction.
Note that $\operatorname{\rm Ker}\nolimits m\sb 3$ is
generated by
$$
3\bar e\sp{\vee}=f/3 \;\bmod S \quad\textrm{and}\quad 3\bar f\sp{\vee}=e/3 \;\bmod S.
$$
By~\eqref{eq:inKer2}, we have
\begin{equation}\label{eq:inF}
H\sp{\perp}\;\;\subset\;\; \Disc{R} \oplus \operatorname{\rm Ker}\nolimits m\sb 3.
\end{equation}
Then $H$ is also contained in $ \Disc{R} \oplus \operatorname{\rm Ker}\nolimits m\sb 3$.
Suppose that $H$ is generated by
$$
g\sp{(\nu)}=\xi\sp{(\nu)}\sb 1 \gamma\sb 1 +\cdots + \xi\sp{(\nu)}\sb{10} \gamma\sb{10} +
\eta\sp{(\nu)}\sb 1 (3\bar e\sp{\vee} ) + \eta\sp{(\nu)}\sb 2 (3\bar f\sp{\vee})
\qquad(\nu=1, \dots, r)
$$
where $\xi\sp{(\nu)}\sb i, \eta\sp{(\nu)}\sb j\in \mathord{\mathbb F}\sb 3$.
We put
$$
M:=\left[
\begin{array}{cccc|cc}
\xi\sb{1}\sp{(1)} & \dots & \dots & \xi\sb{10}\sp{(1)} &\eta\sp{(1)}\sb 2 & \eta\sp{(1)}\sb 1\\
& \dots & \dots & & & \\
& \dots & \dots & & & \\
\xi\sb{1}\sp{(r)} & \dots & \dots & \xi\sb{10}\sp{(r)} &\eta\sp{(r)}\sb 2 & \eta\sp{(r)}\sb 1\\
\end{array}
\right].
$$
From~\eqref{eq:discform10At} and~\eqref{eq:discformUm}, an element
$$
x\sb 1\gamma\sb 1 +\cdots + x\sb{10} \gamma\sb{10} +
y\sb 1 \bar e\sp{\vee} + y\sb 2 \bar f\sp{\vee} \qquad
(x\sb 1, \dots, x\sb{10} \in \mathord{\mathbb F}\sb 3, \;\; y\sb 1, y\sb 2 \in \mathord{\mathbb Z}/9\mathord{\mathbb Z})
$$
of $\Disc{R} \oplus \Disc{S}$ is contained in $H\sp{\perp}$ if and only if
the vector
$\mathord{\bf x}:=[x\sb 1, \dots, x\sb{10}, y\sb 1, y\sb 2]$ satisfies the equation
\begin{equation}\label{eq:lineq}
M\cdot {}\sp T \mathord{\bf x} \equiv \mathord{\bf 0} \;\bmod 3.
\end{equation}
We consider~\eqref{eq:lineq} as a system of linear equations over $\mathord{\mathbb F}\sb 3$.
The property~\eqref{eq:inF} of $H\sp{\perp}$ implies that every solution of~\eqref{eq:lineq} in $\mathord{\mathbb F}\sb 3$
must satisfy
\begin{equation}\label{eq:yzero}
y\sb 1 =y\sb 2=0.
\end{equation}
Because of~\eqref{eq:HcapS} and hence $H \cong H^R$,
we can choose generators $g\sp{(1)}$, \dots, $g\sp{(r)}$ of $H$ in such a way that,
after suitable permutations of $10$ coordinates of $\Disc{R}=\mathord{\mathbb F}\sb 3\sp{10}$ if necessary,
the $r\times 12$ matrix $M$ is of the form
$$
M=\left[
\begin{array}{ccc|c}
&&& \\
&I\sb r &&* \\
&&& \\
\end{array}
\right],
$$
where $I\sb r$ ($r\le 10$) is a diagonal matrix
whose diagonal entries are $1$.
Now non-zero elements of
the subgroup of $H\cong H^R $ of $ H^{\perp}$
should be solutions of~\eqref{eq:lineq} in $\mathord{\mathbb F}\sb 3$, but
do not satisfy~\eqref{eq:yzero}.
Thus we get a contradiction.
Hence $S$ is isomorphic to $U(1)$ or $U(3)$.
If $S\cong U(1)$, then
$2\sigma =10 -2 \dim\sb{\mathord{\mathbb F}\sb 3} H\le 10 $,
while
if $S\cong U(3)$, then
$2\sigma =10+2 -2 \dim\sb{\mathord{\mathbb F}\sb 3} H\le 12 $.
\end{proof}
\begin{example}\label{example:quartic}
Let $[w:x:y:z]$ be homogeneous coordinates of $\P\sp 3$.
For homogeneous polynomials $f(y, z)$, $g(y, z)$ and $h(y, z)$ of degrees $3$, $3$ and $4$,
we consider the quartic surface $X$ defined in $\P\sp 3$ by
$$
w^3 y + x^3z + w f(y, z) + x g(y, z) + h(y, z) = 0.
$$
When $X$ is smooth,
$X$ is a supersingular $K3$ surface,
because $X$ contains a configuration of lines
as in ~\cite[Section 6]{Shimada92}.
It was shown in~\cite[Section 6]{Shimada92} that,
when $f$, $g$ and $h$ are general,
the Artin invariant of $X$ is $6$,
and hence the orthogonal complement $R\sp{\perp}$ of the sublattice $R\subset \operatorname{{\it NS}}\nolimits (X)$
generated by the classes of the lines $C\sb 1, D\sb 1, \dots, C\sb{10}, D\sb{10}$
(10 pairs of intersecting lines in 10 singular fibres of type IV)
is isomorphic to $U(3)$.
When $f$, $g$ are general and $h=0$,
the Artin invariant of $X$ is $5$ by~\cite[Section 4]{Shioda77}.
In this case,
the line $\ell$ defined by $w=x=0$ is contained in $X$.
Since $\ell^2=-2$ and $[\ell]\in R\sp{\perp}$,
$R\sp{\perp}$ is isomorphic to $U(1)$.
\end{example}
\section{Proof of Theorem~\ref{thm:U3}}\label{sec:U3}
The discriminant group $D$ of $\mathord{\mathbb Z}[10A\sb 2]\oplus U(3)$ is equal to
$$
\mathord{\mathbb F}\sb 3 \gamma\sb 1\oplus \cdots \oplus \mathord{\mathbb F}\sb 3 \gamma\sb{10} \oplus \mathord{\mathbb F}\sb 3 \bar e\sp{\vee} \oplus \mathord{\mathbb F}\sb 3 \bar f\sp{\vee},
$$
and the discriminant form $q$ of $\mathord{\mathbb Z}[10A\sb 2]\oplus U(3)$ is given by
$$
q(x\sb 1, \dots, x\sb{10}, y\sb 1, y\sb 2)=-2(x\sb 1\sp 2 + \cdots + x\sb{10}\sp 2 )/3 + 2 y\sb 1 y\sb 2 /3
\;\;\in\;\;\mathord{\mathbb Q}/2\mathord{\mathbb Z}.
$$
We consider subgroups of $D$ as ternary codes.
Recall from \S\ref{sec:disc} that
the Hamming weight of a word $\mathord{\bf x}=(x\sb 1, \dots, x\sb{10})\in \Disc{\mathord{\mathbb Z}[10A\sb 2]}$ is defined by
$$
\operatorname{\rm wt}\nolimits (\mathord{\bf x}):=|\set{i}{x\sb i\ne 0}|.
$$
Then a ternary code $\mathord{\mathcal C}\subset D$ is isotropic with respect to
$q$ if and only if
\begin{equation}\label{eq:isot3}
\operatorname{\rm wt}\nolimits (\mathord{\bf x})\equiv y\sb 1 y\sb 2 \;\bmod 3\;\;\textrm{for any \;\;$(\mathord{\bf x}, y\sb 1, y\sb 2)\in \mathord{\mathcal C}$}
\end{equation}
holds.
A ternary code $\mathord{\mathcal C}\subset D$ satisfying~\eqref{eq:isot3} is therefore called an \emph{isotropic code}.
For an isotropic code $\mathord{\mathcal C}$,
we denote by $N\sb{\mathord{\mathcal C}}$ the overlattice of $\mathord{\mathbb Z}[10A\sb 2]\oplus U(3)$
corresponding to $\mathord{\mathcal C}$ by Proposition~\ref{prop:nikulin}.
By Theorem~\ref{thm:NRS},
$N\sb{\mathord{\mathcal C}}$ is isomorphic to the lattice $N\sb{3, \sigma}$, where $\sigma =6-\dim \mathord{\mathcal C}$.
It is easy to see that the following conditions for an isotropic code $\mathord{\mathcal C}$ are equivalent:
\begin{itemize}
\item[(i)] $\operatorname{\rm wt}\nolimits (\mathord{\bf x})>0 \;\;\; \textrm{for any non-zero word \;$(\mathord{\bf x}, y\sb 1, y\sb 2)\in \mathord{\mathcal C}$}$,
\item[(ii)] $U(3)$ is primitive in $N\sb{\mathord{\mathcal C}}$, and
\item[(iii)] $\mathord{\mathbb Z} [10A\sb 2]\sp{\perp}=U(3)$ in $N\sb{\mathord{\mathcal C}}$.
\end{itemize}
We say that an isotropic code $\mathord{\mathcal C}$ is \emph{admissible} if $\mathord{\mathcal C}$ satisfies the conditions above.
Let $h=a e+ bf$ be a vector of $U(3)$ with $a\ge 1$ and $b\ge 1$.
We have
$h^2=6ab$.
\begin{lemma}\label{lemma:abU3}
Let $\mathord{\mathcal C}$ be an admissible isotropic code.
{\rm (1)} There exists a vector $u\in N\sb{\mathord{\mathcal C}}$ satisfying $hu=1$ or $2$ and $u^2=0$ if and only if
the following hold:
\begin{itemize}
\item[{\rm ($\alpha$)}] $a=b=1$, and
\item[{\rm ($\beta$)}] there exists $(\mathord{\bf x}, y\sb 1, y\sb 2)\in \mathord{\mathcal C}$ such that $\operatorname{\rm wt}\nolimits (\mathord{\bf x})=1$.
\end{itemize}
{\rm (2)} The set of roots
$\operatorname{{\rm Roots}}\nolimits (h\sp{\perp}):=\shortset{r\in N\sb{\mathord{\mathcal C}}}{rh=0,\; r^2=-2}$ in $h\sp{\perp}$
is strictly larger than $\operatorname{{\rm Roots}}\nolimits(\mathord{\mathbb Z}[10A\sb 2])=\{ \pm c\sb i, \pm d\sb i, \pm (c\sb i + d\sb i)\}$
if and only if one of the following holds:
\begin{itemize}
\item[{\rm (a)}]
there exists $(\mathord{\bf x}, y\sb 1, y\sb 2)\in \mathord{\mathcal C}$ such that $\operatorname{\rm wt}\nolimits (\mathord{\bf x})=3$ and $y\sb 1=y\sb 2=0$, or
\item[{\rm (b)}]
$a=b$,
and
there exists $(\mathord{\bf x}, y\sb 1, y\sb 2)\in \mathord{\mathcal C}$ such that $\operatorname{\rm wt}\nolimits (\mathord{\bf x})=2$, or
\item[{\rm (c)}]
{\rm (}$a=2b\;\;\textrm{or}\;\;b=2a${\rm )}
and
there exists $(\mathord{\bf x}, y\sb 1, y\sb 2)\in \mathord{\mathcal C}$ such that $\operatorname{\rm wt}\nolimits (\mathord{\bf x})=1$.
\end{itemize}
\end{lemma}
\begin{proof}
We prove (1) first. Suppose that a vector
\begin{equation}\label{eq:u}
u=r\sb u + \eta\sb 1 e\sp{\vee} + \eta\sb 2 f\sp{\vee} \quad (r\sb u \in \mathord{\mathbb Z}[10A\sb 2]\sp{\vee}, \;\; \eta\sb 1, \eta\sb 2\in \mathord{\mathbb Z})
\end{equation}
of $N\sb{\mathord{\mathcal C}}$ satisfies $hu=1$ or $2$ and $u^2=0$.
Then we have
\begin{eqnarray}
&& a \eta\sb 1 + b \eta\sb 2 = 1 \;\textrm{or}\; 2, \label{eq:1or2} \\
&& r\sb u\sp 2 + 2 \eta\sb 1 \eta\sb 2/3 =0. \label{eq:rueta}
\end{eqnarray}
Note that $(\eta\sb 1, \eta\sb 2)\not\equiv (0,0) \bmod 3$ by~\eqref{eq:1or2}.
Since $\mathord{\mathcal C}$ is admissible, we have $r\sb u\ne 0$ by , and hence
$\eta\sb 1\eta\sb 2>0$ by~\eqref{eq:rueta}.
From~\eqref{eq:1or2},
we obtain
$$
a=b=1, \qquad \eta\sb 1=\eta\sb 2=1,
$$
and hence, from~\eqref{eq:rueta}, we have
$$
r\sb u\sp 2 =-2/3.
$$
By~\eqref{eq:wtr}, the word
$$
\bar u=u \; \bmod (\mathord{\mathbb Z}[10A\sb 2]\oplus U(3))=(\bar{r}\sb u, \bar\eta\sb 1, \bar\eta\sb 2)\qquad
(\textrm{where $\bar{r}\sb u= r\sb u \bmod \mathord{\mathbb Z}[10A\sb 2]$})
$$
of $\mathord{\mathcal C}$ has the property $\operatorname{\rm wt}\nolimits (\bar{r}\sb u)=1$.
\par
Conversely, suppose that $a=b=1$ and that
there exists a word $(\bar r , y\sb 1, y\sb 2)\in \mathord{\mathcal C}$ such that $\operatorname{\rm wt}\nolimits (\bar r)=1$.
Replacing $(\bar r , y\sb 1, y\sb 2)$ by $(-\bar r , -y\sb 1, -y\sb 2)$
if necessary, we can assume that $y\sb 1=y\sb 2=1$ by
~\eqref{eq:isot3}.
Then, by
~\eqref{eq:wtrinv}, there exists a vector
$$
u=r+ e\sp{\vee} + f\sp{\vee}\qquad(r\in \mathord{\mathbb Z}[10A\sb 2]\sp{\vee})
$$
in $N\sb{\mathord{\mathcal C}}$ satisfying $r^2=-2/3$.
This vector $u$ satisfies $hu=2$ and $u^2=0$.
Thus the assertion (1) is proved.
\par
We now prove (2).
Suppose that a vector $u\in N\sb{\mathord{\mathcal C}}$ given by~\eqref{eq:u} satisfies $hu=0$, $u^2=-2$ and $u\notin \operatorname{{\rm Roots}}\nolimits(\mathord{\mathbb Z}[10A\sb 2])$.
Then we have
\begin{eqnarray}
&& a \eta\sb 1 + b \eta\sb 2 = 0, \label{eq:eta0} \\
&& r\sb u\sp 2 + 2 \eta\sb 1 \eta\sb 2/3 =-2. \label{eq:rueta2}
\end{eqnarray}
Suppose that $\eta\sb 1=0$ or $\eta\sb 2 =0$.
Then~\eqref{eq:eta0} implies $\eta\sb 1=\eta\sb 2=0$ and hence
$\operatorname{\rm wt}\nolimits (\bar{r}\sb u)\equiv 0 \bmod 3$ holds because $\mathord{\mathcal C}$ is isotropic.
By~\eqref{eq:wtr} and~\eqref{eq:rueta2}, we have $\operatorname{\rm wt}\nolimits (\bar{r}\sb u)\le 3$.
If $\operatorname{\rm wt}\nolimits (\bar{r}\sb u)=0$, then $u=r\sb u$ is contained in $\operatorname{{\rm Roots}}\nolimits (\mathord{\mathbb Z}[10A\sb 2])$.
Hence we have $\operatorname{\rm wt}\nolimits (\bar{r}\sb u)=3$,
and therefore the condition (a) is satisfied.
Suppose that $\eta\sb 1 \ne 0$ and $\eta\sb 2\ne 0$.
By~\eqref{eq:eta0}, we have $\eta\sb 1\eta\sb 2<0$.
By~\eqref{eq:wtr} and~\eqref{eq:rueta2},
we see that the pair $(\eta\sb 1\eta\sb 2, \operatorname{\rm wt}\nolimits (\bar{r}\sb u))$ is either $(-1, 2)$ or $(-2, 1)$.
In the former case, we have $a=b$ by~\eqref{eq:eta0} and hence (b) is satisfied.
In the latter case, we have $a=2b$ or $b=2a$ by~\eqref{eq:eta0} and hence (c) is satisfied.
\par
Conversely, suppose that (a) is fulfilled.
Using~\eqref{eq:wtrinv}, we have a lift
$$
u=r+0+0\in N\sb{\mathord{\mathcal C}}\qquad (r\in \mathord{\mathbb Z} [10A\sb 2]\sp{\vee})
$$
of the word $(\bar r , 0, 0)\in \mathord{\mathcal C}$ with $\operatorname{\rm wt}\nolimits(\bar r)=3$
such that $r^2=-2$. Then $u\in \operatorname{{\rm Roots}}\nolimits(h\sp{\perp})\setminus \operatorname{{\rm Roots}}\nolimits (\mathord{\mathbb Z}[10A\sb 2])$.
Suppose that (b) is satisfied.
A vector
$$
u=r+e\sp{\vee}-f\sp{\vee}\;\;\in\;\; N\sb{\mathord{\mathcal C}}
$$
with $\operatorname{\rm wt}\nolimits(\bar r)=2$ and $r^2=-4/3$
satisfies $u\in \operatorname{{\rm Roots}}\nolimits(h\sp{\perp})\setminus \operatorname{{\rm Roots}}\nolimits (\mathord{\mathbb Z}[10A\sb 2])$.
Suppose that (c) is satisfied and assume that $a=2b$.
A vector
$$
u=r+e\sp{\vee}-2 f\sp{\vee}\;\;\in\;\; N\sb{\mathord{\mathcal C}}
$$
with $\operatorname{\rm wt}\nolimits(\bar r)=1$ and $r^2=-2/3$
satisfies $u\in \operatorname{{\rm Roots}}\nolimits(h\sp{\perp})\setminus \operatorname{{\rm Roots}}\nolimits (\mathord{\mathbb Z}[10A\sb 2])$.
Thus the assertion (2) is proved.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:U3}]
The implication $\textrm{(iii)}\Longrightarrow\textrm{(ii)}$ is obvious.
Since every vector $h$ of $U(3)$ satisfies $h^2\equiv 0 \bmod 6$,
the implication $\textrm{(ii)}\Longrightarrow\textrm{(i)}$ is also obvious.
Using computer,
we can prove the following Claim~\ref{claim:sevencodes}.
See Remark~\ref{remark:sevencodes} and Table~\ref{table:sevencodes7}.
\begin{claim}\label{claim:sevencodes}
There exists an isotropic admissible code $\mathord{\mathcal C}\subset D$ of dimension $5$ with the following property:
\begin{equation}\label{eq:theproperty}
\parbox{10cm}{
every non-zero word $(\mathord{\bf x}, y\sb 1, y\sb 2)\in \mathord{\mathcal C}$ satisfies the following:
(i) $\operatorname{\rm wt}\nolimits (\mathord{\bf x})\ge 3$, and (ii) if $\operatorname{\rm wt}\nolimits (\mathord{\bf x})=3$, then $(y\sb 1, y\sb 2)\ne(0, 0)$.
}
\end{equation}
\end{claim}
We now prove $\textrm{(i)}\Longrightarrow\textrm{(iii)}$
Suppose that an integer $d=6m \;\; (m\in \mathord{\mathbb Z}\sb{>0})$ is given.
Let $X$ be a supersingular $K3$ surface in characteristic $3$
with Artin invariant $\sigma\le 6$.
For the basis $e, f$ of $U(3)$ at the end of Section 2, we put
$$
h:=e+mf.
$$
Then $h^2=d$.
Let $\mathord{\mathcal C} (\sigma)$ be a linear subspace of the code $\mathord{\mathcal C}$
in Claim~\ref{claim:sevencodes}
with $\dim \mathord{\mathcal C}(\sigma)=6-\sigma$.
Since $\mathord{\mathcal C}(\sigma)$ is isotropic,
the corresponding overlattice $N\sb{\mathord{\mathcal C}(\sigma)}$
of $\mathord{\mathbb Z}[10A\sb 2]\oplus U(3)$ is isomorphic to $N\sb{3, \sigma}$ by Theorem~\ref{thm:NRS}.
Hence there exists an isometry
$$
\phi : N\sb{\mathord{\mathcal C}(\sigma)}\smash{\mathop{\;\to\;}\limits\sp{\sim\,}} \operatorname{{\it NS}}\nolimits(X)
$$
by Theorem~\ref{thm:ARS}.
Since every word of $\mathord{\mathcal C} (\sigma)$ satisfies the conditions (i) and (ii) in~\eqref{eq:theproperty},
Lemma~\ref{lemma:abU3} implies that
there exist no vectors $u$ in $N\sb{\mathord{\mathcal C}(\sigma)}$ satisfying $hu=1$ or $2$ and $u^2=0$,
and that the set of roots in the orthogonal complement $h\sp{\perp}$ of $h$ in $N\sb{\mathord{\mathcal C}(\sigma)}$
coincides with $\operatorname{{\rm Roots}}\nolimits(\mathord{\mathbb Z}[10A\sb 2])$.
By Proposition~\ref{prop:h} and Remark~\ref{rem:degree8},
we can choose the isometry
$\phi : N\sb{\mathord{\mathcal C}(\sigma)}\smash{\mathop{\;\to\;}\limits\sp{\sim\,}} \operatorname{{\it NS}}\nolimits(X)$ in such a way that $\phi (h)$ is the class $[L]$
of a line bundle $L$ very ample modulo $(-2)$-curves
such that $\Phi\sb{|L|}$ induces a contraction $\rho\sb L : X\to Y\sb{{(X, L)}}$
of an $ADE$-configuration of $(-2)$-curves of type $10A\sb 2$.
Since $\mathord{\mathcal C} (\sigma)$ is admissible,
we see that $R\sp{\perp} \sb{{(X, L)}}\subset \operatorname{{\it NS}}\nolimits (X)$ is isomorphic to $U(3)$.
Thus $X$ admits a polarization $L$ of degree $d$ with the hoped-for properties.
\end{proof}
\begin{table}
$$
\mathord{\mathcal C}\sb 1\quad : \quad
\left [\begin {array}{cccccccccccc}
1&0&0&0&0&0&0&0&1&1&0&1\\
0&1&0&0&0&0&0&1&0&1&2&0\\
0&0&1&0&0&0&1&0&1&0&2&0\\
0&0&0&1&0&0&1&1&0&0&0&1\\
0&0&0&0&1&0&1&1&1&1&1&2
\end {array}\right ]
$$
\par\smallskip
$$
\mathord{\mathcal C}\sb 2\quad : \quad \left [\begin {array}{cccccccccccc}
1&0&0&0&0&0&0&0&1&1&0&1\\
0&1&0&0&0&0&0&1&0&1&2&0\\
0&0&1&0&0&0&1&0&1&0&2&0\\
0&0&0&1&0&0&1&1&0&0&0&1\\
0&0&0&0&1&1&1&2&2&1&0&0
\end {array}\right ]
$$
\par\smallskip
$$
\mathord{\mathcal C}\sb 3\quad : \quad \left [\begin {array}{cccccccccccc}
1&0&0&0&0&0&0&0&1&1&0&1\\
0&1&0&0&0&0&1&1&0&0&0&1\\
0&0&1&0&0&1&0&1&0&1&2&2\\
0&0&0&1&0&1&1&0&1&0&2&2\\
0&0&0&0&1&1&1&2&2&1&0&1
\end {array}\right ]
$$
\par\smallskip
$$
\mathord{\mathcal C}\sb 4\quad : \quad \left [\begin {array}{cccccccccccc}
1&0&0&0&0&0&0&0&1&1&0&1\\
0&1&0&0&0&0&1&1&0&0&0&1\\
0&0&1&0&0&1&0&1&0&1&2&2\\
0&0&0&1&0&1&1&0&1&0&2&2\\
0&0&0&0&1&1&2&2&2&2&2&0
\end {array}\right ]
$$
\par\smallskip
$$
\mathord{\mathcal C}\sb 5\quad : \quad \left [\begin {array}{cccccccccccc}
1&0&0&0&0&0&0&1&1&1&1&1\\
0&1&0&0&0&0&1&0&1&1&2&2\\
0&0&1&0&0&1&0&1&0&1&2&2\\
0&0&0&1&0&1&1&0&0&1&1&1\\
0&0&0&0&1&1&1&1&1&1&0&0
\end {array}\right ]
$$
\par\smallskip
$$
\mathord{\mathcal C}\sb 6\quad : \quad \left [\begin {array}{cccccccccccc}
1&0&0&0&0&0&0&1&1&1&1&1\\
0&1&0&0&0&0&1&0&1&1&2&2\\
0&0&1&0&0&1&0&1&0&1&2&2\\
0&0&0&1&0&1&1&0&0&1&1&1\\
0&0&0&0&1&1&2&2&1&0&1&2
\end {array}\right ]
$$
\par\smallskip
$$
\mathord{\mathcal C}\sb 7\quad : \quad \left [\begin {array}{cccccccccccc}
1&0&0&0&0&0&1&1&1&1&1&2\\
0&1&0&0&0&1&0&1&1&2&2&1\\
0&0&1&0&0&1&1&0&2&1&2&1\\
0&0&0&1&0&1&1&2&0&2&1&2\\
0&0&0&0&1&1&2&1&2&0&1&2
\end {array}\right ]
$$
\par\medskip
\caption{Bases of the codes
$\mathord{\mathcal C}\sb 1$, \dots, $\mathord{\mathcal C}\sb 7$}\label{table:sevencodes7}
\end{table}
\begin{table}
\begin{eqnarray*}
\operatorname{\rm we}\nolimits(\mathord{\mathcal C}\sb 1) &=& 1+12\,{z}^{3}+18\,{z}^{4}+36\,{z}^{5}+108\,{z}^{6}+36\,{z}^{7}+18\,{z}^{8}+14\,{z}^{9}, \\
\operatorname{\rm we}\nolimits(\mathord{\mathcal C}\sb 2) &=& 1+8\,{z}^{3}+10\,{z}^{4}+24\,{z}^{5}+86\,{z}^{6}+40\,{z}^{7}+30\,{z}^{8}+40\,{z}^{9}+4\,{z}^{10},\\
\operatorname{\rm we}\nolimits(\mathord{\mathcal C}\sb 3) &=& 1+4\,{z}^{3}+8\,{z}^{4}+24\,{z}^{5}+94\,{z}^{6}+44\,{z}^{7}+30\,{z}^{8}+36\,{z}^{9}+2\,{z}^{10}, \\
\operatorname{\rm we}\nolimits(\mathord{\mathcal C}\sb 4) &=& 1+6\,{z}^{3}+6\,{z}^{4}+18\,{z}^{5}+102\,{z}^{6}+42\,{z}^{7}+36\,{z}^{8}+26\,{z}^{9}+6\,{z}^{10}, \\
\operatorname{\rm we}\nolimits(\mathord{\mathcal C}\sb 5) &=& 1+30\,{z}^{4}+60\,{z}^{6}+120\,{z}^{7}+20\,{z}^{9}+12\,{z}^{10}, \\
\operatorname{\rm we}\nolimits(\mathord{\mathcal C}\sb 6) &=& 1+18\,{z}^{4}+18\,{z}^{5}+96\,{z}^{6}+36\,{z}^{7}+36\,{z}^{8}+38\,{z}^{9}, \\
\operatorname{\rm we}\nolimits(\mathord{\mathcal C}\sb 7) &=& 1+72\,{z}^{5}+60\,{z}^{6}+90\,{z}^{8}+20\,{z}^{9}.
\end{eqnarray*}
\par\medskip
\caption{Weight-enumerators}\label{table:wes}
\end{table}
\begin{remark}\label{remark:sevencodes}
Let $G$ denote the group of linear automorphisms of $D\cong \mathord{\mathbb F}\sb 3 \sp{10}\oplus \mathord{\mathbb F}\sb 3 \sp 2 $
generated by
\begin{eqnarray*}
(x\sb 1, \dots, x\sb{10}, y\sb 1, y\sb 2 ) &\mapsto &
(x\sb{\sigma(1)}, \dots, x\sb{{\sigma(10)}}, y\sb {\tau(1)}, y\sb {\tau(2)} )
\qquad (\sigma \in \mathord{\hbox{\mathgot S}}\sb{10}, \tau\in \mathord{\hbox{\mathgot S}}\sb 2), \quad\textrm{and}\\
(x\sb 1, \dots, x\sb{10}, y\sb 1, y\sb 2 ) &\mapsto &
((-1)\sp{\alpha\sb 1}x\sb 1, \dots, (-1)\sp{\alpha\sb {10}}x\sb {10}, (-1)\sp\beta y\sb 1, (-1)\sp\beta y\sb 2 )\\
&& \qquad (\alpha\sb 1, \dots, \alpha\sb{10}\in \mathord{\mathbb F}\sb 2, \beta\in \mathord{\mathbb F}\sb 2 ).
\end{eqnarray*}
Note that, if $\mathord{\mathcal C}\subset D$ is an isotropic admissible code,
then so is $g (\mathord{\mathcal C})$ for any $g\in G$.
We define the weight enumerator of a ternary code $\mathord{\mathcal C}$ by
$$
\operatorname{\rm we}\nolimits (\mathord{\mathcal C}):=\sum\sb{(\mathord{\bf x}, y\sb 1, y\sb 2)\in \mathord{\mathcal C}} z\sp{\operatorname{\rm wt}\nolimits (\mathord{\bf x})}.
$$
Using computer, we have proved that there exist at least seven isomorphism classes
of isotropic admissible codes of dimension $5$ with the property~\eqref{eq:theproperty}.
The representative codes $\mathord{\mathcal C}\sb 1, \dots, \mathord{\mathcal C}\sb 7$ of these classes are given in
Table~\ref{table:sevencodes7}.
Their weight-enumerators are given in Table~\ref{table:wes}.
\end{remark}
\begin{corollary}\label{cor:seven}
Let $X$ be a supersingular $K3$ surface in characteristic $3$
with Artin invariant $1$.
Then there exist at least seven line bundles $L\sb 1, \dots, L\sb 7$ of degree $6$
on $X$ that are mutually non-isomorphic and that induce contractions
of $10 A\sb 2$-configurations of $(-2)$-curves on $X$.
\end{corollary}
See Example~\ref{example:sigmaG0}.
\section{Proof of Theorem~\ref{thm:U1}}\label{sec:U1}
The proof of Theorem~\ref{thm:U1} is similar to and simpler than
that of Theorem~\ref{thm:U3}.
\par
\medskip
The discriminant group $D$ of $\mathord{\mathbb Z}[10A\sb 2]\oplus U(1)$ is equal to
$$
\mathord{\mathbb F}\sb 3 \gamma\sb 1\oplus \cdots \oplus \mathord{\mathbb F}\sb 3 \gamma\sb {10}.
$$
A ternary code $\mathord{\mathcal C}\subset D$ is isotropic with respect to
the discriminant form $q$ of $\mathord{\mathbb Z}[10A\sb 2]\oplus U(1)$ if and only if
\begin{equation}\label{eq:isot1}
\operatorname{\rm wt}\nolimits (\mathord{\bf x})\equiv 0 \;\bmod 3\;\; \textrm{for any \;\;$\mathord{\bf x}\in \mathord{\mathcal C}$}
\end{equation}
holds.
For an isotropic code $\mathord{\mathcal C}$,
we denote by $N\sb{\mathord{\mathcal C}}$ the overlattice of $\mathord{\mathbb Z}[10A\sb 2]\oplus U(1)$
corresponding to $\mathord{\mathcal C}$.
By Theorem~\ref{thm:NRS},
$N\sb{\mathord{\mathcal C}}$ is isomorphic to the lattice $N\sb{3, \sigma}$, where $\sigma =5-\dim \mathord{\mathcal C}$.
Let $h=a e+ bf$ be a vector of $U(1)$ with $a\ge 1$ and $b\ge 1$.
We have $h^2=2ab$.
\begin{lemma}\label{lemma:ab}
Let $\mathord{\mathcal C}$ be an isotropic code in $D\cong \mathord{\mathbb F}\sb 3 \sp{10}$.
{\rm (1)} There exists a vector $u\in N\sb{\mathord{\mathcal C}}$ satisfying $hu=1$ or $2$ and
$u^2=0$ if and only if $a\le 2$ or $b\le 2$.
{\rm (2)} The set of roots
$\operatorname{{\rm Roots}}\nolimits (h\sp{\perp}):=\shortset{r\in N\sb{\mathord{\mathcal C}}}{rh=0,\; r^2=-2}$ in $h\sp{\perp}$
is strictly larger than $\operatorname{{\rm Roots}}\nolimits(\mathord{\mathbb Z}[10A\sb 2])$
if and only if one of the following holds;
\begin{itemize}
\item[{\rm (a)}]
there exists $\mathord{\bf x}\in \mathord{\mathcal C}$ such that $\operatorname{\rm wt}\nolimits (\mathord{\bf x})=3$, or
\item[{\rm (b)}]
$a=b$.
\end{itemize}
\end{lemma}
\begin{proof}
We prove (1) first. Suppose that a vector
\begin{equation}\label{eq:uU1}
u=r\sb u + \eta\sb 1 f + \eta\sb 2 e \quad (r\sb u \in \mathord{\mathbb Z}[10A\sb 2]\sp{\vee}, \;\; \eta\sb 1, \eta\sb 2\in \mathord{\mathbb Z})
\end{equation}
of $\mathord{\mathbb Z}[10A\sb 2]\sp{\vee} \oplus U(1)\sp{\vee}=\mathord{\mathbb Z}[10A\sb 2]\sp{\vee} \oplus U(1)$ satisfies $hu=1$ or $2$ and $u^2=0$.
Then we have
\begin{eqnarray}
&& a \eta\sb 1 + b \eta\sb 2 = 1 \;\textrm{or}\; 2, \label{eq:1or2U1} \\
&& r\sb u\sp 2 + 2 \eta\sb 1 \eta\sb 2 =0. \label{eq:ruetaU1}
\end{eqnarray}
By~\eqref{eq:ruetaU1},
we have $\eta\sb 1\eta\sb 2 \ge 0$.
Using~\eqref{eq:1or2U1}, we have $a\le 2$ or $b\le 2$.
Conversely, if $a\le 2$, then $u=f$ satisfies $hu=a=1$ or $2$ and $u^2=0$.
Thus (1) is proved.
\par
Next we prove (2).
Suppose that a vector $u$ given by~\eqref{eq:uU1} satisfies $hu=0$, $u^2=-2$ and $u\notin \operatorname{{\rm Roots}}\nolimits(\mathord{\mathbb Z}[10A\sb 2])$.
Then we have
\begin{eqnarray}
&& a \eta\sb 1 + b \eta\sb 2 = 0, \label{eq:eta0U1} \\
&& r\sb u\sp 2 + 2 \eta\sb 1 \eta\sb 2 =-2. \label{eq:rueta2U1}
\end{eqnarray}
Because $\mathord{\mathcal C}$ is isotropic,
$\operatorname{\rm wt}\nolimits (\bar{r}\sb u)\equiv 0 \bmod 3$ holds.
If $\eta\sb 1=0$ or $\eta\sb 2 =0$, then~\eqref{eq:eta0U1} implies $\eta\sb 1=\eta\sb 2=0$.
By~\eqref{eq:wtr} and~\eqref{eq:rueta2U1}, we have $\operatorname{\rm wt}\nolimits (\bar{r}\sb u)\le 3$.
If $\operatorname{\rm wt}\nolimits (\bar{r}\sb u)=0$, then $u=r\sb u$ is contained in $\operatorname{{\rm Roots}}\nolimits (\mathord{\mathbb Z}[10A\sb 2])$.
Hence we have $\operatorname{\rm wt}\nolimits (\bar{r}\sb u)=3$,
and therefore the condition (a) is satisfied.
Suppose that $\eta\sb 1 \ne 0$ and $\eta\sb 2\ne 0$.
By~\eqref{eq:eta0U1}, we have $\eta\sb 1\eta\sb 2<0$.
By~\eqref{eq:rueta2U1},
we have $r\sb u=0$ and $\eta\sb 1\eta\sb 2=-1$,
and hence $a=b$ follows from~\eqref{eq:eta0U1}.
\par
Conversely, suppose that (a) is fulfilled.
Using~\eqref{eq:wtrinv}, we have a lift
$$
u=r+0+0\in N\sb{\mathord{\mathcal C}}\qquad (r\in \mathord{\mathbb Z} [10A\sb 2]\sp{\vee})
$$
of the word $\bar r \in \mathord{\mathcal C}$ with $\operatorname{\rm wt}\nolimits(\bar r)=3$
such that $r^2=-2$. Then $u$ is contained in $\operatorname{{\rm Roots}}\nolimits(h\sp{\perp})\setminus \operatorname{{\rm Roots}}\nolimits (\mathord{\mathbb Z}[10A\sb 2])$.
Suppose that (b) is satisfied.
The vector
$$
u=e-f\;\;\in\;\; N\sb{\mathord{\mathcal C}}
$$
satisfies $u\in \operatorname{{\rm Roots}}\nolimits(h\sp{\perp})\setminus \operatorname{{\rm Roots}}\nolimits (\mathord{\mathbb Z}[10A\sb 2])$.
\end{proof}
In order to prove Theorem~\ref{thm:U1}, it is therefore enough to show the following:
\begin{claim}
There exists an isotropic code $\mathord{\mathcal C}\subset D\cong \mathord{\mathbb F}\sb 3\sp{10}$
of dimension $4$
such that $\operatorname{\rm wt}\nolimits (\mathord{\bf x})\ge 6$ holds for any $\mathord{\bf x} \in \mathord{\mathcal C}$.
\end{claim}
The code $ \mathord{\mathcal C}$ generated by the row vectors of
$$
\left [\begin {array}{cccccccccc}
1&0&0&0&0&1&1&1&1&1\\
0&1&0&0&1&0&1&1&2&2\\
0&0&1&0&1&1&0&2&1&2\\
0&0&0&1&1&1&2&0&2&1
\end {array}\right ]
$$
satisfies $\operatorname{\rm wt}\nolimits (\mathord{\bf x})\ge 6$ for any $\mathord{\bf x} \in \mathord{\mathcal C}$.
The weight-enumerator $\sum\sb{\mathord{\bf x}\in \mathord{\mathcal C}} z\sp{\operatorname{\rm wt}\nolimits (\mathord{\bf x})}$
of this code $\mathord{\mathcal C}$ is
$$
1+60 z^6+20 z^9.
$$
\begin{remark}
The code $\mathord{\mathcal C}$ above is obtained as
a subcode of the extended ternary Golay code in $\mathord{\mathbb F}\sb 3\sp{12}$. See~\cite[Chapter 5, Section 2]{E}.
\end{remark}
\section{Proof of Theorem~\ref{thm:insep}}\label{sec:insep}
Let $(X, L)$ be a polarized $K3$ surface of degree $6$.
Then $Y\sb{(X, L)}$ is
a complete intersection of multi-degree $(2, 3)$ in $\P\sp 4$ by \cite[Theorem 6.1]{SD}.
Let $\widetilde Q\sb{(X, L)}$ denote the unique quadric hypersurface in $\P\sp 4$
containing $Y\sb{(X, L)}$.
\begin{proposition}
Suppose that
$\mathord{\mathcal R}\sb{(X, L)}=10A\sb 2$ and
$R\sb{{(X, L)}}\sp{\perp} \cong U(3)$.
Then $\widetilde Q\sb{(X, L)}$ is a cone over a non-singular quadric surface $Q=\P\sp 1\times \P\sp 1$,
and $Y\sb{(X, L)}$ does not pass through the vertex $P$ of the cone $\widetilde Q\sb{(X, L)}$.
\end{proposition}
\begin{proof}
By the assumption,
$R\sb{{(X, L)}}\sp{\perp}$ is generated by the numerical equivalence classes $[E]$ and $[F]$
of divisors $E$ and $F$ satisfying
\begin{equation}\label{eq:EF}
E^2=F^2=0, \quad EF=3, \quad [L]=[E]+[F].
\end{equation}
By the Riemann-Roch theorem, we can assume that $E$ and $F$ are effective.
Suppose that $|E|$ has a fixed component.
Let $M+\Gamma$ be a general member of $|E|$,
where $\Gamma$ is the fixed part of $|E|$.
Because $\rho\sb L: X\to Y\sb{{(X, L)}}$ is birational,
$\rho\sb L$ induces a birational map from $M$ to $\rho\sb L (M)$.
Note that $\rho\sb L (M+\Gamma)$ is a cubic curve.
If $\rho\sb L (\Gamma)$ is of dimension $1$,
then $\rho\sb L (M)$ is a line or a plane conic,
and hence contradicts $\dim |M|>0$.
Therefore, $\rho\sb L$ contracts every irreducible component of
$\Gamma$ to a point,
and hence $[\Gamma]\in R\sb{{(X, L)}}$.
From $[E]\in R\sb{{(X, L)}}\sp{\perp}$,
we obtain $E\Gamma=0$ and hence
$M^2=E^2+\Gamma^2<0$.
Thus we get a contradiction again.
Hence $|E|$ has no fixed components.
In particular, $E$ is nef.
Since $\rho\sb L$ is birational and $E$ is primitive in $R\sb{{(X, L)}}\sp{\perp}$
(being part of its basis),
a general member $E$ of $|E|$ is mapped by $\rho\sb L$ birationally
to a plane cubic curve in $\P\sp 4$.
Therefore a general member of $|E|$ is irreducible,
and hence $|E|$ is a (quasi-)elliptic pencil by~\cite[Proposition 0.1]{Nikulin79}.
Therefore the quadric hypersurface $\widetilde Q\sb{(X, L)}$ contains a one-dimensional family
$\{\Pi\sp E\sb t\}$
of planes
such that
$$
|E|=\{\rho\sb L \sp * (\Pi\sp E\sb t\cap Y\sb{{(X, L)}})\}.
$$
Hence $\widetilde Q\sb{(X, L)}$ is singular.
Since $\widetilde Q\sb{(X, L)}$ contains two irreducible families $\{\Pi\sp E\sb t\}$ and $\{\Pi\sp F\sb t\}$ of planes
corresponding to $|E|$ and $|F|$,
we have $\dim \operatorname{\rm Sing}\nolimits \widetilde Q\sb{(X, L)}=0$,
and $\widetilde Q\sb{(X, L)}$ is a cone over a non-singular quadric surface $Q=\P\sp 1\times \P\sp 1$.
If $Y\sb{(X, L)}$ passed through the vertex $P$ of the cone $\widetilde Q\sb{(X, L)}$,
then the linear system $|E|$
would have a fixed component that is contracted to the point $P$.
Hence $P$ is not contained in $Y\sb{(X, L)}$.
\end{proof}
Note that a non-ordered pair of the numerical equivalence classes $[E]$ and $[F]$ in $R\sb{{(X, L)}}\sp{\perp}$
satisfying~\eqref{eq:EF} is unique.
The following has been shown in the proof above:
\begin{corollary}\label{cor:EFnef}
The divisors $E$ and $F$ are nef.
The complete linear systems $|E|$ and $|F|$ are {\rm (}quasi-{\rm )}elliptic pencils.
\end{corollary}
We denote by
$$
\pi\sb P : Y\sb{{(X, L)}}\to Q=\P\sp 1\times \P\sp 1
$$
the projection from the vertex $P$ of the cone $\widetilde Q\sb{(X, L)}$.
Let $x$ and $y$ be affine coordinates of the two factors of $\P\sp 1\times \P\sp 1$.
The surface $Y\sb{(X, L)}$ is defined by an equation
\begin{equation}\label{eq:F}
\Psi\;:=\; W^3\,+\, a(x, y)\, W^2\,+ \,b(x, y)\, W \, + \,c(x, y)\;=\;0,
\end{equation}
where $W$ is a fiber coordinate of the affine line bundle $\widetilde Q\sb{(X, L)}\setminus \{P\}\cong L\sb Q(1,1)$
on $Q=\P\sp 1\times \P\sp 1$,
and $a$, $b$, $c$ are polynomials of degrees $1$, $2$ and $3$, respectively.
\par
\medskip
Let us consider the fibrations
\begin{eqnarray*}
\Phi\sb{|E|} = \operatorname{\rm pr}\nolimits\sb 1 \circ \pi\sb P \circ \rho \sb L &:& X\to \P\sp 1, \qquad\textrm{and} \\
\Phi\sb{|F|} = \operatorname{\rm pr}\nolimits\sb 2 \circ \pi\sb P \circ \rho \sb L &:& X\to \P\sp 1,
\end{eqnarray*}
where $\operatorname{\rm pr}\nolimits\sb i: \P\sp 1\times \P\sp 1\to \P\sp 1$ is the projection onto the $i$-th factor.
Because $Y\sb{(X, L)}$ has ten cusps,
the classification of fibers of (quasi-)elliptic fibrations and
the criterion~\cite[Section 4]{RS} for quasi-ellipticity imply the following:
\begin{proposition}\label{cor:QE}
The fibrations $\Phi\sb{|E|}$ and $\Phi\sb{|F|}$
are quasi-elliptic.
Let $\Theta$ be a fiber of the quasi-elliptic fibration $\Phi\sb{|E|}$.
Then
$\Theta$ is either of type $\mathord{\rm II}$,
of type $\mathord{\rm IV}$
or of type $\four\sp *$.
Moreover, we have
\begin{eqnarray*}\label{eq:QE}
\textrm {$\Theta$ is of type $\mathord{\rm II}$}
&\Longleftrightarrow& \textrm{$\rho\sb L (\Theta)$ does not pass through any cusps of $Y\sb{(X, L)}$}, \\
\textrm {$\Theta$ is of type $\mathord{\rm IV}$}
&\Longleftrightarrow& \textrm{$\rho\sb L (\Theta)$ passes through exactly one cusp of $Y\sb{(X, L)}$}, \\
\textrm {$\Theta$ is of type $\four\sp *$}
&\Longleftrightarrow& \textrm{$\rho\sb L (\Theta)$ is a line with multiplicity $3$ passing through} \\
&& \textrm{exactly three cusps of $Y\sb{(X, L)}$}.
\end{eqnarray*}
Same hold for fibers of $\Phi\sb{|F|}$.
\end{proposition}
\begin{proof}[Proof of Theorem~\ref{thm:insep}]
Let $X$ be a supersingular $K3$ surface with $\sigma (X)\le 6$.
We choose a subcode $\mathord{\mathcal C}$ of the isotropic admissible code $\mathord{\mathcal C}\sb 7$
in Table~\ref{table:sevencodes7}
with
$$
\dim \mathord{\mathcal C}=6-\sigma (X),
$$
and consider the corresponding overlattice $N\sb{\mathord{\mathcal C}}$ of $\mathord{\mathbb Z} [10A\sb 2]\oplus U(3)$.
There exists an isometry
$$
\phi : N\sb{\mathord{\mathcal C}} \smash{\mathop{\;\to\;}\limits\sp{\sim\,}} \operatorname{{\it NS}}\nolimits (X)
$$
such that $\phi (e+f)$ is the class $[L]$ of a line bundle $L$ that is very ample
modulo $(-2)$-curves, where $e, f$ form the canonical basis of $U(3)$; see the proof of Theorem 1.5.
Then $Y\sb{(X, L)}$ is a complete intersection in $\P\sp 4$ with multi-degree $(2, 3)$
that has ten cusps as its only singularities.
We will prove Theorem~\ref{thm:insep} by showing that,
for this polarized supersingular $K3$ surface $(X, L)$,
the morphism $\pi\sb P$ from $Y\sb{(X, L)}$ to $\P\sp 1\times \P\sp 1$ is purely inseparable;
that is, the polynomials $a$ and $b$ in~\eqref{eq:F} are zero.
\par
\medskip
We assume that $\pi\sb P$ is separable, and derive a contradiction.
\par
\medskip
For $i=1, \dots, 10$,
let $C\sb i$ and $ D\sb i $ be the $(-2)$-curves
contracted by $\rho\sb L$
satisfying
$$
C\sb i ^2= D\sb i ^2=-2,
\qquad
C\sb i D\sb i =1,
\qquad
\langle [C\sb i], [D\sb i] \rangle \perp \langle [C\sb j], [D\sb j]\rangle \quad (i\ne j),
$$
and let $E$, $F$ be divisors
such that $\phi (e)=[E]$ and $\phi (f)=[F]$.
Then $E$ and $F$ satisfy $[E], [F]\in R\sb{(X, L)}\sp{\perp}$
and~\eqref{eq:EF}.
We put
\begin{eqnarray*}
\gamma\sb i &:=& ([C\sb i] + 2[D\sb i])/3\;\bmod (R\sb{{(X, L)}}\oplus R\sb{{(X, L)}}\sp{\perp}),\\
\bar f \sp{\vee} &:=& [E]/ 3 \;\bmod (R\sb{{(X, L)}}\oplus R\sb{{(X, L)}}\sp{\perp}),\\
\bar e \sp{\vee} &:=& [F]/ 3 \;\bmod (R\sb{{(X, L)}}\oplus R\sb{{(X, L)}}\sp{\perp}).
\end{eqnarray*}
The code $\mathord{\mathcal C}\sb{(X, L)}$ defined by
$$
\mathord{\mathcal C}\sb{(X, L)}:= \operatorname{{\it NS}}\nolimits (X)/ (R\sb{{(X, L)}}\oplus R\sb{{(X, L)}}\sp{\perp}) \;\;\subset \;\;
\Disc{R\sb{{(X, L)}}\oplus R\sb{{(X, L)}}\sp{\perp}}\cong \mathord{\mathbb F}\sb 3\sp{10}\oplus \mathord{\mathbb F}\sb 3\sp 2
$$
is isomorphic to the subcode $\mathord{\mathcal C}$ of $\mathord{\mathcal C}\sb 7$ chosen above.
Let $G$ be a divisor on $X$.
Then $[G]\in \operatorname{{\it NS}}\nolimits (X)$ is written as
$$
\frac{1}{3}\sum\sb{i=1}\sp{10} (s\sb i [C\sb i] + t\sb i [D\sb i]) +\frac{\alpha}{3} [E] + \frac{\beta}{3} [F],
$$
where $s\sb i, t\sb i, \alpha, \beta$ are integers satisfying $s\sb i+t\sb i\equiv 0 \bmod 3$.
We denote by
$$
\wdc{G}:= [G] \;\bmod (R\sb{{(X, L)}}\oplus R\sb{{(X, L)}}\sp{\perp})
$$
the word of $\mathord{\mathcal C}\sb{(X, L)}$ corresponding to $[G]$,
which is written as
$$
(\mathord{\bf x}( G), \bar\alpha, \bar\beta)=
\sum \sb{i=1}\sp{10} x\sb i \gamma\sb i + \bar\alpha \bar f \sp{\vee} + \bar \beta \bar e \sp{\vee},
$$
where $\bar\alpha = \alpha \bmod 3$, $\bar\beta=\beta \bmod 3$, and
$$
x\sb i=\begin{cases}
0 & \textrm{if $(s\sb i, t\sb i) \equiv (0, 0)\;\bmod 3$,} \\
1 & \textrm{if $(s\sb i, t\sb i) \equiv (1, 2)\;\bmod 3$,} \\
2 & \textrm{if $(s\sb i, t\sb i) \equiv (2, 1)\;\bmod 3$.}
\end{cases}
$$
We put
\begin{eqnarray*}
s(G) &:=& \set{i}{ (s\sb i, t\sb i)\ne (0,0)}=\set{i}{ C\sb i G \ne 0 \;\;\textrm{or}\;\; D\sb i G\ne 0}, \\
s\sb 1(\mathord{\bf x}( G))&:=& \set{i}{ x\sb i \ne 0}=\set{i}{(s\sb i, t\sb i)\not\equiv (0,0) \;\bmod 3 }, \\
s\sb 2(G) &:=& \set{i}{ (s\sb i, t\sb i)\ne (0,0) \;\;\textrm{and}\;\; (s\sb i, t\sb i)\equiv (0,0) \;\bmod 3 }.
\end{eqnarray*}
By definition, we have
$$
s (G)=s\sb 1 (\mathord{\bf x} ( G)) \sqcup s\sb 2 (G).
$$
\begin{lemma}\label{lem:s2}
Suppose that $G$ is a reduced irreducible curve on $X$.
Then the following holds:
\begin{equation}\label{eq:Gsq}
|s\sb 2 (G)| \le \frac{1}{3} (\alpha \beta -|s\sb 1 (\mathord{\bf x} (G))|)+1.
\end{equation}
In particular, we have $\alpha \beta -|s\sb 1 (\mathord{\bf x} (G))| \ge -3$.
\end{lemma}
\begin{proof}
Let $s$ and $t$ be integers such that $s+t\equiv 0\;\bmod 3$.
If $(s, t)\ne (0,0)$, then
$$
\left( \nfrac{s C\sb i + t D\sb i }{3}\right)^2\le -2/3
$$
holds. If $(s, t)\ne (0,0)$ and $(s, t)\equiv (0,0)\;\bmod 3$, then
$$
\left( \nfrac{s C\sb i + t D\sb i }{3}\right)^2\le -2
$$
holds. Therefore we have
$$
G^2\le -\frac{2}{3} |s\sb 1 (\mathord{\bf x} (G))| - 2 |s\sb 2 (G)| +\frac{2}{3} \alpha\beta.
$$
On the other hand, we have $G^2\ge -2$.
Hence we get the inequality~\eqref{eq:Gsq}.
\end{proof}
Let us denote by $\bar T$ the Cartier divisor on $Y\sb{{(X, L)}}$ cut out by the equation
\begin{equation}\label{eq:T}
\frac{\partial \Psi}{\partial W}=-aW+b=0,
\end{equation}
and let $T$ be the proper transform of $\bar T$ by $\rho \sb L$.
By the assumption that $\pi\sb P$ is separable,
$\bar T$ is a divisor and
$\pi\sb P$ is \'etale outside $\bar T$.
Hence the divisor $\bar T$ contains the ten cusps of $Y\sb{(X, L)}$.
Therefore we have
\begin{equation}\label{eq:sT}
s(T)=\{ 1, 2, \dots, 10\}.
\end{equation}
From the defining equation~\eqref{eq:T} of $\bar T$ on $Y\sb{(X, L)}$, we have
\begin{equation}\label{eq:ETFT}
ET=FT=6.
\end{equation}
We denote by $C\sb E$ the closure of the locus
$$
\set{x\in X}{\textrm{the fiber of $\Phi\sb{|E|}$ passing through $x$ is of type $\mathord{\rm II}$ and is singular at $x$}},
$$
and equip $C\sb E$ with the reduced structure.
A general member $E$ of $|E|$ intersects $C\sb E$ at one point with multiplicity $3$~(\cite{BM}).
See Figure~\ref{fig:CE}.
We define $C\sb F$ in the same way.
Both of $C\sb E$ and $C\sb F$ are irreducible,
and
we have
\begin{equation}\label{eq:CEECFF}
C\sb E E =C\sb F F=3.
\end{equation}
Because $\operatorname{\rm pr}\nolimits\sb 1 :\P\sp 1\times \P\sp 1\to\P\sp 1$ is smooth,
if $\operatorname{\rm pr}\nolimits\sb 1\circ \pi\sb P$ is not smooth at a non-singular point of $Y\sb{(X, L)}$,
then $\pi\sb P$ is not smooth at that point.
Therefore
the divisor $T$ contains $C\sb E$ as a reduced irreducible component.
Same holds for $C\sb F$.
\begin{claim}\label{claim:distinct}
The two curves $C\sb E$ and $C\sb F$ are distinct.
\end{claim}
\begin{proof}
Suppose that $C\sb E=C\sb F$ holds.
Let $x$ be a general point of $C\sb E=C\sb F$.
Since the fibers $E\sb x$ of $\Phi\sb{|E|}$ and $F\sb x$ of $\Phi\sb{|F|}$
passing through $x$ are both singular at $x$,
we have $E\sb x F\sb x\ge 4$,
which contradicts $EF=3$.
\end{proof}
Let
$$
T=C\sb E + C\sb F + T\sb 1+\cdots+T\sb{t}
$$
be the decomposition of $T$ into reduced irreducible components.
We put
\begin{eqnarray*}
[C\sb E] &=& \sum (s\sb{E,i} [C\sb i] + t\sb {E, i} [D\sb i] )/3 + (\alpha \sb E [E] +\beta\sb E [F])/3,\cr
[C\sb F] &=& \sum (s\sb{F,i} [C\sb i] + t\sb {F, i} [D\sb i] )/3 + (\alpha \sb F [E] +\beta\sb F [F])/3, \cr
[T\sb \nu] &=& \sum (s\sb{\nu,i} [C\sb i] + t\sb {\nu , i} [D\sb i] )/3 + (\alpha \sb \nu [E] +\beta\sb \nu [F])/3
\quad (\nu=1, \dots, t) .
\end{eqnarray*}
Since $E$ and $F$ are nef,
we have
\begin{equation}\label{eq:nef}
\alpha \sb E\ge 0,\;\;
\beta \sb E\ge 0,\;\;
\alpha \sb F\ge 0,\;\;
\beta \sb F\ge 0,\;\;
\alpha \sb \nu\ge 0,\;\;
\beta \sb \nu\ge 0\quad (\nu=1, \dots, t).
\end{equation}
Since $\pi\sb P$ is finite, $\pi\sb P\circ \rho\sb L$ maps each irreducible component of $T$ to a curve on $\P\sp 1\times \P\sp 1$.
Therefore we have
\begin{equation}\label{eq:finite}
\alpha \sb \nu> 0\;\;\textrm{or}\;\;
\beta \sb \nu> 0\qquad (\nu=1, \dots, t).
\end{equation}
By~\eqref{eq:CEECFF}, we have
\begin{equation}\label{eq:33}
\beta\sb E=3, \quad \alpha\sb F=3.
\end{equation}
Then, from~\eqref{eq:ETFT}, we have
\begin{equation}\label{eq:sum3}
\alpha\sb E +\sum\sb{\nu=1}\sp t \alpha\sb{\nu}=3, \quad\textrm{and}\quad \beta\sb F+\sum\sb{\nu=1}\sp t \beta\sb{\nu}=3.
\end{equation}
Consider the words
$$
\wdc{C\sb E}=(\mathord{\bf x}\sb E, \bar{\alpha}\sb E, 0), \quad
\wdc{C\sb F}=(\mathord{\bf x}\sb F, 0, \bar{\beta}\sb F), \quad
\wdc{T\sb{\nu}}=(\mathord{\bf x}\sb {\nu}, \bar{\alpha}\sb \nu , \bar{\beta}\sb \nu)\qquad (\nu=1, \dots, t)
$$
in the code $\mathord{\mathcal C}\sb{(X, L)}$.
From Lemma~\ref{lem:s2}, we have
\begin{equation}\label{eq:effv}
-\operatorname{\rm wt}\nolimits(\mathord{\bf x}\sb\nu) + \alpha \sb \nu \beta\sb \nu \ge -3\qquad (\nu=1, \dots, t).
\end{equation}
\begin{claim}\label{eq:EFzero}
$\mathord{\bf x}\sb E=\mathord{\bf x}\sb F=\mathord{\bf 0}$.
\end{claim}
\begin{proof}
Let $\Theta$ be a fiber of $\Phi\sb{|E|}$ such that
$\rho\sb L (\Theta)$ passes through a cusp $q\sb i:=\rho\sb L (C\sb i)=\rho\sb L (D\sb i)$
of $Y\sb{{(X, L)}}$.
Then $\Theta$ is of type $\mathord{\rm IV}$ or $\four\sp *$.
Suppose that $\Theta$ is of type $\mathord{\rm IV}$.
Then $\Theta$ consists of three irreducible components of multiplicity one,
two of which are $C\sb i$ and $D\sb i$,
that intersect at one point.
The curve $C\sb E$ passes through the intersection point.
Since $\Theta C\sb E=3$,
we have $C\sb E C\sb i = C\sb E D\sb i =1$,
and therefore $s\sb {i, E}= t\sb{i, E}=-3$ holds.
Suppose that $\Theta$ is of type $\four\sp *$.
Then $C\sb E$ passes through a point of the multiplicity $3$ component
of $\Theta$, and does not intersect other irreducible components.
This fact can be proved by considering the pull-back of the quasi-elliptic fibration $\Phi\sb{|E|}$
by the base change $\P\sp 1\to\P\sp 1$ of degree $2$ branching at the point $\Phi\sb{|E|} (\Theta)$,
which makes the fiber $\Theta$ into type $\mathord{\rm IV}$.
Then it follows that $s\sb {i, E}=t\sb {i, E}=0$.
In any case,
we have $i\notin s\sb 1 (\mathord{\bf x}\sb E)$.
Since this holds for any cusp $q\sb i$ of $Y\sb{{(X, L)}}$,
we have $s\sb 1 (\mathord{\bf x}\sb E)=\emptyset$.
\end{proof}
Since $\bar T$ is a Cartier divisor of $Y\sb{(X, L)}$,
the total transform $\rho\sb L\sp *({\bar T}) $ is contained in $R\sb{{(X, L)}}\sp{\perp}=\mathord{\mathbb Z}[E]\oplus \mathord{\mathbb Z} [F]$,
and hence $\wdc{T}=0$. Therefore we obtain
\begin{equation}\label{eq:Cartier}
\mathord{\bf x}\sb 1 +\cdots +\mathord{\bf x}\sb {t} =\mathord{\bf 0}.
\end{equation}
By~\eqref{eq:sT},
we have
$$
s(C\sb E) \cup s (C\sb F) \cup s (T\sb{1}) \cup \dots \cup s (T\sb {t})=\{1, 2, \dots, 10\}.
$$
Using Lemma~\ref{lem:s2},
we obtain
\begin{multline}\label{eq:theineq}
t+2+
\frac{1}{3} \Bigl (\alpha\sb E\beta\sb E + \alpha\sb F \beta\sb F +
\sum\sb{\nu=1}\sp{t} (\alpha\sb \nu \beta\sb \nu - |s\sb 1 (\mathord{\bf x} \sb \nu )|)\Bigr) \ge\\
10 - |s\sb 1 (\mathord{\bf x} \sb{1}) \cup \dots \cup
s\sb 1 (\mathord{\bf x} \sb {t})|.
\end{multline}
Because $\mathord{\mathcal C}\sb{(X, L)}$ is isomorphic to a subcode of $\mathord{\mathcal C}\sb 7$,
we have shown that
there exist integers $\alpha\sb E$, $\beta\sb F$, $\alpha\sb \nu, \beta\sb \nu$ $(\nu=1, \dots, t)$
and
words
$$
(\mathord{\bf 0}, \bar{\alpha}\sb E, 0), \quad
(\mathord{\bf 0}, 0, \bar{\beta}\sb F), \quad
(\mathord{\bf x}\sb {\nu}, \bar{\alpha}\sb \nu , \bar{\beta}\sb \nu)\qquad (\nu=1, \dots, t)
$$
in the code $\mathord{\mathcal C}\sb 7$ satisfying~\eqref{eq:nef}-\eqref{eq:theineq}.
Using computer, however, we can show that such integers and words do not exist.
Thus we get a contradiction.
\end{proof}
Instead of the code $\mathord{\mathcal C}\sb 7$,
we can use the codes $\mathord{\mathcal C}\sb 3, \dots, \mathord{\mathcal C}\sb 6$ in Table~\ref{table:sevencodes7}.
However, we cannot use $\mathord{\mathcal C}\sb 2$ or $\mathord{\mathcal C}\sb 1$.
Indeed, in $\mathord{\mathcal C}\sb 1$,
for example,
we have the following integers and words:
\begin{eqnarray*}
&&[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0], \\
&&[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3], \\
&&[1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1], \\
&&[0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 1, 0], \\
&&[0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 1, 0], \\
&&[0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1], \\
&&[2, 1, 1, 2, 0, 0, 0, 0, 0, 0, 1, 1].
\end{eqnarray*}
Nevertheless, we can ask the following:
\begin{question}\label{question1}
Is $\pi\sb P: Y\sb{{(X, L)}}\to\P\sp 1\times \P\sp 1$ inseparable
for any polarized supersingular $K3$ surface $(X, L)$ of degree $6$ with $\mathord{\mathcal R} \sb{(X, L)}= 10 A\sb 2$
and $R\sp{\perp}\sb{(X, L)}\cong U(3)$?
\end{question}
\begin{example}\label{example:sigmaG0}
Consider the purely inseparable triple cover of $\P\sp 1\times \P\sp 1$ defined by
$$
W^3=(x^3-x)(y^3-y),
$$
and the corresponding polarized supersingular $K3$ surface $(X, L)$.
We will show that the Artin invariant of $X$ is $1$, and that the $5$-dimensional ternary code
$\mathord{\mathcal C}\sb{{(X, L)}}$ is isomorphic to $\mathord{\mathcal C}\sb 1$.
For $\alpha\in \mathord{\mathbb F}\sb 3$,
let $l\sb\alpha$ and $m\sb\alpha$ be the lines on $\P\sp 1\times \P\sp 1$
defined by
$x=\alpha$ and $y=\alpha$, respectively.
The strict transforms of $l\sb\alpha$ and $m\sb\alpha$ by $\pi\sb P\circ \rho\sb L$
are written as $3 \tilde l\sb{\alpha}$ and $3\tilde m\sb\alpha$,
respectively.
Numbering the twenty $(-2)$-curves $C\sb 1, D\sb 1, \dots, C\sb{10}, D\sb{10}$
in an appropriate way,
we can write the numerical equivalence classes
$[\tilde l\sb{\alpha}]$,
$[\tilde m\sb{\alpha}]$
as follows:
\begin{eqnarray*}
[\tilde l\sb{0}] &=& A\sb 1 + A\sb 2 + A\sb 3 + [E]/3, \cr
[\tilde m\sb{0}] &=& A\sp{\prime}\sb 1 + A\sp{\prime}\sb 4 + A\sp{\prime}\sb 7 + [F]/3, \cr
[\tilde l\sb{1}] &=& A\sb 4 + A\sb 5 + A\sb 6 + [E]/3, \cr
[\tilde m\sb{1}] &=& A\sp{\prime}\sb 2 + A\sp{\prime}\sb 5 + A\sp{\prime}\sb 8 + [F]/3, \cr
[\tilde l\sb{2}] &=& A\sb 7 + A\sb 8 + A\sb 9 + [E]/3, \cr
[\tilde m\sb{2}] &=& A\sp{\prime}\sb 3 + A\sp{\prime}\sb 6 + A\sp{\prime}\sb 9 + [F]/3,
\end{eqnarray*}
where
$$
A\sb i= -([C\sb i] + 2 [D \sb i])/3,
\qquad
A\sp{\prime} \sb i = -(2 [C\sb i] + [D \sb i])/3.
$$
The discriminant of the sublattice of $\operatorname{{\it NS}}\nolimits (X)$ generated by the classes $[E], [F]$, the classes of
the twenty exceptional curves, and
the $6$ classes above
is equal to $-9$.
Hence these classes span $\operatorname{{\it NS}}\nolimits (X)$,
and the Artin invariant of $X$ is $1$.
The $6$ words
$\wdc{\tilde l\sb{\alpha}}$,
$\wdc{\tilde m\sb{\alpha}}$
generate a $5$-dimensional ternary code isomorphic to $\mathord{\mathcal C}\sb 1$.
\end{example}
\begin{question}\label{question2}
Find the defining equations
of purely inseparable triple covers of $Q=\P\sp 1\times \P\sp 1$
corresponding to the other ternary codes $\mathord{\mathcal C}\sb 2, \dots, \mathord{\mathcal C}\sb 7$
of dimension $5$
in Table~\ref{table:sevencodes7}.
(See Corollary~\ref{cor:seven}.)
\end{question}
In~\cite{DK},
Dolgachev and Kondo gave various defining equations of the supersingular $K3$ surface
in characteristic $2$ with Artin invariant $1$,
and determined the full automorphism group of this $K3$ surface.
We expect that various defining equations of
the supersingular $K3$ surface
in characteristic $3$ with Artin invariant $1$
would be also helpful in the study of the automorphism group of this surface.
\bibliographystyle{amsplain}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
|
{
"timestamp": "2004-11-19T04:42:35",
"yymm": "0411",
"arxiv_id": "math/0411425",
"language": "en",
"url": "https://arxiv.org/abs/math/0411425"
}
|
\section{Introduction}
Explicit metrics of special holonomy are of interest in both
physics and mathematics. With the introduction of M-theory, seven and
eight dimensional manifolds with $G_2$ and Spin(7) holonomy become
particularly important since they provide natural candidates for
minimally supersymmetric compactifications. (See for example,
\cite{Atiyahi,Atiyahii}.) Explicit complete metrics for such compact
manifolds are unlikely, since they do not have continuous isometries.
However, for non-compact manifolds explicit metrics do exist, and
many $G_2$ and Spin(7) examples have been found
\cite{Hitchin,Bryant,GPP,Spin7i,%
0106026,Brandhuberi,Kannoi,Gukov,Kannoii,0112138,%
Brandhuberii,Chong,Cvetic}.
The $G_2$ metrics with principal orbits $S^3 \times S^3$ are of
special interest because of their rich structure. The first non-singular
example of
this kind was obtained in \cite{Bryant,GPP}, in which the two $S^3$ are
round.
In \cite{0106026} a generalization of the metric ansatz depending on nine
unknown functions was given. An ansatz with four unknown functions
was proposed in \cite{Brandhuberi}. More general metrics of this kind
were given in \cite{0112138,Brandhuberii}. For a review see
\cite{Cvetic}.
The first example of a non-singular Spin(7) metric was given in
\cite{Bryant,GPP}, along with $G_2$ metrics including the one described
above. The principal orbits of
this Spin(7) example are $S^7$, described as an $S^3$ bundle over $S^4$. A
generalization of this metric was given in \cite{Spin7i}, by allowing
the $S^3$ fibres to be ``squashed". This generalization was shown to be
a special case in \cite{Gukov}, in which a new family of Spin(7)
metrics on a certain ${{\mathbb R}}^4$ bundle over ${{\mathbb C}{\mathbb P}}^2$. For other
constructions see \cite{Kannoi,Kannoii}.
In all the above examples, the metrics are of cohomogeneity one.
Hitchin \cite{Hitchin} gave a general practical tool for calculating
such special holonomy metrics of cohomogeneity one. The examples
given there reproduce the Spin(7) metrics in \cite{Bryant,GPP,Spin7i}
and the $G_2$ metrics in \cite{Bryant,GPP,Brandhuberi}. This construction
was used in \cite{Chong} to obtain more general metrics of $G_2$
holonomy. In this paper, we again employ this method to obtain
larger classes of metrics with $G_2$ and Spin(7) holonomy.
The paper is organized as follows. In section 2, We consider
the most general cohomogeneity one $G_2$ metric with $S^3\times S^3$
principal orbits. We obtain first-order equations using the
Hitchin approach, which guarantees the existence of $G_2$
holonomy. The metric is described by 18 functions satisfying 18
first-order differential equations, together with 7 consistent
algebraic constraints. In section 3, we use the same approach to
study $G_2$ metrics with principal orbits $S^3 \times T^3$ \cite{Yau}, which
can be obtained by taking a contraction \cite{Chong} of one of the
$S^3$ factors to $T^3$. We
obtain an analytic solution that has more non-trivial parameters
than those obtained in \cite{Yau}. This demonstrates explicitly that our
first-order system gives rise to a more general class of $G_2$ metrics than
any known previously. In section 4, we apply this technique to
constructing a more general class of Spin(7) metrics whose principal
orbits are $S^7$. We obtain a system of first-order equations,
which guarantees the existence of Spin(7) holonomy. We conclude
our paper in section 5.
\section{General $G_2$ holonomy metric with $S^3\times S^3$
principal orbits}
In this section we use the technique of \cite{Hitchin,Chong}
to obtain the most general cohomogeneity one $G_2$ metric with
$S^3\times S^3$ principal orbits. A $G_2$ manifold is
characterised by its associative 3-form $\Phi_{\sst{(3)}}$, which has the
structure
\begin{equation}
\Phi_{\sst{(3)}} = dt\wedge \omega + \rho\,,\label{g23form}
\end{equation}
where $\omega$ and $\rho$ are invariant 2-forms and 3-forms that do not
involve $dt$, satisfying the necessary condition
\begin{equation}
\omega\wedge\rho=0\,.\label{omegarho}
\end{equation}
Since $S^3$ is an $SU(2)$ group manifold, we can write the
vielbein for the $S^3\times S^3$ in terms of two sets of
left-invariant $SU(2)$ 1-forms $\sigma_i$ and $\Sigma_i$, satisfying
\begin{equation}
d\sigma_i=\fft12\epsilon_{ijk} \sigma_j\wedge \sigma_k\,,\qquad
d\Sigma_i=\fft12\epsilon_{ijk} \Sigma_j\wedge \Sigma_k\,.
\end{equation}
We consider the most general 3-form $\rho$
constructed from $\sigma_i$ and $\Sigma_i$
\footnote{Note that here we break the anti-invariance
\cite{Hitchin} under the $\mathbf {Z}/2$ that interchanges the two
$S^3$ factors.
This anti-invariance can be restored by setting $m=n$, $x_4=x_5$, $x_6=x_7$,
and $x_8=x_9$. It is similar for the following 4-form $\sigma$.}, given by
\begin{eqnarray}
\rho&=&n\Sigma_1\Sigma_2\Sigma_3-m\sigma_1\sigma_2\sigma_3+x_1
d(\sigma_1\Sigma_1)+x_2d(\sigma_2\Sigma_2)+x_3d(\sigma_3\Sigma_3)\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&+x_4d(\sigma_1\Sigma_2)
+x_5d(\sigma_2\Sigma_1)+x_6d(\sigma_2\Sigma_3)+x_7d(\sigma_3\Sigma_2)+
x_8d(\sigma_3\Sigma_1)+
x_9d(\sigma_1\Sigma_3)\,.
\end{eqnarray}
where $m$ and $n$ are constants, and $x_i$ are nine functions
depending on $t$. In order to obtain $\omega$, we first consider the
most general 4-form $\sigma$, involving nine $t$-dependent
functions $y_i$:
\begin{eqnarray}
\sigma&=&y_1\sigma_2\Sigma_2\sigma_3\Sigma_3+
y_2\sigma_3\Sigma_3\sigma_1\Sigma_1+y_3\sigma_1\Sigma_1
\sigma_2\Sigma_2+y_4\sigma_2\Sigma_3\sigma_3\Sigma_1+
y_5\sigma_3\Sigma_2\sigma_1\Sigma_3\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&+y_6\sigma_3\Sigma_1\sigma_1\Sigma_2+
y_7\sigma_1\Sigma_3\sigma_2\Sigma_1+y_8\sigma_1\Sigma_2
\sigma_2\Sigma_3+y_9\sigma_2\Sigma_1\sigma_3\Sigma_2\,.
\label{g2sig}
\end{eqnarray}
Note that in this paper, in many complex equations when there is no
confusion, we shall drop the $\wedge$ notation for wedge products of
differential forms. Following the approach of \cite{Hitchin}, we take
the ``square root'' of the 4-form, writing it as $\sigma=\ft12
\omega^2$. Then we can write $\omega$ as
\begin{equation}
\omega=a\sigma_1\Sigma_1+b\sigma_2\Sigma_2+
c\sigma_3\Sigma_3+e\sigma_2\Sigma_1+f\sigma_1\Sigma_2+
g\sigma_2\Sigma_3+h\sigma_3\Sigma_2+
j\sigma_3\Sigma_1+k\sigma_1\Sigma_3,
\end{equation}
where
\begin{eqnarray}
a=\frac{y_2y_3-y_6y_7}{W},\quad b=\frac{y_1y_3-y_8y_9}{W},\quad
c=\frac{y_1y_2-y_4y_5}{W},\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
e=\frac{y_7y_9-y_3y_4}{W},\quad f=\frac{y_6y_8-y_3y_5}{W},\quad
g=\frac{y_4y_8-y_1y_7}{W},\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
h=\frac{y_5y_9-y_1y_6}{W},\quad j=\frac{y_4y_6-y_2y_9}{W},\quad
k=\frac{y_5y_7-y_2y_8}{W}.
\end{eqnarray}
The condition (\ref{omegarho}) now implies the algebraic constraints
\begin{eqnarray}
-ax_4+bx_5-ex_2+fx_1-jx_7+hx_8=0,\!\!\!&&\!\!\!
-bx_6+cx_7-fx_9+gx_2-hx_3+kx_4=0,\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
ax_9-cx_8+ex_6-gx_5+jx_3-kx_1=0,\!\!\!&&\!\!\!
-ax_5+bx_4+ex_1-fx_2+gx_9-kx_6=0,\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
-bx_7+cx_6-ex_8-gx_3+hx_2+jx_5=0, \!\!\!&&\!\!\!
ax_8-cx_9+fx_7-hx_4-jx_1+kx_3=0.\label{g2con1}
\end{eqnarray}
Having obtained the ansatz for the associative 3-form $\Phi_{\sst{(3)}}$,
we can write down the metric for the $G_2$ manifold. We define the
symmetric tensor density
\begin{equation}
B_{AB} = -\ft1{144} \Phi_{A C_1 C_2}\, \Phi_{B C_3 C_4}\,
\Phi_{C_5 C_6 C_7}\, {\varepsilon}^{C_1\cdots C_7}\,,\label{dendef}
\end{equation}
where ${\varepsilon}^{C_1\cdots C_7}$ is the Levi-Civita tensor density in
seven dimensions (with values $\pm1$ and 0). The metric tensor is then
given by
\begin{equation}
g_{AB} = \det(B)^{-1/9}\, B_{AB}\,.\label{metric}
\end{equation}
The Hamiltonian of the system can be written as $H=V(\rho) -
2W(\sigma)$, where $V(\rho)$ depends only on the tensor $\rho$, and
$W(\sigma)$ depends only on $\sigma$. The function $V(\rho)$ is
defined by
\begin{equation}
V(\rho) = \sqrt{-\ft16 K_a{}^b\, K_b{}_{\phantom{\Sigma}}^a}\,,
\end{equation}
where
\begin{equation}
K_a{}^b\equiv \ft1{12} \rho_{c_1 c_2 c_3}\, \rho_{c_4 c_5 a}\,
{\varepsilon}^{c_1 c_2 c_3 c_4 c_5 b}\,,
\end{equation}
with ${\varepsilon}^{c_1\cdots c_6}$ being the Levi-Civita
tensor density in 6-dimensions.
The function $W(\sigma)$ is calculated from
\begin{equation}
W(\sigma)^2 = \ft1{48}\, {\varepsilon}_{c_1\cdots c_6}\, \tilde\sigma^{c_1 c_2}\,
\tilde\sigma^{c_3c_4}\,\tilde\sigma^{c_5c_6}\,,
\end{equation}
where
\begin{equation}
\tilde \sigma^{ab} \equiv \ft1{24}\, {\varepsilon}^{ab c_1c_2c_3c_4}\,
\sigma_{c_1c_2c_3c_4}\,.
\end{equation}
For our specific example, we find that
\begin{eqnarray}
V&\equiv&\sqrt{-U}\,,\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
U&=&m^2n^2-2mn\sum_{i=1}^{9}x_{i}^2+\sum_{i=1}^{9}x_{i}^{4}\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&-4(m+n)(x_1x_2x_3-x_3x_4x_5-x_1x_6x_7+x_4x_6x_8+x_5x_7x_9-x_2x_8x_9)\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&-2x_{1}^{2}(x_{2}^{2}+x_{3}^{2}-x_{4}^{2}-x_{5}^{2}+x_{6}^{2}+x_{7}^{2}
-x_{8}^{2}-x_{9}^{2})\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&+2x_{2}^{2}(-x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+
x_{7}^{2}-x_{8}^{2}-x_{9}^{2})+
2x_{3}^{2}(-x_{4}^{2}-x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}+x_{9}^{2})\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&+2x_{4}^{2}(-x_{5}^{2}-x_{6}^{2}+x_{7}^{2}-x_{8}^{2}+
x_{9}^{2})+2x_{5}^{2}(x_{6}^{2}-x_{7}^{2}
+x_{8}^{2}-x_{9}^{2})\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&+2x_{6}^{2}(-x_{7}^{2}-x_{8}^{2}+x_{9}^{2})+2x_{7}^{2}(x_{8}^{2}-
x_{9}^{2})+2x_{8}^{2}(-x_{9}^{2})\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&+8x_1(x_2x_4x_5+x_4x_7x_8+x_5x_6x_9+x_3x_8x_9)+8x_3x_5x_6x_8\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&+8x_2x_4x_6x_9+8x_3x_4x_7x_9+8x_2x_3x_6x_7+8x_2x_5x_7x_8\,,\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
W&=&(y_1y_2y_3+y_4y_6y_8+y_5y_7y_9-y_3y_4y_5-y_2y_8y_9-y_1y_6y_7)^{\fft12}\,.
\end{eqnarray}
The manifold with $G_2$ holonomy is then governed by a set of
first-order differential equations following from the Hamiltonian
flow \cite{Hitchin}
\begin{equation}
\dot x_{i}=-\frac{\partial H}{\partial y_i},\quad \dot
y_i=\frac{\partial H}{\partial x_i},\label{hamflow}
\end{equation}
where the dot denotes a derivative with respect to the ``time''
variable $t$, together with the Hamiltonian constraint $H=0$. Thus we
have
\begin{eqnarray}
\dot x_1=\frac{y_2y_3-y_6y_7}{W},\quad \dot
x_2=\frac{y_1y_3-y_8y_9}{W},\quad \dot
x_3=\frac{y_1y_2-y_4y_5}{W}, \nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\ \dot
x_4=\frac{y_6y_8-y_3y_5}{W},\quad \dot
x_5=\frac{y_7y_9-y_3y_4}{W},\quad
\dot x_6=\frac{y_4y_8-y_1y_7}{W},\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
\dot x_7=\frac{y_5y_9-y_1y_6}{W},\quad \dot
x_8=\frac{y_4y_6-y_2y_9}{W},\quad \dot x_9=
\frac{y_5y_7-y_2y_8}{W}.\end{eqnarray} \begin{eqnarray} \dot
y_1&=&[mnx_1+(m+n)(x_2x_3-x_6x_7)+x_1(x_{2}^{2}+x_{3}^{2}-
x_{1}^{2}-x_{4}^{2}-x_{5}^{2}+x_{6}^{2}
+x_{7}^{2}-x_{8}^{2}-x_{9}^{2})\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&-2(x_2x_4x_5+x_4x_7x_8+x_5x_6x_9+x_3x_8x_9)]/W,\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
\dot
y_2&=&[mnx_2+(m+n)(x_3x_1-x_8x_9)+x_2(x_{3}^{2}+x_{1}^{2}-
x_{2}^{2}-x_{4}^{2}-x_{5}^{2}-x_{6}^{2}
-x_{7}^{2}+x_{8}^{2}+x_{9}^{2})\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&-2(x_1x_4x_5+x_4x_6x_9+x_5x_7x_8+x_3x_6x_7)]/W,\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
\dot
y_3&=&[mnx_3+(m+n)(x_1x_2-x_4x_5)+x_3(x_{1}^{2}+x_{2}^{2}-
x_{3}^{2}+x_{4}^{2}+x_{5}^{2}-x_{6}^{2}
-x_{7}^{2}-x_{8}^{2}-x_{9}^{2})\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&-2(x_1x_8x_9+x_5x_6x_8+x_4x_7x_9+x_2x_6x_7)]/W,\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\end{eqnarray}
\begin{eqnarray}
\dot
y_4&=&[mnx_4+(m+n)(x_6x_8-x_3x_5)+x_4(-x_{1}^{2}-x_{2}^{2}+
x_{3}^{2}-x_{4}^{2}+x_{5}^{2}+x_{6}^{2}
-x_{7}^{2}+x_{8}^{2}-x_{9}^{2})\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&-2(x_1x_2x_5+x_1x_7x_8+x_2x_6x_9+x_3x_7x_9)]/W,\nonumber} \def\bd{\begin{document}} \def\ed{\end{document} \\
\dot
y_5&=&[mnx_5+(m+n)(x_7x_9-x_3x_4)+x_5(-x_{1}^{2}-x_{2}^{2}+
x_{3}^{2}+x_{4}^{2}-x_{5}^{2}-x_{6}^{2}
+x_{7}^{2}-x_{8}^{2}+x_{9}^{2})\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&-2(x_1x_2x_4+x_1x_6x_9+x_3x_6x_8+x_2x_7x_8)]/W,\nonumber} \def\bd{\begin{document}} \def\ed{\end{document} \\
\dot
y_6&=&[mnx_6+(m+n)(x_4x_8-x_1x_7)+x_6(x_{1}^{2}-x_{2}^{2}-
x_{3}^{2}+x_{4}^{2}-x_{5}^{2}-x_{6}^{2}
+x_{7}^{2}+x_{8}^{2}-x_{9}^{2})\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&-2(x_1x_5x_9+x_3x_5x_8+x_2x_4x_9+x_2x_3x_7)]/W,\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
\dot{y_7}&=&[mnx_7+(m+n)(x_5x_9-x_1x_6)+x_7(x_{1}^{2}-x_{2}^{2}-
x_{3}^{2}-x_{4}^{2}+x_{5}^{2}+x_{6}^{2}
-x_{7}^{2}-x_{8}^{2}+x_{9}^{2})\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&-2(x_1x_4x_8+x_2x_3x_6+x_3x_4x_9+x_2x_5x_8)]/W,\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
\dot
y_8&=&[mnx_8+(m+n)(x_4x_6-x_2x_9)+x_8(-x_{1}^{2}+x_{2}^{2}-
x_{3}^{2}+x_{4}^{2}-x_{5}^{2}+x_{6}^{2}
-x_{7}^{2}-x_{8}^{2}+x_{9}^{2})\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&-2(x_1x_4x_7+x_1x_3x_9+x_3x_5x_6+x_2x_5x_7)]/W,\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
\dot
y_9&=&[mnx_9+(m+n)(x_5x_7-x_2x_8)+x_9(-x_{1}^{2}+x_{2}^{2}-
x_{3}^{2}-x_{4}^{2}+x_{5}^{2}-x_{6}^{2}
+x_{7}^{2}+x_{8}^{2}-x_{9}^{2})\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&-2(x_1x_5x_6+x_1x_3x_8+x_2x_4x_6+x_3x_4x_7)]/W\,.
\end{eqnarray}
The Hamiltonian constraint implies that
\begin{equation}
U=-4\,(y_1y_2y_3+y_4y_6y_8+y_5y_7y_9-y_3y_4y_5-
y_2y_8y_9-y_1y_6y_7)\,,\label{hamilton}
\end{equation}
Finally we present the explicit form of the metric, which is given by
\begin{eqnarray}
ds^2&=&dt^2+g_{11}\sigma_{1}^2+2g_{12}\sigma_{1}\sigma_{2}+
2g_{13}\sigma_{1}\sigma_{3}+2g_{14}\sigma_{1}\Sigma_{1}+
2g_{15}\sigma_{1}\Sigma_{2}+2g_{16}\sigma_{1}\Sigma_{3}\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&+g_{22}\sigma_{2}^2+2g_{23}\sigma_{2}\sigma_{3}+
2g_{24}\sigma_{2}\Sigma_{1}+2g_{25}\sigma_{2}\Sigma_{2}+
2g_{26}\sigma_{2}\Sigma_{3}\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&g_{33}\sigma_{3}^2+2g_{34}\sigma_{3}\Sigma_{1}+
2g_{35}\sigma_{3}\Sigma_{2}+2g_{36}\sigma_{3}\Sigma_{3}\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&g_{44}\Sigma_{1}^2+2g_{45}\Sigma_{1}\Sigma_{2}+
2g_{46}\Sigma_{1}\Sigma_{3}+g_{55}\Sigma_{2}^2+
2g_{56}\Sigma_{2}\Sigma_{3}+g_{66}\Sigma_{3}^2\,,
\end{eqnarray}
where $g_{ij}$ can be calculated in a straightforward way from
(\ref{metric}). Owing to the complexity of the structures, we shall
not present the explicit results here. We did verify that the system
of first-order equations does imply the closure and co-closure of the
associative 3-form, which demonstrates that the metric indeed has
holonomy $G_2$.
By the above construction we have obtained $G_2$ metrics involving
18 functions $x_i$
and $y_i$, and two constants $m$ and $n$, governed by 7 algebraic
equations (\ref{g2con1},\ref{hamilton}), and $(18-7)=11$ independent
first-order equations.
\section{$G_2$ holonomy metric with $S^{3}\times T^{3}$ principal orbits}
The $SU(2)$ group associated with an $S^3$ can be contracted in three
different ways, namely the Euclidean, Heisenberg, and Abelian contractions
(see, for example, \cite{Chong}). Here we consider the Abelian contraction
for $\sigma_i$. To do this, we define $\sigma_i=\lambda\,\alpha_i$, and then
send $\lambda \rightarrow 0$. Thus we have $d\alpha_i=0$, and
correspondingly the $S^3$ becomes (locally) $T^3$.
We start with the 3-form $\rho$ and 4-form $\sigma$
\begin{eqnarray}
\rho&=&n\Sigma_{1}\Sigma_{2}\Sigma_{3}-m\alpha_{1}\alpha_{2}\alpha_{3}+x_{1}d(\Sigma_{1}\alpha_{1})+x_{2}d(\Sigma_{2}\alpha_{2})+x_{3}d(\Sigma_{3}\alpha_{3})\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&+x_{4}d(\Sigma_{1}\alpha_{2})+x_{5}d(\Sigma_{2}\alpha_{1}),\label{3formrho}\\
\sigma&=&y_{1}\Sigma_{2}\alpha_{2}\Sigma_{3}\alpha_{3}+y_{2}\Sigma_{3}\alpha_{3}\Sigma_{1}\alpha_{1}+y_{3}\Sigma_{1}\alpha_{1}\Sigma_{2}\alpha_{2}\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&+y_{4}\Sigma_{2}\alpha_{3}\Sigma_{3}\alpha_{1}+y_{5}\Sigma_{3}\alpha_{2}\Sigma_{1}\alpha_{3}\label{sigma}
\end{eqnarray}
Note that a 3-form $\rho$ without the $x_4$ and $x_5$ terms, and
correspondingly a 4-form $\sigma$ without the $y_4$ and $y_5$
terms, were considered in \cite{Yau}. We will see later that
the more general 3-form $\rho$ and 4-form $\sigma$ considered here will
give rise to an off-diagonal term in the metric.
The Hamiltonian is given by
\begin{eqnarray}
H=V-2W=\sqrt{-U}-2W
\end{eqnarray}
where
\begin{eqnarray}
U=m^2 n^2 +4m(x_{1}x_{2}x_{3}-x_{3}x_{4}x_{5}),\ \ \
W=(y_{1}y_{2}y_{3}-y_{3}y_{4}y_{5})^{\frac{1}{2}}.
\end{eqnarray}
The co-associative 3-form is $\Phi_{(3)}=dt\wedge\omega+\rho$, where
\begin{equation}
\omega=\frac{y_{2}y_{3}}{W}\Sigma_{1}\alpha_{1}+\frac{y_{3}y_{1}}{W}\Sigma_{2}\alpha_{2}+\frac{W}{y_{3}}\Sigma_{3}\alpha_{3}-\frac{y_{3}y_{4}}{W}\Sigma_{2}\alpha_{1}-\frac{y_{3}y_{5}}{W}\Sigma_{1}\alpha_{2}.
\end{equation}
A $G_2$ holonomy metric is obtained if $x_i$ and $y_i$ satisfy
the Hamiltonian flow equation
\begin{eqnarray} \dot{x_i}=-\frac{\partial
H}{\partial y_{i}}, \quad \dot{y_i}=\frac{\partial H}{\partial
x_{i}}\,,
\end{eqnarray}
which results in
\begin{eqnarray}
\dot{x_1}&=&\frac{y_{2}y_{3}}{W},\quad \ \ \ \ \
\dot{x_2}=\frac{y_{3}y_{1}}{W},\quad \ \ \ \
\dot{x_3}=\frac{y_{1}y_{2}-y_{4}y_{5}}{W},
\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
\dot{x_4}&=&-\frac{y_{3}y_{5}}{W},\quad \quad \dot{x_5}=-\frac{y_{3}y_{4}}{W},\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
\dot{y_1}&=&\frac{2mx_{2}x_{3}}{\sqrt{-U}},\quad \
\dot{y_2}=\frac{2mx_{3}x_{1}}{\sqrt{-U}},\quad\dot{y_3}=\frac{2m(x_{1}x_{2}-x_{4}x_{5})}{\sqrt{-U}},\quad
\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
\dot{y_4}&=&-\frac{2mx_{3}x_{5}}{\sqrt{-U}},\ \
\dot{y_5}=-\frac{2mx_{3}x_{4}}{\sqrt{-U}}.
\end{eqnarray}
A simpler system of equations can be obtained by a change of variable from
$t$ to $\tilde t$ \cite{Yau},
\begin{eqnarray}
\frac{dt}{d\tilde{t}}=4\sqrt{y_{1}y_{2}y_{3}-y_{3}y_{4}y_{5}}, \label{tchange}
\end{eqnarray}
which is equivalent to considering the Hamiltonian flow
\begin{eqnarray}
\tilde{H}=-m^2n^2+4m(x_{1}x_{2}x_{3}-x_{3}x_{4}x_{5})-4(y_{1}y_{2}y_{3}-y_{3}y_{4}y_{5})=-m^2n^2-4mX-4Y
\end{eqnarray}
with
\begin{eqnarray} X=x_{1}x_{2}x_{3}-x_{3}x_{4}x_{5},
Y=y_{1}y_{2}y_{3}-y_{3}y_{4}y_{5}\,.
\end{eqnarray}
The flow equation becomes
\begin{eqnarray}
x_1'&=&4y_{2}y_{3},\ \ \ \ \ \ x_2'=4y_{3}y_{1},\quad \quad x_3'=4(y_{1}y_{2}-y_{4}y_{5}),\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
x_4'&=&-4y_{3}y_{5},\ \ \ \ x_5'=-4y_{4}y_{3},\quad\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\nn\\
y_1'&=&4mx_{2}x_{3},\ \ \ y_2'=4mx_{3}x_{1},\quad y_3'=4m(x_{1}x_{2}-x_{4}x_{5}),\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
y_4'&=&-4mx_{3}x_{5},\quad y_5'=-4mx_{3}x_{4}.
\end{eqnarray}
where the prime denotes a derivative with respect to $\tilde t$.
In addition $x_i$ and $y_i$ satisfy the constraint from the
requirement $\omega\wedge\rho=0$, namely
\begin{eqnarray}
y_{2}x_{4}-y_{1}x_{5}-y_{4}x_{2}+y_{5}x_{1}=0.
\end{eqnarray}
Suggested by \cite{Yau} we find the following conserved quantities
\begin{eqnarray} x_2
y_2 -x_1 y_1 =k_1 ,\quad
x_5 y_5 -x_4 y_4 &=&k_2 ,\quad x_1 y_1 +x_4 y_4 -x_3 y_3 =k_3\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
x_5 y_1 +x_2 y_4 &=&x_1 y_5+x_4 y_2 =\lambda\,,
\end{eqnarray}
where $k_1$, $k_2$, $k_3$ and $\lambda$ are constants. Defining
$z_3 =x_3 y_3$, we find that
\begin{eqnarray}
\frac{dz_3}{d\tilde{t}}&=&4Y-4mX=m^2n^2+8Y\\
\frac{d^2z_3}{d\tilde{t}^2}&=&8\frac{dY}{d\tilde{t}}=-32m[3z_{3}^{2}+2(k_1
+k_2+ 2k_3)z_3 +k_{3}(k_1 +k_2 +k_3 )-\lambda^2]\,.
\end{eqnarray}
This can be integrated explicitly, and $z_3$ can be written in terms
of Weierstrass
function, which has a second order pole. Near a pole
$x_{i}(\tilde{t})$ and $y_{i}(\tilde{t})$ takes the approximate form
$x_{i}\sim
y_{i}\sim \frac{1}{\tilde{t}}$ \cite{Yau}. Written in terms of $t$ through
the relation (\ref{tchange}), we have
\begin{eqnarray} x_1&=&A_1
t^2,\quad y_1 =4A_2A_3t^2,\quad \nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
x_2&=&A_2 t^2,\quad y_2 =4A_3A_1 t^2,\quad \nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
x_3&=&A_3 t^2,\quad y_3 =4(A_1A_2-A_4A_5) t^2,\quad \nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
x_4&=&A_4 t^2,\quad y_4 =-4A_3A_5 t^2,\quad \nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\ x_5&=&A_5
t^2,\quad y_5 =-4A_3A_4t^2.\quad
\end{eqnarray}
If we set $A_1 A_2 A_3 -A_3A_4 A_5 =\frac{m}{64}$, a $G_2$ metric of topology ${{\mathbb R}}^4 \times
T^3$ can be obtained.
\begin{equation}
ds^2=dt^2+\frac{1}{4}t^2(\Sigma_{1}^2+\Sigma_{2}^2+\Sigma_{3}^2)+16[(A_{1}^2-A_{5}^2)\alpha_{1}^2+2(A_1
A_4 +A_2 A_5)\alpha_1 \alpha_2
+(A_{2}^2-A_{4}^2)\alpha_{2}^2+A_{3}^2\alpha_{3}^2]
\end{equation}
In the construction of \cite{Yau}, the $G_2$ metric had two free
parameters. Here, however,
the generalized metric has four independent parameters. Also we note that
there is an off-diagonal term in the metric which results from the
extra $x_4$ and $x_5$ terms in the 3-form (\ref{3formrho}). This shows that
the metric considered here is more general, though it has the same
topology as that in \cite{Yau}.
\section{Spin(7) metric with principle orbit $S^7$}
In this section we will consider new Spin(7) metric with principal
orbits $S^7$ described as an $SU(2)$ bundle over $S^4$. The three
invariant $SU(2)$ connection 1-forms are $\alpha_i$ ($i$=1,2,3), and
$\omega_i$ ($i$=1,2,3) are the curvature 2-forms. The structure equations
for the principal orbits are
\begin{eqnarray} d\Sigma_1&=&\omega_1 -2\Sigma_2 \Sigma_3 ,\quad \quad \ \ d\Sigma_2
=\omega_2 -2 \Sigma_3 \Sigma_1, \quad \quad \ \ \ d\Sigma_3
=\omega_3 -2 \Sigma_1
\Sigma_2,\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
d\omega_1 &=&2(\omega_2 \Sigma_3 -\omega_3 \Sigma_2 ),\quad
d\omega_2 =2(\omega_3 \Sigma_1 -\omega_1 \Sigma_3 ),\quad
d\omega_3 =2(\omega_1 \Sigma_2 -\omega_2 \Sigma_1 )
\end{eqnarray}
with
\begin{eqnarray} \omega_1 =-(\Sigma_0 \Sigma_1 +\Sigma_2 \Sigma_3 ),\quad
\omega_2 =-(\Sigma_0 \Sigma_2 +\Sigma_3 \Sigma_1 ),\quad \omega_3
=-(\Sigma_0 \Sigma_3 +\Sigma_1 \Sigma_2 )\,,
\end{eqnarray}
where the basis $\Sigma_\mu$ ($\mu$=0,1,2,3) give the
standard metric $ds^2=\Sigma_0^2+\Sigma_1^2+\Sigma_2^2+\Sigma_3^2$ on $S^4$.
We consider the following exact 4-form constructed from $\alpha_i$ and $\omega_i$
\begin{eqnarray} \rho&=&x_1 d(\alpha_1 \omega_1)+x_2 d(\alpha_2
\omega_2)+x_3 d(\alpha_3 \omega_3)+x_4 d(2\alpha_1 \alpha_2
\alpha_3)+x_5 d(\alpha_1 \omega_2)\label{rho}\\ &=&2(x_1 +x_2
+x_3)\Sigma_0 \Sigma_1 \Sigma_2 \Sigma_3-2(-x_1 +x_2 +x_3
+x_4)\alpha_2 \alpha_3 (\Sigma_0 \Sigma_1 +\Sigma_2 \Sigma_3)\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&-2(x_1 -x_2 +x_3 +x_4)\alpha_3 \alpha_1 (\Sigma_0 \Sigma_2
+\Sigma_3 \Sigma_1)\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
&&-2(x_1 +x_2 -x_3 +x_4)\alpha_1 \alpha_2 (\Sigma_0 \Sigma_3
+\Sigma_1 \Sigma_2)\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\ &&+2x_5 [\alpha_2 \alpha_3 (\Sigma_0
\Sigma_2 +\Sigma_3 \Sigma_1)+\alpha_3 \alpha_1 (\Sigma_0 \Sigma_1
+\Sigma_2 \Sigma_3)]\label{4formrho}.
\end{eqnarray}
Note that the above 4-form $\rho$ but without the $x_5$ term was
considered in \cite{Hitchin}, reproducing the Spin(7) metrics obtained
in \cite{Spin7i}. Also note that the most general 4-form constructed
from $\alpha_i$ and $\Sigma_\mu$ was written down in \cite{Kannoiii}.
In order to follow Hitchin's procedure, one should then calculate the
metric that is implied by the choice of 4-form, and solve the
equations following from the Hamiltonian flow. In \cite{Kannoiii} the
form of the metric was instead imposed as an additional constraint,
and in fact this was more restrictive than was implied by the choice
of 4-form, leading to a highly constrained solution set. In this
paper, by contrast, we consider a more modest generalization of
previous choices for the 4-form ansatz (by including the $x_5$ term in
(\ref{4formrho})), but we do follow the Hitchin procedure and {\it
derive} the form of the metric, rather than imposing it as an
additional ansatz. We shall see that, as in Section 3, the extra
$x_5$ term will give rise to an off-diagonal term in the Spin(7)
metric. (Such terms were not included in the metric ansatz considered
in \cite{Kannoiii}.) In consequence, we obtain new Spin(7) metrics
that were not found in the analysis in \cite{Kannoiii}. The extension
to most general 4-form ansatz considered in \cite{Kannoiii}, and
with the metric derived from this ansatz, gives a more complicated system of
first-order equations, analogous to those obtained for $G_2$ metrics in
section 2. We shall not present these here, since they are rather
involved but straightforward to derive.
To calculate the metric we first construct the dual tensor density
\begin{eqnarray}
\tilde{\rho}^{abc}=\frac{1}{4!}\varepsilon^{abcd_1 d_2 d_3
d_4}\rho_{d_1 d_2 d_3 d_4}.
\end{eqnarray}
Then, by defining the symmetric tensor density
\begin{eqnarray}
H^{ab}=-\frac{1}{144}\tilde{\rho}^{ac_1 c_2}\tilde{\rho}^{bc_3
c_4}\tilde{\rho}^{c_5 c_6 c_7}\varepsilon_{c_1 c_2 \ldots
c_7}\,,
\end{eqnarray}
we can calculate the volume from $V=|\det H|^{1/12}$, finding
\begin{eqnarray}
V=a^2(a^4b_{1}^2b_{2}^2b_{3}^2-b_1 b_2 v_{1}^2 )^{\fft12}\,,
\end{eqnarray}
where
\begin{eqnarray}
a^4&=&2(x_1 +x_2 +x_3),\quad \quad a^2b_2 b_3=2(-x_1 +x_2 +x_3
+x_4),\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
a^2 b_3 b_1&=&2(x_1 -x_2 +x_3 +x_4), a^2b_1 b_2 =2(x_1
+x_2 -x_3 +x_4), \ \ v_1 =-2x_5\label{abx}.
\end{eqnarray}
The gradient flow equation is given by
\begin{eqnarray} \frac{\partial V}{\partial
x_1}&=&2(-\dot{x_1}+\dot{x_2}+\dot{x_3}+\dot{x_4}),\quad\frac{\partial
V}{\partial x_2}=2(\dot{x_1}-\dot{x_2}+\dot{x_3}+\dot{x_4}),\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
\frac{\partial V}{\partial
x_3}&=&2(\dot{x_3}+\dot{x_2}-\dot{x_3}+\dot{x_4}),\ \
\quad\frac{\partial
V}{\partial x_4}=2(\dot{x_1}+\dot{x_2}+\dot{x_3}),\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
\frac{\partial V}{\partial x_5}&=&-2\dot{x_5}.
\end{eqnarray}
It can be rewritten as
\begin{eqnarray}
\frac{da^4}{dt}&=&\frac{a^6}{V}b_1 b_2 b_3 (b_1 +b_2 +
b_3)-\frac{a^2v_{1}^2}{V},\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
\frac{da^2b_2
b_3}{dt}&=&\frac{V}{2a^4}+\frac{a^6}{V}b_1 b_2 b_3 (-b_1 +b_2
+b_3)-\frac{a^2v_{1}^2}{V},\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
\frac{da^2b_3
b_1}{dt}&=&\frac{V}{2a^4}+\frac{a^6}{V}b_1 b_2 b_3 (b_1 -b_2
+b_3)-\frac{a^2v_{1}^2}{V},\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
\frac{da^2b_1
b_2}{dt}&=&\frac{V}{2a^4}+\frac{a^6}{V}b_1 b_2 b_3 (b_1 +b_2
-b_3)-\frac{a^2v_{1}^2}{V},\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\
\frac{dv_1}{dt}&=&2\frac{a^4b_1 b_2}{V}v_1.\label{Spin7ODE}
\end{eqnarray}
The metric is obtained from
\begin{eqnarray}
h_{ab}=|\det H|^{1/6}H_{ab}
\end{eqnarray}
and it can be written as
\begin{eqnarray}
ds^2=dt^2+h_{11}\sigma_{1}^2+h_{22}\sigma_{2}^2+h_{33}\sigma_{3}^2+
2h_{12}\sigma_{1}\sigma_{2}+h_{\Sigma}^{2}(\Sigma_{1}^2+
\Sigma_{2}^2+\Sigma_{3}^2+\Sigma_{4}^2)\,,
\label{Spin7Metric}
\end{eqnarray}
where
\begin{eqnarray} h_{11}&=&\frac{b_1 b_2
(a^4b_{1}^2b_{3}^2+v_{1}^2)}{a^4 b_1 b_2 b_{3}^2-v_{1}^2},\quad \
h_{22}=\frac{b_1 b_2 (a^4b_{2}^2b_{3}^2+v_{1}^2)}{a^4 b_1 b_2
b_{3}^2-v_{1}^2},\quad h_{33}=b_{3}^2-\frac{v_{1}^2}{a^4b_1
b_2}\nonumber} \def\bd{\begin{document}} \def\ed{\end{document}\\ h_{12}&=&\frac{a^2b_1 b_2 (b_1 +b_2)b_3 v_1}{a^4 b_1 b_2
b_{3}^2-v_{1}^2},\quad h_{\Sigma}=a^2.
\end{eqnarray}
This metric together with the system of first-order
equations (\ref{Spin7ODE}) guarantees
\begin{eqnarray} \frac{\partial
\rho}{\partial t}=d(*\rho).
\end{eqnarray}
which shows that the metric determined by (\ref{Spin7ODE}) has Spin(7)
holonomy. We see that $h_{12}$ gives rise to an off-disgonal
term in (\ref{Spin7Metric}), which is proportional to $x_5$. Through the
relation in (\ref{abx}), this off-diagonal term results from the extra
$x_5$ term in (\ref{4formrho}).
\section{Discussion}
In this paper we have obtained a class of cohomogeneity one
$G_2$ metrics with $S^3\times S^3$ principal orbits, derived from
the most general invariant 3-forms $\rho$ and
4-forms $\sigma$ which break the anti-invariance under the ${{\mathbb Z}}/2$ action
interchanging the two $S^3$ factors. This should be the most general
cohomogeniety one
$G_2$ metric that one can obtain with principal orbits $S^3 \times
S^3$. Although we cannot obtain the explicit solution to the highly coupled
system of non-linear first-order differential equations, we luckily obtained
an analytic solution to the
metric with principal orbits $S^3 \times T^3$, which is arises as
a group contraction \cite{Chong} of one of the two $S^3$ factors. This
type of contraction was considered in \cite{Yau}. By showing that
our solution has more free parameters than the solutions obtained in
\cite{Yau}, we can conclude that this
generalization is a non-trivial one.
We also considered Spin(7) metrics with principal orbits $S^7$,
with a similar generalization of previous results obtained by writing
down a more general exact 4-form, constructed from the
$SU(2)$-connection 1-form $\alpha_i$ and curvature 2-form $\omega_i$.
It was shown there that the most general exact 4-forms one can
consider has ten parameters. In this paper we only considered a five
parameter exact 4-form, which is already sufficient to imply the new
feature of off-diagonal elements in the Spin(7) metrics. We
concentrated on how the generalization modified the system of
differential equations which determined the Spin(7) metrics. We also
showed how the metrics looks. The formalism presented here can also be
applied in a straightforward way to Spin(7) metrics whose principal
orbits are Aloff-Wallach spaces, i.e. $SU(3)/U(1)$ cosets.
\section*{Acknowledgments}
The author is grateful to Gary Gibbons, Hong L\"u and Chris Pope for useful discussions. He is supported in
part by DOE grant DE-FG03-95ER40917.
|
{
"timestamp": "2004-11-12T02:01:59",
"yymm": "0411",
"arxiv_id": "hep-th/0411125",
"language": "en",
"url": "https://arxiv.org/abs/hep-th/0411125"
}
|
\section{Introduction}
Many properties of metals depend crucially on the type and
concentration of defects that perturb the ideal crystal structure. Of
these, the simplest are point defects, such as self-interstitials and
vacancies. Since their formation energies are of order electron
volts, their equilibrium concentrations tend to be very low. They do
form in abundance, however, in radiation environments due to the
collisions between the irradiating species (electrons, heavy ions or
neutrons) and the atoms of the host crystal \cite{johnson}. When the
energy of the impinging particle is close to, but above, the
displacement threshold, the collision typically produces a single
Frenkel pair. Particles with higher kinetic energy, for instance
neutrons produced in fusion reactions, create collision cascades which
can produce not only Frenkel pairs, but an ensemble of mobile and
immobile self-interstitial and vacancy clusters of different
sizes. The ensuing evolution of the point defect distributions due to
diffusion \cite{flynn} determines the long time scale degradation of
the mechanical properties of the material, in addition to volume
swelling at large doses.
For body centered cubic (bcc) metals, such as $\alpha$-Fe, V and Mo,
molecular dynamics simulations have revealed a detailed microscopic
picture of the diffusive motion of individual defects and
vacancies\cite{delarubia97}. While the migration barrier $\Delta E_v$
for vacancy migration is rather high $(\sim0.5$ eV), both
self-interstitial atoms and small self-interstitial cluster are highly
mobile and easily diffuse along particular crystallographic directions
($\langle 111\rangle$-directions). In V, the lowest energy
self-interstitial configuration is a $\langle 111\rangle$-oriented
dumbell, which migrates easily with a crowdion transition state
\cite{han1}. However, this easy migration, with barriers as low as
$\Delta E_i\sim 0.02$ eV \cite{zepeda04}, leads to long
one-dimensional self-interstitial diffusional trajectories. The
self-interstitial dumbbell can change direction by rotating into other
$\langle111\rangle$-directions. The barrier associated with such
rotations $\Delta E_r$ is of the same order as $\Delta E_v$ (i.e.,
$\Delta E_r\gg\Delta E_i$).
In $\alpha$-Fe, by contrast, the ground state of the dumbbell
self-interstitial is the $\langle 110\rangle$ configuration, and
accessing the easy-glide $\langle 111\rangle$ configuration requires
overcoming a qualitatively similar rotational barrier that separates
the $\langle 110\rangle$ from the $\langle 111\rangle$ configurations
\cite{wirth97}. Although the migration barriers for the crowdion
mechanism in $\alpha$-Fe are also believed to be very small $(<0.04$
eV), the observed effective migration barrier (including rotation into
the $\langle 111\rangle$ orientation and migration along the $\langle
111\rangle$-direction) are higher than in V
\cite{marian01,soneda01}. Similar arguments apply to Mo and Ta. Both
diffusion mechanisms imply that interstitial transport in bcc metals
takes place in the form of long one-dimensional trajectories with
occasional directional changes that become more frequent as the
temperature increases. In fcc metals such as Cu, for instance,
simulations showed that self-interstitial diffusion occurs through
much more conventional, isotropic diffusion mechanisms
\cite{osetsky01}. In most metals, however, large separations in time
scale between self-interstitial and vacancy motion exist.
The intent of the present study is to illuminate the consequences of
the intriguing microscopic diffusion mechanisms in bcc metals on the
evolution of the point defect population using simple, but readily
generalizable models. Our models shall minimally include local
self-interstitial/vacancy mutual annihilation and absorption into
(unsaturable) sinks which, in a real system, are in the forms of
dislocations or grain boundaries. Together with the continuous
production of point defects, these processes form the fundamental
events in metals under irradiation conditions.
On the continuum level, point defect dynamics can be described by a
set of kinetic master or rate equations that treat the point defects
as a continuous density whose temporal evolution is governed by
various gain and loss processes. These equations do not take into
account spatial correlations and are only analytically tractable in
the simplest situations. Although we focus the discussion specifically
on bcc metals, the continuum theory makes no reference to the
underlying crystal structure and could be readily applied to fcc
metals as well. In many cases, single interstitials and
vacancies combine with defects of the same type to form stable
clusters. The evolution of the cluster size distribution and other
microscopic variables is more conveniently studied in a particle based
model that explicitly represents defects and their diffusion
mechanisms (translation, rotation) on a lattice. The competition
of events occuring with different rates can then be followed using a
kinetic Monte Carlo (kMC) scheme.
Both master equation and Monte Carlo approaches have frequently been
employed to illuminate the physics of damage evolution in irradiated
materials. Starting from the elementary processes mentioned above
\cite{brailsford72}, steady-state rate equations have been used to
study the effects of preferential absorption of self-interstitial
atoms at sinks (absorption bias) \cite{woo96,singh97}, the formation
of interstitial cluster during the cascade phase (production bias)
\cite{woo96,singh97,golubov00,trinkaus00}, and the one-dimensional
motion of these clusters in bcc metals
\cite{singh97,golubov00,trinkaus00}. kMC models were first used to
evolve the primary damage state obtained from ns-long MD simulations
to macroscopic time scales, but have increasingly been used to study
the evolution of defect structure during continuous irradiation in
copper \cite{heinisch96,heinisch97,heinisch99}, vanadium \cite{alonso}
and iron \cite{soneda03}.
Although many treatments have included a high level of atomistic
detail from the start, we begin by analyzing the simplest situation
that includes only production of Frenkel pairs, point defect
recombination and absorption at sinks in Section
\ref{simple-sec}. Even though this simplified case has been studied
many times before \cite{brailsford72}, we shall see that the inclusion
of specific features of bcc metals leads to a surprisingly rich
evolutionary picture. This approach allows us to gradually increase
the level of complexity of the model in a verifiable, controlled
manner. We introduce first interactions between vacancies that lead to
vacancy cluster formation in Section \ref{vaccluster-sec} and then
interactions between interstitials in Section \ref{intclust-sec}. In
particular, we shall discuss how interstitial cluster mobility affects
vacancy cluster formation. While the discussion in the text is
applicable to bcc metals broadly, we apply our findings to the
specific cases of V and $\alpha$-Fe in Section \ref{real-sec}.
\section{Point defect dynamics in simple situations}
\label{simple-sec}
\subsection{Atomistic details and kMC model}
\label{model-sec}
The physics of radiation damage evolution is governed by the
production of defects due to irradiation and their subsequent
elimination from the population through diffusional processes. Typical
values for the defect production rate $\sigma F$ range between
$10^{-3}$ dpa/s, in ion irradiation experiments \cite{iwai96}, and
$10^{-10}$ dpa/s, for neutron irradiation \cite{heinisch99}, where $F$
is the irradiation flux and $\sigma$ is the cross-section. A
``displacement per atom'' (dpa) refers to the production of one
Frenkel defect pair per lattice site.
Self-interstitial transport is composed of two parts, a
one-dimensional diffusive motion along one of the 4 distinct $\langle
111 \rangle$ directions and dumbbell rotations from one $\langle 111
\rangle$ direction to another. The temperature dependence of the
(one-dimensional) diffusivity $D_i$ is usually described through the
Arrhenius form in most cases except for vanadium, where the unusually
low activation barrier can lead to more complicated behavior at higher
$T$ (see Ref.~\onlinecite{zepeda04}). For $T<600K$, however, the
interstitial diffusivity is well described by the Arrhenius expression
$D_i/a^2=f\nu_i\exp{[-\Delta E_i/k_BT]}$, where $\Delta E_i$ is the
activation energy barrier, $\nu_i$ an attempt frequency and
correlations are expressed through the correlation factor $f$ ($f=1$
when the diffusion is uncorrelated). The temperature dependence of the
rotation rate $\gamma_r$, by contrast, is always of the Arrhenius
form, $\gamma_r=\nu_r\exp{[-\Delta E_r/k_BT]}$, where $\Delta E_r$ the
characteristic rotation barrier and $\nu_r$ is an attempt
frequency. Values for both $\nu_i$ and $\nu_r$ range between
$10^{12}s^{-1}$ and $10^{13}s^{-1}$. Finally, vacancies diffuse
three-dimensionally with rate $D_v/a^2=\nu_v\exp{[-\Delta E_v/k_BT]}$,
where $\Delta E_v$ and $\nu_r$ are the activation barriers and attempt
frequencies, respectively. The defects can undergo two basic
reactions: recombination once an interstitial and vacancy defect are
within a certain ``recombination volume'', or absorption at sinks.
Typical dislocation densities in metals are of the order $10^{12} -
10^{14}\,{\rm m}^{-2}$, which translates into dislocation sink
densities (per lattice site) of $n_s \sim 10 ^{-4}-10 ^{-6}$.
Since the relevant time scales (ns-sec) for the evolution of the point
defect distributions far exceed those of molecular dynamics
simulations, we employ a coarse grained description, in which we
represent vacancies, dumbbell interstitial configurations and sinks as
pointlike objects that occupy ideal lattice sites of a bcc lattice
with lattice parameter $a$. In order to mimic constant irradiation
conditions, new pairs of defects (self-interstitial and vacancy) are
introduced randomly onto defect-free lattice sites with rate $\sigma
F$. The microscopic diffusion process is replaced by instantaneous
hops of point defects to vacant neighboring lattice sites with rates
$D_i/a^2$ and $D_v/a^2$, respectively. A self-interstitial is
constrained to forward-backward hops along one of the four $\langle
111 \rangle$ directions, but can also rotate to another $\langle 111
\rangle$ direction with rate $\gamma_r$. Vacancies diffuse
isotropically. A self-interstitial or vacancy recombines if it finds
itself next to a vacancy or self-interstitial, respectively, or is
absorbed if one of its 8 neighboring sites contains a sink. At each
time step, an event is chosen according to its probability and then
executed. Time is advanced according to the usual continuous time
algorithm \cite{binder92}, where the time increment is chosen from an
exponential distribution.
\subsection{Basic rate equations}
The elementary processes described in the kMC model can,
alternatively, be described within a rate equation formalism. Before
the advent of large scale computer simulations, this method
represented the only viable theoretical approach for simulating long
time radiation damage evolution. The rate equations can be solved by
direct numerical integration or direct analysis in limiting cases. In
the minimal model discussed above, the time evolution of the number
densities of the interstitals $n_i(t)$ and vacancies $n_v(t)$ is given
by the coupled nonlinear equations \cite{sizmann68,brailsford72,sizmann74}
\begin{eqnarray}\nonumber
\frac{dn_i}{dt}&=&\sigma
F-\kappa_v\omega_{iv} n_i-\kappa_i\omega_{vi} n_v-\kappa_{is}\omega_{is} n_i\\
\frac{dn_v}{dt}&=&\sigma
F-\kappa_v\omega_{iv}n_i-\kappa_i\omega_{vi} n_v-\kappa_{vs}\omega_{vs} n_v
\label{rate-eq}
\end{eqnarray}
Defect pairs are added to the population at a rate proportional to the
particle flux $F$ and a cross-section $\sigma$. Loss can occur through
a diffusing interstitial recombining with a vacancy with rate
$\omega_{iv}$ and a diffusing vacancy recombining with an interstitial
with rate $\omega_{vi}$. The recombination rates are all proportional
to the diffusion constant of the moving defect, but depend on the
dimensionality of the diffusion process. $\kappa_i$ and $\kappa_v$ are
dimensionless capture numbers that represent the spatial extent of the
defects and their effective (possibly long ranged) interactions.
Losses can also occur through absorption at sinks with rates
$\omega_{is}$ and $\omega_{vs}$ and corresponding capture numbers
$\kappa_{is}$ and $\kappa_{iv}$ for interstitials and vacancies,
respectively. An alternative representation of Eqs.~(\ref{rate-eq})
is to define sink strengths $k_x^2$ via the relation
$k_{x}^2D_{x}=\kappa_{x}\omega_x$, where the subscript $x$ refers to any
of the combination of indices used above.
The encounter rates of the defects are given by the number of distinct
sites visited by a random walker per unit time. Since the mean squared
displacement $\langle R^2 \rangle=l^2N$ for a random walk with step
length $l$, the number of sites visited is $s=[\langle R^2
\rangle/l^2]^{1/2}\sim N^{1/2}$ in one dimension. By contrast, a
detailed analysis of random walks on three-dimensional cubic lattices
shows that a random walker visits $\mathcal{O}(N)$ distinct sites
after $N$ hops. For a given density of target sites $n$, the typical
collision time $\tau_c$ is given by the condition $D\tau_c/a^2=1/n$
(3D) and $(D\tau_c/a^2)^{1/2}=1/n$ (1D), from which we deduce the
encounter rates
\begin{equation}
\omega_{3D}\sim nD/a^2 \qquad {\rm and} \qquad \omega_{1D}\sim n^2D/a^2.
\end{equation}
In a similar manner, the capture numbers $\kappa$ are also affected by
the dimensionality of the random walk. Assuming ideal spherical
defects of linear dimension $r$, $\kappa \sim r/a$ for a 3D random
walk in a mean-field approximation \cite{brailsford81}, but in
general, capture numbers can also depend on spatial fluctuations and
dose. For a 1D random walk, the scaling with defect size becomes much
stronger \cite{barashev01}, $\kappa \sim (r/a)^4$.
These expressions apply to strictly 1D or 3D random walks. As
discussed in the Introduction, we encounter an intermediate case in
bcc metals, where rotations interrupt 1D random walks and lead to 3D
trajectories. The encounter rate of this random walk must, therefore,
be larger than the purely 1D case. For a rotation rate $\gamma_r$, the
random walker performs on average $D/\gamma_r a^2$ hops along a
particular direction before rotating into a new direction. There are
$\gamma_r\tau$ of these segments during time $\tau$. The mixed 1D/3D
collision time thus follows from the condition $(D/\gamma_r
a^2)^{1/2}\gamma_r \tau_c=1/n$, which implies
\begin{equation}
\omega_{1D/3D}=\omega_{3D}\sqrt{\gamma_ra^2/D}
\label{om-mixed-eq}
\end{equation}
The mixed encounter rate $\omega_{1D/3D}$ scales like the 3D encounter
rate $\omega_{3D}$, but is reduced by the square root of the ratio of
the number of rotations to hops. This reaction rate has also been
derived in ref.~\onlinecite{barashev01}. This work views the kinetics
in the intermediate 1D/3D regime as an enhanced 1D reaction rate, but
the resulting expressions agree up to numerical prefactors. These
authors also showed that the size dependence of the mixed capture
number $\kappa \sim (r/a)^2$, and ref.~\onlinecite{trinkaus02}
provides an interpolation formula between the limiting cases using a
continuum description. While the encounter rates and reaction kinetics
of random walkers decrease when the dimensionality of the random walk
changes from three to one, the diffusivity does not, since the mean
squared displacement of a random walk of $N$ steps of length $l$ is
$\langle {\bf R}(N)^2\rangle=l^2N$ independent of dimensionality.
Note that these encounter rates are derived under the assumption of
collisions with a stationary target. The case of several colliding 1D
random walkers becomes equivalent to a 3D random walk because, from
the rest frame of a given walker, the other walkers appear to be
executing a 3D random walk. This case would be relevant for describing
the collisions of interstitials with each other, but not with the
vacancies or sinks.
Inserting these encounter rates into Eqs.~(\ref{rate-eq}), assuming
mixed 1D/3D encounter for diffusing interstitials, yields rate
eqations of the form
\begin{eqnarray}\nonumber
\frac{dn_i}{dt}&=&\sigma
F-n_i n_v(\kappa_v \sqrt{\beta}D_i/a^2-\kappa_i D_v/a^2)\\&-&\kappa_{is} n_s n_i \sqrt{\beta}D_i/a^2\\
\frac{dn_v}{dt}&=&\sigma \nonumber
F-n_i n_v(\kappa_v \sqrt{\beta}D_i/a^2-\kappa_i D_v/a^2)\\&-&\kappa_{vs} n_s n_v D_v/a^2,
\label{rate2-eq}
\end{eqnarray}
where $\beta=\gamma_ra^2/D_i$ is a dimensionless ratio that describes
the relative frequency of dumbbell rotations and diffusional hops.
\subsection{Simple scaling analysis}
We now specialize these results to the common case of bcc metals,
where $\gamma_ra^2/D_i\ll 1$ and $D_i/D_v\gg 1$. Before performing the
kMC simulations and solving the full rate equations numerically, it is
instructive to analyse the limiting behaviors of this system
\cite{sizmann68}. Inititally, there are no defects in the metal, and
only the first term in the rate equations is important. In this regime
(regime I), defect densities increase linearly with time,
\begin{equation}
n_i^{\rm I}=n_v^{\rm I}=\sigma F t.
\end{equation}
Once a sufficient density of defects has been produced such that the
encounter times between defects becomes smaller than the time between
creation events, loss through recombination ($2^{nd}$ and $3^{rd}$
terms) becomes important \cite{sizmann64}. As we shall see below,
typical parameter ranges lead to a situation in which defect
recombination becomes important before sink loss. The balance between
creation and recombination makes a steady state possible:
$dn_i/dt=dn_v/dt=0$. Ignoring the sink terms, one readily finds for
this Regime II
\begin{equation}
\label{regII-eq}
n_i^{\rm II}=n_v^{\rm
II}=\left(\frac{\kappa_v\sqrt{\beta}D_i+\kappa_iD_v}{\sigma
Fa^2}\right)^{-1/2}\sim\Gamma^{-1/2},
\end{equation}
which only depends on the dimensionless ratio
$\Gamma=\frac{(\sqrt{\beta}D_i+D_v)}{\sigma Fa^2}$. The crossover from
Regime I to Regime II occurs at time $t_{\text{I/II}}=F\sigma
t_{\text{I/II}} \simeq \Gamma^{-1/2}$, i.e., $t_{\text{I/II}}\sim
n_i^{-1}(D_i/a^2)^{-1}$. Note that this scaling regime is bound from
above by the following condition: if $n_v$ becomes so large that the
time between interstitial-vacancy encounters is on average shorter
than the time between two subsequent rotations (i.e., if
$\Gamma<\beta^{-1}$), the scaling of the encounter time becomes that
of a 1D random walk. In this case, the scaling of the steady state
defect density with $\Gamma$ changes from $n_i^{\rm II}=n_v^{\rm
II}\sim\Gamma^{-1/2}$ to $n_i^{\rm II}=n_v^{\rm II}\sim
\Gamma^{-1/3}$. This regime is unlikely to be of experimental
relevance.
Eventually, loss through sinks becomes important and the steady state
Regime II ends. Clearly, there exists a terminal steady state (see
below) in which all loss terms in Eqs.~(\ref{rate-eq}) balance the
creation of defects. If $D_i\gg D_v$ as in the bcc metal case,
however, the system will initially loose mostly interstitials and very
few vacancies. This breaks the symmetry between interstitials and
vacancies and $n_v\gg n_i$. The second steady state is, therefore,
reached through a transient Regime III with distinct
scaling. Subtracting the two rate equations (and neglecting vacancy
loss at sinks),
\begin{equation}
\frac{d(n_v -n_i)}{dt}\simeq\frac{dn_v}{dt}=\kappa_{is}n_s\sqrt{\beta}\frac{D_i}{a^2}n_i,
\end{equation}
where $n_s$ denotes the sink density. This equation allows us to
eliminate $n_i$ in the rate equation for $n_v$, so that
\begin{equation}
\frac{dn_v}{dt}\simeq\sigma F -\frac{\kappa_vn_v}{\kappa_{is}n_s}\frac{dn_v}{dt}.
\label{regIII-eq}
\end{equation}
Integrating Eq.~(\ref{regIII-eq}), we
obtain (to leading order) power-law growth of $n_v$ with time,
\begin{equation}
n_v^{\rm III}\simeq (\kappa_sn_{is}\sigma F t/\kappa_v)^{1/2}.
\end{equation}
In the case of extremely large sink densities $n_s$, the encounter
rates $\omega_{is}$ between interstitials and sinks would not be 3D as
assumed above, but 1D-like. In this case, $n_s$ would have to be
replaced by $n_s^2$, but again, such high densities are unrealistic.
The crossover time $t_{\text{II/III}}$ between Regimes II and III can
be obtained from the condition $\Gamma\simeq \kappa_{is}n_s\sigma F
t_{\text{II/III}}/\kappa_v$, which implies $t_{\text{II/III}}\sim
(\kappa_v/\kappa_{is}n_s)^{1/2}/(D_i/a^2)$. The crossover occurs at
constant time, independent of the defect densities.
If there are only sinks for one species of defect and no sinks for the
other, Regime III will simply continue. In all other cases, a final
steady state will occur when all loss terms are important. Setting
again $\frac{dn_i}{dt}=\frac{dn_v}{dt}=0$ as in
Ref.~\onlinecite{sizmann74}, we obtain a condition for steady state,
\begin{equation}
\label{regIV-1}
\frac{n_v^{\rm IV}}{n_i^{\rm IV}}=\frac{\sqrt{\beta}D_i}{D_v}=\frac{\kappa_{is}}{\kappa_{vs}}\alpha\sqrt{\beta},
\end{equation}
where we have introduced a third dimensionless ratio $\alpha=D_i/D_v$.
The interstitial and vacancy populations will, in general, be
different. The steady state values are the roots of the quadratic
equation
\begin{eqnarray}
\label{regIV-2}
(n_v^{\rm IV})^2 \frac{\kappa_{vs}\Gamma}{\kappa_{is}\alpha\sqrt{\beta}}-n_v^{\rm IV}\frac{\kappa_{vs}n_sD_v}{\sigma Fa^2}=1.
\end{eqnarray}
Since typically $n_s\ll 1$, the linear term can be neglected relative
to the quadratic term, and we obtain the scaling of the vacancy
density in Regime IV,
\begin{equation}
n_v^{\rm IV}\sim \left(\frac{\Gamma}{\alpha\sqrt{\beta}}\right)^{-1/2}.
\end{equation}
The crossover time $t_{\text{III/IV}}$ follows again from the
condition $\kappa_{is}n_s\sigma F t_{\text{III/IV}}/\kappa_v\simeq
(\Gamma/\alpha\sqrt{\beta})^{-1}$, i.e.,~$t_{\text{III/IV}}\sim
(\alpha\sqrt{\beta}\kappa_v/\kappa_{is}n_s)^{1/2}/(D_i/a^2)$ is again
independent of $n_i$ or $n_v$.
The volume swelling is proportional to the excess vacancy population
in this regime, $S=n_v-n_i$. Note that the imbalance arises here due
to the very different diffusivities of the two types of defects. Upon
termination of the irradiation, both the $n_i$- and $n_v$-populations
would relax exponentially to zero in this simple model.
\subsection{Numerical integration and kMC}
A full solution of the rate equations (\ref{rate-eq}) is only possible
numerically. In the following, we show a series of such numerical and
kinetic Monte Carlo simulation results for a representative choice of
parameters $D_i=1000 D_v$, $\gamma_r=0.01 D_i/a^2$, so that
$\alpha=1000$ and $\beta=0.01$ (see Section \ref{real-sec} for typical
experimental regimes). In addition, we set the density of sinks to
$n_s=10^{-4}$. Figure~\ref{time-fig} shows the evolution of the
average interstitial and vacancy density (symbols) based on the above
parameters obtained from the kMC simulation and numerical integration
of the corresponding rate equations (solid lines) using the expression
Eq.~(\ref{om-mixed-eq}) for the mixed encounter rates. The
dimensionless parameter $\Gamma$ was varied by changing the defect
creation rate $F\sigma$. As predicted by the scaling analysis, four
distinct regimes appear with increasing time. After first rising
linearly with time (Regime I), $n_i$ and $n_v$ reach the steady state
plateau of Regime II. Once loss through sinks becomes important
(Regime III), $n_v$ increases as $t^{1/2}$ while $n_i$
decreases. Finally, all curves reach the steady-state Regime IV, where
$n_i$ and $n_v$ are given by Eqs.~(\ref{regIV-1}) and
(\ref{regIV-2}). At the highest defect densities, the kMC simulations
where not carried out into the final steady state regimes, because the
computational effort becomes very large.
\begin{figure}[t]
\includegraphics[width=8cm]{fig1finsm}
\caption{\label{time-fig}Self-interstitial ($\blacksquare$), $n_i$,
and vacancy ($\lozenge$), $n_v$, densities as a function of time for
$\Gamma=10^2, 10^3, 10^4, 10^5,$ and $10^6$ ($\Gamma$ increases from
the top to bottom of the figure). Time is measured in units of the
inverse interstitial hopping rate $a^2/D_i$. The solid lines show the
result of direct numerical integration of Eqs.~(\ref{rate-eq}) with
$\kappa_v=\kappa_i=\kappa_{is}=\kappa_{vs}=21$, and the symbols correspond to the
results of the kMC simulations in a periodic simulation box. The
straight solid lines have slope $1$ and $1/2$. }
\end{figure}
Excellent agreement between continuum theory and kMC model for the 3
largest values of $\Gamma$ is achieved by adjusting all capture
numbers to a single numerical constant. In the simple situation
examined here, kMC and rate theory are completely equivalent. Since we
have used $\omega_{\rm 1D/3D}$ in the rate equations, this agreement
also validates the scaling argument for the mixed 1D/3D encounter
rate. For the two smallest $\Gamma$-values, we observe increasing
discrepancies between rate equations and kMC. As discussed above, we
expect a transition from the mixed 1D/3D encounter rates $\omega_{iv}$
to 1D dominated encounter kinetics when the interstitial density
becomes so high that they typically collide with vacancies or sinks
before rotating ($\Gamma<100$). This transition is not faithfully
reproduced by the rate theory which assumes only the limiting cases of
either 1D or mixed 1D/3D encounter rate scaling, but is properly
captured by the kMC which includes explicit 1D/3D trajectories.
Note also that with increasing $\Gamma$, Regime II begins to shrink
and Regime I crosses over directly into Regime III. This happens
because as the defect densities decrease below the sink density, point
defect loss at sinks will become dominant before recombination plays a
significant role. Since the crossover time $t_{\text{I/III}}\simeq
\kappa_sn_s/\kappa_v\sigma F$ only depends on the point defect
production rate, a further decrease of the production rate will
eventually lead to a direct crossover from Regime I to Regime IV. A
smaller sink density would push the appearance of Regimes II and III
to higher values of $\Gamma$.
\section{Vacancy cluster formation}
\label{vaccluster-sec}
\subsection{Irreversible aggregation}
To this point, neglected interactions between vacancies. Experimental
vacancy-vacancy binding energies $E_b$ can be of order the single
vacancy migration energy. This suggests that it is not unreasonable
to expect vacancies to form stable clusters or microvoids once they
meet. A relatively simple approximation for this situation is to
consider the limiting case of ``irreversible aggregation'', which
neglects any dissociation of a vacancy from a cluster and thus sets
$E_b=\infty$. The realistic situation, which includes finite binding
energies, can be viewed as an intermediate case between irreversible
aggregation and the case of $E_b=0$ of the previous section.
In the ``irreversible aggregation'' model, vacancies bind to form
stable divacancies, leaving the population of free vacancies, while
interstitials recombining with a divacancy recreate a single mobile
vacancy. Stable cluster grow or shrink under the influence of arriving
vacancies and interstitials. Denoting the number density of clusters
of size $m$ as $n_c(m)$, we can write the following set of rate
equations\cite{ghoniem89}:
\begin{eqnarray}\nonumber
\frac{dn_i}{dt}&=&\sigma
F-\kappa_i\omega_{vi} n_v\\\nonumber
&-&(\kappa_v\omega_{iv}+\kappa_{is}\omega_{is}+\sum_m\kappa_i^m\omega_{ic(m)})n_i\\\nonumber
\frac{dn_v}{dt}&=&\sigma
F-(\kappa_i\omega_{vi}+\kappa_{vs}\omega_{vs}+\sum_m\kappa_v^m\omega_{vc(m)}\\\nonumber
&-&2 \kappa_v \omega_{vv})n_v-(\kappa_v\omega_{iv}-\kappa_i^2\omega_{is(2)})n_i\\\nonumber
\frac{dn_c(m)}{dt}&=&(\kappa_v^{m-1}\omega_{vc(m-1)}-\kappa_v^{m}\omega_{vc(m)})n_v\\
&+&(\kappa_i^{m+1}\omega_{ic(m+1)}-\kappa_i^{m}\omega_{ic(m)})n_i ,
\label{rate-irr-eq}
\end{eqnarray}
where $n_v\equiv n_c(1)$ is now understood to refer to the free
vacancy density (or cluster of size 1). The additional terms in the
equations for the evolution of $n_i$ and $n_v$ account for
interstitial/vacancy-cluster encounters, vacancy/vacancy-cluster
encounters, vacancy-cluster nucleation and divacancy
decomposition. $\omega_{ic(m)}= n_c(m)\sqrt{\beta}D_i/a^2$,
$\omega_{vc(m)}= n_c(m)D_v/a^2$ and $\omega_{vv}=n_vD_v/a^2$ denote
the corresponding encounter rates. The last expression in
(\ref{rate-irr-eq}) describes the evolution of stable, immobile
vacancy clusters of size $m>1$ due to the diffusive arrival of
interstitials and vacancies. The hierarchy of rate equations
(\ref{rate-irr-eq}) is, in principle, amenable to a semi-analytical
treatment via numerical integration, if all constants $\kappa^m_{i/v}$ are
specified. However, such a solution requires termination of the set of
equations at a finite cluster size $m_{max}$, i.e., one imposes a
boundary condition $n_c(m_{max})=0$. The corresponding kMC model does
not require any ad hoc assumptions and takes all these processes
naturally into account.
Before examining this model in kMC, we can draw some premliminary
conclusions. In Regime I, only nucleation of divacancies will be
important. Since here the vacancy density grows linearly with time, we
predict the scaling regime
\begin{equation}
n_c(2)^I\sim(\sigma Ft)^3.
\end{equation}
The divacancy density grows cubically with time. In the steady-state
Regime II, the vacancy cluster density is slaved to the free vacancy
density and, therefore, is constant as well. For $\alpha\gg1$, we
expect the total cluster density $n_c=\sum_m n_c(m)$ to be much
smaller than $n_v$ and $n_i$, so that the presence of vacancy
clustering does not yet strongly alter the interstitial and vacancy
population. Using Eq.~(\ref{regII-eq}), we find
\begin{equation}
n_c^{II}\simeq \Gamma^{-1/2}/(\alpha\sqrt{\beta}).
\end{equation}
Once past Regime II, the cluster density will begin to rise again as
$n_v$ increases. $n_v$ will then increasingly fall below the total
number of vacancies in the system $n_{vtot}$, because increasing
numbers of vacancies $\sum_{m>1} mn_c(m)$ become immobilized in
clusters. These clusters act as additional sinks for the diffusing
vacancies, but do not remove them from the system. The higher the
cluster density, the smaller the loss of vacancies through sinks and
interstitial recombination. We therefore expect that Regime III will
extend to larger times and the presence of the immobile vacancy
cluster delays the onset of the final steady-state Regime IV.
For the total number of vacancies $n_{vtot}$, we can write the rate
equations as
\begin{eqnarray}\label{nvtot-scaling-eq}
\frac{dn_{vtot}}{dt}&=&\sigma
F-\kappa_i\omega_{vi}n_v-\sum_{m=1}\kappa_i^{m}\omega_{ic(m)}n_i\\
\nonumber
\frac{dn_i}{dt}&=&\sigma
F-\kappa_i\omega_{vi} n_v-(\sum_{m=1}\kappa_i^m\omega_{ic(m)}-\kappa_{is}\omega_{is})n_i,
\end{eqnarray}
where we have neglected vacancy loss at sinks. This is the Regime III
situation discussed earlier. Subtraction of
Eqs.~(\ref{nvtot-scaling-eq}) and elimination of $n_i$, as in
Eq.~(\ref{regIII-eq}), leads to
\begin{equation}
\frac{dn_{vtot}}{dt}\simeq \sigma F -
\frac{\kappa_v}{\kappa_{is}n_s}\frac{dn_{vtot}}{dt}\sum_{m=1}\frac{\kappa_i^m}{\kappa_v}n_c(m).
\end{equation}
At relatively early times, when only small clusters are present,
$\sum_{m=1}\frac{\kappa_m}{\kappa_v}n_c(m)\simeq n_{vtot}$ and
$n_{vtot}$ will initially increase as $n_{vtot}\sim (\kappa_{is}n_s\sigma
F t)^{1/2}$. The subsequent behavior depends on the detailed form of
the capture numbers $\kappa_{is}$.
In the absence of vacancy clustering, Regime III ends when the vacancy
density has become so large that vacancy loss through sinks balances
interstitial loss through sinks. This condition, Eq.~(\ref{regIV-1}),
must hold for a final steady state to appear. In the present simple
model, this condition depends only on the ratio of the diffusivities,
but in a real system, Eq.~(\ref{regIV-1}) also depends on sink
concentrations and capture numbers, which may be different for
interstitial and vacancies. Eventually this condition can also occur
as a result of vacancy clustering, because vacancy clusters act as
additional sinks for the interstitials and remove them symetrically
from the system. At high cluster densities, the low mobile vacancy
concentration can therefore be compensated. In the steady-state regime
IV, we expect the hierarchy
$n_i^{IV}<n_v^{IV}<n_c^{IV}<n_{vtot}^{IV}$. Because of the later entry
into Regime IV, $S=n_{vtot}-n_i$ (the total amount of volume swelling)
will be larger than in the absence of clustering.
\begin{figure}[t]
\includegraphics[width=8cm]{fig2finsm}
\caption{\label{cluster-fig}Interstitial densities $n_i$ (
$\blacksquare$), free vacancy density $n_v$ ($\lozenge$) as well
as the total vacancy density $n_{vtot}$ ($\star$) and vacany cluster
density $n_{c}$ ($\blacktriangle$) as a function of time for (a)
$\Gamma=10^5$, (b) $\Gamma=10^7,$ and (c) $\Gamma=10^{9}$. The thin
solid lines show the result of direct numerical integration of
Eqs.~(\ref{rate-irr-eq}) with $\kappa^m_i=\kappa_v^m=25$ and a $m_{max}=20$ (see
text). }
\end{figure}
In Regime IV, we can obtain an approximate scaling relation for the
total cluster density. The density is determined by the competition
between nucleation of clusters out of the vacancy gas and
decomposition of divacancies due to arriving interstitials. Denoting
the fraction of clusters that represent divacancies as $f$, we can
write the rate equation
\begin{equation}
\frac{dnc^{{\text
IV}}}{dt}=\kappa_vn_v^2D_v/a^2-f(t)\kappa_i^2n_c^{\text
IV}n_i\sqrt{\beta}D_i/a^2,
\end{equation}
which implies $n_c^{\text
IV}=\kappa_vn_v^2/f\kappa^2_i\alpha\sqrt{\beta}n_i=\kappa_vn_v/f\kappa_2$,
where we have used Eq.~(\ref{regIV-1}). The cluster density is
therefore of order the free vacancy density with a numerical prefactor
that depends on the mean cluster size.
\subsubsection{Average cluster density}
The effect of irreversible vacancy aggregation on the defect dynamics
as seen through kMC is shown in Fig.~\ref{cluster-fig} for three
values of $\Gamma=10^5$, $\Gamma=10^7$ and $\Gamma=10^{9}$ with all
other parameters as before. In this figure, we also plot the total
cluster density $n_c$ and the total vacancy density $n_{vtot}$. In
Regimes I and II, $n_c$ and $n_i$ are nearly unchanged. The defect
density $n_c$ rises first $\sim t^3$ and then remains constant during
Regime II as discussed above. In Regime III, $n_v$ begins to fall
below the result from Fig.~\ref{time-fig} because of trapping of
vacancies into clusters. At the same time, $n_{vtot}$ rises with an
ideal $t^{1/2}$ law, and Regime III extends to larger times. $n_c$
also rises again due to new cluster nucleation events out of the
vacancy gas. All quantities reach constant values in regime IV. As one
might expect, the total number of vacancies in the system is higher
than in the absence of clustering. However, the general sequence of
regimes remains unchanged.
In order to gain additional insight into the growth dynamics, we also
numerically integrate the full set of rate Eqs.~(\ref{rate-irr-eq})
(solid lines in Fig.~\ref{cluster-fig}). This requires specification
of the capture numbers $\kappa^m$. In the simplest mean field model
for nucleation dynamics, the capture numbers do not depend on cluster
size $m$ and $\kappa^1_{i,v}=\kappa^m_{i,v}=const.$ (i.e., the point cluster
model). Interestingly, we find surprisingly good agreement between the
rate equations (solid lines in Fig.~\ref{cluster-fig}) and the full
kMC simulations when using this simple approximation. This suggests
that for relatively small $m$, $\kappa^m$ is only very weakly
size-dependent. The rate equations faithfully reproduce the sequence
of regimes, but begin to show deviations at late times when larger
clusters become more prominent and the point cluster assumption is no
longer accurate. Here, $\kappa^m$ begins to show some size
dependence. The success of this comparison shows, however, that a
mapping between kMC and rate equations is possible even with void
nucleation and growth if the $\kappa^m$'s are accurately parametrized.
Note that in the present model, the volume swelling rate $dS/dt$ is
zero in Regime IV. This is possible because we have chosen the same
capture radii $\kappa_s$ for the sinks for both interstitials and
vacancies, i.e., the sinks are symmetric with respect to defect
type. $S\sim t$ would be valid, if, for example, the capture radius
for interstitials is larger than that for vacancies. This situation
can arise with the introduction of dislocation sinks (known
as``dislocation bias'') and has been invoked to explain unusually
large swelling rates (see also Section \ref{conc-sec}) at times later
than considered here \cite{woo96,singh97}.
\subsubsection{Cluster size distribution}
A full description of the evolution of the point defects in the
material includes a characterization of the cluster size
distribution. The growth of these clusters is the result of the
diffusive arrival of interstitial and vacancy defects at already
nucleated clusters. The full dynamics of this process is described by
the last equation in (\ref{rate-irr-eq}). Consider this equation in
the following simplified notation:
\begin{eqnarray}\nonumber
\frac{dn_c(m)}{dt}&=&n_v(\kappa_v^{m-1}n_c(m-1)-\kappa_v^m n_c(m))D_v/a^2\\
&+&n_i(\kappa_i^{m+1} n_c(m+1)-\kappa_i^{m} n_c(m))\sqrt{\beta}D_i/a^2.
\label{clustsize-eq1}
\end{eqnarray}
If $n_i$ and $n_v$ change very slowly in time, the distribution
$n_c(m)$ is given by the steady state solution to
Eq.~(\ref{clustsize-eq1}), i.e., $dn_c(m)/dt=0$. This equation is
supplemented by the detailed balance condition
\begin{equation}
n_v\kappa_v^m n_c(m)D_v/a^2=n_i \kappa_i^{m+1}n_c(m+1)\sqrt{\beta}D_i/a^2,
\end{equation}
which should also hold for $m\ge 2$. From this condition, we can
deduce the distribution $n_c(m)$ by induction. Since $n_v\equiv
n_c(1)$, we have $n_c(2)=n_v^2\kappa_1/(\kappa^2_i\alpha\sqrt{\beta}
n_i)$ and consequently for all $m>1$
\begin{equation}
n_c(m)=\frac{n_v^m}{(\alpha\sqrt{\beta} n_i)^{m-1}}\frac{\prod_{l=1}^{m-1}\kappa_v^l}{\prod_{l=2}^m\kappa_i^l}.
\label{ncdist-eq}
\end{equation}
Equation~(\ref{ncdist-eq}) is tested against the kMC results in
Fig.~\ref{clusterdist-fig}, where we show the cluster size
distributions at four different times. All curves fall along straight
lines in a semilogarithmic plot, and the slope decreases with
increasing mean cluster size. A comparison with Eq.~(\ref{ncdist-eq})
requires, again, knowledge of the capture numbers $\kappa^m$. As in
the previous section, use of the point cluster approximation
$\kappa^l_{i,v}/\kappa^1_{i,v}=1$ yields excellent agreement
between between the kMC data and Eq.~(\ref{ncdist-eq}) upon inserting
the values for $n_i(t)$ and $n_v(t)$ at the appropriate times.
\begin{figure}[t]
\includegraphics[width=8cm]{fig3finsm}
\caption{\label{clusterdist-fig} Plot of the cluster size distribution
$n_{c}(m)$ found from the kMC simulations with $\Gamma=10^5$ of
Fig.~\ref{cluster-fig}(a) at four different times $tD_i/a^2=3\times
10^4$ $(\blacksquare)$, $3\times 10^5$ $(\blacklozenge)$, $3\times
10^6$ $(\star)$, and $3\times 10^7$ $(\blacktriangle)$ in Regimes III
and IV. The straight lines correspond to
$n_v(t)^m/(\alpha\sqrt{\beta}n_i(t))^{m-1}$.}
\end{figure}
Note that the distributions shown in Fig.~\ref{clusterdist-fig} are
not peaked, i.e., the frequency with which different clusters appear
decrease with increasing cluster size and single vacancies occur most
frequently. This situation is in sharp contrast to other cluster
growth situations as, e.g., found in submonolayer island growth during
vapor deposition \cite{amar95}, where the distribution $n_c(m)$ peaks
at the mean cluster size $\langle m \rangle$. In this problem,
however, clusters cannot shrink, since vacancies are absent. One
expects Eq.~(\ref{ncdist-eq}) to hold as long as the assumption of
quasi-stationary values for $n_i$ and $n_v$ is valid.
\begin{figure}[t]
\includegraphics[width=8cm]{fig4afinsm}
\includegraphics[width=8cm]{fig4bfinsm}
\caption{\label{clusterrev-fig}(a) Interstitial density $n_i$
($\blacksquare$), free vacancy density $n_v$ ( $\lozenge$) as
well as the total vacancy density ($(\blacktriangle)$) and vacancy
cluster density $n_{c}$ $(\star)$ as a function of time for
$\Gamma=10^5$, and $10^{7}$ for reversible aggregation, where
$\delta=0.01$. (b) Cluster size distribution $n_{c}(m)$ for the case
of $\Gamma=10^5$ at four different times in Regimes III and IV. The
straight lines correspond to
$n_v(t)^m/(\alpha\sqrt{\beta}n_i(t)+\kappa_d^1/\kappa_i^1\alpha^\prime)^{m-1}$
with $\kappa_d^1/\kappa_i^1=0.25$.}
\end{figure}
\subsection{Reversible attachment}
The above discussion immediately raises the question of whether the
results for $n_c(t)$ and its size distribution Eq.~(\ref{ncdist-eq})
survive in the more realistic situation of reversible cluster growth,
i.e., vacancies have a finite probability to attach to and to leave
the cluster. Let us introduce ``detachment rates'' from a cluster of
size $m$, $\gamma_{det}(m)/a^2$ for all vacancies, independent of
their local environment. Equation~(\ref{clustsize-eq1}) needs to be
generalized by the addition of two terms,
\begin{eqnarray}\nonumber
\frac{dn_c(m)}{dt}&=&n_v(\kappa_v^{m-1}n_c(m-1)-\kappa_v^mn_c(m))D_v/a^2\\
&+&n_i(\kappa_i^{m+1}n_c(m+1)-\kappa_i^mn_c(m))\sqrt{\beta}D_i/a^2,\\
&+&(\kappa^{m+1}_d n_c(m+1)-\kappa^{m}_dn_c(m))\gamma_{det}(m)/a^2,
\label{clustsize-rev-eq}
\end{eqnarray}
where $\kappa^m_d$ represent ``detachment numbers'' in analogy to the
capture numbers $\kappa^m_{i,v}$. Consequently, the condition for detailed
balance now reads
\begin{eqnarray}\nonumber
\kappa_v^{m}n_vn_c(m)D_v/a^2&=&\kappa_i^{m+1}n_in_c(m+1)\sqrt{\beta}D_i/a^2\\
&+&\kappa^{m+1}_dn_c(m+1)\gamma_{det}(m)/a^2
\end{eqnarray}
and Eq.~(\ref{ncdist-eq}) generalizes to
\begin{equation}
n_c(m)=\frac{\prod_{l=1}^{m-1}\kappa_v^l
n_v^m}{\prod_{l=2}^{m}(\kappa_i^l\alpha\sqrt{\beta}
n_i+\kappa_l^d\alpha^\prime(m))},
\label{ncdist-rev-geneq}
\end{equation}
where $\alpha^{\prime}(m)=\gamma_{det}(m)a^2/D_v$. We see that the
general form of the distribution is the same, but the prefactor
changes due to the additional growth and shrinkage probabilities.
From a statistical point of view, interstitial arrival and vacancy
detachment are equivalent. Eq.~(\ref{ncdist-rev-geneq}) can be easily
evaluated for any functional form of the size-depedent capture numbers
and detachment rates. In the point cluster approximation, where all
$\kappa^m_{i,v,d}/\kappa^1_{i,v,d}=1$ and
$\alpha^\prime=\alpha^{\prime}(m)$ is size-independent, it predicts an
exponential distribution as in the case of irreversible attachment,
\begin{equation}
n_c(m)=\frac{n_v^m}{(\alpha\sqrt{\beta} n_i+\alpha^\prime\kappa^1_d/\kappa_i^1)^{m-1}}.
\label{ncdist-rev-eq}
\end{equation}
Figure~\ref{clusterrev-fig} presents a numerical investigation of
reversible vacancy cluster growth using kMC. Although detachment rates
may be cluster size dependent in general, we only introduce one
size-independent rate $\gamma_{det}/a^2=0.01D_{v}/a^2$ for vacancy
detachment for simplicity. This model would be most relevant for
faceted cluster shapes with one dominant detachment rate from the
faces. The other parameters are that of Fig.~\ref{cluster-fig}. We see
in Fig.~\ref{clusterrev-fig}(a) that the cluster density behaves in a
qualitatively similar manner as in the irreversible case, but $n_c$ is
reduced and the final steady state is reached at earlier times. As in
the irreversible case, the cluster size distribution $n_c(m)$ shown in
Fig.~\ref{clusterrev-fig}(b) is well described by
Eq.~(\ref{ncdist-rev-eq}). The present discussion is of course only
relevant when void coarsening can be neglected.
\begin{figure}[t]
\includegraphics[width=8cm]{fig5finsm}
\caption{\label{intclustimm-fig} Interstitial densities $n_i$ (
$\blacksquare$), free vacancy density $n_v$ ($\lozenge$), total
vacancy density $n_{vtot}$ ($\blacktriangle$), vacany cluster density
$n_{c}$ ($\star$) and density of immobile interstitial cluster
$(\square)$ as a function of time for $\Gamma=10^6$. }
\end{figure}
\section{Interstitial interactions}
\label{intclust-sec}
One of the fascinating aspects of point defect dynamics in bcc metals
is the high mobility of interstitial clusters \cite{osetsky01}. The
interaction between single interstitial atoms is even stronger than
that between vacancies, and the interstitial clusters, such as
dislocation loops \cite{soneda98}, are stable at all relevant
temperatures. Unlike the immobile vacancy cluster, however, the
interstitial cluster migrate easily for clusters with up to 50 or 100
interstitial atoms \cite{soneda01,soneda03}. We now modify the
preceding analysis to account for this behavior.
In this analysis, we return to the irreversible aggregation limit,
i.e., interstitials never separate after encounter. This implies a
reduction of the density of mobile interstitials due to nucleation of
interstitial clusters, i.e., interstitials act as sinks for other
interstitials. However, this creates a population of larger clusters
with a larger crosssection and eventually larger capture numbers. The
reaction kinetics will be affected by the competition of these two
effects.
The rate equations (\ref{rate-irr-eq}) can now be expanded to include
terms representing the above processes, but become even more complex.
In our kMC model, we begin by studying the effect of interstitial
cluster formation by considering completely immobile interstitial
clusters in complete analogy to the vacancy cluster. This situation is
rarely realistic, but provides an upper bound on the magnitude of the
effects. Figure~\ref{intclustimm-fig} shows the evolution of the free
interstitial and vacancy densities as well as the interstitial and
vacancy cluster densities for the same parameters as before. As
expected, the immoblization of interstitials increases the total
number of vacancies in the system. In the final steady state,
$n_{vtot}$ is about three times as large as in the situation in
Fig.~\ref{cluster-fig}. The interstitial clusters (open squares)
nucleate earlier than the vacancy cluster, but their density later
drops below that of the vacancy clusters, since the single
interstitial density is much lower than the single vacancy density.
The situation in Fig.~\ref{intclustimm-fig} can be favorably
contrasted with that of completely mobile interstitial clusters, i.e.,
the cluster diffuse with the same rates as the single interstitials
regardless of their size. This case is again not fully realistic,
since interstitial clusters seldom rotate into other $\langle 111
\rangle$ directions once they contain several interstitials (i.e.,
they diffusive one-dimensionally \cite{marian01,soneda01}). For small
clusters, however, the completely mobile interstitial cluster case
represents a good approximation. Figure~\ref{intclustmob-fig} shows
the corresponding results for the various desities introduced
above. As in the case of immobile interstitial clusters, the mobile
interstitial cluster density first rises due to nucleation
events. However, the highly mobile interstitial clusters also collide
with the sinks, and the cluster density decreases rapidly. Since the
free interstitial density also declines, nucleation events become
rare. Once steady state has been achieved, the interstitial density
has become so low that the nucleation of new interstitial clusters due
to diffusion is almost completely absent. Consequently, the vacancy
densities and the total swelling rates are the same as for the case of
non-interacting interstitials.
\begin{figure}[t]
\includegraphics[width=8cm]{fig6finsm}
\caption{\label{intclustmob-fig}Interstitial, vacancy, interstitial
cluster and vacancy cluster densities for $\Gamma=10^5$ and
$\Gamma=10^7$ (symbols as in Fig.~\ref{intclustimm-fig}). Interstitial
clusters $(\square)$ diffuse and rotate with the same rate as single
interstitials. The interstitial cluster density first rises due to
nucleation events, but then rapidly drops as mobile cluster collide
with sinks. }
\end{figure}
\section{Application to vanadium and iron}
\label{real-sec}
In this section, we apply our model to the V and $\alpha-$Fe
cases. Vanadium is particularly interesting, because here the effect
of mixed 1D/3D diffusion is most pronounced. First principles
calculations and classical MD simulations of V have yielded estimates
of $\Delta E_i=0.018$eV\cite{zepeda04}, $\Delta E_r=0.44$eV
\cite{zepeda04} and $\Delta E_v\approx 0.5$eV \cite{han1}. For Fe,
the different ground state of the self-interstitial ($\langle
110\rangle$ instead of $\langle 111\rangle$) leads to a higher
effective activation barrier for 1D migration, $\Delta E_i$, which
ranges between 0.12 eV \cite{marian01} and 0.17 eV
\cite{soneda01}. The rotation barrier was estimated as $\Delta
E_r=0.16$ eV \cite{soneda01}, and the vacancy migration energy is
assumed to be of the same order \cite{soneda03} as in V (the
prefactors for all processes tend to vary by less than an order of
magnitude). One therefore expects that the self-interstitial
trajectories in $\alpha-$Fe will be much more isotropic than in V.
\begin{table}[b]
\begin{center}
\caption{\label{dimless-table}Dimensionless parameters
$\alpha=D_i/D_v$, $\beta=\gamma_r/D_i$,
$\Gamma=(\sqrt{\beta}D_i+D_v)/\sigma F$, and
$\alpha^\prime=\alpha\sqrt{\beta}$ for V and Fe at 300 K and 600 K.}
\begin{tabular}{|c|c|c|c|c|}
\hline
& V-300K & V-600K & Fe-300K & Fe-600K\\
\hline
$\alpha$ & $10^{8}$ & $10^4$ & $10^{6}$ & $10^{3}$ \\
$\beta$ & $10^{-8}$ & $10^{-4}$ & $10^{0}$ & $10^{0}$ \\
$\Gamma$ & $10^{11}$ & $10^{13}$ & $10^{11}$ & $10^{13}$ \\
$\alpha^\prime$ & $10^4$ & $10^2$ & $10^{6}$ & $10^{3}$ \\
\hline
\end{tabular}
\end{center}
\end{table}
The two metals are best compared in terms of the relevant
dimensionless parameter $\alpha$, $\beta$ and $\Gamma$. Table
\ref{dimless-table} summarizes the values for these quantities using
the above energy barriers and the production rate for ion irradiation
$\sigma F=10^{-3}s^{-1}$ for two representative temperatures $T=300$K
and $T=600$K. We first note that $\Gamma$ is typically larger than
$10^{10}$ and would in fact reach $10^{20}$ when typical production
rates for neutron irradiation are used. A kMC simulation with such
large values of $\Gamma$ would require very long simulation times,
since diffusion and recombination occur much more often than the
introduction of new defects. It also requires large system sizes,
because the defect densities become very small. In addition, the very
small value of $\beta$ implies very long 1D segments of the
interstitial trajectories between rotations. The period of the kMC
simulation box should be several times larger than the typical length
of those segments in order to properly reproduce the continuum theory
values of the encounter rates. If the box period is much smaller than
the 1D segment, the trajectory will wrap around the box many times,
but is not necessary space-filling.
The last of these issues can be addressed by using a result from
Section \ref{simple-sec}, where we showed that the mixed 1D/3D
encounter rates scale like the ideal 3D rates that are reduced by a
factor $\sqrt{\beta}$. We can therefore replace the explicit 1D/3D
trajectories of the interstitials and interstitial cluster with ideal
3D random walks, but reduce the hopping rate by a factor of
$\sqrt{\beta}$. This procedure leaves the reaction kinetics invariant
(up to small corrections from the capture numbers) and implies that
the effective ratio of time scales between interstitial and vacancy
migration is given by
$\alpha^\prime=D_i\sqrt{\beta}/D_v=\sqrt{D_i\gamma_r/a^2}$. Interestingly,
this ratio is very similar for both V and Fe at 300K and 600K (see
Table \ref{dimless-table}), even though the values of $\alpha$ and
$\beta$ are very different. At a given temperature, the
self-interstitial trajectories in $\alpha$-Fe are much more isotropic
than in V, but the encounter rates with vacancies and sinks only
depend on the product of diffusivity $D_i$ and rotation rate
$\gamma_r$. Within the present model, one therefore expects that the
point defect reaction kinetics in these two metals is very similar.
\begin{figure}[t]
\includegraphics[width=7.7cm]{fig7finsm}
\caption{\label{van-fig}Interstitial, vacancy, interstitial cluster
and vacancy cluster densities (symbols as in
Fig.~\ref{intclustimm-fig}) with parameters $\alpha^\prime=10^2$,
$n_s=10^{-5}$, for (a) $\Gamma=10^9$ (b) $\Gamma=10^{11}$ and (c)
$\Gamma=10^{13}$ representative for V or $\alpha$-Fe at 600K. Thin
solid lines show the result of numerically integrating
Eqs.~(\ref{rate-irr-eq}) again for $\kappa_m=25$ and $m_{max}=20$. The
solid lines have slope 1 and 1/2, respectively.}
\end{figure}
In Fig.~\ref{van-fig}, we use a ratio of time scales representative
for V/$\alpha$-Fe at $T=600$K and show results for $\Gamma=10^9$,
$\Gamma=10^{11}$ and $\Gamma=10^{13}$. Here, we have multiplied the
time axis with the production rate, which is a more convential
presentation of the data in experimental studies. Rescaled by the
production rate, all curves initially coincide and start out with
near-constant slopes. For the present parameters, Regime II is
practically absent, and the vacancy density crosses over immediately
from Regime I (where it rises linearly with dose) into Regime III. The
crossover dose is roughly constant and depends on the sink
density. Note that the growth of $n_{vtot}$ in Regime III is well
described by a $(\sigma Ft)^{1/2}$ power law over several decades
before the final steady-state Regime IV is reached. Since the onset of
Regime IV is independent of time (see Section \ref{simple-sec}),
larger values of $\Gamma$ push the beginning of steady state to
smaller doses. Here, steady state is reached at about 1 dpa for
$\Gamma=10^9$ and earlier for $\Gamma=10^{11}$ and
$\Gamma=10^{13}$. There is some initial nucleation of interstitial
clusters at $\Gamma=10^9$, but as discussed before, all of these
rapidly disappear due to fast collisions with sinks. For higher values
of $\Gamma$, the interstitial cluster density becomes negligibly
small.
In the same figure, we also show the predictions of a numerical
integration of the full set of rate equations (\ref{rate-irr-eq})
using the point cluster model (size independent capture numbers). As
in Fig.~\ref{cluster-fig}, the agreement with the ``exact'' kMC
simulation is satisfactory. Since the computational effort for direct
kMC simulations for $\Gamma>10^{10}$ becomes tremendous, the rate
equation approach, when properly parametrized, is clearly preferable.
\label{van-sec}
\section{Conclusions}
\label{conc-sec}
We have studied the early-time evolution of point defect populations
with specific reference to migration mechanisms in bcc metals under
constant irradiation conditions. Only very simplified models that
incorporate the most important processes were employed in order to
identify the rate-limiting events. These models were solved using
scaling arguments, direct numerical integration of kinetic rate
equations and full kMC simulations.
In the simplest case, which only considers homogeneous defect
production, recombination and sink absorption, but no interactions
between defects of the same type, the kMC model and the corresponding
rate equations were shown to be in near perfect agreement. We employed
simple random walk arguments to derive a mixed encounter rate that
describes one-dimensional diffusion with occasional rotations. This
encounter rate scales linearly with target density as the isotropic 3D
encounter rate, but is reduced by the square root of the ratio of
rotation rate and hopping rate. As in ref.~\onlinecite{sizmann68},
four distinct scaling regimes for the point defect density with time
where identified. First, the point defect densities increase linearly
in time due to production (Regime I), then saturate as defect
recombination sets in (Regime II). Sink absorption then begins to
reduce the interstitial density and the vacancy density grows in time
with a characteristic $\sim t^{1/2}$ behavior (regime III). A final
steady state (Regime IV) is reached when all loss processes are taken
into account. The full sequence of Regimes I - IV is most visible when
the production rate is not much smaller than the interstitial hopping
rates. Regimes II and III shrink with decreasing production rate, and
in the limit of very small production rates, Regime I crosses over
directly into Regime IV. These results were based on parameters
typical for bcc metals, but the continuum level reaction kinetics
equally applies to fcc metals or other crystal structures provided
they exhibit similar diffusion mechanisms.
The introduction of vacancy reactions to form immobile vacancy
clusters does not change the general sequence of the scaling regimes,
but has a profound effect on the population dynamics. In steady state,
the density of free vacancies and interstitials is reduced relative to
the non-interacting case, but the total number of vacancies $n_{vtot}$
is enhanced. In Regime III, $n_{vtot}\sim t^{1/2}$, and the steady
state Regime IV is reached at later times. Reasonable agreement
between rate equations and kMC could still be achieved in a point
cluster approximation, where the capture numbers are
size-independent. Of particular interest is the size distribution of
vacancy clusters, which grow and shrink under the diffusive arrival of
interstitials and vacancies. For irreversible vacancy aggregation, we
derived a new expression for the cluster size distribution,
$n_c(m)=\kappa_1 n_v^m/\kappa_m(\alpha\sqrt{\beta} n_i)^{m-1}$. The
general form of this distribution remains when also allowing vacancy
detachment from the cluster.
Immobile interstitial clusters were shown to further increase the
vacancy population. However, interstitial cluster that are as mobile
as single interstitials decay rapidly in relevant parameter
regimes. We therefore expect that nucleation of interstitial clusters
through diffusion plays a negligible role in the microstructural
damage evolution of pure bcc metals. It can become important, however,
if trapping of self-interstitials near impurities occurs.
Because of their importance in applications and the availablitity of
detailed molecular dynamics studies, we applied the model for
parameters suitable for V and $\alpha$-Fe. Within our model, both
metals exhibit similar defect kinetics and therefore similar swelling
rates at a given temperature. For production rates and diffusivities
in typical experimental situations, our calculations revealed a
sequence of growth regimes for the total vacancy density $n_{vtot}$
$\sim t^1$, $\sim t^{1/2}$ and $\sim t^0$. The volume swelling rate
follows the same scaling sequence in the sub-1 dpa
regime. Interestingly, growth of the vacany cluster density as $\sim
t^{1/2}$ over several decades in time has been observed in the much
more detailed kMC simulation of Ref.~\onlinecite{soneda03}, but the
origin of this growth law had not been previously understood. All our
simulations reach the steady state regime at damage levels of less
than 1 dpa. In the experimentally relevant regime of damage levels
between 1 and 100 dpa, it will therefore be more efficient to use
steady state rate equations rather than explicit kMC to predict the
long time point defect distribution evolution. Nonetheless, it is
important to understand the onset of damage evolution and the
conditions of applicability of the steady state theory.
The fact that experimental swelling rates of V and $\alpha$-Fe are
very different is of course an indication that our present model does
not yet include all relevant phenomena of radiation damage. One
obvious refinement of the models developed here would be a more
detailed parametrization of the detachment rates (binding energies) of
interstitial and vacancy cluster. However, the intent of the present
study has been to focus on general trends rather than quantitative
comparisons with experiments, and no new physics is expected to appear
from these additional details. For cascade damage conditions
\cite{bullough75}, however, two other processes not included in this
study are known to have a crucial impact on the defect evolution (in
particular void swelling rates). Frenkel pair production is only
assumed to be homogeneous for energies right above the displacement
threshold, while higher energies lead to the formation of mobile
interstitial and immobile vacancy clusters during the cascade
phase. Our present model is therefore most relevant to low particle
energies. Inclusion of intracascade clustering processes will change
results qualitatively and quantitatively, but requires reliable
information about the cluster size distribution during cascades from
MD simulations.
In addition to this ``production bias'', ``absorption bias'' also
usually exists in bcc metals. ``Absorption bias'' leads to a
preferred absorption of self-interstitials at sinks and is known to be
an essential driving force for swelling. The origin of this bias,
which leads to increased capture numbers, are long range elastic
interactions between point defects and sinks such as dislocations
\cite{kamiyama94,bullough70} or grain boundaries \cite{samaras03}.
Extentions of the model to include these effects should provide a
fruitful topic for future work. The combination of kMC and rate theory
can then be used to determine which parameters are most important in
radiation damage, so that atomistic simulation resources can be better
focused.
The authors gratefully acknowledge useful discussions with
L.~Zepeda-Ruiz, B.~Wirth, and N.~Ghoniem as well as the support of the
Office of Fusion Energy Sciences (Grant DE-FG02-01ER54628).
|
{
"timestamp": "2004-11-28T03:56:48",
"yymm": "0411",
"arxiv_id": "cond-mat/0411680",
"language": "en",
"url": "https://arxiv.org/abs/cond-mat/0411680"
}
|
\section*{Introduction}
The universal enveloping algebra of $\mathfrak{sl}_n$ and its
finite-dimensional highest weight representations have been
constructed geometrically in two different ways by Ginzburg
\cite{G91} and Nakajima \cite{N98} (Nakajima's construction works
for more general Kac-Moody algebras). Both constructions use a
convolution product in homology. In Ginzburg's construction, the
varieties involved are flag varieties and their cotangent bundles
while in Nakajima's construction they are varieties attached to the
quiver (oriented graph) whose underlying graph is the Dynkin graph
of $\mathfrak{sl}_n$. Both realizations produce a natural basis of
the representations given by the fundamental classes of the
irreducible components of the varieties involved. In \cite{N94}
Nakajima conjectured a specific relationship between the two
varieties and this conjecture was later proved by Maffei \cite{M00}.
In the current paper we review this relationship and use it to
examine the representation theoretic constructions in the two
settings and show that while the quotients of the universal
enveloping algebra obtained are different, there is a natural
homomorphism between the two and the natural bases in
representations produced by the two constructions are in fact the
same. Nakajima's construction using the convolution product was in
fact motivated by Ginzburg's construction and thus it is not
surprising that we find that the quiver variety construction is in
some sense a generalization of the flag variety construction to
arbitrary (simply-laced) type. It was certainly expected by experts
that the two bases obtained are the same. However, the author is
not aware of a proof in the literature of the coincidence of the two
bases and the precise relationship between the different
constructions of the universal enveloping algebra (which are, in
fact, slightly different in the two cases).
Finally, we use the relation between the two constructions to define
the structure of a crystal graph on the irreducible components of
the Spaltenstein varieties appearing in Ginzburg's construction by
analogy with the already existing theory for quiver varieties
developed by Kashiwara and Saito. In doing this, we recover the
crystal structure on irreducible components of Spaltenstein
varieties introduced by Malkin in \cite{Mal02}. We now explain the
contents of the paper in some detail.
Fix a positive integer $d$ and let
\[
\mathcal{F} = \{0 = F_0 \subset F_1 \subset \dots \subset F_n = \mathbb{C}^d\}
\]
be the set of all $n$-step flags in $\mathbb{C}^d$. Let $N = \{x \in
\End(\mathbb{C}^d)\ |\ x^n = 0 \}$. The cotangent bundle to $\mathcal{F}$ is
isomorphic to
\[
M = \{(x,F) \in N \times \mathcal{F}\ |\ x(F_i) \subset F_{i-1}\}.
\]
We have the natural projection $\mu : M \to N$ and for $x \in N$
we define
\begin{gather*}
Z = M \times_N M = \{(m_1,m_2) \in M \times M\ |\ \mu(m_1) =
\mu(m_2)\}, \\
\mathcal{F}_x = \mu^{-1}(x).
\end{gather*}
Using the convolution product (see Section~\ref{sec:convolution}),
we give the top-dimensional Borel-Moore homology $H_\textrm{top}(Z)$ the
structure of an algebra and $H_\textrm{top}(\mathcal{F}_x)$ the structure of a module
over this algebra. Let $I_d$ be the annihilator of
$(\mathbb{C}^n)^{\otimes d}$, a two-sided ideal of finite codimension in
the enveloping algebra $U(\mathfrak{sl}_n)$. Here $\mathbb{C}^n$ is the
natural $\mathfrak{sl}_n$-module. Then in \cite{CG,G91} it is
shown that $H_\textrm{top}(Z) \cong U(\mathfrak{sl}_n)/I_d$ and that under
this isomorphism, $H_\textrm{top}(\mathcal{F}_x)$ is the irreducible highest weight
$\mathfrak{sl}_n$-module of highest weight $w_1 \omega_1 + \dots +
w_{n-1} \omega_{n-1}$ where the $\omega_i$ are the fundamental
weights of $\mathfrak{sl}_n$ and $w_i$ is the number of $(i \times
i)$-Jordan blocks in the Jordan normal form of $x$.
Now, in \cite{N98}, Nakajima constructs the same representations
in a similar way using a convolution product in the homology of
quiver varieties. In \cite{M00}, Maffei showed that the varieties
of Nakajima's construction are isomorphic to the following. Let
$S_x$ be a transversal slice in $N$ to the $GL(\mathbb{C}^d)$-orbit
through $x$ (see Section~\ref{sec:isom}). Then let
\begin{gather*}
M' = \mu^{-1}(S_x), \\
Z' = M' \times_{S_x} M'.
\end{gather*}
Then, translated via the isomorphism of \cite{M00}, a result of
\cite{N98} is that, under the convolution product we have
$H_\textrm{top}(Z') \cong U(\mathfrak{sl}_n)/J$ and $H_\textrm{top}(M_x)$ is the same
irreducible highest weight module as in Ginzburg's construction
(see Theorems~\ref{thm:nak-hom} and~\ref{thm:nak-quotient}). Here
$J$ is a certain ideal of finite codimension in
$U(\mathfrak{sl}_n)$ that is different from $I_d$ in general. Thus
the two constructions yield different quotients of the universal
enveloping algebra but the same representation.
Since $Z' \subset Z$ and $M' \subset M$, we have a natural
restriction with support morphism $H_\textrm{top}(Z) \to H_\textrm{top}(Z')$. The
main result of this paper (see Theorem~\ref{thm:quiver-flag}) is
that the following diagram is commutative
\[
\begin{CD}
H_\textrm{top}(Z) \otimes H_\textrm{top}(\mathcal{F}_x) @>>> H_\textrm{top}(Z') \otimes H_\textrm{top}(\mathcal{F}_x)
@>\cong>> \oplus_{\mathbf{v}^1,\mathbf{v}^2} H_\textrm{top}(Z(\mathbf{v}^1,\mathbf{v}_2;\mathbf{w})) \otimes
\oplus_\mathbf{v} H_\textrm{top}(\mathfrak{L}(\mathbf{v},\mathbf{w}))
\\
@VVV @VVV @VVV \\
H_\textrm{top}(\mathcal{F}_x) @>>> H_\textrm{top}(\mathcal{F}_x) @>\cong>> \oplus_\mathbf{v} H_\textrm{top}(\mathfrak{L}(\mathbf{v},\mathbf{w}))
\end{CD}
\]
Here the rightmost term in each row involves the Nakajima quiver
varieties (see Section~\ref{sec:nak} for definitions). We are
also able to conclude that the natural bases of representations
produced by both Ginzburg's and Nakajima's constructions coincide.
We thus obtain a precise relation between the two approaches.
Recently, a relation has been established between a construction
closely related to that of Ginzburg and another geometric approach
of Mirkovi\' c-Vilonen in terms of the affine Grassmannian
\cite{BGV04}. It would be interesting to examine the connection
between the quiver variety and Mirkovi\' c-Vilonen realizations of
finite-dimensional representations of Lie algebras.
The organization of the paper is as follows. In
Sections~\ref{sec:prelim} and \ref{sec:convolution} we recall the
definition of $\mathfrak{sl}_n$ and the convolution product in
Borel-Moore homology. In Sections~\ref{sec:ginz-def} and
\ref{sec:nak} we review Ginzburg's and Nakajima's constructions of
$U(\mathfrak{sl}_n)$ and its representations. Then in
Section~\ref{sec:isom} we describe the precise relationship
between the two constructions. Finally, in
Section~\ref{sec:crystal} we define the structure of a crystal on
the irreducible components of $\mathcal{F}_x$.
The author would like to thank O. Schiffmann for many useful
discussions and suggestions and K. McGerty for very helpful
comments on the properties of the convolution product.
\section{Preliminaries}
\label{sec:prelim}
Let $\ensuremath{\mathfrak{g}}=\mathfrak{sl}_n$ be the Lie algebra of type $A_{n-1}$.
Then \ensuremath{\mathfrak{g}}\ is the space of all traceless $n \times n$ matrices.
Let $\{e_k,f_k\}_{k=1}^{n-1}$ be the set of Chevalley generators.
The Cartan subalgebra $\mathfrak{h}$ is spanned by the matrices
\[
h_k = e_{k,k} - e_{k+1,k+1},\ 1 \le k \le n-1,
\]
where $e_{k,l}$ is the matrix with a one in entry $(k,l)$ and
zeroes everywhere else. Thus the dual space $\mathfrak{h}^*$ is
spanned by the simple roots
\[
\alpha_k = \epsilon_k - \epsilon_{k+1},\ 1\le k \le n-1,
\]
where $\epsilon_k(e_{l,l}) = \delta_{kl}$ and the fundamental
weights are given by
\[
\omega_k = \epsilon_1 + \dots + \epsilon_k,\ 1 \le k \le n-1.
\]
Consider a dominant weight $\mathbf{w} = w_1 \omega_1 + \dots + w_{n-1}
\omega_{n-1}$. Then
\[
\mathbf{w} = \lambda_1 \epsilon_1 + \dots + \lambda_{n-1} \epsilon_{n-1}
\]
where $\lambda_k = w_k + \dots + w_{n-1}$ and so $\mathbf{w}$ corresponds to
a partition $\lambda(\mathbf{w}) = (\lambda_1 \ge \dots \ge \lambda_{n-1})$.
We say that a highest weight $\mathbf{w}$ is a partition of $d$ if
$|\lambda(\mathbf{w})| = \lambda_1 + \dots + \lambda_{n-1} = d$ or,
equivalently, if $\sum_{k=1}^n k w_k = d$.
\section{Convolution algebra in homology}
\label{sec:convolution}
In this section we give a brief overview of the convolution algebra
in homology. The reader interested in further details should consult
\cite{CG}.
In this paper $H_*(Z)$ will denote the Borel-Moore homology with
$\mathbb{C}$-coefficients of a locally-compact space $Z$. Thus, by
definition, if $Z$ is a closed subset of a smooth, oriented
manifold $M$, then
\[
H_k(Z) = H^{\dim_{\mathbb{R}} M - k}(M,M\backslash Z).
\]
If $Z$ and $Z'$ are closed subsets of a smooth variety $M$, we
have a $\cup$-product map
\[
H^k(M, M\backslash Z) \times H^l(M, M\backslash Z') \to H^{k+l}(M,
M \backslash (Z \cap Z')).
\]
Thus we construct the intersection pairing in Borel-Moore homology
\[
\cap : H_k(Z) \times H_l(Z') \to H_{k+l-d}(Z \cap Z'),\quad d=
\dim_{\mathbb{R}} M.
\]
Let $M_1$, $M_2$ and $M_3$ be smooth, oriented manifolds and
$p_{kl} : M_1 \times M_2 \times M_3 \to M_k \times M_l$ be the
obvious projections. Let $Z \subset M_1 \times M_2$ and $Z'
\subset M_2 \times M_3$ be closed subvarieties and assume that the
map
\[
p_{13} : p_{12}^{-1}(Z) \cap p_{23}^{-1}(Z') \to M_1 \times M_3
\]
is proper and denote its image by $Z \circ Z'$. The operation of
convolution
\[
\star: H_k(Z) \times H_l(Z') \to H_{k+l-d}(Z \circ Z'),\quad
d=\dim_{\mathbb{R}} M_2,
\]
is defined by
\[
c \star c' = (p_{13})_* (p_{12}^*c \cap p_{23}^*c'),
\]
where $p_{12}^*c$ means $c \boxtimes [M_3]$, etc.
Now, let $M$ be a smooth manifold and $\mu: M \to N$ be a proper
morphism. Let
\[
Z = M \times_N M = \{(m_1,m_2) \in M \times M\ |\ \mu(m_1) =
\mu(m_2)\} \subset M \times M.
\]
Then $Z \circ Z = Z$ and so convolution makes $H_*(Z)$ a
finite-dimensional associative $\mathbb{C}$-algebra with unit.
For $x \in N$, let $M_x = \mu^{-1}(x)$. We also identify $M_x$
with the variety $M_x \times \text{pt}$. Then setting $M_1 = M_2
= M$ and $M_3 = \text{pt}$, we have $Z \circ M_x = M_x$ and
convolution makes $H_*(M_x)$ a $H_*(Z)$-module.
\section{Ginzburg's construction}
\label{sec:ginz-def}
We recall here Ginzburg's construction of the enveloping algebra
$U(\mathfrak{sl}_n)$ and its irreducible highest weight
representations. Proofs omitted here can be found in \cite{G91} or
\cite{CG}.
Fix an integer $d \ge 1$. Let
\[
\mathcal{F} = \{ 0 = F_0 \subset F_1 \subset \dots \subset F_n = \mathbb{C}^d\}
\]
be the set of all $n$-step partial flags in $\mathbb{C}^d$. The space $\mathcal{F}$
is a disjoint union of smooth compact manifolds with connected
components parameterized by compositions
\[
{\mathbf{d}} = (d_1 + d_2 + \dots + d_n = d), \quad d_i \in \mathbb{Z}_{\ge 0}.
\]
The connected component of $\mathcal{F}$ corresponding to ${\mathbf{d}}$ is
\[
\mathcal{F}_{\mathbf{d}} = \{F = (0 = F_0 \subset \dots \subset F_n = \mathbb{C}^d)\ |\ \dim
F_i/F_{i-1} = d_i\},
\]
and
\[
\dim_\mathbb{C} \mathcal{F}_{\mathbf{d}} = \frac{d!}{d_1! d_2! \cdots d_n!}.
\]
Let
\[
N = \{x \in \End(\mathbb{C}^d)\ |\ x^n=0\}.
\]
Then
\[
T^*\mathcal{F} \cong M = \{(x,F) \in N \times \mathcal{F}\ |\ x(F_i) \subset
F_{i-1},\ 1 \le i \le n\}.
\]
The above decomposition of $\mathcal{F}$ yields a decomposition of $M$ given
by $M = \sqcup_{\mathbf{d}} M_{\mathbf{d}}$ where $M_{\mathbf{d}} = T^*\mathcal{F}_{\mathbf{d}}$ for an $n$-step
composition ${\mathbf{d}}$ of $d$.
The natural projections give rise to the diagram
\[
N \stackrel{\mu}{\longleftarrow} M
\stackrel{\pi}{\longrightarrow} \mathcal{F}.
\]
We have a natural action of $GL_d(\mathbb{C})$ on $\mathcal{F}$, $N$ (by conjugation)
and $M$ and the projections commute with this action.
For $x \in N$, let $\mathcal{F}_x = \mu^{-1}(x)$. It has connected
components $\mathcal{F}_{{\mathbf{d}},x}$ given by $\mathcal{F}_{{\mathbf{d}},x} = \mathcal{F}_{\mathbf{d}} \cap \mathcal{F}_x$.
Define
\[
Z = M \times_N M = \{(m_1,m_2) \in M \times M\ |\ \mu(m_1) =
\mu(m_2) \} \subset M \times M.
\]
We use the convention that under the isomorphism
\[
T^*\mathcal{F} \times T^*\mathcal{F} \cong T^*(\mathcal{F} \times \mathcal{F}),
\]
the standard symplectic form on the right hand side corresponds to
$\omega_1 - \omega_2$ where $\omega_1$ and $\omega_2$ are the
symplectic forms on the first and second factors of the left hand
side respectively.
\begin{prop}
The variety $Z$ is the union of the conormal bundles to the
$GL_d(\mathbb{C})$-orbits in $\mathcal{F} \times \mathcal{F}$. The closures of these
conormal bundles are precisely the irreducible components of $Z$.
\end{prop}
\begin{prop}
We have $Z \circ Z = Z$. Thus $H_*(Z)$ is an associative algebra
with unit and $H_*(\mathcal{F}_x)$ is an $H_*(Z)$-module for any $x \in
N$.
\end{prop}
\begin{prop}
All irreducible components of $Z$ contained in $M_{{\mathbf{d}}^1} \times
M_{{\mathbf{d}}^2}$ are half dimensional. That is, they have complex
dimension
\begin{gather*}
\frac{1}{2} \dim_\mathbb{C}(M_{{\mathbf{d}}^1} \times M_{{\mathbf{d}}^2}) = \frac{1}{2} \left(
2\frac{d^1!}{d^1_1! d^1_2! \cdots d^1_n!} + 2\frac{d^2!}{d^2_1!
d^2_2! \cdots d^2_n!} \right) \\
= \frac{d^1!}{d^1_1! d^1_2! \cdots d^1_n!} + \frac{d^2!}{d^2_1!
d^2_2! \cdots d^2_n!}.
\end{gather*}
\end{prop}
Let $H_\textrm{top}(Z)$ be the vector subspace of $H_*(Z)$ spanned by the
fundamental classes of the irreducible components of $Z$ and let
$H_\textrm{top}(\mathcal{F}_x)$ be the vector subspace of $H_*(\mathcal{F}_x)$ spanned by the
fundamental classes of the irreducible components of $\mathcal{F}_x$.
\begin{prop}
The homology group $H_\textrm{top}(Z)$ is a subalgebra of $H_*(Z)$ and
$H_\textrm{top}(\mathcal{F}_x)$ is an $H_\textrm{top}(Z)$-stable subspace of $H_*(\mathcal{F}_x)$.
\end{prop}
Now, for a composition ${\mathbf{d}}$ we have the diagonal subvariety $\Delta
\subset \mathcal{F}_{\mathbf{d}} \times \mathcal{F}_{\mathbf{d}}$ which is a $GL_d(\mathbb{C})$-orbit. We define
\[
H_k = \sum_{\mathbf{d}} (d_k - d_{k+1}) [T^*_\Delta (\mathcal{F}_{\mathbf{d}} \times \mathcal{F}_{\mathbf{d}})],
\]
where $T_O^*(\mathcal{F}_{\mathbf{d}} \times \mathcal{F}_{\mathbf{d}})$ denotes the conormal bundle
to a $GL_d(\mathbb{C})$-orbit $O \subset \mathcal{F}_{\mathbf{d}} \times \mathcal{F}_{\mathbf{d}}$.
Note that under the sign convention for the symplectic form
mentioned above, the conormal bundle $T^*_\Delta (\mathcal{F}_{\mathbf{d}} \times
\mathcal{F}_{\mathbf{d}})$ is the diagonal in $T^*\mathcal{F}_{\mathbf{d}} \times T^*\mathcal{F}_{\mathbf{d}}$.
Now, for a composition ${\mathbf{d}} = (d_1 + \dots + d_n)$ and $1 \le k \le
n-1$, let
\begin{align*}
{\mathbf{d}}_k^+ &= d_1 + \dots + d_{k-1} + (d_k+1) + (d_{k+1}-1) + d_{k+2}
+ \dots + d_n, \\
{\mathbf{d}}_k^- &= d_1 + \dots + d_{k-1} + (d_k-1) + (d_{k+1}+1) + d_{k+2}
+ \dots + d_n,
\end{align*}
provided that these are compositions (that is, all terms are $\ge
0$). Otherwise, we define ${\mathbf{d}}_k^{\pm} = \nabla$, the ghost
composition.
If $1 \le k \le n-1$ and ${\mathbf{d}} = (d_1 + \dots + d_n)$ is a composition
such that ${\mathbf{d}}_k^+ \ne \nabla$, resp. ${\mathbf{d}}_k^- \ne \nabla$, we define
\begin{align*}
Y_{{\mathbf{d}}_k^+,{\mathbf{d}}} &= \{ (F',F) \in \mathcal{F}_{{\mathbf{d}}_k^+} \times \mathcal{F}_{\mathbf{d}} \ |\ F_l =
F'_l \ \forall \ l\ne k,\, F_k \subset F_k', \dim (F_k'/F_k) = 1 \},\\
Y_{{\mathbf{d}}_k^-,{\mathbf{d}}} &= \{ (F',F) \in \mathcal{F}_{{\mathbf{d}}_k^-} \times \mathcal{F}_{\mathbf{d}} \ |\ F_l =
F'_l \ \forall \ l\ne k,\, F_k' \subset F_k, \dim (F_k/F_k') = 1
\}.
\end{align*}
Note that each $Y_{{\mathbf{d}}_k^{\pm},{\mathbf{d}}}$ is a $GL_d(\mathbb{C})$-orbit in
$\mathcal{F}_{{\mathbf{d}}_k^{\pm}} \times \mathcal{F}_{\mathbf{d}}$ of minimal dimension and thus is a
smooth closed subvariety. Let
\begin{align}
E_k &= \sum_{\mathbf{d}} [T^*_{Y_{{\mathbf{d}}_k^+,{\mathbf{d}}}}(\mathcal{F}_{{\mathbf{d}}_k^+} \times \mathcal{F}_{\mathbf{d}})],\\
\label{def:ginz-Fk}
F_k &= \sum_{\mathbf{d}}
(-1)^{s_k({\mathbf{d}}_k^+,{\mathbf{d}})}[T^*_{Y_{{\mathbf{d}}_k^-,{\mathbf{d}}}}(\mathcal{F}_{{\mathbf{d}}_k^-} \times
\mathcal{F}_{\mathbf{d}})],
\end{align}
where $s_k({\mathbf{d}}_k^+,{\mathbf{d}}) = \frac{1}{2}\left( \dim_\mathbb{C} M_{{\mathbf{d}}_k^+} -
\dim_\mathbb{C} M_{\mathbf{d}} \right)$.
\begin{theo}[\cite{G91}]
\label{thm:ginz}
The map
\[
e_k \mapsto E_k,\ f_k \mapsto F_k,\ h_k \mapsto H_k,
\]
extends to a surjective algebra homomorphism $U(\mathfrak{sl}_n)
\twoheadrightarrow H_\textrm{top}(Z)$. Under this homomorphism,
$H_\textrm{top}(\mathcal{F}_x)$ is the irreducible highest weight module of highest
weight $w_1 \omega_1 + \dots + w_{n-1} \omega_{n-1}$ where
$\omega_i$ are the fundamental weights and $w_i$ is the number of
($i \times i$)-Jordan blocks in the Jordan normal form of $x$.
\end{theo}
\begin{rem}
Note that the sign appearing in \eqref{def:ginz-Fk} does not
appear in \cite{CG,G91}. This arises from the fact that
Theorem~2.7.26 (iii) in \cite{CG} should read $[Z_{12}] \star
[Z_{23}] = (-1)^{\dim F} \chi(F) \cdot [Z_{13}]$ (see \cite[Lemma
8.5]{N98}).
\end{rem}
Let $I_d$ be the annihilator of $(\mathbb{C}^n)^{\otimes d}$, a two-sided
ideal of finite codimension in the enveloping algebra
$U(\mathfrak{sl}_n)$. Here $\mathbb{C}^n$ is the natural
$\mathfrak{sl}_n$-module.
\begin{theo}[{\cite[Proposition~4.2.5]{CG}}]
\label{thm:ginz-quotient}
The homomorphism of
Theorem~\ref{thm:ginz} yields an algebra isomorphism
\[
U(\mathfrak{sl}_n)/I_d \cong H_\textrm{top}(Z).
\]
\end{theo}
It is known that the simple $\mathfrak{sl}_n$ modules that occur
with non-zero multiplicity in the decomposition of
$(\mathbb{C}^n)^{\otimes d}$ are precisely those modules whose highest
weight is a partition of $d$.
\section{Nakajima's construction}
\label{sec:nak}
In this section, we will review the description of the quiver
varieties presented in \cite{N98}. Further details may be found in
\cite{N94} and \cite{N98}. We only discuss the case corresponding
to the Lie algebra $\mathfrak{sl}_n$. Note that we use a different
stability condition that the one used in \cite{N94} and \cite{N98}
and so our definitions differ slightly from the ones that appear
there. One can translate between the two stability conditions by
taking transposes of the maps appearing in the definitions of the
quiver varieties. See \cite{N96} for a discussion of various choices
of stability condition.
As before, let $\mathfrak{g}=\mathfrak{sl}_n$ be the simple Lie
algebra of type $A_{n-1}$. Let $I=\{1,\dots,n-1\}$ be the set of
vertices of the Dynkin graph of $\ensuremath{\mathfrak{g}}$ with the set of oriented
edges given by
\begin{gather*}
H=\{h_{k,l} \ |\ k,l \in I,\ |k-l|=1\}.
\end{gather*}
For two adjacent vertices $k$ and $l$, $h_{k,l}$ is the oriented
edge from vertex $k$ to vertex $l$. We denote the outgoing and
incoming vertices of $h\in H$ by $\out(h)$ and $\inc(h)$
respectively. Thus $\out(h_{k,l}) = k$ and $\inc(h_{k,l})=l$.
Define the involution $\bar{\ } : H \to H$ as the function that
interchanges $h_{k,l}$ and $h_{l,k}$. Fix the orientation $\Omega
= \{h_{k,k-1}\ |\ 2 \le k \le n-1 \}$. We picture this quiver as
in Figure~\ref{fig:quiver_an}.
\begin{figure}
\centering \epsfig{file=quiver_an.eps,width=0.5\textwidth}
\caption{The quiver of type $A_{n-1}$. \label{fig:quiver_an}}
\end{figure}
Let $V = \bigoplus_{k \in I} V_k$ and $W = \bigoplus_{k \in I}
W_k$ be two finite dimensional complex $I$-graded vector spaces with graded
dimensions
\begin{align*}
\mathbf{v} &= (\dim V_1,\dim V_2,\dots,\dim V_{n-1}),\\
\mathbf{w} &= (\dim W_1,\dim W_2,\dots,\dim W_{n-1}).
\end{align*}
Then we define
\[
\mathbf{M}(\mathbf{v},\mathbf{w}) = \bigoplus_{h \in H}
\Hom(V_{\out(h)},V_{\inc(h)}) \oplus \bigoplus_{k \in I}
\Hom(W_k,V_k) \oplus \bigoplus_{k \in I} \Hom(V_k,W_k).
\]
The above three components of an element of $\mathbf{M}(\mathbf{v},\mathbf{w})$
will be denoted by $B=(B_h)$, $i=(i_k)$ and $j=(j_k)$. We
associate elements in the weight lattice of $\ensuremath{\mathfrak{g}}$ to the dimensions
vectors $\mathbf{v} = (v_1,\dots,v_{n-1})$ and $\mathbf{w}=(w_1,\dots,w_{k-1})$ as
follows.
\[
\alpha_\mathbf{v} = \sum_{k \in I} v_k \alpha_k,\quad \omega_\mathbf{w} = \sum_{k
\in I} w_k \omega_k,
\]
where $\alpha_k$ and $\omega_k$ are the simple roots and
fundamental weights respectively.
Now, let
\[
G_\mathbf{v} = \prod_{k \in I} GL(V_k)
\]
act on $\mathbf{M}(\mathbf{v},\mathbf{w})$ by
\[
g(B,i,j) = (gBg^{-1},gi,jg^{-1}),
\]
where $gBg^{-1} = (B'_h) = (g_{\inc(h)}B_h g_{\out(h)}^{-1})$, $gi
= (i'_k) = (g_k i_k)$ and $jg^{-1} = (j'_k) = (j_k g_k^{-1})$. Let
$\epsilon : H \to \{\pm 1\}$ be given by
\[
\epsilon(h) = \begin{cases}
+1 & \text{if } h \in \Omega \\
-1 & \text{if } h \in {\bar \Omega}
\end{cases}.
\]
Define a map $\mu : \mathbf{M}(\mathbf{v},\mathbf{w}) \to \bigoplus_{k \in I} \End (V_k,
V_k)$ with $k$th component given by
\[
\mu_k(B,i,j) = \sum_{h \in H\, :\, \inc(h)=k} \epsilon(h) B_h
B_{\bar h} + i_k j_k.
\]
Let $A(\mu^{-1}(0))$ be the coordinate ring of the affine
algebraic variety $\mu^{-1}(0)$ and define
\[
\mathfrak{M}_0(\mathbf{v},\mathbf{w}) = \mu^{-1}(0)//G = \Spec A(\mu^{-1}(0))^G.
\]
This is the affine algebro-geometric quotient of $\mu^{-1}(0)$ by
$G$. It is an affine algebraic variety and its geometric points
are closed $G_\mathbf{v}$-orbits.
We say that a collection $S=(S_k)$ of subspaces $S_k \subset V_k$
is $B$-stable if $B_h(S_{\out(h)}) \subset S_{\inc(h)}$ for all $h
\in H$. We say that a point of $\mu^{-1}(0)$ is stable if any
$B$-stable collection of subspaces $S$ containing the image of $i$
is equal to all of $V$. We let $\mu^{-1}(0)^s$ denote the set of
stable points.
\begin{prop}
The stabilizer in $G_\mathbf{v}$ of any point in $\mu^{-1}(0)^s$ is
trivial.
\end{prop}
We then define
\[
\mathfrak{M}(\mathbf{v},\mathbf{w}) = \mu^{-1}(0)^s/G_\mathbf{v},
\]
which is diffeomorphic to an affine algebraic manifold. We know
(see \cite[Cor 3.12]{N98}) that
\[
\dim_\mathbb{C} \mathfrak{M}(\mathbf{v},\mathbf{w}) = \mathbf{v} \cdot (2\mathbf{w} - C \mathbf{v}),
\]
where $C$ is the Cartan matrix of $\mathfrak{sl}_n$.
For $(B,i,j) \in \mu^{-1}(0)^s$, we denote the corresponding orbit
in $\mathfrak{M}(\mathbf{v},\mathbf{w})$ by $[B,i,j]$ and if the orbit through $(B,i,j)$ is
closed, we denote the corresponding point of $\mathfrak{M}_0(\mathbf{v},\mathbf{w})$ by the
same notation.
We have a map
\[
\pi : \mathfrak{M}(\mathbf{v},\mathbf{w}) \to \mathfrak{M}_0(\mathbf{v},\mathbf{w})
\]
which sends an orbit $[B,i,j]$ to the unique closed orbit
$[B_0,i_0,j_0]$ contained in the closure of $G(B,i,j)$. Let
$\mathfrak{L}(\mathbf{v},\mathbf{w}) = \pi^{-1}(0)$.
\begin{prop}
The subvariety $\mathfrak{L}(\mathbf{v},\mathbf{w}) \subset \mathfrak{M}(\mathbf{v},\mathbf{w})$ is half-dimensional
and is homotopic to $\mathfrak{M}(\mathbf{v},\mathbf{w})$.
\end{prop}
Actually, under a natural symplectic form on $\mathfrak{M}(\mathbf{v},\mathbf{w})$, the
subvariety $\mathfrak{L}(\mathbf{v},\mathbf{w})$ is Lagrangian. It will be useful in the
sequel to also consider the following direct construction of
$\mathfrak{L}(\mathbf{v},\mathbf{w})$. Let
\[
\Lambda(\mathbf{v},\mathbf{w}) = \{(B,i,j) \in \mu^{-1}(0)\ |\ j=0,\, B \text{ is
nilpotent}\}
\]
where $B$ nilpotent means that there exists an $N \ge 1$ such that
for any sequence $h_1, h_2, \dots, h_N$ in $H$ satisfying
$\inc(h_k) = \out(h_{k+1})$, the composition $B_{h_N} \cdots B_{h_2}
B_{h_1} : V_{\out(h_1)} \to V_{\inc(h_N)}$ is zero.
Furthermore, define
\[
\Lambda(\mathbf{v},\mathbf{w})^s = \{(B,i,j) \in \Lambda(\mathbf{v},\mathbf{w})\ |\ (B,i,j) \in
\mu^{-1}(0)^s\}.
\]
Then we have the following Lemma.
\begin{lem}
We have
\[
\mathfrak{L}(\mathbf{v},\mathbf{w}) = \Lambda(\mathbf{v},\mathbf{w})^s/G_\mathbf{v}.
\]
\end{lem}
If $V'=(V'_k)$ is a collection of subspaces of $V=(V_k)$, we have
a natural inclusion map $\mathfrak{M}_0(\mathbf{v}',\mathbf{w}) \hookrightarrow
\mathfrak{M}_0(\mathbf{v},\mathbf{w})$. Thus, for vector spaces $V^1,V^2,W$, we can consider
the projections $\pi : \mathfrak{M}(\mathbf{v}^k,\mathbf{w}) \to \mathfrak{M}_0(\mathbf{v}^k,\mathbf{w})$ as maps to
$\mathfrak{M}_0(\mathbf{v}^1+\mathbf{v}^2,\mathbf{w})$. We then define
\[
Z(\mathbf{v}^1,\mathbf{v}^2;\mathbf{w}) = \{(x^1,x^2) \in \mathfrak{M}(\mathbf{v}^1,\mathbf{w}) \times \mathfrak{M}(\mathbf{v}^2,\mathbf{w})\
|\ \pi(x_1) = \pi(x_2)\}.
\]
Since $Z(\mathbf{v}^1,\mathbf{v}^2;\mathbf{w}) \circ Z(\mathbf{v}^2,\mathbf{v}^3;\mathbf{w}) \subset
Z(\mathbf{v}^1,\mathbf{v}^3;\mathbf{w})$, we have the convolution product
\[
H_*(Z(\mathbf{v}^1,\mathbf{v}^2;\mathbf{w})) \otimes H_*(Z(\mathbf{v}^2,\mathbf{v}^3;\mathbf{w})) \to
H_*(Z(\mathbf{v}^1,\mathbf{v}^3;\mathbf{w})).
\]
All of the irreducible components of $Z(\mathbf{v}^1,\mathbf{v}^2;\mathbf{w})$ have the same dimension.
Let $H_\textrm{top}(Z(\mathbf{v}^1,\mathbf{v}^2;\mathbf{w}))$ denote the top degree part of
$H_*(Z(\mathbf{v}^1,\mathbf{v}^2;\mathbf{w}))$. It has a natural basis $\{[X]\}$ where
$X$ runs over the irreducible components of $Z(\mathbf{v}^1,\mathbf{v}^2;\mathbf{w})$.
\begin{prop}
The convolution product makes the direct sum
$\bigoplus_{\mathbf{v}^1,\mathbf{v}^2} H_*(Z(\mathbf{v}^1,\mathbf{v}^2;\mathbf{w}))$ into an associative
algebra, and $\bigoplus_\mathbf{v} H_*(\mathfrak{L}(\mathbf{v},\mathbf{w}))$ is a left
$\bigoplus_{\mathbf{v}^1,\mathbf{v}^2} H_*(Z(\mathbf{v}^1,\mathbf{v}^2;\mathbf{w}))$-module. In addition,
the top degree part $\bigoplus_{\mathbf{v}^1,\mathbf{v}^2} H_\textrm{top}(Z(\mathbf{v}^1,\mathbf{v}^2;\mathbf{w}))$
is a subalgebra, and $\bigoplus_\mathbf{v} H_\textrm{top}(\mathfrak{L}(\mathbf{v},\mathbf{w}))$ is a
$\bigoplus_{\mathbf{v}^1,\mathbf{v}^2} H_\textrm{top}(Z(\mathbf{v}^1,\mathbf{v}^2;\mathbf{w}))$-stable submodule.
\end{prop}
Let $\Delta(\mathbf{v},\mathbf{w})$ denote the diagonal in $\mathfrak{M}(\mathbf{v},\mathbf{w}) \times
\mathfrak{M}(\mathbf{v},\mathbf{w})$. Then its fundamental class $[\Delta(\mathbf{v},\mathbf{w})]$ is in
$H_\textrm{top}(Z(\mathbf{v},\mathbf{v};\mathbf{w}))$. Left and right multiplication by
$[\Delta(\mathbf{v},\mathbf{w})]$ define projections
\begin{gather*}
[\Delta(\mathbf{v},\mathbf{w})] \cdot : \bigoplus_{\mathbf{v}^1,\mathbf{v}^2}
H_\textrm{top}(Z(\mathbf{v}^1,\mathbf{v}^2;\mathbf{w}))
\to \bigoplus_{\mathbf{v}^2} H_\textrm{top}(Z(\mathbf{v},\mathbf{v}^2;\mathbf{w})), \\
\cdot [\Delta(\mathbf{v},\mathbf{w})] : \bigoplus_{\mathbf{v}^1,\mathbf{v}^2}
H_\textrm{top}(Z(\mathbf{v}^1,\mathbf{v}^2;\mathbf{w}))
\to \bigoplus_{\mathbf{v}^1} H_\textrm{top}(Z(\mathbf{v}^1,\mathbf{v};\mathbf{w})). \\
\end{gather*}
For $k \in I$, define the \emph{Hecke correspondence}
$\mathfrak{B}_k(\mathbf{v},\mathbf{w})$ to be the variety of all $(B,i,j,S)$ (modulo the
$G_\mathbf{v}$-action) such that $(B,i,j) \in \mu^{-1}(0)^s$ and $S$ is a
$B$-invariant subspace contained in the kernel of $j$ such that
$\dim S = \mathbf{e}^k$ where $\mathbf{e}^k$ has $k$-component
equal to one and all other components equal to zero. We consider
$(B,i,j,S)$ as a point in $Z(\mathbf{v}-\mathbf{e}^k,\mathbf{v};\mathbf{w})$ by taking the
quotient by the subspace $S$ in the first factor. Then
$\mathfrak{B}_k(\mathbf{v},\mathbf{w})$ is an irreducible component of
$Z(\mathbf{v}-\mathbf{e}^k,\mathbf{v};\mathbf{w})$. Let $\omega : \mathfrak{M}(\mathbf{v}^1,\mathbf{w}) \times
\mathfrak{M}(\mathbf{v}^2,\mathbf{w}) \to \mathfrak{M}(\mathbf{v}^2,\mathbf{w}) \times \mathfrak{M}(\mathbf{v}^1,\mathbf{w})$ be the map that
interchanges the two factors. Then define
\begin{gather}
E_k = \sum_{\mathbf{v}} [\mathfrak{B}_k(\mathbf{v},\mathbf{w})] \in \bigoplus_{\mathbf{v}^1,\mathbf{v}^2}
H_\textrm{top}(Z(\mathbf{v}^1,\mathbf{v}^2;\mathbf{w})), \\
\label{def:nak-Fk}
F_k = \sum_{\mathbf{v}} (-1)^{r_k(\mathbf{v},\mathbf{w})} [\omega(\mathfrak{B}_k(\mathbf{v},\mathbf{w}))] \in
\bigoplus_{\mathbf{v}^1,\mathbf{v}^2} H_\textrm{top}(Z(\mathbf{v}^1,\mathbf{v}^2;\mathbf{w})), \\
H_k = \sum_{\mathbf{v}} \left<h_k, \omega_\mathbf{w} - \alpha_\mathbf{v}\right>
[\Delta(\mathbf{v},\mathbf{w})],
\end{gather}
where $r_k(\mathbf{v},\mathbf{w}) = \frac{1}{2}(\dim \mathfrak{M}_\mathbb{C}(\mathbf{v} - \mathbf{e}^k,\mathbf{w}) - \dim_\mathbb{C}
\mathfrak{M}(\mathbf{v},\mathbf{w})) = -\mathbf{e}^k \cdot (\mathbf{w} - C\mathbf{v})-1$. Here $C$ is the
Cartan matrix of $\mathfrak{sl}_n$. Note that since we are
restricting ourselves to the Lie algebra $\mathfrak{sl}_n$, the
varieties $\mathfrak{M}(\mathbf{v},\mathbf{w})$ are only nonempty for a finite number of
$\mathbf{v}$ and so the above elements are well-defined.
\begin{theo}[\cite{N98}]
\label{thm:nak-hom} There exists a unique surjective algebra
homomorphism
\[
\Phi : U(\mathfrak{sl}_n) \twoheadrightarrow \bigoplus_{\mathbf{v}^1,\mathbf{v}^2}
H_\textrm{top}(Z(\mathbf{v}^1,\mathbf{v}^2;\mathbf{w}))
\]
such that
\[
\Phi(h_k) = H_k, \quad \Phi(e_k) = E_k,\quad \Phi(f_k) = F_k.
\]
Under this homomorphism, $\bigoplus_\mathbf{v} H_\textrm{top}(\mathfrak{L}(\mathbf{v},\mathbf{w}))$ is the
irreducible integrable highest weight module with highest weight
$\omega_\mathbf{w}$. The class $[\mathfrak{L}(\mathbf{0},\mathbf{w})]$ is a highest weight
vector.
\end{theo}
\begin{rem}
The result in \cite{N98} is actually in terms of the modified
universal enveloping algebra. In the more general case of a
Kac-Moody algebra with symmetric Cartan matrix, this language is
more natural. However, in our case of $\mathfrak{sl}_n$, since
for a fixed $\mathbf{w}$ the quiver varieties $\mathfrak{M}(\mathbf{v},\mathbf{w})$ are non-empty
only for a finite number of $\mathbf{v}$, we can avoid the use of
the modified universal enveloping algebra.
\end{rem}
Let $J_\mathbf{w}$ be the annihilator in $U(\mathfrak{sl}_n)$ of
$\bigoplus_\mathbf{v} L(\omega_\mathbf{w} - \alpha_\mathbf{v})$, where the sum is over all
$\mathbf{v}$ such that $\omega_\mathbf{w} - \alpha_\mathbf{v}$ is dominant integral and is a
weight of $L(\omega_\mathbf{w})$. Here $L(\lambda)$ is the irreducible
integrable highest weight representation of highest weight
$\lambda$.
\begin{theo}[{\cite[Theorem~10.15]{N98}}]
\label{thm:nak-quotient}
The homomorphism of
Theorem~\ref{thm:nak-hom} yields an algebra isomorphism
\[
U(\mathfrak{sl}_n)/J_\mathbf{w} \cong \bigoplus_{\mathbf{v}_1,\mathbf{v}_2}
H_\textrm{top}(Z(\mathbf{v}^1,\mathbf{v}^2;\mathbf{w})).
\]
\end{theo}
\section{A comparison of the two constructions}
\label{sec:isom}
We now describe the precise relationship between the constructions
of Ginzburg and Nakajima.
We begin by recalling a result of Maffei \cite{M00}. Let $x \in
N$ and let $\{x,y,h\}$ be an $\mathfrak{sl}_2$ triple in
$GL(\mathbb{C}^d)$. We define the \emph{transversal slice} to the orbit
$O_x$ of $x$ in $N$ at the point $x$ to be
\[
S_x = \{u \in N\ |\ [u-x,y]=0\}.
\]
We allow $\{0,0,0\}$ to be an $\mathfrak{sl}_2$ triple. Thus we
have $S_0 = N$.
Now, the orbits of the action of $GL(\mathbb{C}^d)$ on $N$ are determined
by partitions of $d$. Corresponding to a partition $\lambda$ is
the orbit consisting of all those matrices whose Jordan blocks
have sizes $\lambda_i$. We let $O_\lambda$ denote the orbit
corresponding to the partition $\lambda$.
Let $\mu_{\mathbf{d}} : M_{\mathbf{d}} \to N$ denote the restriction of the map $\mu$
to $M_{\mathbf{d}}$. Then let $\alpha = (\alpha_1 \ge \alpha_2 \ge \dots
\ge \alpha_n)$ be a permutation of ${\mathbf{d}}$ and define the partition
$\lambda_{\mathbf{d}} = 1^{\alpha_1 - \alpha_2} 2^{\alpha_2-\alpha_3} \cdots
n^{\alpha_n}$. Then $\lambda_{\mathbf{d}}$ is a partition of $d$ and if
$(x,F) \in M_{\mathbf{d}}$, then $x \in \overline{O}_{\lambda_{\mathbf{d}}}$.
Furthermore, the map $\mu_{\mathbf{d}} : M_{\mathbf{d}} \to \overline{O}_{\lambda_{\mathbf{d}}}$
is a resolution of singularities and is an isomorphism over
$O_{\lambda_{\mathbf{d}}}$. Define
\begin{gather*}
S_{{\mathbf{d}},x} = S_x \cap \overline{O}_{\lambda_{\mathbf{d}}},\quad \widetilde
S_{{\mathbf{d}},x} = \mu_{\mathbf{d}}^{-1}(S_{{\mathbf{d}},x}) = \mu_{\mathbf{d}}^{-1}(S_x).
\end{gather*}
Now, for $\mathbf{v},\mathbf{w} \in (\mathbb{Z}_{\ge 0})^{n-1}$ define $\mathbf{a} = \mathbf{a}(\mathbf{v},\mathbf{w}) =
(a_1,\dots,a_n)$ by
\begin{gather}
\label{eq:a-def}
a_1 = w_1 + \dots + w_{n-1} - v_1,\quad a_n=v_{n-1}, \\
a_k = w_k + \dots + w_{n-1} - v_k + v_{k-1},\ 2 \le k \le n-1
\nonumber.
\end{gather}
Note that $\sum_{k=1}^n a_k = d = \sum_{k=1}^{n-1} kw_k$ and that
for a fixed $d$ and $\mathbf{w}$, the above map is a bijection between
$(n-1)$-tuples of integers $\mathbf{v}$ and $n$-tuples of integers $\mathbf{a}$
such that $\sum_i a_i = d$. Furthermore, let $\mathfrak{M}^1(\mathbf{v},\mathbf{w}) = \pi
(\mathfrak{M}(\mathbf{v},\mathbf{w}))$.
\begin{theo}[\cite{M00}]
\label{thm:maffei} Let $\mathbf{v}$, $\mathbf{w}$, $d$ and $\mathbf{a}=\mathbf{a}(\mathbf{v},\mathbf{w})$ be as
above and let $x \in N$ be a nilpotent element of type $1^{w_1}
2^{w_2} \cdots (n-1)^{w_{n-1}}$. Then there exists an isomorphism
$\theta : \mathfrak{M}(\mathbf{v},\mathbf{w}) \stackrel{\cong}{\longrightarrow} \widetilde
S_{\mathbf{a},x}$ and $\theta_1 : \mathfrak{M}^1(\mathbf{v},\mathbf{w})
\stackrel{\cong}{\longrightarrow} S_{\mathbf{a},x}$ such that
$\theta_1(0)=x$ and the following diagram commutes.
\[
\begin{CD}
\mathfrak{M}(\mathbf{v},\mathbf{w}) @>\theta>> \widetilde S_{\mathbf{a},x} \\
@V\pi VV @V \mu_\mathbf{a} VV \\
\mathfrak{M}^1(\mathbf{v},\mathbf{w}) @>\theta_1>> S_{\mathbf{a},x}
\end{CD}
\]
\end{theo}
Note that by Theorem~\ref{thm:maffei}, if we restrict $\theta$ to
$\mathfrak{L}(\mathbf{v},\mathbf{w})$, we obtain an isomorphism $\mathfrak{L}(\mathbf{v},\mathbf{w}) \cong \mathcal{F}_{\mathbf{a},x}$
which we will also denote by $\theta$. This restriction is fairly
simple to describe as we now show.
We define a \emph{path} to be an ordered set of edges
$(h_1,\dots,h_N)$ such that $\inc(h_i) = \out(h_{i+1})$. Then let
$\mathcal{P}$ be the set of all paths that head left and then
right. That is,
\[
\mathcal{P} =
\{(h_{k,k-1},h_{k-1,k-2},\dots,h_{l+1,l},h_{l,l+1},\dots,
h_{m-1,m})\ |\ 1 \le l \le m,k \le n-1\}.
\]
For $p = (h_1,\dots,h_N) \in \mathcal{P}$, let $\inc(p) =
\inc(h_N)$ be the incoming vertex of the last edge in $p$ and let
$\out(p) = \out(h_1)$ be the outgoing vertex of the first edge in
$p$. We define $\ord(p)$ to be the number of edges heading to the
left. That is, $\ord(p) = \#\{h_i \in p \ |\ h_i \in \Omega\}$.
Furthermore we let $B_p = B_{h_N} \dots B_{h_1}$ be the obvious
composition of maps.
Now, for $1 \le m \le k \le n-1$, let $\iota_k^m : W_k^{(m)} \cong
W_k$ be an isomorphism to a copy of $W_k$. Then for $1 \le k \le
n-1$, let
\begin{equation}
\label{eq:isom-map} \phi_k = \bigoplus_{p \in \mathcal{P},\,
\inc(p) = k} B_p i_{\out(p)} \iota_{\out(p)}^{\out(p)-\ord(p)} :
\bigoplus_{l=1}^{n-1} \bigoplus_{m\le k,l} W_l^{(m)} \to V_k.
\end{equation}
Let $d = \sum_{k=1}^{n-1} k w_k$ and identify $\bigoplus_{m,k:\, m
\le k} W_k^{(m)}$ with $\mathbb{C}^d$. Then $\theta : \sqcup_\mathbf{v} \mathfrak{L}(\mathbf{v},\mathbf{w})
\to \mathcal{F}$ sends the point $[B,i,j]$ to the flag $F = (0 = F_0
\subset \dots \subset F_n = \mathbb{C}^d)$ where $F_k = \ker \phi_k$. Note
that $\theta$ is well-defined since the kernel of $\phi_k$ does
not change under the action of $G_\mathbf{v}$. For $1 \le k \le n-1$, define
\[
W^{\le k} = \bigoplus_{m,l\, :\, m \le l,k} W_l^{(m)}.
\]
Note that we always have
\[
F_k = \ker \phi_k \subset W^{\le k}.
\]
\begin{cor}
\label{cor:lang-isom} The image of the map $\theta : \sqcup_\mathbf{v}
\mathfrak{L}(\mathbf{v},\mathbf{w}) \to \mathcal{F}$ lies in $\mathcal{F}_x$ where $x \in N$ is the map
given in block form by
$W_k^{(m)} \stackrel{\cong}{\to} W_k^{(m-1)}$ (and
$x(W_k^{(1)})=0$). Furthermore $\theta : \sqcup_\mathbf{v} \mathfrak{L}(\mathbf{v},\mathbf{w}) \to
\mathcal{F}_x$ is an isomorphism and $\theta (\mathfrak{L}(\mathbf{v},\mathbf{w})) = \mathcal{F}_{\mathbf{a},x}$ where
$\mathbf{a} = \mathbf{a}(\mathbf{v},\mathbf{w})$ is defined by \eqref{eq:a-def}.
\end{cor}
\begin{prop}
\label{prop:ginz-res}
Let $\mathbf{v},\mathbf{w} \in (\mathbb{Z}_{\ge 0})^{n-1}$, $\mathbf{a}=\mathbf{a}(\mathbf{v},\mathbf{w})$, $x \in N$ a nilpotent
element of type $1^{w_1} 2^{w_2} \cdots (n-1)^{w_{n-1}}$, and $1 \le k
\le n-1$. Then
\begin{equation}
\label{eq:ginz-res} (\theta \times \theta) (\mathfrak{B}_k(\mathbf{v},\mathbf{w}))
= \left( T^*_{Y_{\mathbf{a}_k^+,\mathbf{a}}}(\mathcal{F}_{\mathbf{a}_k^+} \times \mathcal{F}_\mathbf{a}) \right) \cap ({\tilde
S}_{\mathbf{a}_k^+,x} \times {\tilde S}_{\mathbf{a},x}).
\end{equation}
\end{prop}
\begin{proof}
The right side of \eqref{eq:ginz-res} is equal to
\begin{equation} \label{eq:subflag}
\{(F',F) \in {\tilde S}_{\mathbf{a}_k^+,x} \times {\tilde S}_{\mathbf{a},x}\ |\
F_l = F'_l\ \forall \ l \ne k,\, F_k \subset F'_k\,
\dim(F'_k/F_k) = 1 \}.
\end{equation}
Recall that
\[
\mathfrak{B}_k(\mathbf{v},\mathbf{w}) = \{(B,i,j,S)\ |\ (B,i,j) \in
\mu^{-1}(0)^s,\, S \subset V,\, j(S)=0,\, S \text{ $B$-invariant},\, \dim
S=\mathbf{e}^k\}/G_\mathbf{v}.
\]
We consider this as a subset of $\mathfrak{M}(\mathbf{v}-\mathbf{e}^k,\mathbf{w}) \times \mathfrak{M}(\mathbf{v},\mathbf{w})$
by taking the quotient by the subspace $S$ in the first factor. We
know by Theorem~\ref{thm:maffei} that
\[
\theta : \mathfrak{M}(\mathbf{v},\mathbf{w}) \stackrel{\cong}{\longrightarrow} {\tilde
S}_{\mathbf{a},x},\quad \theta : \mathfrak{M}(\mathbf{v}-\mathbf{e}^k,\mathbf{w})
\stackrel{\cong}{\longrightarrow} {\tilde S}_{\mathbf{a}_k^+,x}.
\]
Thus, it suffices to show that a choice of $B$-invariant subspace
$S$ of $V_k$ corresponds to a choice of $F'_k$ such that $F_k
\subset F'_k \subset x^{-1}(F_{k-1})$. We first do this for the
case where $W = W_1$. Then $i=i_1$ and $j=j_1$.
In this case, the isomorphism between
quiver varieties and flag varieties is particularly simple (see
\cite{N94} and \cite{M00}). The isomorphism is given by $\theta :
[B,i,j] \mapsto (x,F)$ where
\begin{align*}
x=ji,\quad F=(0 \subset \ker i \subset \ker B_{12}i \subset \dots
\subset \ker B_{n-2,n-1} \dots B_{12} i \subset W).
\end{align*}
That is, $F_l = \ker B_{l-1,l} \cdots B_{12} i$. Now, let $S
\subset V_k$ be a $B$-invariant subspace contained in the kernel
of $j$ with $\dim S = 1$ and let $(B',i',j')$ be the point of
$\mathfrak{M}(\mathbf{v}-\mathbf{e}^k,\mathbf{w})$ obtained from $(B,i,j)$ by taking the quotient by
the subspace $S$. Now, since $S$ is $B$-invariant, we have that
$S \in \ker B_{k,k-1} \cap \ker B_{k,k+1}$. Here we adopt the
convention that $B_{1,0}=0$ and $B_{n-1,n}=0$. Let $p : V_k \to
V_k/S$ be the canonical projection. Then $\theta([B',i',j']) =
(x,F')$ where $x=ji$ and
\begin{align*}
F'_l &= \ker B_{l-1,l} \dots B_{12} i = F_l,\ l < k,\\
F'_l &= \ker B_{l-1,l} \dots B_{k,k+1} p B_{k-1,k} \dots B_{12}
i,\ l \ge k.
\end{align*}
Now, since $S \subset \ker B_{k,k+1}$, we have that $B_{k,k+1}p$ =
$B_{k,k+1}$. Thus, for $l >k$, $F'_l = F_l$. Also,
\[
F_k' = \ker p B_{k-1,k} \cdots B_{12} i \supset \ker B_{k-1,k}
\cdots B_{12} i = F_k.
\]
Thus it remains to show
that $F'_k \subset x^{-1} (F_{k-1})$. Now,
\begin{align*}
x^{-1}(F_{k-1}) &= x^{-1}(\ker B_{k-2,k-1} \dots B_{12} i) \\
&= \ker (B_{k-2,k-1} \dots B_{12} i x) \\
&= \ker (B_{k-2,k-1} \dots B_{12} iji).
\end{align*}
Now, since $(B,i,j) \in \mu^{-1}(0)$, we have that $ij =
B_{21}B_{12}$ and $B_{l-1,l} B_{l,l-1} = B_{l+1,l} B_{l+1,l}$ for
$2 \le l \le n-2$. Thus,
\begin{align*}
B_{k-2,k-1} \dots B_{12} iji &= B_{k-2,k-1} \dots B_{12} B_{21}
B_{12} i \\
&\ \vdots \\
&=B_{k,k-1} B_{k-1,k} B_{k-2,k-1} \dots B_{12} i.
\end{align*}
Thus
\[
x^{-1}(F_{k-1}) = \ker (B_{k,k-1} B_{k-1,k} B_{k-2,k-1} \dots
B_{12} i).
\]
Now, since $S \subset \ker B_{k,k-1}$, we have
\[
F_k' = \ker(p B_{k-1,k} \dots B_{12}i) \subset \ker (B_{k,k-1}
B_{k-1,k} \dots B_{12} i) = x^{-1}(F_{k-1}).
\]
We have shown that every choice of subspace $S$ corresponds to a
flag $F'$ satisfying the conditions in \eqref{eq:subflag}. It is
easy to see that such a flag $F'$ comes from a subspace $S$ as
follows. We have that $F_k \subset F'_k$. We take $S$ to
be the subspace of $V_k$ such that
\[
\ker (p B_{k-1,k} \dots B_{12} i) = F'_k
\]
for the projection $p : V_k \to V_k/S$. Thus we have proven the
proposition in the special case $W = W_1$.
For the general case, we recall Maffei's construction in
\cite{M00}. For general $W$, Maffei constructs a map
$\Lambda(\mathbf{v},\mathbf{w}) \to \Lambda({\tilde \mathbf{v}},{\tilde \mathbf{w}})$, denoted
$(B,i,j) \mapsto ({\tilde B},{\tilde i},{\tilde j})$, where
${\tilde \mathbf{w}} = c\mathbf{e}^1$ for some $c \in \mathbb{Z}_{\ge 0}$. Thus, if we
show that a choice of a $B$-stable subspace $S$ such that $\dim S
= \mathbf{e}^k$ corresponds to a choice of $\tilde B$-stable subspace
$\tilde S$ such that $\dim {\tilde S} = \mathbf{e}^k$ then we reduce the
proof to the special case considered above. Now,
\begin{align*}
{\tilde V}_k &= V_k \oplus W'_k, \\
\text{where } W'_k &= \bigoplus_{l,m\, :\, 1 \le m \le l-k,\, k+1
\le l \le n-1} W_l^{(m)},
\end{align*}
and $W_l^{(m)}$ is an isomorphic copy of $W_l$. For $1 \le m \le
l-k$ and $k+1 \le l \le n-1$, we have (see \cite{M00})
\begin{gather*}
\mathrm{pr}_{W_l^{(m)}} {\tilde B}_{k,k-1} |_{W_l^{(m)}} = \Id_{W_l}, \\
\mathrm{pr}_{W_l^{(m)}} {\tilde B}_{k,k-1} |_{V_k} = 0,
\end{gather*}
where $\mathrm{pr}_{W_l^{(m)}}$ denotes the projection onto the subspace
$W_l^{(m)}$. In particular, $\ker {\tilde B}_{k,k-1} \subset
V_k$. Thus, since the subspace ${\tilde S} \subset {\tilde V}_k$
must be contained in $\ker {\tilde B}_{k,k-1}$, it must lie in
$V_k$. The result then follows from Remark~19 of \cite{M00}.
\end{proof}
We now compare the Lie algebra action in the two settings. By
\cite[\S 3.7.14]{CG}, $S_x$ is transverse to the orbit $O_x$ in
$N$. Thus, there is an open neighborhood $U \subset N$ of $S$
such that
\[
U \cong (O_x \cap U) \times S.
\]
Let $\tilde U_{\mathbf{d}} = \mu_{\mathbf{d}}^{-1}(U)$ and $M_{\mathbf{d}}' = \mu_{\mathbf{d}}^{-1}(S_x) =
\mu_{\mathbf{d}}^{-1}(S_{{\mathbf{d}},x}) = \tilde S_{{\mathbf{d}},x}$.
Then $\tilde U_{\mathbf{d}} \subset M_{\mathbf{d}}$ is an open
neighborhood of $M_{\mathbf{d}}'$. Let $D = O_x \cap U$, a small
neighborhood of $x$ in $O_x$. By \cite[Cor 3.2.21]{CG},
\[
\tilde U_{\mathbf{d}} \cong (O_x \cap U) \times M_{\mathbf{d}}'.
\]
Then the two commutative diagrams
\[
\begin{CD}
M_{\mathbf{d}}' @>>> M_{\mathbf{d}} \\
@VV{\mu_{\mathbf{d}}}V @VV{\mu_{\mathbf{d}}}V \\
S_x @>>> N
\end{CD}
\qquad \text{and} \qquad
\begin{CD}
M_{\mathbf{d}}' @>{p \mapsto (x,p)}>> D \times M_{\mathbf{d}}' @>{\cong}>> \tilde U_{\mathbf{d}} \\
@VV{\mu_{\mathbf{d}}}V @VV{\mathbf{1}_D \times \mu_{\mathbf{d}}}V @VV{\mu_{\mathbf{d}}}V \\
S_x @>{y \mapsto (x,y)}>> D \times S_x @>{\cong}>> U
\end{CD}
\]
are isomorphic, where the horizontal arrows in the left diagram
are given by the natural inclusions. If we let $\tilde U =
\mu^{-1}(U)$ and $M' = \mu^{-1}(S_x)$ then $\tilde U = \sqcup_{\mathbf{d}}
\tilde U_{\mathbf{d}}$, $M' = \sqcup_{\mathbf{d}} M_{\mathbf{d}}'$ and $\tilde U \subset M$ is
an open neighborhood of $M'$. Thus we have that the two
commutative diagrams
\[
\begin{CD}
M' @>>> M \\
@VV{\mu}V @VV{\mu}V \\
S_x @>>> N
\end{CD}
\qquad \text{and} \qquad
\begin{CD}
M' @>{p \mapsto (x,p)}>> D \times M' @>{\cong}>> \tilde U \\
@VV{\mu}V @VV{\mathbf{1}_D \times \mu}V @VV{\mu}V \\
S_x @>{y \mapsto (x,y)}>> D \times S_x @>{\cong}>> U
\end{CD}
\]
are isomorphic. Let $Z' = M' \times_{S_x} M'$. Then by
Theorem~\ref{thm:maffei},
\[
Z' \cong \sqcup_{\mathbf{v}^1,\mathbf{v}^2} Z(\mathbf{v}^1,\mathbf{v}^2;\mathbf{w}).
\]
We then have the commutative diagram
\begin{equation}
\label{eq:comm-slice}
\begin{CD}
Z' = M' \times_{S_x} M' @>i>> M \times_N M = Z \\
@VVV @VVV \\
M' \times M' @>i>> M \times M
\end{CD}
\end{equation}
where the maps are the obvious inclusions.
Diagram~\eqref{eq:comm-slice} is isomorphic to
\begin{equation}
\begin{CD}
Z' = M' \times_{S_x} M' @>i={p \mapsto (x,p)}>> D_\Delta \times
(M' \times_{S_x} M')
\cong Z \cap (\tilde U \times \tilde U) \\
@V{j}VV @V{\Delta \times j}VV \\
M' \times M' @>i={p \mapsto ((x,x),p)}>> (D \times D) \times (M'
\times M')
\end{CD}
\end{equation}
where $\Delta : D_\Delta \to D \times D$ is the embedding of the
diagonal.
\begin{lem}
\label{lem:irrcomp-restrict}
The inverse image in $M' \times M'$ of an irreducible component of
the variety $Z$ is either empty or else is an irreducible
component of the variety $Z'$.
\end{lem}
\begin{proof}
Let $X$ be a (closed) irreducible component of $Z$. If $X$ does
not intersect the open subset $\tilde U \times \tilde U \subset M
\times M$, then $i^{-1}(X) = \emptyset$, since $i(Z') \subset
\tilde U \times \tilde U$. Now assume that $X_U = X \cap (\tilde
U \times \tilde U)$ is non-empty. Then $X_U$ is an irreducible
component of $Z \cap (\tilde U \times \tilde U)$. Thus is must be
of the form $X_\mathbf{U} \cong D_\Delta \times X'$ where $X'$ is an
irreducible component of $M' \times_{S_x} M' = Z'$. We then have
$i^{-1}(X) = X'$ and the result follows.
\end{proof}
The diagram~\eqref{eq:comm-slice} gives rise to a restriction with
support morphism
\[
i^* : H_*(Z) \to H_*(Z'),\quad c \mapsto c \cap [M' \times M'].
\]
By Lemma~\ref{lem:irrcomp-restrict}, $i^*$ takes $H_\textrm{top}(Z)$
to $H_\textrm{top}(Z')$. Furthermore, by
Proposition~\ref{prop:ginz-res} we have that
\begin{equation}
\label{eq:action-isom}
i^*([T^*_{Y_{\mathbf{a}_k^+,\mathbf{a}}}(\mathcal{F}_{\mathbf{a}_k^+} \times \mathcal{F}_\mathbf{a})]) = [(\theta
\times \theta)(\mathfrak{B}_k(\mathbf{v},\mathbf{w}))]
\end{equation}
where $\mathbf{a} = \mathbf{a}(\mathbf{v},\mathbf{w})$.
Now, $\mathcal{F}_x = \mu^{-1}(x)$ can be viewed
as a subvariety of $M'$ or $M$. If $i : M' \to M$ is the
inclusion, then the restriction with supports morphism $i^* :
H_\textrm{top}(\mathcal{F}_x) \to H_\textrm{top}(\mathcal{F}_x)$ is an isomorphism, where the first and second
$H_\textrm{top}(\mathcal{F}_x)$ are $H^0(M,M\backslash {\mathcal{F}_x})$ and $H^0(M',M'\backslash
{\mathcal{F}_x})$ respectively.
\begin{theo}
\label{thm:quiver-flag}
\begin{enumerate}
\item The morphism $i^* : H_\textrm{top}(Z) \to H_\textrm{top}(Z') \cong \oplus_{\mathbf{v}^1,\mathbf{v}^2}
H_\textrm{top}(Z(\mathbf{v}^1,\mathbf{v}^2;\mathbf{w}))$ is an algebra homomorphism (with respect to
the convolution product).
\item The following diagram, where $x \in N$ is a nilpotent element of type
$1^{w_1} 2^{w_2} \cdots (n-1)^{w_{n-1}}$ and whose vertical maps are given by
convolution, commutes
\[
\begin{CD}
H_\textrm{top}(Z) \otimes H_\textrm{top}(\mathcal{F}_x) @>{i^* \otimes i^*}>> H_\textrm{top}(Z') \otimes
H_\textrm{top}(\mathcal{F}_x) @>\cong>> \oplus_{\mathbf{v}^1,\mathbf{v}^2} H_\textrm{top}(Z(\mathbf{v}^1,\mathbf{v}_2;\mathbf{w}))
\otimes \oplus_\mathbf{v} H_\textrm{top}(\mathfrak{L}(\mathbf{v},\mathbf{w}))
\\
@VVV @VVV @VVV \\
H_\textrm{top}(\mathcal{F}_x) @>{i^*}>> H_\textrm{top}(\mathcal{F}_x) @>\cong>> \oplus_\mathbf{v}
H_\textrm{top}(\mathfrak{L}(\mathbf{v},\mathbf{w}))
\end{CD}
\]
\end{enumerate}
\end{theo}
\begin{proof}
Note that the two rightmost horizontal maps are the isomorphisms induced
by the map $\theta$ of Theorem~\ref{thm:maffei}.
We prove only the first part of the theorem. The second part in
analogous. We have a sequence of embeddings
\[
M' \times M' \hookrightarrow \tilde U \times \tilde U
\hookrightarrow M \times M
\]
So $i^*$ factors as
\[
i^* : H_\textrm{top}(Z) \to H_\textrm{top}(Z \cap (\tilde U \times \tilde U)) \to H_\textrm{top}(Z')
\]
The first map is the restriction to an open subset and thus
commutes with convolution by base locality (cf. \cite[\S
2.7.45]{CG}). The second map is induced by the embedding
\[
Z' \hookrightarrow Z \cap (\tilde U \times \tilde U).
\]
By the above results, this is isomorphic to the natural embedding
\[
Z' \hookrightarrow D_\Delta \times Z',\quad z \mapsto (x,z).
\]
The corresponding map
\[
i^* : H_\textrm{top}(D_\Delta \times Z') \to H_\textrm{top}(Z')
\]
commutes with convolution by the K\" unneth formula for
convolution (cf. \cite[\S 2.7.16]{CG}).
\end{proof}
\begin{cor}
\label{cor:ginz=nak}
If $c \in H_\textrm{top}(F_x)$ and $c'$ is the corresponding class in
$\oplus_\mathbf{v} H_\textrm{top}(\mathfrak{L}(\mathbf{v},\mathbf{w}))$ (under the isomorphism $\theta$) then
we have
\begin{align*}
E^\text{Gin}_k c &= E^\text{Nak}_k c', \text{ and} \\
F^\text{Gin}_k c &= F^\text{Nak}_k c' \text{ for all $k$}.
\end{align*}
Here the superscripts Gin and Nak correspond to the actions
defined by Ginzburg and Nakajima respectively.
\end{cor}
\begin{proof}
The result follows from \eqref{eq:action-isom} and the fact that
since $\tilde U_{\mathbf{d}} \cong (O_x \cap U) \times M_{\mathbf{d}}'$ we have
\begin{align*}
\dim_\mathbb{C} M_{{\mathbf{d}}^1} - \dim_\mathbb{C} M_{{\mathbf{d}}^2} &= (\dim_\mathbb{C} (O_x \cap U) +
\dim_\mathbb{C} M_{{\mathbf{d}}^1}') -( \dim_\mathbb{C} (O_x \cap U) + \dim_\mathbb{C} M_{{\mathbf{d}}^2}') \\
&= \dim_\mathbb{C} M_{{\mathbf{d}}^1}' - \dim_\mathbb{C} M_{{\mathbf{d}}^2}'.
\end{align*}
Thus the signs appearing in \eqref{def:ginz-Fk} and
\eqref{def:nak-Fk} are the same.
\end{proof}
We see from Corollary~\ref{cor:ginz=nak} that the Ginzburg and
Nakajima constructions yield the same representations, with the
same bases, given by the fundamental classes of the irreducibles
components of $\mathcal{F}_x \cong \sqcup_\mathbf{v} \mathfrak{L}(\mathbf{v},\mathbf{w})$. However, note that
the corresponding quotients of the universal enveloping algebra
constructed via convolution is different (compare
Theorems~\ref{thm:ginz-quotient} and \ref{thm:nak-quotient}). To
see that these two quotients are indeed different, it suffices to
consider the case of $\mathfrak{sl}_3$ with $\mathbf{w} = (1,1)$ (so
$\omega_\mathbf{w} = \omega_1 + \omega_2$ and $d=3$). Then the weight
$3\omega_1$ corresponds to a partition of $d$ but is not a weight
of $L(\omega_\mathbf{w})$ (since the tableau of shape $(21)$ with all
three entries equal to 1 is not semistandard).
\section{Crystal structure on flag varieties}
\label{sec:crystal}
Kashiwara and Saito have introduced the structure of a crystal on
the set of irreducible components of Nakajima's quiver varieties. In
this section, we recall this construction and use the isomorphism of
Section~\ref{sec:isom} to define a crystal structure on the flag
varieties (or, more precisely, on the set of irreducible components
of the Spaltenstein varieties $\mathcal{F}_x$). In this way we recover the
crystal structure defined by Malkin (see \cite{Mal02}). In fact,
Malkin and Nakajima have defined a tensor product quiver variety
(see \cite{Mal03} and \cite{Nak01}). One would expect that the
relationship between the two constructions examined in this paper
could be extended to this setting and one would recover the tensor
product crystal structure defined in \cite{Mal02}. However, we will
restrict ourselves to the case of a single representation here.
We first review the realization of the crystal graph via quiver
varieties. See \cite{KS97,S02} for proofs omitted here. Note that,
as mentioned in Section~\ref{sec:nak}, we are using a different
stability condition and thus our definitions differ slightly from
those in \cite{KS97,S02}.
Let $\mathbf{w, v, v', v''} \in (\mathbb{Z}_{\ge 0})^I$ be such that $\mathbf{v} =
\mathbf{v'} + \mathbf{v''}$. Consider the maps
\begin{equation}
\label{eq:diag_action} \Lambda(\mathbf{v}'',\mathbf{0}) \times
\Lambda(\mathbf{v}',\mathbf{w}) \stackrel{p_1}{\longleftarrow} \mathbf{\tilde F
(v,w;v'')} \stackrel{p_2}{\longrightarrow} \mathbf{F(v,w;v'')}
\stackrel{p_3}{\rightarrow} \Lambda(\mathbf{v},\mathbf{w}),
\end{equation}
where the notation is as follows. A point of
$\mathbf{F(v,w;v'')}$ is a point $(B,i) \in \Lambda(\mathbf{v},\mathbf{w})$
together with an $I$-graded, $B$-stable subspace $S$ of $V$ such
that $\dim S = \mathbf{v}''$. A point of $\mathbf{\tilde
F (v,w;v'')}$ is a point $(B,i,S)$ of $\mathbf{F(v,w;v'')}$
together with a collection of isomorphisms $R''_k : V''_k \cong
S_k$ and $R'_k : V'_k \cong V_k / S_k$ for each $k \in I$. Then we
define $p_2(B,i,S, R',R'') = (B,i,S)$, $p_3(B,i,S) = (B,i)$ and
$p_1(B,i,S,R',R'') = (B'',B',i')$ where $B'', B', i'$ are
determined by
\begin{align*}
R''_{\inc(h)} B''_h &= B_h R''_{\out(h)} : V''_{\out(h)} \to
S_{\inc(h)}, \\
R'_k i'_k &= {\bar i}_k : W_k \to V_k/S_k, \\
R'_{\inc(h)} B'_h &= B_h R'_{\out(h)} : V'_{\out(h)} \to
V_{\inc(h)}/S_{\inc(h)},
\end{align*}
where ${\bar i}_k$ denotes the composition of the map $i_k$ with
the canonical projection $V_k \to V_k/S_k$. It follows that $B'$
and $B''$ are nilpotent.
\begin{lem}[{\cite[Lemma 10.3]{N94}}]
One has
\[
(p_3 \circ p_2)^{-1} (\Lambda(\mathbf{v},\mathbf{w})^s) \subset p_1^{-1}
(\Lambda(\mathbf{v}'',\mathbf{0}) \times \Lambda(\mathbf{v}',\mathbf{w})^s).
\]
\end{lem}
Thus, we can restrict \eqref{eq:diag_action} to stable points,
forget the $\Lambda(\mathbf{v}'',\mathbf{0})$-factor and consider the
quotient by $G_\mathbf{v}$, $G_{\mathbf{v}'}$. This yields the diagram
\begin{equation}
\label{eq:diag_action_mod} \mathcal{L}(\mathbf{v}', \mathbf{w})
\stackrel{\pi_1}{\longleftarrow} \mathcal{L}(\mathbf{v}, \mathbf{w}; \mathbf{v} -
\mathbf{v'}) \stackrel{\pi_2}{\longrightarrow} \mathcal{L}(\mathbf{v},
\mathbf{w}),
\end{equation}
where
\[
\mathcal{L}(\mathbf{v}, \mathbf{w}; \mathbf{v} - \mathbf{v'}) \stackrel{\text{def}}{=} \{
(B,i,S) \in \mathbf{F(\mathbf{v},\mathbf{w};\mathbf{v}-\mathbf{v}')}\,
|\, (B,i) \in \Lambda(\mathbf{v},\mathbf{w})^s \} / G_\mathbf{v}.
\]
For $k \in I$ define $\varepsilon_k : \Lambda(\mathbf{v}, \mathbf{w}) \to \mathbb{Z}_{\ge
0}$ by
\[
\varepsilon_k((B,i)) = \dim_\mathbb{C} \ker \left( V_k
\stackrel{(B_h)}{\longrightarrow} \bigoplus_{h\, :\, \out(h)=k}
V_{\inc(h)} \right).
\]
Then, for $c \in \mathbb{Z}_{\ge 0}$, let
\[
\mathcal{L}(\mathbf{v},\mathbf{w})_{k,c} = \{[B,i] \in \mathcal{L}(\mathbf{v},\mathbf{w})\ |\
\varepsilon_k((B,i)) = c\}
\]
where $[B,i]$ denotes the $G_\mathbf{v}$-orbit through the point $(B,i)$.
$\mathcal{L}(\mathbf{v},\mathbf{w})_{k,c}$ is a locally closed subvariety of
$\mathcal{L}(\mathbf{v},\mathbf{w})$.
Assume $\mathcal{L}(\mathbf{v},\mathbf{w})_{k,c} \ne \emptyset$ and let $\mathbf{v}' = \mathbf{v}
- c\mathbf{e}^k$ where $\mathbf{e}^k_l = \delta_{kl}$. Then
\[
\pi_1^{-1}(\mathcal{L}(\mathbf{v}',\mathbf{w})_{k,0}) =
\pi_2^{-1}(\mathcal{L}(\mathbf{v},\mathbf{w})_{k,c}).
\]
Let
\[
\mathcal{L}(\mathbf{v},\mathbf{w};c\mathbf{e}^k)_{k,0} =
\pi_1^{-1}(\mathcal{L}(\mathbf{v}',\mathbf{w})_{k,0}) =
\pi_2^{-1}(\mathcal{L}(\mathbf{v},\mathbf{w})_{k,c}).
\]
We then have the following diagram.
\begin{equation}
\label{eq:crystal-action} \mathcal{L}(\mathbf{v}',\mathbf{w})_{k,0}
\stackrel{\pi_1}{\longleftarrow} \mathcal{L}(\mathbf{v},\mathbf{w};
c\mathbf{e}^k)_{k,0} \stackrel{\pi_2}{\longrightarrow}
\mathcal{L}(\mathbf{v},\mathbf{w})_{k,c}
\end{equation}
The restriction of $\pi_2$ to $\mathcal{L}(\mathbf{v},\mathbf{w};
c\mathbf{e}^k)_{k,0}$ is an isomorphism since the only possible
choice for the subspace $S$ of $V$ is to have $S_l = 0$ for $l \ne
k$ and $S_k$ equal to the intersection of the kernels of the $B_h$
with $\out(h)=k$. Also, $\mathcal{L}(\mathbf{v}',\mathbf{w})_{k,0}$ is an open
subvariety of $\mathcal{L}(\mathbf{v}',\mathbf{w})$.
\begin{lem}[\cite{S02}]
\begin{enumerate}
\item For any $k \in I$,
\[
\mathcal{L}(\mathbf{0},\mathbf{w})_{k,c} =
\begin{cases}
pt & \text{if $c=0$} \\
\emptyset & \text{if $c >0$}
\end{cases}.
\]
\item Suppose $\mathcal{L}(\mathbf{v},\mathbf{w})_{k,c} \ne \emptyset$ and $\mathbf{v}' =
\mathbf{v} - c\mathbf{e}^k$. Then the fiber of the restriction of $\pi_1$
to $\mathcal{L}(\mathbf{v}, \mathbf{w}; c\mathbf{e}^k)_{k,0}$ is isomorphic to a
Grassmannian variety.
\end{enumerate}
\end{lem}
\begin{cor}
\label{cor:irrcomp-isom} Suppose $\mathcal{L}(\mathbf{v},\mathbf{w})_{k,c} \ne
\emptyset$. Then there is a 1-1 correspondence between the set of
irreducible components of $\mathcal{L}(\mathbf{v} - c\mathbf{e}^k,
\mathbf{w})_{k,0}$ and the set of irreducible components of
$\mathcal{L}(\mathbf{v}, \mathbf{w})_{k,c}$.
\end{cor}
Let $\mathcal{B}(\mathbf{v},\mathbf{w})$ denote the set of irreducible components
of $\mathcal{L}(\mathbf{v},\mathbf{w})$ and let $\mathcal{B}(\mathbf{w}) = \bigsqcup_\mathbf{v}
\mathcal{B}(\mathbf{v},\mathbf{w})$. For $X \in \mathcal{B}(\mathbf{v},\mathbf{w})$, let
$\varepsilon_k(X) = \varepsilon_k((B,i))$ for a generic point
$[B,i] \in X$. Then for $c \in \mathbb{Z}_{\ge 0}$ define
\[
\mathcal{B}(\mathbf{v},\mathbf{w})_{k,c} = \{X \in \mathcal{B}(\mathbf{v},\mathbf{w})\ |\
\varepsilon_k(X) = c\}.
\]
Then by Corollary~\ref{cor:irrcomp-isom}, $\mathcal{B}(\mathbf{v} -
c\mathbf{e}^k,\mathbf{w})_{k,0} \cong \mathcal{B}(\mathbf{v}, \mathbf{w})_{k,c}$.
Suppose that ${\bar X} \in \mathcal{B}(\mathbf{v} -
c\mathbf{e}^k,\mathbf{w})_{k,0}$ corresponds to $X \in
\mathcal{B}(\mathbf{v},\mathbf{w})_{k,c}$ by the above isomorphism. Then we define
maps
\begin{gather*}
{\tilde f}_k^c : \mathcal{B}(\mathbf{v} - c\mathbf{e}^k,\mathbf{w})_{k,0} \to
\mathcal{B}(\mathbf{v},\mathbf{w})_{k,c},\quad {\tilde f}_k^c({\bar X})
= X, \\
{\tilde e}_k^c : \mathcal{B}(\mathbf{v},\mathbf{w})_{k,c} \to \mathcal{B}(\mathbf{v} -
c\mathbf{e}^k,\mathbf{w})_{k,0},\quad {\tilde e}_k^c(X) = {\bar X}.
\end{gather*}
We then define the maps
\[
{\tilde e}_k, {\tilde f}_k : \mathcal{B}(\mathbf{w}) \to \mathcal{B}(\mathbf{w}) \sqcup \{0\}
\]
by
\begin{gather*}
{\tilde e}_k : \mathcal{B}(\mathbf{v},\mathbf{w})_{k,c}
\stackrel{{\tilde e}_k^c}{\longrightarrow} \mathcal{B}(\mathbf{v} -
c\mathbf{e}^k, \mathbf{w})_{k,0} \stackrel{{\tilde f}_k^{c-1}}{\longrightarrow}
\mathcal{B}(\mathbf{v} -
\mathbf{e}^k, \mathbf{w})_{k,c-1}, \\
{\tilde f}_k : \mathcal{B}(\mathbf{v},\mathbf{w})_{k,c}
\stackrel{{\tilde e}_k^c}{\longrightarrow} \mathcal{B}(\mathbf{v} -
c\mathbf{e}^k, \mathbf{w})_{k,0} \stackrel{{\tilde f}_k^{c+1}}{\longrightarrow}
\mathcal{B}(\mathbf{v} + \mathbf{e}^k, \mathbf{w})_{k,c+1}.
\end{gather*}
We set ${\tilde e}_k(X)=0$ for $X \in \mathcal{B}(\mathbf{v},\mathbf{w})_{k,0}$ and
${\tilde f}_k(X)=0$ for $X \in \mathcal{B}(\mathbf{v},\mathbf{w})_{k,c}$ with
$\mathcal{B}(\mathbf{v},\mathbf{w})_{k,c+1} = \emptyset$. We also define
\begin{gather*}
\wt : \mathcal{B}(\mathbf{w}) \to P,\quad \wt(X) = \omega_\mathbf{w} - \alpha_\mathbf{v}
\text{
for } X \in \mathcal{B}(\mathbf{v},\mathbf{w}), \\
\varphi_k(X) = \varepsilon_k(X) + \left< h_k, \wt(X) \right>.
\end{gather*}
\begin{prop}[\cite{S02}]
\label{prop:quiver-crystal}
$\mathcal{B}(\mathbf{w})$ is a crystal and is isomorphic to the crystal of
the highest weight $U_q(\ensuremath{\mathfrak{g}})$-module with highest weight
$\omega_\mathbf{w}$.
\end{prop}
We now translate this structure to the language of flag varieties.
We need the following results. We adopt the convention that
$B_{1,0}=0$ and $B_{n-1,n}=0$.
\begin{prop}
\label{prop:comm1} We have
\[
B_{k,k-1} \circ \phi_k = \phi_{k-1} \circ x.
\]
\end{prop}
\begin{proof}
Recall that
\begin{align*}
\phi_{k-1} &= \bigoplus_{p \in \mathcal{P},\,
\inc(p) = k-1} B_p i_{\out(p)} \iota_{\out(p)}^{\out(p)-\ord(p)} \\
\Rightarrow \phi_{k-1} \circ x &= \bigoplus_{p \in \mathcal{P},\,
\inc(p) = k-1,\, \ord(p) \ge 1} B_p i_{\out(p)}
\iota_{\out(p)}^{\out(p)-\ord(p)+1}.
\end{align*}
And
\begin{align*}
\phi_k &= \bigoplus_{p \in \mathcal{P},\,
\inc(p) = k} B_p i_{\out(p)} \iota_{\out(p)}^{\out(p)-\ord(p)} \\
\Rightarrow B_{k,k-1} \circ \phi_k &= \bigoplus_{p \in
\mathcal{P},\, \inc(p) = k} B_{k,k-1} B_p i_{\out(p)}
\iota_{\out(p)}^{\out(p)-\ord(p)}.
\end{align*}
Now, since $j=0$, $\mu(B,i,j)=0$ implies that
\begin{gather*}
B_{l-1,l} B_{l,l-1} = B_{l+1,l} B_{l,l+1} \text{ for } 2 \le l \le
n-2, \\
B_{2,1} B_{1,2} = 0, \quad B_{n-2,n-1} B_{n-1,n-2} = 0.
\end{gather*}
Using these equations, one can see that
\[
\{B_{k,k-1} B_p\ |\ p \in \mathcal{P},\, \inc(p)=k\} = \{B_p\ |\ p \in
\mathcal{P},\, \inc(p) = k-1,\, \ord(p) \ge 1\}.
\]
Therefore,
\[
B_{k,k-1} \circ \phi_k = \bigoplus_{p \in \mathcal{P},\, \inc(p)=k-1,\,
\ord(p) \ge 1} B_p i_{\out(p)}
\iota_{\out(p)}^{\out(p)-(\ord(p)-1)} = \phi_{k-1} \circ x.
\]
\end{proof}
\begin{prop}
\label{prop:comm2} We have
\[
B_{k,k+1} \circ \phi_k = \phi_{k+1} |_{W^{\le k}}.
\]
\end{prop}
\begin{proof}
Let $\mathcal{P}'$ be the subset of $\mathcal{P}$ consisting of those paths that
contain at least one edge belonging to $\bar \Omega$. Then
\begin{align*}
B_{k,k+1} \circ \phi_k &= \bigoplus_{p \in \mathcal{P},\,
\inc(p) = k} B_{k,k+1} B_p i_{\out(p)} \iota_{\out(p)}^{\out(p)-\ord(p)} \\
&= \bigoplus_{p \in \mathcal{P}',\,
\inc(p) = k+1} B_p i_{\out(p)} \iota_{\out(p)}^{\out(p)-\ord(p)} \\
&= \phi_{k+1} |_{W^{\le k}}.
\end{align*}
\end{proof}
\begin{prop}
\label{prop:flag-subspace}
One has
\begin{equation}
\label{eq:flag-subspace} \phi_k^{-1} \left( \ker B_{k,k-1} \cap
\ker B_{k,k+1} \right) = x^{-1}(F_{k-1}) \cap F_{k+1}.
\end{equation}
\end{prop}
\begin{proof}
Since $F_{k-1} \subset W^{\le k-1}$, we have that $x^{-1}(F_{k-1})
\subset W^{\le k}$. Thus, using propositions~\ref{prop:comm1} and
\ref{prop:comm2},
\begin{align*}
x^{-1}(F_{k-1}) \cap F_{k+1} &= x^{-1}(F_{k-1}) \cap
\ker(\phi_{k+1}) \\
&= x^{-1}(F_{k-1}) \cap \ker(\phi_{k+1}|_{W^{\le k}}) \\
&= x^{-1}(\ker \phi_{k-1}) \cap \ker(\phi_{k+1}|_{W^{\le k}}) \\
&= \ker (\phi_{k-1}\circ x) \cap \ker(\phi_{k+1}|_{W^{\le k}}) \\
&= \ker (B_{k,k-1} \circ \phi_k) \cap \ker (B_{k,k+1} \circ
\phi_k) \\
&= \phi_k^{-1} \left( \ker B_{k,k-1} \cap \ker B_{k,k+1} \right)
\end{align*}
\end{proof}
Note that
\begin{equation}
\label{eq:stable-subspace} \ker B_{k,k-1} \cap \ker B_{k,k+1} =
\ker \left( V_k \stackrel{(B_h)}{\longrightarrow} \bigoplus_{h\,
:\, \out(h)=k} V_{\inc(h)} \right),
\end{equation}
and that a collection of subspaces $S_l \subset V_l$ such that
$S_l = 0$ for $l \ne k$ is $B$-stable if and only if $S_k$ is
contained in the righthand side of
equation~\eqref{eq:stable-subspace}. Thus, the flag variety
analogue of the diagram~\eqref{eq:crystal-action} (for $\mathbf{v} - \mathbf{v}' =
c\mathbf{e}^k$) is
\[
\mathcal{F}_{\mathbf{a}^{k,c},x} \stackrel{\pi_1}{\longleftarrow} \mathcal{F}_{\mathbf{a},x}(k,c)
\stackrel{\pi_2}{\longrightarrow} \mathcal{F}_{\mathbf{a},x},
\]
where $\mathbf{a} = \mathbf{a}(\mathbf{v},\mathbf{w})= (a_1,\dots,a_n)$ and $\mathbf{a}^{k,c} =
(a_1,\dots,a_{k-1},a_k+c, a_{k+1}-c,a_{k+2},\dots,a_n)$, and
\[
\mathcal{F}_{\mathbf{a},x}(k,c) = \{(F,S)\ |\ F \in \mathcal{F}_{\mathbf{a},x},\, F_k \subset S
\subset F_{k+1} \cap x^{-1}(F_{k-1}),\, \dim S/F_k = c\}.
\]
In particular
\[
\mathcal{L}(\mathbf{v},\mathbf{w};c\mathbf{e}^k) \cong \mathcal{F}_{\mathbf{a},x}(k,c).
\]
Let $\mathcal{B}(\mathbf{a},x)$ denote the set of irreducible components
of $\mathcal{F}_{\mathbf{a},x}$ and let $\mathcal{B}(x) = \sqcup_\mathbf{a} \mathcal{F}_{\mathbf{a},x}$. Let
\[
\varepsilon_k(F) = \dim (F_{k+1} \cap x^{-1}(F_{k-1})) - \dim F_k,
\]
and for $X \in \mathcal{B}(\mathbf{a},x)$ define $\varepsilon_k(X) =
\varepsilon_k(F)$ for a generic flag $F \in X$. Then for $c \in
\mathbb{Z}_{\ge 0}$ define
\[
\mathcal{B}(\mathbf{a},x)_{k,c} = \{X \in \mathcal{B}(\mathbf{a},x)\ |\
\varepsilon_k(X)=c\}.
\]
Then just as for quiver varieties, we have
$\mathcal{B}(\mathbf{a}^{k,c},x)_{k,0} \cong \mathcal{B}(\mathbf{a},x)_{k,c}$ and
we define ${\tilde f}_k$ and ${\tilde e}_k$ just as before. We also define
\begin{gather*}
\wt(X) : \mathcal{B}(x) \to P,\quad \wt(X) = \sum_{k \in I} a_k
\epsilon_k
\text{ for } X \in \mathcal{B}(\mathbf{a},x), \\
\varphi_k(X) = \varepsilon_k(X) + \left<h_k,\wt(X)\right>.
\end{gather*}
Then, by translating Proposition~\ref{prop:quiver-crystal} into
the language of flag varieties, we have the following theorem.
\begin{theo}
$\mathcal{B}(x)$ is a crystal and is isomorphic to the crystal of
the highest weight $U_q(\mathfrak{sl}_n)$-module with highest
weight $w_1 \omega_1 + \dots + w_{n-1} \omega_{n-1}$ where
$\omega_i$ are the fundamental weights of $\mathfrak{sl}_n$ and
$w_i$ is the number of $(i \times i)$-Jordan blocks in the Jordan
normal form of $x$.
\end{theo}
\bibliographystyle{abbrv}
|
{
"timestamp": "2009-09-02T01:55:13",
"yymm": "0411",
"arxiv_id": "math/0411105",
"language": "en",
"url": "https://arxiv.org/abs/math/0411105"
}
|
\section{Experimental technique}
The magnetic spectrometer COSY-11 \cite{brauksiepe:96} at the COoler SYnchrotron COSY \cite{maier:97nim} is shown in its principal layout in figure \ref{cosy11}.
\begin{figure}[h]
\epsfig{file=cosy11text,width=0.65\textwidth}
\caption{\label{cosy11} Schematic view of the COSY-11 setup for an exemplary event in \mbox{$pp\rightarrow\,ppK^+K^-$}, where the kaon decays before reaching the stop scintillator S3. The not shown cluster target is located in front of the left dipole magnet.}
\end{figure}
The reaction takes place in a cluster target mounted in front of one of the ring dipoles and is operated with hydrogen or deuterium. The positively charged ejectiles are bent to the interior of the ring and their four momenta are reconstructed using the information of a set of drift chambers and a subsequent time of flight measurement. The $K^-$ for the \mbox{$pp\rightarrow\,ppK^+K^-$} reaction is then identified via the missing mass method. In case of $pp\to nK^+\Sigma^+$, additionally the neutron is detected in a lead-scintillator arrangement and the unregistered $\Sigma^+$ is again deduced from the missing mass technique. Details on the explicit experimental technique and the analysis can be found elsewhere \cite{brauksiepe:96,moskal:01}.
\section{Hyperon production}
Close-to-threshold production data \cite{balewski:98-2,sewerin:99,kowina:04} in $pp\to pK^+\Lambda/\Sigma^0$ revealed a cross section ratio \mbox{$R=\sigma_{tot}(\Lambda)/\sigma_{tot}(\Sigma^0)\approx28$} exceeding that at high energies by an order of magnitude. Different models \cite{gasparian:99,shyam:04,sibirtsev:97} within the framework of one-boson exchange do not reproduce the energy dependence too well. In order to have further constraints to the theoretical descriptions, the COSY-11 collaboration collected new data in the different isospin channel $pp\to nK^+\Sigma^+$ at $Q=$\,13 and 60\,MeV.\\
\begin{figure}[H]
\epsfig{file=MM_last,width=0.6\textwidth}
\caption{Left: Squared missing mass spectrum for the $nK^+$-system. Monte Carlo simulations for the main background channels $pp\to pK^+\Lambda$ and $pp\to pK^+\Lambda\gamma$ are shown by the dotted curves. Right: Subtraction of the background Monte Carlo spectra from the data.\label{missmass-nK}}
\end{figure}
For now, a missing mass spectrum for the lower excess energy with respect to the $nK^+$-system was elaborated \cite{rozek:04} and is presented in figure \ref{missmass-nK} left side. The dominant background channels were identified to be $pp\to pK^+\Lambda$ and $pp\to pK^+\Lambda\gamma$. This background was subtracted from the data (right plot in fig. \ref{missmass-nK}). Therefore, the known cross section for $pp\to pK^+\Lambda$ was incorporated and then the $pp\to pK^+\Lambda\gamma$ channel was fitted such that the experimental missing mass spectrum below $1.3\,$GeV$^2$/c$^4$ was reproduced. Although a peak around the $\Sigma^+$ mass is seen, a lot more studies have to be performed before conclusive results on the cross section will be drawn.
\section{Elementary kaon production}
Exclusive data on the $pp\to ppK^+K^-$ reaction have been taken at COSY-11 at several excess energies motivated by the ongoing discussion about the nature of the scalar resonances $a_0$ and $f_0$. Besides the interpretation as a $q\bar{q}$ state \cite{morgan:93}, these resonances are also proposed to be $qq\bar{q}\bar{q}$ states \cite{jaffe:77}, $K\bar{K}$ molecules \cite{weinstein:90,lohse:90}, a hybrid $q\bar{q}$/meson-meson system \cite{beveren:86} or even quarkless gluonic hadrons \cite{jaffe:75}. In addition, final state interaction effects occur -- if existent -- strongly at low excess energies. Therefore, the elementary production process is a good tool to learn more about a possible $pK^-$ FSI or additional degrees of freedom in this four body final state.\\
A first total cross section $\sigma=1.80\pm0.27^{+0.28}_{-0.35}$\,nb for the excess energy $Q=17\,$MeV was determined in a former measurment at COSY-11 \cite{quentmeier:01-2} while further data at two other $Q$-values of 10 and 28\,MeV were taken. After the identification of two protons and a $K^+$, a clear signal in the missing mass spectrum at the kaon mass is observed for both energies (as an example at $Q=28\,$MeV see figure \ref{missmass}). Here, a hit in a scintillator mounted inside the dipole gap, where the $K^-$ is going to, was required.
\begin{figure}[h]
\rotatebox{-90}{\epsfig{file=kpkm,scale=0.5}}
\caption{\label{missmass}Left: Missing mass of the $ppK^+$-system with an additional demand for a hit in the dipole scintillator. Right: Excitation function of the total cross section for \mbox{$pp\rightarrow\,ppK^+K^-$}. New data at $Q=10$ and $28\,$MeV are preliminary and include only statistical errors so far.}
\end{figure}
There is no doubt that the background will be described by the hyperon channels $pp\to pK^+\Lambda(1405) / \Sigma(1385)$ and some misidentified pions like it was shown in the case of the data at $Q=17\,$MeV. Therefore, the full analysis (for details see \cite{quentmeier:01-2}) will most likely give also for the new data sets a background free missing mass spectrum. Within the near future, the final cross sections will be extracted. This however requires much deeper studies of the detection efficiency.
\bibliographystyle{aipproc}
|
{
"timestamp": "2004-11-20T15:58:11",
"yymm": "0411",
"arxiv_id": "hep-ex/0411064",
"language": "en",
"url": "https://arxiv.org/abs/hep-ex/0411064"
}
|
\section{ Introduction}
Group classification of differential equations is one of the
central problems of group analysis. It specifies non-equivalent
classes of equations and open the way to applications of symmetry
tools such as constructing and group generation of exact
solutions, separation of variables, etc. One of the goals of group
classification is a priori description of mathematical models with
a desired symmetry (e.q., relativistic invariance).
The first (and very impressive) achievements in group
classification belong to S. Lie who who had classified second
order ordinary differential equations and specified all cases when
such equations can be integrated in quadratures \cite{lie1}. Lie
had presented also a group classification of an entire class of
partial differential equations, namely, linear equations
with two independent variables. In particular, it was Lie who for
the first time describes group properties of the linear heat
equation \cite{lie2}.
The next step in classification of heat equations was made by
Dorodnitsyn \cite{OV} who had classified nonlinear diffusion equations
\be u_t -u_{xx}=f(u)\label{d}\ee where $f$ is a function of
$u=u(t,x)$ and subscripts denote derivations w.r.t. the
corresponding variables \footnote{In paper \cite{OV} a more general equations with nonlinear diffusion were
classified which include (\ref{d}) as a particular case}
. This result was extended by Fushchych ,
Serov~ \cite{FU} and Clarkson and Mansfield~ \cite{CL} to the case
of non-classical (conditional) symmetries.
The results of group classification of equations (\ref{d}) play an
important role in constructing of their exact solutions and
qualitative analysis of the nonlinear heat equation, refer, e.g.
to \cite{sam}.
In the present paper we perform the group
classification of systems of the nonlinear reaction-diffusion
equations \be \label{1.1}\ba{l}
\displaystyle u_t-\Delta(a u- v)=f^1(u,v),\\
\displaystyle v_ t-\Delta( u+a v)=f^2(u,v) \ea \ee
where $u$ and $v$ are function of $t, x_1, x_2, \ldots , x_m$,
$a$ is a real constant
and $\Delta$ is the Laplace operator in $R^m$. We shall write
(\ref{1.1}) also in the matrix form:
\be \label{1.2} U_t- A\Delta U=f(U) \ee where $A$ is a matrix whose elements are
$A^{11}=A^{22}=a, \ A^{12}=-A^{21}=b$, $U=\left(\ba{l}u\\v\ea\ro$ and $f=\left(\ba{l} f^1 \\
f^2\ea \right)$.
Mathematical models based on equations (\ref{1.1}) are widely used
in mathematical physics, biology, chemistry, etc. Here
we present only two significant examples.
\begin{itemize}
\item The nonlinear Schr{\"{o}}dinger (NS) equation in
$m-$dimensional space:
\begin{equation}
\left( i\partial_ t+\Delta\right) \psi =F(\psi
,\psi ^{*}) \label{d6}
\end{equation}
is a particular case of (\ref{1.1}). If we denote $ \psi=u+iv$,
$F=f_1+if_2$ then (\ref{d6}) reduces to the form (\ref{1.2}) with
$A=\left(\ba{rr}0&-1\\1&0\ea\ro$.
Equations (\ref{d6}) with various nonlinearities $F$ are used in
nonlinear optics, non-linear quantum mechanics \cite{doebner},
they serve as one of basic models of inverse scattering problem
\cite{faddeev}. The most popular models are connected with the
following nonlinearities \cite{FU1}:
\[
F=F(\psi ^{*}\psi )\psi,\quad F=(\psi ^{*}\psi )^k\psi,\qquad
F=(\psi ^{*}\psi )^{\frac 2m}\psi,\quad F=\ln (\psi ^{*}\psi )\psi
\] One more interesting particular case of the NS equation corresponds to
\[\left( i{\partial_ t}+\Delta\right) \psi =(\psi-\psi^*)^2;\]
in this case (\ref{d6}) is a potential equation for the Boussinesq
equation for function $V={\partial_ t}(\psi-\psi^*)$.
Group classification of the NS equation has been performed in paper \cite{pop}.
\item Generalized complex Ginzburg-Landau (CGL) equation
\be W_\tau-(1+i\beta)\Delta W=F(W,W^*)\label{la}\ee
also can be treated as a particular case of system (\ref{1.2}).
Indeed, representing $W$ and $F$ as $W=(u+iv), F=\beta(f^1+if^2)$
and changing independent variable $\tau \to t=\beta\tau$ we
transform (\ref{la}) to the form (\ref{1.2}) with
$A=\left(\ba{ll}\beta^{-1}&-1\\1&\beta^{-1}\ea\ro$. The standard CGL
equation corresponds to the case $F=W-(1+i\alpha)W|W|^2.$
\end{itemize}
Thus the symmetry analysis of equations (\ref{1.1}) has a large
application value and can be used, e.g., to construct exact
solutions for a very extended class of physical and biological
systems. The comprehensive group analysis of systems (\ref{1.1})
is also a nice "internal" problem of the Lie theory which admits
exact general solution for the case of {\it arbitrary} number of
independent variables $x_1, x_2, \ldots , x_m$.
We notice that group classification of equations (\ref{1.1}) by no
means is a standard problem of group analysis of partial differential equations
which can be
solved with direct application of well-known algorithms. Because
of presence of two arbitrary elements, i.e., $f^1$ and $f^2$, this
classification needs a rather nontrivial generalization of the
approach \cite{OV} used for classification of equation (\ref{d}).
Equations (\ref{1.2}) with arbitrary invertible matrix $A$ were
classified in paper \cite{nikwil1}. To our great a pity, mainly due
to typographical errors (made during the editing procedure),
presentation of results in \cite{nikwil1} was not satisfactory
\footnote{The tables presenting the results of group classification
have been deformed and cut off. It is necessary to stress that it
was the authors fault, one of whom signed the paper proofs without
careful reading.}.
The present paper is the first from the series in which we present
the completed group classification of coupled reaction-diffusion
equations (\ref{1.2}) with {\it arbitrary} (i.e., invertible or singular) matrix $A$. Moreover,
we present a straightforward and easily verified procedure of
solution of the determining equations which guarantees the
completeness of the obtained results.
We also indicate clearly the equivalence relations used in the
classification procedure, i.e., present explicitly the equivalence
groups for for all classified equations. In addition, we extend
the results obtained in \cite{nikwil1} to the important case of
{\it non-invertible} matrix $A$ and more general equations
including both the first and second order derivatives with respect to spatial variables.
Let us note that there are three ad hoc non-equivalent classes of
equations (\ref{1.2}) corresponding to the following forms of
matrices $A$
\be\label{2.1} \ba{l} I. \quad A=\left(\ba{cc} a & -1 \\ 1 & a \ea
\right);\quad II.\quad A=\left(\ba{cc} 1 & 0 \\ 0 & a \ea
\right); \quad \ III. \quad A=\left(\ba{cc} a & 0 \\ 1 & a \ea
\right)\ea \ee where $a$ is an arbitrary parameter. Moreover, any
$ 2\times 2$ matrix $A$ can be reduced to one of the forms
(\ref{2.1}) using linear transformations of dependent variables
and scaling independent variables in (\ref{1.2}).
The NS and CGL equations correspond to matrices $A$ of form $I$.
The general equations (\ref{1.2}) with such matrices (i.e.,
generalized CGL equations) are the main subject of group
classification carried out in the present paper while the cases
$II$ and $III$ will be considered in the following publications.
Nevertheless till an appropriate
moment we will consider equations with all types of matrices $A$
enumerated in (\ref{2.1}).
\section{Determining equations and equivalence transformations}
In the first stage we restrict ourselves to group classification
of equations (\ref{1.2}) with invertible matrix $A$. Moreover, till an appropriate moment
we consider
equations (\ref{1.2}) with arbitrary number $n$ of dependent variables.
Using the
standard Lie algorithm \cite{olver} (or its specific version
proposed in \cite{nikwil1}) one can find determining equations for
the functions $\eta ,\xi^{a}$ and $\pi^{a}$ which specify
generator $X$ of the symmetry group admitted by equation
(\ref{1.2}):
\begin{equation}
X=\eta {\frac{\partial }{\partial t}}+\xi ^{\nu}{\frac{\partial
}{\partial x_{\nu}}}-\pi^{b}{\frac{\partial }{\partial
u_{b}}}\equiv \eta\partial_t+\xi^\nu\partial_{x_\nu}-\pi^b\partial_{u_b} \label{3.105}
\end{equation}
where a summation from $1$ to $m$ and from $1$ to $2$ is assumed
over repeated indices $\nu$ and $b$ respectively, and a temporary
notation $u=u_1, v=u_2$ is used. In a more general case of $n$ dependent variables
$U=column(u_1,u_2,\cdots,u_n)$ the repeated indices $b$ run over the values $1,2,\cdots, n$.
We shall not reproduce the deduction of the determining equations
here (refer to \cite{nikwil1}) but present them directly.
Dependence of $\eta, \xi^\nu$ and $\pi^b$ on $U$ is defined by the following relations:
\begin{equation}
\eta_{ u_{a}}=0,\ \xi ^{\nu}_{u_{b}}=0,\ \ \pi ^{a}_{u_{c}u_{b}}=0. \label{3.4a}
\end{equation}
So from (\ref{3.4a}) $\eta $ and $\xi ^{\nu}$ are functions of $t$ and $x_{\mu}$
and, $\pi ^{\nu}$ is linear in $u_{a}$. Thus:
\begin{equation}
\pi ^{a}=N^{ab}u_{b}+B ^{a} \label{3.5a}
\end{equation}
where $N^{ab}, B^a$ are functions of $t $ and $x_{\nu}$ only.
The remaining equations are \cite{nikwil1}:
\begin{equation}
2A\xi _{x_\mu}^{\nu}=-\delta ^{\mu\nu}(\eta_t A+[A,N ]),\qquad {\eta}
_{x_\nu t}=0, \label{4.2a}
\end{equation}
\begin{equation}\ba{l}
{\xi}^\nu_t-2AN_{x_\nu}-A\Delta\xi^{\nu}=0, \\
\eta_tf^k+N^{kb}f^{b}+(N^{kb}_t-\Delta A^{ks}N^{sb})u_b
+B^k_t-\Delta A^{kc}B^c
=(B ^{a}+N^{ab}u_{b})f^{k}_{u_a}.\ea
\label{3.8a}
\end{equation}
Here $N$ and $A$ are matrices whose elements are $N^{ab}$ and $A^{ab}$, $\delta^{ab}$ is the
Kronecker symbol.
In accordance with (\ref{3.5a})--(\ref{3.8a})
the general form of the related generator
(\ref{3.105}) is \cite{nikwil1}: \be \label{2.4} \ba{l}
X=\lambda K+\sigma_\mu G_\mu+\omega_\mu \hat G_\mu+\mu
D-(C^{ab}u_b+B^a){\partial_{ u_a}}\\
+\Psi^{\mu \nu} x_\mu \partial_{x_\nu}+\nu \partial_t+\rho_\mu\partial_{x_\mu} \ea
\ee where the Greek letters denote arbitrary constants, $B^a$ are
functions of $t,x$, and $C^{ab}$ are functions of $t$ satisfying
\be \label{2.5} C^{ab}A^{bk}-A^{ab}C^{bk}=0 \ee and \be
\label{2.6} \ba{l} K=2t(t\partial_t+x_\mu
\partial_{x_\mu})-\frac{x^2}{2}(A^{-1})^{ab}
u_b\partial_{u_a}-tmu_a\partial_{u_a}
,\\
G_\mu=t\partial_{x_\mu}+\frac{1}{2}x_\mu(A^{-1})^{ab}u_b\partial_{u_a},\\
\hat G_\mu=e^{\gamma t}\left(\partial_{x_\mu}+\frac{1}{2}\gamma
x_\mu(A^{-1})^{ab}
u_b {\partial}_{u_a}\ro,\\
D=t\partial_t+\frac12 x_\mu \partial_{x_\mu}. \ea \ee
Here $A^{ab}$ and $(A^{-1})^{ab}$ are elements of matrix $A$ and
matrix inverse to $A$ respectively.
In accordance with (\ref{3.8a}) equation (\ref{1.2}) admits symmetry operator (\ref{2.4})
iff the following classifying equations for $f^1$ and
$f^2$ are satisfied:
\be \label{2.7} \ba{l} (\lambda t(m+4)
+\mu)f^a+\left(\frac{\lambda}{2} x^2+\sigma_\mu x_\mu+\gamma e^{\gamma t}
\omega_\mu x_\mu\right) (A^{-1})^{ab}f^b\\
+C^{ab}f^b+C^{ab}_t u_b +B_t^a-\Delta A^{ab}B^b\\
=(B^s+C^{sb} u_b+\lambda mtu_s +\left(\frac{\lambda}2 x^2+\sigma_\mu x_\mu+\gamma e^{\gamma t}\omega_\mu
x_\mu\right) (A^{-1})^{sk}u_k) f^a_{u_s}. \ea \ee
Thus the group classification of equations (\ref{1.2}) with a
non-singular matrix $A$ reduces to solving equation (\ref{2.7})
where $\lambda, \mu, \sigma_\mu, \omega_\mu, \gamma $ are arbitrary
parameters, $B^a$ and $C^{ab}$ are functions of $(t,x)$ and $t$
respectively. Moreover, matrix $C$ with elements $C^{ab}$ should
commute with $A$.
We notice that relations (\ref{2.4})-(\ref{2.7}) are valid for
group classification of systems (\ref{1.2}) of coupled
reaction-diffusion equations including {\it arbitrary} number $n$
of dependent variables $U=(u_1, u_2, \ldots u_n)$ provided the
related $n \times n$ matrix $A$ be invertible \cite{nikwil1}. In
this case indices $a, b, s, k$ in (\ref{2.4})-(\ref{2.7}) run over
the values $1,2 \ldots n$.
We will solve classifying equations (\ref{2.7}) up to equivalence
transformations $ U \to \tilde U=G(U,t,x)$, $t \to \tilde
t=T(U,t,x)$, $x \to \tilde x=X(U,t,x)$ and $ f \to \tilde
f=F(U,t,x,f)$ which keep the general form of equations (\ref{1.2})
but can change functions $f^1$ and $f^2$. The group of equivalence
transformations for equation (\ref{1.2}) can be found using the
classical Lie approach and treating $f^1$ and $f^2$ as additional
dependent variables. In addition to the obvious symmetry
transformations \be \label{2.2} t \to t'=t+a, \quad x_\mu \to
x'_\mu=R_{\mu \nu} x_\nu +b_\mu \ee where $a, b_\mu$ and $R_{\mu
\nu}$ are arbitrary parameters satisfying $R_{\mu \nu} R_{\mu
\lambda}=\delta_{\mu\lambda}$, this group includes the following
transformations \be \label{x.2} \ba{l}
u_a\to K^{ab}u_b+b_a, \quad f^a \to \lambda^2 K^{ab}f^b,\\
t \to \lambda^{-2} t, \quad x_a \to \lambda^{-1}x_a \ea \ee where
$K^{ab}$ are elements of an invertible constant matrix $K$
commuting with $A$, $\lambda \not=0$ and $b_a$ are arbitrary
constants.
For the case when $n=2$ and matrix $A$ belongs to type $I$ (\ref{2.1}) the
transformation matrix has the following form
\be \label{x.5}
K=\left(\ba{cc} K_{1} & - K_{2}\cr K_{2} & K_{1}\ea
\right), \quad K^2_1 + K^2_2 \not = 0. \ee
It is possible to show that there is no more extended equivalence
relations valid for arbitrary nonlinearities $f^1$ and $f^2$.
However, if functions $f^1, f^2$ are specified, the invariance
group can be more extended. In addition to transformations
(\ref{x.2}) it includes symmetry transformations generated by
infinitesimal operator (\ref{2.4}), and can include additional
equivalence transformations (AET). We will specify AET in the
following.
\section{Basic, main and extended symmetries}
Thus to describe Lie symmetries of equation (\ref{1.2}) (whose
generators have the general form (\ref{2.4})) it is necessary to
find all non-linearities $f^1$ and $f^2$ which satisfy the
corresponding classifying equations (\ref{2.7}). To solve these
rather complicated equations we use the main algebraic property of
the related symmetries, i.e., the fact that they should form a Lie
algebra. In other words, instead of going throw all non-equivalent
possibilities arising via separation of variables in the
classifying equations we first specify all non-equivalent
realizations of the invariance algebra for our equations. Then
using the one-to-one correspondence between these algebras and
classifying equations (\ref{2.7}) we easily solve the group
classification problems for equations (\ref{1.2}).
Equation (\ref{2.7}) does not include parameters $\Psi^{\mu \nu},
\nu$ and $\rho_\nu$ present in (\ref{2.4}) thus for any $f^1$ and
$f^2$ equation (\ref{1.2}) admits symmetries generated by the
following operators \be \label{4.1} P_0=\partial_t, \quad
P_\lambda=\partial_{x_\lambda}, \quad J_{\mu \nu}=x_\mu\partial_{x_\nu}-x_\nu
\partial_{x_\mu}. \ee
Infinitesimal operators (\ref{4.1}) generate the evident symmetry
transformations (\ref{2.2}) which form the kernel of invariance
groups of equation (\ref{1.2}). For some classes of nonlinearities
$f^1$ and $f^2$ the invariance algebra of equation (\ref{1.2}) is
more extended but includes (\ref{4.1}) as a subalgebra. We will
refer to (\ref{4.1}) as to {\it basic symmetries}.
Let us specify one more subclass of symmetries of equation
(\ref{1.2}) which we call {\it main symmetries}. The related
generator $\tilde X$ has the form (\ref{2.4}) with
$\Psi^{\mu\nu}=\nu=\rho_\nu=\sigma_\nu=\omega_\nu=0$, i.e., \be
\label{4.2} \tilde X=\mu D+C^{ab}u_b \partial_{
u_a}+B^a\partial_{ u_a}. \ee
The classifying equation for symmetries
(\ref{4.2}) can be obtained from (\ref{2.7}) by setting
$\mu=\sigma^a=\omega^a=0$. As a result we obtain \be \label{4.3} (\mu
\delta^{ab}+C^{ab})f^b+C^{ab}_t u_b+B^a_t-\Delta A^{ab}B^b=
(C^{nb}u_b+B^n)f^a_{u_n}. \ee
It is easily verified that operators (\ref{4.2}) and (\ref{4.1})
form a Lie algebra which is a subalgebra of symmetries for
equation (\ref{1.2}) (this algebra can be either finite of
infinite dimensional).
On the other hand, if equation (\ref{1.2}) admits a more general
symmetry (\ref{2.4}) with $\sigma_a\neq 0$ or (and) $\lambda\neq 0,\
\omega^\mu\neq 0$ then it has to admit symmetry (\ref{4.2}) also. To
prove this we calculate multiple commutators of (\ref{2.4}) with the
basic symmetries (\ref{4.1}) and use the fact that such commutators
have to belong to symmetries of equation (\ref{1.2}), i.e., generate
their own classifying equation (\ref{2.7}).
Let equation (\ref{1.2}) admits symmetry (\ref{2.4}) with
$\sigma_\mu\not=0, \Psi^{\mu\nu}=\rho_\mu=\nu=\lambda=\omega^k=0$,
i.e., \be \label{4.4} X=\sigma_\nu G_\nu+\mu
D+(C^{ab}u_b+B^a)\partial_{ u_b}. \ee
Commuting $Y$ with $P_\mu$ we obtain one more symmetry \be
\label{4.5} Y_\mu=-\frac{\sigma_\mu}{2}(A^{-1})^{ab} u_b
\partial_{ u_a}+B^a_\mu \partial_{ u_a}+\mu P_\mu. \ee
The last term belongs to the basic symmetry algebra (\ref{4.1})
and so can be omitted. The remaining terms are of the type
(\ref{4.2}).
Thus supposing the extended symmetry (\ref{4.4}) is admissible we
conclude that equation (\ref{1.2}) has to admit the main symmetry
also.
Commuting (\ref{4.5}) with $P_0$ and $P_\lambda$ we come to the
following symmetries: \be \label{4.6}
Y_{\mu\nu}=B^a_{\mu\nu}\partial_{ u_a}, \ Y_{\mu t}=B^a_{\mu
t}\partial_{ u_a}. \ee
Any symmetry (\ref{4.4})-(\ref{4.6}) generates this own system
(\ref{2.7}) of classifying equations. After strait forward but
rather cumbersome calculations we conclude that all these systems
are compatible provided the following condition is satisfied \be
\label{4.8} (A^{-1})^{ab}f^b=(A^{-1})^{nb}u_b f^a_{ u_n}. \ee
If (\ref{4.8}) is satisfied equation (\ref{1.2}) admits symmetry
(\ref{4.4}) with $\mu=C^{ab}=B^a=0$, i.e., Galilei generators
$G_\nu$ of (\ref{2.6}).
Analogously, supposing that equation (\ref{1.2}) admits
extended symmetry (\ref{2.4}) with $\lambda \not=0$
and $\omega^a=0$ we conclude that it has to admit symmetry
(\ref{4.4}) with $\mu\not=0$ and $\sigma_\nu\not=0$ also. The related
functions $f^1$ and $f^2$ should satisfy relations (\ref{4.8}) and
(\ref{4.3}). Moreover, analyzing possible dependence of $C^{ab}$ and $B^a$
in the corresponding relations (\ref{2.7}) on $t$ we conclude that they
should be ether scalars or linear in $t$, i.e.,
$ C^{ab}=\mu^{ab}t+\nu^{ab}$.
Moreover, up to equivalence transformations (\ref{x.2}) we can choose $ B^a=0$
and reduce the related equation system
(\ref{2.7}), (\ref{4.3}) to the following equations:
\be\label{!!}\ba{l}(m+4)f^a+\mu^{ab}f^b=(\mu^{kb}u_b+mu_k)\frac{\partial f^a}{\partial u_k},
\\\nu^{ab}f^b+\mu^{ab}u_b=\nu^{kb}u_b\frac{\partial f^a}{\partial u_k}\ea\ee
where constants $\nu^{ab}$ and $\mu^{ab}$ are non-trivial in the case
of the diagonal diffusion matrix only.
Finally for general symmetry (\ref{2.4}) it is not difficult to
show that the condition $\omega^a\not=0$ leads to the following
equation for $f^a$ \be \label{4.10} (A^{-1})^{kb}(f^b+\gamma
u^b)=(A^{-1})^{ab}u_b f^k_{u_a}. \ee
We notice that relations (\ref{4.8}) and (\ref{4.10}) are
particular cases of (\ref{4.3}) for $ \mu=0 $,
$C^{ab}=(A^{-1})^{ab}$ and $ \mu=0 $, $C^{ab}=e^{\gamma
t}(A^{-1})^{ab}$ respectively. Thus if relation (\ref{4.8}) is
valid then, in addition to $G_\mu$ (\ref{2.6}) equation
(\ref{1.2}) admits the symmetry \be \label{4.11}
X=(A^{-1})^{ab}u_b \partial_{ u_a}. \ee
Alternatively, if (\ref{4.10}) is satisfied, equation (\ref{1.1})
admits symmetry $\hat G_\mu$ (2.6) and also the following one \be
\label{4.12} X=e^{\gamma t}(A^{-1})^{ab}u_b \partial_{ u_a}, \quad
\gamma \not =0 .\ee
Thus it is reasonable first to classify equations (\ref{1.2})
which admit main symmetries (\ref{4.2}) and then specify all cases
when these symmetries can be extended.
The conditions when system (\ref{1.2}) admits extended symmetries
are given by relations (\ref{4.8})-(\ref{4.10}).
Results of Section 3 are valid for equations (\ref{1.2}) with
arbitrary invertible diffusion matrix. In the following we
restrict ourselves to the case when $n=2$ and matrix $A$ has the form $I$
given in (\ref{2.1}), i.e., to the case of generalized CLG
equations. In other words, we will classify the following systems
of coupled reaction-diffusion equations:
\be
\label{LG}\ba{l}
\displaystyle u_t-\Delta(a u- v)=f^1(u,v),\\
\displaystyle v_ t-\Delta(a v +u)=f^2(u,v) \ea \ee which are
particular cases of general systems (\ref{1.2}) when matrix $A$ has the form $I$ (\ref{2.1}).
\section{ Algebras of main symmetries}
In accordance with the plane outlined above we start with
investigation of main symmetries (\ref{4.2}) admitted by equation
(\ref{LG}).
The first step of our analysis is to describe non-equivalent Lie
algebras $\cal A$ of operators (\ref{4.2}) which can be admitted
by this equation. We shall consider consequently one-, two-,...,
n-dimensional algebras $\cal A$.
For any type of matrix $A$ enumerated in (\ref{2.1}) we specify
all non-equivalent "tails" of operators (\ref{4.2}), i.e., the
terms
\be \label{8.1}
N=C^{ab}u_b\partial_{ u_a} + B^a \partial_{ u_a}. \ee
These terms can either be a constituent part of a more general
symmetry (\ref{4.2}) or represent a particular case of (\ref{4.2})
corresponding to $\mu=0$. Thus the problem of classification of
algebras $\cal A$ includes a subproblem of classification of
algebras of operators (\ref{8.1}).
Let equation (\ref{1.2}) admits a one-dimensional invariance
algebra whose basis element have the form (\ref{8.1}), and does
not admit a more extended algebra of the main symmetries. Then
commutators of $N$ with the basic symmetries $P_0$ and $P_a$
should be equal to a linear combination of $N$ and operators
(\ref{4.1}). It is easily verified that there are three
possibilities: \be\ba{l} \label{8.4} 1.\ C^{ab}=\mu^{ab}, \quad
B^a=\mu^a,\\ 2.\ C^{ab}=e^{\lambda t} \mu^{ab}, \quad B^a=e^{\lambda t}
\mu^a, \\ 3.\ C^{ab}=0, \quad B^a= e^{\lambda t+\omega \cdot x} \mu^a
\ea\ee where $\mu^{ab}, \mu^a, \lambda $, and $\omega$ are constants,
and matrix with elements $\mu^{ab}$ should commute with $A$. In
the case when $A$ is of the form (\ref{matrix})
constants $\mu^{ab}$ are restricted by the following relations:
$\mu^{11}=\mu^{22},\ \mu^{12}=-\mu^{21}$.
To specify all non-equivalent operators (\ref{8.1}), (\ref{8.4})
we use the isomorphism of (\ref{8.1}) with $3 \times 3$ matrices
of the following form
\be
\label{8.5} g=\left( \ba{ccc}
0 & 0 & 0\\
B^1 & C^{11} & C^{12}\\
B^2 & C^{21} & C^{12} \ea \right) \sim \left( \ba{ccc}
0 & 0 & 0\\
\mu^1 & \mu^{11} & \mu^{12}\\
\mu^2 & -\mu^{12} & \mu^{11} \ea \right). \ee
Equations (\ref{LG}) admit equivalence transformations (\ref{x.2})
which change the term $N$ (\ref{8.1}) and can be used to
simplify it. The corresponding transformation for matrix
(\ref{8.5}) can be represented as \be \label{8.6} g \to g'=U g
U^{-1} \ee where $U$ is a $3 \times 3$ matrix of the following
special form \be \label{8.7} U=\left( \ba{ccc}
1 & 0 & 0\\
b^1 & K^{1} & K^{2}\\
b^2 & -K^{2} & K^{1} \ea \right). \ee
Up to equivalence transformations (\ref{8.6}), (\ref{8.7}) there
exist three matrices $g$, namely \be\label{8.28}
g_1=\left(\ba{ccc}0&0&0\\0&1&0\\0&0&1\ea\ro,\
g_2=\left(\ba{ccc}0&0&0\\1&0&0\\0&0&0\ea\ro,\
g_3=\left(\ba{ccc}0&0&0\\0&\alpha&-1\\0&1&\alpha\ea\ro.\ee
In accordance with (\ref{8.1}), (\ref{8.4}) the related symmetry
operator can be represented in one of the following forms \be
\label{8.11}\ba{l} X_1=\mu D-2(g_a)_{bc}\tilde u_c \partial_{
u_b},\ X_2=e^{\lambda t}(g_a)_{bc}\tilde u_c \partial_{ u_b} \ea
\ee or \be \label{8.12} X_3=e^{\lambda t+\omega \cdot x} \partial_{
u_2}. \ee Here $(g_a)_{bc}$ are elements of a chosen matrix
(\ref{8.28}), $b,c$ =0, 1, 2, $\tilde u=$ column $(1, u, v)$.
Formulae (\ref{8.11}) and (\ref{8.12}) give the principal
description of one-dimension algebras $A$ for equation (\ref{LG}).
To describe two-dimension algebras $\cal A$ we classify matrices
$g$ (\ref{8.5}) forming two-dimension Lie algebras. Choosing one
of the basis elements in the forms given in (\ref{8.28}) and the
other element in the general form (\ref{8.5}) we find that up to
equivalence transformations (\ref{8.6}) there exist three
two-dimension algebras of matrices $g$ \be\label{8.29}
A_{2,1}=\{g_1,g_3\}, \ A_{2,2}=\{g_2, g_4\},\ A_{2,3}=\{g_1
,g_2\}\ee where $g_1, g_2, g_3$ are matrices (\ref{8.28}), and
$$g_4=\left( \ba{ccc}
0 & 0 & 0\\
0 & 0 & 0 \\
1 & 0 & 0 \ea \right). $$
Algebras $A_{2,1}$ and $A_{2,2}$ are Abelian while the basis
elements of $A_{2,3}$ satisfy $[g_1, g_2]= g_2$.
Using (\ref{8.29}) we easily find pairs of operators (\ref{4.2})
forming two-dimension Lie algebras. Denoting
\[
\hat e_\alpha=(e_\alpha)_{ab} {\tilde u}_b \partial_{ u_a}, \quad
\alpha=1,2
\]
we represent them as follows: \be \label{8.17}\ba{l} < \mu D+ \hat
e_1+\nu t \hat e_2, \hat e_2>,\ \ \ < \mu D+ \hat e_2+\nu t \hat
e_1, \hat e_1>,\\<\mu D-\hat e_1,
\nu D-\hat e_2>,\ \ \ <F_1 \hat e_1+ G_1 \hat e_2,\ F_2\hat e_1+G_2
\hat e_2>\ea \ee for $e_1, e_2$ belonging to algebras $A_{2,1}$,
$A_{2,2}$, and \be \label{8.18} <\mu D- \hat e_1, \hat e_2>, \ \ \
< \mu D+ \hat e_1+\nu t \hat e_2, \hat e_2>\ee for
$e_1$ and $e_2$ belonging to $A_{2,3}$.
Here $\{F_1, G_1\}$ and $\{F_2,G_2\}$ are fundamental solutions of
the following system \be \label{8.20} F_t=\lambda F+\nu G, \quad G_t
= \sigma F+\gamma G \ee with arbitrary parameters $\lambda, \nu, \sigma,
\gamma$.
The list (\ref{8.17})-(\ref{8.18}) does not includes algebras of
the types \be\label{ag}< F\hat e,\ G\hat e>\ \ \text{and}\ \
<\mu D+e^{\nu t +\omega \cdot x}\hat e,\ \
e^{\nu t +\omega \cdot x}\hat e>\ee
(with $F,\ G$ satisfying (\ref{8.20}))
which are either incompatible with classifying equations (\ref{2.7}) or reduce to
one-dimension algebras. All the other two-dimension algebras $\cal A$
can be reduced to one the form given in (\ref{8.17}), (\ref{8.18})
using equivalence transformations (\ref{x.2}).
In analogous way we can find three- and four-dimensional algebras
of operators (\ref{4.2}).
Thus we have two three-dimension algebras of matrices (\ref{8.5})
\be\ba{c}
A_{3,1}:\ \ e_1=g_1,\ e_2=g_2,\ e_3= g_4;\\
A_{3,2}:\ \ e_1=g_2,\ e_2=g_3,\ e_3=g_4\ea\label{n1}\ee
and the only four-dimension algebra:
\be A_{4,1}:\ \ e_1=g_1,\ e_2=g_3,\ e_3=\tilde g_4,\
e_4=g_2.\label{n2}\ee
Algebras (\ref{n1}) can be generalized to the following algebras $\cal A$
\be\label{8.24}\ba{l}<\mu D-2\hat e_1,\ \hat e_2,\ \hat e_3>,
\
<D+2\hat e_1+2\nu t\hat e_2,\ \hat e_2,\ \hat e_3>,\\
<D+2\hat e_1+2\nu t\hat e_3,\ \hat e_3,\ \hat e_2>,\ <\hat e_1,\
\ F_1 \hat e_2+G_1 \hat e_3,\ F_2\hat e_2+G_2 \hat e_3>\ea\ee
while (\ref{n1}) generate the following algebras:
\be\label{8.23}\ba{l}<\mu D-2\hat e_1,\ \hat e_2, \hat
e_3>, \
<\hat e_1, \ D+2\hat e_2+2\mu t \hat e_3,\ \hat e_3>.\ea\ee
Finally, $A_{4,2}$
generates the following algebras $\cal A$
\be\ba{l}
<\mu D-2\hat e_1, \ \nu D-2\hat e_2, \ \hat e_3, \ \hat e_4>,\ \
<\hat e_1, \ \hat e_2, \ \hat e^{\mu t+\nu \cdot x}e_3,\ \hat
e^{\mu t+\nu \cdot x}e_4>.\ea\label{n3}\ee
The list (\ref{8.24})-(\ref{n3}) does not include algebras which
have subalgebras (\ref{ag}) (which are incompatible with classifying equations (\ref{4.3})).
\section{Classification results}
Using results presented in previous section we easily perform
group classification of equations (\ref{LG}). To do it we solve
the classifying equations (\ref{4.3}) with their known
coefficients $C^{ab}$ and $B^a$ which are defined comparing
(\ref{4.2}) with the found realizations of algebras $\cal A$.
If equation (\ref{LG}) admits one-dimensional algebra $\cal A$, i.e.,
one of algebras (\ref{8.11}), (\ref{8.12}) the related functions $f^1$ and $f^2$
have to satisfy the corresponding determining equation (\ref{4.3} ) which
define $f^1$ and $f^2$ up to arbitrary
functions. In the case of two dimension algebras whose
realizations are given by relations (\ref{8.17}), (\ref{8.18}) we
have a system of two determining equations corresponding to
two basis elements which usually define $f^1$ and $f^2$ up to
arbitrary parameters. In addition, we control the cases when
equation (\ref{LG}) admit extending symmetries $G_\mu$, $\widehat
G_\mu$ and $K$ (\ref{2.6}), i.e., when the found functions $f^1$
and $f^2$ satisfy conditions (\ref{4.8}), (\ref{4.10}) and
(\ref{!!}) respectively.
The next important step is to find additional equivalence transformations admitted by
equations (\ref{1.2}) with specified non-linearities $f^1$ and $f^2$. These transformations
are relatively easy calculated using the standard Lie algorithm and treating $f^1$ and $f^2$ as additional
variables.
We will not reproduce here the related routine calculations but
present their results in Tables 1-3, where non-linearities $f^1,
f^2$ and the related symmetries are specified. Additional
equivalence transformations are given here also.
The list of possible
AETs is present in the following formulae
\be\label{eqv}\ba{l}1.\ u\to\exp(\omega t)u, \ \ v\to \exp(\omega t)v,\\
2.\ u\to u\cos\omega t-v\sin\omega t,\ v\to v\cos\omega
t+u\sin\omega t,\\
3.\ u\to \exp(\omega t)u,\ v\to v+\omega \frac{t^2}{2},\\
4.\ u\to u+\omega t,\ v\to v,\\5.\ u\to u, \ v\to v+\omega t,\\
6.\ u\to \exp({\nu\omega t})\left( u\cos(\sigma\omega t)+v\sin(\sigma\omega t)\ro,\\
\ \ \ v\to \exp({\nu\omega t})\left( v\cos(\sigma\omega t)-u\sin(\sigma\omega t)\ro,
\\7.\ u\to \exp({2\omega t})\left( u\cos(\sigma\omega t^2)-v\sin(\sigma\omega t^2)\ro,\\
\ \ \ v\to \exp({2\omega t})\left( v\cos(\sigma\omega t^2)+u\sin(\sigma\omega t^2)\ro,\\
8.\ u\to \exp({\lambda\omega t^2})\left( u\cos(2\omega t)+v\sin(2\omega t)\ro,\\
\ \ \ v\to \exp({\lambda\omega t^2})\left( v\cos(2\omega t)-u\sin(2\omega t)\ro
\ea\ee
where the Greek letters denote parameters whose values will be
specified in the tables. In contrast with (\ref{2.2}), (\ref{x.2}) these transformations
are valid only for equations with some special non-linearities $f^1, f^2$
specified in the Tables.
Table 1 presents non-equivalent equations which admit the main or the main
and extended symmetries.
The conditions for non-linearities which extend the symmetries are
specified in the fourth column. The additional equivalence transformations
are presented in the last (fifth) column.
In Table 2 the non-linearities are collected which correspond to the main
symmetries only. The additional equivalence transformations are specified
in the fifth column.
In Table 3 symmetries of a subclass of equations
(\ref{LG}) are specified. The additional equivalence transformations
are given in square brackets and placed in the fourth column.
In Tables 1-3 $G_\mu$, $\widehat G_\mu$ and $K$ are operators (\ref{2.6})
were $A$ is matrix of type $I$ (\ref{2.1}), $ \Psi_\mu(x)$ is an arbitrary
solutions of the Laplace
equation:
$$\Delta \Psi_\mu=
\mu\Psi_\mu.$$ The Greek letters denote arbitrary parameters,
moreover, up to equivalence transformations we restrict ourselves to
$\varepsilon=\pm 1$ and $\kappa=0, \pm 1$.
In Table 1 $\ba{l}R=\left(
u^2+v^2\ro^\frac12,\ z=\tan^{-1}\left(\frac{v}{u}\ro
\end{array}$, $F_1$ and $F_2$ are arbitrary functions of $R \exp(\mu z)$.
\newpage
\begin{center}
{\bf Table 1. Non-linearities and extendible symmetries for equations
(\ref{LG})}
\end{center}
\begin{tabular}{|l|l|l|l|l|}
\hline \text{No} & \text{Nonlinear terms}
&$
\begin{array}{l}
\text
{Main}\\\text{symmetries} \\
\end{array}$
&$
\begin{array}{l}
\text{Additional} \\
\text{symmetries}
\end{array}$&$
\ba{l}\text {AET}\\(\ref{eqv}) \ea$
\\
\hline 1. &$
\begin{array}{l}
f^1=uF_1+vF_2 \\
-\kappa z\left( \mu u+v\right),\ea$ &$
\begin{array}{l}
e^{\kappa t}\left( \mu R\partial_R-\partial_z\ro \ea$&$
\begin{array}[t]{l}
\widehat{G}_\alpha ,\ \text{if }\\ \mu =a,
\kappa \neq 0; \ea$&$\ba{l}\\ 2, \text{if}\ \mu\ea$\\
\cline{4-4} &$\ba{c}f^2=vF_1-uF_2\\+\kappa z\left(u- \mu v
\right)\ea$ & &$\ba{l} G_\alpha ,\ \text{if }\\ \mu =a,\kappa =0
\end{array}$&$\ba{l}=\kappa=0 \\ \ea$\\
\hline
2. &$
\begin{array}{l}
f^1=e^{\nu z}R^\sigma (\lambda u-\mu v),\\f^2=e^{\nu z}R^\sigma (\lambda
v+\mu u)\ea$&$
\begin{array}{l}
\sigma D-u\partial_u-v \partial_v,\\\nu D-u
\partial_v+v\partial_u \ea$&$\ba{l}
G_\alpha \text{ if }\nu =a\sigma\\ \text{\&}\ K \
\text{if}\ \sigma =\frac 4m
\end{array}
$&$\ba{c}\ \ \ 6\ea$\\
\hline
\end{tabular}
\vspace{2mm}
In Table 2 $F_1$ and $F_2$ are arbitrary functions whose arguments
are specified in Column 3, $R$ and $z$ have the same meaning as in Table 1.
\begin{center}
{\bf Table 2. Non-linearities and non-extendible symmetries for
equations (\ref{LG})}
\end{center}
\begin{tabular}{|l|l|c|l|l|}
\hline \text{No} & \text{Nonlinear terms} &
\begin{tabular}{l}
Argu- \\
ments\\ of \
$F_\alpha$
\end{tabular}
&$
\begin{array}{l}
\text
{Symmetries}\\
\end{array}$
&$ \ba{l}\text {AET}\\(\ref{eqv}) \ea$
\\
\hline
1. & $\begin{array}{l}
f^1=u^{\nu +1}F_{1,} \\
f^2=u^{\nu +1 }F_2
\end{array}$
& $\frac{u}{v}$ &$\begin{array}{l} \nu D-u{\partial_ u}- v{
{\partial_ v}}
\end{array}$&$\ba{l}1\ \text{if}\\
\nu=0\ea$
\\
\hline
2. &$\ba{l} f^1=\beta u+F_1,\\ f^2=-\kappa u+F_2 \ea$&$ v
$&$ e^{(\beta +a\kappa
)t}\Psi _\kappa(x) \partial_u $&$\ba{c}4 \ \texttt{if}\\\kappa=\beta=0\ea$\\
\hline 3. &$
\begin{array}{l}
f^1=e^{\kappa v}F_1, \\
f^2=e^{\kappa v}F_2
\end{array}$
& $\ba{l}u\ea$ & $\kappa D-\partial_v$& \\
\hline 4.&$\ba{l}f^1=u(F_1+\varepsilon\ln u),\\ f^2=v(F_2+\varepsilon\ln
u)\ea$&$\frac vu$&$e^{\varepsilon t}(u\partial_u+v\partial_v)$& $\ba{c}\ea$\\\hline
5.&$\ba{l}f^1=e^{\lambda z}(uF_1+vF_2),\\f^2=e^{\lambda
z}(vF_1-uF_2)\ea$& $Re^{\nu z}$&$\lambda D+\nu R\partial_R-\partial_z$& $\ba{c}
6, \ \sigma=1 \\ \texttt{if}\ \lambda=0\ea$\\\hline 6.
&$\ba{l}f^1=\kappa v^{\nu +1},\\ f^2=\beta v^{\nu +1}\ea $&&$\ba{l}
\nu D-u
\partial_u-v\partial_v,\\ \Psi_0
(x)\partial_u \ea$&$\ \ \ \ 4$\\
\hline 7. &$\ba{l} f^1=\kappa e^{v},\\ f^2=\beta e^{v} \ea$&&$
\ba{l}D-\partial_v,\ \Psi_0 (x)\partial_u\ea$&$\ \ \ \ 4$\\
\hline 8.&$ \ba{l} f^1=\mu \ln v,\\ f^2=\varepsilon \ln v \ea$&&$
\begin{array}{l}
\Psi_0 (x)\partial_u ,\\
D+u\partial_u+v\partial_v\\
{+}\left( (\mu-\varepsilon a )t\right.\\\left.-\frac
{\varepsilon}{2 m}x^2\right)\partial_u
\end{array}
$&$\ \ \ \ 4$\\
\hline
9.&$\ba{l} f^1=\lambda,\\ f^2=\ln u \ea$&&$
\begin{array}{l}
D+u\partial_u+v\partial_v+ t\partial_v,\\
\Psi _0(x) \partial_v
\end{array}
$&$\ \ \ 3, 5$\\
\hline
\end{tabular}
\vspace{3mm}
\newpage
\begin{center}
{\bf Table 3. Symmetries of equations (\ref{LG}) with non-linearities
$f^1=(\mu u-\sigma v)\ln R+z(\lambda u- \nu v)$, $f^2=(\mu
v+\sigma u)\ln R+z(\lambda v+\nu u)$}
\end{center}
\begin{tabular}{|l|l|l|l|}
\hline
No & Conditions & Main symmetries & Additional \\
& for coefficients &and AET (\ref{eqv}) & symmetries \\ \hline $1$ &
$\ba{l}\lambda =0,\\\mu =\nu\ea$ & $e^{\mu t}\partial_ z,\
e^{\mu t}\left( R\partial_ R+\sigma
t\partial_z\ro$ &
$\ba{l}
\hat{G}_\alpha \ \text {if}\ a\sigma=0,\\\mu \neq 0\ea$ \\
\cline{4-4} & & [AET 7 if $\mu=0$] &$\ba{l}
G_\alpha\
\text{if} \\ a=\nu =0,\sigma\neq 0\ea$ \\
\hline
$2$ & $\ba{l}\lambda =0,\\\mu \neq \nu,\ea$ & $\ba{l}e^{\nu
t}\partial_ z
,\\
e^{\mu t}\left( \sigma \partial_z+\left( \mu -\nu \right)
R\partial_ R\right) \ea$ &
$\ba{l}\hat{G}_\alpha\ \text{ if }\mu \neq 0\\ a\sigma =\nu -\mu ,\\
\text{or}\
a=0,\ \mu \neq 0\ea$\\
\cline{4-4} & &[AET 6 if $\mu\nu=0$] &$\ba{l} G_\alpha\ \text{if}\\
a\sigma =\nu
,\ \mu =0\ \ea$ \\
\hline
$3$ & $\ba{l}\delta=\frac14(\mu-\nu)^2\\+\lambda\sigma=0,\ea$ &
$\ba{l}X_3=e^{\omega _0t}\left( 2\lambda R\partial_
R+\left( \nu -\mu \right) \partial_ z \right),
\\2e^{\omega _0t}\partial_ z +tX_3\
\ea$ &$\ba{l} {\hat G}_\alpha\ \text{if} \ \omega_0 \neq 0
\\ a(\mu -\nu
)=2\lambda \ea$ \\
\cline{4-4} &$\ba{l}\mu+\nu=2\omega_0\\
\lambda\neq 0 \ea$&$\ba{l} \texttt{[AET 6 if} \mu+\nu=0{]},\\
\texttt{{[}AET 8}\ \&\ 1 \texttt{ if }\ \mu=\nu=0{]}\ea$ &$\ba{l}
G_\alpha\ \text{if}\\ a\nu =-\lambda
,\ \omega_0=0 \ea$ \\
\hline
$4$ & $\ba{l}\lambda\neq 0,\ \delta=1,\\\omega_\pm=\omega_0\pm1\ea $ &
$\ba{l}e^{\omega _{+}t}\left( \lambda R\partial_ R+\left( \omega
_{+}-\mu \right) \partial_ z \right),\\
e^{\omega _{-}t}\left( \lambda R\partial_R+\left( \omega
_{-}-\mu \right)
\partial_ z \right) \ea$ &$\ba[t]{l}\widehat G_\alpha\ \text{if}\ \mu\nu\neq\lambda\sigma,\\
\lambda=a(\mu-\nu+a\sigma)\ea$ \\
\cline{4-4} & & $\ba{l}{[}\texttt{AET 6 } \texttt{if
}\mu\nu=\lambda\sigma, \\ \&\ 1 \ \texttt{if }\mu=\sigma=0 {]}\ea$
&$\ba{l}G_\alpha\
\text{if} \ \nu\mu=\lambda\sigma,\\
\lambda=a\mu\ea$
\\
\hline $5$ & $\delta=-1$ & $\ba{l} \exp (\omega_0t)\left[2
\lambda\cos tR\partial_ R\right.\\ \left.+\left( \left( \nu-\mu\right)
\cos t- 2\sin t\right)\partial_ z \right],\\\exp
(\omega_0t)\left[2 \lambda\sin tR{\partial_ R}\right.\\
\left.+\left( \left( \nu-\mu\right) \sin t+ 2\cos t\right){\partial_ z}
\right] \ea $ & none \\ \hline
\end{tabular}
\vspace{2mm}
\section{Discussion}
We perform group classification of generalized CLG equations
(\ref{LG}), i.e., find all non-equivalent equations of the
considered type and describe their symmetries. The obtained results
can be used to construct exact solutions for equations which admit
sufficiently extended symmetries, using the standard Lie algorithm
\cite{olver}. The other application is to search for models with
{\it a priory} required symmetry, e.g., Galilei-invariance.
In Tables 1-3 all non-equivalent generalized CLG equations are
listed together with their symmetries and additional equivalence
transformations. More exactly, we specify here only extensions of
the basic symmetries (\ref{4.1}) and did not consider linear equations.
First we notice that the usual CGL equation appears as a particular case of
the classification procedure. The related non-linearities and symmetries are present in
Table 1, Item 2 when $\mu=0$. In addition to the basic symmetries (\ref{4.1}) this equation admits
the dilatation symmetry and symmetry $u\partial_v-v\partial_u$, which correspond to scaling of
dependent and independent variables and multiplying solutions of the CLG equation by a phase factor.
In accordance with our classification there exist eight types of generalized CLG
equations defined up to arbitrary functions $F_1$ and $F_2$
depending on variables indicated in the third column of Table 1,
Item 1 and Table 2, Items 1-5. Nonlinearities corresponding to the
most extended symmetries are collected in Table 1. In particular
there are nonlinearities corresponding to Galilei-invariant
equations (\ref{LG}) (refer to Table 1, Item 1 for $a=\mu,
\nu=0$):
$$
\begin{array}{l}
u_t-\Delta(au-v)=uF_1+vF_2, \\
v_t-\Delta(av+u)=vF_1-uF_2 \ea$$ where $F_1$ and $F_2$ are
arbitrary functions of $R{}e^{a z}$. This system can be rewritten
as a single equation for a complex function $W=u+iv$:
\be\label{exz1}W_t-(a+i)\Delta W={\cal F}W\ee where ${\cal
F}=F_1-iF_2$ is a complex function of real variable
$\xi=\ln|W|+az$ with $z$ being a phase of $W$.
The standard CLG equation does not belong to the class
(\ref{exz1}) and so is not Galilei invariant. On the other hand,
setting in (\ref{exz1}) $a=0$ we come to the Galilei-invariant
subclass of the NS equations (\ref{d6}).
The non-linearities enumerated in Table 3, Item 2 of Table 1 and
Items 6-9 of Table 2 are defined up to arbitrary parameters. The
most extended symmetry is indicated in Item 2 of Table 1 and
corresponds to the following equation for complex function
$W=u+iv$:
\be\label{exz2}W_t-(a+i)\Delta W=\alpha \left(
e^{az}|W|\ro^{\rho}W\ee where $\alpha= \lambda+i\sigma$ is a complex
parameter and $\rho=\frac{4}{m}$.
In accordance with the above equation (\ref{exz2}) admits Lie
algebra of the Shr\"odinger group including operators $P_\mu,
J_{\mu\nu}$ (\ref{4.1}) and also generators of dilatation $D$,
Galilean transformations $G_\mu$ and conformal transformations $K$
(\ref{2.6}). Setting in (\ref{exz2}) $a=0$ we reduce (\ref{exz2})
to the very popular NS equation with critical power $4/m$
non-linearity.
If $\rho\neq \frac{4}{m}$ then equation (\ref{exz2}) admit all
the above mentioned symmetries except the generator $K$ of conformal
transformations.
In general we indicate six classes of equations (\ref{LG}) which
admit symmetries $G_\mu$ and so are invariant with respect to
Galilei group. Namely, in addition to (\ref{exz1}), (\ref{exz2})
there are the following Galilei-invariant equations:
\be\label{exz3} iW_t+\Delta W=-\sigma W\ln|W|\ee
which corresponds to Item 1, $\nu=0$ of Table 3, and
\be\label{exz4}W_t-(a+i)\Delta W=cW(\ln |W|+az)\ee
where $c$ is a complex number equal to $i\sigma,\ \sigma(i-a)$ or
$\mu+i\sigma$ for versions indicated in Items 2, 3 or 4 of Table 3 correspondingly.
Non-linearities collected in Table 2 correspond to equations
(\ref{LG}) which admit the main and basic symmetries only.
Thus we
present completed group classification of systems of
reaction-diffusion equations (\ref{1.2}) with square diffusion
matrix of type $I$ (\ref{2.1}).
The additional aim of this paper is to present an effective
approach for solving classifying equations (\ref{2.7}). It was
demonstrated in Sections 3 and 4 that the problem of group
classification of equations (\ref{1.2}) can be effectively reduced
to searching for the main symmetries (\ref{1.2}). We also make a
priori specification of these symmetries using the fact that they
should form a basis of a Lie algebra. The idea of such a
specification was proposed in papers \cite{wint} and
\cite{zhdan1}.
In our following publications we use the approach described here
to classify reaction-diffusion equations with diagonal and
triangular diffusion matrix. In other words we plane to complete
classification of equations (\ref{1.2}) with general diffusion
matrix whose non-equivalent versions are given by formulae
(\ref{2.1}).
|
{
"timestamp": "2006-09-19T16:52:48",
"yymm": "0411",
"arxiv_id": "math-ph/0411027",
"language": "en",
"url": "https://arxiv.org/abs/math-ph/0411027"
}
|
\section{The Most Metal-rich dSph Stars}
The most metal-rich stars in dwarf spheroidals (dSph) have been shown to have
significantly lower even-Z abundance ratios than stars of similar metallicity
in the Milky Way (MW). In addition, the most metal-rich dSph stars are
dominated by an s-process abundance pattern in comparison to stars of
similar metallicity in the MW.
This has been interpreted as excessive contamination by Type Ia super-novae (SN)
and asymptotic giant branch (AGB) stars
( Bonifacio et al. 2000, Shetrone et al. 2001, Smecker-Hane \& McWilliam 2002).
By comparing these results to MW chemical evolution,
Lanfranchi \& Matteucci (2003) conclude that the dSph galaxies have had a
slower star formation rate
than the MW (Lanfranchi \& Matteucci 2003). This slow star formation, when
combined with an efficient galactic wind, allows
the contribution of Type Ia SN and AGB stars to be incorporated into the ISM
before the Type II SN can bring the metallicity up to MW thick disk
metallicities.
Recent abundance ratio work in this field falls into two
categories. The first category has been investigations into
aspects of metal-poor AGB and Type Ia SN yields and their relationship to
the chemical evolution in the dSph galaxies, e.g. McWilliam et al. (2003),
Venn et al. (2004), McWilliam \& Smecker-Hane (2005). In these works
the abundances of specific elements are compared to predictions of
yields of low metallicity SN and AGB stars. While the origins of these
elements, such as Mn, Cu
and Y, may seem slightly esoteric, these types of analyses will help constrain
future models of SN yields.
The second catagory of dSph abundance investigations has been attempts to
gain large enough samples to accurately
model the extent of the chemical evolution, the relative contributions
the Type Ia and AGB yields and to what extent galactic winds have played a
roll in the chemical evolution.
The new instruments that have come on-line in the last year have increased
the multiplexing capabilities of these surveys. As an example, Figure 1
shows preliminary results from the Dwarf Abundance and Radial velocity Team
(DART) ESO large program; using UVES FLAMES$+$GIRAFFE on a sample of Sculptor
dSph giants, Hill (private communication, 2005) collected nearly 100 stellar
spectra,
a sample larger than all of the literature high resolution dSph surveys
combined. This survey will be able to show subtle declines and trends that the
other surveys could never detect. The decline seen in [Ca/Fe] with
increasing [Fe/H] reported by Shetrone et al. (2003) and Geisler et al. (2004)
are easily detected. The spread in [Ca/Fe] at a given metallicity is being
investigated by DART.
\begin{figure}[1]
\begin{center}
\includegraphics[width=0.81\textwidth]{shetroneF1.eps}
\end{center}
\caption[]{An example of the new data sets coming available with the VLT$+$FLAMES.
This data, from Hill (private communication, 2005) exhibits the steep
decline of [Ca/Fe] with
increasing [Fe/H] seen in all dSph. In addition, the most metal-poor stars
do not exhibit the typical MW halo ratio near 0.3 dex.}
\label{shetrone1}
\end{figure}
\section{The Most Metal-poor dSph Stars}
In a closed box or leaky-box chemical evolution model the most metal-poor
stars would have formed before the majority of the Type Ia SN or the AGB
stars were significant contributors to the the ISM. Thus, the abundances of
these very metal-poor stars should be excellent surrogates of the Pop III
and very metal-poor Pop II Type II SN products held in the dSph gravitational
potential. This last point is important because if the yields of some
masses of Type II SN are lost from the dSph gravitational potential then
this might significantly impact abundance ratios found in the next
generation of dSph stars.
The first papers in this field suggested that the overall alpha abundances
found in the most metal-poor dSph stars are not similar to those found in the
halo. However, a more detailed analysis of the individual elements
including corrections for differences in log gf values has shown that the
O and Mg abundances are consistent with those found in the MW halo, while
the Ca and Ti abundances are systematically lower than those in the MW
halo, see figure 1, 2 and 6 in Shetrone (2004). This can also be seen
in Figure 1 which shows that the most metal-poor Sculptor stars have
[Ca/Fe] ratios less than
0.2 dex while the halo median at this metallicity is roughly 0.15 dex larger.
The difficulty with studies of the most metal-poor dSph stars is actually
finding the most metal-poor stars. Not only are these stars less numerous
than their more metal-rich counter parts, but their spatial distribution is
larger than the more metal-rich stars, e.g. Tolstoy et al. (2004),
Palma et al. (2003), Harbeck et al. (2001), Majewski et al. (1999). The
use of high resolution multi-object spectrographs such as FLAMES becomes
less efficient in the search for the most-metal poor stars because of the
large spatial extent, rarity and huge background contamination. For these
types of studies targeting single star high resolution spectral
follow-up to photometric or low resolution surveys can be more appropriate.
One star analyzed in the Shetrone et al. (1998) survey was found to be very
metal-poor but with fairly low overall-alpha abundances. Unfortunately, due to
the low S/N and low metallicity many of the elements had upper limits and large
error bars. This single star, Draco 119, was re-observed by Fulbright
et al. (2004) with much higher S/N. They were able to confirm that the
Mg abundance was halo like while the Ca and Ti abundances were lower than
those found in the halo by a few tenths of dex. Even more remarkable were
the upper limits found for the neutron capture elements. Fulbright et al.
found upper limits for [Ba/Fe] 1 dex lower than MW halo giants of similar
metallicity, and, even more amazingly, the upper limit for [Sr/Fe] was
found to be nearly 2 dex lowerthan similiar MW giants.
This begged the question: is the Draco 119
abundance pattern unique, ie. due to some strange inhomogeneous mixing event,
or is this pattern found in all very metal-poor Draco stars.
A search for equally metal-poor Draco stars, using the Hobby-Eberly telescope,
did not turn up any Draco giants as metal-poor as Draco 119; but it
did turn up a few stars just
a few tenths more metal-rich. By integrating long enough to detect the
strong red Ba lines in these stars using the HRS on the Hobby-Eberly telescope,
Shetrone et al. (2005) did not find extremely
neutron capture poor stars, see Figure 2.
The abundance pattern of Draco 119 appears to be a due to inhomogeneous mixing
and is not found in all very metal-poor Draco stars.
I would like to thank Kim Venn, Andy McWilliam, Verne Smith, Jon Fulbright
and the DART for preprints and invaluable discussions. I would also like
to thank the NSF for support through AST-0306884; and summer REU intern John
Moore and the team at the HET for their assistance in bringing these results to
press.
\begin{figure}[2]
\begin{center}
\includegraphics[width=0.81\textwidth]{shetroneF2.eps}
\end{center}
\caption[]{The abundance ratio of Ba from the Shetrone et al. (1998), squares,
Fulbright et al. (2004), circle, and Shetrone et al. (2005), triangles shown
plotted against their derived metallicities. The upper limit from
Fulbright et al. (2004) is more than an order of magnitude lower than that
of the slightly more metal rich Draco dSph stars or comparable metallicity
MW halo stars, small symbols.}
\label{shetrone2}
\end{figure}
|
{
"timestamp": "2004-11-01T13:57:26",
"yymm": "0411",
"arxiv_id": "astro-ph/0411030",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411030"
}
|
\section{Introduction}
{\sc Planck} represents the third generation of mm-wave instruments
designed for space observations of the cosmic microwave background (CMB)
anisotropies within the new Cosmic Vision 2020 ESA Science Program.
Following the present NASA's mission Wilkinson Microwave Anisotropy
Probe (Bennett et al. 2003)
{\sc Planck} will map the whole sky with unprecedented sensitivity (the average sensitivity on a pixel of {\sc fwhm} side in the measurement of the temperature anisotropy is about two parts per million), angular resolution ({\sc fwhm} from about 30 down to 5 arcmin), and frequency coverage, and it likely leads us to the final comprehension of the CMB anisotropies
(J. Tauber 2004, {\it this Meeting}).
The Low Frequency Instrument (LFI, Mandolesi et al. 1998), operating in the 30 $\div$ 70 GHz range, is one of the two instruments onboard {\sc Planck} satellite, sharing the focal region of a 1.5 meter off-axis dual reflector telescope together with the High Frequency Instrument (HFI, Puget et al. 1998) o\-perating at 100 $\div$ 857 GHz.
LFI consists of four main units: the front end unit
(FEU, Sect.~2),
the back end unit (BEU,
Sect.~3),
the radiometer electronics box assembly
(REBA,
Sect.~4),
and the sorption cooler system
(SCS, Sect.~5).
We describe how these units have been conceived,
as well as the major instrumental systematic effects that could
degrade the measurements and the solutions adopted in the
design phase in order to adequately reduce and control them.
In addition, the removal of systematic effects and the LFI data processing
center (DPC) are mentioned and described in
Sects.~6 and~7.
\begin{figure}[h]
\centering
{\includegraphics[width=6.5cm]{lfis_3.eps}}
\caption{The Low Frequency Instrument: the front end unit is at the top, whereas the back end unit, the radiometer electronics box assembly, and sorption cooler system are in the lower part.}
\label{lfis}
\end{figure}
\section{The Front End Unit}
\label{feu}
\subsection{Feed horns and OMTs}
LFI is coupled to the {\sc Planck} telescope by an array of conical dual profiled
corrugated feed horns
(Villa et al. 2002).
Dual profiled corrugated horns have been selected as the best design in terms of shape of the main lobe, very low level of cross polarization and sidelobe, control of the phase centre location, low weight and compactness.
In addition, the electromagnetic field inside the horn propagates with low attenuation and low return loss.
The ortho mode transducers (OMTs) separate the orthogonal polarizations with minimal losses and cross-talk.
The straylight, or the unwanted off-axis radiation contributing to the observed signal, represents one of the major sistematic effects in {\sc Planck}/LFI and can be controlled optimizing the feed horn design, since from the latter depends the illumination of the mirrors. A trade-off between angular resolution (the more the primary mirror is illuminated, the best is the angular resolution) and straylight rejection (the more the mirrors are illuminated, the worst is the straylight rejection) has been obtained for each feed horn coupled with the {\sc Planck} telescope. This optimization has been performed computing the full optical response of several realistic feed horn patterns
(Sandri et al. 2003)
and convolving the full pattern with the sky signal by considering the observational stra\-tegy
(Burigana et al. 2003)
in order to calculate the straylight contamination for each feed model analized. As a result, the LFI feed horns have different designs (i.e. inner corrugation profile), depending from their location on the focal surface, and the corresponding angular resolution achieved is the best one, satisfying the straylight rejection requirement.
\subsection{Hybrids, phase swithes and amplifiers}
The OMTs are followed by blocks conta\-i\-ning hybrid couplers and amplifiers (inclu\-ding phase switches and output hybrids), all cooled to 20K by the H$_2$ sorption cooler system.
This front-end is designed to minimize the $1/f$ noise in the radiometer
(one of the most important potential source of systematic effects) while
maintaining low thermal noise
(Seiffert et al. 2002, Mennella et al. 2003). Each block contains two
hybrid
couplers: each hybrid has two inputs, one of which sees the sky, the other one looks at the 4K reference load through a small rectangular horn. The hybrid coupler combines the signals from the sky and cold load with a fixed phase offset of either 90$^\circ$ or 180$^\circ$ between them. It has the necessary bandwidth, low loss, and amplitude balance needed at the output to ensure adequate signal isolation. The low-noise amplifiers use InP HEMTs in cascaded gain stages. Of all transistors, InP HEMTs have the highest frequency response, lowest noise, and lowest power dissipation. The amplifiers at 30 and 44 GHz use discrete InP HEMTs incorporated into a microwave integrated circuit (MIC). At these frequences, cryogenic MIC amplifiers have demonstrated noise figures of about 10K, with 20\% bandwidth. At 70 GHz, MMICs (Monolithic Microwave Integrated Circuits) architectures, which incorporate all circuit elements and the HEMT transistors on a single InP chip, are used. The LFI will fully exploit both MIC and MMIC technologies at their best.
For all frequencies, 30--40 dB of gain are sufficient to guarantee that the overall noise is dominated by the front end amplifiers. If additional gain were located in the front end, the power dissipated would grow significantly, putting too much load on the cooler.
Following amplification the signals are passed through a phase switch. The switch consumes microwatts of power, it is broad band, and it works at cryogenic temperatures with switch rates in excess of 1 kHz. The phase switch adds 90$^\circ$ or 180$^\circ$ of phase lag to the signals, thus selecting the input source as either the sky or the reference load at the radiometer output. The phase lagged pair of signals is then passed into a second hybrid coupler, separating the signals. The signals are then transitioned to high performance two meter long bent twisted composite (copper -- stainless steel -- gold-plated stainless steel) rectangular waveguides that carry the double chain signal to the BEU. The waveguides are thermically connected to the three V-grooved shields of the payload at 50, 90, and 140K from the FEU to the BEU, respectively.
\section{The Back End Unit}
\label{beu}
\subsection{Back end modules}
Each back end module (BEM) comprises two parallel chains of amplification, filtering, detection, and integration. The detected signals are amplified and a low-pass filter reduces the variance of the random signal, providing in each channel a DC output voltage related to the average value. Post-detection amplifiers are integrated into the BEMs to avoid data transmission problems between the radiometer and the electronics box. The sky and reference signals are at different levels, which are equalized after detection and integration by modulating the gain synchronously with the phase switch. Because of the phase switching in the front end modules, a given detector alternately sees the sky and the reference signals. Differences between the detectors are therefore common-mode in the output signals and have no effect on the final difference, which is calculated in the signal processing unit. Each back end module will be packaged, including analog-to-digital converters, into a box of a few centimeters on a side, including the biasing circuitry and the input and output connectors.
\subsection{Data acquisition electronics}
The data acquisition electronics (DAE) comprises the analog conditioning electronics, the multiplexers, the analog-to-digital converters, the parallel-to-serial converters, the control electronics, the communication interface, and the power conditioning and distribution electronics. It performs the following functions: communication with the data processing unit (DPU), including command reception and status transmission; acquisition, conditioning, and multiplexing of the signals; control of the data acquisition chain; transmission of raw data to the signal processing unit (SPU); power supply conditioning and distribution; DC biasing of the FEU and BEU amplifiers; synchronous control of the FEU phase switches; and ON/OFF control of FEU and BEU amplifiers.
\section{The REBA}
\label{reba}
The radiometer electronics box assembly (REBA) comprises three sub-units: the SPU, the DPU, and the power supply unit (PSU).
The REBA supplies all the telemetry and telecommand communication interfaces with the spacecraft, controls the radiometer array assembly through its interface to the BEU, and processes all the radiometer outputs which have been analogue to digital converted in the BEU into science telemetry.
\section{The H$_2$ Sorption Cooler}
\label{scs}
The LFI FEU is cooled to 20K by the hydrogen sorption cooler developed at the JPL (NASA). The operating cooler also provides 18K precooling to the HFI 4K cooler. Each cooler is a Joule-Thomson cooler in which ~0.0065 g/s of hydrogen expands from 5 MPa to ~0.03 MPa through a Joule-Thomson (J-T) expander. The high and low gas pressures are maintained by the fact that the equilibrium pressure of gas above the sorbent bed is a strong function of temperature.
\section{Removal of systematic effects}
\label{syst}
The extreme accurate control of all instrumental systematic effects
that could in principle affect the measurements (and in practice do
it) is crucial at {\sc Planck} sensitivity levels. On the other hand,
just the high {\sc Planck} sensitivity allows to successfully apply
several kinds of algorithms to remove systematic effects.
$1/f$ noise and thermal drifts, previously reduced through an accurate
design and realisation of 4~K reference loads and sorption coolers,
will be further subtracted during the data analysis through dedicated
destriping (see, e.g.,
Keihanen et al. 2003
and references therein)
and map making (see, e.g.,
Natoli et al. 2001
and references therein)
codes respectively in the
TOD domain and on the sky map domain. The first method is blind by
contruction while the second one requires an accurate noise
parametrization available both from ground testing and from the
in-flight analysis of the TOD. The correction of the effect on the
{\sc Planck} data from optical distortions, previousloy reduced
through an appropriate optical design study, requires the use of
specific algorithms. External planet transits on the {\sc Planck}
field of view represents the best way for an accurate main beam
reconstruction in flight
(Burigana et al. 2001)
and the main beam distortion effect
on the CMB angular power spectrum recovery can be blindly subtracted
through dedicated deconvolution algorithms jointed to Monte Carlo
simulations for the deconvolved noise subtraction
(Burigana \& S\'aez 2003)
The straylight effect
due to the beam far sidelobes is non negligible for {\sc Planck}.
Dedicated ground measures will represent the first step of the
software reduction of the straylight contamination.
The study of a semi-blind approach to a joint far beam
estimate and straylight subtraction is on-going.
\section{The LFI DPC}
\label{dpc}
From the mission operation center (MOC, that will control the {\sc Planck} spacecraft), the scientific data produced by {\sc Planck} will be piped daily to two data processing centres (DPCs). These DPCs will be responsible for all levels of processing of the {\sc Planck} data, from raw telemetry to deliverable scientific products. Although contributions to the LFI DPC, in terms of information on instrument characteristics and prototype software, come from a variety of geographically distributed sites, the operations are mainly centralized in Trieste, where they are run jointly by the OAT and SISSA.
The structure of the LFI (and HFI) DPC has been divided into five levels: LS (during the pre-launch phase, simulation of data acquired from the {\sc Planck} mission on the basis of a software system agreed upon across Consortia), L1 (telemetry processing and instrument control), L2 (data reduction and calibration), L3 (component se\-paration and optimization), and L4 (ge\-neration of final products).
\begin{acknowledgements}
LFI is funded by the national space agencies of the Institutes of the {\sc Planck} Consortium.
In particular the Italian participation is funded by ASI.
\end{acknowledgements}
\bibliographystyle{aa}
|
{
"timestamp": "2004-11-15T17:16:22",
"yymm": "0411",
"arxiv_id": "astro-ph/0411412",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411412"
}
|
\section{Introduction and Statement of Results}
We study the structure of the scattering amplitude associated with the
semi-classical Schr\"{o}dinger operator with a short range potential on $\mathbb{R}^{n}.$
We prove that, when restricted away from the diagonal on
$\mathbb{S}^{n}\times\mathbb{S}^{n},$ the natural scattering amplitude quantizes the scattering
relation in the sense of semi-classical Fourier integral operators.
The scattering relation at energy $\lambda>0$ here is given roughly by the Hamiltonian flow of the symbol $p$ of the operator between two hypersurfaces ``at infinity'' inside the energy surface $\{p=\lambda\}.$
\subsection{A Survey of Earlier Results}
The structure of the scattering matrix has been of significant interest
to researchers in mathematical physics.
Earlier results have focused primarily on establishing asymptotic expansions of the
scattering amplitude.
In this section we describe briefly only those asymptotic expansions most relevant to our
work and refer to \cite{AI} for a more comprehensive survey.
We begin by introducing some notation.
Let $P(h)=-\frac{1}{2}h^{2}\Delta+V,$ $0<h<<1,$ where
\begin{equation}\label{potential}
\left|\partial^{\alpha}V(x)\right|\leq
C_{\alpha}\langle x\rangle^{-\rho-|\alpha|},\;
x\in\mathbb{R}^{n},
\rho>1,
\end{equation}
where $\langle x\rangle=(1+\|x\|^{2})^\frac{1}{2}.$
Let $\lambda>0$ and for $\omega\in\mathbb{S}^{n-1}$ and
$z\in\omega^{\perp}$ we denote by
\[\gamma_{\infty}\left(\cdot; z, \sqrt{2\lambda}\omega\right)=\left\{q_{\infty}\left(\cdot; z, \sqrt{2\lambda}\omega\right),
p_{\infty}\left(\cdot; z, \sqrt{2\lambda}\omega\right)\right\}\]
the unique phase
trajectory, i.~e. the integral curve of the Hamiltonian vector field of
$p(x, \xi)=\frac{1}{2}\|\xi\|^{2}+V(x),$ such that
\begin{gather*}
\lim_{t\to -\infty}\left\|q_{\infty}\left(t; z,
\sqrt{2\lambda}\omega\right)-\sqrt{2\lambda}\omega
t-z\right\|=0,\\
\lim_{t\to -\infty}\left\|p_{\infty}\left(t; z, \sqrt{2\lambda}\omega\right)-\sqrt{2\lambda}\omega\right\|=0
\end{gather*}
in the $C^{\infty}$ topology for the impact parameter $z$ and $\omega.$
If $\lim_{t\to\infty}\left\|q_{\infty}\left(t; z, \sqrt{2\lambda}\omega\right)\right\|=\infty,$ then, setting $\mathbb{S}^{n-1}_{2\lambda}=\{x\in\mathbb{R}^{n}: \|x\|=2\lambda\},$ we have that there exist $U\subset T^{\star}\mathbb{S}^{n-1}_{2\lambda}$ open, $\left(\sqrt{2\lambda}\omega, z\right)\in U,$ where $\mathbb{S}^{n-1}_{2\lambda}=\{x\in\mathbb{R}^{n}: \|x\|=2\lambda\},$ $\xi_{\infty}\in C^{\infty}\left(T^{*}\mathbb{S}^{n-1}_{2\lambda}\cap U; \mathbb{S}^{n-1}\right),$
and $x_{\infty}\in C^{\infty}\left(T^{*}\mathbb{S}^{n-1}_{2\lambda}\cap U; \mathbb{R}^{n}\right)$
such that
\begin{gather*}
\lim_{t\to\infty}\left\|q_{\infty}\left(t; z,
\sqrt{2\lambda}\omega\right)-\sqrt{2\lambda}\xi_{\infty}\left(z,
\sqrt{2\lambda}\omega\right)t-x_{\infty}\left(z,
\sqrt{2\lambda}\omega\right)\right\|_{C^{\infty}(U)}=0,\\
\lim_{t\to\infty}\left\|q_{\infty}\left(t;
z, \sqrt{2\lambda}\omega\right)-\sqrt{2\lambda}\xi_{\infty}\left(z,
\sqrt{2\lambda}\omega\right)\right\|_{C^{\infty}(U)}=0.
\end{gather*}
The trajectory $\gamma_{\infty}\left(\cdot; z, \sqrt{2\lambda}\omega\right)$ is
then said to have initial
direction $\omega$ and final direction $\theta=\xi_{\infty}\left(z,
\sqrt{2\lambda}\omega\right).$
We also make the following
\begin{Def}
The outgoing direction $\theta\in\mathbb{S}^{n-1}$ is called {\it non-degenerate}, or
{\it regular}, for
the incoming direction $\omega\in\mathbb{S}^{n-1}$ if $\theta\ne\omega$ and for all
$z'\in\omega^{\perp}$ with
$\xi_{\infty}\left(z', \sqrt{2\lambda}\omega\right)=\theta,$ the map $\omega^{\perp}\ni z\mapsto \xi_{\infty}\left(z, \sqrt{2\lambda}\omega\right)\in\mathbb{S}^{n-1}$ is non-degenerate at $z'.$
\end{Def}
Several authors, working under the assumption that a certain final direction $\theta$ is non-degenerate for a given initial direction $\omega,$ have proved asymptotic expansions of the scattering amplitude $A$ of the form
\begin{equation}\label{vexpansion}
K_{A(\lambda, h)}(\omega, \theta)=\sum_{j=1}^{l}\hat{\sigma}\left(z_{j}, \omega;
\lambda\right)^{-1/2}\exp\left(ih^{-1}S_{j}-i\mu_{j}\pi/2\right)+\mathcal{O}\left(h\right),
\end{equation}
where $\left(z_{j}\right)_{j=1}^{l}\equiv\left(\xi_{\infty}^{-1}\left(\cdot,
\sqrt{2\lambda}\omega\right)\right)
(\theta_0),$ $\hat{\sigma}\left(z_{j}, \omega;
\lambda\right)=\det\left(\J\xi_{\infty}\left(\cdot, \sqrt{2\lambda}\omega\right)\right)\left(z_j\right),$ with $\J$ denoting the Jacobian matrix,
\begin{equation}\label{modaction}
S_j=\int_{-\infty}^{\infty}\left(\frac{1}{2}\left|p_{\infty}\left(t; z, \sqrt{2\lambda}\omega\right)\right|^{2}-V\left(q_{\infty}\left(t; z,
\sqrt{2\lambda}\omega\right)\right)-\lambda\right)dt-\left\langle x_{\infty}\left(z, \sqrt{2\lambda}\omega\right),
\sqrt{2\lambda}\theta\right\rangle
\end{equation}
is a modified action along the $j-$th $(\omega, \theta)$ trajectory,
and $\mu_{j}$ is the path index of that trajectory.
Vainberg \cite{V} has studied smooth compactly supported potentials $V$ at energies
$\lambda>\sup V$ and has proved such an asymptotic expansion with the error term
estimated uniformly over a sufficiently small neighborhood containing the final direction
while the initial direction is held a constant.
Guillemin \cite{G} has established a similar asymptotic expansion in the setting of smooth
compactly-supported metric perturbations of the Laplacian for fixed incoming and outgoing
directions.
Working with non-trapping potential perturbations of the Laplacian satisfying
\eqref{potential} with $\rho>\max\left(1, \frac{n-1}{2}\right),$ Yajima
\cite{Y} has proved such an asymptotic expansion in the $L^{2}$ sense.
For non-trapping short-range ($\rho>1$) potential perturbations of the Laplacian, Robert and
Tamura \cite{RT} have established a pointwise asymptotic expansion of this form for
constant initial and final directions.
This result has been extended to the case of
trapping energies by
Michel \cite{M} under an additional assumption on the distribution of the resonances of $P(h).$
In \cite{AI} we have proved, without making the non-degeneracy assumption, that the scattering
amplitude for smooth compactly supported potential and metric perturbations of the Euclidean
Laplacian at both trapping and non-trapping energies is a semi-classical Fourier integral
operator associated to the scattering relation.
We have further showed how the expansion \eqref{vexpansion} follows from the general
theory of semi-classical Fourier integral operators developed in \cite{Afio},
once the non-degeneracy assumption on the initial and final directions is made.
Here we extend these results to the case of short-range perturbations of the Laplacian
when the scattering amplitude is restricted away from the diagonal in
$\mathbb{S}^{n-1}\times\mathbb{S}^{n-1}.$
\subsection{Statement of Main Theorem}
We consider the semi-classical Schr\"{o}dinger operator
$P(h)=-\frac{1}{2}h^{2}\Delta+V,$ on $\mathbb{R}^{n}$ for $n\geq 2,$
$0<h\leq1,$ with the potential $V\in C^{\infty}(\mathbb{R}^{n}; \mathbb{R})$ satisfying \eqref{potential}.
Let $P_{0}(h)=-\frac{1}{2}h^{2}\Delta.$
Then $P(h)$ and $P_0(h)$ admit unique self-adjoint realizations
on $L^{2}(\mathbb{R}^{n})$ with common domain $H^2(\mathbb{R}^{n}).$
It is well-known that the wave operators
\begin{equation*}
W_{\pm}=\text{s-}\lim_{t\to\pm\infty}U(t)U_0(-t),
\end{equation*}
where
\[U(t)=e^{-\frac{i}{h}tP(h)}, \; U_0(t)=e^{-\frac{i}{h}tP_{0}(h)},\; t\in\mathbb{R}.\]
We can therefore define the scattering operator
\begin{equation*}
S=W_{+}^{*}W_{-}=\mathcal{F}^{-1}\int_{\lambda>0}\bigoplus S(\lambda, h)
d\lambda\,\mathcal{F},
\end{equation*}
where $\mathcal{F}$ denotes the unitary Fourier transform on $L^{2}(\mathbb{R}^{n}).$
The operator $S(\lambda, h)$ is called the scattering matrix at energy
$\lambda>0$ and is a unitary operator on
$L^{2}(\mathbb{S}^{n-1}).$
The scattering amplitude $A(\lambda, h)$ is defined by
$A(\lambda, h)=c(n, \lambda, h)T(\lambda, h),$ where $T(\lambda,
h)=-i(2\pi)^{-1}\left(I-S(\lambda, h)\right)$ and
\[c(n, \lambda, h)=-2\pi(2\lambda)^{-\frac{n-1}{4}}(2\pi
h)^{\frac{n-1}{2}}e^{-i\frac{(n-3)\pi}{4}}.\]
To state our Main Theorem, we let $R(\lambda+i0, h)=\lim_{\epsilon\downarrow
0}\left(P(h)-\lambda-i\epsilon\right)^{-1},$ where the limit is taken in
the space $\mathcal{B}(L^{2}_{\alpha}(\mathbb{R}^{n}),
L^{2}_{-\alpha}(\mathbb{R}^{n})),$
$\alpha>\frac{1}{2},$ with $L^{2}_{\alpha}(\mathbb{R}^{n})=\{f:
\langle\cdot\rangle^{\alpha}f\in L^{2}(\mathbb{R}^{n})\}.$
We further refer the reader to Section \ref{sgeom} for the definitions non-trapped trajectories and the scattering relation $SR_{\bar{U}}(\lambda).$
The class of semi-classical Fourier integral operators $\mathcal{I}^{r}_{h}$ is defined in Appendix \ref{scanal},
where we also review the notion of pseudodifferential operators of principal type.
We are now ready to prove our
\medskip
\noindent
{\bf Main Theorem.}
{\it Let $\lambda>0$ be such that the operator $P(h)-\lambda$ is of principal type.
Let also
\begin{equation}\label{resest}
\left\|R(\lambda+i 0, h)\right\|_{\mathcal{B}\left(L^{2}_{\alpha}(\mathbb{R}^{n}),
L^{2}_{-\alpha}(\mathbb{R}^{n})\right)}=\mathcal{O}(h^{s}), \;s\in\mathbb{R}, \;\alpha>\frac{1}{2}.
\end{equation}
Let $(\omega, z)\in T^{*}\mathbb{S}^{n-1}$ be such that
$\gamma_{\infty}\left(\cdot; z, \sqrt{2\lambda}\omega\right)$ is a
non-trapped trajectory.
Then there exist open sets $U\subset T^{*}\mathbb{S}^{n-1}$ with
$(\omega, z)\in U$ such that
{\em
\[A(\lambda, h)\in \mathcal{I}_{h}^{\frac{n}{2}+2}\left({\mathbb
S}^{n-1} \times{\mathbb S}^{n-1}\backslash\diag(\mathbb{S}^{n-1}\times\mathbb{S}^{n-1}),
SR_{\bar{U}}(\lambda)\right).\]}}
We remark that \cite[Theorem 1]{A} gives a characterization of semi-classical Fourier integral distributions as oscillatory integrals.
Applied to the scattering amplitude here this characterization says approximately that for every non-degenerate phase function $\phi$ which locally parameterizes $SR_{\bar{U}}(\lambda)$ we can find a symbol $a$ admitting an asymptotic expansion in $h$ such that $CK_{A(\lambda, h)},$ where $C$ is a microlocal cut-off to $SR_{\bar{U}}(\lambda)$ (see Appendix \ref{scanal}), can be represented as an oscillatory integral with phase $\phi$ and symbol $a.$
From the discussion in \cite[Section 4.1]{A} we further know that such a non-degenerate
phase function always exists, and therefore we can always express $CK_{A(\lambda, h)}$ as an oscillatory integral admitting an asymptotic expansion in $h.$
In the special case when the non-degeneracy assumption holds, we recover the phases \eqref{maction} in \eqref{vexpansion} -- see
Theorem \ref{tmicrol} below.
We expect that a finer analysis based on our method would give a precise description of the
amplitudes as well.
What is different here is the fact that we can handle the cases in which the non-degeneracy
assumption fails.
We now introduce some of the notation we shall use below.
For a
sequentially continuous operator
$T:C^{\infty}_{c}(\mathbb{R}^{m})\to\mathcal{D}'(\mathbb{R}^{n})$ we shall
denote by $K_{T}$ its Schwartz kernel.
On any smooth manifold $M$ we denote by $\sigma$ the canonical symplectic
form on $T^{*}M$ and everywhere below we work with the canonical
symplectic structure on $T^{*}M.$
We shall denote by $H_{p}$ the Hamiltonian vector field of $p.$
The integral curve of $H_{p}$ with initial conditions $(x_0, \xi_0)\in
T^{*}\mathbb{R}^{n}$ will
be denoted by
$\gamma(\cdot; x_0, \xi_0)=(x\left(\cdot; x_0, \xi_0\right), \xi(\cdot;
x_0, \xi_0)).$
If $C\subset T^{*}M_1\times T^{*}M_2,$ where $M_j,$ $j=1, 2,$ are smooth
manifolds, we will use the notation $C'=\{(x, \xi; y,
-\eta): (x, \xi; y, \eta)\in C\}.$
We shall also use $\|\cdot\|_{\pm\gamma, \mp\gamma}$ to denote the norm of a linear operator
between the spaces $L^{2}_{\pm\gamma}(\mathbb{R}^{n})$ and
$L^{2}_{\mp\gamma}(\mathbb{R}^{n}).$
Lastly, we set $B(0, r)=\{x\in\mathbb{R}^{n}: \|x\|<r\}$ and $B(0, r,
r+1)=\{x\in\mathbb{R}^{n}: r<\|x\|<r+1\}.$
This paper is organized as follows.
We review the definition of semi-classical Fourier integral distributions and
operators in the Appendix, where we also recall the relevant part of
semi-classical analysis.
Isozaki-Kitada's representation of the short-range scattering amplitude which
we will use in this article is presented in Section \ref{sreprsa}.
Two preliminary lemmas giving additional information on the structure of
the semi-classical amplitude, are given in Section \ref{2ls}.
The scattering relation is defined in Section \ref{sgeom}, where we also prove that it can
be parameterize by the modified actions when the non-degeneracy assumption is made.
The proof of the Main Theorem is presented in Section \ref{pmain} and its applications to
non-trapping and trapping perturbations are discussed in Section \ref{snt} and Section
\ref{str}, respectively.
Finally, the theorem giving the microlocal representation of the scattering amplitude
as an oscillatory integral under
the
non-degeneracy assumption is proved in Section \ref{smicrol}.
\section{Preliminaries}
In this section we introduce some of the preliminary results we shall use throughout this article.
\subsection{Representation of the Scattering Amplitude}\label{sreprsa}
Here we present the representation of the short range scattering amplitude
developed by Isozaki and Kitada \cite{IK}.
This is the representation we shall use in the proof of our Main
Theorem.
\begin{Def}
Let $\Omega\subset T^{*}\mathbb{R}^{n}$ be an open subset.
We denote by $A_{m}(\Omega)$ the class of symbols $a$ such that $(x, \xi)\mapsto a(x, \xi,
h)$ belongs to $C^{\infty}(\Omega)$ and
\begin{equation*}
\left|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x, \xi)\right|\leq C_{\alpha\beta}\langle
x \rangle^{m-|\alpha|}\langle\xi\rangle^{-L}, \text{ for all } (x, \xi)\in\Omega, (\alpha,
\beta)\in\mathbb{N}^{n}\times\mathbb{N}^{n}, L>0.
\end{equation*}
We denote $A_{m}(T^{*}\mathbb{R}^{n})$ by $A_{m}.$
\end{Def}
We also use the notation
\[\Gamma_{\pm}(R, d, \sigma)=\left\{(x, \xi)\in\mathbb{R}^{n}\times\mathbb{R}^{n}:|x|>R,
\frac{1}{d}<|\xi|<d, \pm cos(x, \xi)>\pm\sigma\right\}\] with $R>1,$
$d>1,$ $\sigma\in(-1,
1),$ and $cos(x, \xi)=\frac{\langle x, \xi\rangle}{|x||\xi|},$ for the
outgoing and incoming
subsets of phase space, respectively.
For $\alpha>\frac{1}{2},$ we denote the bounded operator $F_{0}(\lambda, h):
L^{2}_{\alpha}(\mathbb{R}^{n})\rightarrow L^{2}(\mathbb{S}^{n-1})$ by
\begin{equation*}
\left(F_{0}(\lambda, h)f\right)(\omega)=(2\pi
h)^{-\frac{n}{2}}(2\lambda)^{\frac{n-2}{4}}\int_{\mathbb{R}^{n}}
e^{-\frac{i}{h}\sqrt{2\lambda}\langle \omega, x\rangle}f(x)dx, \lambda>0.
\end{equation*}
Let $R_{0}>>0,$ $1<d_4<d_3<d_2<d_1<d_0,$ and
$0<\sigma_4<\sigma_3<\sigma_2<\sigma_1<\sigma_0<1.$
Using the WKB method, Isozaki and Kitada \cite{IK} have constructed
parametrices for the wave operators with phase functions
$\Phi_{\pm}$ and symbols $a_{\pm}$ and $b_{\pm}$ such that:
\begin{enumerate}
\item $\Phi_{\pm}\in C^{\infty}(T^{*}\mathbb{R}^{n})$ solve the eikonal equation
\begin{equation}\label{eikonal}
\frac{1}{2}\left|\nabla_{x}\Phi_{\pm}(x, \xi)\right|^{2}+V(x)=\frac{1}{2}|\xi|^{2}
\end{equation}
in $(x, \xi)\in\Gamma_{\pm}(R_0, d_0, \pm\sigma_0),$ respectively.
\item $\Phi_{\pm}(\cdot, \cdot\cdot)-\langle\cdot, \cdot\cdot\rangle\in
A_{0}\left(\Gamma_{\pm}(R_0, d_0, \pm\sigma_{0})\right).$
\item For all $(x, \xi)\in T^{*}\mathbb{R}^{n}$
\begin{equation}\label{phinondeg}
\left|\frac{\partial^{2}\Phi_{\pm}}{\partial_{x_{j}}\partial_{\xi_{k}}}(x,
\xi)-\delta_{jk}\right|<\epsilon(R_0),
\end{equation}
where $\delta_{jk}$ is the Kronecker delta and $\epsilon(R_0)\to R_{0}$ as $R_{0}\to\infty.$
\item $a_{\pm}\sim\sum_{j=0}^{\infty}h^j a_{\pm j},$ where $a_{\pm j}\in
A_{-j}(\Gamma_{\pm}(3R_0, d_1, \mp\sigma_1)),$ $\supp a_{\pm j}\subset\Gamma_{\pm}(3R_0,
d_1, \mp\sigma_1),$ $a_{\pm j}$ solve
\begin{equation}\label{aeq1}
\langle \nabla_{x}\Phi_{\pm}, \nabla_{x}a_{\pm 0}\rangle
+\frac{1}{2}\left(\Delta_{x}\Phi_{\pm}\right)a_{\pm 0}=0
\end{equation}
\begin{equation}\label{aeq2}
\langle \nabla_{x}\Phi_{\pm}, \nabla_{x}a_{\pm
j}\rangle+\frac12\left(\Delta_{x}\Phi_{\pm}\right)a_{\pm j}=\frac{i}{2} \Delta_{x}a_{\pm
j-1}, j\geq 1,
\end{equation}
with the conditions at infinity
\begin{equation}\label{condinfty}
a_{\pm 0}\to 1, a_{\pm j}\to 0, j\geq 1, \text{ as } |x|\to \infty.
\end{equation}
in $\Gamma_{\pm}(2R_0, d_2, \mp\sigma_2),$
and solve \eqref{aeq1} and \eqref{aeq2} in $\Gamma_{\pm}(4R_0, d_1, \mp\sigma_2).$
\item $b_{\pm}\sim\sum_{j=0}^{\infty}h^j b_{\pm j},$ where $b_{\pm j}\in
A_{-j}(\Gamma_{\pm}(5R_0, d_3, \pm\sigma_{4}),$ $\supp b_{\pm j}\subset\Gamma_{\pm}(5R_0,
d_3, \pm\sigma_{4}),$ $b_{\pm j}$ solve \eqref{aeq1} and \eqref{aeq2} with the conditions
at infinity \eqref{condinfty} in $\Gamma_{\pm}(6R_0, d_4, \pm\sigma_3),$ and solve
\eqref{aeq1} and \eqref{aeq2} in $\Gamma_{\pm}(6R_0, d_3, \pm\sigma_{3}).$
\end{enumerate}
For a symbol $c$ and a phase function $\phi$, we denote by $I_{h}(c, \phi)$ the oscillatory integral
\begin{equation*}
I_{h}(c, \phi)=\frac{1}{(2\pi h)^{n}}\int_{\mathbb{R}^{n}} e^{\frac{i}{h}(\phi(x, \xi)-\langle y,
\xi\rangle)}c(x, \xi) d\xi
\end{equation*}
and let
\begin{equation*}
\begin{aligned}
K_{\pm a}(h) & =P(h)I_{h}(a_{\pm}, \Phi_{\pm})-I_{h}(a_{\pm}, \Phi_{\pm})P_{0}(h)\\
K_{\pm b}(h) & =P(h)I_{h}(b_{\pm}, \Phi_{\pm})-I_{h}(b_{\pm}, \Phi_{\pm})P_{0}(h).
\end{aligned}
\end{equation*}
The operator $T(\lambda, h)$ for $\lambda\in\left(\frac{1}{2d_{4}^{2}},
\frac{d_{4}^{2}}{2}\right)$ is then given by (see \cite[Theorem 3.3]{IK})
\begin{equation*}
T(\lambda, h)=T_{+1}(\lambda, h)+T_{-1}(\lambda, h)-T_{2}(\lambda, h),
\end{equation*}
where
\begin{equation*}
T_{\pm 1}(\lambda, h)=F_{0}(\lambda, h)I_{h}(a_{\pm}, \Phi_{\pm})^{*}K_{\pm b}(h)
F_{0}^{*}(\lambda, h)
\end{equation*}
and
\begin{equation*}
T_{2}(\lambda, h)=F_{0}(\lambda,
h)K_{+a}^{*}(h)R(\lambda+i0,
h)\left(K_{+b}(h)+K_{-b}(h)\right)F_{0}^{*}(\lambda, h),
\end{equation*}
\subsection{Two Preparatory Lemmas}\label{2ls}
The following two lemmas will be useful in studying the structure of the scattering amplitude.
\begin{Lem}\label{tempop}
Let $W=\mathcal{O}_{\mathcal{B}(L^{2}(\mathbb{R}^{n}))}(h^{s}),$ $h\to
0,$ or $W=\mathcal{O}_{\mathcal{B}(L^{2}_{\alpha}(\mathbb{R}^{n}),
L^{2}_{-\alpha}(\mathbb{R}^{n}))}(h^{s}),$ $h\to 0,$ for some $s\in\mathbb{R}.$
Then $K_W\in\mathcal{D}_{h}'(\mathbb{R}^{2n}).$
\end{Lem}
\begin{proof}
By Schwartz Kernel Theorem, for some $h_0>0$ and every $h\in(0, h_0],$ there
exists $w_h\in\mathcal{D}'(\mathbb{R}^{2n})$ such that $\langle T \varphi,
\psi\rangle=\langle w_h, \varphi\otimes\psi\rangle,$ $\varphi, \psi\in
C_{c}^{\infty}(\mathbb{R}^{n}).$
Let $\chi\in C_{c}^{\infty}(\mathbb{R}^{2n})$ and let $c_1>c_2>0$ be such that $\supp\chi\subset K_1(c_2)\times K_2(c_2),$ where $K_j(d)=\{x\in\mathbb{R}^{n}: |x_l|<d, l=1,
\dots, n\},$ $j=1, 2,$ $d>0.$
Let also $\rho_j\in
C_{c}^{\infty}(K_j(c_1)),$ $j=1, 2,$ be such that $\rho_1\times\rho_2=1$ on $K_1(c_2)\times K_2(c_2).$
Then, by the proof of Schwartz Kernel Theorem \cite[Theorem 6.1.1]{F}, we have that
\begin{equation*}
\left\langle w_h, \chi e^{-\frac{i}{h}\left(\langle \cdot,
\xi\rangle+\langle\cdot\cdot,
\eta\rangle
\right)}\right\rangle=\Sigma_{\mathbb{Z}^{n}\times\mathbb{Z}^{n}}\hat{\chi}_{m, k}\left\langle T\rho_1
E_{h}(\langle m, \cdot \rangle), \rho_2 E_{h}(\langle k, \cdot\cdot \rangle)\right\rangle,
\end{equation*}
where $E(t)=e^{\frac{2\pi i t}{b}},$ $t\in\mathbb{R},$ and $\hat{\chi}_{m,
k}=\frac{1}{b^{2n}}\int_{K_{1}\times K_{2}}\chi(x, y)e^{-\frac{i}{h}\left(\langle x, \xi\rangle+\langle y, \eta\rangle\right)}E(-m\cdot x- k\cdot y)
dx dy.$
Integration by parts now gives
\begin{equation}\label{est1}
(1+|m|)^{M}(1+|k|)^{M}\hat{\chi}_{m, k}\leq C_{1}h^{-2M}\langle (\xi,
\eta)\rangle^{M}\sum_{|\alpha|\leq M, |\beta|\leq M}\left\|\partial_{x}^{\alpha}\partial_{y}^{\beta}\chi\right\|_{L^{\infty}(\mathbb{R}^{2n})}, m, k\in \mathbb{Z}^{n},
M\in\mathbb{N}_{0}.
\end{equation}
We also have
\begin{equation}\label{est2}
\left|\left\langle T \rho_1 E_{h}(\langle m, \cdot \rangle), \rho_2 E_{h}(\langle k,
\cdot\cdot
\rangle)\right\rangle\right|\leq C_{2} h^{s}.
\end{equation}
From estimates \eqref{est1} and \eqref{est2} we obtain
\begin{equation}\label{kernelestim}
\left|\Sigma_{\mathbb{Z}^{n}\times\mathbb{Z}^{n}}\hat{\chi}_{m, k}\left\langle T\rho_1
E_{h}(\langle m, \cdot \rangle), \rho_2 E_{h}(\langle k, \cdot\cdot \rangle)\right\rangle
\right|\leq C_{3}\sum_{|\alpha|\leq M, |\beta|\leq M}\left
\|\partial_{x}^{\alpha}\partial_{x}^{\beta}\chi\right\|_{L^{\infty}(\mathbb{R}^{2n})} h^{s-2M}\langle (\xi,
\eta)\rangle^{M},
\end{equation}
with
\begin{equation*}
C_{3}=C_1 C_2 \sum_{\mathbb{Z}^{n}\times\mathbb{Z}^{n}}(1+|m|)^{-M}(1+|k|)^{-M}<\infty,
\end{equation*}
if $M$ is taken large enough.
Therefore $K_T\in\mathcal{D}'_{h}(\mathbb{R}^{2n}).$
\end{proof}
\begin{Lem}\label{rsym}
Let $\nu:\mathbb{R}^{2n}\to\mathbb{R}^{2n}$ be given by $\nu(x, y)=(y,
x).$
Then $\nu^{*}K_{R(\lambda+i0, h)}=K_{R(\lambda+i0, h)}$ for every
$\lambda>0.$
\end{Lem}
\begin{proof}
For $u, v\in L^{2}(\mathbb{R}^{n})$ let $\left\langle u, v\right\rangle=\int u v.$
Let $u$ and $v$ further satisfy $u, v\in C_{c}^{\infty}(\mathbb{R}^{n})$ and
let $z\in\mathbb{C}$ be such that $\Im z>0.$
We then have
\begin{equation}\label{symR}
\begin{aligned}
\left\langle R\left(z, h\right)u, v\right\rangle
& =\left\langle R\left(z, h\right)u, \left(P\left(h\right)-z\right)R\left(z,
h\right)v\right\rangle\\
&=\left\langle \left(P(h)-z\right)R\left(z,
h\right)u, R\left(z, h\right)v\right\rangle\\
&=\left\langle u, R\left(z, h\right)v\right\rangle.
\end{aligned}
\end{equation}
Let, now, $\lambda\in\mathbb{R}\backslash\left\{0\right\}$ and let
$\left(z_{k}\right)_{k\in\mathbb{N}}\subset\mathbb{C}$ satisfy $\Im z_{k}\downarrow 0,$
$k\to\infty,$ and $\Re z_{k}=\lambda,$ $k\in\mathbb{N}.$
Then, from (\ref{symR}) we have that for every $k$
\begin{equation}\label{zk}
\left\langle R\left(z_{k}, h\right)u, v\right\rangle
=\left\langle u, R\left(z_{k}, h\right)v\right\rangle.
\end{equation}
Letting $k\to\infty$ in (\ref{zk}) and using the fact that
\[R\left(\lambda+i0, h\right)=\lim_{\epsilon\downarrow 0} R(\lambda+i\epsilon,
h) \text{ in } \mathcal{B}\left(L^{2}_{\alpha}(\mathbb{R}^{n}),
L^{2}_{-\alpha}(\mathbb{R}^{n})\right),\,\alpha>\frac{1}{2},\]
we obtain
\begin{equation*}
\left\langle R\left(\lambda+i0, h\right)u, v\right\rangle
=\left\langle u, R\left(\lambda+i0, h\right)v\right\rangle.
\end{equation*}
Since $C_{c}^{\infty}(\mathbb{R}^{n})\otimes C^{\infty}_{c}(\mathbb{R}^{n})$ is dense in
$C^{\infty}_{c}(\mathbb{R}^{2n}),$ this completes the proof of the lemma.
\end{proof}
\section{Scattering Geometry}\label{sgeom}
In this section we describe the scattering relation and prove that it can be
parameterized by the modified actions \eqref{modaction} when the non-degeneracy assumption
holds.
The scattering relation is a Lagrangian submanifold of $T^{*}\mathbb{S}^{n-1}\times
T^{*}\mathbb{S}^{n-1},$ which relates the incoming and the outgoing data in the way
suggested by Figure \ref{fig:sr}.
\begin{figure}[t]
\begin{center}
\input{srsr.pstex_t}
\end{center}
\caption{The scattering relation consists of the points
$\left(\omega, z; \theta, -w\right)$ related as in this figure.}
\label{fig:sr}
\end{figure}
To make this precise, we first give the following
\begin{Def}\label{NTlambda}
The trajectory $\gamma\left(\cdot; x_{0}, \xi_{0}\right)$ is non-trapped if for every $r>0$,
there exists $T>0$ such that $\|x_{0}\|<r$ implies that
$\|x\left(s; x_{0}, \xi_{0}\right)\|>r$ for $|s|>T.$
The energy $\lambda>0$ is non-trapping if for every $r>0$ there exists $T>0$ such
that for every $(x_{0}, \xi_{0})\in T^{*}\mathbb{R}^{n}$ with
$\frac{1}{2}\|\xi_{0}\|^{2}+V(x_{0})=\lambda$ and $\|x_{0}\|<r$ we have $\|x\left(s;
x_{0}, \xi_{0}\right)\|>r$ for $|s|>T.$
We also introduce the notation $T(r)$ for the infimum over $s$
with this property.
\end{Def}
Let, now, $\lambda>0$ be such that the operator $P(h)-\lambda$ is of principal type.
Then $\Sigma_{\lambda}=p^{-1}(\lambda)$ is a smooth $2n-1$-dimensional submanifold of $T^{*}\mathbb{R}^{n}.$
Let, further, $\left(\omega_0, z_0\right)\in T^{*}\mathbb{S}^{n-1}$ be
such that
$\gamma_{\infty}\left(\cdot; z_0, \sqrt{2\lambda}\omega_0\right)$ is a
non-trapped
trajectory with $\xi_{\infty}\left(z_0,
\sqrt{2\lambda}\omega_0\right)\ne\omega_0.$
Then there exists $U\subset T^{*}\mathbb{S}^{n-1},$ open, $\left(\omega_0,
z_0\right)\in U,$ such
that for every $(\omega, z)\in U$ the trajectory $\gamma_{\infty}\left(\cdot; z,
\sqrt{2\lambda}\omega\right)$ is non-trapped and $\xi_{\infty}\left(z,
\sqrt{2\lambda}\omega\right)\ne\omega.$
By decreasing $U,$ if necessary, we therefore have that
\begin{equation}\label{defsr}
\begin{aligned}
SR_{\bar{U}}(\lambda)=\left\{\left(\omega, z; \xi_{\infty}\left(z,
\sqrt{2\lambda}\omega\right), x_{\infty}\left(z, \sqrt{2\lambda}\omega\right)\right): (\omega, z)\in \bar{U}\right\}'
\end{aligned}
\end{equation}
is a closed Lagrangian submanifold of $\left(T^{*}\mathbb{S}^{n-1}\times
T^{*}\mathbb{S}^{n-1}, \pi_{1}^{*}\sigma+\pi_{2}^{*}\sigma\right),$ which
we call a scattering relation at energy $\lambda$ (see Figure \ref{fig:sr}).
If $\lambda$ is a non-trapping energy level, we define the scattering relation at energy
$\lambda>0$ as
\begin{equation*}
SR(\lambda)=\Big\{\left(\omega, z; \xi_{\infty}\left(z, \sqrt{2\lambda}\omega\right), x_{\infty}\left(z, \sqrt{2\lambda}\omega\right)\right): \;(\omega, z)\in T^{*}\mathbb{S}^{n-1}, \omega\ne\xi_{\infty}\left(z, \sqrt{2\lambda}\omega\right)\Big\}'
\end{equation*}
We now show how, under the assumption that a certain outgoing direction is
regular for a given incoming direction, we can find a non-degenerate phase function which
parameterizes the scattering relation.
We begin with the following
\begin{Lem}\label{regL}
Let $\theta_0\in\mathbb{S}^{n-1}$ be regular for
$\omega_0\in\mathbb{S}^{n-1}.$
Then there exist $O_j\subset\mathbb{S}^{n-1},$ $j=1, 2,$ open,
$\omega_0\in O_1,$ $\theta_0\in O_2,$ and $L\in\mathbb{N}$ such
that for every $(\omega, \theta)\in O_1\times O_2$ the number of $(\omega,
\theta)$ trajectories is at least $L.$
\end{Lem}
\begin{proof}
By \cite[Remark 1.1]{M} and the discussion following it, we have
that
there exists $L\in\mathbb{N}$ such that the number of $(\omega_0,
\theta_0)$ trajectories is $L.$
Let $\left(z_{l}\right)_{l=1}^{L}\equiv\left(\xi_{\infty}^{-1}\left(\cdot,
\sqrt{2\lambda}\omega_0\right)\right)(\theta_0),$
By the Implicit Function Theorem, since $\theta_0$ is regular for $\omega_0,$ we have
that there exist open sets $O_1, O_2\subset\mathbb{S}^{n-1}$ with
$\theta_{0}\in O_1$ and $\omega_{0}\in O_2$ and functions $z_{l}\in
C^{\infty}(O_1\times O_2; \mathbb{R}^{n-1}),$ $l=1,
\dots, L,$ such that
$z_{l}(\omega_0, \theta_0; \lambda)=z_l$ and $\xi_{\infty}\left(z_{l}(\omega, \theta; \lambda), \sqrt{2\lambda}\omega\right)=\theta,$ $(\omega, \theta)\in O_1\times O_2,$ which completes the
proof.
\end{proof}
Let, now, $w_{l}(\omega, \theta; \lambda)=x_{\infty}\left(z_l\left(\omega, \theta;
\lambda\right),
\sqrt{2\lambda}\omega\right).$
As in \cite[Lemma 4]{AI}, we have the following
\begin{Lem}\label{nondeg2}
Let $\theta_0\in\mathbb{S}^{n-1}$ be regular for
$\omega_0\in\mathbb{S}^{n-1}.$
Then there exist $O_j\subset\mathbb{S}^{n-1},$ $j=1, 2,$ open,
$\omega_0\in O_1,$ $\theta_0\in O_2,$ such that the map
\[\theta^{\perp}\ni w\mapsto\xi_{\infty}\left(w,
-\sqrt{2\lambda}\theta\right)\in\mathbb{S}^{n-1}\] is
non-degenerate at $w_{l}\left(\omega, \theta\right),$ $\left(\omega, \theta\right)\in O_1\times O_2,$ $l=1, \dots, L.$
\end{Lem}
We now choose $O_{1}$ and $O_{2}$ in such a way that the conclusions of Lemmas \ref{regL}
and \ref{nondeg2} hold in some open neighborhoods of $\bar{O}_{1}$ and $\bar{O}_{2}$ and $\bar{O}_1\cap\bar{O}_2=\emptyset.$
We set
\begin{equation}\label{lagrl}
SR_l(\lambda)=\left\{\left(\omega, \theta, z_l(\omega, \theta; \lambda), -w_l(\omega,
\theta; \lambda)
\right): (\omega, \theta)\in \bar{O}_1 \times \bar{O}_2 \right\}.
\end{equation}
The same proof as in \cite[Lemma 3.2]{RT} now shows that there exist $\bar{R}>>0,$
$T_0>T\left(\bar{R}\right),$ and open sets $U^{l}_{\omega, \theta}\subset\omega^{\perp},$
$z_{l}\left(\omega, \theta; \lambda\right)\in U^{l}_{\theta, \omega},$
$l=1, \dots, L,$ $(\omega, \theta)\in \bar{O}_1\times \bar{O}_2,$ such that
\begin{equation}\label{candefaction}
\det\left(\frac{\partial x\left(t; \cdot,
\nabla_{x}\Phi_{-}\left(\cdot, \sqrt{2\lambda}\omega\right)(y)\right)}{\partial
y}\left(y\right)\right)\ne 0
\end{equation}
for $y\in\left\{x_{\infty}\left(s; z,
\sqrt{2\lambda}\omega\right)\cap B\left(0, \bar{R}, \bar{R}+1\right): z\in
U^{l}_{\omega, \theta}, s<0\right\},$ $t>T_0.$
Let, now, $t_0>T_0$ be fixed.
From \eqref{candefaction} it follows that for $(\theta, \omega)\in \bar{O}_1\times\bar{O}_2$ we define the
(modified)
action along the segment of the $\left(\omega,
\theta\right)$-trajectory $\gamma_l(\omega, \theta, \lambda)=\left(x_l(\omega, \theta, \lambda), \xi_l(\omega, \theta, \lambda)\right)=\gamma_{\infty}\left(\cdot; z_l(\omega, \theta; \lambda),
\sqrt{2\lambda}\omega\right),$
between the points
\[y_{l}\left(s; \omega, \theta, \lambda\right)=x_{\infty}\left(s; z_l\left(\omega, \theta; \lambda\right),
\sqrt{2\lambda}\omega\right)\cap B\left(0, \bar{R}, \bar{R}+1\right)\] for
some $s<0$ and $x_{l}(t_0; s, \omega, \theta, \lambda)
=x\left(t_0; y_l\left(s; \omega, \theta, \lambda\right), \nabla_{x}\Phi_{-}\left(y_l\left(s;
\omega, \theta, \lambda\right), \sqrt{2\lambda}\omega\right)\right)$ and we set
\begin{equation}\label{maction}
S_{l}\left(\omega, \theta\right)=\Phi_{-}\left( y_{l}\left(s;
\omega, \theta, \lambda\right),
\sqrt{2\lambda}\omega\right)+\int_{0}^{t_0} L\left(x, \dot{x}\right)dt-\Phi_{+}\left( x_l\left(t_0; s, \omega, \theta,
\lambda\right),
\sqrt{2\lambda}\theta\right)+\lambda t_0,
\end{equation}
where $L(x, \dot{x})=\frac{1}{2}\left\|\dot{x}\right\|^{2}_{g}-V(x)$ is
the Lagrangian, and the integral is taken over the segment of the bicharacteristic curve $x_l(\omega, \theta, \lambda)$
connecting $y_{l}\left(s; \omega, \theta, \lambda\right)$ and
$x_{l}\left(t_0; s, \omega, \theta, \lambda\right).$
From the representations \cite[(4.5)]{RT}
\begin{equation}\label{tail-}
\Phi_{-}(x, \sqrt{2\lambda}\omega)=2\tau\lambda+\int_{-\infty}^{\tau}\left(\frac{1}{2}\left|p_{\infty}
\left(t; z, \sqrt{2\lambda}\omega\right)\right|^{2}-V\left(q_{\infty}\left(t; z,
\sqrt{2\lambda}\omega\right)\right)-\lambda\right)dt
\end{equation}
for $x=q_{\infty}\left(\tau; z, \sqrt{2\lambda}\omega\right)\in B\left(0, \bar{R},
\bar{R}+1\right)$ and \cite[(4.4)]{RT}
\begin{equation}\label{tail+}
\begin{aligned}
\Phi_{+}(x, \xi)=2\lambda\tau &+\left\langle x_{\infty}\left(z, \sqrt{2\lambda}\omega\right),
\xi\right\rangle\\
& -\int_{\tau}^{\infty}\left(\frac{1}{2}\left|p_{\infty}\left(t; z,
\sqrt{2\lambda}\omega\right)
\right|^{2}-V\left(q_{\infty}\left(t; z,
\sqrt{2\lambda}\omega\right)\right)-\lambda\right)dt
\end{aligned}
\end{equation}
for $(x, \xi)\in\Gamma_{+}\left(R_0, d_0, -\sigma_0\right)$ with
$x=q_{\infty}\left(\tau; z, \sqrt{2\lambda}\omega\right),$
$\xi=\lim_{t\to\infty}p_{\infty}\left(t; z, \sqrt{2\lambda}\omega\right),$ we see that $S_l(\omega, \theta)$ is independent of the choice of $s$ with the specified properties.
We now have the following
\begin{Lem}\label{actionsr}
Let $\omega_{0}\in\mathbb{S}^{n-1}$ be regular for
$\theta_{0}\in\mathbb{S}^{n-1}.$
Then $SR_{l}(\lambda)=\Lambda_{S_{l}},$ where $\Lambda_{S_{l}}=\left\{\left(\omega, \theta,
d_{\omega}S_{l}, d_{\theta}S_{l}\right): (\omega, \theta)\in \bar{O}_1\times
\bar{O}_2\right\},$ $l=1, \dots, L.$
\end{Lem}
\begin{proof}
We consider
\begin{equation}\label{stheta}
\begin{aligned}
d_{\theta}S_{l}(\omega, \theta) & =d_{\theta}\left(\Phi_{-}\left(
y_l\left(s; \omega, \theta, \lambda\right),
\sqrt{2\lambda}\omega\right)+\int_{0}^{t_{0}} L\left(x,
\dot{x}\right)dt\right)-d_{\omega}\Phi_{+}\left(x_l\left(t_0; s, \omega, \cdot, \lambda\right), \sqrt{2\lambda}\cdot\right)(\theta)\\
& =\left\langle\xi\left(t_0; y_l\left(s; \omega, \theta, \lambda\right),
\nabla_{x}\Phi_{-}\left(y_{l}(s, \omega, \theta, \lambda), \sqrt{2\lambda}\omega\right)\right), d_{\theta}x_l(t_0; s, \omega, \cdot,
\lambda)(\theta)\right\rangle\\
&\quad\quad -\left\langle\nabla_{x}\Phi_{+}\left(x_{l}(t_0; s, \omega, \theta, \lambda), \sqrt{2\lambda}\theta\right), d_{\theta}x_l(t_0; s, \omega, \cdot,
\lambda)(\theta)\right\rangle\\
& \quad\quad -d_{\theta}\left\langle \nabla_{\xi}\Phi_{+}\left(x_l(t_0; s, \omega, \theta, \lambda), \sqrt{2\lambda}\theta\right), \sqrt{2\lambda}\cdot\right\rangle(\theta)\\
& = -d_{\theta}\left\langle \nabla_{\xi}\Phi_{+}\left(x_l(t_0; s, \omega, \theta, \lambda), \sqrt{2\lambda}\theta\right), \sqrt{2\lambda}\cdot\right\rangle(\theta),
\end{aligned}
\end{equation}
where \eqref{candefaction} has allowed us to use \cite[Theorem 46.C]{A}
to obtain the second equality.
Lastly, we recall from \cite[Lemma 4.1]{RT} that
\begin{equation}\label{xiphi+}
\lim_{t\to\infty}\left|x_{\infty}\left(t; z_l(\omega, \theta,
\lambda),\sqrt{2\lambda}\omega\right)-\sqrt{2\lambda}\theta
t-\nabla_{\xi}\Phi_{+}\left(x_l(t_0; s, \omega, \theta, \lambda),
\sqrt{2\lambda}\theta\right)\right|=0.
\end{equation}
To compute $d_{\omega}S_l$ we first reparameterize the phase
trajectories in the reverse direction, which is equivalent to
considering the reverse of the initial and final directions.
Using \eqref{tail-} and \eqref{tail+} we further re-write
$S_l\left(\omega, \theta\right)$ in the
following way
\begin{equation*}
S_l\left(\omega, \theta\right)=-\Phi_{+}\left(x_l\left(s; \omega, \theta,
\lambda\right), \sqrt{2\lambda}\theta\right) +\int_{0}^{t_{0}} L\left(x_l,
\dot{x}_l\right)dt+\Phi_{-}\left( y_l\left(t_0; s, \omega, \theta, \lambda\right),
\sqrt{2\lambda}\omega\right)+\lambda t_0,
\end{equation*}
where $x_l\left(s; \omega, \theta, \lambda\right)=x_{\infty}\left(s; z_{l}\left(\omega, \theta; \lambda\right), \sqrt{2\lambda}\omega\right)\cap B\left(0, \bar{R}, \bar{R}+1\right)$ for some $s>0,$
\[y_l\left(t_0; s, \omega, \theta, \lambda\right)=x\left(t_0;
x_l\left(s; \omega, \theta, \lambda\right),-\nabla_{x}\Phi_{+}\left(x_l\left(s; \omega, \theta, \lambda\right), \sqrt{2\lambda}\theta\right)\right),\]
and the integral is taken over the segment of the bicharacteristic curve $x_l(\omega,
\theta, \lambda)$ connecting
$x_l\left(s; \omega, \theta, \lambda\right)$ and $y_l\left(t_0; s, \omega, \theta, \lambda\right).$
We observe that this bicharacteristic curve is uniquely defined by Lemma \ref{nondeg2} and
\eqref{candefaction}.
Lemma \ref{nondeg2} and \eqref{candefaction} further allow us to proceed as in
(\ref{somega})
and we obtain
\begin{equation}\label{somega}
\begin{aligned}
d_{\omega}S_l\left(\omega,
\theta\right) & = d_{\omega}\left(-\Phi_{+}\left( x_l\left(s; \omega, \theta,
\lambda\right), \sqrt{2\lambda}\theta\right) +\int_{0}^{t_{0}} L\left(x,
\dot{x}\right)dt\right)\\
&\quad +d_{\omega}\Phi_{-}\left( y_l\left(t_{0}; s, \cdot,
\theta, \lambda\right), \sqrt{2\lambda}\cdot\right)(\omega)\\
& =d_{\omega}\left\langle \nabla_{\xi}\Phi_{-}\left(y_l(t_{0}; s, \omega, \theta, \lambda), \sqrt{2\lambda}\omega\right), \sqrt{2\lambda}\cdot\right\rangle(\omega).
\end{aligned}
\end{equation}
As above, we have that
\begin{equation}\label{xiphi-}
\lim_{t\to -\infty}\left|x_{\infty}\left(t; z_l(\omega, \theta,
\lambda),\sqrt{2\lambda}\omega\right)-\sqrt{2\lambda}\theta
t-\nabla_{\xi}\Phi_{-}\left(y_l(t_0; s, \omega, \theta, \lambda),
\sqrt{2\lambda}\omega\right)\right|=0.
\end{equation}
From (\ref{stheta}), \eqref{xiphi+}, \eqref{somega}, and \eqref{xiphi-} we
therefore have
that $S_l$ is
a non-degenerate phase function such that $SR_l(\lambda)=\Lambda_{S_{l}}.$
\end{proof}
We remark that \eqref{tail-} and \eqref{tail+} allow us to rewrite $S_l(\omega, \theta)$ in the following way
\begin{equation}\label{laction}
\begin{aligned}
S_l(\omega, \theta)=\int_{-\infty}^{\infty}\left(\frac{1}{2}\left|p_{\infty}\left(t; z_l(\omega, \theta), \sqrt{2\lambda}\omega\right)\right|^{2}-V\left(q_{\infty}\left(t; z_l(\omega, \theta), \sqrt{2\lambda}\omega\right)\right)-\lambda\right)dt\\
-\left\langle x_{\infty}\left(z_l(\omega, \theta), \sqrt{2\lambda}\omega\right),
\sqrt{2\lambda}\theta\right\rangle,
\end{aligned}
\end{equation}
which is the same as the modified actions given by \eqref{modaction}.
\section{Proof of Main Theorem}\label{pmain}
We now turn to the proof of the Main Theorem.
\begin{proof}
Since $S(\lambda, h)$ is a unitary operator on $L^{2}(\mathbb{S}^{n-1}),$
we have, by Lemma \ref{tempop}, that $K_{S(\lambda,
h)}\in\mathcal{D}'_{h}(\mathbb{S}^{n-1}\times\mathbb{S}^{n-1})$ and
therefore $K_{T(\lambda,
h)}\in\mathcal{D}'_{h}(\mathbb{S}^{n-1}\times\mathbb{S}^{n-1}).$
Since we are working away from the diagonal in $\mathbb{S}^{n-1}\times\mathbb{S}^{n-1}$
we can use integration by parts, as in \cite{RT} and \cite{M}, and obtain
\begin{equation*}
K_{T_{\pm 1}}=\mathcal{O}_{L^{2}(\mathbb{S}^{n-1}\times\mathbb{S}^{n-1}\backslash\diag
(\mathbb{S}^{n-1}\times\mathbb{S}^{n-1}))}(h^{\infty}).
\end{equation*}
Therefore, by \eqref{kernelestim}, we obtain
\begin{equation}\label{Tpm1}
WF_{h}^{f}\left(K_{T_{\pm 1}}\right)=\emptyset.
\end{equation}
We now observe that the proof of \cite[Lemma 2.1]{RT} depends only on the estimate
\eqref{resest} and the support properties of the symbols $a_{\pm}$ and $b_{\pm}$ and
therefore its assertion holds here as well and we have the following
estimates for $\gamma>\frac{n}{2}$ close to $\frac{n}{2}$
\begin{equation}\label{cutest}
\begin{aligned}
& \left\|K_{+a}^{*}(h)R(\lambda+i0, h)K_{+b}(h)\right\|_{-\gamma,
\gamma}=\mathcal{O}(h^{\infty})\\
& \left\|K_{+a}^{*}(h)R(\lambda+i0, h)(1-\chi_b)K_{-b}(h)\right\|_{-\gamma,
\gamma}=\mathcal{O}(h^{\infty})\\
& \left\|((1-\chi_a)K_{+a})^{*}(h)R(\lambda+i0, h)K_{-b}(h)\right\|_{-\gamma,
\gamma}=\mathcal{O}(h^{\infty}),
\end{aligned}
\end{equation}
where $\chi_a\in C_{c}^{\infty}\left(B\left(0, 20R_0+1\right)\right), \chi_a(x)=1,
|x|<20R_0$ and $\chi_b\in C_{c}^{\infty}\left(B\left(0, 10R_0+1\right)\right), \chi_b(y)=1,
|y|<10R_0.$
From \eqref{Tpm1}, \eqref{cutest}, and \eqref{kernelestim} we then
conclude, as in \cite[Corollary]{RT}, that
\begin{equation}\label{G0}
WF_{h}^{f}\left(\chi\left(K_{A(\lambda, h)}-c_1(n, \lambda, h)K_{G_{0}}\right)\right)=\emptyset,
\end{equation}
for every $\chi\in
C^{\infty}\left(\mathbb{S}^{n-1}\times\mathbb{S}^{n-1}\backslash
\diag(\mathbb{S}^{n-1}\times\mathbb{S}^{n-1})\right),$ where
\begin{equation*}
G_0(\theta, \omega; \lambda, h)
=\left\langle e^{-\frac{i}{h}\Phi_{+}\left(\cdot, \sqrt{2\lambda}\theta\right)}g_{+a}(\cdot,
\theta; h)\otimes e^{\frac{i}{h}\Phi_{-}\left(\cdot\cdot,
\sqrt{2\lambda}\omega\right)}g_{-b}(\cdot\cdot, \omega; h), K_{R(\lambda+i0,
h)}\right\rangle,
\end{equation*}
\begin{equation*}
g_{+a}(x, \theta; h)=e^{-\frac{i}{h}\Phi_{+}\left(x, \sqrt{2\lambda}\theta\right)}
[\chi_{a}, P_0(h)]a_{+}\left(x,
\sqrt{2\lambda}\theta; h\right)
e^{\frac{i}{h}\Phi_{+}\left(x, \sqrt{2\lambda}\theta\right)},
\end{equation*}
\begin{equation*}
g_{-b}(y, \omega; h)=e^{-\frac{i}{h}\Phi_{-}\left(y, \sqrt{2\lambda}\omega\right)}[\chi_{b}, P_0(h)]b_{-}\left(y, \sqrt{2\lambda}\omega; h\right)
e^{\frac{i}{h}\Phi_{-}\left(y, \sqrt{2\lambda}\omega\right)},
\end{equation*}
and
\begin{equation*}
c_1(n, \lambda, h)=2\pi(2\lambda)^{\frac{n-3}{4}}{(2\pi
h)^{-\frac{n+1}{2}}e^{-\frac{i(n-3)\pi}{4}}}.
\end{equation*}
Let, now, $\bar{p}\in SR_{U}(\lambda)$ be such that
$\tilde{\pi}_{1}\left(\bar{p}\right)=(\omega, z),$ where
$\tilde{\pi}_{1}: T^{*}\mathbb{S}^{n-1}\times
T^{*}\mathbb{S}^{n-1}\to T^{*}\mathbb{S}^{n-1}$ is the canonical projection onto the first
factor.
Let $A_{j}\in\Psi_{h}^{0}(1,
\mathbb{S}^{n-1}\times\mathbb{S}^{n-1}\backslash\diag(\mathbb{S}^{n-1}\times\mathbb{S}^{n-1})),$
$j=0, \dots, N,$ have compactly supported symbols near $\bar{p}$ and satisfy
$\sigma_{0}(A_j)|_{SR_{\bar{U}}(\lambda)}=0,$ $j<N.$
We also set $\varphi_{+}(x, \theta)=\Phi_{+}\left(x, \sqrt{2\lambda}\theta\right),$
$\left(x,
\sqrt{2\lambda}\theta\right)\in\Gamma_{+}(R_0, d_0, \sigma_0),$ and
$\varphi_{-}(y, \omega)=\Phi_{-}\left(y, \sqrt{2\lambda}\omega\right),$ $(y,
\omega)\in\Gamma_{-}(R-0, d_0, -\sigma_0).$
First, we shall prove that the generalization of Egorov's
Theorem to manifolds of unequal dimensions \cite[Lemma 7]{Afio} can be
applied to the
semi-classical Fourier integral operator $F$ given by the Schwartz kernel
\begin{equation*}
K_{F}=e^{-\frac{i}{h}\varphi_{+}}g_{+a}\otimes e^{\frac{i}{h}\varphi_{-}}g_{-b}.
\end{equation*}
For that, let
\begin{equation*}
\begin{aligned}
\Lambda_{F}=\Big\{\Big( & x, y, -\nabla_{x}\varphi_{+}\left(x, \theta\right),
\nabla_{y}\varphi_{-}\left(y, \omega\right);\, \theta, \omega,
-\nabla_{\theta}\varphi_{+}\left(x, \theta\right), \nabla_{\omega}\varphi_{+}\left(y, \omega\right):\\
& \left(x, \sqrt{2\lambda}\theta\right)\in\Gamma_{+}(R_0, d_0, \sigma_0)\cap
\left(T^{*}(\supp\nabla\chi_{a})\times T^{*}\mathbb{S}^{n-1}_{2\lambda}\right),\\
& \left(y, \sqrt{2\lambda}\omega\right)\in\Gamma_{-}(R_0, d_0,
-\sigma_0)\cap\left(T^{*}(\supp\nabla\chi_{b})\times T^{*}\mathbb{S}^{n-1}_{2\lambda}\right)\Big)\Big\}.
\end{aligned}
\end{equation*}
For every
$(x, \xi)\in\Gamma_{\pm}(R_0, d_0, \pm\sigma_0)$ there exist unique phase trajectories
$(q_{\pm}(\cdot; x, \xi), p_{\pm}(\cdot; x, \xi))$ such that $q_{\pm}(0; x, \xi)=x$ and
$\lim_{t\to\pm\infty}p_{\pm}(t; x, \xi)=\xi,$ respectively (see \cite[Subsection 4.1]{RT} as
well as the discussion following \cite[Definition 1.10]{I}).
Furthermore, by the construction of $\Phi_{\pm},$
\begin{equation*}
\nabla_{x}\Phi_{\pm}(q_{\pm}(t; x, \xi), \xi)=p_{\pm}(t; x, \xi).
\end{equation*}
By \cite[Lemma 4.1]{RT}, we also have that
\begin{equation*}
\lim_{t\to\pm\infty}\left|q_{\pm}(t; x, \xi)-\xi t-\nabla_\xi\Phi_\pm(x, \xi)\right|=0.
\end{equation*}
These considerations imply that
\begin{equation}\label{immerse}
\pi_{1}|_{\Lambda_{F}} \text{ is an immersion,}
\end{equation}
where
\begin{equation*}
\pi_{1}:T^{*}\mathbb{R}^{n}\times T^{*}\mathbb{R}^{n}\times T^{*}\mathbb{S}^{n-1}\times T^{*}\mathbb{S}^{n-1}\to
T^{*}\mathbb{R}^{n}\times T^{*}\mathbb{R}^{n}
\end{equation*}
is the canonical projection.
With \eqref{immerse} the hypotheses of the generalization of
Egorov's
Theorem to manifolds of unequal dimensions \cite[Lemma 7]{Afio} are
satisfied and applying \cite[Lemma 7]{Afio} we obtain that there exist
$B_{j}\in\Psi_{h}^{0}(1,
\mathbb{R}^{n}\times\mathbb{R}^{n}),$ $j=0, \dots, N,$ satisfying the
following conditions
\begin{enumerate}
\item $\sigma(B_j),$ $j=0, \dots, N,$ have compact support near
a point $\bar{q}\in T^{*}\mathbb{R}^{n}\times T^{*}\mathbb{R}^{n}$ such that
$\hat{\pi}_{1}\left(\bar{q}\right)\in\gamma_{\infty}\left(\cdot; z, \sqrt{2\lambda}\theta\right),$ where
$\hat{\pi}_{1}: T^{*}\mathbb{R}^{n}\times T^{*}\mathbb{R}^{n}\to T^{*}\mathbb{R}^{n}$
is the canonical projection onto the first factor.
\item $\sigma_{0}(B_j)|_{\Lambda_{R}(\lambda)}=0, j<N,$ where
$\Lambda_R(\lambda)=\cup_{t>0}\left(\graph\exp(tH_{p})|_{\Sigma_{\lambda}}\right)'.$
\item Near $\left(\bar{p}, \bar{q}\right),$
\begin{equation}\label{interwine}
\left(\prod_{j=0}^{N}A_{j}\right)\left(e^{-\frac{i}{h}\phi_{+}}g_{+a}\otimes
e^{\frac{i}{h}\phi_{-}}g_{-b}\right)\equiv
\left(e^{-\frac{i}{h}\phi_{+}}g_{+a}\otimes e^{\frac{i}{h}\phi_{-}}g_{-b}\right)
\left(\prod_{j=0}^{N}B_j\right).
\end{equation}
\end{enumerate}
Assumption \eqref{resest} and Lemma \ref{tempop}, now, imply that
$K_{R(\lambda+i0, h)}\in\mathcal{D}_{h}'(\mathbb{R}^{2n}).$
From \eqref{interwine} we therefore obtain
\begin{equation}\label{interw2}
\left(\prod_{j=0}^{N}A_{j}\right)K_{A(\lambda, h)}\equiv
c_{1}\left(n, \lambda, h\right)
\left(e^{-\frac{i}{h}\phi_{+}}g_{+a}\otimes e^{\frac{i}{h}\phi_{-}}g_{-b}\right)
\left(\prod_{j=0}^{N}B_j\right)\left(\chi_{2}\otimes\chi_{1}\right)K_{R(\lambda+i0, h)},
\end{equation}
near $\left(\bar{p}, \bar{q}\right),$ where $\chi_j\in
C_{c}^{\infty}(\mathbb{R}^{n}; \mathbb{R}),$ $j=1, 2,$ are such that
$\chi_2=1$ on $\supp g_{+a},$ $\chi_1=1$ on $\supp g_{-b},$ and
$\supp\chi_1\cap\supp\chi_2=\emptyset.$
Estimate \eqref{resest}, Lemma \ref{rsym}, and the same proof as in
\cite[Theorem 1]{AI} further give that there exists an open set
$V\subset\Lambda_{R}(\lambda),$ $\bar{q}\in V,$ such that
$\left(\chi_{2}\otimes\chi_{1}\right)K_{R(\lambda+i0, h)}\in
I_{h}^{1}\left(\mathbb{R}^{2n}, \Lambda_{R}(\lambda)\cap\bar{V}\right).$
(We recall here that the fact that $\chi_1$ and $\chi_2$ have disjoint support is crucial in
the proof of \cite[Theorem 1]{AI}.)
Therefore
\begin{equation}\label{estR}
\left(\prod_{j=0}^{N}B_j\right)\left(\chi_{2}\otimes\chi_{1}\right)K_{R(\lambda+i0,
h)}
=\mathcal{O}_{L^{2}(\mathbb{R}^{2n})}\left(h^{N-1-\frac{n}{2}}\right), h\to 0.
\end{equation}
Since $g_{+b}, g_{-a}\in S^{-1}_{2n-1}(1)\cap
C_{c}^{\infty}(\mathbb{R}^{n}\times\mathbb{S}^{n-1}),$ we easily find that
\begin{equation}\label{estF}
\left\|\left(e^{-\frac{i}{h}\phi_{+}}g_{+a}\otimes e^{\frac{i}{h}\phi_{-}}g_{-b}\right)
\right\|_{\mathcal{B}(L^{2}(\mathbb{R}^{n}), L^{2}(\mathbb{S}^{n-1}))}=\mathcal{O}(h).
\end{equation}
Estimates \eqref{estR} and \eqref{estF} together with \eqref{G0} and
\eqref{interw2} now imply that
\begin{equation*}
\left(\prod_{j=0}^{N}A_{j}\right)K_{A(\lambda, h)}=
\mathcal{O}_{L^{2}(\mathbb{S}^{n-1}\times\mathbb{S}^{n-1})}\left(h^{N-n-\frac{3}{2}}\right),
\end{equation*}
and therefore
\begin{equation*}
A(\lambda, h)\in \mathcal{I}_{h}^{\frac{n}{2}+2}\left({\mathbb
S}^{n-1} \times{\mathbb S}^{n-1}\backslash\diag(\mathbb{S}^{n-1}\times\mathbb{S}^{n-1}),
SR_{\bar{U}}(\lambda)\right).\qedhere
\end{equation*}
\end{proof}
\section{Applications}
In this section we discuss two applications of our Main Theorem to trapping and non-trapping
energies, respectively.
\subsection{Non-Trapping Energies}\label{snt}
\begin{Co}\label{ntfio}
Let $\lambda>0$ be a non-trapping energy level for $P$ and such that $P(h)-\lambda$ is of
principal type.
Then $A(\lambda, h)\in \mathcal{I}_{h}^{\frac{n}{2}+2}\left({\mathbb
S}^{n-1} \times{\mathbb S}^{n-1}\backslash\diag(\mathbb{S}^{n-1}\times\mathbb{S}^{n-1}),
SR(\lambda)\right).$
\end{Co}
\begin{proof}
From \cite[Lemma 2.2]{RT} we have that $\left\|R(\lambda+i0, h)\right\|_{\alpha,
-\alpha}=\mathcal{O}\left(\frac{1}{h}\right), \alpha>\frac{1}{2}.$
The result now follows from the Main Theorem.
\end{proof}
\subsection{Trapping Energies}\label{str}
\begin{Co}\label{tfio}
Let $\lambda>0$ be a trapping energy level for $P$ and such that $P(h)-\lambda$ is of
principal type.
Let also
\begin{enumerate}[(i)]
\item there exist $\theta_0\in [0, \pi),$ $R>0$ such that the potential $V$
extends holomorphically to the
domain $D_{R, \theta_{0}}=\{z\in\mathbb{C}^{n}: |z|>R, |\Im z|\leq \tan\theta_0|\Re z|\}$ and $|V(x)|\leq C|x|^{-\beta}$ for all $x\in D_{R, \theta_{0}}$ and some $\beta>0,$ $C>0,$
and
\item $Res(P(h))\cap([\lambda-\epsilon, \lambda+\epsilon]+i[0, Ch^{M}])=\emptyset$ for some $\epsilon>0,$ $C>0,$ and $M>0.$
\end{enumerate}
Lastly, let there exist $(\theta, z)\in T^{*}\mathbb{S}^{n-1}$ such that
$\gamma_{\infty}\left(\cdot; z, \sqrt{2\lambda}\theta\right)$ is a
non-trapped
trajectory.
Then there exists an open set $U\subset T^{*}\mathbb{S}^{n-1}$ such that
\begin{equation*}
A(\lambda, h)\in
\mathcal{I}_{h}^{\frac{n}{2}+2}\left({\mathbb S}^{n-1} \times{\mathbb
S}^{n-1}\backslash\diag(\mathbb{S}^{n-1}\times\mathbb{S}^{n-1}),
SR_{\bar{U}}(\lambda)\right).
\end{equation*}
\end{Co}
\begin{proof}
We choose $U$ as in Definition \ref{defsr}.
From \cite[Proposition 4.1]{M}, we have that
there exists
$m\in\mathbb{N}$ such that
\begin{equation*}
\left\|R(\lambda+i0, h)\right\|_{\alpha, -\alpha}
=\mathcal{O}\left(\frac{1}{h^{m}}\right), \alpha>\frac{1}{2}.
\end{equation*}
The assertion of the Corollary now follows from the Main
Theorem.
\end{proof}
\subsection{Microlocal Representation of the Scattering Amplitude}\label{smicrol}
Here we show how under the non-degeneracy assumption the expansion \eqref{vexpansion}
follows from the results we have proved in this article and the
characterization of semi-classical
Fourier integral distributions as oscillatory integrals, which we have
developed in \cite[Theorem 1]{AI}.
More precisely, we have the following
\begin{Th}\label{tmicrol}
Let $\omega_0\in\mathbb{S}^{n-1}$ be regular for
$\theta_0\in\mathbb{S}^{n-1}$ and
$L\in\mathbb{N}$ be the number of $(\theta_0, \omega_0)$ phase
trajectories.
Let $\lambda>0$ be such that $P-\lambda$ is of principle type and
$\left\|R(\lambda+i0, h)\right\|_{\alpha, -\alpha}=\mathcal{O}(h^{m}),$
$m\in\mathbb{R},$ $\alpha>\frac{1}{2}.$
Then, if $P_l\in\Psi_{h}^{0}(1, \mathbb{S}^{n-1}\times\mathbb{S}^{n-1}),$ $l=1,
\dots, L,$ are microlocal cut-offs to the Lagrangian submanifolds
$SR_l(\lambda)$ defined by
\eqref{lagrl},
respectively,
\begin{equation*}
P_l K_{A(\lambda, h)}=e^{\frac{i}{h}S_{l}}a_{l},\: l=1, \dots, L,
\end{equation*}
where $S_{l},$ $l=1, \dots, L,$ are as given by
(\ref{maction}) and $a_l\in S_{2n-2}^{n+\frac{3}{2}}(1),$ $l=1, \dots, L,$
have compact support.
\end{Th}
\begin{proof}
By our Main Theorem, $A(\lambda,
h)\in\mathcal{I}_{h}^{\frac{n}{2}+2}\left(\mathbb{S}^{n-1}\times\mathbb{S}^{n-1},
\cup_{l=1}^{L}SR_{l}(\lambda)\right).$
From \cite[Lemma 5]{AI} it follows that $P_l K_{A(\lambda,
h)}\in I_{h}^{\frac{3}{2}}\left(\mathbb{S}^{n-1}\times\mathbb{S}^{n-1},
SR_{l}(\lambda)\right),$ $l=1, \dots, L.$
With this and Lemma \ref{actionsr} the hypotheses of \cite[Theorem 1]{AI} are
satisfied and we obtain that there exist $a_l\in S_{2n-2}^{n+\frac{3}{2}}(1),$ $l=1, \dots,
L,$ such that $P_l K_{A(\lambda, h)}=e^{\frac{i}{h}S_{l}}a_{l}$ microlocally near
$SR_{l}(\lambda),$ $l=1, \dots, L.$
\end{proof}
We remark that the conclusion of this theorem holds whenever we have polynomial bound on the
resolvent.
We also remark that this theorem recovers the phases \eqref{modaction} in
\eqref{vexpansion}, due to \eqref{laction}.
|
{
"timestamp": "2004-11-26T18:07:11",
"yymm": "0411",
"arxiv_id": "math/0411599",
"language": "en",
"url": "https://arxiv.org/abs/math/0411599"
}
|
\section{Introduction}
In 1970 Jacob T.\ Schwartz launched the computable set theory
longterm project \cite{Sch}, which aimed to merge set theory and
theoretical computer science with reciprocal benefits.
Since then,
this research field revealed its pure combinatorial behavior.
Ten years later, M.\ Breban (cf.\ \cite{BreFe}) made an attempt to solve
the decidability problem for the language consisting of
the conjuctions of literals of
the following forms:
\begin{equation}
\begin{array}{llll}
v=w, & v\neq w, & v=\emptyset, & v=u \cup w,\\
v=u \cap w, & v=u \setminus w, & v \subseteq u, & v
\not\subseteq u,\\
v\in w, & v\notin w, & v=\pow{w}, &
v=\{w_{0},w_{1},\dots,w_{H}\},
\end{array} \label{literals}\tag{$\mathbf{\dagger}$}
\end{equation}
Breban was able to solve the problem allowing at most one occurrence
of the powerset operator.
Indeed, this unquantified language, known as MLSSP (i.e.,\
\emph{Multi-Level Syllogistic with Singleton and Powerset
operators}),
shows how drastically the complexity of combinatorics increases,
as one enriches the language
with new strong set constructors.
In \cite{Fe} Ferro solved the problem with two occurrences
of the powerset constructor;
whereas Cantone (see \cite{C1}),
exploiting a more sophisticated approach, solved the whole
decidability problem for MLSSP,
without any restriction on the number of occurrences.
However, any attempt to use the same
simple combinatorial approach to lengthen the list of set constructors
(in a non trivial way), crashed against the
fact that such languages build formulas which force
any model to be infinite.
Therefore, one of the main goals in solving advanced decidability problems
is to find a way to overcome the impossibility
to find finite models not exceeding a fixed size.
Recently (see \cite{CU97}),
the use of formative processes as a \emph{history} of a set assignment
gave a new perspective to solve this kind of problems.
Indeed, it makes use of the history (or trace) of the model
to obtain new information in order to decrease
the size of the model up to a suitable one.
This observation
motivated our interest to the study of a small model property for
languages which contain MLSSP.
In \cite{Cantone-Omodeo-Ursino02}, we
discovered the small model property
for MLSSP, and,
by means of this result, we built a satisfiability decision algorithm.
If we add to MLSSP particular set constructors, the small model property fails to hold.
A rather explicit example is the finiteness operator $\mathit{Finite}(x)$
(meaning that the cardinality of the set designated by $x$ is
smaller than $\aleph_0$).
Of course, since we admit negation
among propositional connectives, we must also take into account
literals of the form ${\bf\neg}\,\mathit{Finite}(x)$.
Thus MLSSP, extended with
the monadic relator $\mathit{Finite}$, ``forces the infinity"
(informally, a language forces the infinity whenever
has inside formulas whose models must be of infinite size).
The same happens allowing
the unitary union operator $\bigcup (x)$.
As consequence, languages which allow the use of
this type
of operators cannot satisfy the small model property.
This gave us the suggestion to focus on the structure of infinite models
(in particular, to their combinatorial features).
Hence we formulate
\begin{myproblem}
Which combinatorial properties two assignments
have to share, in order to satisfy the same
{\rm MLSSP}-like literals?
\end{myproblem}
Corollary \ref{MsatisfiesII} below gives a satisfactory
answer to this question. In Corollary~\ref{upwards},
we provide an analogous result, but referred to the
formative processes of the assignments.
These two corollaries are the tool to prove how
a finite assignment can be equipped with a special structure
that allows to increase some variables,
without affecting the validity of the formula.
We agree to denote such variables as \emph{potential infinite variables},
and we find a condition for this property to hold.
The above results allow us to investigate
\begin{myproblem}
Even if a language forces the infinity, is it still possible,
for any satisfiable formula, to
exhibit a finite assignment that
witnesses this satisfiability or, in other terms, to show a finite representation of an infinite model?
\end{myproblem}
This kind of property of languages is here introduced as
\emph{witness-small model property}.
Theorem~\ref{pumping1}
demonstrates how a combinatorial property of a finite assignment to a formula of MLSSPF can
witness the satisfiability of literals which require an infinite assignment.
More generally, this paper shows how
the formative processes can be used in order to prove the witness-small model property
in some cases.
This result
leads to the solution of
some open problems, such as the decidability
of languages which allow the use of the above-cited set constructors, namely,
MLSSP extended with
the monadic relator $\mathit{Finite}$ (the so-called MLSSPF) \cite{COU2}, and
MLSSP extended with
the monadic operator $\bigcup (x)$ (known as MLSSPU) \cite{CU}.
Our method is based on a specific analysis, both of the model and of a
formative process which generates it.
An detailed treatment of the general features of computable
set theory can be found in \cite{CFO} and \cite{COP}.
\section{Basic notations and background}
For the reader's convenience, we provide in this section
brief description of the standard
tools used in set computable theory.
For usual set theoretic notion we refer to any textbook of the field
(see \cite{JEC}, for example), instead
a complete survey of the specific notions mentioned in the sequel
may be found in \cite[\S 2]{Cantone-Omodeo-Ursino02}.
\subsection{Assignments and models}
Fix allowed forms for literals. A propositional combinations of
literals of such forms is said a
\emph{formula}.
It is customary to denote \emph{language of set theory}
the family of
all formulas built with assigned forms of literals.
Assume $\Phi$ is a formula and
let ${\mathcal M}\in\{\:\mbox{\rm sets}\:\}^{\mathcal
X}$ be a set-valued assignment
defined on the collection ${\mathcal X}_{\Phi}$
of variables in $\Phi$. If ${\mathcal M}$ satisfies all the literals,
it is said to be
a \emph{model} for $\Phi$.
A model is \emph{rank-bounded} by $k$ if the rank
of any set involved
in the assignment does not exceed $k$.
\begin{mydef}\label{}\rm
A language satisfies the \emph{small model property} if there exists
a computable natural function $f$ such that for any given
formula $\Phi$ of that language
and any model
${\mathcal M}$ of $\Phi$ there is a finite model ${\mathcal M}'$
rank-bounded by $f(\vert {\mathcal X}_{\Phi}\vert)$.
\end{mydef}
Assume that $\Phi$ is a formula of a language, and
${\mathcal A}$ is a set assignment to its variables ${\mathcal X}_{\Phi}$.
We say that ${\mathcal A}$ \emph{witnesses the satisfiability of}
$\Phi$ (even if ${\mathcal A}$ is not a model for $\Phi$),
provided that the structure of ${\mathcal A}$ allows to infer the satisfiability of
$\Phi$.
A formula of set theory \emph{forces the infinity} if it possesses
a variable $x$
such that, for any model ${\mathcal M}$ which satisfies the formula,
${\mathcal M}(x)$ is of infinite size.
From this point of view, a formula which forces the infinity cannot
have a finite model, but it could have a finite assignment which witnesses its
satisfiability. Hence the following
definition makes sense:
\begin{mydef}\label{}\rm
A language satisfies the \emph{witness-small model property}
if there exists
a computable natural function $f$ such that for any given
formula $\Phi$ of that language
and any model
${\mathcal M}$ of $\Phi$
there exists a finite assignment ${\mathcal A}$
rank-bounded by $f(\vert {\mathcal X}_{\Phi}\vert)$ which witnesses the
satisfiability of $\Phi$.
\end{mydef}
\subsection{Transitive partitions and syllogistic boards}
\begin{mydef}\label{defPart}\rm
A family $\Sigma$ of pairwise nonempty disjoint sets
is called a \emph{partition} (of
$\bigcup\Sigma$). Its members are the \emph{blocks} of
$\Sigma$.
The set
$\varsigma_{\Sigma}\stackrel{_{\tiny {\rm def}}}{_{\normalsize =}}\pow{\bigcup\Sigma}\setminus\bigcup\Sigma$
(often denoted simply by $\varsigma$) will occasionally be
treated as a block of the partition too. In this case, it is called the
\emph{outer block} of $\Sigma$.
\end{mydef}
As is well known, the function
$$\Sigma\stackrel{\sim_{\Sigma}}{\longmapsto}\{[X,Y]\,
\mbox{\tt|}\:(\exists\,b\in\Sigma)( X\in b\:\&\: Y\in b)\}
$$
establishes
a one-to-one correspondence between the partitions of a given set $S$
and the equivalence relations on $S$.
A useful relation $\sqsubseteq$ on $\pow{\pow{ S}}$ is defined by
setting
$${\mathcal B}\sqsubseteq{\mathcal
A}\;\iffAs\;(\forall\,a\in{\mathcal A})(\exists\,B\subseteq{\mathcal
B})\:a=\bigcup B\:.$$
The relation ${\mathcal B}\sqsubseteq{\mathcal A}$ reads
`` $\mathcal B$ is {\em finer} than $\mathcal A$ ", or
`` $\mathcal A$ is {\em coarser} than $\mathcal B$". This obviously is a
{\em preorder} relation that, when restricted to the set
$\varpi( S )$ of all partitions of
$S$, $\sqsubseteq$, becomes a {\em partial ordering}.
\begin{mydef}\label{transitivePartition}\rm
A partition $\Sigma$ is
said to be \emph{transitive} if $\bigcup\Sigma$ is transitive.
\end{mydef}
We consider a finite set
$\mathcal{P}$, whose elements are called \emph{places} and whose
subsets are called \emph{nodes}. Places and nodes will be the
vertices of a directed bipartite graph $\mathcal{G}$ of a special
kind, called a \emph{$\mathcal{P}$-board}. The edges issuing from
each place $q$ are, mandatorily, all pairs $q,B$ such that $q\in
B\subseteq \mathcal{P}$. The remaining edges of $\mathcal{G}$ must
lead from nodes to places. Hence, $\mathcal{G}$ is fully
characterized by the so called \emph{target function}
$$T\:\in\:\pow{\mathcal{P}}^{\pow{\mathcal{P}}},$$
associating with each node $A$ the set of all places $t$ such
that $\langle A,t\rangle$ is an edge of $\mathcal{G}$. The
elements of $\Targets{A}$ are called the \emph{targets} of
$A$. We will usually represent $\mathcal{G}$ simply by
$T$.
Places and nodes of a $\mathcal{P}$-board are meant to represent the
blocks $\sigma$, and the subsets $\Gamma$ (or, quite often, their
unionsets $\bigcup\Gamma$), of a transitive partition $\Sigma$,
respectively. Moreover, in this case,
there is a quite natural way to define the above-mentioned
directed bipartite graph structure.
For our convenience we define the further operator
$$A\ni\in B=_{def}A\cap B\ne\emptyset.$$
For any set $X$, we put
$$\powast{ X }\stackrel{_{\tiny {\rm def}}}{_{\normalsize =}}\{\,Y\,\mbox{\tt|}\:Y\subseteq \bigcup X\:\&\:
(\forall\,z\in X)(\:z\ni\in Y\:)\,\}\:,$$
that is, the elements of the family $\powast{X}$ are all the
sets $Y$ that can be obtained by extracting from each $z\in X$ a nonnull
$W_{z}\subseteq z$, so forming $Y=\bigcup_{z\in X}W_{z}$.
\begin{mydef}\rm
A transitive partition $\Sigma$ is said to
\emph{comply with} $\mathcal{G}$ via $q\mapsto
q^{(\bullet)}$, where $\mathcal{G}$ is
$\mathcal{P}$-board,
$q\mapsto q^{(\bullet)}$ belongs to
$\Sigma^{\mathcal{P}}$
and $T (A)=\{q\mid\powast{A^{(\bullet)}}
\ni\in q^{(\bullet)}\}$, if the function $T$ satisfies all the
properties required by $\mathcal{G}$, as indicated above (in
particular, this requires $q\mapsto q^{(\bullet)}$ to be
injective).
\end{mydef}
Any such board is said to be \emph{induced} by $\Sigma$
(for short, a \emph{$\Sigma$-board}).
We denote a transitive $\Sigma$-board by a couple $(\Sigma,\mathcal{G})$,
where $\Sigma$ is a transitive partition and
$\mathcal{G}$ is the induced $\mathcal{P}$-board.
For the purposes of this paper, some additional structure must be
superimposed on $\mathcal{P}$-boards:
\begin{mydef}\rm
A $\mathcal{P}$-board $\mathcal{G}=(T,\mathcal{F},\mathcal{Q})$
is said to be \emph{colored} when it has
\begin{itemize}
\item a designated set $\mathcal{F}$ of places,
\item a designated set $\mathcal{Q}$ of nodes, such that
$D\in\mathcal{Q}$ holds whenever $D\subseteq B\in\mathcal{Q}$ \\ (in
short, $\bigcup\mbox{\textclg{P}}[\mathcal{Q}]\subseteq\mathcal{Q}$ ), and
\item a target function $T$.
\end{itemize}
The places in $\mathcal{F}$ are said to be \emph{red}, the ones in
$\mathcal{P}\setminus\mathcal{F}$ are said to be \emph{green}; the nodes
in $\mathcal{Q}$ are called \emph{$\mbox{\textclg{P}}$-nodes}.
A node is red if all places in it are red, and
green otherwise; a list of vertices is
green if all vertices lying on it are green.
\end{mydef}
\begin{mydef}\label{SimBoard}\rm
Let ${\mathcal G}$ be a colored transitive \emph{$\Sigma$-board}. Then
$\widehat{\Sigma}$ is said to \emph{simulates}
$(\Sigma,{\mathcal G})$ \emph{upwards},
when there is a bijection
$\beta\in\widehat{\Sigma}^{\Sigma}$ such that
\begin{itemize}
\item $\widehat{\Sigma}$ $\in$-simulates $\Sigma$ via
$\beta$. That is, $\bigcup\beta[X]\in\bigcup\beta[Y]$ if and only if
$\bigcup X\in\bigcup Y$, for $X,Y\subseteq\Sigma$;
\item $\widehat{\Sigma}$ $\mbox{\textclg{P}}$-simulates $\Sigma$ via
$\beta$. That is, $\bigcup\beta[X]=\pow{\bigcup\beta[Y]}$ if
$\bigcup
X=\pow{\bigcup Y}$, for $Y\in\mathcal{Q}$ $X,Y\subseteq\Sigma$.
\item $\widehat{\Sigma}$ $Red$-simulates $\Sigma$ via
$\beta$. That is,
if $\sigma\in\mathcal{F}$, then $\card{\beta(\sigma)}=\card{\sigma}$;
\end{itemize}
\end{mydef}
As far as the Boolean constructs
$\emptyset,\cap,\setminus,\cup,=,\neq,\subseteq, \not\subseteq$
are concerned, all relevant information about a family of sets is
conveyed by the following structure:
\begin{mydef}\rm
Given a family $\mathcal F$, the \emph{Venn partition} of $\mathcal F$
is the coarsest partition $\Sigma$ of ~$\bigcup{\mathcal F}$ which
fulfill the condition
$$(\forall\,x\in{\mathcal
F})(\forall\,p\in\Sigma)(\:p\ni\in x\;\rightarrow\;p\subseteq
x\:).$$
\end{mydef}
Assume that $\Phi$ is a collection of literals which have one of the forms
$($\ref{literals}$)$, and let
${\mathcal M}\in\{\:\mbox{\rm sets}\:\}^{{\mathcal X}_{\Phi}}$
be a set-valued assignment defined on the collection
${\mathcal X}_{\Phi}$
of variables in $\Phi$.
We denote by $\Sigma_{{\mathcal X}_{\Phi}}$
the Venn partition of the set ${\mathcal M}[{\mathcal X}_{\Phi}]$, and by $\Im_{{\mathcal M}}$
the function $\Im_{{\mathcal M}}\in\pow{\Sigma_{{\mathcal X}_{\Phi}}}^{{\mathcal X}_{\Phi}}$ such that
${\mathcal M}(v)= \bigcup\Im_{{\mathcal M}}(v)$ holds for every $v$ in ${\mathcal X}_{\Phi}$.
\begin{myremark}\rm
Observe that any formula $\Phi$ with variables ${\mathcal X}_{\Phi}$
of a language resulting from an extension of Multi Level Syllogistic can be modified,
without affecting its satisfiability, in such a way any model ${\mathcal M}$ generates a transitive
$\Sigma_{{\mathcal X}_{\Phi}}$ \cite[pp.195-196]{C1}.
Because of that, from now on we shall assume that
$\Sigma_{{\mathcal X}_{\Phi}}$ is transitive, for any model ${\mathcal M}$
of a formula $\Phi$
with variables ${\mathcal X}_{\Phi}$.
\end {myremark}
Whenever literals as $v=\pow{w}$ and $Finite(v)$
appear in $\Phi$, $\Sigma_{{\mathcal X}_{\Phi}}$ can
be naturally transformed into a colored \emph{$\Sigma_{{\mathcal X}_{\Phi}}$-board}
${\mathcal G}=(T,\mathcal{F},\mathcal{Q})$
(i.e., the $\Sigma$-board ${\mathcal G}$
induced by $\Sigma_{{\mathcal X}_{\Phi}}$), in the following way.
\begin{itemize}
\item[(a)] $\mathcal{F}=\bigcup\{\Im(v)\mid \mbox{ for all literals of the
form $v=\{w_{1},\dots,w_{H}\}$ and $Finite(v)$ in $\Phi$}\}$;
\item[(b)] $\mathcal{Q}$ is equal to the minimal collection
of nodes such that
\begin{itemize}
\item $\Im(u)\in\mathcal{Q}$ for all literals of the form $u=\pow{w}$ in $\Phi$, and
\item $\bigcup\mbox{\textclg{P}}[\mathcal{Q}]\subseteq\mathcal{Q}$.
\end{itemize}
\end{itemize}
In the above case we refer to such a $\Sigma_{{\mathcal X}_{\Phi}}$-board as
\emph{ the canonical board of the assignment ${\mathcal M}$ to the $MLSSPF$ formula $\Phi$}.
\begin{mylemma}
\label{MsatisfiesI}
Consider a formula $\Phi\in {\rm MLSSPF}$,
a set-valued assignment ${\mathcal M}\in\{\:\mbox{\rm sets}\:\}^{{\mathcal X}_{\Phi}}$
defined on the collection ${\mathcal X}_{\Phi}$
of variables in $\Phi$, together with the
colored transitive $\Sigma_{{\mathcal X}_{\Phi}}$-board
${\mathcal G}=(T,\mathcal{F},\mathcal{Q})$.
Define $\Phi^-$ as the formula $\Phi$ without literals of the type $Finite(x)$
or $\neg Finite(x)$.
Moreover, let be $\widehat{\Sigma}$ a partition and $\beta$ a
bijection between $\Sigma_{{\mathcal X}_{\Phi}}$ and $\widehat{\Sigma}$ such that
$\widehat{\Sigma}$ simulates
$(\Sigma,{\mathcal G})$ upwards via
$\beta$, and let ${\mathcal M}'(v)=\bigcup\beta[\Im_{{\mathcal M}}(v)]$.
Then, for every literal in $\Phi^-$, the following conditions are
fulfilled:
\begin{des}
\item if the literal is satisfied by $\mathcal M$, then it is satisfied
by ${\mathcal M}'$ too;
\item if the literal is satisfied by ${\mathcal M}'$, and does not
involve $\mbox{\textclg{P}}$~ or the construct $\{\anonymous,\dots,\anonymous\}$,
then it is satisfied by ${\mathcal M}$ too;
\item if the literal $Finite(x)$ appears in $\Phi$ and is satisfied by
$\mathcal M$, then it is satisfied
by ${\mathcal M}'$ too.
\end{des}
\end{mylemma}
\begin{proof}
The thesis can be recast as follows. For $u,v,w$ and $w_{i}$ in
${\mathcal X}_{\Phi}$, the following conditions hold for all literals in $\Phi$:
\begin{itemize}
\item[(1)] $\bigcup\Im(v)\;\Re\;\bigcup\Im(w)$ iff
$\bigcup\beta[\Im(v)]\;\Re\;\bigcup\beta[\Im(w)]$,
for $\Re$ in $\{\:=\,,\:\in\,,\:\subseteq\:\}$;
\item[(2)] $\bigcup\Im(v)=\bigcup\Im(u)\:\star\:\bigcup\Im(w)$ iff
$\bigcup\beta[\Im(v)]=\bigcup\beta[\Im(u)]\:\star\:
\bigcup\beta[\Im(w)]$,
for $\star$ in $\{\:\cap\,,\:\setminus\,,\:\cup\:\}$, and
$\bigcup\Im(v)=\emptyset$ iff
$\bigcup\beta[\Im(v)]=\emptyset$;
\item[(3)] if $\bigcup\Im(v)=\pow{\bigcup\Im(w)}$, then
$\bigcup\beta[\Im(v)]=\pow{\bigcup\beta[\Im(w)]}$;
\item[(4)] if
$\bigcup\Im(v)=\{\,\bigcup\Im(w_{1}),\dots,\bigcup\Im(w_{H})\,\}$,
then\\ $\bigcup\beta[\Im(v)]=\{\,\bigcup\beta[\Im(w_{1})],\dots,
\bigcup\beta[\Im(w_{H})]\,\}$.
\item[(5)] if $Finite(v)$ appears in $\Phi$ then
$\vert\bigcup\Im(v)\vert=\vert\bigcup\beta[\Im(v)]\vert$
\end{itemize}
Property $(1)_{\in}$ (here $\Re$ is meant to be $\in$) follows from $\in$-simulates in
\defn{SimBoard}. $(3)$ follows from the assumption $\Im(v)\in\mathcal{Q}$ and the
notion of $\mbox{\textclg{P}}$-simulates given in the same definition.
Condition $(5)$ plainly follows from definition of $Red$-simulates.
We are left to prove that $(4)$ hold.
Observe that $\Im(v)\subseteq\mathcal{F}$, then consider $\Im(v)$
as the set $X$ and $Y_i$ as the sets $\Im(w_{i})$. Hence we can assume that
$\bigcup X=\{\bigcup Y_1,\dots, \bigcup Y_L\}$, $X\subseteq\mathcal{F}$, and
$Y_1,\dots, Y_L$ are distinct. We must check that
$\bigcup\beta[X]=\{\bigcup\beta[Y_1],\dots,$ $\bigcup\beta[Y_L]\}$.
Since $\widehat{\Sigma}$ $Red$-simulates
$(\Sigma,{\mathcal G})$ and $X\subseteq\mathcal{F}$,
and $\card{\beta(\sigma)}=\card{\sigma}$ for
each $\sigma\in X$, the desired conclusion easily follows.
Indeed, by property (1) of Def.~\ref{SimBoard},
$\bigcup\beta[Y_i]\in\beta(\sigma)$
if and only if
$\bigcup Y_i\in\sigma$, and $\beta[Y_1],\dots,\beta[Y_L]$
(and, accordingly, $\bigcup\beta[Y_1],\dots,\bigcup\beta[Y_L]$)
are pairwise distinct.
The proofs of remaining bi-implications go exactly as in \cite[Lemma 10.1]
{Cantone-Omodeo-Ursino02}
\end{proof}
\begin{mydef}
\label{defImitation}\rm
Consider a colored \emph{$\Sigma$-board} ${\mathcal G}=(T,\mathcal{F},\mathcal{Q})$
A partition $\widehat{\Sigma}$ is said to \emph{imitate}
$(\Sigma,{\mathcal G})$ when there is a bijection $\beta\in
\widehat{\Sigma}^{\Sigma}$ such that, for
$\Gamma\subseteq\Sigma$, $\sigma\in\Sigma$,
\begin{itemize}
\item[(1)] $\beta(\sigma)\ni\in\powast{\beta[\Gamma]}$ holds [if
and]
only if $\sigma\ni\in\powast{\Gamma}$;
\item[(2)] $\bigcup\beta[\Gamma]\in\beta(\sigma)$ holds if and only if $\bigcup
\Gamma\in\sigma$;
\item[(3)] if $\Gamma\in\mathcal{Q}$ holds, then
$\powast{\beta[\Gamma]}\subseteq\bigcup\widehat{\Sigma}$;
\item[(4)] if
$\sigma\in\mathcal{F}$ holds, then $\card{\beta(\sigma)}<\aleph_0$.
\end{itemize}
We will say that $\widehat{\Sigma}$
\emph{imitates} $(\Sigma,{\mathcal G})$
\emph{upwards} when the following
additional condition holds, for all $\sigma\in\Sigma$:
\begin{itemize}
\item[(4$'$)] if
$\sigma\in\mathcal{F}$, then $\card{\beta(\sigma)}=\card{\sigma}$;
\end{itemize}
\end{mydef}
\begin{mylemma}\label{imitaSimula}
Consider a colored \emph{$\Sigma$-board} ${\mathcal G}=(T,\mathcal{F},\mathcal{Q})$
assume that a transitive partition $\widehat{\Sigma}$ {imitates} $(\Sigma,{\mathcal G})$
{upwards}
then it {simulates}
$(\Sigma,{\mathcal G})$ {upwards}.
\end{mylemma}
\begin{proof}
Let $\Sigma$ and $\widehat{\Sigma}$ be transitive partitions, and
let ${\mathcal G}$ be a colored \emph{$\mathcal{P}$-board} induced by
$\Sigma$. Assume that $\widehat{\Sigma}$ {imitates}
$(\Sigma,{\mathcal G})$ {upwards}
via the bijection $\beta\in
(\widehat{\Sigma})^{\Sigma}$.
Finally, let $X, Y \subseteq \Sigma$.
Then we have: $\bigcup\beta[X]\in\bigcup\beta[Y]$ iff
$(\exists\,\widehat{\sigma}\in\beta[Y])(\bigcup\beta[X]\in\widehat{\sigma})$
iff $(\exists\,\sigma\in
Y)(\bigcup\beta[X]\in\beta(\sigma))$
iff $(\exists\,\sigma\in
Y)(\bigcup X\in\sigma)$ iff $\bigcup X\in\bigcup Y$.
Assuming now that $\bigcup X=\pow{\bigcup Y}$, $Y\in\mathcal{Q}$, let us
prove that
$\pow{\bigcup\beta[Y]}\subseteq\bigcup \beta[X]$. Indeed, suppose
$t\subseteq\bigcup\beta[Y]$ and let $\widehat{\Sigma}_{t}$ be the
subset of $\widehat{\Sigma}$ for which
$t\in\powast{\widehat{\Sigma}_{t}}$ (so that
$\widehat{\Sigma}_{t}\subseteq \beta[Y]$, which implies
$\widehat{\Sigma}_{t}\in\mathcal{Q}$ by the hereditarily closedness by inclusion
of $\mathcal{Q}$). As
$\beta^{-1}[\widehat{\Sigma}_{t}] \subseteq Y$,
it follows that
$\powast{\beta^{-1}[\widehat{\Sigma}_{t}]} \subseteq
\pow{\bigcup Y} =
\bigcup X \subseteq \bigcup \Sigma$.
Therefore, by the fact that
$\widehat{\Sigma}$ {imitates}
$(\Sigma,{\mathcal G})$ {upwards} and $\widehat{\Sigma}_{t}\in\mathcal{Q}$,
it follows that
$\powast{\widehat{\Sigma}_{t}} \subseteq \bigcup \widehat{\Sigma}$,
so
that $t \in \bigcup \widehat{\Sigma}$. Let $\widehat{\sigma}_{t}$
be the block in $\widehat{\Sigma}$ to which $t$ belongs, and let
$\sigma_{t}$ be the block in $\Sigma$ for which
$\beta(\sigma_{t})=\widehat{\sigma}_{t}$. Then, since
$\powast{\widehat{\Sigma}_{t}}\ni\in\widehat{\sigma}_{t}$, we
have that
$\powast{\beta^{-1}[\widehat{\Sigma}_{t}]}\ni\in\sigma_{t}$, which
yields $\bigcup X=\pow{\bigcup Y}\supseteq\powast{\beta^{-1}[
\widehat{\Sigma}_{t}]}\ni\in\sigma_{t}$, so that $\bigcup
X\ni\in\sigma_{t}$, $\sigma_{t}\in X$, and hence
$t\in\widehat{\sigma}_{t} \in\beta[X]$, which in turn yields $t \in
\bigcup \beta[X]$.
Next, assuming again
$\bigcup X=\pow{\bigcup Y}$, let us prove that $\bigcup
\beta[X]\subseteq\pow{\bigcup\beta[Y]}$. Indeed, for each $t\in
\bigcup\beta[X]$ there is a unique $\sigma_{t}\in X$ such that
$t\in\beta(\sigma_{t})$; moreover, by the transitivity of
$\bigcup\widehat{\Sigma}$, there is a unique $\Gamma\subseteq\Sigma$ for
which $t\in\powast{\beta[\Gamma]}$. Moreover, since
$\powast{\beta[\Gamma]}\ni\in\beta(\sigma_{t})$, we also have that
$\powast{\Gamma}\ni\in\sigma_{t}$.
Thus we can take
$t'\in\sigma_{t}\cap\powast{\Gamma}$ that, as
$\sigma_{t}\subseteq\bigcup X=\pow{\bigcup Y}$, fulfills
$t'\in\powast{Z}$ for a suitable $Z\subseteq Y$. In conclusion,
$\Gamma=Z$, and therefore
$t\subseteq\bigcup\beta[\Gamma]=\bigcup\beta[Z]\subseteq\bigcup\beta[Y]$.
\end{proof}
As an immediate consequence, we have
\begin{mycorollary}\label{MsatisfiesII}
Consider a formula $\Phi\in {\rm MLSSPF}$,
a set-valued assignment ${\mathcal M}\in\{\:\mbox{\rm sets}\:\}^{{\mathcal X}_{\Phi}}$
defined on the collection ${\mathcal X}_{\Phi}$
of variables in $\Phi$, together with the
colored transitive {$\Sigma_{{\mathcal X}_{\Phi}}$-board}
${\mathcal G}=(T,\mathcal{F},\mathcal{Q})$.
Moreover, let $\widehat{\Sigma}$ and $\beta$ be a partition and a
bijection, respectively,
such that $\widehat{\Sigma}$ {imitates} $(\Sigma,{\mathcal G})$
{upwards} via
$\beta$, and let ${\mathcal M}'(v)=\bigcup\beta[\Im_{{\mathcal M}}(v)]$,
where $\Im$ is
the function $\Im\in\pow{\Sigma}^{\mathcal X_{\Phi}}$ such that
${\mathcal M}(v)= \bigcup\Im(v)$ holds for every $v$ in $\mathcal X$.
Then, for every literal in $\Phi^-$ and literals of the type
$Finite(x)$, the following conditions are
fulfilled:
\begin{des}
\item if the literal is satisfied by $\mathcal M$, then it is satisfied
by ${\mathcal M}'$ too;
\item if the literal is satisfied by ${\mathcal M}'$, and does not
involve $\mbox{\textclg{P}}$~ or the construct $\{\anonymous,\dots,\anonymous\}$,
then it is satisfied by ${\mathcal M}$ too.
\end{des}
\end{mycorollary}
\subsection{Formative processes}
We now formalize the concept of ``history'' of a model
by a transfinite construction.
Using the transitivity of any transitive partition,
it is possible to single out a process that
builds it, having the empty partition as starting point.
The following notions are introduced to specify this concept.
\begin{mydef}\label{prolongation}\rm
Let $\Sigma$ and $\Sigma'$ be two partitions, and let $\Gamma
\subseteq \Sigma$. We say that $\Sigma'$
{\em prolongates} $\Sigma$ via $\Gamma$ when the following
conditions hold:
\begin{enumerate}
\item
for all $\sigma\in\Sigma$, there is one and only one
$\sigma'\in\Sigma'$ such that $\sigma\subseteq\sigma'$;
\item
$\bigcup\Sigma'\setminus\bigcup\Sigma\subseteq\powast{\Gamma}$;
\item $\Sigma\neq\Sigma'$.
\end{enumerate}
When just condition (1) is met, possibly without (2) or (3),
we say that $\Sigma'$ {\em extends} $\Sigma$. If
both (1) and (3) hold true, then $\Sigma'$ is said to extend $\Sigma$
{\em properly}.
\end{mydef}
\begin{mydef}{\bf [Coherence requirement]}
\label{cohExtends}\rm
Let $\Gamma$, $\Sigma'$ and $\Sigma''$ be partitions, with
$\Sigma'$ extending $\Gamma$ (typically,
$\Gamma\subseteq\Sigma'$) and $\Sigma''$ extending $\Sigma'$.
Then $\Sigma''$ is said to extend $\Sigma'$ {\em coherently} with
$\Gamma$ if no element of $\bigcup\Sigma''$ belongs to
$\powast{\Gamma}\setminus\bigcup\Sigma'$.
\end{mydef}
\begin{mydef}\label{formativeProcess}\rm
Let $\xi$ be an ordinal and let
$\left(\{q^{(\mu)}\}_{q\in\mathcal{P}}\right)_{\mu\leqslant\xi}$ be a
$(\xi+1)$-sequence of functions, all defined on the
same domain $\mathcal{P}$. Put
$B^{(\mu)}\stackrel{_{\tiny {\rm def}}}{_{\normalsize =}}\{\,q^{(\mu)}\,\mbox{\tt|}\:q\in B\,\}$ for all
$B\subseteq P$, and let
$\Sigma_{\mu}=\mathcal{P}^{(\mu)}\setminus\{\emptyset\}$, for all
$\mu\leqslant\xi$.
Assume that the following conditions are fulfilled:
\begin{itemize}
\item $q^{(\mu)}\cap p^{(\mu)}=\emptyset$ when $p,q\in P$, $p\neq q$,
and $\mu\leqslant\xi$;
\item $q^{(\nu)}\subseteq q^{(\nu+1)}$ for all $q\in
\mathcal{P}$ when $\nu<\xi$;
\item
$q^{(\lambda)}=\bigcup_{\nu<\lambda}q^{(\nu)}$ for every $q\in
\mathcal{P}$ and every limit ordinal $\lambda\leqslant \xi$;
\item
$q^{(0)}=\emptyset$ and $\emptyset\neq q^{(\xi)}$, for all
$q\in\mathcal{P}$.
\end{itemize}
In particular, $\Sigma_{0}=\emptyset$ and, for every $\mu\leqslant\xi$,
$\Sigma_{\mu}$ is a partition of the subset $\bigcup P^{(\mu)}$
of $\bigcup P^{(\xi)}$.
Assume moreover that to each $\nu<\xi$ corresponds
$\Gamma_{\nu}\subseteq\Sigma_{\nu}$ such that
\begin{itemize}
\item $\Sigma_{\nu+1}$ prolongates $\Sigma_{\nu}$ via
$\Gamma_{\nu}$ (cf.\ \defn{prolongation});
\item $\Sigma_{\xi}$ extends $\Sigma_{\nu+1}$ coherently with
$\Gamma_{\nu}$ (cf.\ \defn{cohExtends}).
\end{itemize}
Then the sequence
$\left(\{q^{(\mu)}\}_{q\in\mathcal{P}}\right)_{\mu\leqslant\xi}$
(occasionally, $\big(\Sigma_\mu\big)_{\mu\leqslant\xi}$) is
called a ({\em strong}) {\em formative process} for
$\Sigma_{\xi}$.
Furthermore, the $\xi$-sequences $(A_{\nu})_{\nu<\xi}$ and
$(A_{\nu},T_{\nu})_{\nu<\xi}$, with $ A_{\nu}, T_{\nu}\subseteq
\mathcal{P}$, satisfying for each $\nu$ the conditions
\begin{itemize}
\item $A_{\nu}^{(\nu)}=\Gamma_{\nu}$,
\item $\{\,q^{(\nu+1)}\setminus q^{(\nu)}\,\mbox{\tt|}\:q\in
T_{\nu}\}$ is a partition of $\bigcup\Sigma_{\nu+1}\setminus
\bigcup\Sigma_{\nu}$ ~
($=\mbox{\sf(}\,\powast{\Gamma_{\nu}}\setminus\bigcup\Sigma_{\nu}\,\mbox{\sf)}
\cap\bigcup\Sigma_{\nu+1}$)
\end{itemize}
are called the {\em trace} of the
formative process, and a {\em history} of $\Sigma_{\xi}$,
respectively.
A {\em weak formative process} is like a formative process, except
that the coherence requirement is withdrawn from the definition.
A {\em weak trace} is defined similarly.
\end{mydef}
In the sequel it will be helpful the following simplified notation.
\begin{mydef}\rm
Let $(\{q^{(\mu)}\}_{q\in\mathcal{P}})_{\mu\leqslant\xi}$ be a weak
formative process. Then, for $q\in \mathcal{P}$, $B\subseteq
\mathcal{P}$ and $\nu<\xi$, we set
$$
q^{(\bullet)}\stackrel{_{\tiny {\rm def}}}{_{\normalsize =}} q^{(\xi)},\qquad
B^{(\bullet)}\stackrel{_{\tiny {\rm def}}}{_{\normalsize =}} B^{(\xi)},\qquad
\Delta^{(\nu)}(q) \stackrel{_{\tiny {\rm def}}}{_{\normalsize =}} q^{(\nu+1)}\setminus\bigcup\mathcal{P}^{(\nu)}.
$$
\end{mydef}
If we take, along with a colored $\mathcal{P}$-board
$(T,\mathcal{F},\mathcal{Q})$, a bijection $q\mapsto q^{(\bullet)}$
from the places $\mathcal{P}$ to the final partition $\Sigma_\xi$ of
a formative process, and if moreover $\Sigma_\xi$ complies with
$T,\mathcal{F},\mathcal{Q}$, we get what we call a
{\em colored $\mathcal{P}$-process}: namely, the quintuple
$((\{q^{(\mu)}\}_{q\in\mathcal{P}})_{\mu\leqslant\xi},(\bullet),
T,\mathcal{F},\mathcal{Q})$.
\begin{mydef}\rm
$e\in\bigcup\mathcal{P}^{(\bullet)}$ is said to be \em{unused}
at $\mu\leqslant\xi$ if $e\notin
\bigcup\bigcup\mathcal{P}^{(\mu)}$, i.e., if $e\notin z$ for any
$q\in\mathcal{P}$ and any $z\in q^{(\mu)}$.
\end{mydef}
\begin{mydef}\rm
An $e\in\bigcup\mathcal{P}^{(\bullet)}$ is said to be \textsc{new}
at $\mu\leqslant\xi$ if $e\in\Delta^{(\mu)}(q)$ for some $q\in\mathcal{P}$.
\end{mydef}
Obviously a new element is, in particular, unused.
\begin{mylemma}\label{unused}
If $b$ is a set made of unused elements only, the same is
$\powast{\{b\}\cup A}$.
\end{mylemma}
\subsection{Grand events and local trash}
We begin with the following easy remark.
The block at place $s$ belonging to a
$\mbox{\textclg{P}}$-node $A$ cannot become infinite during a colored process,
unless $A$ has a green place among its targets. To see that,
assume that $s\in A\in\mathcal{Q}$ and $\infinite{s^{(\bullet)}}$.
Consequently, $\card{\powast{\bigcup A^{(\bullet)}}}>\aleph_0$ and
$\powast{\bigcup
A^{(\bullet)}}\subseteq\bigcup\mathcal{P}^{(\bullet)}$. Hence
there must be a place $g$ such that $\card{\powast{\bigcup
A^{(\bullet)}}\cap g^{(\bullet)}}>\aleph_0$, since
$\card{\mathcal{P}^{(\bullet)}}=\card{\mathcal{P}}<\aleph_0$. This
obviously implies that $g\in\Targets{A}\setminus\mathcal{F}$.
In light of generalizing the above remark, recalling the notion
of grand move, and noticing that such an event occurs, in a colored
process, at most once for each node $A$, we give the following
definition of \emph{grand event} $GE(A)$ associated with $A$.
\begin{mydef}\rm
For every node $A$ and every $\nu$ such that
$0\leqslant{\nu}<\xi$
$$
GE(A)\;\stackrel{_{\tiny {\rm def}}}{_{\normalsize =}}\;\left\{\begin{array}{ll}
\mbox{\rm the ordinal $\nu$ for which $\bigcup
A^{(\bullet)}\in\bigcup\mathcal{P}^{(\nu+1)}\setminus\bigcup\mathcal{P}^{(\nu)}$,}
& \mbox{\rm if any exists}, \\
\mbox{\rm the length $\xi$ of the process}, & \mbox{\rm otherwise}.
\end{array}\right.
$$
Moreover, for any given collection ${\mathcal A}$ of nodes, we put
$$
GE({\mathcal A})\;\stackrel{_{\tiny {\rm def}}}{_{\normalsize =}}\; \min \{GE(A)\mid A \in {\mathcal A}\}\,.
$$
\end{mydef}
Notice that this Definition implies that for any node $A$ and any
$\nu$ such that $0\leqslant\nu<\xi$,
$$\begin{array}{lcl} \nu\leqslant GE(A) & {\bf\leftrightarrow} &
\bigcup A^{(\bullet)}\notin\bigcup\mathcal{P}^{(\nu)},\\
\nu= GE(A) & {\bf\leftrightarrow} & \bigcup
A^{(\bullet)}\in\bigcup\mathcal{P}^{(\nu+1)}\setminus\bigcup\mathcal{P}^{(\nu)},
\\
\nu> GE(A) & {\bf\leftrightarrow} & \bigcup
A^{(\bullet)}\in\bigcup\mathcal{P}^{(\nu)}.
\end{array}
$$
Further elementary properties, whose proofs are left to the reader, are
stated in the next lemma.
\begin{mylemma}
Let
$\big(\Sigma_\mu\big)_{\mu\leqslant\xi},(\bullet),T,\mathcal{F},\mathcal{Q}$
be a colored $\mathcal{P}$-process and let $A \subseteq \mathcal{P}$ be
a node. Then
\begin{itemize}
\item $A^{(\alpha)} = A^{(\bullet)}$, where $\alpha = GE(A)$;
\item if $q^{(\nu+1)} \supsetneq q^{(\nu)}$, for some $q \in A$
and some $\nu < \xi$, then $GE(A) > \nu$.
\end{itemize}
\end{mylemma}
Other important related definitions are the following.
\begin{mydef}\rm
A place $g$ is said to be a {\em local trash} for a node $A$ if
\begin{itemize}
\item $g\in\Targets{A}\setminus\mathcal{F}$, i.e., $g$ is a green
target of $A$;
\item there holds $GE(A)<GE(B)$, for every node $B$ such that $g\in B$.
\end{itemize}
\end{mydef}
\begin{mydef}
A set $\mathcal{W}$ of places is said to be {\em closed} if
\begin{itemize}
\item all of its elements are green;
\item every $\mbox{\textclg{P}}$-node which intersects $\mathcal{W}$ has a local
trash which belongs to $\mathcal{W}$.
\end{itemize}
\end{mydef}
\subsection{Minus-Surplus refinement}
In this section we recall some technical notions
to refine the original transitive partition.
This procedure stores some elements
(the {\em Surplus} portion of a block) in order to trigger off a
construction which is supposed to ``pump''
elements inside fixed bocks. Conversely, the remaining
collection of elements (the {\em Minus} portion of a block)
will be used to copy the original formative
process.
We shall adopt the following notation. For a couple of
ordinals $\beta ',\beta ''$ we denote by
$[\beta ',\beta '']$ the collection of ordinals
$\{\beta\mid \beta '\le\beta\le\beta ''\}$.
We say that a transitive partition $\Sigma$ is equipped of a Minus-Surplus
partitioning if each block $q$ is partitioned into two sets,
namely, $Surplus(q)$ and
$Minus(q)$. Consistently, we can extend this notation to a
formative process $\big(\Sigma_\mu\big)_{\mu\leqslant\xi}$.
Given a node $\Gamma$, we indicate by $Minus(\Gamma ^{(\mu)})$
the collection
of sets
$$\{Minus(q^{(\mu)})\mid q\in\Gamma\}.$$
Define now a Minus-Surplus partitioning for $\Sigma _0$, and assume that
for each step $\mu$ of the process a refinement of the partition
$\{\Delta ^{(\mu)}(q)\}_{q\in\Sigma}$ is decided
in the following way:
for each $q\in\Sigma$ the set $\Delta ^{(\mu)}(q)$
is partitioned into two sets
$\Delta ^{(\mu)}Minus(q)\subseteq\mbox{\textclg{P}}~^{\ast} (Minus(A_{\mu}^{(\mu)}))$
and $\Delta ^{(\mu)}Surplus(q)\subseteq(\mbox{\textclg{P}}~^{\ast}(A_{\mu}^{(\mu)})\setminus
\mbox{\textclg{P}}~^{\ast} (Minus(A_{\mu}^{(\mu)}))$.\\
Then define inductively $$Surplus(q^{(\mu+1)})=Surplus(q^{(\mu)})\cup
\Delta ^{(\mu)}Surplus(q)$$ and
$$Minus(q^{(\mu+1)})=Minus(q^{(\mu)})\cup\Delta ^{(\mu)}Minus(q).$$
As far as $\xi$ limit are concerned, we put
$$Minus(q^{(\xi)})=\bigcup_{\mu <\xi}Minus(q^{(\mu)})$$
and, analogously,
$$Surplus(q^{(\xi)})=\bigcup_{\mu <\xi}Surplus(q^{(\mu)})$$
If $\Gamma$ is a subset of $\Sigma$, we denote by $Surplus(\Gamma)$ the
set
$$\{q\mid q\in\Gamma\wedge Surplus(q)\neq\emptyset\}.$$
\begin{mydef}\label{}\rm
Whenever a Surplus-Minus partition is defined for all blocks of a
transitive partition $\Sigma$, we say that $\Sigma$ is equipped of
a \emph{Minus-Surplus partitioning},
and we denote by $Surplus$-$Minus(\Sigma )$ the following
refinement of the original one:
$$\{Minus(q),Surplus(q)\mid q\in\Sigma\}.$$
It is rather obvious that $Surplus$-$Minus(\Sigma )\sqsubseteq\Sigma$.
\end{mydef}
\begin{myremark}\label{powdisj}\rm
Easy combinatorial arguments (see \cite[Lemma 3.1 5(b)]
{Cantone-Omodeo-Ursino02})
show that $\powast\_$ of Surplus
and Minus nodes are mutually disjoint.
\end {myremark}
The next definition says which structural properties
a formative process has to fulfill in order to copy
the history of a transitive partition.
\begin{mydef}\label{formimit}\rm
Let
$(\left(\{q^{(\mu)}\}_{q\in\mathcal{P}}\right)_{\mu\leqslant\xi},(\bullet),
T,\mathcal{F},\mathcal{Q})$
be a {colored $\mathcal{P}$-process}. Besides, let
$\big(\{\widehat{q}^{[\alpha]}\}_{\widehat{q}\in\widehat{\mathcal{P}}}\big)_
{\alpha\in [\alpha',\alpha '']}$
a formative processes equipped of a Minus-Surplus
partitioning. Assume that
$q\rightarrow \widehat{q}$ is a bijection from $\mathcal{P}$ to
$\widehat{\mathcal{P}}$,
$\beta ''\le\xi$, and $\gamma$ is an order preserving
injection from
$[\beta ',\beta '']$ to $[\alpha',\alpha '']$.
Let ${\mathcal C}$ be a closed collection of green blocks,
and $q\rightarrow \widehat{q}$ be a bijection from $\mathcal{P}$ to
$\widehat{\mathcal{P}}$.
We
say that $\big(\{\widehat{q}^{[\alpha]}\}_{\widehat{q}
\in\widehat{\mathcal{P}}}\big) _{\alpha\in\gamma
[[\beta ',\beta '']]}$ \emph{imitates}
the segment $[\beta ',\beta '']$ of the process
$(\left(\{q^{(\mu)}\}_{q\in\mathcal{P}}\right)_{\mu\leqslant\xi}$
if the following hold
for all $\beta$ in $[\beta ',\beta '']$:
\begin{itemize}
\item[(i)]$\vert q^{(\beta)}\vert=\vert Minus^{[\gamma(\beta)]}(\widehat{q})\vert$;
\item[(ii)] $\vert \Delta ^{(\beta)}(q)\vert=
\vert\Delta ^{[\gamma (\beta)]} Minus(\widehat{q})\vert$;
\item[(iii)]$\Delta ^{[\gamma (\beta)]}Surplus(\widehat{q})\neq\emptyset$
implies
$\beta=GE(A_{\beta})$, $q$ local trash for $A_{\beta}$ and
$q\in {\mathcal C}$;
\item[(iv)] If $\Gamma\in\mathcal{Q}$ holds, then
$\powast{\widehat{\Gamma}^{[\gamma (GE(\Gamma))]} }
\subseteq\bigcup\widehat{\Sigma}^{[\gamma (GE(\Gamma)+1)]}$;
\item[(v)]For all $\beta\ne GE(\Gamma)$
$\bigcup\Gamma^{(\beta)}\in\Delta ^{(\beta)}(q)$ iff
$\bigcup Minus\widehat{\Gamma}^{[\gamma (\beta)]}\in\Delta ^{[\gamma (\beta)]}(
\widehat{q})$;
\item[(vi)] If $\beta =GE(\Gamma)$ then $\bigcup\Gamma^{(\beta)}\in
\Delta ^{(\beta)}(q)$ iff
$\bigcup\Gamma^{[\gamma (\beta)]}\in\Delta ^{[\gamma (\beta)]}(\widehat{q})$;
\item[(vii)] For all $q\in\mathcal{F}$ $\widehat{q}^{[\gamma (\beta)]}=
Minus^{[\gamma(\beta)]}(\widehat{q})$;
\item[(viii)] For all ordinals $\beta$ $\{q\mid
\widehat{q}\in Surplus(\widehat{\Sigma})^{[\gamma (\beta)]}\}
\subseteq{\mathcal C}$;
\item[(ix)] $\vert\mbox{\textclg{P}}~^{\ast} (Minus (\widehat{\Gamma}^{[\gamma(k)]}))
\setminus \bigcup _{q\in\Sigma}q^{[\gamma(k)]}\vert=
\vert\mbox{\textclg{P}}~^{\ast} (\Gamma) ^{(k)}\setminus \bigcup _{q\in\Sigma}q^{(k)}\vert$;
\item[(x)] $\vert\mbox{\textclg{P}}~^{\ast} (Minus(\widehat{\Gamma}^{[\gamma(k-1)]})) \cap q^{
[\gamma(k)]}\vert
=\vert \mbox{\textclg{P}}~^{\ast} (\Gamma^{(k-1)})\cap q^{(k)}\vert$.
\end{itemize}
\end{mydef}
\smallskip
\begin{myremark}\rm
\label{rem1}
We make some simple observations.
\begin{itemize}
\item $\mbox{\textclg{P}}~^{\ast} (\Gamma^{(k-1)})\cap q^{(k)}=\mbox{\textclg{P}}~^{\ast} (\Gamma^{(k)})\cap q^{(k)}$. Hence,
whenever $\gamma (k)$ is the successor of $\gamma (k-1)$, (x) can be rephrased as
$$\vert\mbox{\textclg{P}}~^{\ast} (Minus(\widehat{\Gamma}^{[\gamma(k)]})) \cap q^{
[\gamma(k)]}\vert =\vert \mbox{\textclg{P}}~^{\ast} (\Gamma^{(k)})\cap q^{(k)}\vert.$$
\item Naturally, (ix) belongs to the structural properties that a
formative process has to fulfill in order to
simulate another one, although it can be obtained from (i) and (x).
\item Assume that (viii) holds at the beginning of the process.
Then (iii) entails (viii), therefore, whenever
one has to prove inductively
the previous properties, it suffices to show that (viii) holds only
in the starting step.
The same argument holds for (x).
Indeed, it can be obtained from (ii), (iii) and (x) of the preceding step.
\end{itemize}
\end {myremark}
The following requirements set are to be satisfied by
the initial conditions of a transitive partition
in order to
play the role of starting point of an imitation process
(as it is easily seen, they are purely combinatorial).
\begin{mydef}\label{weakimit}\rm
Let
$(\left(\{q^{(\mu)}\}_{q\in\mathcal{P}}\right)_{\mu\leqslant\xi},
(\bullet),T,\mathcal{F},\mathcal{Q})$
be a {colored $\mathcal{P}$-process},
$(\widehat{\Sigma},\widehat{\mathcal{G}})$ be a {$\widehat{\Sigma}$-board}
equipped with a Minus-Surplus
partitioning,
$q\rightarrow \widehat{q}$ be a bijection from $\mathcal{P}$
to $\widehat{\mathcal{P}}$, and
${\mathcal C}$ be a closed collection of green blocks.
Assume $k'<\xi$,
such that (i), (vii), (viii) and (x) of Def.\ref{formimit} hold in the version
$\widehat{\Sigma}_{\gamma (k')}=
\widehat{\Sigma}$.
We say that $\widehat{\Sigma}$ \emph{weakly imitates}
$\Sigma$ \emph{upwards}, provided that
the following conditions are satisfied:
\begin{itemize}
\item [(a)] for all $\Gamma\subseteq\Sigma$ and $q\in\Sigma$,
$$\bigcup Minus(\widehat{\Gamma})\in\mbox{\textclg{P}}~^{\ast} (Minus(\widehat{\Gamma}))
\setminus \bigcup _{q\in\Sigma}\widehat{q}\quad {\rm iff}
\quad\bigcup\Gamma^{(k')}\in
\mbox{\textclg{P}}~^{\ast} (\Gamma^{(k')})\setminus \bigcup _{q\in\Sigma}q^{(k')};$$
\item [(b)] $q\in\Gamma\wedge Surplus(q)\neq\emptyset\wedge GE(\Gamma)\ge k'$
implies $\bigcup\widehat{\Gamma}\in\mbox{\textclg{P}}~^{\ast}
(\widehat{\Gamma})
\setminus \bigcup _{q\in\Sigma}\widehat{q}$;
\item [(c)] if $GE(\Gamma)<k'$, then $\bigcup\Gamma^{(k')}\in q^{(k')}$ iff
$\bigcup \widehat{\Gamma}\in \widehat{q}$ and $\Gamma\in\mathcal{Q}$
implies $\mbox{\textclg{P}}~^{\ast} (\widehat{\Gamma})\subseteq\bigcup\widehat{\Sigma}$.
\end{itemize}
\end{mydef}
\section{Two structural results concerning Minus-Surplus partition}
The following Lemma relates Definition \ref{weakimit}
with the notion of imitating a formative process.
\begin{mylemma}\label{pasting}
Let
$(\left(\{q^{(\mu)}\}_{q\in\mathcal{P}}\right)_{\mu\leqslant\xi},
(\bullet),T,\mathcal{F},\mathcal{Q})$
be a {colored $\mathcal{P}$-process},
$(\widehat{\Sigma},\widehat{\mathcal{G}})$ be a {$\widehat{\Sigma}$-board},
the latter equipped of a Minus-Surplus
partitioning,
$q\rightarrow \widehat{q}$ be a bijection from $\mathcal{P}$
to $\widehat{\mathcal{P}}$, and
${\mathcal C}$ be a closed collection of green blocks.
Assume that $k'\le\xi$, and that
$\widehat{\Sigma}$ weakly imitates upward $\Sigma_{k'}$.
Define $\widehat{\Sigma}=\widehat{\Sigma}_{\gamma(k')}$ and, for all
$q\in\widehat{\mathcal{P}}$,
$\widehat{q}=\widehat{q}^{[\gamma(k')]}$.
Then for all ordinals
$k''$ such that $k''\le\xi$ and $\vert [k',k'']\vert <\omega$
it can be constructed a formative process
$\big(\{\widehat{q}^{[\alpha]}\}_{\widehat{q}
\in\widehat{\mathcal{P}}}\big)_{\gamma(k')\leqslant\mu\leqslant\gamma(k'')}$
which imitates the
segment $[k',k'']$ of the process $(\left(\{q^{(\mu)}\}_{q\in\mathcal{P}}\right)_{\mu\leqslant\xi},
(\bullet),T,\mathcal{F},\mathcal{Q}$.
\end{mylemma}
\begin{proof}
We construct a formative process by induction
satisfying the requested properties (i)-(x).
Concerning the base case $\mu =\gamma (k')$,
(i),(vii),(viii)(x) hold by hypothesis, and
(ix) holds by Remark~\ref{rem1}, since (i) and (x) hold.
Assume $k'\ne GE(A_{k'})$.
Using (ix) and hypothesis (a) we can define a partition
$\bigcup_{q\in\Sigma}(\Delta
^{[\gamma(k')]}(q)$ of
$$\mbox{\textclg{P}}~^{\ast} (Minus
^{[\gamma(k')]}(\widehat{A_{k'}}))\setminus\bigcup _{q\in\Sigma}q^{[\gamma(k')]}$$
such that (ii) and (v) hold, as well.
If $k'=GE(A_{k'})$ and $Surplus(\widehat{q}^{[\gamma(k')]})\neq\emptyset
$ for some $q\in A_{k'}$
(otherwise we proceed as before, and condition
(vi) is automatically satisfied), then,
using (b), interchanging $\bigcup Minus(\widehat{A_{k'}}^{[\gamma(k')]})$ with
$\bigcup A^{[\gamma(k')]}$, (vi) is satisfied.
If $A_{k'}\in\mathcal{Q}$ and $\widehat{A_{k'}}=Minus(\widehat{A_{k'}})$, proceed as
before (in this case (iv) holds by a straight checking
of cardinality starting from (ix)).
Otherwise, since (viii) holds,
there must exist a local trash
$q\in {\mathcal C}$ for $A_{k'}$.
Then, construct the partition as before, except
for $\Delta^{[\gamma(k')]}Surplus(\widehat{q})$, in which we put
the whole remainder
$$(\mbox{\textclg{P}}~^{\ast} (\widehat{A_{k'}}^{[\gamma(k')]})
\setminus \bigcup _{q\in\Sigma}q^{[\gamma(k')]})\setminus
\bigcup_{q\in\Sigma}\Delta^{[\gamma(k')]} Minus(\widehat{q}),$$
so satisfying
(iii) and (iv).
Now, assume all the inductive hypotheses for $\gamma(k)$. Our aim
is to demonstrate
the case $\gamma(k+1)$.
By Remark~\ref{rem1}, provided that (iii)[$\gamma(k+1)$] is proven,
(viii) automatically holds.
Plainly (i)[$\gamma(k)$] and (ii)[$\gamma(k)$] entail
(x)[$\gamma(k+1)$] and (i)[$\gamma(k+1)$]. The latter
in turns implies the following for all $\Gamma\subseteq\Sigma$
\begin{equation}
\label{pow2}
\vert\mbox{\textclg{P}}~^{\ast} (Minus ^{[\gamma(k+1)]}\widehat{\Gamma}))\vert=\vert
\mbox{\textclg{P}}~^{\ast} (\Gamma^{(k+1)})\vert.
\end{equation}
In order to show (ix) we observe that,
since
\begin{align*}
&\mbox{\textclg{P}}~^{\ast} (Minus(\widehat{\Gamma}^{[\gamma(k+1)]})\setminus
\bigcup _{q\in\Sigma}q^{[\gamma(k+1)]}\\
&= (\mbox{\textclg{P}}~^{\ast} (Minus(\widehat{\Gamma}^{[\gamma(k+1)]})
\setminus\mbox{\textclg{P}}~^{\ast} (Minus(\Gamma^{[\gamma(k)]}))
\setminus
\bigcup _{q\in\Sigma}q^{[\gamma(k+1)]})\cup \\
&\quad \cup(\mbox{\textclg{P}}~^{\ast} (Minus(\widehat{\Gamma}^{[\gamma(k)]}))
\setminus
\bigcup _{q\in\Sigma}q^{[\gamma(k+1)]}),
\end{align*}
it follows that
\begin{align*}
&\mbox{\textclg{P}}~^{\ast} (Minus(\widehat{\Gamma}^{[\gamma(k+1)]}))\setminus
\mbox{\textclg{P}}~^{\ast} (Minus(\widehat{\Gamma}^{[\gamma(k)]}))
\setminus
\bigcup _{q\in\Sigma}q^{[\gamma(k+1)]}\\
&=\mbox{\textclg{P}}~^{\ast} (Minus(\widehat{\Gamma}^{[\gamma(k+1)]}))
\setminus\mbox{\textclg{P}}~^{\ast} (Minus(\Gamma^{[\gamma(k)]})).
\end{align*}
Therefore,
\begin{align*}
&\mbox{\textclg{P}}~^{\ast} (Minus(\widehat{\Gamma}^{[\gamma(k+1)]}))\setminus
\bigcup _{q\in\Sigma}q^{[\gamma(k+1)]}\\
&=\mbox{\textclg{P}}~^{\ast} (Minus(\widehat{\Gamma}^{[\gamma(k+1)]}))
\setminus\mbox{\textclg{P}}~^{\ast} (Minus(\widehat{\Gamma}^{[\gamma(k)]}))
\cup\mbox{\textclg{P}}~^{\ast} (Minus(\widehat{\Gamma}^{[\gamma(k)]}))
\setminus
\bigcup _{q\in\Sigma}q^{[\gamma(k+1)]}.
\end{align*}
Reasoning in the same way, we obtain
\begin{align*}
&\mbox{\textclg{P}}~^{\ast} (\Gamma)^{(k+1)}\setminus\bigcup_{q\in\Sigma}q^{(k+1)}\\
&= \mbox{\textclg{P}}~^{\ast} (\Gamma)^{(k+1)}\setminus\mbox{\textclg{P}}~^{\ast} (\Gamma)^{(k)}\cup\mbox{\textclg{P}}~^{\ast} (\Gamma)^{(k)}
\setminus\bigcup _{q\in\Sigma}q^{(k+1)}.
\end{align*}
By the induction hypothesis (i)[$\gamma(k)$] we have
$\vert\mbox{\textclg{P}}~^{\ast} (Minus ^{[\gamma(k)]}\widehat{\Gamma}))\vert=\vert
\mbox{\textclg{P}}~^{\ast} (\Gamma^{(k)})\vert$, and by
equation~\eqref{pow2},
$$\vert\mbox{\textclg{P}}~^{\ast} (Minus ^{[\gamma(k+1)]}\widehat{\Gamma}))\vert=\vert
\mbox{\textclg{P}}~^{\ast} (\Gamma^{(k+1)})\vert,$$
which in turns implies
$$\vert\mbox{\textclg{P}}~^{\ast} (Minus(\widehat{\Gamma}^{[\gamma(k+1)]}))\setminus\mbox{\textclg{P}}~^{\ast}
(Minus(\widehat{\Gamma}^{[\gamma(k)]}))\vert=\vert\mbox{\textclg{P}}~^{\ast} (\Gamma)^{(k+1)}\setminus
\mbox{\textclg{P}}~^{\ast} (\Gamma)^{(k)}\vert.$$
Hence we are left to prove the equality
\begin{equation}
\label{pow3}
\vert\mbox{\textclg{P}}~^{\ast} (Minus(\widehat{\Gamma}^{[\gamma(k)]}))
\setminus
\bigcup _{q\in\Sigma}q^{[\gamma(k+1)]}\vert=\vert\mbox{\textclg{P}}~^{\ast} (\Gamma)^{(k)}
\setminus\bigcup _{q\in\Sigma}q^{(k+1)}\vert.
\end{equation}
Observe that
$$\mbox{\textclg{P}}~^{\ast} (Minus(\widehat{\Gamma}^{[\gamma(k)]}))
\setminus
\bigcup _{q\in\Sigma}q^{[\gamma(k+1)]}=\mbox{\textclg{P}}~^{\ast} (Minus(\widehat{\Gamma}^{[\gamma(k)]}))
\setminus\bigcup _{q\in\Sigma}
q^{[\gamma(k)]}\setminus\bigcup _{q\in\Sigma}\Delta^{[\gamma (k)]}Minus(q).$$
If $\Gamma\neq A_{k}$, by the disjointness of $\mbox{\textclg{P}}~^{\ast}$ we get
$$\mbox{\textclg{P}}~^{\ast} (Minus(\widehat{\Gamma}^{[\gamma(k)]}))
\setminus\bigcup _{q\in\Sigma}
q^{[\gamma(k)]}\setminus\bigcup _{q\in\Sigma}\Delta^{[\gamma (k)]}Minus(q)=
\mbox{\textclg{P}}~^{\ast} (Minus(\widehat{\Gamma}^{[\gamma(k)]}))\setminus\bigcup _{q\in\Sigma}
q^{[\gamma(k)]}.$$
Plainly, the same is true in the $\_ ^{()}$ version, thus \eqref{pow3}
holds for $\gamma(k)$, by virtue of (ix).
Otherwise, since $\bigcup _{q\in\Sigma}\Delta^{[\gamma (k)]}Minus(q)$
is a partition of a subset extract from
$$\mbox{\textclg{P}}~^{\ast} (Minus(\widehat{\Gamma}^{[\gamma(k)]}))
\setminus
\bigcup _{q\in\Sigma}q^{[\gamma(k)]},$$
we have that
\begin{align*}
&\vert\mbox{\textclg{P}}~^{\ast} (Minus(\widehat{\Gamma}^{[\gamma(k)]}))
\setminus\bigcup _{q\in\Sigma}
q^{[\gamma(k)]}\setminus\bigcup _{q\in\Sigma}\Delta^{[\gamma (k)]}
Minus(q)\vert\\
&=\vert\mbox{\textclg{P}}~^{\ast} (Minus(\widehat{\Gamma}^{[\gamma(k)]}))
\setminus\bigcup _{q\in\Sigma}
q^{[\gamma(k)]}\vert -\sum _{q\in\Sigma}\vert\Delta^{[\gamma (k)]}
Minus(q)\vert.
\end{align*}
Again, the same holds in the $\_ ^{()}$ version, and
\eqref{pow3} is reached by (i)[$\gamma(k)$] and (ii)[$\gamma(k)$].
This concludes the proof of (ix)[$\gamma(k+1)$].
Concerning (vii)[$\gamma(k+1)$], observe that
$q^{[\gamma(k+1)]}=q^{[\gamma(k)]}\cup\Delta^{[\gamma (k)]}(q)$.
By the induction hypothesis (vii)[$\gamma(k)$],
$$q^{[\gamma(k)]}=Minus(q^{[\gamma(k)]}).$$
On the other side,
since (iii)[$\gamma(k)$] holds and ${\mathcal C}$
is composed of green places only,
$$\Delta^{[\gamma (k)]}(q)=\Delta^{[\gamma (k)]}Minus(q),$$
which implies (vii)[$\gamma(k+1)$].
Regarding (ii)[$\gamma(k+1)$]-(vi)[$\gamma(k+1)$], the argument goes like
in the base case.
\end{proof}
\begin{mylemma}\label{upward}
Let
$(\left(\{q^{(\mu)}\}_{q\in\mathcal{P}}\right)_{\mu\leqslant\xi},
(\bullet),T,\mathcal{F},\mathcal{Q})$
be a {colored $\mathcal{P}$-process}. Moreover, let
$\big(\{\widehat{q}^{[\alpha]}\}_{\widehat{q}\in
\widehat{\mathcal{P}}}\big)_{\alpha\leqslant\xi '}$ be another formative process,
equipped of a Minus-Surplus partitioning.
Assume that, for some $k'\le\xi$ and $m\le\xi '$,
\begin{itemize}
\item $\widehat{\Sigma}_{m}$ {weakly imitates}
$\Sigma_{k'}$ {upwards};
\item the process
$\big(\{\widehat{q}^{[\alpha]}\}_{\widehat{q}\in\widehat{\mathcal{P}}}
\big)_{\alpha\in\gamma [k',\xi]}$ {imitates}
$\big(\{q^{(\mu)}\}_{q\in\mathcal{P}}\big)_{k'\le\mu\leqslant\xi}$, where $\gamma$ is an
injective map from $[k',\xi]$ to $[m,\xi ']$;
\item $\widehat{\Sigma}_{\xi '}$ has the same targets of $\Sigma_{\xi}$;
\item for all $\mu >m\wedge\mu\notin\gamma [k',\xi]$ the following holds:
$\Delta^{[\mu]}(\widehat{q})\subseteq\Delta^{[\mu]}Surplus(\widehat{q})$;
\item
if $\beta$ is the greatest ordinal
such that
$\beta\in\gamma[k',\xi]\wedge\beta\le\mu$, if
$q$ is a local trash of ${A}_{\mu}$, and if $GE(A_{\mu})>\gamma^{-1}(\beta)$,
then $\bigcup\widehat{A_\mu}^{[\xi ']}\notin
\Delta^{[\mu]}Surplus(\widehat{q})$.
\end{itemize}
Then $\widehat{\Sigma}_{\xi '}$ {imitates} $\Sigma_{\xi}$ {upwards}.
\end{mylemma}
\begin{proof}
We prove that the resulting partition $\widehat{\Sigma}_{\xi '}$
fulfills the
conditions:
\begin{itemize}
\item[(0)] $q^{(\xi)}\ni\in\powast{\Gamma}^{(\xi)}$
holds if and
only if $\widehat{q}^{[\xi ']}\ni\in\powast{\widehat{\Gamma}^{[\xi ']}}$;
\item[(1)] $\bigcup\widehat{\Gamma}^{[\xi ']}\in\widehat{q}^{[\xi ']}$
if and only if $\bigcup \Gamma^{(\xi)}\in q^{(\xi)}$;
\item[(2)] if $\Gamma\in\mathcal{Q}$ holds, then
$\powast{\widehat{\Gamma}^{[\xi ']}}\subseteq\bigcup\widehat{\mathcal{P}}^{[\xi ']}
$;
\item[(3$'$)] if
$q\in\mathcal{F}$, then $\card{\widehat{q}^{[\xi ']}}=\card{q^{(\xi)}}$.
\end{itemize}
Along the verification of properties (0)-(3$'$) we refer to (i)-(x)
of Def.~\ref{formimit}.
\begin{itemize}
\item[(0)] By the fact that the two partitions have the same targets;
\item[(1)]
In case $\bigcup\widehat{\Gamma}^{[\xi ']}\in\widehat{q}^{[\xi ']}$,
assuming that it
is distributed strictly before $m$, then $GE(\Gamma)<k'$. Indeed,
if not so, by (vi) Def.~\ref{formimit}, since $\bigcup\Gamma^{(GE(\Gamma))}\in
\Delta ^{(GE(\Gamma))}(q)$,
$$\bigcup\widehat{\Gamma}^{[\xi ']}\in\widehat{q}^{[\xi ']}=
\bigcup\widehat{\Gamma}^{[\gamma (GE(\Gamma))]}
\in\Delta^{[\gamma (GE(\Gamma))]}\widehat{q},$$
which is impossible, due to the fact
that $\bigcup\widehat{\Gamma}^{[\xi ']}\in\widehat{q}^{[\xi ']}$
is already in $\widehat{q}^{[\gamma (GE(\Gamma))]}$, and
$\Delta^{[\gamma (GE(\Gamma))]}\widehat{q}$,
by definition, is made of elements of
$\mbox{\textclg{P}}~^{\ast} (\widehat{\Gamma}^{[\gamma(GE(\Gamma)]})
\setminus
\bigcup _{q\in\Sigma}q^{[\gamma(GE(\Gamma))]}$.
Then, using the fact that
$\widehat{\Sigma}_{m}$
weakly simulates $\Sigma_{k'}$, the result follows.
Concerning the right implication, we are left to prove the case when
$\bigcup\widehat{\Gamma}^{[\xi ']}$
is distributed after or in $m$.
Let $j$ be such an index. By hypothesis,
$j$ cannot be outside $\gamma [k',\xi]$, and so
$j=\gamma (k)$ for some $k$.
We show that $k=GE(\Gamma)$. By contradiction, let us assume $k>GE(\Gamma)$.
Then, by (vi) Def.\ref{formimit},
$$\bigcup\widehat{\Gamma}^{[\gamma (GE(\Gamma))]}
\in\Delta^{[\gamma (GE(\Gamma))]}\widehat{q}.$$
Observe that, after $\gamma (GE(\Gamma))$, $\widehat{\Gamma}$ cannot change inside
the range of $\gamma$, on
account of (ii) and (iii) of Def.\ref{formimit}. It
it cannot change for
an index $j$ outside, since
$GE(\Gamma)$ is greater than the greatest ordinal $\beta$ such that
$\beta\in\gamma[k',\xi]\wedge\beta\le j$.
On the other hand,
$k$ cannot be strictly less than $GE(\Gamma)$, since in this case the same argument
used for $\bigcup\widehat{\Gamma}^{[\xi ']}$ distributed before $m$ and $GE(\Gamma)\ge k'$
applies.
Therefore $k=GE(\Gamma)$, and we are done.
We now show the left implication in the case $GE(\Gamma)< k'$.
The hypothesis implies that
$\bigcup\widehat{\Gamma}^{[m]}\in\widehat{q}^{[m]}$. Reasoning as before,
we conclude that $\widehat{\Gamma}$ cannot change along the process after $m$.
Finally, assuming $GE(\Gamma)\ge k'$, by (vi) \eqref{formimit} there holds
$$\bigcup\widehat{\Gamma}^{[\gamma (GE(\Gamma))]}\in
\Delta^{[\gamma (GE(\Gamma))]}\widehat{q}.$$
Again $\widehat{\Gamma}$ cannot change in the sequel of the
process, either along
the imitated process, or outside.
\item[(2)]Follows plainly from (iv) \eqref{formimit}. Indeed,
$\Gamma\in\mathcal{Q}$, therefore
$$\powast{\widehat{\Gamma}^{[\gamma (GE(\Gamma))]}}
\subseteq\bigcup\widehat{\mathcal{P}}^{[\gamma (GE(\Gamma))]}.$$
As observed in the previous point,
after $[\gamma (GE(\Gamma))]$, $\widehat{\Gamma}$ cannot change either
along the imitating process, by (ii) and (iii) \eqref{formimit},
or outside, by hypothesis.
Thus $\powast{\widehat{\Gamma}^{[\xi ']}}\subseteq\bigcup\widehat{\mathcal{P}}^{[\xi ']}$.
\item[(3$'$)]The red places cannot belong to
${\mathcal C}$. Hence, by the property (viii),
they cannot have Surplus part, which in turns implies that
$Minus(\widehat{q}^{[\xi ']})=\widehat{q}^{[\xi ']}$.
This, combined with
$\vert Minus^{[\gamma(\xi)]}(q)\vert=\vert q^{(\xi)}\vert$, due to
(i) \eqref{formimit},
leads to the thesis.
\end{itemize}
\end{proof}
The following theorem summarizes the previous results and
shows which properties two formative processes have to share in order to model the same literals.
The proof is a straight application of Corollary \ref{MsatisfiesII}
\begin{mytheorem}\label{upwards}
Let
$(\left(\{q^{(\mu)}\}_{q\in\mathcal{P}}\right)_{\mu\leqslant\xi},
(\bullet),T,\mathcal{F},\mathcal{Q})$
be a {colored $\mathcal{P}$-process}. Moreover, let
$\big(\{\widehat{q}^{[\alpha]}\}_{\widehat{q}\in
\widehat{\mathcal{P}}}\big)_{\alpha\leqslant\xi '}$ be another formative process,
equipped of a Minus-Surplus partitioning.
Assume that, for some $k'\le\xi$ and $m\le\xi '$,
\begin{itemize}
\item $\widehat{\Sigma}_{m}$ {weakly imitates}
$\Sigma_{k'}$ {upwards};
\item the process
$\big(\{\widehat{q}^{[\alpha]}\}_{\widehat{q}\in\widehat{\mathcal{P}}}
\big)_{\alpha\in\gamma [k',\xi]}$ {imitates}
$\big(\{q^{(\mu)}\}_{q\in\mathcal{P}}\big)_{k'\le\mu\leqslant\xi}$, where $\gamma$ is an
injective map from $[k',\xi]$ to $[m,\xi ']$;
\item $\widehat{\Sigma}_{\xi '}$ has the same targets of $\Sigma_{\xi}$;
\item for all $\mu >m\wedge\mu\notin\gamma [k',\xi]$ the following holds:
$\Delta^{[\mu]}(\widehat{q})\subseteq\Delta^{[\mu]}Surplus(\widehat{q})$;
\item
if $\beta$ is the greatest ordinal
such that
$\beta\in\gamma[k',\xi]\wedge\beta\le\mu$, if
$q$ is a local trash of ${A}_{\mu}$, and if $GE(A_{\mu})>\gamma^{-1}(\beta)$,
then $\bigcup\widehat{A_\mu}^{[\xi ']}\notin
\Delta^{[\mu]}Surplus(\widehat{q})$.
\end{itemize}
Consider a formula $\Phi\in {\rm MLSSPF}$,
a set-valued assignment ${\mathcal M}\in\{\:\mbox{\rm sets}\:\}^{{\mathcal X}_{\Phi}}$
defined on the collection ${\mathcal X}_{\Phi}$
of variables in $\Phi$
assuming that $(\left(\{q^{(\mu)}\}_{q\in\mathcal{P}}\right)_{\mu\leqslant\xi},
(\bullet),T,\mathcal{F},\mathcal{Q})$
is a {colored $\mathcal{P}$-process} for the \emph{$\Sigma_{{\mathcal X}_{\Phi}}$-board}
then, letting ${\mathcal M}'(v)=\bigcup[\Im_{{\mathcal M}}(\widehat{v})]$,
for every literal in $\Phi$, the following conditions are
fulfilled:
\begin{des}
\item if the literal is satisfied by $\mathcal M$, then it is satisfied
by ${\mathcal M}'$ too;
\item if the literal is satisfied by ${\mathcal M}'$, and does not
involve $\mbox{\textclg{P}}$~ or the construct $\{\anonymous,\dots,\anonymous\}$,
then it is satisfied by ${\mathcal M}$ too.
\end{des}
\end{mytheorem}
\begin{myremark}\rm
\label{salient}
The same result holds even in more relaxed conditions, revealing its
strength when we are looking for small models.
Namely, when we prune
the process instead of prolongate it.
In fact, the previous theorem holds, with an identical proof,
provided that the domain of $\gamma$
contains the following two collections of
{\em salient} ordinals:
$$M_{arrow}=\{ \mu\mid k'\le\mu<\xi\wedge
\exists q\in \mathcal{P}
q^{(\mu)} \cap \powast{A^{(\mu)}_{\mu}} =\emptyset\wedge
\Delta^{(\mu)}(q)\neq\emptyset\}$$
and
$$M_{GE}=\{\mu\mid k'\le\mu<\xi
\wedge
\bigcup A_{\mu}^{(\mu)} =\bigcup A_{\mu}^{(\bullet)}
\in\bigcup P^{(\bullet)}\}.$$
\end {myremark}
\section{Using the Two Structural Lemmas into Set Computable Examples}
Assume that ${\mathcal M}$ is a finite set assignment to the
variables of an assigned formula $\Phi$ of MLSSPF, which
contains literals of the type $\neg Finite(x)$.
Obviously, ${\mathcal M}$ cannot be a model for $\Phi$, although it
could happen that it satisfies every other literal, except those
of that kind. The question is: in this situation could
${\mathcal M}$ witness the satisfiability of $\Phi$?
The answer is positive, as we will show, and
the core argument for proving that
lies inside a possible history of ${\mathcal M}$.
Indeed, given a formative process for the Venn partition
$\Sigma$ inherited from ${\mathcal M}$,
if we can find an ``engine" capable to pump elements inside at least one Venn region
for each variable $x$, such that $\neg Finite(x)$
lies in $\Phi$ without affecting the satisfiability of other literals, we reach
the desired conclusion.
We will be more precise on the exact meaning
of ``engine'', and how profitably the results of the previous
sections can be used in order to preserve the satisfiability of the other literals.
even though the
size of the assignment of some variables is infinitely increased.
\begin{mydef}\rm
\label{prepareForPumpingPaths} In a $\mathcal{P}$-board
$\mathcal{G}$, a \emph{path} is an ordered vertex list
$W_1,\dots,W_k$, in which places and nodes are so alternate that
$W_i,W_{i+1}$ is an edge of $\mathcal{G}$, for $i=1,\dots,k-1$. A
path is said to be \emph{simple} if neither places nor nodes occur
twice (i.e., $W_i\neq W_j$ when $0<i<j\leqslant k$ and
$i\equiv j\;(\mbox{{mod }}2)$ ).
\end{mydef}
\begin{mydef}\rm
\label{pumpingPaths} In a colored $\mathcal{P}$-board
$\mathcal{G}=(T,\mathcal{F},\mathcal{Q})$, a path
$$\mathcal{C} \equiv
C_0,q_0,C_1,\dots,q_n,C_{n+1} $$
where the piece $C_0,q_0,C_1,\dots,q_n,$ is simple and
$n\geqslant 0$, \emph{devoid} of red places, and such
that $C_{n+1} = C_{0}$,
is said to be a
\emph{simple pumping cycle}.
\end{mydef}
Given a path $\mathcal{D}$ in a $\mathcal{P}$-board $\mathcal{G}$, we
denote by $\PLACES{\mathcal{D}}$ and $\NODES{\mathcal{D}}$ the
collections of places and nodes occurring in $\mathcal{D}$, respectively.
Moreover, given a node $B$ in $\mathcal{G}$, we denote
by $\PN{B}$ the collection of all nodes which have nonnull
intersection with $B$.
The following is to be regarded as the engine which increases the size
of some places without affecting the validity of the formula.
\begin{mydef}\rm
\label{pumpchain}
\rm Let $\mathcal{C}$ be a simple pumping cycle
relative to a given colored $\mathcal{P}$-process
$\big(\Sigma_\mu\big)_{\mu\leqslant\ell},[\bullet],T,\mathcal{F},\mathcal{Q}$,
with $\ell$ finite. Then $<q_0,i_0,\mathcal{C}>$ is called a
\emph{simple pumping event} whenever we have
\begin{itemize}
\item[(i)] $q_0^{[i_{0}]} \setminus \bigcup\bigcup
\mathcal{P}^{[i_{0}]} \neq \emptyset$, $q_0\in
\PLACES{\mathcal{C}}$;
\item[(ii)] $GE(\PN{\PLACES{\mathcal{C}}})\ge i_0$;
\item[(iii)] $\mbox{\textclg{P}}~^{\ast}(B^{[i_{0}]}) \neq \emptyset$
(i.e., $\emptyset\notin B^{[i_{0}]}$), for $B
\in\NODES{\mathcal{C}}$.
\end{itemize}
\end{mydef}
If $\Sigma$ is a particular Venn partition $\Sigma_{{\mathcal X}_{\Phi}}$,
the variables that contain the places involved in the pumping cycle can be considered
\emph{potential infinite variables}.
\begin{mytheorem}\label{pumping1}
Assume that ${\mathcal M}$ is a finite transitive set
assignment to the variables ${\mathcal X}_{\Phi}$ of an assigned formula
$\Phi$ of {\rm MLSSPF}, that
satisfies every other literals except those
of the type $\neg Finite(x)$. Consider the transitive
\emph{$\Sigma_{{\mathcal X}_{\Phi}}$-board} ${\mathcal G}=(T,\mathcal{F},\mathcal{Q})$,
and an associated colored $\mathcal{P}$-process
$\big(\Sigma_\mu\big)_{\mu\leqslant\ell},(\bullet),T,\mathcal{F},\mathcal{Q}$,
with $\ell$ finite.
Then there exists a model for $\Phi$,
provided there is a {simple pumping event}
$<q,i_0,\mathcal{C}>$ such that $\PLACES{\mathcal{C}}$ is
contained in a closed set $\overline {\mathcal{C}}$
satisfying the statement:
$$\mbox{ For each variable } x \mbox{ such that }\neg Finite(x)\in\Phi,
\Im_{{\mathcal M}}(x)\cap\PLACES{\mathcal{C}}\mbox{ is not empty. }$$
\end{mytheorem}
\proof
Let $<q_0,i_0,\mathcal{C}>$ be our {simple pumping event}, where
$\mathcal{C}$ is equal to $$\{C_0,q_0\dots q_{n}, C_{n+1}\}.$$
We build a new formative process
$\big(\widehat{\Sigma}_\mu\big)_{\mu\leqslant\ell},[\bullet],T $,
using the original one as an oracle.
In the meanwhile, a Minus-Surplus refinement is done.
We first define the sequence of the nodes to be used in this new process.
Denote by $\seq {\ell}=\{A_0 \dots A_{\ell}\}$ the sequence of nodes
used along the given process $\big(\Sigma_\mu\big)_{\mu\leqslant\ell},
(\bullet),T,\mathcal{F},\mathcal{Q}$.
The following sequence serves to our scope:
$$A_1,\dots A_{i_0 -1}\underbrace{C_1\dots C_{n+1}}_{\aleph_{0} -times},
A_{\gamma(i_0)}\dots A_{\gamma(\ell)},$$
where, for all $j$, $A_{\gamma(j)}=A_j$ and the cycle
$\mathcal{C}$ are repeated $\aleph_{0} -times$.
In order to define a formative process, we just need to exhibit the way to
distribute all the elements produced at each stage.
Our strategy consists to follow the old formative process up to the
stage $i_0 -1 =\gamma (i_0-1)$,
setting $\left(\{\widehat{q}^{[j]}\}_{q\in\mathcal{P}}\right)_{j\leqslant i_0-1}$
$\left(\{q^{(j)}\}_{q\in\mathcal{P}}\right)_{j\leqslant i_0-1}$.
Along this segment,
we define $\gamma$ as the identity map; then, we ``pump'' the cycle in order to create
new elements and distribute them.
This procedure by transfinite induction
increases the cardinality of the blocks inside the cycle, preserving the cardinality
of all the blocks not involved in the pumping procedure.
In order to do that, we
distinguish the elements reserved for the pumping procedure (Surplus portion) from those
used for mimicking the old process (Minus portion). The
Minus-Surplus refinement that we are about
to define will serve such a scope.
Without loss of generality,
we assume that at each step the cycle can distribute at least
three new elements (otherwise, we can pump the cycle
to give at least two elements to every block involved in the cycle).
By Definition of {simple pumping event}, $q^{(i_{0})}\setminus\bigcup\bigcup
\mathcal{P}^{(i_{0})} \neq \emptyset$, which means that in $q^{(i_{0})}$ there are unused elements.
Let $t_0$ be one of these, and
define the partitions Surplus and Minus as follows:
\begin{itemize}
\item For all $q\neq q _0$ put\\
$Surplus ^{[\gamma (i_0-1)+1]}(\widehat{q}) =\emptyset$ and
$Minus^{[\gamma (i_0-1)+1]}(\widehat{q})=q^{(i_0)}$;
\item For $q _0$ put\\
$Surplus ^{[\gamma (i_0-1)+1]} (\widehat{q} _0)=\{t_0\}$
$ Minus^{[\gamma (i_0-1)+1 ]}(\widehat{q} _0)=
q_0^{(i_0)}\setminus\{ t_0 \} $;
\end{itemize}
Since every block involved in the cycle has at least two elements,
the set
$$\mbox{\textclg{P}}~^{\ast} \Big( \big\{ Surplus^{[\gamma (i_0-1)+1]}(\widehat{q}_0)\big\}
\cup \widehat{C_1}^{[\gamma (i_0-1)+1]}\Big)\setminus
\Big\{ \bigcup \widehat{C_1}^{[\gamma (i_0-1)+1]}\Big\}$$
is not empty.
Moreover, by Lemma \ref{unused}, it is made of unused elements only.
Thus,
\begin{align*}
&\mbox{\textclg{P}}~^{\ast} ( \{ Surplus^{[\gamma (i_0-1)+1]}(\widehat{q}_0)\}
\cup\widehat{C_1} ^{[\gamma (i_0-1)+1]})\setminus
\{ \bigcup\widehat{C_1} ^{[\gamma (i_0-1)+1]}\}\\
&=(\mbox{\textclg{P}}~^{\ast} ( \{ Surplus^{[\gamma (i_0-1)+1]}(\widehat{q}_0)\}
\cup\widehat{C_1} ^{[\gamma (i_0-1)+1]})\setminus
\{ \bigcup\widehat{C_1} ^{[\gamma (i_0-1)+1]}\})\setminus
\bigcup _{q\in\Sigma}\widehat{q}^{[\gamma(i_0-1)+1]},
\end{align*}
so that the position
$$\Delta ^{[\gamma (i_0-1)+1]}(Surplus(\widehat{q}_1))=
\mbox{\textclg{P}}~^{\ast} ( \{ Surplus^{[\gamma (i_0-1)+1]}(\widehat{q}_0)\}
\cup \widehat{C_1}^{[\gamma (i_0-1)+1]})\setminus
\{ \bigcup \widehat{C_1}^{[\gamma (i_0-1)+1]}\}$$
makes sense.
The other $\Delta$-set are left empty.
Observe that, in particular, for all $q\ne q_0$ this yields
$$Minus^{[\gamma (i_0-1)+2]}(\widehat{q})
=Minus^{[\gamma (i_0-1)+1]}(\widehat{q})=q^{(i_0)}.$$
We then continue defining
\begin{align*}
&\Delta ^{[\gamma (i_0-1)+2]}(Surplus(\widehat{q}_2))\\
&=
\mbox{\textclg{P}}~^{\ast} ( \{\Delta ^{[\gamma (i_0-1)+1]}(Surplus(\widehat{q}_1)) \}
\cup\widehat{C_2} ^{[\gamma (i_0-1)+2]})\setminus
\{ \bigcup\widehat{C_2} ^{[\gamma (i_0-1)+2]}\},
\end{align*}
and all the argument used in the previous step can be repeated.
This procedure will prosecuted until the end of the cycle is reached, that is,
the node $C_{n+1}$.
At this step we introduce a slight modification in the construction of the $\Delta$-sets.
Namely, we have to restore the cardinality of $Minus(\widehat{q})$,
which was pertubed moving
$t_0$ from the Minus to the Surplus portion, in order to trigger off the pumping procedure.
Hence, pick an element $t_1$ inside
$$\mbox{\textclg{P}}~^{\ast} (\{\Delta (Surplus^{[\gamma (i_0-1)+n+1]}(\widehat{q}_n)\}
\cup\widehat{C_{n+1}} ^{[\gamma (i_0-1)+n+1]})\setminus
\{\bigcup\widehat{C_{n+1}} ^{[\gamma (i_0-1)+n+1]}\}.$$
Since we are assuming that
at each step the cycle can distribute at least 3 new elements, the set
\begin{align*}
&\Delta ^{[\gamma (i_0-1)+n+1]} (Surplus(\widehat{q}_0))\\
&=\mbox{\textclg{P}}~^{\ast} (\{\Delta (Surplus^{[\gamma (i_0-1)+n+1]}(\widehat{q}_n)\}
\cup\widehat{C_{n+1}} ^{[\gamma (i_0-1)+n+1]})\setminus
\{\bigcup\widehat{C_{n+1}} ^{[\gamma (i_0-1)+n+1]}\}
\setminus \{t_1\}
\end{align*}
is certainly not empty.
Then define
$$ \Delta ^{[\gamma (i_0-1)+1]}(Minus(\widehat{q}_0))=\{t_1\}.$$
Notice that $t_1$ is unused, and so will be kept along the entire pumping
procedure of pumping, since
it lies in the Minus portion of $\widehat{q}_0$,
which is untouched in this segment of the new formative process.
As before, the procedure can prosecute $\aleph _0$-times.
Since $q^{(\lambda)}=\bigcup_{\nu<\lambda}q^{(\nu)}$ for every $q\in
\mathcal{P}$ and every limit ordinal $\lambda\leqslant \xi$, it is clear that
$\widehat{q}^{[\omega =\gamma (i_0)]}$ is equal to
$\bigcup_{i\in{\mathbb N}}\widehat{q}^{[(i_0-1)+i]}$ for all $q\in
\mathcal{P}$, consistently the Minus-Surplus partition is defined for the stage $\omega$.
By construction, for all $q\in\mathcal{P}$ such that $q\neq q_0$
$(Minus^{\gamma(i_0)}(\widehat{q}))$ is equal to $q^{(i_0)}$ while
$(Minus^{[\gamma(i_0)]}(\widehat{q}_0))$
is equal to $(q_0^{(i_0)}\setminus\{ t_0 \})\cup\{ t_1 \}$.
Our aim is to show that the transitive partitions
$\Sigma_{i_0}$ and $\widehat{\Sigma}_{\gamma(i_0)}$ verify the
conditions to apply subsequently Lemma \ref{pasting} and Corollary \ref{upwards}, so proving the satisfiability of $\Phi$.
Concerning the application of Lemma~\ref{pasting},
we have to show properties (i), (vii), (viii), (x), and (a)-(c).
This is just a bookkeeping argument, and we
detail it in the Appendix.
Now the formative process $[\bullet]$ has copied the original one along
the segment $[i_0,\ell]$.
In order to apply
Lemma \ref{upwards}, we need to show that $\widehat{\Sigma}_{\gamma(\ell)}$
has the same target as $\Sigma_{\ell}$.
We simply observe that, if $q^{(\ell)}$ is a target of $\Gamma^{(\ell)}$,
there must
exist a step $i$ such that $\Gamma=A_{i}$ and $\Delta^{i}(q)\ne\emptyset$.
Since
both the segment $[0,(i_0-1)]$ is equal to $[0,\gamma (i_0-1)]$,
and the segment $[i_0,\ell]$ is imitated by one application of Lemma \ref{pasting},
then
$\Delta^{[\gamma (i)]}(\widehat{q})\ne\emptyset$ too.
On the other side, if $\Delta^{[\alpha]}(\widehat{q})\ne\emptyset$ for some $\alpha$, $q$
has to be a target of $\widehat{A}_{\alpha}$, so that we are done.
At this point Corollary \ref{upwards} applies, therefore
all literals except those of $Finite$-type are satisfied.
Finally, the literals as $Finite(x)$ are satisfied as well.
Indeed, every block $q$ contained in $\Im_{{\mathcal M}}(x)$
lies in $\mathcal{F}$, and the formative process
$[\bullet]$ does not change size of such a block. Also, by hypothesis,
$\mbox{ for each variable } x$ such that $\neg Finite(x)\in\Phi,
\Im_{{\mathcal M}}(x)\cap\PLACES{\mathcal{C}}\mbox{ is not empty }$,
and the blocks in the pumping cycle are infinitely increased during the pumping procedure.
Hence all of them are of
infinite size, as well as all the variables containing at least one of them.
This in turns implies that all $\neg Finite(x)\in\Phi$ are satisfied by the new model.
\qed
The above technique provides a valid tool to
solve problems which require to build an infinite model.
In \cite{CU} it is shown that there is a computable function
$f(n)$ such that, if a formula $\Phi$ in MLSSPF
is satisfiable, then there is an assignment rank bounded by
$f(\vert{\mathcal X}_{\Phi}\vert)$ which satisfies a slight
modification of the properties
described in Theorem \ref{pumping1}.
But then MLSSPF has the witness small property, and is
therefore decidable.
A similar argument it is used to prove the witness small property for MLSSPU.
\section{Open Problems}
\subsection{A Decidability Problem}
Even if all the problems related to the literals which force the infinity
are treatable by the present approach,
the decidability of MLSSP extended by the cartesian product binary
operator [$x=y\times z$] is still an open question.
Observe that this language forces the infinity.
This problem is originally due to M.\ Davis, who proposed it as a set computable
version of the Tenth Hilbert Problem (see \cite{Mat}).
\subsection{ A Complexity Problem}
Decidability of MLSSP is NP-complete, therefore there is no hope to find
a polynomial time bound for our problems.
Nevertheless, the witness small model property furnishes
double exponential decision algorithms. An exponential bound could be a good platform to
perform polynomial time for special cases.
\section*{Appendix}
Here we exhibit a complete verification of the properties
requested for the application of Lemma \ref{pasting}
within the proof of Theorem \ref{pumping1}.
\begin{itemize}
\item[(i)] First assume $q\ne q_{0}$. By construction, only Surplus
sides are increased along pumping procedure.
Therefore $q^{(i_0)}=Minus^{[\gamma(i_0)]}\widehat{q}$. Otherwise, observe that
$Minus^{[\gamma(i_0)]}\widehat{q}=(q^{(i_0)}\setminus\{t_0\})\cup \{t_1\}$, hence
$\vert q^{(i_0)}\vert=\vert Minus^{[\gamma(i_0)]}\widehat{q}\vert$.
\item[(vii)] Observe that $\overline {\mathcal{C}}$ is
composed of green blocks only. Therefore, if $q\in\mathcal{F}$, by
hypothesis $q$ cannot belong to
$\PLACES{\mathcal{C}}$, but the only blocks whose size is increased are inside
$\PLACES{\mathcal{C}}$, hence
$q^{(i_0)}=Minus^{[\gamma(i_0)]}\widehat{q}=\widehat{q}^{[\gamma(i_0)]}$.
\item[(viii)] Trivial.
\item[(x)] Assume $q_{0}\notin\Gamma$.
In this case, $Minus^{[\gamma(i_0)]}\widehat\Gamma=
Minus^{[\gamma(i_0-1)]}\widehat\Gamma$. Therefore, for all block $q$,
$$\mbox{\textclg{P}}~^{\ast} (Minus^{[\gamma (i_0-1)]}(\widehat{\Gamma}))\cap q^{[\gamma (i_0-1)+1]}
=\mbox{\textclg{P}}~^{\ast} (Minus^{[\gamma (i_0)]}(\widehat{\Gamma}))\cap q^{[\gamma (i_0-1)+1]}.$$
Along the pumping procedure, only the Surplus nodes are used.
Since $\mbox{\textclg{P}}~^{\ast}$ of the Surplus nodes
are always disjoint from the Minus ones, we can prolongate the previous chain
of equalities with
$$\mbox{\textclg{P}}~^{\ast} (Minus^{[\gamma (i_0)]}(\widehat{\Gamma}))\cap q^{[\gamma (i_0-1)+1]}=
\mbox{\textclg{P}}~^{\ast} (Minus^{[\gamma (i_0)]}(\widehat{\Gamma}))\cap q^{[\gamma (i_0)]}.$$
On the other hand,
$$\mbox{\textclg{P}}~^{\ast} (\Gamma ^{(i_0-1)})\cap q^{(i_0)}
=\mbox{\textclg{P}}~^{\ast} (\Gamma ^{(i_0)})\cap q^{(i_0)}.$$
Finally, by construction,
$$\mbox{\textclg{P}}~^{\ast} (\Gamma ^{(i_0-1)})\cap q^{(i_0)}
=\mbox{\textclg{P}}~^{\ast} (Minus^{[\gamma (i_0-1)]}(\widehat{\Gamma}))\cap\widehat{q}^{[\gamma (i_0-1)+1]}.$$
In the other case, observe that $t_1$ is new at the step
$\gamma(i_0-1)+n+1$. Thus everything created from $t_1$
cannot be inside any block $q$ before its distribution, neither in the segment
$[\gamma (i_0-1)+n+1,\gamma (i_0)]$, for only the Surplus nodes are used, and $t_1$ is in the
Minus side of block $q_0$. This yields
$$\mbox{\textclg{P}}~^{\ast} (Minus^{[\gamma (i_0-1)]}(\widehat{\Gamma}))\cap q^{[\gamma (i_0-1)+1]}
=\mbox{\textclg{P}}~^{\ast} (Minus^{[\gamma (i_0)]}(\widehat{\Gamma}))\cap q^{[\gamma (i_0-1)+1]}.$$
The prosecution of the argument follows exactly the one of the former case.
\item[(a)] If $q_{0}\in\Gamma$, the property trivially holds since $t_0$ is new at
the step $i_0$; therefore $\bigcup\Gamma$ cannot have been distributed at the stage $i_0$.
On the other hand $t_1$, which belongs to $Minus^{[\gamma (i_0-1)+n+1]}(\widehat{\Gamma}))$,
is new at the step $\gamma(i_0-1)+n+1$.
Hence $\bigcup Minus^{[\gamma (i_0-1)+n]}(\widehat{\Gamma})$
cannot have been distributed at the stage
$\gamma (i_0-1)+n]$. Again, the Minus nodes are unused along the pumping procedure, hence
$\bigcup Minus^{[\gamma (i_0)]}(\widehat{\Gamma}))$ is not distributed
at the limit step $\gamma (i_0)$ as well.
Conversely, if $q_{0}\notin\Gamma$, the result easily follow
by standard arguments from the
fact that the Minus portion of $\Gamma$ and the original $\Gamma$ are
equal at the stage $\gamma (i_0)$, and
the Minus nodes are unused along the pumping procedure.
\item[(b)] $Surplus^{[\gamma (i_0)]}(q)\ne\emptyset$ and $q\in\Gamma$,
therefore the node $\Gamma$
is changed along the pumping procedure.
By construction, $\bigcup\Gamma$ is never distributed along pumping procedure, so
$$\bigcup\widehat{\Gamma}^{[\gamma (i_0)]}
\in\mbox{\textclg{P}}~^{\ast} (\widehat{\Gamma}^{[\gamma (i_0)]})\setminus\bigcup_{q\in\Sigma}
\widehat{q}^{[\gamma (i_0)]}.$$
\item[(c)] Easily follows from the fact that
after a grand event nothing changes in the formative process,
and from (ii) of Def.\ref{pumpchain},
which asserts that $GE(\PN{\PLACES{\mathcal{C}}}))\ge i_0$.
\end{itemize}
|
{
"timestamp": "2004-12-30T10:58:32",
"yymm": "0411",
"arxiv_id": "math/0411226",
"language": "en",
"url": "https://arxiv.org/abs/math/0411226"
}
|
\section{Introduction}
In the past few years determinations of the star formation history of
the universe have allowed us to begin to understand quantitatively
when and how the stars in the universe were formed. Measurements of
the rest-frame ultraviolet luminosities of galaxies have been
particularly useful in this endeavor. In the very local universe,
there is a relative lack of systematic surveys of galaxies in the UV.
Before the launch of GALEX, the most comprehensive survey of galaxies
in the local universe was from the FOCA experiment \citep{milliard92},
a balloon-borne telescope that made measurements in a single band
centered at 2000\AA. Based upon FOCA observations of a total of
$\sim2.2$ deg$^2$, \citet{treyer98} and
\citet{sullivan00} measured the first UV
luminosity function (LF) for a sample of 273 galaxies with
spectroscopic redshifts at $\bar{z}=0.15$. Their LF has a steep faint
end slope and a total UV luminosity density, and corresponding star
formation rate density, larger than most previous estimates. This
higher local UV luminosity density in conjunction with measurements
at larger distances lead \citet{wilson02} to infer a luminosity
density evolution proportional to $(1+z)^{1.7\pm1.0}$, a trend
shallower than had been estimated previously from the CFRS sample
\citep{lilly96}.
In this letter we present the first results regarding the UV LF based
upon measurements from the {\it Galaxy Evolution Explorer} (GALEX) in
conjunction with redshifts from the 2dF Galaxy Redshift Survey (2dFGRS)
\citep{colless01}. The new GALEX data allow us to expand upon the
previous FOCA results using a much larger sample drawn from an area of
56.73 deg$^2$ although to a shallower limiting magnitude of
$m_{UV}=20$. Throughout this paper, we assume $H_0=70~{\rm
km~s^{-1}~Mpc^{-1}}$, $\Omega_{M}=0.3$ and $\Omega_{\Lambda}=0.7$.
\section{Data}
The data analyzed in this paper consist of 133 GALEX All-Sky Survey
(AIS) pointings that overlap the 2dF Galaxy Redshift Survey in the
South Galactic Pole region.
The GALEX field-of-view is circular with diameter of $1.2\arcdeg$ and each
pointing is imaged simultaneously in both the FUV and NUV bands
with effective wavelengths of 1530\AA~and 2310\AA,
respectively. The median exposure time for the fields is 105 seconds,
allowing us to reach a S/N ratio of $\sim5$ for $FUV\approx20.0$ and
$NUV\approx20.5$. See \citet{martin04} and \citet{morissey04}
for details regarding the GALEX instruments and mission.
Sources were detected and measured from the GALEX images using the
program SExtractor \citep{bertin96}. As the NUV images are
substantially deeper than the FUV, we used the NUV images for
detection and measured the FUV flux in the same aperture as for the
NUV. The fields analyzed here were processed using a larger SExtractor
deblending parameter DEBLEND\_MINCONT as the standard GALEX pipeline
processing tends to break apart well-resolved galaxies into more than
one source. We elected to use the MAG\_AUTO magnitudes measured by
SExtractor through an elliptical aperture whose semi-major axis is
scaled to 2.5 times the first moment of the object's radial profile,
as first suggested by \citet{kron80}. All of the apparent magnitudes
were corrected for foreground extinction using the \citet{schlegel98}
reddening maps and assuming the extinction law of
\citet{cardelli89}. The ratio of the extinction in the GALEX bands to
the reddening $E(B-V)$ was calculated by averaging the extinction law
over each GALEX bandpass, resulting in $A_{FUV}/E(B-V) = 8.376$ and
$A_{NUV}/E(B-V)=8.741$. The median extinction correction for the
galaxies in our South Galactic Pole
sample is $0.15$ mag in both bands, with the
corrections ranging from 0.1 to 0.3 mag.
The GALEX catalogs were matched with the 2dFGRS input catalog using a
search radius of $6\arcsec$. To remove any overlap between adjacent
pointings, we only included sources detected within the inner
$0.45\arcdeg$ of each field. In addition, sources likely contaminated
by artifacts from bright stars, with 2dF redshift quality flag less
than three or with effective exposure times less than 60 sec were
removed. Finally, we excluded GALEX sources in regions where the 2dF
redshift completeness was less than 80\%. After applying all of these
cuts to each band, the total area on the sky of GALEX-2dF overlap is
56.73 deg$^{2}$.
The GALEX resolution of $6-7\arcsec$ (FWHM) \citep{morissey04} is not
sufficient to accurately separate stars and galaxies. Furthermore, the
2dFGRS input catalog available from the 2dFGRS web
page\footnote{http://www.mso.anu.edu/2dFGRS/}
only includes galaxies brighter than $b_j=19.45$ and
does not include stars. In order to asses the total completeness of
our 2dF-GALEX matched sample, we normalized our results to the total galaxy
number counts determined by \citet{xu04} based primarily upon 22.64
deg$^{2}$ of GALEX Medium Imaging Survey data overlapping the Sloan
Digital Sky Survey (SDSS) Data Release 1 \citep{abazajian03}. As the
SDSS data include stars and galaxies and reach fainter magnitudes,
they result in a more accurate determination of the galaxy number counts
in the UV.
If we assume that the average galaxy number counts in the SDSS North
Galactic Pole fields are the same as in the GALEX-2dF overlap, then
the redshift completeness of the 2dF matched catalog is given by the
number counts of galaxies with redshifts from 2dF divided by the total
galaxy number counts from the SDSS overlap. This ratio is shown in Figure
\ref{completeness}.
The completeness turns over at 20th mag because the redshift sample
becomes incomplete for galaxies with blue $(FUV - b_j)$ or $(NUV -
b_j)$ colors. We have limited our LF determination in each band to
galaxies brighter than this limit. To avoid problems with photometry
of large bright galaxies, we also imposed a bright magnitude limit of
17. The average completeness weighted by the number counts in the
range $17-20$ mag is 92\% in the FUV and 79\% in the NUV. For objects
with magnitudes brighter than 20.0 in either band, we visually
inspected all of the 2dF spectra and removed a total of 27 objects
with very broad emission lines indicating that the objects are most
likely some sort of AGN. The redshift distributions for the FUV and
NUV samples are shown in Figure \ref{distributions}.
We further restricted our sample to those galaxies with redshifts $z<0.1$
to insure that our sample is not sensitive to evolution.
The average redshifts are 0.055 and
0.058 in the FUV and NUV, respectively.
After applying all of the cuts mentioned in this section,
a total of 896 galaxies in the FUV and 1124 galaxies in the
NUV remained. The luminosity functions for
the objects with $z>0.1$ are presented in \citet{treyer04}.
\begin{figure}
\plotone{f1.ps}
\caption{Completeness of the GALEX-2dF redshift sample defined as the
ratio of the number counts of galaxies with 2dF redshifts to
the number counts of galaxies as derived from GALEX observations that overlap
the SDSS survey \citep{xu04}.
The solid and dashed lines indicate the FUV and NUV redshift completeness,
respectively.\label{completeness}}
\end{figure}
\begin{figure}
\plotone{f2.ps}
\caption{The redshift distributions
of the FUV and NUV selected samples (blue solid and red dashed lines,
respectively) in the range $17\le m_{uv} \le 20$.
\label{distributions}}
\end{figure}
\section{Luminosity Functions}
Using the FUV,
NUV and $b_j$ magnitudes, we assigned a best-fit spectral type
to each galaxy using a representative subset of the SEDs from
\citet{bruzual03} and determined
the K-correction needed to transform the observed UV magnitudes to
rest-frame measurements at $z=0$. The K-corrections are in general
quite small ($\lesssim 0.2$).
We determined the LF $\Phi(M)$ and its error
$\sigma(\Phi(M))$ in each band using the $V_{max}$
method \citep{felten76}:
\begin{equation}
\Phi(M) = \sum{f(m)/V_{max}}
\end{equation}
\begin{equation}
\sigma(\Phi(M)) = \left(\sum{f^2(m)/V_{max}^2}\right)^{1/2}
\end{equation}
where $f(m)$ is the inverse of the redshift completeness as estimated in
\S2 above and $V_{max}$ is the maximum co-moving volume within which
each galaxy
could have been observed given the bright and faint limiting magnitudes
of our sample and its best-fit SED. The resulting LFs
are shown in Figure \ref{lf}.
By minimizing $\chi^2$, we fit the $V_{max}$ LF points
in each band with a Schechter function \citep{schechter76}:
$\Phi(L)dL=\phi^{\ast} (L/L^{\ast})^{\alpha} e^{-L/L^{\ast}} dL/L^{\ast}$
where $\phi^{\ast}$, $M^{\ast}$ and $\alpha$ were free parameters. The best fit
parameters and their errors, calculated using the range of solutions
within 1.0 of the minimum $\chi^2$, are listed in Table \ref{params} along
with the best-fit LF from \citet{sullivan00} converted
to the AB magnitude system and to $H_0=70$. The errors in $\alpha$
and $M^{\ast}$ are highly correlated and the inset of Figure \ref{lf}
shows the $1\sigma$ error contours projected into the $M^{\ast}-\alpha$
plane. Since the $V_{max}$ method
can be biased in the presence of clustering, we also computed the best-fit
Schechter parameters using the maximum likelihood STY method
\citep{sandage79}. The resulting STY values are listed in Table \ref{params}
and are also plotted in the inset of Figure \ref{lf}. The STY values
lie just inside and outside of the $1\sigma$ $V_{max}$ error ellipses
in the FUV and NUV, respectively. We adopt the $V_{max}$ results
in the discussion below.
\begin{figure}
\plotone{f3.ps}
\caption{The FUV (blue circles) and NUV (red triangles) LFs
for $z<0.1$. The solid lines are the Schechter function fits with best-fit
parameters from Table \ref{params}. The dotted green line
shows the LF measured at ${\rm 2000~\AA}$ from FOCA
data by \citet{sullivan00} over the range of absolute magnitudes
explored in that study. The inset plots the $1\sigma$ error contours
of the Schechter function fits projected into the $M^{\ast} - \alpha$ plane
for the FUV (blue) and NUV (red). The dashed contour shows values with
$\chi^2 - \chi^2_{min} = 1.0$ while the solid contour delineates
$\chi^2 - \chi^2_{min} = 2.3$ which corresponds to the joint $1\sigma$
uncertainty on $M^{\ast}$ and $\alpha$. The red and blue stars indicate
the best-fitting values of $M^{\ast}$ and $\alpha$ obtained from the
STY method. \label{lf}}
\end{figure}
\begin{deluxetable}{lcccccccc}
\rotate
\tablecolumns{8}
\tablewidth{0pc}
\tablecaption{Schechter Function Parameters\label{params}}
\tablehead{
\colhead{} & \colhead{} & \multicolumn{4}{c}{$V_{max}$ method} & \colhead{} &
\multicolumn{2}{c}{STY method} \\
\cline{3-6} \cline{8-9} \\
\colhead{band} & \colhead{$z$} & \colhead{$M^{\ast}$} & \colhead{$\alpha$} &
\colhead{$\log{\phi^{\ast}}$ (Mpc$^{-3}$)} &
\colhead{$\log{\rho_L}$ (ergs s$^{-1}$ Hz$^{-1}$ Mpc$^{-3}$)} & \colhead{} &
\colhead{$M^{\ast}$} & \colhead{$\alpha$}
}
\startdata
FUV & $0-0.1$ & $-18.04 \pm 0.11$ & $-1.22 \pm 0.07$ & $-2.37 \pm 0.06$ & $25.55 \pm 0.12$ & & $-18.12$ & $-1.23$ \\
NUV & $0-0.1$ & $-18.23 \pm 0.11$ & $-1.16 \pm 0.07$ & $-2.26 \pm 0.06$ & $25.72 \pm 0.12$ & & $-18.27$ & $-1.10$ \\
FOCA & 0.15 & $-19.10 \pm 0.13$ & $-1.51 \pm 0.10$ & $-2.48 \pm 0.11$ & $26.06 \pm 0.15$ & & \nodata & \nodata \\
\enddata
\end{deluxetable}
\section{Discussion}
As can be seen in Figure \ref{lf}, there are significant differences
between the results presented here and those from
\citet{sullivan00}. The GALEX results have a fainter $M^{\ast}$ in both bands
and have a shallower faint end slope. The FOCA passband is centered
at 2015\AA~with FWHM of 188\AA~and thus one would expect the FOCA
results to lie in between the GALEX FUV and NUV data. However, the
FOCA sample is truly an UV-selected sample while that presented here
relies upon the $b_j$-selected 2dFGRS. This selection could
introduce a bias in our results if the galaxies for which we do not
have redshifts have a different redshift distribution than the
galaxies which are included in our sample. On the other hand, it is
now well established that the UV luminosity density increases with
redshift \citep[e.g.][]{somerville01} and part of the difference is
likely a real effect \citep{treyer04}. However, the difference of
$\sim0.9$ mag between the FOCA and NUV values for $M^{\ast}$ would
require evolution much larger than determined from other surveys as
well as GALEX data at higher redshifts
\citep{arnouts04,schiminovich04}. A preliminary comparison of the
GALEX and FOCA photometry in a couple of overlapping fields indicates
that the FOCA magnitudes are on average brighter with the difference
becoming larger for fainter sources. It appears likely these offsets
and non-linearities in the FOCA photometry account for a major part of
the difference between the FOCA and GALEX LFs with the remainder
likely due to a combination of galaxy evolution and the FOCA sample
selection.
In Table \ref{params} we also list the total luminosity density
calculated from the best-fit Schechter parameters
as $\rho_L = \int_{0}^{\infty}{L \Phi(L) dL} = \phi^{\ast} L^{\ast} \Gamma(\alpha+2)$.
The statistical errors in $\log{\rho_L}$ that take into account the
covariance between the three Schechter function parameters are
0.02 in each band. In addition to this error,
the uncertainty in the GALEX photometric zeropoint
is $\approx 10\%$ in both bands, corresponding to an uncertainty in
$\log{\rho_L}$ of 0.04. A potentially larger source of error is that
due to large scale structure. The variation in the number density of
galaxies $\bar{n}$ in a contiguous volume $V$ is given approximately by
$\delta \bar{n}/\bar{n} \approx (J_3/V)^{1/2}$ \citep{davis82}
where $J_3$ is an integral
over the galaxy 2-point correlation function and has a value of
$\sim10^4~{\rm Mpc^{3}}$ for a correlation function of the
form $\xi(r) = (r/r_0)^{-\gamma}$ with $r_0=7.21 {\rm Mpc}$
and $\gamma=1.67$ \citep{hawkins03}.
The galaxy number counts from \citet{xu04}
used to set the normalization of our LFs were derived
from approximately 22.64 deg${^2}$. For $z<0.1$, the corresponding
rms variation in the number density would be $\delta n/n \approx
0.24$, or an uncertainty in $\delta \log{\rho_L}\approx 0.11$.
Since UV-selected, star-forming galaxies are likely less clustered than
optically-selected
samples, this value is really an upper limit. Adding these uncertainties
due to large scale structure and calibration in quadrature to the
statistical errors results in a total
uncertainty of $\delta \log{\rho_L} \approx 0.12$ in both bands.
The spectral slope $\beta$, defined as $f_{\lambda} \propto \lambda^{\beta}$
with $f_{\lambda}$ in units of
${\rm ergs~s^{-1}~\AA^{-1}~Mpc^{-3}}$,
corresponding to our two luminosity density measurements is
$\beta \approx -1.1$. This is slightly bluer than the slope
of $\beta = -0.9$
determined by \citet{cowie99} at $z=0.7-1.3$ from measurements
at longer rest frame wavelengths spanning ${\rm 1700\AA}$ to
${\rm 2750\AA}$.
The FUV luminosity density can be used to estimate the star formation
rate (SFR) density in the local universe. For a constant star
formation history and a Salpeter IMF, the SFR is related to the UV
luminosity $L_{\nu}$ (in the range $1500-2800{\rm \AA}$)
by ${\rm SFR~(M_{\sun}~yr^{-1}}) =
1.4\times10^{-28} L_{\nu} ({\rm ergs~s^{-1}~Hz^{-1}})$
\citep{kennicutt98}. For the FUV luminosity density in Table \ref{params},
we obtain $\log{{\rm (SFR_{FUV}) (M_{\sun}~yr^{-1}~Mpc^{-3})}}=-2.30\pm0.12$
with no extinction correction. For comparison, the
extinction-corrected ${\rm H\alpha}$ LF at $z \lesssim 0.045$ from
\citet{gallego95} shifted to our assumed Hubble constant corresponds
to $\log{{\rm (SFR_{H\alpha})}}=-1.86\pm0.04$ using the ${\rm H\alpha}$ to SFR
conversion from \citet{kennicutt98}. Based upon ${\rm H\alpha}$
imaging of a subsample of the galaxies used by
\citet{gallego95}, \citet{perez-gonzalez03} argued that the local
${\rm H\alpha}$ luminosity density is $\sim 60\%$ higher due to
uncertainties in the aperture corrections applied to the spectroscopic
data and corresponds to $\log{{\rm (SFR_{H\alpha})}}=-1.6\pm0.2$. Bringing the
FUV SFR into agreement with this result would require an extinction of
$A_{FUV} \simeq 1.8$.
An average extinction of $A_{FUV} \simeq 1.8$ is consistent with a
simple estimate made using the observed $(FUV-NUV)$ colors. While
there is a well-defined relationship between the UV extinction and the
spectral slope for starburst galaxies, more quiescent galaxies tend to
have less extinction for a given UV slope than would be inferred from
nearby starbursts \citep{bell02}. In particular \citet{kong04} used
the population synthesis models of \citet{bruzual03} along with the
prescription described in \citet{charlot00} for determining how
starlight is absorbed by dust in galaxies to show that the smaller
extinction in non-starburst galaxies can be explained by variations in
the galaxies' star formation histories. Based upon a set of Monte Carlo
realizations of these models spanning a range of extinctions, ages and
star formation histories, \citet{kong04} were able to approximate the
dependence of the FUV extinction on the UV spectral slope $\beta$ with
the following formula: $A_{FUV} = 3.87 + 1.87(\beta+0.40\log{b})$
where the variable $b$ parametrizes the star formation history and is
defined as the ratio of current to the past average star formation
rate. Assuming a constant star formation history ($b=1$), an
extinction of $A_{FUV}=1.8$ is obtained for a spectral slope $\beta =
-1.1$, a value consistent with that measured from the FUV and NUV
luminosity densities. On the other hand, the average $(FUV-NUV)$
color of our FUV-selected sample is 0.14, corresponding to a spectral
slope of $\beta=-1.67$. For this $\beta$ and $b=1$, the \citet{kong04}
formula results in $A_{FUV}=0.7$ mag. This extinction is similar to
the results of \citet{buat04} who found that the average extinction
for a local NUV-selected sample is $A_{FUV} \simeq 1$ mag based
upon the FIR to UV
flux ratio. If an extinction of $A_{FUV} \simeq 1$ is more appropriate
for the UV-selected sample presented
here, then the UV-based star formation rate density would be
$\log{{\rm (SFR_{FUV})}}=-1.9\pm0.1$, a value lower than that from
${\rm H\alpha}$ although still consistent to within the errors. In reality
the extinction is likely a function of absolute magnitude and future
GALEX papers will address in more detail correcting UV fluxes for
extinction in a more rigorous way.
In the near future we will continue our investigation of the UV luminosity
function in the local universe using GALEX AIS data covering
${\rm \sim 1000~deg^2}$ of the SDSS. In addition to expanding our sample
to include more galaxies, we will use the SDSS
photometry and spectroscopy to explore the dependence of
UV luminosity on other galaxy characteristics, such as color, surface
brightness, environment, metallicity and stellar mass.
\acknowledgements
GALEX (Galaxy Evolution Explorer) is a NASA Small Explorer, launched
in April 2003. We gratefully acknowledge NASA's support for
construction, operation, and science analysis for the GALEX mission,
developed in cooperation with the Centre National d'Etudes Spatiales
of France and the Korean Ministry of Science and Technology.
|
{
"timestamp": "2004-11-13T02:07:52",
"yymm": "0411",
"arxiv_id": "astro-ph/0411364",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411364"
}
|
\section{introduction}
The three neutrino framework has 9 physical parameters:
3 neutrino masses,
3 mixing angles,
and
3 CP violating phase,
if neutrinos are Majorana particles.
Neutrino oscillation experiments are sensitive to 6 parameters:
2 mass-squared difference,
3 mixing angles,
and
1 CP phase.
Usually, data from the experiments are analyzed
within the two-flavor framework, which is governed by only
1 mass-squared difference and 1 mixing angle.
So far,
for the long base-line
neutrino oscillation experiments,
we have been able to neglect the effect from
the smaller mass-squared difference,
the other mixing angles, and CP phase,
which we call three-flavor effect.
This is because
the error of the larger mass-squared difference,
related to the atmospheric neutrino observations~\cite{atm}
and the on-going long base-line experiment K2K~\cite{K2K},
is larger than the smaller mass-squared difference,
which is obtained from solar neutrino observations~\cite{solar} and
the reactor neutrino experiment KamLAND~\cite{KamLAND}.
The long base-line neutrino oscillation experiments
in future~\cite{T2K,nova,BNL}
plan to measure the mass-squared difference
and mixing parameter precisely.
Because the order of magnitude of the ratio between
the smaller mass-squared difference and the larger one
is supposed to be similar to that of expected
error of the future long base-line experiments,
it is necessary to take into account
the contribution of the three-flavor effect
in
the determination of the mass-squared difference
and mixing angle from the future long base-line
precision measurements.
In this article, we estimate this effect
using the three-flavor framework.
We discard the result of the LSND experiment~\cite{LSND}.
Obviously, this analysis is, in general, not valid,
if the LSND result is confirmed by the
MiniBooNE experiment~\cite{miniboone}.
From the survival probability of $\nu_\mu$,
we find that the larger mass-squared difference is shifted by the
three-flavor effect
and that
the order of magnitude of this shift
depends on the neutrino energy
and is similar to
that of the smaller mass-squared difference.
We obtain the same result from the transition probability
$\nu_\mu^{} \to \nu_e^{}$,
and also find the CP phase dependence for the
transition probability.
A lot of groups have analyzed the experimental data with
the three-flavor framework numerically \cite{nume}.
However, these analyses cannot point out the specific reason
for the value of parameters.
In this paper, we point out the specific contribution to the
parameters from the three-flavor effect.
We think that
these formulations are useful to study in the numerical analysis.
This article is organized as follows.
In section 2, we will show the useful notations and
the convenient form of the probability for easy estimating
the contribution of the three-flavor effect.
In section 3, we will estimate the three-flavor effect for the
$\nu_\mu$ disappearance mode.
We will also estimate the three-flavor effect for the
$\nu_\mu \to \nu_e$ transition mode, in section 4.
Finally, we will be devoted to the summary in the last section.
\section{notations}
In the three neutrino framework,
and in the basis in which the charged leptons are diagonal,
the three weak interaction
eigenstates, $\nu_\alpha^{}$ ($\alpha = e,\mu,\tau$) are
expressed as
\begin{equation}
\nu_\alpha^{} =
\sum^{3}_{i=1} (V_{\rm MNS})_{\alpha i}~\nu_i^{}\,,
\end{equation}
where $\nu_i^{}$ are the three mass eigenstates and $V_{\rm MNS}$ is
the Maki-Nakagawa-Sakata (MNS) matrix~\cite{MNS}.
We adopt the following parameterization~\cite{PDB}
\begin{equation}
V_{\rm MNS}^{} = U_{}^{} {\cal P}
= U_{}^{}~{\rm diag}(e^{i\alpha_1/2},e^{i\alpha_2/2},1)\,,
\end{equation}
where ${\cal P}$ cannot be determined from the neutrino oscillation
experiment.
The matrix $U$, which has three mixing angles and one phase,
can be parameterized in the same way as the Cabibbo-Kobayashi-Maskawa
matrix~\cite{CKM}.
Because the present neutrino oscillation experiments
constrain directly the elements
$U_{e2}$, $U_{e3}$, and $U_{\mu 3}$,
we find it most convenient to adopt the parameterization~\cite{HO1,H2B},
where these three elements in the upper-right
corner of the $U$ matrix are the independent parameters.
Without losing generality, we can take
$U_{e2}$ and $U_{\mu 3}$ to be real and non-negative.
By allowing $U_{e3}$ to have the complex phase,
\begin{equation}
U_{e2}\,,U_{\mu 3} \geq 0,~~~
U_{e3}\equiv \left|U_{e3}\right| e^{-i\phi}
~~(0 \leq \phi < 2\pi )\,,
\end{equation}
these $U_{e2}, U_{\mu 3}, |U_{e3}|$, and $\phi$
are the four independent parameters.
All the other elements of $U$
are then determined by the unitary
conditions,
\begin{subequations}
\begin{eqnarray}
U_{e1} &=& \sqrt{1-|U_{e3}|^2-|U_{e2}|^2}\,,
~~~~~U_{\tau 3}
=
\sqrt{1-|U_{e3}|^2-|U_{\mu 3}|^2}\,,
\\
U_{\mu 1} &=& - \frac{U_{e2}U_{\tau 3} + U_{\mu 3}U_{e1}U_{e3}^{\ast} }
{1-|U_{e3}|^2}\,,
~~U_{\mu 2} = \frac{U_{e1}U_{\tau 3} - U_{\mu 3}U_{e2}U_{e3}^{\ast} }
{1-|U_{e3}|^2}\,,\\
U_{\tau 1} &=& \frac{U_{e2}U_{\mu 3} - U_{\tau 3}U_{e1}U_{e3}^{\ast} }
{1-|U_{e3}|^2}\,,
~~~~U_{\tau 2} = - \frac{U_{\mu 3}U_{e1} + U_{e2}U_{\tau 3}U_{e3}^{\ast} }
{1-|U_{e3}|^2}\,.
\end{eqnarray}
\eqlab{def_MNS}
\end{subequations}
For the convenience, the independent parameters in the MNS matrix
are rewritten as
\begin{equation}
U_{e3}^{} \equiv \sin \theta_{13}\,,
~~~~~~
U_{e2}^{} \equiv \sin \theta_{12} \cos \theta_{13}\,,
~~~~~~
U_{\mu3}^{} \equiv \sin \theta_{23} \cos \theta_{13}\,.
\eqlab{another_MNS}
\end{equation}
The atmospheric neutrino oscillation experiments, which measure the $\nu_\mu$
survival probability determine the absolute values of the larger
mass-squared differences and one-mixing angle \cite{atm} as
\begin{equation}
1.5\times 10^{-3}<
\left| \delta m^2_{\rm atm} \right|
< 3.4 \pm 0.5 \times 10^{-3} \mbox{{eV}}^2\,,
~~~~\mbox{{and}}~~~
\sin^22\theta_{\rm atm} >0.92 \,,
\eqlab{atm}
\end{equation}
at the 90\% C.L.
The K2K experiment \cite{K2K} confirms the above results.
These values are planed to measure more precisely,
a few percent order by the future long base-line experiments
\cite{T2K,nova,BNL}.
The solar neutrino experiments, which measure the $\nu_e$ survival
probability in the sun \cite{solar},
and the KamLAND experiment which measure the $\bar{\nu}_e$ survival
probability from the reactors \cite{KamLAND},
determine the smaller mass-squared difference and another mixing angle
as
\begin{equation}
\delta m^2_{\rm sol} = 8.2 ^{+0.6}_{-0.5} \times 10^{-5} \mbox{{eV}}^2\,,
~~~~\mbox{{and}}~~~
\tan^2\theta_{\rm sol} = 0.40 ^{+0.09}_{-0.07}\,.
\eqlab{sol}
\end{equation}
It is remarkable point that the order of the smaller mass
squared-difference is as same as that of expected error of the future
long base-line experiments \cite{T2K,nova,BNL}.
Thus, it is necessary to take into account the contribution of the
three-flavor effect in the determination of the mass-squared difference
and mixing angle analytically,
because experimental data are analyzed ordinary
in the two-flavor framework.
The CHOOZ reactor experiment \cite{CHOOZ} gives the upper bound of the
third mixing angle as
\begin{eqnarray}
\sin^22\theta_{\rm rct} &<& 0.20
~~~\mbox{{for}}~~~\delta m^2 = 2.0 \times 10^{-3} \mbox{{eV}}^2\,,\nonumber\\
\sin^22\theta_{\rm rct} &<& 0.16
~~~\mbox{{for}}~~~\delta m^2 = 2.5 \times 10^{-3} \mbox{{eV}}^2\,,\nonumber\\
\sin^22\theta_{\rm rct} &<& 0.14
~~~\mbox{{for}}~~~\delta m^2 = 3.0 \times 10^{-3} \mbox{{eV}}^2\,,
\eqlab{rct}
\end{eqnarray}
at the 90\% C.L.
Since we can always take
$\left| \Delta_{12} \right| < \left| \Delta_{13} \right|$
without loosing generality,
we assume that
$\Delta_{12}$ is from the results of solar neutrino and reactor
anti-neutrino observations
and
$\Delta_{13}$ is related to the atmospheric and long base-line
neutrino experiments.
The sign of $\Delta_{12}$ is determined form the solar neutrino
observation, $\Delta_{12} > 0$.
However, that of $\Delta_{13}$ cannot be determined by any
observations.
In this article, we call $\Delta_{13} > 0$ ``normal hierarchy''
and $\Delta_{13} < 0$ ``inverted hierarchy''.
Under the
$\left| \Delta_{12} \right| < \left| \Delta_{13} \right|$ relation,
$\theta_{\rm atm}$, $\theta_{\rm sol}$, and
$\theta_{\rm rct}$ are related to the MNS matrix elements as :
\begin{eqnarray}
2 \cU{\mu 3}{2} &=&
1-\sqrt{1-\sin^2 2\theta_{\rm atm}}\,,\nonumber\\
2 \left|\cU{e3}{}\right|^2 &=&
1-\sqrt{1-\sin^2 2\theta_{\rm rct}}\,,\nonumber\\
2\cU{e2}{2} &=&
1-|\cU{e3}{}|^2 -
\sqrt{\left(1-|\cU{e3}{}|^2\right)^2-\sin^22\theta_{\rm sol}}
\,.
\end{eqnarray}
Starting from the flavor eigenstate $\alpha$,
the probability of finding the
flavor eigenstate $\beta$
at the base-line length $L$
is, in vacuum,
\begin{subequations}
\begin{eqnarray}
P_{\nu_\alpha^{} \to \nu_\beta^{}} &=&
\left|\sum_{j=1}^3 (V_{\rm MNS})_{\beta j}^{}
\exp\left(-i\frac{m_j^2}{2E_\nu}L\right)
(V_{\rm MNS})_{\alpha j}^{\ast}
\right|^2\eqlab{P_ex_00} \\
&=&
\delta_{\alpha\beta}
-4{\rm Re}\left\{
\cU{\alpha 3}{}\cU{\beta 3}{\ast}\cU{\beta 1}{}\cU{\alpha 1}{\ast}
+
\cU{\alpha 2}{}\cU{\beta 2}{\ast}\cU{\beta 3}{}\cU{\alpha 3}{\ast}
\right\}
\sin^2\displaystyle\frac{\Delta_{13}^{}}{2}\nonumber\\
&&~~~~
-4{\rm Re}\left\{
\cU{\alpha 2}{}\cU{\beta 2}{\ast}\cU{\beta 1}{}\cU{\alpha 1}{\ast}
+
\cU{\alpha 2}{}\cU{\beta 2}{\ast}\cU{\beta 3}{}\cU{\alpha 3}{\ast}
\right\}
\sin^2\displaystyle\frac{\Delta_{12}^{}}{2}\nonumber\\
&&~~~~
+2{\rm Re}
\left(
\cU{\alpha 2}{}\cU{\beta 2}{\ast}\cU{\beta 3}{}\cU{\alpha 3}{\ast}
\right)
\left(
\sin\Delta_{12}\sin\Delta_{13}
+ 4\sin^2\displaystyle\frac{\Delta_{12}}{2}\sin^2\displaystyle\frac{\Delta_{13}}{2}
\right)
\nonumber\\
&&~~~~
- 4J_{\rm MNS}^{(\alpha,\beta)}\left(
\sin^2\displaystyle\frac{\Delta_{13^{}}}{2}\sin\Delta_{12} -
\sin^2\displaystyle\frac{\Delta_{12^{}}}{2}\sin\Delta_{13}
\right)\,,
\eqlab{P_ex_03}
\end{eqnarray}
\eqlab{P_ex_0}
\end{subequations}
where $J_{\rm MNS}^{(\alpha,\beta)}$ is the
Jarlskog parameter~\cite{JarP}:
\begin{eqnarray}
J_{\rm MNS}^{(\alpha,\beta)}~
&\equiv&~{\rm Im}
\left((V_{\rm MNS})_{\alpha i}
(V_{\rm MNS})_{\beta i}^{\ast}
(V_{\rm MNS})_{\beta j}
(V_{\rm MNS})_{\alpha j}^{\ast}\right) \nonumber\\
&=& {\rm Im} \left({U_{\alpha i} U_{\beta i}^{\ast} U_{\beta j}
U_{\alpha j}^{\ast}}\right)
= - \frac{U_{e1}U_{e2}U_{\mu 3}U_{\tau 3}}{1-\left|U_{e3}\right|^2}
{\rm Im}\left({U_{e3}}\right)
~\equiv~A \sin \phi\,,
\eqlab{Jmns}
\end{eqnarray}
which is defined to be positive
for $(\alpha,\beta)=(e,\mu)$, $(\mu,\tau)$, $(\tau,e)$
and $(i,j)=(1,2),(2,3),(3,1)$.
$A$ is the absolute value of the Jarlskog parameter.
In addition, $\Delta_{ij}$ is
\begin{equation}
\Delta_{ij}
\equiv
\displaystyle\frac{m^2_j - m^2_i}{2E_\nu}L
=
\displaystyle\frac{\delta m_{ij}^2}{2E_\nu}L
\simeq
2.534 \displaystyle\frac{\delta m_{ij}^2 ({\rm eV}^2)}{E_\nu({\rm GeV})}L({\rm km}
)\,,
\end{equation}
where $E_\nu$ is the neutrino energy.
We rewrite \eqref{P_ex_0} in a form convenient to estimate the
contribution of the smaller mass-squared difference,
\begin{equation}
P_{\nu_\alpha^{} \to \nu_\beta^{}}
\equiv
P_0(\alpha,\beta) + P_1(\alpha,\beta) \times \sin \Delta_{12}^{}
+ P_2(\alpha,\beta) \times 4\sin^2\displaystyle\frac{\Delta_{12}}{2}\,,
\eqlab{P_ex_11}
\end{equation}
where each term is
\begin{subequations}
\begin{eqnarray}
P_0(\alpha,\beta) &=&
\delta_{\alpha\beta}-4{\rm Re}\left\{
\cU{\alpha 3}{}\cU{\beta 3}{\ast}\cU{\beta 1}{}\cU{\alpha 1}{\ast}
+
\cU{\alpha 2}{}\cU{\beta 2}{\ast}\cU{\beta 3}{}\cU{\alpha 3}{\ast}
\right\}
\sin^2\displaystyle\frac{\Delta_{13}^{}}{2}\,,
\eqlab{P_ex_p0}\\
P_1(\alpha,\beta) &=&
2{\rm Re}
\left(
\cU{\alpha 2}{}\cU{\beta 2}{\ast}\cU{\beta 3}{}\cU{\alpha 3}{\ast}
\right)
\sin\Delta_{13}^{}
-
4J_{\rm MNS}^{(\alpha,\beta)} \sin^2\displaystyle\frac{\Delta_{13^{}}}{2}\,,
\eqlab{P_ex_p1}
\\
P_2(\alpha,\beta) &=&
-
{\rm Re}\left(
\cU{\alpha 1}{}\cU{\beta 1}{\ast}\cU{\beta 2}{}\cU{\alpha 2}{\ast}
+
\cU{\alpha 2}{}\cU{\beta 2}{\ast}\cU{\beta 3}{}\cU{\alpha 3}{\ast}
\cos \Delta_{13}^{}
\right)
+ J_{\rm MNS}^{(\alpha,\beta)} \sin\Delta_{13}^{}\,.
\eqlab{P_ex_p2}
\end{eqnarray}
\eqlab{P_ex_p}
\end{subequations}
All the above formulas remain valid
when replacing the mass-squared differences and
the MNS matrix elements with the ``effective'' ones,
\begin{equation}
\Delta_{ij} \to \widetilde \Delta_{ij}\,,
~~~
U_{\alpha i} \to \widetilde U_{\alpha i}\,,
~~~
J_{\rm MNS}^{(\alpha,\beta)} \to \widetilde J_{\rm MNS}^{(\alpha,\beta)}\,,
\end{equation}
as long as the matter density remains the same along the base-line.
The effective parameters
$\widetilde U_{\alpha i}$ are
defined from the following Hamiltonian
\begin{equation}
{{\cUm{}{}}}
\left(\begin{array}{ccc}
{\lambda_1} & 0 & 0 \\
0 & {\lambda_2} & 0 \\
0 & 0 & {\lambda_3}
\end{array}\right)
{{\cUm{}{\dagger}}}
=
{\cU{}{}}
\left(\begin{array}{ccc}
0 & 0 & 0 \\
0 & \delta m^2_{12} & 0 \\
0 & 0 & \delta m^2_{13}
\end{array}\right)
{\cU{}{\dagger}}
+
\left(\begin{array}{ccc}
a & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}\right)\,,
\eqlab{H_mat}
\end{equation}
and $\widetilde \Delta_{ij}$ is defined as
\begin{equation}
\widetilde{\Delta}_{ij}
\equiv
\displaystyle\frac{\delta \widetilde{m}_{ij}^{2}}{2E_\nu}L
=
\displaystyle\frac{{\lambda_j}-{\lambda_i}}{2E_\nu}L\,.
\end{equation}
The term $a$ in \eqref{H_mat}
stands for the matter effect~\cite{MSW},
\begin{eqnarray}
a(E_\nu^{})
=2\sqrt{2}G_F n_e E_\nu^{}
={7.56}\times 10^{-5}({\rm eV}^2)\left(\frac{\rho}{{\rm g/cm}^{3}}\right)
\left(\frac{E_\nu}{\rm GeV}\right)\,,
\eqlab{matter_a}
\end{eqnarray}
where
$n_e$ is the electron density of the matter,
$G_F$ is the Fermi constant,
and $\rho$ is the matter density.
When $\delta m^2_{12} < a < \delta m^2_{13}$
and $ U_{e3}^{} \ll O(1)$,
the effective mass-squared differences
are written as
\begin{equation}
\widetilde{\Delta}_{13} \simeq \Delta_{13}^{}-\Delta_{12}^{}\cos^2\theta_{12}^{}\,,
~~~~~
\widetilde{\Delta}_{12} \simeq \displaystyle\frac{a}{2E}L-\Delta_{12}^{}\cos2\theta_{12}^{}\,,
\eqlab{ele_app1}
\end{equation}
and the mixing angles become
\begin{equation}
\widetilde{\theta}_{23}^{} \simeq \theta_{23}\,,
~~~~~
\widetilde{\theta}_{13}^{} \simeq \left(1+\displaystyle\frac{a}{\delta m^2_{13}}\right)
\theta_{13}^{}\,,
~~~~~
\tan 2 \widetilde{\theta}_{12}^{} \simeq
\displaystyle\frac{\delta m^2_{12} \sin 2\theta_{12}^{}}
{\delta m^2_{12} \cos 2 \theta_{12}^{}- a}\,.
\eqlab{ele_app2}
\end{equation}
Hereafter, we use $U_{\alpha i}^{}$ and $\Delta_{ij}$ instead
of $\widetilde U_{\alpha i}^{}$ and $\widetilde \Delta_{ij}$, because of
simplicity.
But we have to keep in mind that these values depend on the
neutrino energy.
\section{survival probability of $\nu_\mu$}
From \eqsref{P_ex_11} and\eqvref{P_ex_p},
the survival probability of $\nu_\mu^{}$
is written as
\begin{eqnarray}
P_{\nu_\mu^{} \to \nu_\mu^{}}=
P_0(\mu,\mu)
+ P_1(\mu,\mu) \times \sin{\Delta_{12}}
+ P_2(\mu,\mu) \times 4\sin^2\displaystyle\frac{\Delta_{12}}{2}\,,
\eqlab{Pmm}
\end{eqnarray}
where
\begin{subequations}
\begin{eqnarray}
P_0(\mu,\mu) &=& 1-4\left|\cU{\mu 3}{}\right|^2
\left(1-\left|\cU{\mu 3}{}\right|^2\right)
\sin^2\displaystyle\frac{\Delta_{13}}{2}\,,
\eqlab{Pmm0}\\
P_1(\mu,\mu)&=&
2\left|\cU{\mu 2}{}\right|^2\left|\cU{\mu 3}{}\right|^2
\sin\Delta_{13}\,,
\eqlab{Pmm1}\\
P_2(\mu,\mu)&=&
-\left|\cU{\mu 2}{} \right|^2
\left(\left|\cU{\mu 1}{}\right|^2
+\left|\cU{\mu 3}{}\right|^2 \cos{\Delta_{13}}
\right)\,.
\eqlab{Pmm2}
\end{eqnarray}
\eqlab{Pmm_ex}
\end{subequations}
The survival probability of $\nu_\mu^{}$ in the two-flavor framework is
written as
\begin{equation}
P_{\nu_\mu^{}\to\nu_\mu^{}}^{(2)}
=
1 - \sin^22\theta_{\mu\mu} \sin^2 \displaystyle\frac{\Delta_{\mu\mu}}{2}\,,
\eqlab{Pmm2f}
\end{equation}
where $\theta_{\mu\mu}$ is the mixing angle
and
$\Delta_{\mu\mu}$ is the mass-squared difference,
which are obtained from the two-flavor analysis.
We expect that all these parameters, especially $\Delta_{\mu \mu}$,
to be shifted by the three-flavor effect.
In order to estimate this effect,
we rewrite them as
\begin{equation}
\sin^22\theta_{\mu\mu} = 1.0 - \varepsilon\,,
\hspace*{5ex}
\Delta_{\mu\mu} = \Delta_{13} + 2\delta \,,
\eqlab{app1}
\end{equation}
where $\delta$ denotes the three-flavor effect,
and $\varepsilon$ stands for the difference from
the maximal mixing.
Both of them depend on the neutrino energy,
base-line length, and the oscillation parameters.
We assume that $\delta$ is smaller than
$\Delta_{13}$ and $\Delta_{13}\sim O(1)$ for long base-line experiments.
From the atmospheric neutrino observations and K2K experiment,
we already know that $\nu_\mu^{}$ oscillate to
another flavor maximally at the first-dip,
$\Delta_{\mu\mu}(E_\nu=E_{\rm dip})=\pi$
for normal hierarchy.
By using \eqref{app1},
{\it i.e.,~} $\Delta_{13} = \pi-2\delta^{\rm dip}$,
\eqref{Pmm_ex} becomes
\begin{subequations}
\begin{eqnarray}
P_0(\mu,\mu) &=& 1-4\left|\cU{\mu 3}{\rm dip}\right|^2
\left(1-\left|\cU{\mu 3}{\rm dip}\right|^2\right)
\cos^2\delta^{\rm dip} \,,
\eqlab{Pmmdip0}\\
P_1(\mu,\mu)&=&
2\left|\cU{\mu 2}{\rm dip}\right|^2\left|\cU{\mu 3}{\rm dip}\right|^2
\sin2\delta^{\rm dip}\,,
\eqlab{Pmmdip1}\\
P_2(\mu,\mu)&=&
-\left|\cU{\mu 2}{\rm dip} \right|^2
\left(\left|\cU{\mu 1}{\rm dip}\right|^2
-\left|\cU{\mu 3}{\rm dip}\right|^2 \cos{2\delta^{\rm dip}}
\right)\,,
\eqlab{Pmmdip2}
\end{eqnarray}
\eqlab{Pmmdip_ex}
\end{subequations}
and \eqref{Pmm2f}
\begin{equation}
P_{\nu_\mu^{}\to\nu_\mu^{}}^{(2)}
= \varepsilon^{\rm dip}\,,
\eqlab{Pmmdip2f}
\end{equation}
where
the label ``dip'' in $|\cU{\mu i}{}|$,
$\delta$, and $\varepsilon$ means
that these quantities take some fixed value
at the first-dip energy $E_{\rm dip}$.
From \eqsref{Pmmdip_ex} and\eqvref{Pmmdip2f},
we obtain
\begin{eqnarray}
\delta^{\rm dip}
\simeq
\displaystyle\frac
{4\left|\cU{\mu3}{\rm dip}\right|^2
\left(1-\left|\cU{\mu3}{\rm dip}\right|^2\right)
-\left(1^{}_{}-\varepsilon^{\rm dip}_{} \right)
}
{4\Delta_{12}^{\rm dip}\left|\cU{\mu 2}{\rm dip}\right|^2
\left|\cU{\mu 3}{\rm dip}\right|^2}
+
\displaystyle\frac
{ 1
- \left|\cU{\mu 2}{\rm dip}\right|^2
-2\left|\cU{\mu 3}{\rm dip}\right|^2}
{4\left|\cU{\mu 3}{\rm dip}\right|^2}
\Delta_{12}^{\rm dip}\,.
\eqlab{dip_ex1}
\end{eqnarray}
The first term of \eqref{dip_ex1} has to be zero
because of the assumption $O(\delta) < 1$,
and therefore we obtain
\begin{equation}
\left|\cU{\mu 3}{\rm dip}\right|^2
=\displaystyle\frac{\left(1 \pm \sqrt{\varepsilon^{\rm dip}}\right)}{2}\,.
\eqlab{cond_dip1}
\end{equation}
When we take a negative sign in \eqref{cond_dip1},
$\delta^{\rm dip}$ becomes
\begin{equation}
\delta^{\rm dip}
\simeq
-\displaystyle\frac{\Delta_{12}^{\rm dip}}{2}
\left\{
\left|\cU{\mu 2}{\rm dip}\right|^2
+
\left(1-\left|\cU{\mu 2}{\rm dip}\right|^2\right)
\sqrt{\varepsilon^{\rm dip}}
-
\varepsilon^{\rm dip}
\right\}\,.
\end{equation}
Since the best fit value of the
mixing angle is maximum from the experiments~\cite{atm,K2K},
we take $\varepsilon^{\rm dip} = 0$.
Thus, $\delta^{\rm dip}$ simplifies to
\begin{eqnarray}
\delta^{\rm dip}
\simeq -\displaystyle\frac{\Delta_{12}^{\rm dip}}{2}
\left|\cU{\mu 2}{\rm dip}\right|^2\,.
\eqlab{dip1}
\end{eqnarray}
From \eqref{app1}, the larger mass-squared difference
at $E_{\rm dip}$ is
\begin{equation}
\delta m^2_{13}(E_{\rm dip})
=
\delta m^2_{\rm dip}
+
\delta m^2_{12}(E_{\rm dip}) \left|\cU{\mu 2}{\rm dip}\right|^2\,,
\eqlab{dip2}
\end{equation}
where
\begin{equation}
\delta m^2_{\rm dip} \equiv \displaystyle\frac{2\pi E_{\rm dip}}{L}\,,
\end{equation}
is from the two-flavor analysis.
\Eqref{dip2} denotes that
the order of magnitude of the three-flavor effect is roughly
the same as that of the expected error of future experiments.
Since we know that both $\delta m^2_{12}$
and $\left|\cU{\mu 2}{\rm dip}\right|^2$ are positive,
the mass-squared difference from two-flavor analysis
is slightly smaller than the larger mass-squared difference.
We also obtain the relation between $\delta$ and
the smaller mass-squared difference
at the first peak $E_{\rm peak}$,
where $\Delta_{\mu\mu}(E_{\rm peak})=2\pi$,
\begin{eqnarray}
\delta^{\rm peak}
\simeq - \displaystyle\frac{\Delta_{12}^{\rm peak}}{2}
\left(1-\left|\cU{\mu 2}{\rm peak}\right|^2\right)
\,.
\eqlab{peak1}
\end{eqnarray}
Thus, the larger mass-squared difference at $E_{\rm peak}$ is
\begin{equation}
\delta m^2_{13}(E_{\rm peak})
=
\delta m^2_{\rm peak}
+
\delta m^2_{12}(E_{\rm peak})
\left( 1- \left|\cU{\mu 2}{\rm peak}\right|^2\right)\,,
\eqlab{peak2}
\end{equation}
where
\begin{equation}
\delta m^2_{\rm peak} \equiv \displaystyle\frac{4\pi E_{\rm peak}}{L}\,,
\end{equation}
is also from the two-flavor analysis.
We obtain the same results for the inverted hierarchy.
Since $\delta m^2_{13}$ is not changed by matter effect at
$E_{\nu}<$10 GeV,
we obtain the relation between \eqref{dip2} and \eqref{peak2}:
\begin{equation}
\delta m^2_{\rm dip}
+
\delta m^2_{12}(E_{\rm dip}) \left|\cU{\mu 2}{\rm dip}\right|^2
=
\delta m^2_{\rm peak}
+
\delta m^2_{12}(E_{\rm peak})
\left( 1- \left|\cU{\mu 2}{\rm peak}\right|^2\right)\,.
\eqlab{rel_dip_peak}
\end{equation}
By using the definition of $\delta m^2_{\rm dip, peak}$,
we find
\begin{equation}
\displaystyle\frac{L}{2\pi}
\displaystyle\frac{
\delta m^2_{12}\left(E_{\rm dip}\right)\left|\cU{\mu2}{\rm dip}\right|^2
-
\delta m^2_{12}\left(E_{\rm peak}\right)
\left(1-\left|\cU{\mu2}{\rm peak}\right|^2\right)
}
{2E_{\rm peak} - E_{\rm dip}}
=\pm 1\,,
\eqlab{IorIII}
\end{equation}
where the sign of the right-hand side corresponds to the type of
the mass hierarchy,
the positive sign being that of the normal hierarchy.
From the \eqsref{dip2} and \eqvref{peak2}, we easily understand
that the three-flavor effect for the larger mass-squared difference
depends on the energy.
When $\rho = 2.5$(g/cm$^3$) and $E_\nu^{}\simeq O(1)$GeV,
$\widetilde{\theta}_{12}^{}$ is to shift away from $\theta_{12}^{}$
towards $90^\circ$, which is obtained from \eqref{ele_app1}.
From \eqsref{def_MNS} and \eqvref{another_MNS},
the value of $|\cU{\mu 2}{}|$ becomes $0$.
Because $\delta m^2_{12}(E_{\nu})$ is also changed by the matter effect,
which is shown in \eqref{ele_app1},
\eqref{IorIII} becomes
\begin{equation}
\displaystyle\frac{L}{2\pi}
\displaystyle\frac{
a(E_{\rm peak}) - \delta m^2_{12} \cos 2\theta_{12}^{}
}
{E_{\rm dip}-2E_{\rm peak}}
=\pm 1\,,
\eqlab{IorIII-2}
\end{equation}
where the value of $\delta m^2_{12}$ and $\theta_{12}^{}$ is
that of vacuum one, which are listed in \eqref{sol}.
This relation suggest us that
we can determine the sign of the $\delta m^2_{13}$ from the
long base-line experiments,
when we measure the energy of ``peak'' and ``dip'' precisely
and we know the value of the smaller mass-squared difference,
mixing angle with small errors.
This result cannot be obtained from the numerical analysis.
This fact points out that we can pick up the three-flavor effect
from the fitting function of the survival probability which is
obtained from the experimental data.
\section{transition probability}
From \eqsref{P_ex_0} and\eqvref{P_ex_p}, the transition probability,
$\nu_\mu^{} \to \nu_e^{}$ is written as
\begin{eqnarray}
P_{\nu_\mu^{} \to \nu_e^{}}&=&
P_0(\mu,e)
+ P_1(\mu,e) \times \sin{\Delta_{12}}
+ P_2(\mu,e) \times 4\sin^2\displaystyle\frac{\Delta_{12}}{2}\,,
\eqlab{Pme}
\end{eqnarray}
where
\begin{subequations}
\begin{eqnarray}
P_0(\mu,e) &=&
4\left|\cU{\mu 3}{}\cU{e 3}{}\right|^2
\sin^2\displaystyle\frac{\Delta_{13}}{2}\,,
\eqlab{Pme0}\\
P_1(\mu,e) &=&
2\left\{
{\rm Re}\left(
\cU{e 2}{\ast}\cU{\mu 2}{}
\cU{e 3}{}\cU{\mu 3}{\ast}
\right)
\sin\Delta_{13}
+
2J_{\rm MNS}^{(\mu,e)} \sin^2\displaystyle\frac{\Delta_{13}}{2}
\right\}\,,\\
P_2(\mu,e)&=&
-{\rm Re}\left(
\cU{\mu 1}{}\cU{e 1}{\ast}\cU{e 2}{}\cU{\mu 2}{\ast}
+
\cU{\mu 2}{}\cU{e 2}{\ast}\cU{e 3}{}\cU{\mu 3}{\ast}
\cos \Delta_{13}
\right)
-
J_{\rm MNS}^{(\mu,e)} \sin \Delta_{13}\,.
\eqlab{Pme2}
\end{eqnarray}
\eqlab{Pme_ex}
\end{subequations}
Here,
$J_{\rm MNS}^{(\mu,e)} = -A \sin \phi \equiv - J$.
Under the two-flavor framework,
this transition probability, $\nu_\mu^{}\to \nu_e^{}$ is written as
\begin{equation}
P^{(2)}_{\nu_\mu \to \nu_e} =
\sin^2\theta_{\mu e} \sin^2\displaystyle\frac{\Delta_{\mu e}}{2}\,,
\eqlab{Pme20}
\end{equation}
where $\Delta_{\mu e}$
is the mass-squared difference
and
$\theta_{\mu e}$ is the unknown mixing angle.
We suppose that
these two parameter, especially $\Delta_{\mu e}$,
are changed by the three-flavor effect.
As in the case of the survival probability,
we rewrite these parameters as
\begin{equation}
\sin^2 \theta_{\mu e} = h\,,~~~~~
\Delta_{\mu e} = \Delta_{13} + 2 \delta\,,
\eqlab{setH}
\end{equation}
where $\delta$ is smaller than $\Delta_{13}$
and $h$, in general, can take an arbitrary value.
Before estimating the three-flavor effect,
let us calculate the value of $h$ and $\delta$
for $\Delta_{12}=0$.
The transition probability becomes
\begin{equation}
P_{\nu_\mu^{} \to \nu_e^{}}=
4\left|\cU{\mu 3}{}\cU{e 3}{}\right|^2
\sin^2\displaystyle\frac{\Delta_{13}}{2}\,,
\eqlab{Pme00}
\end{equation}
with $\Delta_{12}=0$.
At the first peak of transition probability,
$\Delta_{\mu e}(E_{\rm peak})=\pi$,
\eqsref{Pme20} and\eqvref{Pme00} become
\begin{eqnarray}
P^{(2)}_{\nu_\mu \to \nu_e}
(\Delta_{\mu e} = \pi) &=&
h\,, \nonumber\\
P_{\nu_\mu^{} \to \nu_e^{}}
(\Delta_{\mu e} = \pi) &=&
4\left|\cU{\mu 3}{}\cU{e 3}{}\right|^2
\cos^2\delta\,,
\eqlab{Delta120}
\end{eqnarray}
respectively.
From these equations,
$h$ and $\delta$ are solved as
\begin{eqnarray}
h_0^{}
&\equiv&
h(\Delta_{12}=0,\Delta_{\mu e}=\pi)
=
4\left|\cU{\mu 3}{}\cU{e 3}{}\right|^2\,,\nonumber\\
\delta_0^{}
&=&\delta(\Delta_{12}=0,\Delta_{\mu e}=\pi)
=0\,,
\eqlab{h0}
\end{eqnarray}
when we suppose that $\Delta_{\mu e}$ is same as
$\Delta_{13}$ without three-flavor effect.
When we assume that $\Delta_{12}$ is nonvanishing, but small,
\eqref{Pme} becomes
\begin{eqnarray}
P_{\nu_\mu^{} \to \nu_e^{}}&=&
P_0(\mu,e)
+ P_1(\mu,e) \times \Delta_{12}
+ P_2(\mu,e) \times \Delta_{12}^{2}
+O(\Delta_{12}^3)\,,
\eqlab{Pme4}
\end{eqnarray}
where
$P_0(\mu,e)$, $P_1(\mu,e)$ and $P_2(\mu,e)$ are
\begin{subequations}
\begin{eqnarray}
P_0(\mu,e) &=&
4\left|\cU{\mu 3}{}\right|^2
\left|\cU{e 3}{}\right|^2
\cos^2\delta^{} \nonumber\\
&=&
4\left|\cU{\mu 3}{}\right|^2
\left|\cU{e 3}{}\right|^2 + O(\delta^2)\,,
\eqlab{Pme0d}\\
P_1(\mu,e) &=&
2{\rm Re}\cA{23}{}
\sin 2\delta^{}
-
4J\cos^2\delta^{}\nonumber\\
&=&
4\delta{\rm Re}\cA{23}{}
-
4J+O(\delta^2)\,,
\eqlab{Pme1d}\\
P_2(\mu,e)&=&
-{\rm Re}\cA{12}{}
+{\rm Re}\cA{23}{}\cos{2\delta^{}}
+
J\sin2\delta^{}\nonumber\\
&=&
-{\rm Re}\cA{12}{}
+{\rm Re}\cA{23}{}
+2\delta J+O(\delta^2)\,,
\eqlab{Pme22}
\end{eqnarray}
\eqlab{Pmed_ex}
\end{subequations}
at the first peak $E_{\rm peak}$.
Here, the symbols $\cA{ij}{}$ are defined as
\begin{equation}
\cA{ij}{} = \cU{\mu i}{} \cU{e i}{\ast} \cU{e j}{} \cU{\mu j}{\ast}\,.
\end{equation}
The value of $\cU{ij}{}$, $\cA{ij}{}$, and $\delta$
is fixed at $E_{\rm peak}$\footnote{{%
We drop here the label ``peak'' for
$\cU{}{\alpha j}$, $\cA{}{ij}$, and $\delta$,
for simplicity.
}}.
We assume that $h$ at $E_{\rm peak}$
is function of $\Delta_{12}$,
\begin{equation}
h = h(\Delta_{12}) = h_0 + \sum_{k}a_k^{}\Delta_{12}^k\,,
\eqlab{defHF}
\end{equation}
where $h_0$ is equal to
$4\left|\cU{\mu 3}{}\cU{e 3}{}\right|^2$
and $a_k^{}$ are independent of the neutrino energy.
From \eqsref{Pmed_ex} and \eqvref{defHF},
$\delta$ can be solved as
\begin{equation}
\delta \simeq \displaystyle\frac
{
\left(a_1^{} + 4J\right)
+
\left({\rm Re}\cA{12}{} - {\rm Re}\cA{23}{} + a_2^{}\right)\Delta_{12}^{}
+
a_3^{} \Delta_{12}^2
}
{
4{\rm Re}\cA{23}{} + 2 J \Delta_{12}^{}
}\,.
\eqlab{delta1}
\end{equation}
When we take the limit $\Delta_{12} \to 0$,
$\delta$ must vanish
because of \eqref{h0}.
Thus,
\begin{equation}
a_1^{} = -4J\,,
\end{equation}
and
\begin{equation}
\delta \simeq
\displaystyle\frac{\left({\rm Re}\cA{12}{} - {\rm Re}\cA{23}{} + a_2^{}\right)
+a_3^{} \Delta_{12}}
{4{\rm Re}\cA{23}{} + 2 J \Delta_{12}^{}}
\Delta_{12}^{}
\,.
\eqlab{delta2}
\end{equation}
The denominator of \eqref{delta2} becomes 0
under some conditions that are related to the value
of the MNS matrix elements,
\begin{equation}
2{\rm Re}\hat\cA{23}{} + \hat{J} \hat\Delta_{12}^{}
= 0\,,
\eqlab{deno1}
\end{equation}
where $\hat\cA{23}{}$, $\hat{J}$, and $\hat\Delta_{12}^{}$
denote some fixed value of them.
By using the mixing angles $\theta_{ij}$ and CP phase $\phi$,
$\hat\Delta_{12}^{}$ is written as
\begin{equation}
\hat\Delta_{12} = - \displaystyle\frac{2}{\sin \hat \phi}
\left(\cos \hat \phi -
\tan \hat \theta_{12}
\tan \hat \theta_{23}
\sin \hat \theta_{13}
\right)\,.
\end{equation}
Since $\delta$ does not diverge at the first peak,
the numerator also has to be 0
\begin{equation}
\left({\rm Re}\hat\cA{12}{} - {\rm Re}\hat\cA{23}{} + a_2^{}\right)
+ a_3^{} \hat\Delta_{12} = 0 \,,
\eqlab{num1}
\end{equation}
under the same condition.
Using \eqref{deno1}, $a_2^{}$ and $a_3^{}$ become
\begin{equation}
a_2^{} = -{\rm Re}\hat\cA{12}{}\,,
~~~\mbox{ and }~~~
a_3^{} = -\displaystyle\frac{1}{2} \hat{J}
\,.
\eqlab{as}
\end{equation}
Finally,
we obtain
\begin{equation}
\delta(E_{\rm peak}) \simeq -
\displaystyle\frac{\Delta_{12}^{}}{4}
\displaystyle\frac{
2\left(
{\rm Re}\hat\cA{12}{}
-{\rm Re}\cA{12}{}
+{\rm Re}\cA{23}{}
\right)
-\hat{J} \Delta_{12}}
{2{\rm Re}\cA{23}{} + J \Delta_{12}^{}}\,,
\eqlab{delta_a}
\end{equation}
and
\begin{equation}
h(E_{\rm peak})
= 4\left|\cU{\mu 3}{}\right|^2 \left|\cU{e 3}{}\right|^2
-4 A \sin\phi~ \Delta_{12}^{}
-{\rm Re}\hat\cA{12}{}\Delta_{12}^2
+O(\Delta_{12}^3)\,.
\eqlab{ha}
\end{equation}
Because the order of $\cA{ij}{}$ is less than 1,
the shift of the first peak for the transition probability
is not large.
\Eqref{ha} shows that
the first-peak of the transition probability with
$\delta_{\rm MNS}=90^\circ$
is smaller than that with $\delta_{\rm MNS}=270^\circ$.
Since the value of Re$\hat \cA{12}{}$ is negative for
$\rho \simeq 3.0 $ (g/cm$^3$),
the first-peak of transition probability for the CP-conserved case
is slightly larger than that with $\Delta_{12}=0$.
These features remain unchanged for the inverted hierarchy.
But the shifting direction of the first-peak is different
between the normal hierarchy and inverted one.
\section{summary}
In this article, we have estimated the three-flavor effect for
the determination of the mass-squared difference and the mixing angle
from the survival and transition probability of $\nu_\mu^{}$.
From both probabilities,
the larger mass-squared difference is changed
by the three-flavor effect.
The order of magnitude of the difference between
the larger mass-squared difference $\delta m^2_{13}$
and
the mass-squared difference of the two-flavor analysis
$\delta_{\rm dip,peak}$
is not only proportional to $\delta m^2_{12}$ but also
to the MNS matrix elements.
We also find the CP phase dependence for
the transition probability.
If there is no three-flavor effect for $\nu_\mu$ survival and
transition probabilities, the first-dip energy of $\nu_\mu$ survival
probability and the first-peak energy of $\nu_\mu$ transition one are
as same as that from $\Delta_{13}$.
However,
each of them is different from the value of $\Delta_{13}$,
which are shown from \eqref{dip1} and \eqref{delta_a},
because of three-flavor effect.
This means that the value of $\delta m^2_{13}$ from $\nu_\mu$
survival probability is slightly different from that from
$\nu_\mu\to\nu_e$ transition one.
The order of them are not so smaller than that of the expected
error for the $\delta m^2_{13}$ in the future long base-line neutrino
oscillation experiment.
These results are useful to estimate the three-flavor
effects for the value of the parameters
which are obtained from the numerical analysis.
{\it Acknowledgment}
I would like to thank
F.~Borzumati,
K.~Hagiwara,
K.~Nishikawa,
K.~Senda,
and
T.~Takeuchi,
for useful discussions and comments.
I am also grateful to H.~Kai
for warmhearted supports.
|
{
"timestamp": "2005-09-16T05:17:16",
"yymm": "0411",
"arxiv_id": "hep-ph/0411388",
"language": "en",
"url": "https://arxiv.org/abs/hep-ph/0411388"
}
|
\section{Main Results}
\noindent
Consider the complex untwisted affine Lie algebra of type $A_{n-1}^{(1)}$ :
$$\mathfrak{g}=\widehat{\mathfrak{sl}}_n=
\sl_n({\C[t^{\pm 1}]})\ \oplus\ \mathbb{C} K\ \oplus\ \mathbb{C} d
,$$
where $\sl_n({\C[t^{\pm 1}]})$ denotes the traceless $n\times n$ matrices with entries in the Laurent polynomials ${\C[t^{\pm 1}]}=\mathbb{C}[t,t^{-1}]$,\ $K$ is a central element of $\mathfrak{g}$, and $d=t\frac{d}{dt}$ is a derivation (see \cite[Ch 7]{kac1}). We have the Cartan decomposition $\mathfrak{g}=\mathfrak{n}\oplus\mathfrak{h}\oplus\mathfrak{n}_{-}$, where $\mathfrak{h}$ is the Cartan subalgebra
$$
\mathfrak{h}=\bigoplus_{1\leq i\leq n-1}
\!\!\! \mathbb{C}(E_{ii}\sh-E_{i+1,i+1})\
\oplus\ \mathbb{C} K \ \oplus\ \mathbb{C} d\,;
$$
and $\mathfrak{n}$ is the maximal nilpotent subalgebra
$$
\mathfrak{n}:=\mathop{\bigoplus_{1\leq i<j\leq n}}_{k\geq 0}\!\!\!\mathbb{C} t^kE_{ij}\ \oplus\mathop{\bigoplus_{1\leq i<j\leq n}}_{k\geq 1}\!\!\!\mathbb{C} t^k E_{ji}
\ \oplus\
\mathop{\bigoplus_{1\leq i\leq n-1}}_{k\geq 1}t^k
(E_{ii}-E_{i+1,i+1})\,.
$$
Here $E_{ij}\in\mathfrak{gl}_n(\mathbb{C})$ denotes a coordinate matrix.
Let $\Lambda_0,\Lambda_1,\ldots,\Lambda_{n-1}$ be the fundamental weights of $\mathfrak{g}$, and let $V(\Lambda_m)$ be the level 1 irreducible $\mathfrak{g}$-module with highest weight $\Lambda_m$. (Thus, $V(\Lambda_0)$ is the basic representation of $\mathfrak{g}$.)\ Let us recall the construction of $V(\Lambda_m)$ inside the fermionic Fock space $\mathcal{F}$ (cf.~\cite[Ch 14]{kac1}, \cite{kac2}). Let $\mathbb{C}^\infty=\bigoplus_{i\in\mathbb{Z}} \mathbb{C} \epsilon_i$ be the $\mathbb{C}$-vector space with basis $\{\cdots,\epsilon_{-2},\epsilon_{-1},\epsilon_0,\epsilon_1,\epsilon_{2},\cdots\}$.
Let $\mathcal{T}$ denote the collection of all subsets $J\subset\mathbb{Z}$ which are comparable to the non-positive integers $\mathbb{Z}_{\leq 0}$, meaning that $J\sh\setminus\mathbb{Z}_{\leq 0}$ and $\mathbb{Z}_{\leq 0}\sh\setminus J$ are both finite:
$$
\mathcal{T}=\{\,J\subset\mathbb{Z}\ \ \ \text{s.t.}\ \ \
|J\sh\setminus\mathbb{Z}_{\leq 0}|,\,|\mathbb{Z}_{\leq 0}\sh\setminus J|<\infty\,\}\,.
$$
We write such a set as:
$$
J=\{\cdots<j_{-2}<j_{-1}<j_0\}\,.
$$
Define the Fock space as the semi-infinite wedge product of $\mathbb{C}^\infty$:
$$\mathcal{F}=\wedge^{\!\infty/2}\,\mathbb{C}^\infty:=\bigoplus_{J\in\mathcal{T}}\mathbb{C} \epsilon_J\,,$$
the $\mathbb{C}$-vector space with basis elements:
$$
\epsilon_J:=\,\cdots\sh\wedge\epsilon_{j_{-2}}\sh\wedge\epsilon_{j_{-1}}\sh\wedge\epsilon_{j_{0}}\,.
$$
Thus, the $J\in\mathcal{T}$ play the role of tableaux indexing the basis vectors of the Fock space.
For $i,j\in\mathbb{Z}$, let $E_{ij}'$ denote a coordinate
matrix acting on $\mathbb{C}^\infty$ by $E_{ij}'(\epsilon_j)=\epsilon_i$,\ \, $E_{ij}'(\epsilon_k)=0$ for $k\neq j$.
Then for $i<j$,\ $E_{ij}'$ acts on the Fock space in the expected way:
$$
E_{ij}'(\epsilon_J) = \left\{\begin{array}{cl}
\pm \epsilon_{J\setminus j\cup i}&\text{if } j\in J,\ i\not\in J\\[.5em]
0& \text{otherwise,}
\end{array}\right.$$
Here we denote:
$$
J\setminus j\cup i\,:=\,(\,J\sh\setminus\{j\}\,)\,\cup\,\{i\}\,,
$$
the operation which moves the element $j\in J$ to the vacant position $i\not\in J$;
and $\pm=(-1)^\ell$ with $\ell={|J\cap[i,j]|-1}$, the sign of the permutation needed to sort the wedge factors of $\epsilon_{J\setminus j\cup i}$ into increasing order.
We let:
$$
{\widehat{E}}_{pq}:=\sum_{k\in\mathbb{Z}} E_{p+nk,q+nk}'\,,
$$
which is a well-defined operator on $\mathcal{F}$.
Now, if $i<j$ or $k>0$, we let $t^k E_{ij}$ act on $\mathcal{F}$ by the operator
${\widehat{E}}_{pq}$, where $p=i-nk$, $q=j$:
$$
t^k E_{ij}={\widehat{E}}_{i-nk,\,j}:\mathcal{F}\to\mathcal{F}\,.
$$
This defines the action\begin{footnote}{
This action arises naturally
if we identify the free ${\C[t^{\pm 1}]}$-module
${\C[t^{\pm 1}]}^n=\bigoplus_{i=1}^n{\C[t^{\pm 1}]} \epsilon_i$ with the $\mathbb{C}$-vector space
$\mathbb{C}^\infty=\bigoplus_{j\in\mathbb{Z}}\mathbb{C}\epsilon_j$
via: $t^k \epsilon_i\leftrightarrow\epsilon_{i-nk}\,.$
This gives an embedding $\mathfrak{gl}_n({\C[t^{\pm 1}]})\subset\mathfrak{gl}(\mathbb{C}^\infty)$,
so that the natural action of the upper triangular part of $\mathfrak{gl}(\mathbb{C}^\infty)$ on the Fock space restricts to the specified action of $\mathfrak{n}\subset\mathfrak{gl}_n({\C[t^{\pm 1}]})$.
However, this gives only a projective representation of the entire $\mathfrak{gl}_n({\C[t^{\pm 1}]})$, which then lifts to a true representation of the central extension $\widehat{\mathfrak{gl}}_n$.}
\end{footnote}of $\mathfrak{n}$ on $\mathcal{F}$, and we can similarly define the action of $\mathfrak{n}_{-}$ and $\mathfrak{h}$.
Indeed, the Chevalley generators of $\mathfrak{n}_{-}$ are:
$F_i=E_{i+1,i}={\widehat{E}}_{i+1,i}$ for $i=1,\ldots,n\sh-1$ and $F_0=t^{-1}E_{1,n}={\widehat{E}}_{n+1,n}$.
Now let $L_m:=\mathbb{Z}_{\leq m}\in \mathcal{T}$.
It is well known that the $U(\mathfrak{g})$-span of
the highest-weight vector $\epsilon_{L_m}$ is an irreducible $\mathfrak{g}$-module:
$$
U(\mathfrak{g})\cdot \epsilon_{L_m} = U(\mathfrak{n}_{-})\cdot \epsilon_{L_m} \cong
V(\Lambda_m)\,,
$$
where we define $\Lambda_{m}:=\Lambda_{(m \mod n)}$.
Recall that we can realize the Weyl group $W$ of $\mathfrak{g}$ as a permutation group on $\mathbb{Z}$. Indeed, we can write the simple reflection $s_i:\mathbb{Z}\to\mathbb{Z}$ as a product of commuting transpositions:
$$
s_i:=\prod_{k\in\mathbb{Z}} (i\sh+nk,\,i\sh+1\sh+nk)\ ,
$$
so that $s_i(i')=i'\sh+1$ whenever $i'\equiv i\mod n$.
Then $W=\langle s_0,\ldots,s_{n-1}\rangle$ is the corresponding Coxeter group.
The Weyl group $W$ acts on $\mathcal{T}$ via
$w(J):=\{w(j)\}_{j\in J}$.
Indeed, the extremal weight vectors of $V(\Lambda_m)\subset\mathcal{F}$ are just $\epsilon_J$ for $J=w(L_m)$.
Equivalently, a basis vector $\epsilon_J$ is an extremal weight vector whenever $J$ is {\it $n$-stable}: that is, whenever $j-n\in J$ for all $j\in J$.
We define the {\it parabolic Bruhat order} between $K=\{\cdots\sh<k_{-1}\sh<k_0\}$ and $J=\{\cdots\sh<j_{-1}\sh<j_{0}\}$ as:
$$
K\stackrel{\rm B}{\leq} J \quad\Longleftrightarrow\quad
\left\{\begin{array}{cl}
k_i\leq j_i&\text{ for all } i\\
k_i=j_i&\text{ for all } i\sh\ll 0
\end{array}\right.
$$
This induces an order on the $n$-stable $J=w(L_m)$ which is consistent with the usual Bruhat order on $w\in W$.
The {\it Demazure modules} \cite{demazure} of $V(\Lambda)$ are the $\mathfrak{n}$-modules obtained by raising the extremal weights:
$$
V_w(\Lambda_m)\ \cong\ U(\mathfrak{n})\cdot \epsilon_{w(L_m)}\,.
$$
We can get the same modules also by lowering the highest weight:
$$
V_w(\Lambda_m)=\mathop{\rm Span}\nolimits_\mathbb{C}\{F_{i_1}^{k_1}\cdots F_{i_t}^{k_t}
\epsilon_{L_m}\
\mid k_1,\ldots,k_r\geq 0\}\,,
$$
where $w=s_{i_1}\cdots s_{i_t}$ is a reduced word.
Next we describe the sets $J\in\mathcal{T}$ which index basis vectors of $$V_w(\Lambda_m)\subset V(\Lambda_m)\subset\mathcal{F}\,.$$
Let us say that a set $J\in\mathcal{T}$ is
{\it n-bounded} if $j_i-j_{i-1}\leq n$ for all $i$.
Also, we define the {\it order} of a set $J$ by:
$\mathop{\rm ord}\nolimits(J):=|J\sh\setminus \mathbb{Z}_{\leq 0}|-|\mathbb{Z}_{\leq 0}\sh\setminus J|$ ; equivalently, $\mathop{\rm ord}\nolimits(J)=m$ means that
$j_i=m+i$ for all sufficiently large negative $i$.
Now let
$$\begin{array}{rcl}
\mathcal{C}(L_m)&:=&
\{J\in \mathcal{T}\ \mid\ \mathop{\rm ord}\nolimits(J)=m\,\text{ and $J$ is $n$-bounded}\}
\\[.3em]
&=&\left\{J=\{\cdots\sh<j_{-2}\sh<j_{-1}\sh<j_{0}\}\subset\mathbb{Z}\,\left|\begin{array}{c}
j_i=m+i\text{ for }i\ll 0\\
j_i-j_{i-1}\leq n\text{ for all }i
\end{array}\right.\right\}
\end{array}$$
(The reader should be aware of a frequently used alternative notation in terms of ``colored Young diagrams'' instead of subsets.\!\!
\footnote{In \cite{FLOTW} and related literature, the basis
of $V(\Lambda_m)$ is indexed by the set of all
partitions $\lambda=(\lambda_1\sh\geq\lambda_2\sh\geq\cdots)$ with $\lambda_i\geq 0$, \ $\lambda_i=0$ for $i\gg0$, and
$\lambda_{i+1}\sh-\lambda_i\leq n\sh-1$. Namely, a set $J=\{\cdots\sh< j_{-1}\sh<j_0\}$ of order $m$ corresponds to $\lambda$ with $\lambda_{i+1}=m-i-j_{-i}$. It is useful to picture $\lambda$ as a Young diagram colored with a mod-$n$ checkerboard pattern: square $(i,j)$ has color $i-j\in\mathbb{Z}/n\mathbb{Z}$.
})
We can give $\mathcal{C}(L_m)$ a crystal graph structure by defining the crystal lowering operators ${f}_i$
for $i=0,\ldots,n\sh-1$, as recalled in Section 2 below.
If it is defined, the crystal operator $f_i$ on a set $J$ picks out a certain element $r\in J$ with $r\equiv i\mod n$, and replaces it with $r\sh+1\equiv i\sh+1\mod n$: that is,
$f_i(J)=J\setminus r\cup(r\sh+1)\,.$
We define the Demazure crystal as:
$$\mathcal{C}_w(L_m)=\{ {f}_{i_1}^{k_1}\cdots {f}_{i_t}^{k_t} L_m\mid k_1,\ldots,k_t\geq 0\},$$
where $w=s_{i_1}\cdots s_{i_t}$ is again a reduced word.
Our first theorem is a simpler description of the sets $J$ in this Demazure crystal, in analogy with the ``left key'' algorithm of Lascoux-Schutzenberger \cite{LS}.
If $J$ is $n$-bounded but not $n$-stable, define the following up-operation (which is different from the crystal operators):
$$\begin{array}{c}
\mathop{\rm up}\nolimits(J):=J\setminus p\cup q\quad \text{where:}\\[.5em]
p:=\max\{p'\ \mid\ p'\in J,\ \,p'\sh-n\not\in J\},\quad
\\[.5em]
q:=\min\{q'> p\ \mid\ q '\not\in J,\ \ q'\sh-n\in J,\ \
q'\not\equiv p \mod n\}\,.
\end{array}$$
To rephrase this in words, define a {\it seam} as a maximal arithmetic progression $S=\{\cdots<j\sh-2n<j\sh-n<j\}$ contained in $J$. We call the vacant position $j\sh+n\not\in J$ the {\it tight end} of $S$\,; and if $S$ is finite, we call the minimal element $p\in S$ the {\it loose end} of $S$. The up-operation moves $p\in J$ to $q\not\in J$, where $p$ is the maximal loose end in $J$, part of a seam $S=\{p,p\sh+n,\ldots\}$, and $q>p$ is the tight end of a different seam, the minimal such tight end. See the examples below.
Iterating the up-operation ``pulls out" this seam, distributing all the elements of $S$ to the tight ends of different seams; and then the operation starts on another seam. After each seam is pulled out, the number of loose ends decreases by one. Once all the finite seams of $J$ are pulled out, the result is an $n$-stable set which we call the {\it roof} of $J$:
$$
\mathop{\rm roof}\nolimits(J):=\mathop{\rm up}\nolimits^\ell(J)=\mathop{\rm up}\nolimits(\cdots\mathop{\rm up}\nolimits(J)\cdots)=y(L_m)
\quad\text{for some}\ y\in W\,.
$$
{\bf Theorem 1}\quad {\it
Let $\mathcal{C}_w(L_m)$ be the Demazure crystal generated from the highest weight $L_m$ according to a reduced word for $w\in W$. Then:
$$\begin{array}{rcl}
\mathcal{C}_w(L_m) &=&\{J\in\mathcal{C}(L_m) \ |\ \mathop{\rm roof}\nolimits(J)\leq w(L_m)\}
\end{array}$$
}
\\[-1.2em]
This gives a highly efficient algorithm for testing the membership of $J$ in $\mathcal{C}_w(L_m)$. (Later in this section we give a corresponding algorithm for generating all $J\in\mathcal{C}_w(L_m)$.)
Next we give elementary bases of $V(\Lambda_m)$ and its dual which are compatible with the Demazure modules, in analogy to the construction of Raghavan-Sankaran \cite{raghavan} (generalized by Littelmann \cite{littelmann3}).
\\[1em]
{\bf Theorem 2}\quad {\it
(i) Given $J\in\mathcal{C}_w(L_m)$, suppose $\mathop{\rm up}\nolimits^i(J)=\mathop{\rm up}\nolimits^{i-1}(J)\setminus p_i\cup q_i$ for $i=1,\ldots,\ell$, and $\mathop{\rm roof}\nolimits(J)=\mathop{\rm up}\nolimits^\ell(J)=y(L_m)$.
Define $$ v_J:=
{\widehat{E}}_{p_1,q_1}\cdots{\widehat{E}}_{p_\ell,q_\ell}\epsilon_{y(L_m)}\,.$$
Then the irreducible highest-weight module $V(\Lambda_m)$,
a submodule of the Fock space $\mathcal{F}$, has basis $\{v_J\mid J\in\mathcal{C}(L_m)\}$; and the Demazure module $V_w(\Lambda_m)$ has basis $\{v_J\mid J\in\mathcal{C}_w(L_m)\}$.
\\[.5em]
(ii) The irreducible lowest-weight module $V(\Lambda_m)^*$,
a quotient of the dual Fock space $\mathcal{F}^*$, has basis
$\{\epsilon_J^*\mid J\in\mathcal{C}(L_m)\}$; and the dual Demazure module $V_w(\Lambda_m)^*$ has basis
$\{\epsilon_J^*\mid J\in\mathcal{C}_w(L_m)\}$.
Here $\epsilon_J^*$ denotes a dual basis vector of $\mathcal{F}^*$ restricted to $V(\Lambda_m)$ or to $V_w(\Lambda_m)$ respectively.
\\[.5em]
}
In geometric terms, the functions $\epsilon_J^*$ can be considered as Plucker coordinates on the affine Grassmannian embedded in the infinite projective space $\mathbb{P}(\mathcal{F})$.
The vectors $v_J$ possess a triangularity property
with respect to the standard basis of the Fock space.
Define {\it lexicographic order} on sets
$K,J$ as follows:
$$
K\stackrel{\rm lex}{<} J\quad\Longleftrightarrow\quad\left\{\begin{array}{cl}
k_N<j_N&\text{for some $N$}\\
k_i=j_i &\text{for all $i<N$\,.}
\end{array}\right.
$$
{\bf Proposition 3}\quad {\it
Let us write:
$$v_J=\sum_K a_K^J \epsilon_K\quad \text{with coefficients}\quad\ a_K^J\in\mathbb{C}\,.$$
\\[-1em]
(i) We have a non-zero coefficient
$a_K^J\neq 0$ only if $K\stackrel{\rm lex}{\leq} J$.\\
(ii) We have a non-zero leading coefficient $a_J^J\neq 0$ for every $J$, given explicitly as follows.
If $J$ is $n$-stable, then $a_J^J=1$.
If $J$ is $n$-bounded but not $n$-stable, suppose that $\mathop{\rm up}\nolimits^i(J)=\mathop{\rm up}\nolimits^{i-1}(J)\setminus p_i\cup q_i$, and $t$ is maximal such that
$p_1\equiv\cdots\equiv p_t\mod n\,.$
Define $\mu_d:=\#\{\,i\leq t\mid q_i-p_i=d\,\}\,$ and ${\widetilde J}:=\mathop{\rm up}\nolimits^t(J)$. Then:
$$
a_J^J=\pm\left(\prod_{d\geq 1}\mu_d!\right) a_{\widetilde J}^{\widetilde J}\,.
$$
}
\\[0em]
In part (ii), note that the sequence $S=\{p_1<\ldots<p_t\}$ is actually the first seam of $J$ pulled out by the roof algorithm:\
$S=\{p\,,\,p\sh+n,\ldots, p\sh+n(t\sh-1)\}$.
Iterating part (ii), we get a combinatorial formula for
the leading coefficient $a_J^J$ of each $v_J$ depending only on the sequences $p_1,\ldots,p_\ell$ and $q_1,\ldots,q_\ell$ in the roof algorithm.
\\[1em]
{\bf Example}\quad Let $n=5$ and let:
$$
J:=\{\ldots,-4,-3,-2,-1, 0,3,4,7,10,12,14,17,18,23,27,32,33,35,37\}\,.
$$
Then $J\in\mathcal{C}(L_m)$ for $m=14$, since
$L_0\subset J$ and $|J\sh\setminus L_0|=14$, so that
$\mathop{\rm ord}\nolimits(J)=\mathop{\rm ord}\nolimits(L_0)\sh+14=14$.
We sort $J$ into its residue classes mod $n$ to show the seam structure. We mark the maximal loose end with boldface, and the tight end used by the up-operation with $\TT$.
$$\begin{array}{r@{\,}c@{\,}l}
J&=&\!\!
\mbox{\footnotesize$\left[\begin{array}{@{\ }c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\,}}
\cdots&-4&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdots&-3&\cdot&7&12&17&\cdot&27&32&37\\
\cdots&-2&3&\cdot&\cdot&18&23&\cdot&33&\cdot\\
\cdots&-1&4&\cdot&14&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdots&\ \ 0&\cdot&10&\cdot&\cdot&\cdot&\cdot&{35}&\cdot&
\end{array}\!\!\!\right]$}
=
\mbox{\footnotesize$\left[\begin{array}{@{\ }c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\,}}
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&7&12&17&\cdot&27&32&37\\
3&\cdot&\cdot&18&23&\cdot&33&\TT\\
4&\cdot&14&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&10&\cdot&\cdot&\cdot&\cdot&\mathbf{35}&\cdot&
\end{array}\!\!\!\!\right]$}
\\[2.5em]
&\stackrel{\mathop{\rm up}\nolimits}{\to}&
\!\!\mbox{\footnotesize$\left[\begin{array}{@{\ }c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\,}}
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&7&12&17&\cdot&27&32&37&\TT\\
3&\cdot&\cdot&18&23&\cdot&\mathbf{33}&38&\cdot\\
4&\cdot&14&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&10&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\end{array}\!\!\!\right]$}
\stackrel{\ \mathop{\rm up}\nolimits^{\mbox{\scriptsize 2}}}{\longrightarrow}
\mbox{\footnotesize$\left[\begin{array}
{@{\ }c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\,}}
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&7&12&17&\cdot&\mathbf{27}&{32}&{37}&{42}&{47}\\
3&\cdot&\cdot&18&23&\TT&\cdot&\cdot&\cdot&\cdot\\
4&\cdot&14&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&10&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\end{array}\!\!\!\right]$}
\\[2.5em]
&\stackrel{\ \mathop{\rm up}\nolimits^{\mbox{\scriptsize 5}}}{\longrightarrow}&
\!\!\mbox{\footnotesize$\left[\begin{array}
{@{\ }c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\,}}
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&7&12&17&\cdot&\cdot&\cdot&\cdot&\cdot&\\
3&\cdot&\cdot&\mathbf{18}&23&28&33&38&43&48\\
4&\cdot&14&\TT&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&10&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\end{array}\!\!\!\right]$}
\!\!\stackrel{\ \mathop{\rm up}\nolimits^{\mbox{\scriptsize 16}}}{\longrightarrow}\!\!\!\!
\mbox{
\footnotesize$\left[\begin{array}
{@{\ }c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\,}}
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\!\!\mathbf{7}&\!\!12&17&22&27&32&37&42&47&52&57&62\\
3&\TT&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
4&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\end{array}\!\!\!\right]$}
\\[2.5em]
&\stackrel{\ \mathop{\rm up}\nolimits^{\mbox{\scriptsize 12}}}{\longrightarrow}&
\!\!\mbox{
\footnotesize$\left[\begin{array}
{@{\ }c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\,}}
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
3&8&13&18&23&28&33&38&43&48&53&58&63\\
4&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\end{array}\!\!\!\right]$}
=\mathop{\rm roof}\nolimits(J)=y(L_{14})\,.
\end{array}$$
We thus have $(p_1,q_1)\sh=(35,38)$,\
$(p_2,q_2)\sh=(33,42)$,\
$(p_3,q_3)\sh=(38,47)$,\ldots, and:
$$\begin{array}{rcl}
v_J&=& {\widehat{E}}_{35,38}\,{\widehat{E}}_{33,42}\,{\widehat{E}}_{38,47}
\cdots \epsilon_{y(L_{14})}\\[.5em]
&=&{\widehat{E}}_{5,8}
{\widehat{E}}_{3,12}^{\,2}\,
{\widehat{E}}_{2,3}^{\,5}\,
{\widehat{E}}_{3,4}^{\,7}\,
{\widehat{E}}_{4,7}^{\,8}\,
{\widehat{E}}_{5,57}\,
{\widehat{E}}_{2,3}^{\,12}
\ \epsilon_{y(L_14)}\,.
\end{array}$$
Since $J$ has seven loose ends, we must apply Proposition 3(ii) seven times to compute:
$
a_J^J=\pm\ 1!\cdot 2!\cdot 5!\cdot 7!\cdot 8!\cdot
1!\cdot 1!\cdot 12!\,.
$
(Here we have only one factorial for each seam, though in general there will be several.)
To determine a reduced decomposition for the Weyl group element $y$, we start with the extremal weight $K=y(L_{14})$ and perform the simple reflection:
$K\mapsto s_{r}(K)$,
where
$$
r:=\min\,\{\,k\!\not\in\! K\ \mid\ k\sh+1\!\in\! K\,\}\,
$$
is the minimal ``hole'' of $K$, and $s_r:=s_{(r\mod n)}$. This will always give
$K\stackrel{\mathrm{B}}{>}s_r(K)$, and iterating the operation produces a canonical reduced word for $y$.
Indeed, $$
y(L_{14})=s_{2}\,s_{1}\,s_{3}\,s_{2}\,s_{0}\,
(s_{4}s_{3}\,s_{2}\,s_{1}\,s_{0})^{11}\,s_{4}\, L_{14}
\ .$$
See also the Example in the next section.
\quad$\Box$
\\[1em]
{\bf Example}\quad Let $n=2$. Then for fixed $m$, the Bruhat order on the sets $w(L_m)$ reduces to a linear order: for example,
$$L_0 = s_1(L_0)
\stackrel{\rm B}{<}
s_{0}(L_0)
\stackrel{\rm B}{<}
s_{1}s_{0}(L_0)
\stackrel{\rm B}{<}
s_{0}s_{1}s_{0}(L_0)
\stackrel{\rm B}{<}\cdots\,.$$
For $J$ an $n$-bounded set with order $m$, the roof operation reduces to:
$$\begin{array}{rcl}
\mathop{\rm roof}\nolimits(J)&=&\min\{\,w(L_m)\,\mid\, w\in W,\ J\stackrel{\rm B}{\leq} w(L_m)\,\}\\[.5em]
&=&L_a\cup\{a\sh+2,a\sh+4,\ldots,a\sh+2k\}\,,
\end{array}$$
where $a=\max\{a'\mid L_{a'}\subset J\}$ and $k=|J\setminus L_a|$. That is, the Demazure crystal is simply $\mathcal{C}_w(L_m)=\{J\in\mathcal{C}(L_m)\,\mid\,J\stackrel{\rm B}{\leq} w(L_m)\}$.
This case is further considered in the context of completely integrable lattice models in \cite{FMO}. \quad$\Box$
\\[1em]
{\bf Example}\quad Generalizing the previous case, let $n$ be arbitrary and suppose $w(L_m)$ is of the form:
$$
w(L_m) = L_a\cup
\{a\sh+n, a\sh+2n, \ldots, a\sh+kn\}
$$
for some $a$ and $k = m-a$. Then for any $J\stackrel{\rm B}{\leq} w(L_m)$, we have $\mathop{\rm roof}\nolimits(J)\stackrel{\rm B}{\leq} w(L_m)$, so that again $\mathcal{C}_w(L_m)=\{J\in\mathcal{C}(L_m)\,\mid\,J\stackrel{\rm B}{\leq} w(L_m)\}$. This case, which is considered in \cite{MW}, is exceptional: in general, it often happens that $J\stackrel{\rm B}{\leq} w(L_m)$, but $\mathop{\rm roof}\nolimits(J)\not\stackrel{\rm B}{\leq} w(L_m)$.
\quad$\Box$
\\[1em]
Next we consider the modifications which must be made to our theory to generalize it to positive characteristic.
Since the leading coefficients $a_J^J$ are not necessarily $\pm 1$, the vectors $v_J$ could become linearly dependent if we work over $\mathbb{Z}$ and then reduce modulo a prime. To define a characteristic-free basis $\{\,v^{\,\prime}_J\mid J\in\mathcal{C}_w(L_m)\,\}$, we start with $v^{\,\prime}_J:=\epsilon_J$ for $n$-stable $J$, then for a general $n$-bounded $J$ we define:
$$
v^{\,\prime}_J\ \ :=\ \ \frac
{{\widehat{E}}_{p_1,q_1}\cdots{\widehat{E}}_{p_t,q_t}}
{\prod_{d\in\mathbb{Z}}\mu_d!}\cdot v^{\,\prime}_{\widetilde J}
\ \ =\ \
\mathop{\prod_{d\in\mathbb{Z}}}_{d\not\equiv 0}\frac{{\widehat{E}}_{p,p+d}^{\,\mu_d}}{\mu_d!}\cdot v^{\,\prime}_{\widetilde J}\ ,
$$
where $t$ is maximal such that $p:=p_1\equiv\cdots\equiv p_t\mod n$. The second equality follows because $d_i:=q_i-p_i\not\equiv 0\mod n$, so all the operators ${\widehat{E}}_{p,p+d}$ commute with each other. The basis $\{v^{\,\prime}_J\}$ clearly lies in the Kostant $\mathbb{Z}$-form of the Demazure module $V_w(\Lambda_m)$, and it has leading coefficients $\pm1$, so it reduces to a basis over an arbitrary field. (Cf. \cite[Ch. 26]{humphreys}.)
Theorem 1 also gives an alternative ``bottom-up'' algorithm to generate $\mathcal{C}_w(L_m)$, as opposed to the ``top-down'' definition in terms of crystal lowering operators. We write:
$$\begin{array}{rcl}
\mathcal{C}_{=y}(L_m)&:=&\{J\in\mathcal{C}(L_m)\,\mid\,\mathop{\rm roof}\nolimits(J)=y\}\\[.5em]
&=&\{y(L_m)\}\,\cup\,\mathop{\rm up}\nolimits^{-1}(y(L_m))\,\cup\,
\mathop{\rm up}\nolimits^{-1}\mathop{\rm up}\nolimits^{-1}(y(L_m))\,\cup\, \cdots
\end{array}$$
where $\mathop{\rm up}\nolimits^{-1}({\widehat{J}})$ means the set of all $J$ such that $\mathop{\rm up}\nolimits(J)={\widehat{J}}$. To compute this for any given ${\widehat{J}}\in\mathcal{C}(L_m)$, we first find ${\widetilde{p}}<{\widehat{p}}$\,, the two maximal loose ends of ${\widehat{J}}$ (with one or both possibly $=-\infty$). Next we choose any $q>{\widehat{p}}\sh-n$ such that $q\sh+n$ is the tight end of a seam $S\subset{\widehat{J}}$ of length $|S|\geq 2$,
and we let ${\widehat{q}}$ be the maximal tight end of ${\widehat{J}}$
less than $q$. Finally, we define:
$$
P(q):=\left\{\, p\ \left|\begin{array}{c}
p\sh-n,p\not\in{\widehat{J}}\\[.3em]
\max({\widehat{p}},{\widehat{q}}\,)<p<q
\end{array}\right.\right\}
\,\cup\,
\left\{\, p={\widehat{p}}\sh-n\ \left|\begin{array}{c}
p\sh-n\not\in{\widehat{J}}\\[.3em]
\max({\widetilde{p}},{\widehat{q}}\,)<p
\end{array}\right.\right\}\,.
$$
Then we have:
$$
\mathop{\rm up}\nolimits^{-1}({\widehat{J}})=
\left\{J:={\widehat{J}}\setminus q\cup p\ \left|\
\begin{array}{c}
q\sh-n,q\in{\widehat{J}},\,\ q\sh+n\not\in{\widehat{J}}\\[.3em]
q>{\widehat{p}}\sh-n,
\,\ p\in P(q)
\end{array}\right.\right\}\,.
$$
Applying this to all $y\leq w$, we generate all $J\in\mathcal{C}_w(L_m)$.
We will prove Theorem 1 in Section 3 and Proposition 3 in Section 4. Theorem 2 is a corollary of these, as follows.
By Theorem 1 and the definitions, we have:
$$
V':=\mathop{\rm Span}\nolimits_\mathbb{C}\{\,v_J\mid \mathop{\rm roof}\nolimits(J)\leq w\,\}
=\mathop{\rm Span}\nolimits_\mathbb{C}\{\,v_J\mid J\in\mathcal{C}_w(L_m)\,\}
\subset V_w(\Lambda_m)\,.
$$
Proposition 3 implies that the $v_J$ are linearly independent vectors in $\mathcal{F}$ (since they are triangular with respect to the standard basis
$\{\epsilon_J\}$ ), so that
$\dim_\mathbb{C} V'= |\mathcal{C}_w(L_m)|\,;$
but it is well known from crystal graph theory (Section 2 below) that $|\mathcal{C}_w(L_m)|=\dim_\mathbb{C} V_w(\Lambda_m)$, so that
$V'=V_w(\Lambda_m)$. This shows Theorem 2 for the Demazure module $V_w(\Lambda_m)$, and the claims for the irreducible module and the dual modules follow trivially.
We comment on related work which is closest to our point of view. The pioneering paper \cite{DJKMO} by Date, Jimbo, Kuniba, Miwa, and Okado of the Kyoto school defined the tableaux $\mathcal{C}(L_m)$ for $V(\Lambda_m)$ (\,in fact for all $V(\ell\Lambda_m)$\,), and the crystal graph structure was first defined by Misra, Miwa, Jimbo, et al.~in \cite{MM}, \cite{jimbo}. Certain Demazure crystals $\mathcal{C}_w(L_m)$ were considered by Kuniba, Misra, Miwa, Uchiyama and others in \cite{KMOTU},\cite{KMOU},\cite{FMO},\cite{MW}.
A useful survey of related work is \cite{FLOTW}, and \cite{kashiwara} is a fundamental reference.
\\[1em]
{\bf Notation}\quad For a set $J\subset\mathbb{Z}$, we define:
$$
J^{\equiv i}:=\{j\in J\,\mid\, j\equiv i\mod n\}\,,
\qquad
J_{<r}:=\{j\in J\,\mid\, j<r\}\,.
$$
Similarly for\ $J_{>r}$ , for\ \
$J^{\equiv i}_{>r}:=J^{\equiv i} \cap J_{>r}$ ,
for\ \ ${}_{q\leq}J_{\leq r}:=J_{\geq q}\cap J_{\leq r}$ ,
etc.
\section{Crystal Operators}
In this section, we review the necessary facts about the crystal raising and lowering operators acting on $\mathcal{C}(L_m)$.\,\begin{footnote}{These operators are sometimes encoded in the {\it crystal graph} having vertices $J\in\mathcal{C}(L_m)$ and $i$-colored edges $J\stackrel{i}{\to}f_iJ$.}
\end{footnote}These operators were first defined in our case by the Kyoto school \cite{jimbo}, and they can also be derived from Littelmann's path model (as modified for semi-infinite paths in \cite{magyar}). The crystal operators are basically different from the up-operation: indeed, by Theorem 1 the two are in some sense transversal to each other.
If it is defined, the lowering operator $f_i$ for $i=0,1,\ldots n\sh-1$ acting on a set $J\in\mathcal{T}$ picks out a certain element $r\in J$ with $r\equiv i\mod n$, and replaces it with $r\sh+1\equiv i\sh+1\mod n$.
(We say that $f_i(J)$ is ``lower'' than $J$ because it is farther from the highest-weight element $L_m$.)\ \
Similarly, the raising operator $e_i(J)$ picks out a certain element $r'\in J$ with $r'\equiv i\sh+1$ and replaces it with $r'\sh-1\equiv i$. We have:\ \ $f_i(J)=J' \,\Longleftrightarrow\, J=e_i(J')$.
\\[1em]
{\bf Definition}\ \ {\it Given $J\in\mathcal{T}$.\\
(i) Let $$
R:=\{\,r\ \text{\ \ s.t.\ \ for all $k\geq r$}\,,\ \
|{}_{r\leq}J_{\leq k}^{\equiv i}\,|
> |{}_{r\leq}J_{\leq k}^{\equiv (i+1)}\,|\ \}\,.
$$
If $R$ is empty, then $f_i(J)$ is undefined. Otherwise,
$$
f_i(J):=J\setminus r\cup(r\sh+1)\,,\quad\text{where}\quad
r:=\min(R)\,.
$$
(ii) Let
$$
R':=\{\,r'\text{\ \ s.t.\ \ for all $k\leq r'$,}\ \
|{}_{k\leq}J_{\leq r'}^{\equiv (i+1)}\,|
> |{}_{k\leq}J_{\leq r'}^{\equiv i}\,|\ \}\,.
$$
If $R'$ is empty, then $e_i(J)$ is undefined. Otherwise,
$$
e_i(J):=J\setminus r'\cup (r'\sh-1)\,,\quad\text{where}\quad
r':=\max(R')\,.
$$
}
\\[-1em]
The importance of the crystal operators lies in the following
Refined Demazure Character Formula (cf. Jimbo, et al. \cite{jimbo}). Define the {\it weight} of a tableau $J\in\mathcal{C}(L_m)$ by $\mathop{\rm wt}(L_m):=\Lambda_m$
and $\mathop{\rm wt}(f_i(J)):=\mathop{\rm wt}(J)-\alpha_i$.
\\[1em]
{\bf Proposition}\quad {\it The character of the Demazure module $V_w(\Lambda_m)$ is the weight generating function of the crystal graph $\mathcal{C}_w(L_m)$: that is,
$$
\sum_{\mu}\ \dim_\mathbb{C}\! V_w(\Lambda_m)_\mu\ \,e^\mu
=\sum_{J\in\mathcal{C}_w(L_m)} e^{\mathop{\rm wt}(J)}\,.
$$
In particular,\ $\dim_\mathbb{C}\! V_w(\Lambda_m)=\#\mathcal{C}_w(L_m)$.
}
\\[1em]
Let us give a more pictorial way to understand these operators in the spirit of Lascoux-Schutzenberger \cite{LS}: we progressively remove elements of $J$ which are irrelevant to the action.
We call $j\in J^{\equiv i}$ the {\it $i$-elements} of $J$, and we write sets as usual in increasing order: $J=\{\cdots\sh<j_{-1}\sh<j_{0}\}$. We start by removing all $j\in J$ except the $i$- and $(i\sh+1)$-elements. We consider each remaining $i$-element which is immediately followed by an $(i\sh+1)$-element, and we remove these pairs.
Now we look again for remaining $i$-elements followed by $(i\sh+1)$-elements, and remove these pairs.
After finitely many iterations, we are left with
a finite subset
$$J'=\{j'_1<\cdots<j'_s<j''_1<\cdots<j''_t\}\subset J
\quad\text{with all}\ \ j'_k\equiv i\sh+1,\ \
j''_k\equiv i\,.
$$
Then we take $r=j''_1$, the smallest $i$-element, and $r':=j'_s$, the largest $(i\sh+1)$-element of $J'$, so that:
$$
f_i(J)=J\setminus j''_1\cup(j''_1\sh+1)\,,
\qquad
e_i(J)=J\setminus j'_s\cup(j'_s\sh-1)\,.
$$
\noindent {\bf Example}\quad
We exhibit the action of $e_2,f_2$ on the $J$ from our previous example. This time, we write the elements of $J$ reduced modulo $n=5$:
since $J$ is $n$-bounded, this loses no information.
We have underlined the elements to be removed.
$$
\begin{array}{ccr@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}l}
J&=&\{\,\cdots,&\!-3,\,&\!-2,\,&\!-1,\,&\,0,\,&3,\,&4,\,&7,&10,&12,&14,&17,&18,&23,&27,&32,&33,&35,&37\,\}
\\
&=&\cdots\ &2&3&4&5&3&4&2&5&2&4&2&3&3&2&2&3&5&2
\\
&\Rightarrow&\cdots\ &\underline2&\underline3&&&3&&2&&2&&\underline{2}&\underline{3}&3&2&\underline2&\underline3&&2
\\
&\Rightarrow&&&&&&3&&2&&\underline2&&&&\underline3&2&&&&2
\\
J'&=&&&&&&\mathbf3&&\mathbf2&&&&&&&\mathbf2&&&&\mathbf2\\[.5em]
f_2(J)&=&\cdots\ &2&3&4&5&\mathbf3&4&\mathbf3&5&2&4&2&3&3&\mathbf2&2&3&5&\mathbf2 \\
f_2^2(J)&=& \cdots\ &2&3&4&5&\mathbf3&4&\mathbf3&5&2&4&2&3&3&\mathbf3&2&3&5&\mathbf2 \\
f_2^3(J)&=& \cdots\ &2&3&4&5&\mathbf3&4&\mathbf3&5&2&4&2&3&3&\mathbf3&2&3&5&\mathbf3 \\
f_2^4(J)&=&\text{undefined}\hspace{-2.8em}\\[.5em]
e_2(J)&=& \cdots\ &2&3&4&5&\mathbf2&4&\mathbf2&5&2&4&2&3&3&\mathbf2&2&3&5&\mathbf2 \\
e_2^2(J)&=&\text{undefined}\hspace{-2.8em}\\
\end{array}
$$
Note that the irrelevant elements removed from $J$ are the same as those from $e_i(J)$ and $f_i(J)$, so we can easily perform $e_i$ and $f_i$ repeatedly.
In the previous example we computed $\mathop{\rm roof}\nolimits(J)=y(L_m)$,
where:
$$
y=s_{2}\,s_{1}\,s_{3}\,s_{2}\,s_{0}\,
(s_{4}s_{3}\,s_{2}\,s_{1}\,s_{0})^{11}\,s_{4}\,.
$$
By Theorem 1, this means that $J\in\mathcal{C}_y(L_m)$:
$$
J=f_{2}^{\mbox{\tiny $\bullet$}}\,f_{1}^{\mbox{\tiny $\bullet$}}\,f_{3}^{\mbox{\tiny $\bullet$}}\,f_{2}^{\mbox{\tiny $\bullet$}}\,f_{0}^{\mbox{\tiny $\bullet$}}\,(f_{4}^{\mbox{\tiny $\bullet$}}\,f_{3}^{\mbox{\tiny $\bullet$}}\,f_{2}^{\mbox{\tiny $\bullet$}}\,f_{1}^{\mbox{\tiny $\bullet$}}\,f_{0}^{\mbox{\tiny $\bullet$}})^{11}\,f_{4}^{\mbox{\tiny $\bullet$}}\, L_{14}\ ,
$$
where each $f_i^{\mbox{\tiny $\bullet$}}$ represents some non-negative integer power of $f_i$. We see from this the comparative rapidity of the roof algorithm in defining and generating Demazure crystals.\quad$\diamond$
\section{Proof of Theorem 1}
For a set $J\in\mathcal{C}(L_m)$, let $\mathcal{C}_y(L_m)$ be the unique minimal Demazure crystal containing $J$, and define the {\it ceiling of $J$} to be the extremal element of $\mathcal{C}_y(L_m)$:
$$
\ceil(J):=y(L_m)\,.
$$
Thus\
$\mathcal{C}_w(L_m) =\{J\in\mathcal{C}(L_m)\mid\ceil(J)\leq w(L_m)\}\,,$ and we can restate:
\\[1em]
{\bf Theorem 1}\quad {\it We have\ \ $\mathop{\rm roof}\nolimits(J)=\ceil(J)$\ \
for all $J\in\mathcal{C}(L_m)$.}
\\[1em]
For $J\neq L_m$, we define:
$$
a(J):=\max\{a\mid L_a\subset J\}
\quad\text{and}\quad
r(J):=\min J_{>a(J)}\,.
$$
That is, $a(J)<r(J)$ are the smallest consecutive elements of $J$ which are not consecutive integers.
We let $e^{\max}_i(J)$ denote the result of applying the highest possible power of the raising operator $e_i$ to $J$, and we let:
$$K:={e}_{r-1}^{\max}J\,,$$
where $r:=r(J)$. Observe that $r\sh-1\in K$
(and thus $K\neq J$), since in
$J_{<r}=L_{a(J)}$, the pairs of consecutive entries congruent to $r\sh-1$ and $r$ are irrelevant for the crystal operation.
\\[1em]
\noindent{\bf Ceiling Lemma}\\[.3em] {\it
(i) For all $J\in\mathcal{C}(L_m)$, we have:
$$r(\ceil(J))=r(J)
\quad\text{and}\quad
a(\ceil(J))=a(J)\,.$$
(ii) With $J\neq L_m$ and $K$ as above, we have:
$$
\ceil(J)>\ceil(K)=s_{r-1}\,\ceil(J)\,.
$$
}\\[0em]
{\bf Roof Lemma}\ \ {\it With $J\neq L_m$ and $K$ as above, we have:
$$
\mathop{\rm roof}\nolimits(J)>\mathop{\rm roof}\nolimits(K)=s_{r-1}\,\mathop{\rm roof}\nolimits(J)\,.
$$
}\\[0em]
Assuming these two Lemmas, we can immediately prove Theorem 1 by induction on the quantity:
$$
\mathop{\rm height}\nolimits(J):=\sum_{i\leq 0}\,(j_i-i-m)\,,
$$
a sum with finitely many non-zero terms for $J\in\mathcal{C}(L_m)$.
If $\mathop{\rm height}\nolimits(J)=0$, then $J=L_m$ and there is nothing to prove. Otherwise, $\mathop{\rm height}\nolimits(K)<\mathop{\rm height}\nolimits(J)$, and we may assume $\mathop{\rm roof}\nolimits(K)=\ceil(K)$. Then the
Roof and Ceiling Lemmas imply:
$$\qquad\qquad
\mathop{\rm roof}\nolimits(J)=s_{r-1}\mathop{\rm roof}\nolimits(K)=s_{r-1}\ceil(K)=\ceil(J)\,.
\qquad\Box
$$
\\[0em]
{\it Proof of Ceiling Lemma.}\ \
We first prove that $a(\ceil(J))\leq a(J)$.
Let $a:=a(J)$.
Let $\ceil(J)=s_{i_t}\cdots
s_{i_1}$, a reduced decomposition. Then for some
$c_t,\ldots,c_1\geq 0$, $J=f_{i_t}^{c_t}\cdots f_{i_1}^{c_1}L_m$.
The sequence $\{i_t,\ldots,i_1\}$ must contain a subsequence
$\{a\sh+1,\ldots,m\}$. Let $\{j_k,\ldots,j_1\}$ be the rightmost such subsequence: that is, ${j_1}$ is the rightmost occurrence of
$m$ in $\{i_t,\ldots,i_1\}$; and for $k=1,\ldots,m\sh-a\sh-1$, after ${j_k}$ has been determined
let ${j_{k+1}}$ be the rightmost occurrence of $m\sh-k$ in
$\{i_t,\ldots,i_1\}$ to the left of ${j_k}$.
Let $f_{i}^{\max}T$ denote the result of
applying the lowering operator $f_{i}$ as many times as possible
to $T$; thus, for example, $\ceil(J)=f_{i_t}^{\max}\cdots
f_{i_1}^{\max}L_m$. Then
$$
a(f_{i_t}^{\max}\cdots f_{i_1}^{\max}L_m)\leq
a(f_{{j_k}}^{\max}\cdots f_{j_1}^{\max}L_m)=m\sh-k,
$$
for $k=1,\ldots,m\sh-a$. Setting $k=m\sh-a$, we obtain the result.
We prove (i) and (ii) together by induction on $\mathop{\rm height}\nolimits(J)$. Let $r:=r(J)$.
If
$\mathop{\rm height}\nolimits(J)=0$ then $\ceil(J)=J=L_m$, so (i) is true, and (ii) is
vacuously true. Assume $\mathop{\rm height}\nolimits(J)>0$. Note that
$$\ceil(J)\geq\ceil(K)\geq s_{r-1}\,\ceil(J)\,,$$ so (ii) is
equivalent to $\ceil(J)\neq\ceil(K)$.
If $r(J)=a(J)+2$, then $a(K)=a(J)+1$. Since
$\mathop{\rm height}\nolimits(K)<\mathop{\rm height}\nolimits(J)$, by induction, $a(\ceil(K))=a(K)$, thus
$a(\ceil(K))=a(K)>a(J)\geq a(\ceil(J))$, implying
$\ceil(K)\neq\ceil(J)$. Therefore $\ceil(J)=s_{r-1}\ceil(K)$, and
clearly (i) follows.
If $r(J)>a(J)+2$, on the other hand, let $wL_m=\ceil(K)$. Since
$\mathop{\rm height}\nolimits(K)<\mathop{\rm height}\nolimits(J)$, by induction we have $a(wL_m)=a(K)=a$ and
$r(wL_m)=r(K)=r\sh-1$. Define
$$
v=s_{a+1}s_{a+2}\cdots s_{r-2}\,w\,.
$$
Note that $a(vL_m)=a\sh+1$.
Let $v=s_{i_t}\cdots s_{i_1}$ be a reduced decomposition.
Then $w=s_{r-2}\cdots s_{a+1}s_{i_t}\cdots s_{i_1}$,
also a reduced decomposition. Indeed, if we define $k$ by $r(wL_m)=w_k$
(where $wL_m=\{\cdots\sh>w_{-2}\sh>w_{-1}\sh>w_0\}$\,), then $$
(s_{a+(j+1)}\cdots
s_{a+1}s_{i_t}\cdots s_{i_1}L_m)_k =
1+(s_{a+j}\cdots
s_{a+1}s_{i_t}\cdots s_{i_1}L_m)_k
$$
for $j=1,\ldots,r\sh-a\sh-3$, and
$$
(s_{a+1}s_{i_t}\cdots s_{i_1}L_m)_k =
1+(s_{i_t}\cdots
s_{i_1}L_m)_k\,.
$$
In other words, with each successive
multiplication of $v=s_{i_t}\cdots s_{i_1}$ by $s_{a+j}$ for $j=1,\ldots, r\sh-a\sh-2$, the product increases.
Now suppose $\ceil(J)=wL_m$.
Then $J=f_{r-2}^{c_{r-2}}\cdots
f_{a+1}^{c_{a+1}}f_{i_t}^{d_t}\cdots f_{i_1}^{d_1}L_m$ for some
$c_{r-2},\ldots,c_{a+1},d_t,\ldots,d_1\geq 0$. Thus,
$e_{a+1}^{c_{a+1}}\cdots e_{r-2}^{c_{r-2}}J\in C_v(L_m)$. Thus\\
$a(e_{a+1}^{c_{a+1}}\cdots e_{r-2}^{c_{r-2}}J)\geq a(vL_m)=a\sh+1$, a
contradiction. Therefore $\ceil(J)\neq wL_m$, so
$\ceil(J)=s_{r-1}wL_m$, from which (i) follows immediately.
\hfill$\Box$
\\[1em]
{\it Proof of Roof Lemma.} For $i\in\mathbb{Z}$, define
$\mathop{\rm roof}\nolimits_i(J)$ by
$$\mathop{\rm roof}\nolimits_i(J):=\mathop{\rm roof}\nolimits(L_i\cup J).$$ Several properties of
$\mathop{\rm roof}\nolimits_i(J)$ follow easily from the definition:
\begin{itemize}
\item[1.] $\mathop{\rm roof}\nolimits_i(J)=\mathop{\rm roof}\nolimits_{i+1}(J)$ \ if $i+1\in J$.
\item[2.] $\mathop{\rm roof}\nolimits_i(J)=\mathop{\rm roof}\nolimits(\mathop{\rm roof}\nolimits_{i+1}(J)\setminus \{i+1\})$ \ if $i+1\not\in J$.
\item[3.] $\mathop{\rm roof}\nolimits_i(J)=\mathop{\rm roof}\nolimits(J)$ if $J\supset L_i$ ; \ $\mathop{\rm roof}\nolimits_i(J)=L_i$ \ if $L_i\supset J$.
\end{itemize}
For $T\in\mathcal{C}(L_k)$, define
$$\lub(T)=\min_{\stackrel{\rm lex}{\geq}}\left\{J'\in \mathcal{C}(L_k)\Big|
\begin{array}{l}
J'\stackrel{\rm B}{\geq} T,\\
J'\hbox{ is n-stable}
\end{array}
\right\}.
$$
If $T$ has at most one seam, then $\mathop{\rm roof}\nolimits(T)=\lub(T)$. Since
$\mathop{\rm roof}\nolimits_{i+1}(J)\setminus (i\sh+1)$ has at most one seam, property
2 above can be modified:
\begin{itemize}
\item[4.] $\mathop{\rm roof}\nolimits_i(J)=\lub(\,\mathop{\rm roof}\nolimits_{i+1}(J)\setminus (i\sh+1)\,)$ \ if\,
$i\sh+1\not\in J$.
\end{itemize}
For $k\in\mathbb{Z}_{\geq 0}$, let $r[k]:=r+kn$. We will prove the following eight
statements $\bA_k$---$\bH_k$ together by decreasing
induction on $k$. Then we will show that the Roof Lemma is a
consequence of statement $\bC_0$ (i.e., $\bC_k$ for $k=0$).
\begin{itemize}
\item[$(\bA_k)$] Either $\mathop{\rm roof}\nolimits_{r[k]}(K)=\mathop{\rm roof}\nolimits_{r[k]}(J)$ or
$\mathop{\rm roof}\nolimits_{r[k]}(K)=s_{r-1}\mathop{\rm roof}\nolimits_{r[k]}(J)$.
\item[$(\bB_k)$] $\mathop{\rm roof}\nolimits_{r[k]}(K)\stackrel{\rm B}{\leq}\mathop{\rm roof}\nolimits_{r[k]}(J)$.\\[-.5em]
\item[$(\bC_k)$] If $r[k]\in J\setminus K$, $r[k]-1\in K\setminus J$,
then $\mathop{\rm roof}\nolimits_{r[k]-2}(K)=s_{r-1}\mathop{\rm roof}\nolimits_{r[k]-2}(J)$.\\[-.5em]
\item[$(\bD_k)$] If $r[k]\in J\cap K$, but $r[k]-1\not\in J$ or $K$,
then $\mathop{\rm roof}\nolimits_{r[k]-2}(K)=\mathop{\rm roof}\nolimits_{r[k]-2}(J)$.\\[-.5em]
\item[$(\bE_k)$] If $r[k], r[k]-1\not\in J\cup K$, then
$\mathop{\rm roof}\nolimits_{r[k]-2}(K)=\mathop{\rm roof}\nolimits_{r[k]-2}(J)$.\\[-.5em]
\item[$(\bF_k)$] If $r[k]-1\in J\cap K$, but $r[k]\not\in J$ or $K$,
then $\mathop{\rm roof}\nolimits_{r[k]-2}(K)=\mathop{\rm roof}\nolimits_{r[k]-2}(J)$.\\[-.5em]
\item[$(\bG_k)$] Either $\mathop{\rm roof}\nolimits_{r[k]-2}(K)=\mathop{\rm roof}\nolimits_{r[k]-2}(J)$ or
$\mathop{\rm roof}\nolimits_{r[k]-2}(K)=s_{r-1}\mathop{\rm roof}\nolimits_{r[k]-2}(J)$.
\item[$(\bH_k)$] $\mathop{\rm roof}\nolimits_{r[k]-2}(K)\stackrel{\rm B}{\leq}\mathop{\rm roof}\nolimits_{r[k]-2}(J)$.\\[-.5em]
\end{itemize}
Our induction proof will establish
the following implications:
\[
(\bA_{k+1}\sh-\bF_{k+1})\ \Rightarrow\ (\bG_{k+1},\bH_{k+1})
\ \Rightarrow\ (\bA_{k},\bB_k)\ \Rightarrow\ (\bC_k,\bD_k,\bE_k)
\]
\[
(\bD_m, \bE_m, \bF_m\!:\,m\sh>k\,)\ \Rightarrow\ (\bF_k)
\]
For the starting point of induction, select $k$ large enough so
that $r[k-2]>j_0$. For such $k$, by property 3,
$\mathop{\rm roof}\nolimits_i(K)=\mathop{\rm roof}\nolimits_i(L)=L_i$ for $i=r[k\sh-2],r[k\sh-1],r[k]$. Thus
$\bA_k\sh-\bH_k$ are trivially true.
\\[1em]
$(\bA_{k+1}\sh-\bF_{k+1})\ \Rightarrow\ (\bG_{k+1}, \bH_{k+1})$ :
\\[1em]
Let us restate this as:
$(\bA_{k}\sh-\bF_{k})\Rightarrow(\bG_{k}, \bH_{k})$. If any of the
hypotheses of $\bC_k-\bF_k$ are satisfied, then $\bC_k-\bF_k$
imply $\bG_k$ and $\bH_k$. There are two
possibilities omitted from the hypotheses of
$\bC_k-\bF_k$:
\begin{itemize}
\item[(i)] $r[k]-1\in J\setminus K$, $r[k]\in K\setminus J$, and
\item[(ii)] $r[k]-1, r[k]\in J\cap K$.
\end{itemize}
However, (i) cannot occur, since $K$ is obtained from $J$ by
applying the raising operator $e_{r-1}$ several times. If (ii)
occurs, then by property 1, $\mathop{\rm roof}\nolimits_{r[k-2]}(J)=\mathop{\rm roof}\nolimits_{r[k]}(J)$,
and $\mathop{\rm roof}\nolimits_{r[k-2]}(K)=\mathop{\rm roof}\nolimits_{r[k]}(K)$. Thus, in this case as
well, $\bG_k$ and $\bH_k$ follow immediately from $\bA_k$ and
$\bB_k$.
\\[1em]
$(\bG_{k+1}, \bH_{k+1})\Rightarrow (\bA_k,\bB_k)$ :
\\[1em]
Define $t$ by
$\mathop{\rm roof}\nolimits_{r[k]}(J)=\mathop{\rm up}\nolimits^t(\mathop{\rm roof}\nolimits_{r[k+1]-2}(J))$. Then necessarily
$\mathop{\rm roof}\nolimits_{r[k]}(K)=\mathop{\rm up}\nolimits^t(\mathop{\rm roof}\nolimits_{r[k+1]-2}(K))$. Letting
$T=\mathop{\rm roof}\nolimits_{r[k+1]-2}(J)$, $U=\mathop{\rm roof}\nolimits_{r[k+1]-2}(K)$, we show that
\begin{eqnarray}\label{eqn_TU}
&&\hbox{Either }\ \mathop{\rm up}\nolimits^i(U)=\mathop{\rm up}\nolimits^i(T)\ \hbox{ or
}\ \mathop{\rm up}\nolimits^i(U)=s_{r-1}\mathop{\rm up}\nolimits^i(T),\ \hbox{ and}\\
&&\mathop{\rm up}\nolimits^i(U)\stackrel{\rm B}{\leq}\mathop{\rm up}\nolimits^i(T)
\end{eqnarray}
$0\leq i\leq t$, by induction on $i$; the
result is then obtained by setting $i=t$.
\\[1em]
We have $\mathop{\rm up}\nolimits^{0}(T):=T$, $\mathop{\rm up}\nolimits^{0}(U):=U$; thus (1) and (2) hold
for $i=0$. Let $0<i\leq t$, and assume that (1) and (2) hold for
$i-1$. Then either
\\[.5em]
(i) $\mathop{\rm up}\nolimits^{i-1}(U)=\mathop{\rm up}\nolimits^{i-1}(T)$, in which case
$$\mathop{\rm up}\nolimits^i(U)=\mathop{\rm up}\nolimits(\mathop{\rm up}\nolimits^{i-1}(U))=\mathop{\rm up}\nolimits(\mathop{\rm up}\nolimits^{i-1}(T))=\mathop{\rm up}\nolimits^{i}(T),\hbox{ or }$$
(ii) $\mathop{\rm up}\nolimits^{i-1}(U)=s_{r-1}\mathop{\rm up}\nolimits^{i-1}(T)$ and $\mathop{\rm up}\nolimits^{i-1}(U)\stackrel{\rm B}{\leq}
\mathop{\rm up}\nolimits^{i-1}(T)$. In this case, define $p, q$ by
$$\mathop{\rm up}\nolimits^i(T)=\mathop{\rm up}\nolimits^{i-1}
(T)\setminus p\cup q.$$ Then it is easy to see that
$\mathop{\rm up}\nolimits^i(U)=\mathop{\rm up}\nolimits^{i-1}(U)\setminus p\cup q'$, where
\[
q'=
\begin{cases}
q, &\hbox{ if }q\not\equiv r-1, r\mod n\\
q+1, &\hbox{ if }q\equiv r-1\mod n\\
q-1, &\hbox{ if }q\equiv r\mod n\\
\end{cases}.
\]
Thus $\mathop{\rm up}\nolimits^i(U)=s_{r-1}\mathop{\rm up}\nolimits^i(T)$ and $\mathop{\rm up}\nolimits^{i}(U)\stackrel{\rm B}{\leq} \mathop{\rm up}\nolimits^{i}(T)$.
This proves (1) and (2).
\\[1em]
$(\bD_m, \bE_m, \bF_m: m>k) \Rightarrow(\bF_k)$ :
\\[1em]
Let $m>k$ be the minimum integer such that not both $r[m]-1$ and
$r[m]$ are in $K$. Then, by the definition of the raising operator
$e_{r-1}$, it is not possible that $r[m]\in J\setminus K$,
$r[m]-1\in K\setminus J$. Thus either the hypotheses of $\bD_m$,
$\bE_m$, or $\bF_m$ must hold. Thus
$\mathop{\rm roof}\nolimits_{r[m]-2}(J)=\mathop{\rm roof}\nolimits_{r[m]-2}(K)$. Since also
$$
_{r[k]-2\leq}J_{\leq r[m]-2}\ =\ _{r[k]-2\leq }K_{\leq r[m]-2}\,,$$
we have
$\mathop{\rm roof}\nolimits_{x}(J)=\mathop{\rm roof}\nolimits_{x}(K)$, for $r[k]-2\leq x\leq r[m]-2$.
\\[1em]
$(\bA_k,\bB_k)\Rightarrow(\bD_k)$ :
\\[1em]
By Property 1, $\mathop{\rm roof}\nolimits_{r[k]-1}(J)=\mathop{\rm roof}\nolimits_{r[k]}(J)$ and
$\mathop{\rm roof}\nolimits_{r[k]-1}(K)=\mathop{\rm roof}\nolimits_{r[k]}(K)$. Now $\bA_k$, $\bB_k$ imply that
if $J'\in\mathcal{C}(L_m)$ is $n$-stable and $J'\stackrel{\rm B}{\geq}
\mathop{\rm roof}\nolimits_{r[k]-1}(K)\setminus \{r[k]-1\}$, then $J'\stackrel{\rm B}{\geq}
\mathop{\rm roof}\nolimits_{r[k]-1}(J)\setminus \{r[k]-1\}$ (the same statement with $J$
and $K$ switched holds obviously). The result follows from
Property 4.
\\[1em]
$(\bA_k,\bB_k)\Rightarrow(\bC_k)$ :
\\[1em]
Let
$J'=\mathop{\max}_{\stackrel{\rm B}{\geq}}\{\mathop{\rm roof}\nolimits_{r[k]}(J),s_{r-1}\mathop{\rm roof}\nolimits_{r[k]}(J)\}$. By
Property 1, $\mathop{\rm roof}\nolimits_{r[k]-1}(J)=\mathop{\rm roof}\nolimits_{r[k]}(J)$. If $J'\in\mathcal{T}$ is
$n$-stable and $J'\stackrel{\rm B}{\geq} \mathop{\rm roof}\nolimits_{r[k]-1}(J)\setminus \{r[k]-1\}$, then
$J'\stackrel{\rm B}{\geq} \hat{J}\setminus \{r[k]-1\}$; conversely, if $J'\in\mathcal{T}$ is
$n$-stable and $J'\stackrel{\rm B}{\geq} \hat{J}\setminus \{r[k]-1\}$, then $J'\stackrel{\rm B}{\geq}
\mathop{\rm roof}\nolimits_{r[k]-1}(J)\setminus \{r[k]-1\}$. Thus
$\mathop{\rm roof}\nolimits_{r[k]-2}(\hat{J})=\mathop{\rm roof}\nolimits_{r[k]-2}(J)$.
We claim that $K\stackrel{\rm B}{\leq} s_{r-1}K$. Indeed, let $m>k$ be the
minimum integer greater than $k$ such that not both $r[m]-1,r[m]\in
K$. Then by the definition of the raising operator $e_{r-1}$, it
is not possible that $r[m]\in K$. Thus $\mathop{\rm roof}\nolimits_{r[m]-2}(K)\stackrel{\rm B}{\leq}
s_{r-1}\mathop{\rm roof}\nolimits_{r[m]-2}(K)$. Define $t'$ by
$\mathop{\rm roof}\nolimits_{r[k]}(K)=\mathop{\rm up}\nolimits^{t'}(\mathop{\rm roof}\nolimits_{r[m]-2}(K))$. Then it is easy to
see that $\mathop{\rm up}\nolimits^{i}(\mathop{\rm roof}\nolimits_{r[m]-2}(K))\stackrel{\rm B}{\leq}
s_{r-1}\mathop{\rm up}\nolimits^{i}(\mathop{\rm roof}\nolimits_{r[m]-2}(K))$, $1\leq i\leq t'$. This proves
the claim.
Define $t$ by $\mathop{\rm roof}\nolimits_{r[k]-2}(\hat{J})=\mathop{\rm up}\nolimits^t(\hat{J})$. Then by
Property 1,
$\mathop{\rm roof}\nolimits_{r[k]-2}(K)=\mathop{\rm roof}\nolimits_{r[k]-1}(K)=\mathop{\rm up}\nolimits^t(\mathop{\rm roof}\nolimits_{r[k]}(K))$. For
$1\leq i\leq t$, let
$$\mathop{\rm up}\nolimits^i(\hat{J})=\mathop{\rm up}\nolimits^{i-1}
(\hat{J})\setminus p\cup q.$$ Then
$\mathop{\rm up}\nolimits^i(\mathop{\rm roof}\nolimits_{r[k]}(K))=\mathop{\rm up}\nolimits^{i-1}(\mathop{\rm roof}\nolimits_{r[k]}(K))\setminus p\cup
q'$, where
\[
q'=
\begin{cases}
q, &\hbox{ if }q\not\equiv r \mod n\\
q-1, &\hbox{ if }q\equiv r\mod n\\
\end{cases}.
\] The result follows from this.
\\[1em]
$(\bA_k,\bB_k)\Rightarrow(\bE_k)$ :
\\[1em]
We have that
$\mathop{\rm roof}\nolimits_{r[k]-2}(J)=\mathop{\rm roof}\nolimits(\mathop{\rm roof}\nolimits_{r[k]}(J)\setminus\{r[k]-1,r[k]\})$,
$\mathop{\rm roof}\nolimits_{r[k]-2}(K)=\mathop{\rm roof}\nolimits(\mathop{\rm roof}\nolimits_{r[k]}(K)\setminus\{r[k]-1,r[k]\})$. Let
$L=\mathop{\rm roof}\nolimits_{r[k]}(J)\setminus\{r[k]-1,r[k]\}$. If $L$ has only one seam, then
the result is obvious. Thus assume
that $L$ has two seams:
\begin{eqnarray*}
S_1&=&\{r[k]=p_1<\cdots<p_t\}\\
S_2&=&\{r[k]-1=p_{t+1}<\cdots<p_{t+s}\}
\end{eqnarray*}
where $p_{i+1}=p_i+n$, $1\leq i\leq t+s-1$, $i\neq t$. Then
$\mathop{\rm up}\nolimits^t(L)$ has exactly one seam, namely $S_2$ with possibly some
additional elements added to its tight end. Thus
$\mathop{\rm roof}\nolimits_{r[k]-2}(J)=\mathop{\rm roof}\nolimits(\mathop{\rm up}\nolimits^t(L))=\lub(\mathop{\rm up}\nolimits^t(L))$. We claim that
$\lub(\mathop{\rm up}\nolimits^t(L))=\lub(L)$. The claim implies
$\mathop{\rm roof}\nolimits_{r[k]-2}(J)=\lub(L)=\lub(\mathop{\rm roof}\nolimits_{r[k]}(J)\setminus\{r[k]-1,r[k]\})$;
replacing $J$ with $K$, in precisely the same manner we show that
$\mathop{\rm roof}\nolimits_{r[k]-2}(K)=\lub(\mathop{\rm roof}\nolimits_{r[k]}(K)\setminus\{r[k]-1,r[k]\})$. But
it is clear that $\lub(\mathop{\rm roof}\nolimits_{r[k]}(J)\setminus\{r[k]-1,r[k]\})=
\lub(\mathop{\rm roof}\nolimits_{r[k]}(K)\setminus\{r[k]-1,r[k]\})$. Thus the result
follows from the claim.
To prove the claim, note that since $\mathop{\rm up}\nolimits^t(L)\stackrel{\rm B}{\geq} L$,
$\lub(\mathop{\rm up}\nolimits^t(L))\stackrel{\rm lex}{\geq}\lub(L)$. It suffices to show that
$\lub(L)\stackrel{\rm B}{\geq} \mathop{\rm up}\nolimits^t(L)$, since this implies
$\lub(L)\stackrel{\rm lex}{\geq}\lub(\mathop{\rm up}\nolimits^t(L))$. We show something slightly
stronger: if $M\stackrel{\rm B}{\geq} L$ is $n$-stable, then $M\stackrel{\rm B}{\geq} \mathop{\rm up}\nolimits^t(L)$. We
can express $M=L\setminus\{p_1,\ldots,p_{t+s}\}\cup
\{p_1,\ldots,p_{t+s}\}$, where $x_i> p_i$, $i=1,\ldots,t+s$.
Likewise, $\mathop{\rm up}\nolimits^t(L)=L\setminus\{p_1,\ldots,p_{t}\}\cup
\{q_1,\ldots,q_{t}\}$, where $\mathop{\rm up}\nolimits^i(L)=\mathop{\rm up}\nolimits^{i-1}(L)\setminus
p_i\cup q_i$. To show $M\stackrel{\rm B}{\geq} \mathop{\rm up}\nolimits^t(L)$, it suffices to show that
$q_i\leq x_i$, $i=1,\ldots,t$. This is clear from the definition
of the $\mathop{\rm up}\nolimits$ operation. Indeed, let $i_{\max}$ be the largest $i$
for which $q_i-p_i> n$. There are no tight ends in $\mathop{\rm up}\nolimits^t(L)$
between $r[k]$ and $q_{i_{\max}}$; thus $q_i\leq x_i$, $1\leq i\leq
i_{\max}$. If $i_{\max}<t$, then for $i_{\max}<i\leq t$,
$$q_i=\min\{q\not\in L\mid q-p_i\leq n-1,\,
q-n\in\mathop{\rm up}\nolimits^{i-1}(L),\, q\not\equiv r\mod n,\}.$$ Inductively, this
implies that $q_i\leq x_i$.
\\[2em]
This completes the proof of $\bA_k-\bH_k$. Noting that $r[0]=r$, we see that
$\bC_0$ implies $\mathop{\rm roof}\nolimits_{r-2}(K)=s_{r-1}\mathop{\rm roof}\nolimits_{r-2}(J)$. By
property 3, $\mathop{\rm roof}\nolimits(J)=\mathop{\rm roof}\nolimits_{a(J)}(J)$ and
$\mathop{\rm roof}\nolimits(K)=\mathop{\rm roof}\nolimits_{a(J)}(K)$. Using identical arguments as in the
proof of $(\bG_{k+1}, \bH_{k+1})\Rightarrow (\bA_k,\bB_k)$, we see
that $\mathop{\rm roof}\nolimits_{a(J)}(K)=s_{r-1}\mathop{\rm roof}\nolimits_{a(J)}(J)$, which completes the
proof of the Roof Lemma. \hfill$\Box$
\section{Proof of Proposition 3}
\subsection{Proof of Proposition 3(i)}
The result states that:
$$
v_J=\sum_K a_K^J\, \epsilon_K\,,
$$
where the sum runs over $K\stackrel{\rm lex}{\leq} J$.
We use induction on $\ell$, where $\mathop{\rm roof}\nolimits(J)=\mathop{\rm up}\nolimits^\ell(J)$. If $\ell=0$, then $v_J=\epsilon_J$ and there is nothing to prove.
Now let $\ell>0$. We inductively apply the Proposition to
$
{\widehat{J}} :=\mathop{\rm up}\nolimits(J)=J\setminus p\cup q\,,
$\ \ so that:
$$
v_{\widehat{J}} = \sum_{\widehat{K}} a_{\widehat{K}}^{\widehat{J}} \,\epsilon_{\widehat{K}}\,,
$$
where the sum runs over ${\widehat{K}}\stackrel{\rm lex}{\leq}
{\widehat{J}}$. Thus:
$$
v_J={\widehat{E}}_{pq}\, v_{\widehat{J}}
=\sum_{\widehat{K}} \sum_{h\in\mathbb{Z}}
a_{\widehat{K}}^{\widehat{J}}\, E_{p+nh,q+nh}\,\epsilon_{{\widehat{K}}}\,.
$$
It suffices to show the following:
\\[1em]
{\bf Lemma}\quad {\it
Let ${\widehat{J}}:=\mathop{\rm up}\nolimits(J)=J\setminus p\cup q$. Consider any ${\widehat{K}}\stackrel{\rm lex}{\leq}{\widehat{J}}$ and any $h\in\mathbb{Z}$ such that:
$$
p':=p\sh+nh\not\in{\widehat{K}}\quad\text{and}\quad
q':=q\sh+nh\in{\widehat{K}}\,.
$$
Then we have:
$$
K:={\widehat{K}}\setminus q'\cup p'\ \stackrel{\rm lex}{\leq}\ J={\widehat{J}}\setminus q\cup p\,.
$$
}
\\
We prove the Lemma using several facts which follow easily from the definitions. Let $K,J\in\mathcal{C}(L_m)$.
For $J=\{\cdots\sh<j_{-1}\sh<j_0\}$ with $\mathop{\rm ord}\nolimits(J)=m$, recall that:\ \
$
\mathop{\rm height}\nolimits(J):=\sum_{i\leq 0}\,(j_i-i-m)\,.
$
\begin{enumerate}
\item If $a_K^J\neq 0$, then $\mathop{\rm height}\nolimits(K)=\mathop{\rm height}\nolimits(J)$.
\item If $a_K^J\neq 0$,\
then $|K^{\equiv i}_{>N}| = |J^{\equiv i}_{>N}|$ for all $N\ll0$.
\item If ${\widehat{J}}=\mathop{\rm up}\nolimits(J)=J\setminus p\cup q$,\
then ${\widehat{J}}^{\equiv q}\subset {\widehat{J}}_{\leq q}$.
\item
If $J_{>p}$ contains no loose ends of $J$ (that is, $j\sh-n\in J$ for all $j\in J_{>p}$),
\ \ and\ \
$|K^{\equiv i}_{\geq p}|=
|J^{\equiv i}_{\geq p}|$ for some $i$, then
$K^{\equiv i}_{\geq p} \stackrel{\rm B}{\geq}
J^{\equiv i}_{\geq p}$.
\end{enumerate}
Proceeding with the proof of the Lemma, suppose first that ${\widehat{K}}={\widehat{J}}$. By Fact 3 we must have $q'\leq q$, so $p'\leq p$ and clearly
$K={\widehat{J}}\setminus q'\cup p'\,\stackrel{\rm lex}{\leq}\,
{\widehat{J}}\setminus q\cup p =J$.
Now let ${\widehat{K}}\stackrel{\rm lex}{<}{\widehat{J}}$, and let ${\widehat{k}}$ be the {\it split point}, the value such that:
$$
{\widehat{k}}\in{\widehat{K}},\ \ {\widehat{k}}\not\in {\widehat{J}},
\quad\text{and}\quad
K_{<{\widehat{k}}}=J_{<{\widehat{k}}}.
$$
Case (a): ${\widehat{k}}<p$. Then: $$
{\widehat{k}}\in{\widehat{K}},\ \ {\widehat{k}}\not\in J
\quad\text{and}\quad {\widehat{K}}_{<{\widehat{k}}}={\widehat{J}}_{<{\widehat{k}}}=J_{<{\widehat{k}}}\,,
$$
so ${\widehat{K}}\stackrel{\rm lex}{\leq} J$. But clearly $K\stackrel{\rm lex}{\leq} {\widehat{K}}$, so $K\stackrel{\rm lex}{\leq} J$ as desired.
Case (b): $p\leq{\widehat{k}}$. If $p'\leq p$, then clearly
$K\stackrel{\rm lex}{\leq} J$ as desired.
On the other hand, suppose $p<p'$.
Then $K_{<p}={\widehat{K}}_{<p}={\widehat{J}}_{<p}=J_{<p}$ by the definition of ${\widehat{k}}$.
Furthermore, for all $i$ and some $N<p$ we have
$|K^{\equiv i}_{>N}|=|J^{\equiv i}_{>N}|$ by Fact 2,
and thus
$|K^{\equiv i}_{\geq p}|=|J^{\equiv i}_{\geq p}|$. By definition $J_{>p}$ contains no loose ends, so Fact 4 implies that $K^{\equiv i}_{\geq p}\stackrel{\rm B}{\geq} J^{\equiv i}_{\geq p}$ for all $i$. We also have $K_{<p}=J_{<p}$, so $K\stackrel{\rm B}{\geq} J$. If $K\stackrel{\rm B}{>}J$, then clearly $\mathop{\rm height}\nolimits(K)>\mathop{\rm height}\nolimits(J)$, contradicting Fact 1. We conclude that
$K=J$, and we are done.
This proves the Lemma, and hence Proposition 3(i).
\subsection{Proof of Proposition 3(ii)}
To derive the formula relating the leading coefficients $a_J^J$ and $a_{\widetilde J}^{\widetilde J}$, note first that
$$\begin{array}{rcl}
v_J&=&({\widehat{E}}_{p_1q_1}\cdots{\widehat{E}}_{p_tq_t})v_{\widetilde J}\\[.5em]
&=&\displaystyle\sum_K\sum_{h_1,\ldots,h_t\in\mathbb{Z}\!\!\!\!\!}\!\!\!\!
a_K^{\widetilde J}\, (E_{p_1+nh_1,q_1+nh_1}\cdots E_{p_t+nh_t,q_t+nh_t})\,\epsilon_K\\[1.5em]
&=&\displaystyle\sum_K\sum_{h_1,\ldots,h_t\in\mathbb{Z}\!\!\!\!\!}\!\!
\pm\,a_K^{\widetilde J}\ \epsilon_{K{\uparrow}(h_1,\ldots,h_t)}\ ,
\end{array}$$
where we use notation:
$$
K{\uparrow}(h_1,\ldots,h_t):=K\setminus\{q_1\sh+nh_1
,\ldots ,q_t\sh+nh_t\}\cup\{p_1\sh+nh_1,\cdots
p_t\sh+nh_t\}
$$
provided $q_i\sh+nh_i\in K$ and $p_i\sh+nh_i\not\in K$ for all $i\leq t$ ; otherwise $K{\uparrow}(h_1,\ldots,h_t)$ is undefined, and $\epsilon_{K{\uparrow}(h_1,\ldots,h_t)}:=0$.
\\[1em]
{\bf Lemma} {\it (i) If $J=K{\uparrow}(h_1,\ldots,h_t)$ for some $K$ with $a_K^{\widetilde J}\neq 0$, then $K={\widetilde J}$.
\\
(ii) If $J={\widetilde J}{\uparrow}(h_1,\ldots,h_t)$, then there is a unique permutation $\sigma$ of $\{1,2,\ldots,r\}$ such that
$$
p_i\sh+nh_i=p_{\sigma(i)}\,,\qquad q_i\sh+nh_i=q_{\sigma(i)}\,.
$$
We obtain in this way every permutation $\sigma$ satisfying
$q_i-p_i=q_{\sigma(i)}-p_{\sigma(i)}$ for all $i\leq t$.
}
\\[1em]
The Proposition follows easily from (i) and (ii) of the Lemma, since:
$$\begin{array}{rcl}
v_J&=&\displaystyle\sum_K\!\!\!\!
\mathop{\sum_{h_1,\ldots,h_t\in\mathbb{Z}\!\!\!\!\!}}
_{K{\uparrow}(h_1,\ldots,h_t)=J}\!\!\!\!
a_K^{\widetilde J}\, (E_{p_1+nh_1,q_1+nh_1}\cdots E_{p_t+nh_t,q_t+nh_t})\,\epsilon_K
\\[3em]
&\stackrel{(i)}=&\left(a_{\widetilde J}^{\widetilde J}\!\!\!\!
\displaystyle
\mathop{\sum_{h_1,\ldots,h_t\in\mathbb{Z}\!\!\!\!\!}}
_{{\widetilde J}{\uparrow}(h_1,\ldots,h_t)=J}\!\!\!\!
(E_{p_1+nh_1,q_1+nh_1}\cdots E_{p_t+nh_t,q_t+nh_t})
\,\epsilon_{{\widetilde J}}\right)+
\text{lower}
\\[3em]
&\stackrel{(ii)}=&\left(a_{\widetilde J}^{\widetilde J}\,
\displaystyle\sum_{\sigma}
(E_{p_{\sigma(1)},q_{\sigma(1)}}\cdots E_{p_{\sigma(r)},q_{\sigma(r)}})
\,\epsilon_{{\widetilde J}}\right)+
\text{lower}\\[2em]
&\stackrel{(*)}=&\left(a_{\widetilde J}^{\widetilde J}\,
\displaystyle
\mathop{\sum_{\sigma}}
(E_{p_1q_1}\cdots E_{p_tq_t})
\,\epsilon_{{\widetilde J}}\right)+
\text{lower}\\[2em]
&=&\pm\, a_{\widetilde J}^{\widetilde J}\cdot\#\{\sigma\}\cdot\epsilon_J\ +\ \text{lower}\,,
\end{array}$$
where $\sigma$ runs over the set of all permutations of $\{1,\ldots,t\}$ such that $q_i-p_i=q_{\sigma(i)}-p_{\sigma(i)}$ for $i\leq t$ : clearly $\#\{\sigma\}=\prod_{d\geq 1} \mu_d!$ .
Equation $(*)$ holds because the operators $E_{p_iq_i}$ all commute for $i\leq t$. It remains to prove the Lemma.
\\[1em]
{\it Proof of Lemma (i).} Suppose
$J=K{\uparrow}(h_1,\ldots,h_t)={\widetilde J}{\uparrow}(0,\ldots,0)$ and $a_K^{{\widetilde J}}\neq 0$. Let
$$
p':=\min\{p_1\sh+nh_1,\ldots,p_t\sh+nh_t\}\,,\qquad
p:=p_1=\min\{p_1,\ldots,p_t\}\,.
$$
We clearly have $K_{<\min(p,p')}={\widetilde J}_{<\min(p,p')}$.
If $p'<p$, then $p'\not\in K$, $p'\in{\widetilde J}$, and $K_{<p'}={\widetilde J}_{<p'}$, so ${\widetilde J}\stackrel{\rm lex}{<} K$, which contradicts Proposition 3(i).
Thus $p\leq p'$, and $K_{<p}={\widetilde J}_{<p}$. Furthermore, by Fact 2 in the proof of Prop. 3(i), for any $i\leq t$ we have $|K^{\equiv i}_{>N}|=|{\widetilde J}^{\equiv i}_{>N}|$ for some $N<p$. Hence for any $i$,\ \ $|K^{\equiv i}_{\geq p}|=|{\widetilde J}^{\equiv i}_{\geq p}|$ . Since ${\widetilde J}_{>p}$ clearly has no loose ends, Fact 4 implies $K^{\equiv i}_{\geq p}\stackrel{\rm B}{\geq}{\widetilde J}^{\equiv i}_{\geq p}$ for any $i$, and we also know $K_{<p}={\widetilde J}_{<p}$. We conclude that $K\stackrel{\rm B}{\geq}{\widetilde J}$, and a fortiori $K\stackrel{\rm lex}{\geq}{\widetilde J}$. Since $K\stackrel{\rm lex}{\leq}{\widetilde J}$ by Proposition 3(i), we must have $K={\widetilde J}$.
\\[1em]
{\it Proof of Lemma (ii).} Suppose
$J={\widetilde J}{\uparrow}(0,\ldots,0)={\widetilde J}{\uparrow}(h_1,\ldots,h_t)$. Define:
$$
p'_i:=p_i+nh_i\,,\qquad q'_i:=q_i+nh_i\,,\qquad
d_i:=q_i-p_i:=q'_i-p'_i$$
Then we have
$\{p_1,\ldots,p_t\}=\{p'_1,\ldots,p'_t\}$
and $\{q_1,\ldots,q_t\}=\{q'_1,\ldots,q'_t\}$,
so there exist permutations $\alpha,\beta$ of $\{1,\ldots,r\}$ such that:
$$
p_i = p'_{\alpha(i)}\,,\qquad
q_i = q'_{\beta(i)}\,.
$$
We will use the following facts:
\begin{enumerate}
\item We have $p_i=p+n(i-1)$ for $i=1,\ldots,r$,\, and also\ $q_1<q_2<\cdots<q_t$ .
This follows from the seam-pulling action of the up-operation.
\item If $i<j$ and $q_i\equiv q_j\mod n$, then
$d_i\geq d_j$ .
Indeed, for a fixed $k$, the set of all
$q_i\equiv k\mod n$ forms an arithmetic progression
$\{q,q\sh+n,\ldots\}$, whereas the corresponding set of $\{p_i\mid q_i\equiv k\}$ is a subset of the arithmetic progression $\{p_1, p_2,\ldots\}=\{p,p\sh+n,\ldots\}$. Hence, if $i<j$ and $q_i\equiv q_j\mod n$, then $p_j-p_i\geq q_j-q_i$, and so $d_i=q_i-p_i\geq q_j-p_j=d_j\,.$
\item $\alpha(i)=\beta(i)
\quad\Longleftrightarrow\quad
d_i=d_{\alpha(i)}
\quad\Longrightarrow\quad
q_i\equiv q_{\alpha(i)}\mod n\quad
$
If $\alpha(i)=\beta(i)$, then
$
d_i=q_i-p_i=q'_{\beta(i)}-p'_{\alpha(i)}
=q'_{\alpha(i)}-p'_{\alpha(i)}=d_{\alpha(i)}
\,,$
and also $q_i=q'_{\beta(i)}\equiv q_{\beta(i)}=q_{\alpha(i)}$.
\end{enumerate}
\smallskip
\noindent
Assume $\alpha\neq\beta$, and let $j$ be the smallest value such that $\alpha(j)\neq \beta(j)\,.$\
Then $j$ is minimal with $\beta^{-1}\alpha(j)\neq j$, and necessarily:
$$
\beta^{-1}\alpha(j)>j\quad\text{with}\quad
q_{\beta^{-1}\alpha(j)}=q'_{\alpha(j)}\equiv q_{\alpha(j)}\,.
$$
Consider the sequence:\
$
j,\, \alpha(j),\, \alpha^2(j),\, \alpha^3(j),\ldots\,.
$\ \
If $\alpha(j),\alpha^2(j),\cdots,\alpha^{c}(j)\sh<j$, then by the definition of $j$
and Fact 3 we have:
$$\begin{array}{c}
d_j\neq d_{\alpha(j)}=d_{\alpha\alpha(j)}=d_{\alpha\alpha\alpha(j)}=\cdots=d_{\alpha^{c+1}(j)}\\[.7em]
q_{\alpha(j)}\equiv q_{\alpha\alpha(j)}\equiv
q_{\alpha\alpha\alpha(j)}\equiv\cdots\equiv q_{\alpha^{c+1}(j)}\,.
\end{array}$$
But we eventually have $\alpha^{c+1}(j)=j$, so to avoid the contradiction $d_j\neq d_j$, there must exist some $k:=\alpha^{c+1}(j)$ such that:
$$
k>j\quad\text{with}\quad
d_k=d_{\alpha(j)}
\quad\text{and}\quad
q_k\equiv q_{\alpha(j)}\,.
$$
Case (a): $j<\beta^{-1}\alpha(j) \leq k$.
Then by Fact 2, we have $d_{\beta^{-1}\alpha(j)}\geq d_k=d_{\alpha(j)}$. But:
$$\begin{array}{rcl}
d_{\beta^{-1}\alpha(j)}&=&
q_{\beta^{-1}\alpha(j)}-p_{\beta^{-1}\alpha(j)}\\
&<&
q_{\beta^{-1}\alpha(j)}-p_{j}\\
&<&
q'_{\alpha(j)}-p'_{\alpha(j)}=d_{\alpha(j)}
\,,\end{array}$$
so this case is impossible.
Case (b): $j<k<\beta^{-1}\alpha(j)$.
Then by Fact 1, we have $p_j<p_k<q_k<q_{\beta^{-1}\alpha(j)}$.
But:
$$\begin{array}{rcl}
q_{\beta^{-1}\alpha(j)}
&=&q'_{\alpha(j)}\\
&=&p'_{\alpha(j)}+d_{\alpha(j)}\\
&=&p_j+d_{k}\\
&<&p_k+d_k\ =\ q_k\,.
\end{array}$$
Thus, this case is impossible also.
The above contradictions show that $\alpha=\beta$. Hence we have
$$
p+nh_i=p'_i=p_{\sigma(i)}\,,\qquad
q+nh_i=q'_i=q_{\sigma(i)}\,,
$$
where $\sigma=\alpha^{-1}=\beta^{-1}$, which is the first part of Lemma (ii).
To see the second part of Lemma (ii), suppose $p_i,q_i$ given and let $\sigma$ satisfy
$d_i=d_{\sigma(i)}$. Then define $h_i:=(p_{\sigma(i)}-p_i)/n$, so that
$p'_i:=p_i+nh_i=p_{\sigma(i)}$ and:
$$
q'_i:=q_i+nh_i=p'_i+d_i=p_{\sigma(i)}+d_{\sigma(i)}=q_{\sigma(i)}\,.
$$
Thus $\{p_1,\ldots,p_t\}=\{p'_1,\ldots,p'_t\}$
and $\{q_1,\ldots,q_t\}=\{q'_1,\ldots,q'_t\}$,
so $J={\widetilde J}{\uparrow}(h_1,\ldots,h_t)$, as desired.
This proves the Lemma, and hence Proposition 3(ii).
|
{
"timestamp": "2005-02-20T17:59:31",
"yymm": "0411",
"arxiv_id": "math/0411017",
"language": "en",
"url": "https://arxiv.org/abs/math/0411017"
}
|
\section{Introduction}
\label{sec:intro}
The clustering of galaxies provides a window not only to the formation
of inhomogeneities in the early universe but also onto the physics of
galaxy formation. Galaxies with different properties cluster differently
\citep{hubble36,zwicky68,davis76,dressler80,postman84,hamilton88,white88,park94,loveday95,guzzo97,benoist96,Wil98,Bro00,Car01,Nor01,Zeh02,Nor02,Bud03,madgwick03,Hog03,Zeh04b},
and these trends can be connected to their
small-scale environments, notably the masses of their host dark matter
halos ({\it e.g.}, \citealt{Kai84,BBKS,Mo96,Ben00,She01,Ber03}). This path has
been strengthened recently by the discovery of
deviations from the canonical power-law correlation function on small
scales ({\it e.g.}, \citealt{Zeh04a,Zhe04}) and the ease of interpretation of these
features by contemporary models of galaxy and halo clustering, in terms of
the clustering of galaxies within single halos and the clustering between halos
\citep{Kau97,jing98,Kau99,Ben00,Ma00,Pea00,Sel00,Sco01,Ber02,Ber03,Mag03,Kra04,Zeh04a,Zhe04}.
The Sloan Digital Sky Survey (SDSS;\citealt{Yor00}) was designed in scope
and systematic control to permit the study of galaxy clustering over a wide
range of scales and galaxy properties
({\it e.g.}, \citealt{Con02,Zeh02,Bud03,Hog03,Teg04,Zeh04b};
all using the SDSS main galaxy sample).
To improve the precision of clustering measurements on the largest scales,
the SDSS provides a spectroscopic sample of luminous red galaxies (LRG).
These galaxies reach a redshift of 0.5, thereby providing a sample of over
$1\ensuremath{h^{-3}{\rm\,Gpc^3}}$ (see \citealt{Eis01}).
Thus far, over 50,000 spectra of LRGs have been acquired.
In this paper, we will investigate the clustering of these luminous early-type
galaxies on scales between $0.3$ and $40\ensuremath{h^{-1}{\rm\,Mpc}}$. This stretches from
the quasi-linear to the deeply non-linear regime. As massive early-type
galaxies are known to inhabit preferentially rich environments ({\it e.g.},
\citealt{San72,dressler80,Hoe80,Sch83,postman84,Pos95},M.\ Bernardi 2004, in
preparation), this
selection should permit one to study the clustering and internal
structure of massive halos. Models that differ in their association of
LRGs to cluster-sized halos or to the fraction in smaller halos will
vary not only in their predicted correlation length, but also in the
fine structure of the correlation functions.
With the sample size available
within the SDSS LRG sample, we expect to reach the precision necessary to
perform such tests despite the rarity of massive galaxies.
We note that the rapid increase in the clustering
of early-type galaxies at the highest luminosities \citep{Hog03,Zeh04b}
implies that the connections
between the most massive galaxies and their environments is notably
different than even $L_*$ early-types.
The outline of the paper is as follows.
In \S~\ref{sec:data} we present the LRG sample.
In \S~\ref{sec:sclustering} we present the clustering measurements in
redshift-space and in \S~\ref{sec:rclustering} we show the inferred
real-space clustering results. We conclude in \S~\ref{sec:discussion}.
Details of our sample modeling are given in the Appendix.
Throughout the paper, all distances are comoving and quoted in $\ensuremath{h^{-1}{\rm\,Mpc}}$,
where $h \equiv H_0/100 \kmsmpc$. For all distances and absolute magnitude
we use a cosmology of $\Omega_m=0.3$ and $\Lambda=0.7$ and adopt $h=1$ to
compute absolute magnitudes.
\section{Data}
\label{sec:data}
\subsection{The SDSS LRG Sample}
\label{subsec:data}
The SDSS \citep{Yor00} is imaging $10^4$ square degrees away from the Galactic Plane
in 5 passbands, $u$, $g$, $r$, $i$, and $z$ \citep{Fuk96,Gun98}.
Image processing \citep{Lup01,Sto02,Pie03} and calibration
\citep{Hog01,Smi02} allow one to select galaxies, quasars, and stars
for follow-up spectroscopy with twin fiber-fed double-spectrographs.
The spectra cover 3800\AA\
to 9200\AA\ with a resolution of 1800. Targets are assigned to plug
plates with a tiling algorithm that ensures nearly complete samples
\citep{Bla03a}. An operational constraint of using fibers to obtain
spectra is that no two fibers can be closer than $55''$ on the same
plate. This constraint is partly alleviated by having roughly a third
of the sky covered by overlapping plates.
Galaxy spectroscopic target selection proceeds by two algorithms.
The primary sample \citep{Str02}, referred to here as the MAIN sample,
targets galaxies brighter than $r<17.77$. The surface density of such
galaxies is about 90 per square degree.
The LRG algorithm \citep{Eis01} then selects $\sim\!12$ additional galaxies
per square degree, using color-magnitude cuts in $g$, $r$, and $i$
to select galaxies to $r<19.5$ that are likely to be luminous early-types
at redshifts up to $\sim\!0.5$. The selection is extremely efficient,
and the redshift success rate is very high. A few galaxies (3 per square degree
at $z>0.15$) matching the rest-frame color and luminosity properties of the LRGs
are extracted from the MAIN sample; we refer to this combined set as the
LRG sample. In detail, there are two parts to the LRG algorithm, known
as Cut I and Cut II and described in \citet{Eis01}.
We begin from a spectroscopic sample covering 3,836 square degrees.
The exact survey geometry is expressed in terms of spherical polygons
and is known as {\tt lss\_sample14} (M. Blanton 2004, in preparation).
This set contains 55,000 spectroscopic LRGs in the redshift range
$0.15<z<0.55$.
\subsection{Redshift and Magnitude Cuts}
\label{subsec:samples}
The SDSS LRG sample is nearly volume-limited, but not precisely so.
At $z>0.37$, the flux limits of $r<19.2$ (Cut I) and $r<19.5$ (Cut II)
begin to move into the passively-evolving luminosity threshold. In this
paper, we wish to analyze volume-limited samples,
so as to study the clustering properties of well defined populations of
galaxies.
We therefore define three subsamples in passively-evolved luminosity and
restrict the redshift ranges to ensure complete coverage. The subsamples
are $-23.2<M_g<-21.2$ with $0.16<z<0.36$, $-23.2<M_g<-21.8$ with $0.16<z<0.44$,
and $-22.6<M_g<-21.6$ with $0.16<z<0.36$.
The first of these sets is picked to maximize our use of the LRG spectroscopy
for the innately volume-limited portion of the sample. The second is
selected because the Cut II selection creates a knee in the number densities
as a function of redshift that we can exploit. The third is chosen to match
the luminosity range of the $-23<M_r<-22$ volume-limited MAIN galaxy sample
described in \citet{Zeh04b}. We use here only the red subsample of the latter
(as defined in \citealt{Zeh04b}, with $0.10<z<0.23$) to compare to the LRG
clustering. These LRG and MAIN samples overlap nearly completely in the
redshift range in common, $0.16<z<0.23$.
The basic information regarding the three LRG samples is summarized in
Table~\ref{tab:samples}, and their comoving number density as a function
of redshift is shown in Figures~\ref{fig:nz210}-\ref{fig:nz214}.
\begin{table*}[tb]\footnotesize
\caption{\label{tab:samples}}
\begin{center}
{\sc LRG Sample Statistics\\}
\begin{tabular}{cccccc}
\tableskip\hline\hline\tableskip
$M_g$\tablenotemark{a} & Redshift & Number & Density\tablenotemark{b} & $\left<M_g\right>$\tablenotemark{c} & $\left<z\right>$\tablenotemark{d} \\
\tableskip\hline\tableskip
$-23.2<M_g<-21.2$ & $0.16<z<0.36$ & 29298 & $9.7\times10^{-5}$
& -21.63 & 0.28 \\
$-23.2<M_g<-21.8$ & $0.16<z<0.44$ & 12992 & $2.4\times10^{-5}$
& -22.01 & 0.34 \\
$-22.6<M_g<-21.6$ & $0.16<z<0.36$ & 14500 & $4.8\times10^{-5}$
& -21.84 & 0.28 \\
\tableskip\hline\tableskip
\end{tabular}
\end{center}
\tablenotetext{a}{Rest-frame $g$-band absolute magnitudes, passively evolved
to $z=0.3$.}
\tablenotetext{b}{Average comoving densities are in units of $\ensuremath{h^3 {\rm\,Mpc}^{-3}}$.}
\tablenotetext{c}{Average rest-frame $g$-band absolute magnitude, $M_g$}
\tablenotetext{d}{Average redshift}
\end{table*}
\begin{figure}[tb]
\plotone{nz_210_230.eps}
\caption{\label{fig:nz210}
The comoving number density of the $-23.2<M_g<-21.2$ sample. The shaded
histogram is the distribution of the actual data, and the solid continuous
line is our model for the redshift distribution, described in Appendix A.
The sample is close to a constant comoving volume for $z<0.36$, although the
fluctuations are reaching about 30\% peak-to-peak (but one should note that the
lowest redshifts, where the excess is, contain less volume than the redshift
range would suggest).
}
\end{figure}
\begin{figure}[tb]
\plotone{nz_216_230.eps}
\caption{\label{fig:nz216}
As Figure \protect\ref{fig:nz210}, but for the $-23.2<M_g<-21.8$ sample.
The sample is close to a constant comoving volume for $z<0.44$.
}
\end{figure}
\begin{figure}[tb]
\plotone{nz_214_224.eps}
\caption{\label{fig:nz214}
As Figure \protect\ref{fig:nz210}, but for the $-22.6<M_g<-21.6$ sample.
The sample is close to a constant comoving volume for $z<0.36$.
}
\end{figure}
The above luminosities have been $k$-corrected and passively evolved
to rest-frame magnitudes at $z=0.3$ (near the median redshift of the
LRG sample). We use the observed $r$-band to estimate rest-frame $g$,
as this requires minimal $k$-corrections at $z=0.3$.
We have used the ``non-star-forming'' model presented in Appendix B of \citet{Eis01}
but normalized to $M_g$ at $z=0.3$.
The model has relatively mild evolution, only about 1 magnitude per unit redshift,
compared to other measurements \citep{Bla03c}.
Additional details of the samples' modeling are given in Appendix A.
To the extent that our model is appropriate, our selections represent mass
thresholds throughout the sample volume. However, as shown in Figure
\ref{fig:nz210}, the $-23.2<M_g<-21.2$ sample still
has some redshift evolution in the number density.
This is due to small fluctuations in the selection thresholds of the parent sample
in luminosity and rest-frame color as a function of redshift.
The other two samples, being safely more luminous than the LRG sample
selection limits, are much closer
to a constant comoving threshold (Figs.\ \ref{fig:nz216} and \ref{fig:nz214}).
\section{Redshift-Space Clustering}
\label{sec:sclustering}
We calculate the LRG correlation function in redshift space as a
function of the redshift-space separation $s$. To estimate the mean
density and account for the complex survey geometry, we generate random
catalogs, applying the
radial and angular selection functions of the samples. The details of
the radial and angular modeling are given in the Appendix.
We typically use in each random catalog 100-150 times the
number of galaxies in the real sample, and we have verified that changing
the random catalog makes negligible difference to the results.
We estimate the correlation function using the \citet{Lan93} estimator
\begin{equation}
\xi=\frac{DD-2DR+RR}{RR} ,
\label{eq:LS}
\end{equation}
where DD, DR and RR are the suitably normalized numbers of weighted
data-data, data-random and random-random pairs in each separation
bin. We weight the galaxies (real and random) according to the angular
and radial selection functions.
We use a simple weighting by the inverse of the selection function, as
the samples we use are all approximately volume-limited, and we have
verified that our results are insensitive to employing alternative
weighting schemes. We also used the alternative $\xi$ estimators of
\citet{Dav83} and \citet{Ham93} and found no significant differences
in the results.
Here, and throughout the paper,
we estimate statistical errors on our measurements
using jackknife resampling with 104 angular subsamples. Each subsample
excludes roughly 37 square degrees (generally contiguous on the sky),
which is about $90\ensuremath{h^{-1}{\rm\,Mpc}}$ comoving on a side at $z=0.3$.
The 2.5 degree SDSS stripes are $36\ensuremath{h^{-1}{\rm\,Mpc}}$ comoving at $z=0.3$.
\citet{Zeh04b} performed extensive tests with mock catalogs to check the
reliability of the jackknife error estimates over a similar range of
separations (see their Fig.~2). Their tests showed that the jackknife
method is a robust way to estimate the error covariance matrix, especially
for large volumes as probed here.
Figure~\ref{fig:xsis} shows the redshift-space correlation function,
$\xi(s)$,
for the $-23.2<M_g<-21.2$ and $-23.2<M_g<-21.8$ LRG samples introduced in
\S~\ref{subsec:samples}, with errorbars obtained from the jackknife
resampling.
The small difference in amplitude arises from the difference in the average
luminosity of the galaxies in the samples (see Table~\ref{tab:samples}),
reflecting the known trend of stronger clustering with luminosity
({\it e.g.}, \citealt{hamilton88,park94,loveday95,benoist96,guzzo97,Nor01,Zeh02,Hog03,Zeh04b}).
The redshift-space correlation functions values are given in
Table~\ref{tab:results}.
\begin{figure}[tb]
\plotone{xsisl2.ps}
\caption{\label{fig:xsis}
Redshift-space correlation function $\xi(s)$ for the LRG samples.
Bins in $s$ are in logarithmic separation of $0.2$.
}
\end{figure}
\begin{table*}[tb]\footnotesize
\caption{\label{tab:results}}
\begin{center}
{\sc Correlation Functions Measurements\\}
\begin{tabular}{cccccccccc}
\tableskip\hline\hline\tableskip
\multicolumn{1}{c}{ } & \multicolumn{3}{c}{$-23.2<M_g<-21.2$} & \multicolumn{3}{c}{$-23.2<M_g<-21.8$} & \multicolumn{3}{c}{$-22.6<M_g<-21.6$} \\
separation & $\xi(s)$ & ${w_p(r_p)}$ & $\xi(r)$ & $\xi(s)$ & ${w_p(r_p)}$ & $\xi(r)$ & $\xi(s)$ & ${w_p(r_p)}$ & $\xi(r)$ \\
\tableskip\hline\tableskip
0.418 & 87.4 (16.8) & 772.9 (52.7) & 675.7 (80.6) & 74.6 (44.0) & 1281 (243) & 1276 (399) & 39.6 (17.7) & 743.9 (86.8) & 484 (133) \\
0.663 & 37.7 (5.1) & 414.4 (26.0) & 210.3 (23.3) & 46.6 (17.5) & 575.2 (86.9) & 332.8 (84.8) & 30.3 (8.9) & 548.0 (66.8) & 316.9 (59.7) \\
1.051 & 25.1 (1.9) & 238.8 (13.4) & 66.9 (7.9) & 44.2 (8.3) & 286.8 (33.8) & 83.1 (21.8) & 30.9 (4.3) & 268.0 (29.0) & 75.5 (16.7) \\
1.665 & 18.5 (0.9) & 154.9 (7.5) & 24.9 (3.0) & 17.3 (2.3) & 178.5 (20.6) & 26.0 (7.8) & 19.4 (1.8) & 174.6 (14.1) & 30.8 (5.4) \\
2.639 & 11.0 (0.3) & 109.0 (5.1) & 11.7 (1.0) & 12.4 (1.0) & 132.4 (12.6) & 13.0 (3.2) & 12.1 (0.8) & 111.6 (9.0) & 10.1 (2.1) \\
4.182 & 6.32 (0.15) & 74.0 (3.8) & 5.04 (0.47) & 8.45 (0.45) & 95.5 (7.8) & 6.01 (1.11) & 7.35 (0.33) & 84.9 (7.1) & 5.58 (0.88) \\
6.628 & 2.99 (0.08) & 50.5 (2.6) & 2.32 (0.19) & 4.08 (0.19) & 70.8 (4.8) & 3.80 (0.50) & 3.52 (0.14) & 59.2 (4.1) & 2.62 (0.34) \\
10.505 & 1.28 (0.04) & 33.0 (2.2) & 1.04 (0.09) & 1.70 (0.08) & 38.8 (3.6) & 1.16 (0.19) & 1.52 (0.06) & 40.1 (3.1) & 1.25 (0.16) \\
16.650 & 0.54 (0.03) & 20.0 (1.8) & 0.43 (0.05) & 0.62 (0.03) & 25.2 (2.2) & 0.56 (0.09) & 0.59 (0.04) & 25.1 (2.5) & 0.57 (0.08) \\
26.388 & 0.19 (0.02) & 11.0 (1.4) & 0.14 (0.02) & 0.26 (0.02) & 13.2 (2.0) & 0.17 (0.06) & 0.23 (0.02) & 13.2 (1.8) & 0.20 (0.05) \\
\tableskip\hline\tableskip
\end{tabular}
\end{center}
NOTES.---%
Measurements of the redshift-space correlation function, $\xi(s)$, projected
correlation function, ${w_p(r_p)}$, and real-space correlation function, $\xi(r)$,
for the three LRG samples discussed in the paper.
Correlation functions are calculated for each sample over the range for
which it is approximately volume-limited, denoted in Table~\ref{tab:samples}.
Comoving separations and ${w_p(r_p)}$ values are in $\ensuremath{h^{-1}{\rm\,Mpc}}$ units. Redshift-space
$\xi(s)$ and real-space $\xi(r)$ are dimensionless. The diagonal terms
of the measurements error covariance matrices are given in parentheses.
Our radial bins are logarithmically spaced with widths of 0.2 dex
beginning at $10^{-0.49}$. The separations listed in column 1 are the
linear centers of the bins. Strictly speaking, the listed values of $\xi(s)$
and $w_p(r_p)$ are the averages of these correlation functions over the
annuli. However, for reasonable power-law interpolations, the values
of $w_p(r_p)$ and $\xi(r)$ are very nearly ($\ll1\%$) the values at the
linear bin centers. The interpolated value of $\xi(s)$ at the bin centers
would be about $1.5\%$ higher than the values in the table.
\end{table*}
Figure~\ref{fig:xsislg} shows $\xi(s)$ for the $-22.6<M_g<-21.6$ LRG sample
plotted together with $\xi(s)$ obtained for the comparable red
$-23<M_r<-22$ MAIN galaxy subsample of \citet{Zeh04b},
where we restrict the LRG sample to $0.23<z<0.36$, such that we are probing
independent volumes. The LRG
sample contains $\sim\!12400$ galaxies, while the MAIN galaxy sample
includes only $\sim\!2700$. As is obvious from the plot, the agreement
between the samples is excellent, and the LRG results thus extend in essence
the MAIN galaxy clustering results to higher redshifts. The deviations
at small separations are mainly due to shot noise effects arising from the
small number of galaxies in the MAIN sample and are consistent within
the errorbars. The numerical values of this LRG correlation function
are also provided in Table~\ref{tab:results}.
\begin{figure}[tb]
\plotone{xsislg.ps}
\caption{\label{fig:xsislg}
Redshift-space correlation function $\xi(s)$ for the $-22.6<M_g<-21.6$
passively-evolved LRG sample and the comparable one from the MAIN galaxy
sample.
}
\end{figure}
\pagebreak
\section{Real-Space Clustering}
\label{sec:rclustering}
\subsection{Projected Correlation Function}
\label{subsec:projected}
To separate effects of redshift distortions from spatial correlations,
it is customary to estimate the correlation function on a two dimensional
grid of pair separations parallel ($\pi$) and perpendicular ($r_p$) to
the line of sight, termed $\xi(r_p,\pi)$.
One can then learn about the real-space correlation function by
computing the projected correlation function
\begin{equation}
{w_p(r_p)} = 2 \int_0^{\infty} d\pi \, \xi(r_p,\pi).
\label{eq:wp}
\end{equation}
In practice, we integrate up to $\pi_{max}=80\ensuremath{h^{-1}{\rm\,Mpc}}$, which is large enough
to include most correlated pairs and gives a stable result.
The omission of pairs at $\pi>80\ensuremath{h^{-1}{\rm\,Mpc}}$ likely causes an overestimate
of $w_p(r_p)$ by 1--$2\ensuremath{h^{-1}{\rm\,Mpc}}$, as the correlations on such scales are
driven negative by redshift distortions. However, we have varied
$\pi_{max}$ from 50--120$\ensuremath{h^{-1}{\rm\,Mpc}}$ without significant change in $w_p$.
We also checked the robustness to binning in $r_p$ and in the integrated-over
$\pi$ direction, finding the results to be insensitive to either.
The projected correlation function can in turn be related to the
real-space correlation function, $\xi(r)$,
\begin{equation}
{w_p(r_p)}= 2 \int_{r_p}^{\infty} r dr\; \xi(r) (r^2-{r_p}^2)^{-1/2}
\label{eq:wp2}
\end{equation}
\citep{Dav83}.
In particular, fitting a power-law to the ${w_p(r_p)}$ measurement allows
us to infer the best-fit power law for $\xi(r)$.
Figure~\ref{fig:wpl} shows the resulting projected correlation function,
${w_p(r_p)}$, for the two inclusive LRG samples analyzed in this paper. Again,
the differences in amplitude reflect the luminosity bias between the samples.
\begin{figure}[tb]
\plotone{wpl2.ps}
\caption{\label{fig:wpl}
Projected correlation function ${w_p(r_p)}$ for the LRG samples.
}
\end{figure}
Figure~\ref{fig:covarn} shows the normalized (such that the diagonal
is 1) jackknife error covariance matrix of the ${w_p(r_p)}$ measurements
for the $-23.2<M_g<-21.2$ sample. As is apparent, there is
significant correlation between the measurements on different scales,
but the auto-correlation along the diagonal is relatively
strong with the cross-correlation falling off rapidly.
\begin{figure}[tb]
\plotone{covarm.ps}
\caption{\label{fig:covarn}
Normalized error covariance matrix for the ${w_p(r_p)}$ measurement of the
$-23.2<M_g<-21.2$ sample. The normalized covariance matrix is defined as
$C_{ij}/(C_{ii} \cdot C_{jj})^{1/2}$, where $C_{ij}$ are the elements of the
error covariance matrix. Contour spacing is 0.1 going from 1 on the
diagonal (thick line) down to 0. The dashed lined denotes the 0.5 contour.
Tickmarks denote the elements in the covariance
matrix, and the labels denote the corresponding $r_p$ values.
}
\end{figure}
Similar to the comparison to the MAIN galaxy sample results shown
in \S~\ref{sec:sclustering}, Figure~\ref{fig:wplg} compares the projected
correlation function of the analogous LRG and MAIN samples.
The small differences seen in the plot do not appear to be significant.
A $\chi^2$ statistic of the difference, performed with the sum of the error
covariance matrices of the two measurements, is $11.6$
for the $10$ degrees of freedom, consistent with cosmic variance.
\begin{figure}[tb]
\plotone{wplg.ps}
\caption{\label{fig:wplg}
Projected correlation function ${w_p(r_p)}$ for the $-22.6<M_g<-21.6$
LRG sample and the comparable one from the MAIN galaxy sample.
}
\end{figure}
Now that we have demonstrated that the LRG sample extends the MAIN
galaxy sample, it is interesting to compare the amplitude of the LRG
clustering to that of typical $L_*$ galaxies.
Figure~\ref{fig:bias} shows such a comparison for our largest
LRG sample ($-23.2<M_g<-21.2$) and the $L_*$ MAIN galaxy sample
(a volume-limited sample with $-21<M_r<-20$ containing $5700$ galaxies;
\citealt{Zeh04b}). Note that this is in contrast to the previous comparisons
(Fig.~\ref{fig:xsislg} and \ref{fig:wplg}), where the LRG and red MAIN
galaxy samples were chosen to match in luminosity and color.
The quantities plotted are $[{w_p(r_p)}/{w_p}^{fid}(r_p)]^{1/2}$, where
the fiducial ${w_p}^{fid}(r_p)$ corresponds to a power-law correlation
function $\xi(r)=(r/5\ensuremath{h^{-1}{\rm\,Mpc}})^{-1.8}$, and allow to infer the relative bias.
For illustration purposes, we also plot
this for a flat $\Lambda$CDM cosmology (with $\Omega_m=0.3$,
$h=0.7$, $n=1$ and $\sigma_8=0.9$) projected correlation function
computed from the nonlinear power spectrum of \citet{Smi03} (Z.\ Zheng,
private communication). The matter
correlation function is comparable in amplitude to the $L_*$ MAIN correlation
function, but distinct in detail. For the LRG galaxies, this scaled quantity
appears roughly scale-invariant for $1-10\ensuremath{h^{-1}{\rm\,Mpc}}$,
with a notable upturn on smaller scales and a downturn on large scales.
When fitting a constant bias factor between the two samples, taking into
account the error covariance matrices, one obtains
$b_{LRG}/b_* = 1.84 \pm 0.11$ when fitting over $1\ensuremath{h^{-1}{\rm\,Mpc}} < r_p \la 10 \ensuremath{h^{-1}{\rm\,Mpc}}$.
The scale dependence of the LRG inferred bias is in accord with the
steeper correlation functions associated with red galaxies ({\it e.g.},
\citealt{Wil98,Bro00,Zeh02,Zeh04b}). The downturn of the projected
correlation function from a power-law on large separations is similar to that
predicted by CDM models and to what is measured in the SDSS MAIN galaxy
sample and in the 2dF survey \citep{Haw03}.
\begin{figure}[tb]
\plotone{bias.ps}
\caption{\label{fig:bias}
$[{w_p(r_p)}/{w_p}^{fid}(r_p)]^{1/2}$ as a function of separation $r_p$ for the
$-23.2<M_g<-21.2$ LRG sample (solid symbols and line) and for the
$L_*$ MAIN galaxy sample (open symbols and short-dashed line;
\citealt{Zeh04b}), allowing to infer their relative bias.
${w_p}^{fid}(r_p)$ is the projected correlation function corresponding to
a fiducial power-law
$\xi(r)=(r/5\ensuremath{h^{-1}{\rm\,Mpc}})^{-1.8}$. The long-dashed curve shows this relative
quantity for a $\Lambda$CDM cosmology computed from the nonlinear power
spectrum of \citet{Smi03}.
}
\end{figure}
Real-space correlation functions have been historically well described
by power laws
\citep{Tot69,Pee74,Got79,Dav83,Fis94,jing98,jing02,Nor01,Zeh02},
although recent precision measurements
provide evidence for deviations from a power-law and a means of
explaining them ({\it e.g.}, \citealt{Ber03,Mag03,Mal03,Zeh04a,Zhe04}).
Figure~\ref{fig:wpl_pl} shows power-law fits to our
projected correlation functions. The inferred $\xi(r)$ power-law fits
are given in Table~\ref{tab:pl_fits}, while the ${w_p(r_p)}$ measurements themselves
are provided in Table~\ref{tab:results}. Inspection of the values of the
correlation length, $r_0$, show clearly the trend with luminosity. The
power-law slopes, $\gamma$, span the range 1.89 -- 1.94.
The $\chi^2/d.o.f$ values for the power-law fits are in the range $2.3-3.9$,
indicating that a power-law is not a good fit. (The confidence level of a
power-law fit is about 1.5\% in the best case and less than 0.1\% in the worst
case.)
The deviations from a power-law are clearly visible in
Figure~\ref{fig:wpl_relpl} where we divide the clustering measurements
by a representative power-law ${w_p(r_p)}$ corresponding to a $\xi(r)$ with
$r_0=10\ensuremath{h^{-1}{\rm\,Mpc}}$ and $\gamma=1.9$. Similar deviations are also seen in the
complementary analysis of the LRG samples by \citet{Eis04}.
These deviations appear to be of a similar nature to the deviations
detected in the MAIN galaxy samples \citep{Zeh04a,Zeh04b}, which are naturally
explained by contemporary models of galaxy clustering as the transition
from a small-scale regime dominated by galaxy pairs in the same dark matter
halo to a large-scale regime dominated by pairs of galaxies in separate halos.
We delay to future work
detailed halo modeling of this sort and interpretation of our measurements.
There is a hint from the brightest sample of an increase at
small scales ($<1\ensuremath{h^{-1}{\rm\,Mpc}}$) in the luminosity dependence of the bias,
in agreement with the findings of \citet{Eis04}.
\begin{figure}[tb]
\plotone{wpl_pl.ps}
\caption{\label{fig:wpl_pl}
Projected correlation function ${w_p(r_p)}$ for the three LRG samples
discussed in the paper, plotted together with power-law
fits fitted over the range $0.3<r_p<30\ensuremath{h^{-1}{\rm\,Mpc}}$.
}
\end{figure}
\begin{table}[tb]\footnotesize
\caption{\label{tab:pl_fits}}
\begin{center}
{\sc $\xi(r)$ Power Law Fits\\}
\begin{tabular}{cccc}
\tableskip\hline\hline\tableskip
$M_g$ & $r_0$ & $\gamma$ & $\chi^2/d.o.f.$ \\
\tableskip\hline\tableskip
$-23.2<M_g<-21.2$ & $9.80 \pm 0.20$ & $1.94 \pm 0.02$ & $3.9$ \\
$-23.2<M_g<-21.8$ & $11.21 \pm 0.24$ & $1.92 \pm 0.03$ & $3.1$ \\
$-22.6<M_g<-21.6$ & $10.59 \pm 0.29$ & $1.88 \pm 0.03$ & $2.3$ \\
\tableskip\hline\tableskip
\end{tabular}
\end{center}
NOTES.---%
$r_0$ and $\gamma$ are obtained from a fit to ${w_p(r_p)}$ using the full
error covariance matrix. The values of $r_0$ are given in $\ensuremath{h^{-1}{\rm\,Mpc}}$ units.
\end{table}
\begin{figure}[tb]
\plotone{wpl_relpl.ps}
\caption{\label{fig:wpl_relpl}
Projected correlation function ${w_p(r_p)}$ for the LRG samples
shown in Fig.~\ref{fig:wpl_pl}, now all divided out by a fiducial
power-law ${w_p(r_p)}$ corresponding to $\xi(r)=(r/10\ensuremath{h^{-1}{\rm\,Mpc}})^{-1.9}$. The deviations
from a power-law are clearly visible.
}
\end{figure}
\vspace{0.5cm}
\subsection{Luminosity and Redshift Dependences}
\label{subsec:variations}
Figure~\ref{fig:z_dep} shows the redshift dependence of the
$-23.2<M_g<-21.8$ results. One can see small deviations of the results
corresponding to the different redshift ranges. For two independent redshift
shells, we estimate the best-fitting multiplicative factor, $a$, between
the two ${w_p(r_p)}$ measurements, taking into account the error covariance
matrices. This factor would be significantly different than one if redshift
evolution was present and consistent with one otherwise.
The multiplicative factor between the $0.16<z<0.23$ and $0.23<z<0.36$
results is $a=0.84 \pm 0.14$ and between the $0.23<z<0.36$ and $0.36<z<0.44$
measurements it is $a=1.18 \pm 0.12$.
For our longest redshift baseline, we find $a=1.03 \pm 0.17$ between the
$0.16<z<0.23$ and $0.36<z<0.44$ measurements. Converting this to a limit
on $(1+z)^n$, we find $n=-0.2\pm1.1$.
We thus conclude that while some variations between the
different redshift shells are present, these are likely to reflect large-scale
structure variations,
and that no consistent trend of redshift evolution is detected. We note that
for the ${w_p(r_p)}$ measurements for the comparable LRG and MAIN samples shown in
Figure~\ref{fig:wplg}, $a= 1.01 \pm 0.10$, indicating clearly that no
significant redshift evolution is present.
\begin{figure}[tb]
\plotone{wpr_z.ps}
\caption{\label{fig:z_dep}
Redshift dependence of the ${w_p(r_p)}$ clustering results for the
$-23.2<M_g<-21.8$ sample.
}
\end{figure}
We also checked the robustness of our results when calculating
${w_p(r_p)}$ separately for the two large disjoint areas of the northern Galactic
sky covered in the current SDSS samples. The results from these two
independent regions
are very similar and fully consistent, when calculated over the full
redshift ranges of our samples. When looking at narrower redshift
shells the differences tend to be a bit larger, reflecting the slight
variations with redshift seen in Figure~\ref{fig:z_dep}. For example, the
tendency of ${w_p(r_p)}$ to have a slightly lower amplitude for the $0.16<z<0.23$
shell is reproduced
in the off-equatorial region, while ${w_p(r_p)}$ for the equatorial region is
similar to that of the full volume.
This supports our conclusion that these small deviations are sample
variance effects reflecting the large-scale structure fluctuations.
Figure~\ref{fig:L_dep} shows the projected correlation function ${w_p(r_p)}$
obtained for the three LRG samples over an identical volumes
($0.16<z<0.36$).
As mentioned previously, the small differences in clustering amplitude
reflect the increase of clustering with luminosity. Again, we assess
the significance of the increased clustering amplitude by estimating the
best-fit multiplicative factor, $a$, between the measurements. Since
these measurements are not fully independent (they are obtained from the
same volume and thus are susceptible to similar cosmic variance effects),
we cannot treat their individual error covariance matrices as independent.
Instead, we estimate the value of $a$ from the mean and scatter of $a$
obtained from the individual jackknife realizations. The resulting
factor between the $-23.2<M_g<21.8$ and $-23.2<M_g<21.2$ measurements
is $a = 1.34 \pm 0.08$, more than a $4\sigma$ detection of luminosity
bias. For the $-22.6<M_g<-21.6$ versus the $-23.2<M_g<21.2$ measurements,
$a = 1.08 \pm 0.05$. It is clear that we detect a non-negligible luminosity
bias among the different LRG samples.
\begin{figure}[tb]
\plotone{wpr_M.ps}
\caption{\label{fig:L_dep}
Luminosity dependence of ${w_p(r_p)}$ obtained for the three LRG samples for
$0.16<z<0.36$.
}
\end{figure}
\subsection{Real-Space Correlation Function}
\label{subsec:real-space}
It is possible to directly invert ${w_p(r_p)}$ to get $\xi(r)$ independent of
the power-law assumption. This is done by recasting Equation~\ref{eq:wp2}
as
\begin{equation}
\label{eq:xi}
\xi(r) = - \frac{1}{\pi} \int_r^{\infty} dr_p \, {dw_p(r_p)\over{dr_p}} ({r_p}^2-r^2)^{-1/2}.
\end{equation}
({\it e.g.}, \citealt{Dav83}). We calculate the integral analytically
by linearly interpolating between the binned ${w_p(r_p)}$ values, following
\citet{Sau92}. We note that this estimate is only accurate to a few
percent level, due to the inaccuracy of the linear interpolation.
Figure~\ref{fig:xil} presents the real-space correlation function, obtained
in this fashion, for the three LRG samples. The trends with luminosity and
the hints of deviations from a power-law are noticeable here as well.
The $\xi(r)$ values for these samples are given as well in
Table~\ref{tab:results}.
\begin{figure}[tb]
\plotone{xil.ps}
\caption{\label{fig:xil}
Real-space correlation function $\xi(r)$ for the three LRG samples.
}
\end{figure}
It is common to summarize the amplitude of the correlation function
as the rms variation above Poisson
in the counts of galaxies in $R=8\ensuremath{h^{-1}{\rm\,Mpc}}$ comoving radius spheres.
The variance $\sigma_R^2$ can be calculated as
\begin{equation}
\sigma_R^2 = \int_{|\vec{r}_1|<R} d^3r_1 \
\int_{|\vec{r}_2|<R} d^3r_2 \xi(|\vec{r}_2-\vec{r}_1|)
\end{equation}
where the integrals are over the interior of two spheres of
radius $R$. This can be simplified to
\begin{equation}\label{eq:xi2sig}
\sigma_R^2 = \int_0^2 dy\;y^2\xi(yR)
\left( 3 -{9y\over 4} + {3y^3\over 16}\right),
\end{equation}
a useful formula that seems to have dropped out of the standard lore.
Following \citet{Eis03b}, we express this integral in terms of $w_p$ as
\begin{equation}\label{eq:wp2sig}
\sigma_R^2 = {1\over R^3} \int_0^\infty dr_p\;r_p w_p(r_p) g(r_p/R)
\end{equation}
where $g(x)$ is
$$
\left\{\begin{array}{ll}
{1\over 2\pi}\left[3\pi - 9x + x^3\right] & {\rm for\ } x\le 2, \\
{1\over 2\pi}\left[{-x^4+11 x^2 -28\over \sqrt{x^2-4}} + x^3 -9x + 6\sin^{-1}(2/x)\right] & {\rm for\ } x>2.
\end{array}\right.
$$
The kernel $g(x)$ is simpler than it looks: it starts positive, goes through
zero at $x\approx 1.28$, and then returns to zero as $-1/\pi x^3$ at large $x$.
It is differentiable at $x=2$.
We compute $\sigma_8$ by using Equation \ref{eq:wp2sig} and assuming
that $r_p w_p(r_p)$ is constant in each bin. This yields
$\sigma_8 = 1.80 \pm 0.03$ for the $-23.2<M_g<-21.2$ sample and
$\sigma_8 = 2.06 \pm 0.05$ for the $-23.2<M_g<-21.8$ sample,
with the errors obtained from the jackknife subsamples. We stress that this
is the real-space, non-linear $\sigma_8$, which should not be directly
compared with the linear-regime $\sigma_8$ values that are typically
quoted for CDM model normalization.
To facilitate comparison with the cross-correlations between
LRGs and $L^*$ galaxies presented by \citet{Eis04}, we also compute
the following integral of the real-space correlation function:
\begin{equation}\label{eq:Deltadef}
\Delta = {1\over V}\int_0^\infty 4\pi r^2dr\;\xi(r) W(r),
\end{equation}
where
\begin{equation}\label{eq:Wr}
W(r) = {r^2\over a_0^2} \exp\left(-{r^2\over 2a_0^2}\right),
\end{equation}
where $a_0$ is a constant scale factor.
The $\Delta$ statistic isolates about one octave of scale in the real-space
correlation function.
For power-law correlations of the observed slopes,
$\Delta(a_0)\approx \xi(1.75a_0)$.
The $\Delta$ values are computed as a simple non-singular
integral over ${w_p(r_p)}$.
For the $-23.2<M_g<-21.2$ sample, we compute
$\Delta = 66.5\pm3.5$, $15.6\pm0.7$, $4.40\pm0.19$, and $1.31\pm0.06$
for $a_0=0.5\ensuremath{h^{-1}{\rm\,Mpc}}$, $1\ensuremath{h^{-1}{\rm\,Mpc}}$, $2\ensuremath{h^{-1}{\rm\,Mpc}}$, and $4\ensuremath{h^{-1}{\rm\,Mpc}}$ {\it proper} distance
at $z=0.3$. These scales are chosen to match those in
\citet{Eis04}. For the $-23.2<M_g<-21.8$ sample, the
$\Delta$ values are $91.8\pm9.6$, $18.1\pm1.8$, $5.54\pm0.43$, and
$1.73\pm0.12$, respectively.
\section{Conclusions}
\label{sec:discussion}
We have presented a statistical analysis of the intermediate-scale
($0.3$ to $40\ensuremath{h^{-1}{\rm\,Mpc}}$) correlations of $35,000$ luminous red galaxies
from the SDSS, using three nearly volume-limited subsamples. The size of
the sample permits measurements of superb precision for these rare galaxies.
We find clear deviations from power-law models in the projected correlation
functions. Relative to a power-law, there are excesses of clustering on
sub-Mpc scales and at $5-10$ Mpc scales. These match qualitatively the
deviations found in the SDSS MAIN sample galaxies and predicted by halo
modeling \citep{Zeh04a,Zeh04b}.
The SDSS LRG sample reproduces the clustering results obtained with
the SDSS MAIN galaxy sample when matched appropriately in magnitude
and color and thus provides an extension of the galaxy clustering
analyses to higher redshifts, and a better signal-to-noise measurement
of high luminosity galaxies.
We find no evidence for redshift evolution of the correlation functions
(for a fixed passively evolving magnitude range) out
to $z=0.4$. However, the errors are such that dependences even as strong as
$(1+z)^{\pm2}$ cannot be excluded at more than $2\sigma$.
All of the LRG samples are highly clustered, with correlation lengths around
$10\ensuremath{h^{-1}{\rm\,Mpc}}$ comoving, roughly twice that of $L_*$ galaxies
({\it e.g.}, \citealt{Nor01,Zeh02,Zeh04b}). For the $-23.2<M_g<-21.2$ LRGs,
the inferred bias relative to that of
$L_*$ galaxies is $1.84 \pm 0.11$ on scales of $1 - 10 \ensuremath{h^{-1}{\rm\,Mpc}}$.
The bias is roughly scale invariant on these scales and shows stronger
clustering on smaller scales.
We find that more luminous LRGs are yet more clustered; however, none
of our samples reach the correlation levels of rich clusters
\citep{Bah83,Nic92,Pea92,Cro97,Aba98,Lee99,Col00,Gon02,Bah03}.
Note that all the latter estimate the cluster {\it redshift}-space correlation
function, and thus should be compared to our $s_0$ values inferred from
Table~\ref{tab:results} and not the $r_0$ values quoted in
Table~\ref{tab:pl_fits}. The LRG clustering strengths and mean separations
$d$ are comparable to those of the poorest clusters mentioned in these works.
Our measurements are roughly consistent with the
\citet{Bah03} trend of increasing correlation length with mean separation,
$s_0 = 2.6 \sqrt{d}$ (see also their Fig.~2).
The LRG clustering strength we find is comparable as well to that of rich
groups, which again have similar mean separations \citep{Pad04,Yan04}.
The SDSS LRG sample offers an enormous data set for the study of rare
but important massive early-type galaxies.
The interplay of number density, clustering amplitude, correlation function
shapes, redshift distortions, and higher-order correlations will provide a
rich data set for the modeling of the relationship of these galaxies
to their host halos and thereby to the evolution of the extreme end of the
galaxy mass function.
\acknowledgments
We thank Zheng Zheng for useful discussions and for providing the
$\Lambda$CDM projected correlation function curve and Risa Wechsler
for useful comments.
IZ and DJE are supported by grant AST-0098577 from the National Science
Foundation. DJE was further supported by an Alfred P.\ Sloan Research
Fellowship.
Funding for the creation and distribution of the SDSS Archive has been
provided by the Alfred P. Sloan Foundation, the Participating
Institutions, the National Aeronautics and Space Administration, the
National Science Foundation, the U.S. Department of Energy, the
Japanese Monbukagakusho, and the Max Planck Society. The SDSS Web site
is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium (ARC) for
the Participating Institutions. The Participating Institutions are The
University of Chicago, Fermilab, the Institute for Advanced Study, the
Japan Participation Group, The Johns Hopkins University, the Korean
Scientist Group, Los Alamos National Laboratory, the
Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute
for Astrophysics (MPA), New Mexico State University, University of
Pittsburgh, Princeton University, the United States Naval Observatory,
and the University of Washington.
|
{
"timestamp": "2004-11-19T02:17:44",
"yymm": "0411",
"arxiv_id": "astro-ph/0411557",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411557"
}
|
\section{Introduction}\label{sec1}
There has been important progress in the experimental search of
multiquark hadrons since last year. LEPS collaboration reported
evidence of the $\Theta^+$ pentaquark with $S=+1, B=+1$ and the
minimum quark content $uudd\bar s$ \cite{leps}. Such a state is
clearly beyond the conventional quark model if it is established
by future experiments. A recent review of pentaquarks can be found
in Ref. \cite{ijmpa}.
BABAR, CLEO and BELLE collaborations observed two narrow
charm-strange mesons $D_{sJ}(2317), D_{sJ}(2457)$ below threshold
\cite{babar}. These states are 160 MeV below quark model
predictions. Some authors speculate they are four-quark states
\cite{cheng,rev1}. These states may admit a small portion of $D K$
or $D^\ast K$ continuum contribution in their wave functions. But
the dominant component of $D_{sJ}(2317), D_{sJ}(2457)$ should be
$c\bar s$ \cite{dai}. This topic is reviewed in Ref. \cite{rev2}.
BELLE collaboration discovered a new narrow charmonium-like state
X(3872) in the $J/\psi \pi^+\pi^-$ channel \cite{x}. Its mass is
very close to $D{\bar D}^\ast$ threshold. Its production rate is
comparable to those of other excited charmonium states. Very
recently BELLE collaboration observed the same signal in the
$J/\psi$ $\omega$ channel \cite{x1}. Its mass and decay pattern
seems to favor its interpretation as a deuteron-like $D {\bar
D}^\ast$ molecule \cite{swan}. Naively, one would expect the lower
production rate for $D {\bar D}^\ast$ molecules.
This year SELEX collaboration reported a narrow state
$D_{sJ}(2632)$ above threshold \cite{selex}. Its dominant decay
mode is $D_s\eta$. Such an anomalous decay pattern strongly
indicates $D_{sJ}(2632)$ is a four quark state in the $SU(3)_F$
${\bf 15}$ representation with the quark content ${1\over
2\sqrt{2}}(ds\bar{d}+sd\bar{d}+su\bar{u}+us\bar{u}-2ss\bar{s})\bar{c}$
\cite{liu}. Other possible interpretations were discussed in Refs.
\cite{liu2}.
In the light meson sector, $f_0(980)/a_0(980)$ lies 10 MeV below
the $K^+K^-$ threshold. It's difficult to find a suitable position
for them within the framework of quark model. So they were
postulated to be candidates of kaon molecule or four quark states.
Recently there has accumulated some evidence of the four-quark
interpretation of the low lying scalar mesons from lattice QCD
calculation \cite{lattice1,lattice2}.
The study of multiquark states started nearly three decades ago in
the MIT bag model \cite{Jaffe,kfl}. The reaction
$\gamma\gamma\rightarrow\rho\rho$ was suggested in the search of
$q\bar qq\bar q$ resonances. Later ARGUS collaboration found
evidence of a four-quark state in the dominant partial wave
$J^PJ_z=2^+2$ in the reaction
$\gamma\gamma\rightarrow\rho^0\rho^0\rightarrow4\pi$
\cite{Augus2}. Now the observed signal was named as $X(1600\pm
100)$ with the quantum numbers $J^{PC}I^G=2^{++}2^+$ \cite{list}.
In this work, we employ QCD finite energy sum rules (FESR) to
explore whether there exists a resonance in the
$J^{PC}I^G=2^{++}2^+$ channel.
\section{Formalism}\label{sec2}
QCD sum rule approach has proven useful in extracting the masses
of the ground state hadrons \cite{shifman}. QSR approach can yield
the absolute mass scale as the lattice QCD formalism. First one
starts from the correlation function composed of the interpolating
current which strongly couples to the hadron which one wants to
study. The correlation function can be calculated using the
operator product expansion (OPE) technique. As one approaches the
resonance region from large $Q^2$, the nonperturbative power
corrections become important gradually. One gets the spectral
density of the correlation function in terms of quark and gluon
condensates at the quark level. The hadron mass enters the
spectral density of the correlation function at the hadron level.
With the quark hadron duality assumption, one can extract the
hadron mass.
The construction of a suitable interpolating current is crucial.
There are only two independent color structures for a tetraquark.
Let's first focus on a pair of $q\bar q$. Since ${\bar 3}_c \times
3_c =1_c+ 8_c$, there are only two ways to form a color singlet
tetraquark from two pairs of $\bar q q$. Both pairs are either in
the color-singlet or color-octet state simultaneously. We denote
the corresponding interpolating currents as $\eta^1, \eta^8$. If
we focus on the two quarks, $3_c \times 3_c ={\bar 3}_c+ 6_c$. The
anti-quark pair must be in either $3_c$ or ${\bar 6}_c$ color
state in order to get a color-singlet tetraquark. We denote the
corresponding interpolating currents as $\eta^{\bar 3}, \eta^6$.
It's important to note that $\eta^{\bar 3}, \eta^6$ are linear
combinations of $\eta^1, \eta^8$. We refer the reader to the
appendix for details.
We use a general interpolating current for X(1600) which is the
linear combination of $\eta^{\bar 3}$ and $\eta^6$.
\begin{eqnarray}\label{general}
\eta_{\mu\nu}(x)&=& \eta_{\mu\nu}^{\mathbf{\bar 3}} +Y
\eta_{\mu\nu}^{\mathbf{6}}\\
&=&(Y+1)\bar d^l (x) \gamma_\mu u^l (x)
\bar d^m (x) \gamma_\nu u^m(x)\nonumber +(Y-1)\bar d^l (x) \gamma_\mu u^m (x)
\bar d^m (x) \gamma_\nu u^l(x)\nonumber + \left( g_{\mu\nu}
\mbox{terms}\right)
\end{eqnarray}
where $Y=a+bi$ is a complex number. a and b are real numbers. The
decay constant $f_X$ for X(1600) is defined as
\begin{equation}
\langle 0| \eta_{\mu\nu} (0)| X (1600)\rangle = f_X
\varepsilon_{\mu\nu}
\end{equation}
where $\varepsilon_{\mu\nu}$ is the polarization tensor of X(1600)
meson.
We consider the following correlation function
\begin{equation}\label{cor}
i\int d^4xe^{-ipx}<0|T\{\eta_{\mu\nu}(x)
\eta^{+}_{\alpha\beta}(0)\}|0>= \Delta_{\mu\nu; \alpha\beta}(p)
\Pi(p^2) + \cdots
\end{equation}
where
\begin{equation}
\Delta_{\mu\nu; \alpha\beta}(p)= \frac
{1}{2}\left(\Delta_{\mu\alpha}(p)\Delta_{\nu\beta}(p)
+\Delta_{\mu\beta}(p)\Delta_{\nu\alpha}(p)- \frac
{2}{3}\Delta_{\mu\nu}(p)\Delta_{\alpha\beta}(p)\right)\;,
\end{equation}
\begin{equation}
\Delta_{\mu\nu}(p)=g_{\mu\nu}-p_{\mu}p_{\nu}/p^2 \;.
\end{equation}
In Eq. (\ref{cor}), we have kept the unique tensor structure for
$J^{PC}=2^{++}$ mesons: $\Delta_{\mu\nu; \alpha\beta}(p)$. We note
in passing that the $q\bar q$ meson states with $J^{PC}=2^{++}$
have been studied in \cite{reinders,shifman1}, where L=1 orbital
excitation has to be introduced. For any of its four Lorentz
indices, $\Delta_{\mu\nu; \alpha\beta}(p)$ satisfies:
\begin{equation}
q^\mu\Delta_{\mu\nu; \alpha\beta}(p)=0
\end{equation}
\begin{equation}
\Delta^{\mu}_{\mu; \alpha\beta}(p)=0 \; .
\end{equation}
The non-resonant $\rho^+\rho^+$ intermediate states do not
contribute to this tensor structure.
\begin{figure}[hbt]\label{diagram}
\begin{center}
\begin{picture}(500,130)(0,0)
\Curve{(25,50)(55,65)(85,50)}
\Curve{(27,50)(55,55)(85,50)}
\Curve{(27,50)(55,45)(85,50)}
\Curve{(25,50)(55,35)(85,50)}
\Curve{(125,50)(155,65)(185,50)}
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\Curve{(125,50)(155,55)(185,50)}
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\Curve{(125,50)(155,45)(185,50)}
\Curve{(125,50)(155,35)(185,50)}
\Curve{(225,50)(235,51)(245,52)}
\BCirc(245,52){2}
\BCirc(265,52){2}
\Curve{(265,52)(275,51)(285,50)}
\Curve{(225,50)(235,59)(245,62)}
\BCirc(245,62){2}
\BCirc(265,62){2}
\Curve{(265,62)(275,59)(285,50)}
\Curve{(225,50)(255,45)(285,50)}
\Curve{(225,50)(255,35)(285,50)}
\Curve{(325,50)(335,60)(345,64)}
\BCirc(345,64){2}
\BCirc(365,64){2}
\Curve{(365,64)(375,60)(385,50)}
\Curve{(325,50)(335,51)(345,52)}
\BCirc(345,52){2}
\BCirc(365,52){2}
\Curve{(365,52)(375,51)(385,50)}
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\Curve{(325,50)(355,35)(385,50)}
\Curve{(425,50)(435,60)(445,64)}\BCirc(445,64){2}
\BCirc(465,64){2} \Curve{(465,64)(475,60)(485,50)}
\Curve{(425,50)(435,51)(445,52)} \BCirc(445,52){2}
\BCirc(465,52){2} \Curve{(465,52)(475,51)(485,50)}
\BCirc(455,52){2} \Gluon(455,43)(455,52){2}{1}
\Curve{(425,50)(455,43)(485,50)} \Curve{(425,50)(455,35)(485,50)}
\end{picture}
\end{center}
\label{fig1} \caption{The Feynman diagrams in the calculation of
the two-point correlation function in Eq. (\ref{cor}).}
\end{figure}
The scalar complex function $\Pi (p^2)$ satisfies the following
dispersion relation
\begin{equation}\label{disp}
\Pi (p^2)= \int {\rho(s) \over s-p^2 -i \epsilon} ds
\end{equation}
where $\rho (s)$ is the spectral density. For a narrow resonance
\begin{equation}
\rho (s)= f_X^2 \delta (s-M_X^2) + \mbox{higher states}\;.
\end{equation}
At the quark gluon level the correlation function (\ref{cor})
reads
\begin{eqnarray}\label{x}\nonumber
i\int d^4xe^{-ipx}\{|Y+1|^2[{\bf
\mbox{Tr}}[\gamma_{\mu}S^{jn}(x)\gamma_{\alpha}S^{mi}(-x)]\times
{\bf \mbox{Tr}}[\gamma_{\nu}S^{im}(x)\gamma_{\beta}S^{nj}(-x)]\\
\nonumber
-{\bf\mbox{Tr}}[\gamma_{\mu}S^{jn}(x)\gamma_{\alpha}S^{mj}(-x)
\gamma_{\nu}S^{im}(x)\gamma_{\beta}S^{ni}(-x)]\\
\nonumber +2(|Y|^2-1)[{\bf
\mbox{Tr}}[\gamma_{\mu}S^{jm}(x)\gamma_{\alpha}S^{mi}(-x)]\times
{\bf \mbox{Tr}}[\gamma_{\nu}S^{in}(x)\gamma_{\beta}S^{nj}(-x)]\\
\nonumber
-{\bf\mbox{Tr}}[\gamma_{\mu}S^{jm}(x)\gamma_{\alpha}S^{mj}(-x)
\gamma_{\nu}S^{in}(x)\gamma_{\beta}S^{ni}(-x)]\\
\nonumber +|Y-1|^2[{\bf
\mbox{Tr}}[\gamma_{\mu}S(x)\gamma_{\alpha}S(-x)]\times {\bf
\mbox{Tr}}[\gamma_{\nu}S(x)\gamma_{\beta}S(-x)]\\
-{\bf\mbox{Tr}}[\gamma_{\mu}S(x)\gamma_{\alpha}S(-x)\gamma_{\nu}S(x)
\gamma_{\beta}S(-x)]+(\alpha\leftrightarrow\beta)]
\end{eqnarray}
where $iS(x)=\langle 0|T\{q(x) \bar q(0)\}|0\rangle $ is the full
quark propagator in the coordinate space. The $\bf\mbox{Tr}$
denotes the summation of both the color and Lorentz indices.
Throughout our calculation, we assume the up and down quarks are
massless. The first few terms of quark propagator is
\begin{equation}
iS^{ab}(x)=\frac{i\delta^ab}{2\pi^2x^4}\hat{x}+\frac{i}{32\pi^2}\frac{\lambda^n_{ab}}{2}g_cG^n_{\mu\nu}
\frac{1}{x^2}(\sigma^{\mu\nu}\hat{x}+\hat{x}\sigma^{\mu\nu})-\frac{\delta^{ab}}{12}\langle\bar
qq\rangle+\frac{\delta^{ab}x^2}{192}\langle g_s\bar q\sigma
Gq\rangle+\cdots
\end{equation}
The relevant terms that contribute to this correlator are
represented pictorially in Figure 1.
After making Fourier transformation to $\Pi(x)$ we arrive at
$\Pi(p^2)$. From the imaginary part of $\Pi(p^2)$ we extract the
spectral density $\rho(s)$.
\begin{eqnarray}\nonumber
\rho(s) &=& {1\over 2^{12}\cdot 7\cdot
\pi^6}(1+\frac{2}{3}|Y|^2)s^4 -{1\over 2^{14}\cdot 15 \cdot
\pi^6}(23|Y|^2-21(Y+Y^{\ast})+47)s^2\langle g_s^2GG\rangle \\
&& + {5\over 18\cdot \pi ^2}(1-\frac{2}{5}|Y|^2)s
{\langle\bar qq\rangle}^2 -\frac{1}{144\cdot
\pi^2}(13|Y|^2-3(Y+Y^{\ast})-33){\langle\bar qq\rangle}{\langle
g_s\bar q\sigma G q\rangle}
\end{eqnarray}
where we have used the factorization approximation for the
high-dimension quark condensates.
\section{FESR And Numerical Analysis}\label{sec3}
In the Borel sum rule (BSR) analysis there are two parameters: the
continuum threshold $s_0$ and Borel mass $M_B$. FESR contains a
single parameter $s_0$ \cite{fesr}. The dimension of the
interpolating currents of the conventional hadrons like rho and
nucleon is not high. Both BSR and FESR yield roughly the same
results. If the interpolating current is of high dimension, the
working window of $M_B$ does not exist sometimes. In this case,
FESR may have some advantage over BSR \cite{narison}.
With the spectral density, the $n$th moment of FESR is defined as
\begin{equation}
W(n,s_0)=\int_{0}^{s_0}ds s^n \rho(s)
\end{equation}
where $n\ge 0$. With the quark hadron duality assumption we get
the finite energy sum rule
\begin{equation}\label{12}
W(n,s_0)|_{Hadron} =W(n,s_0)|_{QCD}\; .
\end{equation}
The mass can be obtained as
\begin{equation}
M^2={W(n+1,s_0)\over W(n,s_0)} \;.
\end{equation}
In principle, one can extract the threshold self-consistently from
the requirement that the hadron mass has the least dependence on
$s_0$, i.e., $ {d M^2\over d s_0} =0$. However, the threshold
value $s_0$ extracted this way is not necessarily physical
sometimes. The weight function of FESR enhances the continuum part
even more than the weight function $e^{-s/M_B^2}$ in the Borel sum
rule. One must make sure that only the lowest pole contributes to
the FESR below $s_0$. Otherwise the result will be very
misleading. To be more specific, a {\sl naive} stability region in
$s_0$ is no guarantee of a {\sl physically reasonable} value for
$s_0$. For example, the FESR with an extracted threshold
$s_0\approx 10$ GeV$^2$ is certainly irrelevant for the possible
X(1600) state.
Besides the weak dependence on $s_0$, we also require (1) the
zeroth moment $W(0, s_0) >0$ and (2) the convergence of the
operator product expansion (OPE). Higher dimension condensates
should be suppressed in order to ensure a reliable FESR based on
the converging OPE series. This convergence requirement is not
easily satisfied if the interpolating current is of high
dimension. As shown in the appendix, neither $\eta^{\bar 3}$ nor
$\eta^6$ leads to a converging FESR for a reasonable value of
$s_0$. Only with a mixed current which is the linear combination
of $\eta^{\bar 3}$ and $\eta^6$, can we make the OPE series of
FESR converging.
We use the following values of condensates: $\langle\bar
qq\rangle=-(0.24 \mbox{GeV})^3,\langle g_s^2GG\rangle =(0.48\pm
0.14)\mbox{GeV}^4,\langle g_s\bar q\sigma G
q\rangle=-m_0^2\times\langle\bar qq\rangle $,
$m_0^2=(0.8\pm0.2)$GeV$^2$. To simply numerical analysis, we
introduce variables $\epsilon_1, \epsilon_2$ which are defined as:
\begin{eqnarray}\nonumber
a^2+b^2={5\over 2}(1-\epsilon_1) \\ \nonumber a=-{1\over
12}(1+\epsilon_2) \; .
\end{eqnarray}
They satisfy two constraints: $\epsilon_1\le 1, |1+\epsilon_2|\le
6\sqrt{10(1-\epsilon_1)}$. Now the spectral density reads
\begin{eqnarray}\label{ddd}\nonumber
\rho(s) &=& {1\over 2^{12}\cdot 21\cdot \pi^6}(8-5\epsilon_1)s^4
-{1\over 2^{15}\cdot 15 \cdot
\pi^6}(216-115\epsilon_1+7\epsilon_2+7)s^2\langle g_s^2GG\rangle \\
&& + {5\over 18\cdot \pi ^2}\epsilon_1 s
{\langle\bar qq\rangle}^2 +\frac{1}{288\cdot
\pi^2}(65\epsilon_1-\epsilon_2){\langle\bar qq\rangle}{\langle
g_s\bar q\sigma G q\rangle} \; .
\end{eqnarray}
From Eq. (\ref{ddd}) it's clear that we can adjust the variables
$\epsilon_1, \epsilon_2$ to suppress $D=6, 8$ condensates. In fact
there exists a parameter space of $(\epsilon_1, \epsilon_2)$, with
which there exists a stable FESR plateau in the $M_X$ {\sl vs}
$s_0$ curve. The extracted $s_0$ at the stable plateau is
physically reasonable. With this $s_0$ the zeroth FESR moment is
positive. and the OPE is convergent. One typical set of these
variables is $\epsilon_1=-0.02, \epsilon_2=-2$. After we divide
each piece in the FESR by the perturbative term, the zeroth moment
reads
\begin{equation}
W(0,s_0)\sim 1-{3.53\over s^2_0} -{2.79\over s_0^3}-{1.95\over
s^4_0}\; .
\end{equation}
The positivity requirement leads to $s_0>2.3$ GeV$^2$.
\begin{figure}[hbt]
\begin{center}
\scalebox{0.8}{\includegraphics{M.eps}}
\end{center}\label{bi0m}
\caption{The variation of the mass $M$ with the threshold $s_0$
(in unit of $\mbox{GeV}^2$) for the current (\ref{general}). The
long-dashed and solid curves correspond to FESR when truncated at
dimension $D=6$ and $D=8$ respectively.}
\end{figure}
For $s_0>2.3$ GeV$^2$, the OPE also converges well. The variation
of $M_X$ with $s_0$ is presented in Figure 2. The extracted mass
is
\begin{equation}
M_X=(1.65\pm 0.15) \mbox{GeV} \;.
\end{equation}
As can be seen from Figure 2, the value of $M_X$ does not change
much if we truncate the OPE at $D=6$. This is another sign of
convergence of our FESR.
\section{Discussion}\label{sec4}
The state $X(1600)$ with $J^{PC}I^G=2^{++}2^+$ has inspired many
theoretical papers \cite{russia}. The potential scattering of
$\rho^0\rho^0$ via the $\sigma$ meson exchange was proposed to
explain the signal in Ref. \cite{bajc}. In the diquark cluster
model \cite{cluster}, the mass and decay width of $X(1600)$ is
studied together with other tetraquarks assuming the $\mathbf{\bar
3}_c$ color wave function for two quarks. Using the potential
model $X(1600)$ mass was estimated to be $1544$ MeV recently
\cite{vijande2}.
In this work we have investigated this state using QCD finite
energy sum rule. With any single current $\eta^1$, $\eta^8$,
$\eta^{\bar 3}$, $\eta^6$, the operator product expansion of the
corresponding correlator does not converge. After we consider the
linear combination of $\eta^{\bar 3}$ and $\eta^6$, we have
derived a converging FESR and observe a resonance signal at 1.65
GeV, which is close to the experimentally observed X(1600) state.
Our analysis indicates both the "hidden color" and coupled channel
effects may be important in the multiquark system.
From charge, angular momentum, parity, isospin, C parity, and G
parity conservation, we may obtain the allowed decay modes of
$X(1600)$. G parity requires even number of pions in the final
state for pure pion final states. For example, $X^0(1600) \to
\rho^0 \pi^0, \omega \pi^+\pi^-$ is forbidden by G parity. C
parity forbids $X^0(1600) \to \omega (3\pi^0)$. Isospin
conservation forbids the following decay modes: $X^0(1600) \to
\omega \pi^0, \omega\omega$. The possible modes are $X^0(1600) \to
\pi^0 \pi^0, \pi^+\pi^-, 4\pi, \rho^+\rho^-, \rho^0\rho^0$.
Angular momentum conservation requires D-wave decay for the two
pion mode. Therefore, two pion decay modes may be suppressed
compared with S-wave $\rho\rho$ mode.
Filippi et al. reported the possible existence of the
$J^{PC}I^G=0^{++}2^+$ state decaying into $\pi^+\pi^+$ in the
reaction $\bar np\longrightarrow \pi^+\pi^+\pi^-$ \cite{fili}. One
the other hand, some constraint has been obtained on the possible
resonance in the scalar isotensor channel from the phase shift
analysis of $\pi\pi \to \rho\rho\to \pi\pi$ scattering \cite{zou}.
Constraint on the D-wave I=2 resonance from the similar analysis
of the same reactions will be very desirable.
The process $\gamma \gamma\to \rho\rho\to X(1600)\to 4\pi$ was
suggested and used in the search of $X(1600)$. We suggest the
following reactions to look for this charming state.
\begin{itemize}
\item $J/\psi$ decays at BES, CLEO and $\Upsilon$ decays at BELLE,
BABAR
Symmetry considerations requires $X(1600)$ is produced together
with odd number of pions in $J/\psi$ decays. One pion mode is
forbidden by isospin conservation. So the favorable decay chain is
$J/\psi \to X(1600) +3\pi \to 2\rho+3\pi \to 7\pi$. There is also
some chance in $J/\psi \to X(1600) +3\pi \to 2\pi+3\pi \to 5\pi$.
\item Hadron reactions especially charge exchange processes
Charge exchange processes are useful for the production of
$X^{++}(1600)$ if such a state exists. Some of them are $\pi^+ N
\to \pi^- + X^{++}(1600) + N \to 5\pi +N$, $ p + N \to n +
N^\prime + X^{++}(1600)$ where $N$ is either a nucleon or nuclei.
The later process can be studied at CSR facility at Lan Zhou.
\item Anti-proton annihilation on the proton or deuteron targets
Let's take $X(1600)$ production from anti-proton annihilation on
the proton target as an example. Since the isospin of $p \bar p$
is either 0 or 1, $X(1600)$ should be accompanied by one or
several pions. However, kinematics allows only one pion decay
mode: ${\bar p} + p \to X(1600) + \pi^0 \to 5\pi$. So we get
$C_{p\bar p}=(-)^{L_{p\bar p}+S_{p\bar p}}=+, I_{p\bar p}=1$.
(1) When the $p \bar p$ pair is in the S-wave, $L_{p\bar
p}=S_{p\bar p}=J_{p\bar p}=0, P_{p\bar p}=-$. Angular momentum and
parity conservation requires $L_{\pi X(1600)}=2$; (2) When the $p
\bar p$ pair is in the P-wave, $L_{p\bar p}=S_{p\bar p}=1$,
$P_{p\bar p}=+$, $J_{p\bar p}=1$ or 2. We get $L_{\pi X(1600)}=1$;
(3) When the $p \bar p$ pair is in the D-wave, $L_{p\bar p}=2,
S_{p\bar p}=0, J_{p\bar p}=2, P_{p\bar p}=-$. We have $L_{\pi
X(1600)}=0$.
\end{itemize}
\section*{Acknowledgments}
S.L.Z. thanks B.-S. Zou and H.-Q. Zheng for helpful discussions.
This project was supported by the National Natural Science
Foundation of China under Grants 10375003 and 10421003, Ministry
of Education of China, FANEDD, Key Grant Project of Chinese
Ministry of Education (NO 305001) and SRF for ROCS, SEM.
\section{Appendix}
In this appendix we list the currents with different color
structure and their spectral densities.
The color-singlet $\rho^+ (x) \rho^+(x)$ type interpolating
current reads
\begin{equation}\label{current1}
\eta^1_{\mu\nu} (x) =\bar d^i (x)\gamma_\mu u^i (x)\bar d^j (x)
\gamma_\nu u^j(x) -1/4 g_{\mu\nu}\bar d^i(x) \gamma_\rho u^i(x)
\bar d^j(x)\gamma^\rho u^j (x) \;,
\end{equation}
which "i, j" are the color indices.
The current with color octet structure reads
\begin{equation}\label{current2}
\eta^8_{\mu\nu} (x) =\bar d^i (x)\left({\lambda^a\over 2}\right)_{ij}
\gamma_\mu u^j (x)\bar d^m (x) \left({\lambda^a\over 2}\right)_{mn}
\gamma_\nu u^n(x)-{1\over 4} g_{\mu\nu}\bar d^i(x)\left({\lambda^a\over 2}
\right)_{ij}\gamma_\rho u^j(x)\bar d^m(x) \left({\lambda^a\over2}
\right)_{mn}\gamma^\rho u^n (x)
\end{equation}
where ${\lambda^a\over 2}$ is the $SU(3)_c$ generator.
Using the identity of the $\lambda$ matrix
\begin{equation}
\sum\limits_{a}\frac{\lambda_{ij}^a}{2}\cdot\frac{\lambda_{kl}^a}{2}
=\frac{1}{2}(\delta_{il}\delta_{jk}-\frac{1}{3}\delta_{ij}\delta_{kl})
\end{equation}
we can rewrite Eq. (\ref{current2}) as
\begin{equation}
\eta_{\mu\nu}^8 (x)
=-\frac 16\bar d^l (x) \gamma_\mu u^l (x)\bar d^m (x) \gamma_\nu u^m(x)
+\frac 12\bar d^l (x) \gamma_\mu u^m (x)\bar d^m (x) \gamma_\nu u^l(x)
+ \left(g_{\mu\nu} \mbox{terms}\right) \; .
\end{equation}
Alternatively, when two quarks are in the $\mathbf{\bar 3}_c$
state, we have
\begin{equation}\label{diquark-type}
\eta_{\mu\nu}^{\mathbf{\bar 3}_c} (x)=\bar d^l (x) \gamma_\mu u^l (x)
\bar d^m (x) \gamma_\nu u^m(x)\nonumber -\bar d^l (x) \gamma_\mu u^m (x)
\bar d^m (x) \gamma_\nu u^l(x)\nonumber + \left( g_{\mu\nu}
\mbox{terms}\right) \; .
\end{equation}
The current with the $\mathbf{6_c}$ color structure is
\begin{equation}\label{6c-type}
\eta_{\mu\nu}^{\mathbf{6_c}} (x)=\bar d^l (x) \gamma_\mu u^l (x)
\bar d^m (x) \gamma_\nu u^m(x)\nonumber +\bar d^l (x) \gamma_\mu u^m (x)
\bar d^m (x) \gamma_\nu u^l(x)\nonumber + \left( g_{\mu\nu}
\mbox{terms}\right)\; .
\end{equation}
The spectral densities of the above four currents
\begin{equation} \rho^1 (s)={5\over 2^{14}\cdot
21\cdot \pi^6}s^4 -{7\over 2^{12}\cdot 15\cdot \pi ^6}s^2\langle
g_s^2GG\rangle+ {1\over 24\pi ^2}s {\langle\bar
qq\rangle}^2+\frac{7}{288 \cdot \pi^2}{\langle\bar
qq\rangle}{\langle g_s\bar q\sigma G q\rangle}\; .
\end{equation}
\begin{equation} \rho^8 (s)={1\over 2^{13}\cdot
27\cdot \pi^6}s^4 -{127\over 2^{16}\cdot 135 \cdot \pi
^6}s^2\langle g_s^2GG\rangle+ {1\over 36\cdot \pi ^2}s
{\langle\bar qq\rangle}^2+\frac{131}{64 \cdot 81\cdot
\pi^2}{\langle\bar qq\rangle}{\langle g_s\bar q\sigma G
q\rangle}\; .
\end{equation}
\begin{equation} \rho^{\mathbf{\bar 3}_c} (s)={1\over 2^{12}\cdot
7\cdot \pi^6}s^4 -{47\over 2^{14}\cdot 15 \cdot \pi ^6}s^2\langle
g_s^2GG\rangle+ {5\over 18\cdot \pi ^2}s {\langle\bar
qq\rangle}^2+\frac{11}{48\cdot \pi^2}{\langle\bar
qq\rangle}{\langle g_s\bar q\sigma G q\rangle} \; .
\end{equation}
\begin{equation} \rho^{\mathbf{6}_c} (s)={1\over 2^{11}\cdot
21\cdot \pi^6}s^4 -{23\over 2^{14}\cdot 15 \cdot \pi
^6}s^2\langle g_s^2GG\rangle- {1\over 9\cdot \pi ^2}s {\langle\bar
qq\rangle}^2-\frac{13}{144\cdot \pi^2}{\langle\bar
qq\rangle}{\langle g_s\bar q\sigma G q\rangle} \; .
\end{equation}
In order to study the convergence of OPE series in FESR, we divide
each piece in the zeroth moments by the four quark condensate.
\begin{eqnarray}
W^{1}(0,s_0)\sim 7.5\times 10^{-3} s^3_0-4.7\times 10^{-2} s_0
+1-{0.9\over s_0}\; .\\ \nonumber W^{8}(0,s_0)\sim 3.5\times
10^{-3}s^3_0+8.9\times 10^{-3} s_0 +1-{1.5\over s_0}\; .\\
\nonumber W^{\mathbf{\bar 3}}(0,s_0)\sim 2.7\times 10^{-3}
s^3_0+1.2\times 10^{-2} s_0 +1-{1.3\over s_0}\; .\\ \nonumber
W^{\mathbf{6}}(0,s_0)\sim -9.0\times 10^{-3} s^3_0+2.9\times
10^{-2} s_0 +1-{2.6\over s_0}\; .
\end{eqnarray}
Clearly these moments converge only when the threshold parameter
is very large, $s_0>10$ GeV$^2$. Such a large $s_0$ is irrelevant
for X(1600).
|
{
"timestamp": "2006-04-05T09:14:40",
"yymm": "0411",
"arxiv_id": "hep-ph/0411140",
"language": "en",
"url": "https://arxiv.org/abs/hep-ph/0411140"
}
|
\section{Introduction}
It is widely known that the last scattering surface (LSS) has a
finite thickness $\Delta z$ (in redshift space) because the
hydrogen recombination process takes a finite time. This fact has
important consequences for the physics relating theoretical
predictions and observations of the the angular pattern of CMBR
anisotropies. In particular, it implies that anisotropies at
length scales smaller than $\Delta z $ must be naturally
suppressed.
On the other hand, recent observations from type Ia
Supernovae\cite{1} combined with CMBR experiments\cite{2} strongly
suggest an accelerating and nearly flat Universe ($q_o < 0,\
\Omega_{Total}\approx 1$). In the framework of general relativity
both facts are readily accommodated by assuming the existence of
an extra dark energy (DE) component with negative pressure (in
addition to cold dark matter (CDM)). Besides the cosmological
constant\cite{Peebles1} ($\Lambda$) and a scalar field\cite{PAD}
($\Phi$), there are at least 3 distinct dark energy candidates
proposed in the literature, namely: vacuum decay
$\Lambda(t)-models $\cite{Lima}, X-matter\cite{X}, and a Chaplygin
gas\cite{CG}.
In this paper we discuss the width of the LSS for accelerating
world models driven by the last two above quoted DE candidates
(X-matter and Chaplygin gas). Since the dark energy component
present in these models is separately conserved, the consequences
to LSS are completely different to what happens with accelerating
models endowed with any kind of adiabatic photon
creation\cite{Lima,mc}. With basis on the WMAP
observations\cite{W}, in the present calculations we consider
$H_0= 71$ km s$^{-1}$Mpc$^{-1}$, $\Omega_b=0.04$ for baryonic
matter, and $\Omega_{DM}=0.27$ for cold dark matter.
\section{Last Scattering Surface and the Recombination Process}
The probability that a CMBR photon undergone its last scattering
between $z$ and $z+ dz$ is $P = 1 - e^{-\tau (z)}$, where $\tau =
\int\limits_0^z n_e\; \sigma_T\; c\; dt$ is the optical depth to
redshift $z$, $\sigma_T$ is the Thomson cross section and $n_e$ is
the density of electrons. The quantity ${dP(z)}/{dz}$ determines
the visibility function, that is, the probability distribution for
the redshifts where the CMBR photons had their last scattering. By
defining the effective profile of the LSS:
$V(z)=dP(z)/dz=e^{-\tau(z)}d\tau/dz$, and fitting the visibility
curve with a Gaussian form, the standard deviation yields a
reasonable estimate of the LSS width whereas its peak stands for
the beginning of the recombination epoch\cite{4}.
At least 3 physical process are acting during and after
recombination on the baryonic matter: photoionization, cooling
from recombination, and Compton cooling-heating. Hence, neglecting
the helium and treating the hydrogen atom as a two-level system,
the fraction $x_e$ of ionized matter obeys\cite{5}
\begin{equation}
{\frac{dx_e}{dt}=\frac{\Lambda_{2s,1s}}{(\Lambda_{2s,1s}+\beta_e)}\left[\beta_e
e^{-\frac{(B_1-B_2)}{k_\beta T_\gamma}}(1-x_e)-\frac{a_r \rho
x_e^2}{m_p}\right]}\, , \label{xe}
\end{equation}
where $B_1=13.6\ {\mathrm eV}$ is ground state energy, $B_2=3.4\
{\mathrm eV}$ is the first excited state energy, $\beta_e = (2\pi
m_ek_{{}_B}T_\gamma)^{3/2}h^{-3} e^{-(B_2/k_{{}_B}T_\gamma)}a_r\;$
is the photoionization rate, $a_r=2.84\times 10^{-11}\;
T_m^{-1/2}$cm$^3$ sec$^{-1}$ is the recombination coefficient, and
$\Lambda_{2s,1s}=8.272 $ sec$^{-1}$ is the two-photon emission
rate.
After decoupling, the matter temperature ($T_m$) of the neutral
atoms fall faster than the radiation temperature ($T_\gamma$). The
matter temperature decreasing is governed by the equation
\begin{equation}
\frac{dT_{m}}{dt} = T_{m}\left[ -2\frac{\dot R}R - \frac{\dot
x_e}{3\left( 1+x_e\right) } \right] - \frac{8\sigma_T b}{3m_e
c}\frac{T_\gamma^4 x_e} {\left( 1+x_e\right) }\left(T_{m}-T_\gamma
\right) \, , \label{Tm}
\end{equation}
where $\dot R/R$ is the Hubble parameter, $\dot x_e = dx_e/dt$,
and $8\sigma_T b/3 m_e c = 8.02\times10^{-9}$ sec$^{-1}$K$^{-4}$.
\section{Visibility Function: Main Results}
Let us now discuss the visibility function for the accelerating
models (X-matter and Chaplygin gas) quoted in the introduction.
{\bf (i) X-Matter Models:}
In cosmological scenarios driven by X-matter plus cold dark matter
(sometimes called XCDM parametrization) both fluid components are
separately conserved. The equation of state of the dark energy
component is $p_x = w (z) \rho_x$. Unlike to what happens with
scalar field motivated models where $w(z)$ is derived from the
field description, the expression of $w(z)$ for XCDM scenarios
must be assumed a priori. Models with constant $w$ are the
simplest ones and their free parameters can easily be constrained
from the main cosmological tests. In what follows we focus our
attention to this class of models assuming a flat geometry. The
differential time-redshift relation is
\begin{equation}
dt=\frac{1}{H_0}\frac{dx}{ x \left[ \Omega_{M} x^{-3} + (1 -
\Omega_{M}) x^{-3(1+\omega)}\right]^{1/2} } \, , \label{quinta}
\end{equation}
where $\Omega_{M} = 1- \Omega_{X}$ is the density parameter of the
dark matter. Taking the limiting case $\omega=-1$, the
$\Lambda$CDM results are recovered. The basic results are
presented in Table 1. The left panel of Figure 1 shows the
corresponding visibility function.
{\bf (ii) Chaplygin Gas:}
This class of accelerating models refers to an exotic fluid whose
equation of state is given by $p_{C} = -A/\rho_{C}^{\alpha}$,
where $A$ and $\alpha$ are positive parameters. Actually, the
above equation for $\alpha \neq 1$ generalizes the original
Chaplygin equation of state whereas for $\alpha = 0$, the model
behaves like scenarios with cold dark matter plus a cosmological
constant ($\Lambda$CDM). The dynamics of such a fluid is similar
to non-relativistic matter (dark matter) at high redshift and as a
negative-pressure DE component at late times. Two different
pictures are usually considered in the literature: the first is a
flat scenario driven by a non-relativistic matter plus the
Chaplygin gas as a dark energy (GCgCDM), whereas in the second
one, the Chaplygin type gas together with the observed baryonic
content are responsible by the dynamics of the present-day
universe (unifying dark matter with dark energy (UDME or
Quartessence). The differential age-redshift relation as a
function of the observable now reads
\begin{equation}
dt=\frac{1}{H_0}\left\{
\frac{x}{\Omega_{j}+(1-\Omega_{j})x^3[A_s+(1-A_s)
x^{-3(\alpha+1)}]^{\frac{1}{1+\alpha} } }\right\}^{1/2}dx \, ,
\label{Chaply}
\end{equation}
where $A_s=A/ \rho_{c_0}^{1+\alpha}$ and $H_{0}$ is the Hubble
constant. $\Omega_{j}$ stands for baryonic + dark matter density
parameter in GCgCDM models but only to the baryonic matter density
parameter in the UDME (Quartessence) scenarios. The results are
presented in Table 2 and the right panel of Figure 1 show the
corresponding visibility function to both cases.
In summ, the thickness of the LSS has been discussed using two
different accelerating world models. Tables 1 and 2 show the main
conclusions of this work. As we have seen, the X-matter models
present the same behaviour of constant $\Lambda$ models,
regardless of the value of $\omega$. Probably, more important, the
recombination epoch is just the same for all models driven by
X-matter and the Chaplygin gas (it is located at redshift
$z_{rec}=1.127$). Further, the width of the LSS is only weakly
dependent on the kind of dark energy models considered here. As
one can see from Table 2, the UDME models (in which the C-gas
plays the role of both dark matter and dark energy) has a little
influence on the width of the LSS. This is in line with the
visibility function presented in figure (b).
Finally, for the sake of comparison, we have also computed
$z_{rec}$ and the width of the LSS for models with decaying vacuum
energy density\cite{Lima} and adiabatic gravitational creation of
matter and radiation\cite{mc}. Due to the adiabatic creation of
photons, the results concerning the width of the LSS and $z_{rec}$
are strongly modified. This means that the physics of the LSS may
constrain with great accuracy any model endowed with photon
creation because the temperature law of the CMBR is modified. This
problem will be discussed in a forthcoming communication.
{\bf Acknowledgements:}The authors are grateful to J. C. Neves de
Araujo by the numerical code and many helpful discussions.
|
{
"timestamp": "2004-11-24T02:19:03",
"yymm": "0411",
"arxiv_id": "astro-ph/0411657",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411657"
}
|
\section{Introduction}
For the helium liquids, measurements on ions have served as a valuable
probe of liquid properties. As examples we may mention the use of ions
to put in evidence vortex lattices in rotating liquid $^{4}$He and
the measurement of ionic mobilities in superfluid $^{3}$He to elucidate
scattering processes. The first
experiments on ions in an ultracold gas of $^{87}$Rb atoms were reported
by the group in Pisa \cite{Pisa}, who produced ions by irradiating
a rubidium condensate with laser pulses which ionize atoms through
one- and two-photon absorption processes.
Theoretically, the capture of atoms into weakly bound states of the
atom-ion potential has been considered in Ref.\ \cite{Cote'}. In
this paper we consider the structure of a positive ion in a Bose-Einstein
condensate when there is no capture of atoms into bound states, and
in particular we calculate the excess number of atoms associated with
an ion. We shall demonstrate that this number is typically of order 10$^{2}$,
either positive or negative.
The interaction between an atom and a positively charged alkali-metal
ion (charge $e$), which are separated by a distance $r$, is given
at large distances by the polarization potential caused by the electrostatic
field $\mathcal{E}_{\rm es}$ due to the ion, ${\mathcal{E}}_{\rm es}=e/4\pi\epsilon_{0}r^{2}$, which
gives rise to a change in the energy of the neutral atom
given by $V=-\alpha{\mathcal{E}}^{2}/2$,
where $\alpha$ is the polarizability of the atom. Writing the polarizability
as $\alpha=4\pi\epsilon_{0}\tilde{\alpha}$, where $\tilde{\alpha}$
has the dimension of volume, the energy shift of an atom due to the
ion becomes
\begin{equation}
V(r)=-\tilde{\alpha}\frac{e_{0}^{2}}{2r^{4}},
\label{pot}
\end{equation}
where $e_{0}^{2}\equiv e^{2}/4\pi\epsilon_{0}$. At short distances
($r\la10a_{0})$ the potential has a repulsive core. An important
characteristic length, which we denote by $\beta_{4}$, may be
identified by equating the kinetic energy $\hbar^{2}/2m\beta_{4}^{2}$
to the potential energy $V(\beta_{4})$, resulting in
\begin{equation}
\beta_{4}=\sqrt{\frac{\tilde{\alpha}}{a_{0}}\frac{m}{m_{\rm e}}}.
\label{beta4}
\end{equation}
Here $m$ denotes the mass of a neutral atom, $m_{\rm e}$ is the
electron mass and $a_{0}\equiv\hbar^{2}/m_{\rm e}e_{0}^{2}\sim0.53$ \AA \
is the Bohr radius. Using the measured values $\tilde{\alpha}=320\, a_{0}^{3}$
for $^{87}$Rb and $\tilde{\alpha}=163\, a_{0}^{3}$ for $^{23}$Na, one finds $\beta_{4}^{\rm Rb}\approx 7150 a_0$ and $\beta_{4}^{\rm Na}\approx 2620a_0$. The quantity $\beta_{4}$
gives the distance from the ion beyond which the zero-energy atom-ion wave function ceases to
oscillate, and it sets the scale of atom-ion scattering lengths, but their actual values
depend on the details of the potential at short distances.
We begin by deriving from thermodynamics a general expression for the excess number of atoms around an ion and show
that in
dilute systems the excess number depends only on the ratio of the atom-ion and atom-atom scattering
lengths. As we shall see, this approach suggests that the number of
atoms associated with an ion is typically of the order of 10--100,
but that it may be either positive or negative.
In denser systems the
excess number must be obtained from microscopic considerations, and we
shall determine the structure of an ion immersed in a Bose-Einstein
condensate at zero temperature, assuming that atom-atom interactions
may be described within the framework of the Gross-Pitaevskii (GP)
mean-field approach. We present solutions of the GP equation for a
number of potentials which include a hard core repulsion, an
attractive square well, and one which resembles the atom-ion
interaction, a hard core with a $1/r^{4}$ attraction at larger
distances.
For a given inner boundary condition, the Schr\"odinger equation has
only one solution for a given value of the energy. By contrast, the
GP equation, because it is nonlinear, can have more than one solution
for a given chemical potential. For potentials like the atom-ion one
that can support two-body bound states, we shall find that at low
densities there are $2\nu_{\rm S}+1$ solutions of the GP equation,
where $\nu_{\rm S}$ is the number of bound states of the Schr\"odinger
equation for the same potential. With increasing density, pairs of
solutions merge and disappear until there is only a single solution
with no nodes. We shall illustrate this behavior for two potentials,
an attractive square well and one with an attractive $1/r^4$ tail. An
important question is which of these solutions is physically relevant.
At low condensate densities, one expects the wave function close to
the ion to resemble the zero-energy solution of the Schr\"odinger
equation, and to have $\nu_{\rm S}$ nodes. This will be the case
unless inelastic processes can populate lower-lying states. We find
that with increasing condensate density, this solution ceases to
exist. This indicates that the evolution of the state with density
cannot be continuous even in the absence of inelastic processes.
The plan of the paper is as follows. In Sec.\ \ref{sec:thermo} we
present thermodynamic considerations. Section \ref{microscopic}
contains a description of the asymptotic behavior of the condensate
wave function far from the ion. In Sec.\ \ref{sec:model-pot} we
consider two simple potentials to illustrate important general
features of our results, and in Sec.\ \ref{polpot} we analyse the case
of a potential that, like the actual atom-ion potential, behaves as
$r^{-4}$ at large distances. We calculate the excess number of atoms
from numerical solutions of the Gross-Pitaevskii equation for a given
background condensate density. The concluding section, Sec.\ \ref{concl},
discusses our main results. In an appendix, we address the question
of the validity of the Gross-Pitaevskii equation in the present
context.
\section{Thermodynamic considerations}
\label{sec:thermo}
We wish to calculate the excess number of particles associated with
an ion. To define this quantity precisely, we imagine adding an ion
to a condensate. This will generally change the density of atoms far
from the ion by an amount that varies as $1/V$, where $V$ is the
volume of the system. A natural definition of the excess number of
atoms $\Delta N$ associated with the ion is the number of particles that
must be added to keep the atom chemical potential constant, since
this will ensure that the properties of the condensate far from the
ion are unaltered by the addition of the ion.
In terms of the microscopic density of atoms $n(r)$ around the ion, the excess number is given by
\begin{equation}
\Delta N=4\pi\int_0^{\infty}dr\ r^{2}\left[n(r)-n_{0}\right],
\label{DeltaN}
\end{equation}
where $n_0$ is the density of atoms at large distances from the ion.
This is analogous to
what has been done earlier to calculate the excess number of $^{4}$He
atoms associated with a $^{3}$He impurity in liquid $^{4}$He \cite{bbp}.
We shall denote the energy per unit volume as ${\cal E}(n_\mathrm{a},n_\mathrm{i})$,
where $n_\mathrm{a}$ and $n_\mathrm{i}$ are the number densities
of atoms and ions, respectively. The chemical potential of the atoms
is given by
\begin{equation}
\mu_\mathrm{a}=\frac{\partial {\cal E}}{\partial n_\mathrm{a}},
\end{equation}
and therefore the condition that this be unchanged by adding one
ion and $\Delta N$ atoms is
\begin{equation}
\frac{\partial^{2}{\cal E}}{\partial n_\mathrm{a}\partial n_\mathrm{i}}
+\frac{\partial^{2}{\cal E}}{\partial n_\mathrm{a}^2}\Delta N=0,
\end{equation}
or
\begin{equation}
\Delta N=-{\frac{\partial^{2}{\cal E}}{\partial n_\mathrm{a}\partial n_\mathrm{i}}}\left/\frac{\partial^{2}{\cal E}}{\partial n_\mathrm{a}^2}\right. .\label{1stDeltaN}
\end{equation}
When the density of ions is sufficiently low, $\partial {\cal E}/\partial n_\mathrm{i}$
is equal to the energy change $\epsilon_\mathrm{i}$ when one ion
is added to the condensate, and therefore \begin{equation}
\Delta N=-{\frac{\partial\epsilon_\mathrm{i}}{\partial n_\mathrm{a}}}
\left/\frac{\partial^{2}{\cal E}}{\partial n_\mathrm{a}^2}\right. .
\end{equation}
One may also calculate $\Delta N$ from the change $\Delta F$ in the
thermodynamic potential $F=E-\mu_\mathrm{a}N$ when
a single ion is added to the system at constant $\mu_\mathrm{a}$. Here $E$ is the total energy and $N$ the total number of atoms.
Since the number of atoms is given by
\begin{equation}
N=-\frac{\partial F}{\partial\mu_\mathrm{a}},
\end{equation}
it follows immediately that
\begin{equation}
\Delta N=-\frac{\partial\Delta F}{\partial\mu_\mathrm{a}}.
\label{DeltaN-FreeEnergy}
\end{equation}
Provided the volume considered
is large compared with the scale of the atom excess around the ion,
$\Delta F$ will be independent of the volume.
Let us begin by making estimates for a dilute gas. Provided the scattering of atoms by atoms and of atoms by ions may be treated as independent binary events, the energy density may be expressed in terms of the
scattering lengths associated with the atom-atom and atom-ion
interactions. If ion-ion interactions are neglected,
we may write
\begin{equation}
{\cal E}(n_\mathrm{a},n_\mathrm{i})=
\frac{1}{2}U_\mathrm{aa} n_\mathrm{a}^{2}+U_\mathrm{ai}n_\mathrm{a} n_\mathrm{i},
\end{equation}
and therefore from Eq.\ (\ref{1stDeltaN}) we obtain
\begin{equation}
\Delta N=-\frac{U_\mathrm{ai}}{U_\mathrm{aa}}.
\end{equation}
The mean-field interaction constant $U_{jl}$ for species $j$ and $l$,
which may be either atoms (a) or ions (i), is related to the
scattering length $a_{jl}$ by
\begin{equation}
U_{jl}=\frac{2\pi\hbar^{2}a_{jl}}{m_{jl}},
\label{U}
\end{equation}
where
$m_{jl}=m_j m_l/(m_j+m_l)$
is the reduced mass of the two particles. Our result can therefore be expressed as
\begin{equation}
\Delta N=-\frac{m_\mathrm{aa}}{m_\mathrm{ai}}\frac{a_\mathrm{ai}}{a_\mathrm{aa}}.
\label{eq: deficit in the dilute limit1}
\end{equation}
If, as in Ref.\ \cite{Pisa}, the ion is obtained by photoionization
of the condensate itself, the latter expression reduces to
\begin{equation}
\Delta N=-a_\mathrm{ai}/a_\mathrm{aa}.
\label{DeltaNscatlength}
\end{equation}
To obtain an order of magnitude estimate of the excess number of atoms
associated with an ion, we note that the characteristic scale for the
magnitudes of atom-ion scattering lengths
$\left|a_{\mathrm{ai}}\right|$ is set by $\beta_{4}$, given in Eq.\
(\ref{beta4}), while the scale for the magnitudes of atom--atom scattering
lengths $\left|a_{\mathrm{aa}}\right|$ is set by
\begin{equation}
\beta_6= \left(2\frac{m}{m_{\rm e}}C_6 \right)^{1/4} a_0.
\label{beta6}
\end{equation}
Here $C_6$ is the coefficient of $r^{-6}$ in the van der Waals
interaction, expressed in atomic units.
Thus we arrive at the estimate
\begin{equation}
|\Delta N|\sim \frac{\beta_4}{\beta_6} \sim
\left(\frac{m}{2m_{\rm e}} \frac{{\tilde \alpha}^2}{C_6}\right)^{1/4},
\label{DeltaN_thermo}
\end{equation}
which is of order 35 for Rb and 25 for Na.
The fact that the excess number of atoms is so large indicates that it
may well be a poor approximation to regard the ion as a free particle,
with mass equal to the bare ion mass. Rather, the recoil of the ion
will be suppressed by the other atoms surrounding the ion, and if
$\Delta N \gg 1$ it will be a better approximation to regard the ion
as being stationary. In that case the excess number of atoms
will be given by
\begin{equation}
\Delta N= - \frac{a_{\rm ai}(m)}{2 a_{\rm aa}},
\label{DeltaNinfmass}
\end{equation}
where the argument of $a_{\rm ai}$ indicates that the scattering
length is to be evaluated for a reduced mass equal to the atom mass.
Expression (\ref{DeltaNinfmass}) gives a value for $\Delta N$ that is
typically of the same order of magnitude as that given by Eq.\
(\ref{DeltaN_thermo}). However, we stress the fact that the estimate
for $\Delta N$ depends sensitively on the value of the effective mass
of the ion, since the atom-ion potential has many bound states, and
therefore relatively small changes in the reduced mass can result in
large changes in the scattering length. Given that in the limit of
low atom density the magnitude of the excess number of atoms is
expected to be very much greater than unity, the result
(\ref{DeltaNscatlength}) will generally not give a realistic estimate
even in that case.
The perturbation induced by the ionic potential is very
strong. Therefore the question arises of whether the customary
assumption of an essentially zero range for the atom-atom interaction
is valid. We address this point in Appendix A, where we argue that
the corrections to the GP result should not be large for the
properties of interest here.
\section{Microscopic theory}
\label{microscopic}
We now turn to microscopic considerations. Since, as we shall see,
the distortion of the condensate wave function in the vicinity of an
ion extends to large distances from the ion and involves many atoms,
we expect that the effective mass of an ion and its dressing cloud
will be much larger than that of an atom, and we may regard the ion as
being static. To describe the structure of the condensate in the
vicinity of an ion we must therefore calculate the structure of the
condensate in a static external potential given by the atom-ion
interaction. Provided the length scale on which the condensate wave
function $\psi$ varies in space is sufficiently large, we may do this
by employing the Gross-Pitaevskii equation with the interaction
of atoms with the ion included as an external potential,
\begin{equation}
\left[-\frac{\hbar^{2}}{2m}\nabla^{2}+V(r)+U_{0}\left|\psi\right|^{2}\right]\psi=\mu\psi.
\label{gp}
\end{equation}
Here and in what follows we shall denote the chemical potential of an atom by $\mu$, and for simplicity we
have written
$U_{0}\equiv U_{\mathrm{aa}} =4\pi\hbar^{2}a_{\mathrm{aa}}/m$,
since $m_{\rm aa}=m/2$. We wish to find
solutions that tend to a constant at large
distances from the ion, and since the potential is
spherically symmetric, these
solutions depend only on the radial coordinate $r$.
Thus Eq.\ (\ref{gp}) becomes
\begin{equation}
\left[-\frac{\hbar^{2}}{2m}\frac{d^{2}}{d r^2}+V(r)+U_{0}\frac{\left|\chi\right|^{2}}{r^2}\right]\chi=\mu\chi,
\label{gpradial}
\end{equation}
where $\chi=r\psi$.
The behavior of the condensate wave function at large distances depends
on the nature of the potential $V(r)$.
On linearizing the GP equation (\ref{gp}) and making use of the fact that
the chemical potential is related to the condensate wave function
$\psi_0$ at large distances by the relation $\mu = n_0 U_0$ where
$n_0=|\psi_0|^2$, one finds that
the deviation
\begin{equation}
\delta \psi =\psi -\psi_0
\label{deltapsi}
\end{equation}
of the condensate wave
function
from its asymptotic value satisfies the
linearized GP equation
\begin{equation}
\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+V(r) + 2U_0 n_0\right)\delta \psi=-V(r)\psi_0.
\label{linearGP}
\end{equation}
For potentials with a finite range, one may neglect the potential at
large distances from the ion, and the deviation
that vanishes for $r\rightarrow \infty$ is thus given by
\begin{equation}
\delta \psi \propto \frac{{\rm e}^{-k_\xi r}}{r},
\label{deltapsir}
\end{equation}
where $k_\xi=\sqrt 2/\xi$ and $\xi$ is the healing length for
the bulk condensate,
\begin{equation}
\xi=\frac{1}{\sqrt{8\pi a_{\mathrm{aa}}n_{0}}}.
\label{healing}
\end{equation}
For a potential, such as the atom-ion potential, that falls off at large distances less rapidly than the
solution (\ref{deltapsir}),
the behavior is different. The leading term
in the solution for large
$r$ is then
the Thomas--Fermi result $\psi_{\rm TF}$, given by
\begin{equation}
V(r)+U_0|\psi_{\rm TF}|^2=\mu,
\end{equation}
which, for the
atom-ion potential with asymptotic form given by
Eq.\ (\ref{pot}), amounts to
\begin{equation}
n_{\mathrm{TF}}(r)=n_{0}-\frac{V(r)}{U_{0}}=
n_{0}\left(1+\frac{(\xi\beta_{4})^{2}}{r^{4}}\right)\label{TFsolution}
\end{equation}
or, to first order in $V$,
\begin{equation}
\psi_{\rm TF}\approx \psi_0 - \frac{V(r)}{2U_0 \psi_0},
\label{deltapsiTF}
\end{equation}
where we have taken $\psi_0$ to be real. The density perturbation at large distances is seen to be
always positive. Corrections to
this result for smaller $r$ may be calculated from Eq.\
(\ref{linearGP}) by neglecting the potential on the left hand side
of the equation. The resulting differential equation may be
written in terms of a function $\delta\chi$ defined by
$\delta\chi=r\delta\psi$,
\begin{equation}
\left(-\frac{d^2}{dr^2}+k_{\xi}^2\right)\delta\chi=-\frac{2m}{\hbar^2}rV(r)\psi_0.
\label{chieq}
\end{equation}
This differential equation may be solved exactly in terms of
exponential integrals, the two linearly independent solutions of the
homogeneous equation being $\exp(-k_{\xi}r)$ and $\exp(k_{\xi}r)$. By
inspection of (\ref{chieq}) it is evident that the leading term for
large $r$ of the particular solution to the inhomogeneous equation is
given by $\delta \chi=- (2m/\hbar^2k_{\xi}^2)rV(r)\psi_0$, which
yields the Thomas-Fermi expression (\ref{deltapsiTF}). The correction
to this result may be obtained from the exact solution, but it is
simpler to iterate Eq.\ (\ref{chieq}) by moving the term
$d^2\delta\chi/dr^2$ to the right hand side and replacing $\chi$ in it
by the Thomas-Fermi solution. This results in
\begin{equation}
\frac{\delta \psi}{\psi_0}=-\frac{2m}{\hbar^2k_{\xi}^2}V(r)\left(1+\frac{12}{k_{\xi}^2r^2}\right).
\label{chieq1}
\end{equation}
The leading correction to the Thomas-Fermi result for $\delta\psi$
given in (\ref{deltapsiTF}) is thus seen to be proportional to
$r^{-6}$. Since we have already neglected the potential energy on
the left hand side of (\ref{linearGP}), we cannot by this method
obtain higher-order corrections to the particular solution than
the one exhibited in (\ref{chieq1}).
By keeping in the general solution only the exponentially decaying
term we thus get the asymptotic result
\begin{equation}
\frac{\delta \psi}{\psi_0} \sim -\frac{V(r)}{2n_0U_0 }\left(1+\frac{6\xi^2}{r^2}\right)
+C\frac{{\rm e}^{-k_\xi r}}{r}, \label{deltapsilarger}
\end{equation}
where $C$ is an arbitrary constant.
For $r \rightarrow \infty$ the asymptotic behavior of the solution is
always given by the TF result. However, whether or not this behavior
is relevant for determining the structure of most of the cloud of
atoms surrounding the ion depends on the relative size of the two
characteristic lengths, $\beta_4$ and $\xi$. On the one hand, for
$\beta_4\gg\xi$ most of the cloud will be described by the TF
approxmation, and only at distances less than $\sim \xi$ will the
exponential term become important. On the other hand, for
$\xi\gg\beta_4$ (i.e.\ for low external density) the structure will be
dominated by the exponential term, and the TF tail will become
important quantitatively only at very large $r$. At shorter distances
from the ion, the mean-field energy becomes small compared with the
atom-ion potential and the GP equation reduces to a good
approximation to the Schr\"odinger equation.
\section{Simple model potentials}
\label{sec:model-pot}
Before presenting results for the attractive $1/r^{4}$
potential we begin by examining two simpler model
potentials, a repulsive hard-core and a spherical well.
\subsection{Hard-core potential}
Consider an interacting Bose-Einstein condensed gas in the
presence of a repulsive hard-core potential of radius $R$.
This model may be treated analytically in both the small and large core radius limits.
The solution to the GP equation at large distances from the ion is given by Eq.\ (\ref{deltapsir}),
\begin{equation}
\psi\simeq\sqrt{n_{0}}\left(1+C\frac{\exp(-k_\xi r)}{r}\right).
\label{Yuk}
\end{equation}
If one assumes that this expression holds for all $r$ greater than
$R$, we can determine the constant of proportionality $C$ by imposing
the boundary condition $\psi(R)=0$. This gives $C=-R{\rm
e}^{k_{\xi}R}\sqrt{n_{0}}$. For $r$ close to $R$ this has the
form of the scattering solution for the Schr\"odinger equation,
$\psi=1-R/r$. In fact, for $R/\xi\ll 1$ this solution becomes
essentially exact, since this function fails to satisfy the GP
equation only in the region where $r \simeq R$, and in this region the
total change in the slope $d\chi/dr$ of the radial wave function is
small and may be neglected. As an illustration of this fact, we
calculate the excess number of particles, which is given by Eq.\
(\ref{DeltaN}), and find
\begin{eqnarray}
\Delta N\!&\!=\!&\!4\pi n_0 \int_{R}^{\infty}\!\!\!dr\, r^{2}\left(-2R\frac{e^{-k_{\xi}(r-R)}}{r} + R^2\frac{e^{-2k_{\xi}(r-R)}}{r^2}\right) \nonumber\\
\!&\!\!&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-\frac {4\pi n_0 R^3}{3}=\nonumber\\
\!&\!=\!&\!-4\pi n_0\left( R\xi^2+ \frac{3R^2 \xi}{2\sqrt 2}+\frac{R^3}{3} \right).
\end{eqnarray}
For $\xi \gg R$, this reduces to
\begin{equation}
\Delta N=-\frac{R}{2a_{\rm aa}}.
\end{equation}
Let us now compare this result with the one derived on the basis of thermodynamic arguments.
For a hard-core potential the scattering length coincides with
the core radius. Since we have assumed the ion to be stationary, its effective mass is taken to be infinitely large, and therefore the reduced mass for the ion and an atom is $m$, rather than the value $m/2$ one obtains for an ion and an atom with equal masses. Thus, this
result is in precise agreement with Eq.\ (\ref{eq: deficit in the dilute limit1}).
When the core radius is much larger than the healing length,
the wave function reaches its asymptotic value on a length scale that
is short compared to $R$. In Eq.\ (\ref{gpradial}) we can therefore
replace the factor $1/r^{2}$ appearing in the nonlinear
term by the constant $1/R^{2}$ and we are left with an effectively
one-dimensional GP equation whose solution is
\begin{equation}
\psi=\sqrt{n_{0}}\tanh\frac{r-R}{\sqrt{2}\xi}, \;\; r\geq R,
\label{eq: 1Dsoliton}
\end{equation}
and zero otherwise, as may be seen by inspection.
The excess number of particles is given by
\begin{equation}
\Delta N= -\frac{4}{3}\pi R^{3}n_{0} -\frac{R^{2}}{\sqrt{2}a_{\mathrm{aa}}\xi},
\end{equation}
where the leading term is due to exclusion of atoms from the core.
\subsection{Attractive square well}
We next consider a more physical potential, an attractive well:
\begin{equation}
V(r)=-\frac{\hbar^{2}k_{0}^{2}}{2m}, \;\;\;r<R,
\end{equation}
$V(r)=0$ otherwise. Like the actual ion-atom potential, this can have
bound states for the two-body problem. With this potential we shall be
able to examine how solutions of the GP equation disappear as the
condensate density increases. The GP equation (\ref{gpradial}) reads
\begin{equation}
\left[-\frac{1}{2}\frac{d^{2}}{d r^{2}}-\frac{k_{0}^{2}}{2}\theta(R-r)+4\pi a_{\rm aa}\left(\frac{\left|\chi\right|^{2}}{r^{2}}-n_0\right)\right]\chi=0
\end{equation}
(where $\chi=r\psi$ and $\theta(x)$ is the step function)
and the scattering length for this potential is\begin{equation}
a=R\left(1-\frac{\tan k_0 R}{k_0R} \right).\end{equation}
Since this equation is nonlinear, there can be multiple solutions for
the same boundary conditions (i.e.\ the same bulk density $n_{0}$). As
we will show in the following, for small $n_0$ it has $2\nu_{\rm S}+1$
solutions, where $\nu_{\rm S}$ is the number of nodes of the zero-energy
Schr\"{o}dinger solution $\psi_{\rm S}$ or, equivalently, the number of bound states
of the Schr\"odinger equation. In the low background density limit,
inside the well the solution with the maximum number of nodes approaches
$\psi_S$, i.e.\ $\psi(r)\propto\sin(k_{0}r)/r$, while outside it
tends towards the uniform density $n_0$ with the asymptotic behavior
given in Eq.\ (\ref{deltapsir}), $\delta\psi(r)\propto\exp(-k_{\xi}r)/r$.
With increasing $n_{0}$, the mean-field repulsion between the atoms
makes the effective potential shallower, which tends to push nodes of
the wave function outwards. At the same time, the increase in the
chemical potential has the opposite effect on the nodes. What we find
is that if the zero-energy solution of the Schr\"odinger equation has
$\nu_{\rm S}$ nodes, for low condensate densities the GP equation has
one solution with no nodes, and {\it two} solutions with any nonzero
number of
nodes less than or equal to $\nu_{\rm S}$.
To demonstrate this, we analyze separately the behavior of the wave
function inside and outside the well, and match them at some
intermediate point, which for this particular potential we take to be
the edge of the well. Specifically, we integrate out from the origin,
where $\chi=0$, for different choices of the derivative of $\chi$ at
$r=0$ and calculate $\psi$ and $\psi'$ at the boundary $r=R$. These
trace a curve in $\psi -\psi'$ space. Then we integrate inwards from
large distances, where the solution is defined by the proportionality
constant $C$ of the Yukawa asymptotic form, Eq.\ (\ref{deltapsir}).
As $C$ is varied, another curve in $\psi -\psi'$ space is traced out.
If the mean-field interaction could be neglected for $r<R$, the ratio
$\psi'(R)/\psi(R)$ would not depend on the normalization of the
wavefunction, and therefore the curve corresponding to the inner
boundary would be a straight line through the origin. In the presence
of atom-atom interactions, the ratio $\psi'/\psi$ obtained by
integrating outwards traces out a spiral. For low $n_0$ this crosses
the $\psi'$ axis a number of times equal to the number of nodes the
zero-energy solution of the Schr\"odinger equation has inside the
potential. This follows from the observation that for low $\chi'(0)$
the solution will have the same number of nodes inside the potential
as the zero-energy solution of the Schr\"odinger equation, while for
very large values of $\chi'(0)$ the effects of the mean field will be
so strong that the solution has no nodes inside the potential.
The corresponding plot obtained by integrating inwards has two
branches, depending on whether $\psi(r\rightarrow \infty)$ is positive
or negative. Examples of the plots are given in Fig.\
\ref{cap:spirals} for parameters such that $\nu_{\rm S}=3 $. For low
$n_0$, there are $2\nu_{\rm S}+1$ intersections of the two sets of
curves, corresponding to solutions of the GP equation. This is
illustrated in Fig.\ \ref{cap:spirals}a. As $n_0$ increases, pairs of
solutions with the same number of nodes merge and disappear, as shown
in Fig.\ \ref{cap:spirals}b. Eventually, at sufficiently high values
of $n_0$ only the nodeless solution survives.
In Figs.\ \ref{cap:spirals} and \ref{cap:merging} we show how,
increasing the external density, the solutions with the highest number
of nodes actually merge. For densities higher than this critical
value, the only solutions are ones with a smaller number of nodes.
\begin{figure}
\includegraphics[width=\columnwidth,angle=0,clip=]{fig1a.eps}
\includegraphics[width=\columnwidth,angle=0,clip=]{fig1b.eps}
\caption{\label{cap:spirals}(Color online) Behavior of $\psi(R)$ and $\psi'(R)$
for the solution inside the well (solid line) and outside it (dashed
and dot-dashed lines for $\psi (r\rightarrow \infty)=\pm \psi_0$,
respectively). In the plots we have set $k_0 R= 9$, which gives
three bound states for the Schr\"odinger equation. We measure
energies in units of $\hbar^2 k_0^2/2m$ and lengths in units of $R$.
The calculations were performed for $U_0=0.45$ in these units, but
results for other values of $U_0$ may be obtained by scaling, since
for a given chemical potential, $\psi$ and $\psi'$ vary as
$U_0^{-1/2}$. The symbols near intersections indicate the number of
nodes of the solution, and for this case $\nu_{\rm S}=3$. The upper
panel (a) is for $\mu=0.45$, and the lower one (b) for $\mu=2.9$,
just above the value $\mu=2.52$ at which the two solutions with 3
nodes merge and disappear.}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth,clip=]{fig2a.eps}
\includegraphics[width=\columnwidth,clip=]{fig2b.eps}
\caption{\label{cap:merging}(Color online) Two solutions of the Gross-Pitaevskii equation
for the attractive square well potential. $k_0R$ and $U_0$ are the
same as in Fig.\ \ref{cap:spirals}. For the upper panel, the
chemical potential is $0.45$, as in Fig.\ \ref{cap:spirals}a, while
for the lower one it is $2.5$, just below the value at which the
solutions merge. The solutions both have three nodes, and are the
first to merge as the chemical potential increases.}
\end{figure}
Despite its short-range character, the model given above captures the
main features of the solutions of the GP equation for the atom-ion
potential, which is long-ranged. We note that the discontinuous
behavior does not occur in a one-dimensional model.
\section{The $r^{-4}$ potential}
\label{polpot}
We now turn to a more realistic potential with the same $r^{-4}$ behavior
as the actual atom-ion interaction at large distances. For definiteness, we consider parameters
appropriate for a $^{87}$Rb condensate, and we take $a_{\rm aa}=100\,a_0$.
At large distances, we take the atom-ion potential to be given by Eq.\
(\ref{pot}) with $\tilde \alpha=320a_0^3$. The wave functions are
sensitive to the short-range behavior of the potential, but we may
obtain illustrative results by cutting the $1/r^4$ potential off by a
hard core of radius $R$. Since
many atoms are bound to the ion, we assume the ion to be static and
set $m_{\rm ai}= m$. The atom-ion scattering length of such potential may
be calculated in the WKB approximation, and is given by
\cite{GribFlam}
\begin{equation}
a_\mathrm{ai}=\beta_{4}\cot\left[\frac{\beta_{4}}{R}\right].
\label{eq: a(ai)}
\end{equation}
The number of bound states allowed by the potential can be estimated by increasing the potential strength from 0 to its actual value. A bound state appears each time the scattering length diverges, and therefore the number of bound states is given by
\begin{equation}
\nu_{\rm S}={\rm Int}\left(\frac{\beta_4}{\pi R}\right),
\label{bs}
\end{equation}
where ${\rm Int}(x)$ denotes the integer part of $x$.
To model actual atom-ion potentials, a physically reasonable value of
$R$ would be $\sim 10a_0$. However, the properties of the wave
function of most importance here are those at relatively large
distances, $r\ga 10^3 a_0$, so we take $R=300a_0$, since this should
give us the correct physical behavior for the distances of
interest. We do not expect the qualitative behavior of the wave
function to depend on $R$, even though quantities like the scattering
length do, and we have verified this numerically.
We now describe numerical solutions of the GP equation that approach a
constant density $n_{0}$ far from the ion. Just as for the finite-range
potential considered in the previous section, there is generally
more than one solution for a given value of the chemical potential,
and for small external densities one expects $2\nu_{\rm S} +1$. In
Fig.\ \ref{cap: 2 solutions} we show the wave functions
corresponding to the two states with the highest number of nodes,
namely seven for the parameters chosen, in agreement with the
quasiclassical result (\ref{bs}). The free energy, for a given
condensate density $n_0$, is highest for the states with the highest
number of nodes, and decreases as the number of nodes decreases.
In the absence of inelastic processes, we expect only the
uppermost state of the ionic potential to play an important role in
the capture process, since it is the only one with an appreciable
overlap with the continuum wave function representing the unbound atoms
\cite{Cote'}.
\begin{figure}
\includegraphics[width=\columnwidth,clip=]{fig3.eps}
\caption{\label{cap: 2 solutions}
(Color online) Condensate wave functions for the two uppermost states in the $1/r^4$
potential with the parameters given in the text for
$n_0=10^{14}$cm$^{-3}$. Both states have seven nodes, but the
resolution of the figure is inadequate to exhibit the rapid
oscillations for $r$ close to $R$. The state that, in the dilute limit, becomes
the zero-energy solution of the Schr\"odinger equation is given by the solid
line.}
\end{figure}
The excess number of atoms is given in terms of the atomic density distribution by
Eq.\ (\ref{DeltaN}) or,
alternatively, from the free energy $F=E-\mu N$ by Eq.\ (\ref{DeltaN-FreeEnergy}).
In Fig. \ref{cap:Surplus-of-atoms} we show results obtained from our
numerical simulations by both methods. In the limit of very low
condensate density we get values for $\Delta N$ in accord with the
thermodynamic arguments in Sec. \ref{sec:thermo}. The consistency of
the two methods of calculation has been confirmed for core radii that
give scattering lengths in the range $\left|a_{\rm ai}\right|<5000\,
a_{0}$.
\begin{figure}
\includegraphics[width=\columnwidth,clip=]{fig4.eps}
\caption{\label{cap:Surplus-of-atoms}(Color online) Excess number of atoms around
a single ion as a function of the bulk density. The dashed line is
the dilute limit appropriate for a fixed ion, $\Delta N=-a_{\rm
ai}/2a_{\rm aa}$ ($R=300\, a_0$ gives $a_{\rm ai}\approx -1980a_0$
for an infinitely massive ion). Results are shown for the four
uppermost levels for this potential (i.e.\ the two with 7 nodes and
the two with 6 nodes, indicated respectively by the solid and
dot-dashed lines). The lines are obtained from Eq.\ (\ref{DeltaN}),
the circles from Eq.\ (\ref{DeltaN-FreeEnergy}). The inset exhibits
the behavior at lower densities.}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth,clip=]{fig5.eps}
\caption{\label{cap:DeltaFreeEn}
(Color online) Difference in free energy for the states given in the
previous figure: the solid lines are for the two states with 7 nodes
and the dot-dashed line for one of the states with 6 nodes. The free
energy is measured in units of 10$^{-5}\hbar^2/ma_0^2$. The other
state with 6 nodes lies much lower, at around $\Delta F \approx
-3\cdot10^{-4} \hbar^2/ma_0^2$.}
\end{figure}
The figure shows that the excess numbers of atoms for two states with
the same number of nodes become equal at the density above which the
solutions no longer exist. This is to be expected, since the
solutions become identical at this point. In Appendix A we use
quasiclassical arguments to estimate the density at which solutions
merge and disappear, and these are in good agreement with the
numerical results.
In the detailed calculations described so far we have focused
attention on states with close to the maximum number of nodes. In
particular, in the low-density limit and in the absence of inelastic
processes that can cause the system to relax, one would expect the
state of the condensate to be the one that close to the ion resembles
the zero-energy solution of the Schr\"odinger equation. However,
three-body processes can relax the system, thereby populating states
with lower numbers of nodes. To calculate properties of such a
system, one could start with a many-particle wave function of the
Hartree-Fock type in which more than one single-particle state is
occupied, and solve the Hartree-Fock equations. This is, however,
beyond the scope of this paper because the density of atoms rises to
values sufficiently high that the dilute gas approximation for the
interaction energy employed in the GP approach fails at relatively
large distances from the ion. To see this, we note that the density
of atoms far from the ion will be given by the Thomas--Fermi
approximation, Eq.\ (\ref{TFsolution}). The dilute gas approximation
is valid provided $n|a_{\rm aa}|^3\ll 1 $. This condition becomes
\begin{equation} n|a_{\rm aa}|^3 \approx \left|\frac{V(r)}{U_0}a_{\rm aa}^3\right|
=\frac{\beta_4^2 a_{\rm aa}^2}{8 \pi r^4} \ll 1 \end{equation} or \begin{equation} r\gg
(\beta_4 |a_{\rm aa}|)^{1/2}/2, \end{equation} which for rubidium ($a_{\rm
aa}\approx 100 a_0$) implies that the GP equation is valid only for
$r \gg 400 a_0 $ for such states.
\section{Conclusions and discussion}
\label{concl}
In this paper we have investigated solutions of the Gross-Pitaevskii
equation for a Bose--Einstein condensate in the presence of a positive
ion. We find that for low condensate densities, there are $2\nu_{\rm
S}+1$ solutions for a given condensate density, where $\nu_{\rm S}$ is
the number of bound states of the Schr\"odinger equation. With increasing condensate density, pairs of states become
degenerate and disappear, and the state of the system must change
discontinuously. An interesting challenge is to find experimental
evidence for such a behavior.
We have calculated the excess number of atoms around an ion, and find
that for the state that resembles the zero-energy solution of the
Schr\"odinger equation it can be either positive or negative,
depending on the sign of the atom--ion phase shift, and a typical
magnitude is of order $\sim 10^2$ . The spatial size of the density
disturbance around an ion is set by $\beta_4 \sim 1$ $\mu$m. Our
estimates indicate that the Gross-Pitaevskii equation should give a
reliable first approximation for the wave function of such states. For
states with fewer nodes, the density of atoms may reach values high
enough that the GP equation fails.
There are many outstanding problems. In most of the calculations we
have assumed that the state of interest is that with the maximum
possible number of nodes. More study is needed of inelastic processes
that will cause atoms to relax to lower states \cite{Cote'}.
Experimental studies will be valuable in providing guidance for future
work.
\subsubsection*{Acknowledgments}
We are grateful to Halvor Nilsen for valuable discussions. We also
thank Andrew Jackson and Alexander Lande for helpful insights, and
Marco Anderlini and Vasili Kharchenko for useful correspondence.
Part of this work was done while one of the authors (CJP)
participated in the program on Quantum Gases at the Kavli Institute
for Theoretical Physics at the University of California, Santa
Barbara. This research was supported in part by the National Science
Foundation under Grant No. PHY99-07949.
|
{
"timestamp": "2004-11-12T15:51:31",
"yymm": "0411",
"arxiv_id": "cond-mat/0411339",
"language": "en",
"url": "https://arxiv.org/abs/cond-mat/0411339"
}
|
\section{Introduction}
The problem of determining the probability of encountering (matching)
a given string of length $l$ in another string of length $k$,
whose letters have been drawn randomly from
an alphabet of $r$ letters, has a variety of applications
ranging from designing fast algorithms for pattern searching \cite{boyer,knuth},
to problems in genetics such as assessing the likelihood of events
such as the frequency of
occurrence of DNA segments \cite{pevzner,prum}, or that certain DNA segments
align \cite{karlin,dembo}. In each of theses cases, the likelihood
estimates for random sequences can be used as a benchmark against which one
can evaluate the statistical significance of actually observed events.
The problem is non-trivial, because of the possibility of overlapping
occurrences in the string, which introduce correlations that need to be
dealt with. Guibas and Odlyzko \cite{guibas0,guibasI,guibasII}
derived the moment generating functions associated with
the probability for not encountering a given set of words in a random string,
whose letters were distributed independently and identically. The resulting
distributions turn out to depend on a set of correlation functions that
capture the overlap properties of the words with each other.
Building on the work of Guibas and Odlyzko, several authors have studied the
probability distribution for the number of occurrences $n$ of of a given
$l$ letter word in a random string of $k$ letters, under various
assumptions on the distribution of random letters
\cite{chrys,geske,fudos,regszpan,pevzner,goldwater,waterman,schbath,prum,reinertetal}:
The cases where the letters of the random string are independently
and identically distributed (i.i.d.) was treated by Fudos et al. \cite{fudos},
whereas the case where the letter distribution follows the
steady state distribution of a Markov process has been
investigated by several authors \cite{chrys,geske,regszpan,waterman,reinertetal}.
All of these results have been obtained for asymptotic regimes
($k$ large + various assumptions on the length of the word $l$),
where tools of statistics such as the central limit theorem
\cite{fudos,regszpan,prum}, theory of large deviations
\cite{reinertetal,regszpan}, or (compound) poisson
approximations for rare-events \cite{chrys,geske,goldwater,schbath} are
applicable.
The regimes of applicability can be difficult to identify, however.
It has been noted that, even in the case of i.i.d letters, when the
length $l$ of the word to be matched is fixed, and assuming the length
of the random string to be large, the most accurate approximation
to chose (gaussian or compound poisson) still depends on the
word itself that is being matched \cite{robschbath}.
It is therefore desirable to come up with a single, explicit
analytical expression for the
probability distribution that is generally valid, and to obtain the
asymptotic expressions, mentioned above, as special cases by
taking certain limits.
This is what we set out to do in this article. Besides the
obvious advantage of having a single description, such an approach will
naturally identify the regimes of application of the various asymptotic
approximations, while also pointing out when and how they fail.
It turns out that all of these issues are present even in the simplest
case where the letters of the random string are uniformly and independently
distributed. For the sake of simplicity and clarity of presentation, we
will perform the analysis for this case. However, we will point out
in detail how these results carry over to the more general case of
random letter distributions.
Our approach to this problem, which appears to be novel,
can be summarized as follows: We first show that the
problem of calculating the probability distribution for the number of
occurrences $n$ of of a given $l$ letter word in a random string of $k$
letters, can be rigorously mapped into the problem of calculating the
configurational part of the grand-canonical partition function
of a 1d lattice gas. In this
mapping the number of particles correspond to the number occurrences,
the "volume" of the gas is the length of the random string, and the correlations
between subsequent occurrences turn into pairwise interactions whose nature
depends on properties of the word to be matched. It turns out that common to
all interactions is a relatively strong and short-ranged segment of range $l$,
followed by a weak and exponentially decaying tail.
With the help of the lattice gas analogy, and by using techniques of
liquid theory, such as the virial expansion, we are able to obtain
an analytical expression for the probability distribution that
reproduces the known asymptotic limits. We show how the distribution
crosses over into the asymptotic forms of the distribution,
and thereby expose the conditions required
for these limits to be applicable.
More importantly, our method allows us to analytically treat the
intermediate regime of moderate string lengths, $k \gtrsim r^\ell \gtrsim 1$,
as well. This regime is most relevant for biological applications and
turns out to be the hardest one to tackle analytically,
since in this regime the effects of the tail are strong and need to be kept
in the analysis. This is also the reason behind the deviations of the
asymptotic forms from the actual distribution, since, as we will show,
these distributions are obtained by neglecting the tail of the interactions.
These deviations become more pronounced for short words and
small number of letters in the alphabet, small $l$ and $r$, respectively.
Our results are readily generalized to the broader class of letter distributions,
such as non-uniformly distributed letters or letters generated by a Markov
process. Such distributions give rise to a broader class of
effective interactions. In particular, it turns out that these
interactions can have stronger tails than can be achieved by a
uniform letter distribution. This potentially renders our method
of approach even more relevant to such letter distributions.
We would also like to note that our approach is in spirit similar to recent
attempts at solving combinatorial problems, such as the k-SAT problem
\cite{kirkpatrick,mezard,mertens,achlioptas}, using ideas borrowed from
statistical mechanics \cite{fuanderson,monasson}.
The article is organized as follows:
In Section II we introduce our notation and formalism,
rederiving in this setting some of the relevant and known results.
In Section III we establish the partition function analogy.
We derive and study the properties of the effective particle
interactions and then set up a virial expansion for the
"equation of state". From the virial expansion we obtain
the $n$-particle partition function, which in this analogy
corresponds to the $n$-match probability distribution function.
We show how the various know limits
arise and discuss the underlying assumptions.
Section IV discusses the generalization and implications of our
approach to the more general class of letter distributions studied
in the literature and we will discuss our
results in Section V.
\section{The matching probabilities}
In this section we derive some of the known expressions for the
the matching and $n$-match probability, the probability of a
at least one and precisely $n$ occurrences of a given word, respectively.
Besides providing a review of the relevant results, the main
purpose of this section is to introduce our notation and
provide the setting for the statistical mechanics approach
to be taken up in the following Section.
\subsection{Definitions}
Assume that $x$, and $y$ are variables that take
values from an $r$ letter alphabet such that
$x, y \in \{0, \ldots , r-1\} $.
Let ${\bf x} = (x_1,x_2,x_3, \ldots, x_{l})$ and
${\bf y} = (y_1,y_2,y_3, \ldots, y_{l})$,
be two strings of $l$ letters.
Define the match indicator function
$ \Phi({\bf x},{\bf y})$ as
\begin{equation}
\Phi({\bf x},{\bf y }) = \prod_{t=1}^{l} \delta(x_t,y_t)
\label{eqn:Phidef}
\end{equation}
So that we have
\begin{equation}
\Phi({\bf x}, {\bf y}) =
\left \{\begin{array}{ll} 1, & \mbox{if ${\bf x} = {\bf y}$} \\
0, & \mbox{otherwise}. \end{array} \right.
\label{eqn:Phidefetaz}
\end{equation}
Let ${\bf y} = (y_1,y_2, \ldots , y_k) $ be a string of length
$k \ge l$ and denote by ${\bf y}_{a,l} = ( y_{a+1}, y_{a+2}, \ldots , y_{a+l})$
the substring of length $l$ starting at position $a$, $ a = 0, 1, \ldots, k-l$.
Furthermore, let
\begin{equation}
f_a({\bf x},{\bf y}) = \Phi({\bf x},{\bf y}_{a,l}).
\label{eqn:fdef}
\end{equation}
We have
\begin{equation}
f_a({\bf x},{\bf y})
= \left \{\begin{array}{ll} 1, & \mbox{$ {\bf x} = {\bf y}_{a,l}$ }
\\ 0, & \mbox{otherwise}. \end{array} \right.
\label{eqn:fadef}
\end{equation}
In other words, $f_a({\bf x},{\bf y}) = 1$,
if and only if ${\bf x}$ matches ${\bf y}$ at position $a$, and zero otherwise.
\subsection{The matching probability}
Define $p(m; { \bf x})$ to be the the probability that
a given word ${\bf x}$ of length $l$ is contained
{\em at least once} in a randomly
drawn string ${ \bf y}$ of length $m+l$. We will refer to this as the
{\em matching probability}.
Let $I_M({\bf x},{\bf y})$ be the function that takes on the value
one if the $k$-string ${ \bf y}$ contains the given $l$-string ${\bf x}$
at least once, and
zero otherwise. Using Eq.~(\ref{eqn:fadef}), we can write
\begin{equation}
I_M({\bf x},{\bf y}) = 1 - \prod_{a=0}^{k-l}
\left [ 1 - f_a({\bf x},{\bf y}) \right ].
\end{equation}
Since $k \ge l$, it is convenient to define the excess length $m = k-l$.
We thus find
\begin{equation}
p(m; { \bf x}) = 1 - \frac{1}{r^{m+l}} \sum_{{\bf y}}^{} \prod_{a=0}^{m}
\left [ 1 - f_a({\bf x},{\bf y}) \right ],
\label{eqn:p1function}
\end{equation}
where $r^{m+l}$ is the number of distinct $k=m+l$-strings of $r$-letters,
and the summation is over all such strings ${\bf y}$.
In \cite{mungan},
the products on the right hand side of Eq.~(\ref{eqn:p1function})
were expanded into a Mayer-like sum,
\begin{equation}
p(m;{\bf x}) = \frac{1}{r^k} \sum_{{\bf y}}^{} \sum_{a} f_a
- \frac{1}{r^k} \sum_{{\bf y}}^{} \sum_{a < b} f_a f_b
+ \frac{1}{r^k} \sum_{{\bf y}}^{} \sum_{a < b < c} f_a f_b f_c - \ldots,
\label{eqn:meier}
\end{equation}
(arguments of $f_a$ will be suppressed in what follows)
and the terms in the sum where evaluated approximately.
Here we will take a different approach.
The following algebraic identity will be of use in the following:
\begin{equation}
1 - \prod_{a=0}^{m} \left ( 1 - f_a \right ) =
\sum_{b=0}^{m} f_b \prod_{a=0}^{b-1} \left ( 1 - f_a \right ),
\label{eqn:flemma}
\end{equation}
with the convention that when $b=0$,
the product on the right hand side is set to one.
Eq.~(\ref{eqn:flemma}) is readily proven by induction.
Using this identity, $p(m;{\bf x})$ can be written as
\begin{equation}
p(m;{\bf x}) = \frac{1}{r^{m+l}} \sum_{{\bf y}}^{}
\sum_{b=0}^{m} f_b \prod_{a=0}^{b-1} \left ( 1 - f_a \right ).
\end{equation}
Note that for any
given $b$, the expression on the right hand side only involves the variables
$y_1, y_2, \ldots, y_{b+l}$. The sum over the remaining indices yields
$r^{m-b}$ and we find that
\begin{equation}
p(m;{\bf x}) = \sum_{b=0}^{m} \frac{1}{r^{b+l}}
\sum_{y_1 \cdots y_{b+l}}^{} f_b \prod_{a=0}^{b-1} \left ( 1 - f_a \right ).
\end{equation}
Defining the correlator $d(b;{\bf x})$ as
\begin{equation}
d(b;{\bf x}) = \sum_{y_1 \cdots y_{b+l}}^{} f_b \prod_{a=0}^{b-1} \left ( 1 - f_a
\right ),
\label{eqn:ddef}
\end{equation}
$p(m;{\bf x})$ can be therefore written as
\begin{equation}
p(m;{\bf x}) = \sum_{b=0}^{m} \frac{1}{r^{b+l}} d(b;{\bf x}).
\label{eqn:pdef}
\end{equation}
We can obtain a recursion relation for $d(b;{\bf x})$ by factoring
out the $a=0$ term in Eq.~(\ref{eqn:ddef}),
\begin{equation}
d(b;{\bf x}) = \sum_{y_1 \cdots y_{b+l}}^{} f_b \prod_{a=1}^{b-1}
\left ( 1 - f_a \right ) -
\sum_{y_1 \cdots y_{b+l}}^{} f_0 f_b \prod_{a=1}^{b-1} \left ( 1 - f_a \right ).
\end{equation}
The argument of the first sum does not contain the variable $y_1$,
while the sum over the remaining variables yields $d(b-1;{\bf x})$. Thus,
\begin{equation}
d(b;{\bf x}) = r d(b-1;{\bf x}) - h(b;{\bf x}),
\label{eqn:drec}
\end{equation}
with the correlator $h$ defined as
\begin{equation}
h(b;{\bf x}) = \sum_{y_1 \cdots y_{b+l}}^{} f_0 \left [ \prod_{a=1}^{b-1}
\left ( 1 - f_a \right ) \right ] f_b.
\label{eqn:gdef}
\end{equation}
Note that for $m=0$
\begin{equation}
p(0; {\bf x}) = \frac{1}{r^l}.
\end{equation}
Comparing with Eq.~(\ref{eqn:pdef}) this implies that
\begin{equation}
d(0; {\bf x}) = 1.
\end{equation}
Since there are no constraints on $h(0; {\bf x})$, we will define
$h(0; {\bf x}) = 0$.
We next seek a recursion relation for $h$. Using the
identity, Eq.~(\ref{eqn:flemma}), we find from Eq.~(\ref{eqn:gdef})
\begin{equation}
h(b;{\bf x}) =
\sum_{y_1 \cdots y_{b+l}}^{} \left \{ f_0 f_b - \sum_{c=1}^{b-1} f_0
\left [ \prod_{a=1}^{c-1} \left ( 1 - f_a \right ) \right ] f_c f_b \right \}.
\label{eqn:g2}
\end{equation}
Recall from the definition of $f_b({\bf x},{\bf y})$ that $f_b$ is a product of
Kronecker deltas, Eqs.~({\ref{eqn:Phidefetaz}) and
({\ref{eqn:fdef}). The Kronecker deltas enforce a transitive relation between their
arguments, and we can write
$f_b({\bf x},{\bf y};0) = f_b({\bf x},{\bf y};0) f_b({\bf x},{\bf \tilde{y}};0)$,
where we have introduced an auxiliary set of variables ${\bf \tilde{y}}$ over
which a sum is to be performed.
Thus Eq.~(\ref{eqn:g2}) can be rewritten as
\begin{equation}
\sum_{y_1 \cdots y_{b+l}}^{} \sum_{c=1}^{b-1} f_0 \left [ \prod_{a=1}^{c-1}
\left ( 1 - f_a \right ) \right ] f_c f_b =
\sum_{c=1}^{b-1} \left \{ \sum_{y_1 \cdots y_{b+l}}^{} f_0 \left [ \prod_{a=1}^{c-1}
\left ( 1 - f_a \right ) \right ] f_c \right \}
\left \{ \sum_{\tilde{y}c \cdots \tilde{y}_{b+l}}^{}
f_c f_b \right \}.
\label{eqn:grel}
\end{equation}
Defining the correlator $C(b;{\bf x})$ as
\begin{equation}
C(b;{\bf x}) = \sum_{y_1 \cdots y_{b+l}}^{} f_0({\bf x},{\bf y})
f_b({\bf x},{\bf y}),
\label{eqn:cdef}
\end{equation}
and substituting Eq.~(\ref{eqn:grel}) into
Eq.~(\ref{eqn:g2}), we find
\begin{equation}
h(b;{\bf x}) = C(b;{\bf x}) - \sum_{a=1}^{b-1} h(a;{\bf x}) C(b-a;{\bf x}).
\label{eqn:grec}
\end{equation}
Using Eq.~(\ref{eqn:cdef}), it can be easily shown that for $b\ge l$
\begin{equation}
C(b;{\bf x}) = r^{b-l}.
\label{eqn:CCdef}
\end{equation}
Denoting the values of $C(b;{\bf x})$ for $b<l$ by $c_b({\bf x})$, we have
\begin{equation}
c_b({\bf x}) = \sum_{y_1 \cdots y_{b+l}}^{}
f_0({\bf x},{\bf y};0) f_b({\bf x},{\bf y};0), \; \; 0 < b < l,
\label{eqn:cexpl}
\end{equation}
and thus
\begin{equation}
C_b = \left \{ \begin{array}{ll} c_b, & \mbox {$0 < b < l$,} \\ r^{b-l} & \mbox{$b \ge l$.}
\end{array} \right.
\label{eqn:Concemore}
\end{equation}
As evident from Eq.~(\ref{eqn:cexpl}), the set of indices $c_b({\bf x}) \in \{0,1 \}$,
with $b=1,2,\ldots,l-1$ measure the auto-correlations of ${\bf x}$. They
are referred to as the bit-vector ${\bf c} = (c_1,c_2,\ldots,c_{l-1})$
associated with ${\bf x}$, and were studied by Harborth \cite{harborth} and
later in considerable detail by Guibas and
Odlyzko \cite{guibas0,guibasI,guibasII}.
\subsection{Bit-vectors}
\label{bit-sect}
From the definition, Eq.~(\ref{eqn:cexpl}), it is
clear that $c_b = 1$ if and only if the string ${\bf x}$ shifted by an amount
$b$ relative to itself coincides on the overlapping part. Conversely $c_b = 0$,
if the overlapping part does not coincide. It turns out that the set
of $r^l$ possible words ${\bf x}$ of length $l$ are partitioned into equivalence
classes with respect to their bit-vectors ${\bf c} = (c_1,c_2, \ldots c_{l-1})$
and that the possible classes are independent of the number of letters $r$
(as long as $r \ge 2$) \cite{guibasI}.
Tables \ref{bitv1} and \ref{bitv2} list the sets of possible
bit-vectors upto $l=8$ along with the number of elements in their respective
equivalence classes for $r=2,3,4$.
We see that the definition of $c_b({\bf x})$ imposes strong
conditions on the possible values of the $l-1$ bits of a bit-vector and it turns out
that the resulting bit-vectors have interesting properties \cite{guibasI,rivrah},
of which we will mention only the most relevant ones.
For example, if $c_p = c_q = 1$ with $p<q$ this implies that $c_t = 1 $ for all $t$ of the
form $t = p + i (q-p)$ with, $i=0, 1, 2, \ldots$ and $t < l$. This is referred
to as the {\em forward propagation rule} \cite{guibasI}. In particular,
$c_p = 1$ implies that $c_{ip} = 1$ for all $i, 1, 2, \ldots$ such that
$ip < l$. The latter result shows that $p$ can be considered as a period.
We define the {\em fundamental period} of a string ${\bf x}$, $\chi({\bf x})$,
to be the smallest $p$, with $0<p<l$ such that $c_p = 1$. If $ {\bf x}$ is such
that its bit-vector is $000\cdots0$ (all zeroes), we define $\chi({\bf x}) = l$.
\subsection{The $n$-match probability}
Denote by $p(n;m,{ \bf x})$ the probability that
that a randomly drawn $k$-string ${ \bf y}$ contains a given $l$-string
${\bf x}$ {\em precisely} $n$ times. We will refer to this as the
{\em n-match probability}.
If we let the random variable $N({\bf x},{\bf y})$ denote the number
of occurrences of ${\bf x}$ in ${\bf y}$, it follows that
\begin{equation}
N({\bf x},{\bf y}) = \sum_{a=0}^{m} f_a.
\end{equation}
Thus the average number of matches $\langle n \rangle$ and its
second moment $\langle n^2 \rangle$ are readily obtained as
\begin{equation}
\langle n \rangle = \frac{1}{r^{m+l}} \sum_{\bf y} \sum_{a=0}^{m} f_a = \frac{m+1}{r^l}
\label{eqn:nave}
\end{equation}
and
\begin{equation}
\langle n^2 \rangle = \frac{1}{r^{m+l}} \sum_{\bf y} \sum_{a<b}^{} f_a f_b.
\end{equation}
\twocolumn
\begin{table}
\vskip 0.25cm
\begin{tabular}{||r|l|l|l||}
\hline
${\bf c}$ & $r=2$ & $r=3$ & $r=4$ \\
\hline
0 & 2 & 6 & 12 \\
1 & 2 & 3 & 4 \\
\hline
00 & 4 & 18 & 48 \\
01 & 2 & 6 & 12 \\
11 & 2 & 3 & 4 \\
\hline
000 & 6 & 48 & 180 \\
001 & 6 & 24 & 60 \\
010 & 2 & 6 & 12 \\
111 & 2 & 3 & 4 \\
\hline
0000 & 12 & 144 & 720 \\
0001 & 10 & 66 & 228 \\
0010 & 4 & 18 & 48 \\
0011 & 2 & 6 & 12 \\
0101 & 2 & 6 & 12 \\
1111 & 2 & 3 & 4 \\
\hline
00000 & 20 & 414 & 2832 \\
00001 & 22 & 210 & 948 \\
00010 & 6 & 48 & 180 \\
00011 & 6 & 24 & 60 \\
00100 & 4 & 18 & 48 \\
00101 & 2 & 6 & 12 \\
01010 & 2 & 6 & 12 \\
11111 & 2 & 3 & 4 \\
\hline
\end{tabular}
\caption{Equivalence classes of bit-vectors and their number of elements.
The table shows the bit-vectors ${\bf c} = (c_1,c_2,\ldots,c_{l-1})$
associated with strings of length $l=2,3,4,5$ and $6$ and
the number of elements in these equivalence classes
for $r=2,3$ and $4$ letter alphabets.}
\label{bitv1}
\end{table}
\begin{table}
\vskip 0.25cm
\begin{tabular}{||r|l|l|l||}
\hline
${\bf c}$ & $r=2$ & $r=3$ & $r=4$ \\
\hline
000000 & 40 & 1242 & 11328 \\
000001 & 38 & 606 & 3732 \\
000010 & 16 & 162 & 768 \\
000011 & 12 & 72 & 240 \\
000100 & 8 & 54 & 192 \\
000101 & 2 & 12 & 36 \\
000111 & 2 & 6 & 12 \\
001001 & 6 & 24 & 60 \\
010101 & 2 & 6 & 12 \\
111111 & 2 & 3 & 4 \\
\hline
0000000 & 74 & 3678 & 45132 \\
0000001 & 82 & 1866 & 15108 \\
0000010 & 26 & 462 & 3012 \\
0000011 & 22 & 210 & 948 \\
0000100 & 16 & 162 & 768 \\
0000101 & 8 & 54 & 192 \\
0000111 & 6 & 24 & 60 \\
0001000 & 6 & 48 & 180 \\
0001001 & 6 & 24 & 60 \\
0010010 & 4 & 18 & 48 \\
0010011 & 2 & 6 & 12 \\
0101010 & 2 & 6 & 12 \\
1111111 & 2 & 3 & 4 \\
\hline
\end{tabular}
\caption{Equivalence classes of bit-vectors and their number of elements.
The table shows the bit-vectors ${\bf c} = (c_1,c_2,\ldots,c_{l-1})$
associated with strings of length $l=7$ and $8$ and
the number of elements in these equivalence classes
for $r=2,3$ and $4$ letter alphabets.}
\label{bitv2}
\end{table}
\onecolumn
The latter expression can be worked out using Eqs.~(\ref{eqn:cdef}), (\ref{eqn:CCdef}) and
(\ref{eqn:cexpl}) and we find for the variance
\begin{equation}
\sigma_n^2 = \frac{m+1}{r^l} + \frac{1}{r^{2l}} \left [ (m+1)(1-2l) + l(l-1) \right ]
+ \frac{1}{r^l} \sum_{b=1}^{l-1} (m+1-b)c_b({\bf x}).
\label{eqn:sigman}
\end{equation}
Eq.~(\ref{eqn:sigman}) is a special case of a result due to Kleffe and Borodovsky
\cite{kleffe}, who considered general distributions of random letters.
Let
$I_{n,m}(a_1,a_2,\ldots,a_n;{\bf x}, {\bf y})$ be the
function that takes on the value $1$ when $n$ matches occur that are
located at positions
$a_1,a_2,\ldots,a_n$, with $0 < a_1 <a_2 < \cdots < a_n < m$ and zero otherwise,
\begin{eqnarray}
I_{n,m}(a_1,a_2,\ldots a_n;{\bf x}, {\bf y}) &=& \nonumber \\
\left [ \prod_{i_1=1}^{a_1-1}(1-f_{i_1}) \right ] f_{a_1}
\left [ \prod_{i_2=a_1+1}^{a_2-1}(1-f_{i_2}) \right ] f_{a_2}
&\cdots&
\left [ \prod_{i_n=a_{n-1}+1}^{a_n-1}(1-f_{i_n}) \right ] f_{a_n}
\left [ \prod_{i_{n+1}=a_{n}+1}^{m}(1-f_{i_{n+1}}) \right ] . \nonumber \\
\label{eqn:Imndef}
\end{eqnarray}
In terms of $I_{n,m}(a_1,a_2,\ldots a_n;{\bf x}, {\bf y})$ we can
write $p(n;m,{\bf x})$ as
\begin{equation}
p(n;m,{\bf x}) =
\sum_{a_1<a_2<\cdots<a_n}\frac{1}{r^{m+l}}\sum_{\bf y}
I_{n,m}(a_1,a_2,\ldots a_n;{\bf x}, {\bf y}).
\label{eqn:pnmdef}
\end{equation}
Analogously to the reasoning leading from Eq.~(\ref{eqn:g2}) to
Eq.~(\ref{eqn:grel}), it can be shown that the sum over ${\bf y}$ factorizes
$I_{n,m}(a_1,a_2,\ldots a_n;{\bf x}, {\bf y})$ as
\begin{equation}
\frac{1}{r^{m+l}} \sum_{\bf y} I_{n,m}(a_1,a_2,\ldots a_n;{\bf x}, {\bf y})
= \frac{1}{r^{m+l}} d(a_1)
\left [ \prod_{i=1}^{n-1}h(a_{i+1}-a_{i}) \right ] d(m-a_n),
\label{eqn:Gaaexp}
\end{equation}
where $d$ and $h$ are as defined in Eqs.~(\ref{eqn:ddef}) and (\ref{eqn:gdef}). Thus
$p(n;m,{\bf x})$ becomes
\begin{equation}
p(n;m,{\bf c})
= \sum_{a_1<a_2<\cdots<a_n} \frac{1}{r^{m+l}} d(a_1)
\left [ \prod_{i=1}^{n-1}h(a_{i+1}-a_{i}) \right ] d(m-a_n),
\label{eqn:pnmaspartition}
\end{equation}
where we have changed the argument of the distribution function to
$p(n;m,{\bf c})$, to emphasize that the
distribution really depends on the bit-vector ${\bf c}$ only. It is readily seen
that the sum over the positions $a_i$ is an $n+1$ fold convolution of $d$ and $h$.
To simplify the results as well as well as to be able to obtain
asymptotic expressions, we next introduce generating functions.
\subsection{Generating functions}
Define the generating function $g(z)$ associated with a sequence $g(b)$ by
\begin{equation}
g(z) = \sum_{b=0}^{\infty} z^b g(b).
\end{equation}
From Eqs.~(\ref{eqn:cdef}), (\ref{eqn:CCdef}) and (\ref{eqn:cexpl}) we find
\begin{equation}
C(z;{\bf x}) = c(z;{\bf x}) + \frac{z^l}{1-zr},
\label{eqn:czdef}
\end{equation}
where $c(z;{\bf x})$ is a polynomial of degree $l-1$,
\begin{equation}
c(z;{\bf x}) = \sum_{b=1}^{l-1} z^b c_b({\bf x}).
\end{equation}
It is useful to also define the polynomial of degree $l$, $\lambda(z;{\bf c})$, as
\begin{equation}
\lambda(z;{\bf c}) = z^l + r^l(1-z) \left [ 1 + c(z/r;{\bf x}) \right ] .
\label{eqn:lambdazdef}
\end{equation}
Using Eqs.~(\ref{eqn:drec}),(\ref{eqn:grec}), (\ref{eqn:CCdef}) and (\ref{eqn:lambdazdef}),
we see that the generating function of $h$ and $d$ are given in terms of $c(z/r;{\bf x})$ as
\begin{equation}
h(z/r;{\bf c}) = 1 - \frac{1}{1 + C(z/r;{\bf x})} = 1 - \frac{1}{1+ c(z;{\bf x}) +\frac{1}{r^l} \frac{z^l}{1-z}},
\label{eqn:hexpl0}
\end{equation}
which can be written in terms of $\lambda(z;{\bf c})$ as
\begin{equation}
h(z/r;{\bf c}) = 1 - r^l \; \frac{1 - z}{\lambda(z;{\bf c})},
\label{eqn:hzexplicit}
\end{equation}
and likewise,
\begin{equation}
d(z/r;{\bf c}) = \frac{1 - h(z/r;{\bf c})}{1-z} = \frac{r^l}{\lambda(z;{\bf c})}.
\label{eqn:dzexplicit}
\end{equation}
The generating function for $p(m;{\bf c})$ thus becomes
\cite{guibas0,guibasII}
\begin{equation}
p(z;{\bf x}, 0) = \frac{1}{1-z} \; \;
\frac{1}{\lambda(z;{\bf c})}.
\label{eqn:pzex}
\end{equation}
Turning next to the generating function of
$p(n;m,{\bf c})$,
\begin{equation}
p(n;z,{\bf c}) = \sum_{m=n}^{\infty} z^m p(n;m,{\bf c}),
\end{equation}
we obtain (for $n \ge 1$)
\begin{equation}
p(n;z,{\bf c}) = \frac{1}{r^l} d(z/r;{\bf c})^2 h(z/r;{\bf c})^{n-1}.
\label{eqn:pnzdef}
\end{equation}
Eq.~(\ref{eqn:pnzdef}) is a special case of a more general result due to R\'egnier
and Szpankowski \cite{regszpan} who consider a broader class of
letter distributions,
including inhomogeneous letter distributions as well as sequences of
random letters generated from the steady-state of a Markov process.
As an aside, we can alternatively write \cite{regszpan} $p(n;z,{\bf c})$
in terms $c(z/r)$ alone as
\begin{equation}
p(n;z,{\bf c}) = \frac{1}{r^l} \; \; \frac{\left [ r^l(1-z)c(z/r) + z^l \right ]^{n-1}}
{\left [ r^l(1-z)(1+c(z/r)) + z^l \right ]^{n+1}},
\end{equation}
or in terms of the matching probability $p(z;{\bf c})$ as
\begin{equation}
p(n;z,{\bf c}) = r^l (1-z)^2p(z;{\bf c})^2
\left [ 1 - r^l (1-z)^2 p(z;{\bf c}) \right ]^{n-1}.
\end{equation}
Note that from the last expression, we recover again Eq.~(\ref{eqn:pzex})
in terms of the generating functions,
\begin{equation}
\sum_{n=1}^{\infty} p(n;z,{\bf c}) = p(z;{\bf c}).
\end{equation}
In fact for $n=0$ we therefore have
\begin{equation}
p(0;z,{\bf c}) = \frac{1}{1-z} - p(m;{\bf c})
= \frac{1}{1-z} \; \left [ 1 - \frac{1}{\lambda(z;{\bf c})} \right ].
\end{equation}
\subsection{Asymptotic behavior}
\label{asySec}
Once the generating functions have been determined, the original functions can be obtained
by an inverse transformation defined as follows:
If $f(z)$ is the generating function associated with $f(b)$, then
\begin{equation}
f(b) = \frac{1}{2\pi i} \oint_{\partial D} dz \frac{1}{z^{m+1}} \; \; f(z),
\label{eqn:fbycontour}
\end{equation}
where $\partial D$ is the boundary of a domain $D$ in the complex plane that includes
the origin and on which $f(z)$ is analytic \cite{wilf}.
Note that the generating functions of
$h(z;{\bf c})$, $d(z;{\bf c})$, $p(z;{\bf c})$, and $p(n;z,{\bf c})$
are all rational functions, with their denominators involving $\lambda(z;{\bf c})$
or its powers and that they all go to zero as $\| z \| \rightarrow \infty$.
For example, for the matching probability we have
\begin{equation}
p(m,{\bf c}) = \frac{1}{2\pi i} \oint_{\partial D} dz \frac{1}{z^{m+1}} \; \;
\frac{1}{(1-z)\lambda(z;{\bf c})}.
\label{eqn:pmzsol}
\end{equation}
As we will show below, the behavior of $p(m,{\bf c})$ for large $m$ (and likewise for
$h$, $d$ and $p(n;m,{\bf c}$) is dominated by the zeroes of $\lambda(z;{\bf c})$
that are closest to the origin.
A numerical inspection of the zeroes of $\lambda(z;{\bf c})$
for $2< l < 10$ and $r=2,3,4$ shows that: (1) All zeroes $z_i$ of $\lambda$ are distinct,
(2) the zero of smallest magnitude, $z_1$, is real, and greater but
near $1$ and (3) all other zeroes have
magnitudes of the order $\| z_i\| \sim r, i = 2, \ldots l $. Fig. \ref{rootplot}
shows a plot of the zeroes for $l=4,r=2$ and $l=8,r=2,3,4$. In fact, it can be
rigorously proven \cite{guibas0} that $\lambda(z;{\bf c})$ has a single (real)
zero in a circular domain centered at $z=1$ and of sufficiently small
radius $\epsilon$.
The asymptotic behavior of $f(b)$ in Eq.~(\ref{eqn:fbycontour}) can be
obtained by stretching the contour $\partial D$ to infinity while
circling around the zeroes of $f(z)$ without including them. The integral
over the boundary at infinity turns out to yields no contribution since for the
cases of interest $f(z) \rightarrow 0$ and the integrand is
asymptotically of the order of at least $1/z^{m+1}$.
This leaves the contributions from the zeroes of $(1-z)\lambda(z;{\bf c}) $, which
are traversed counter-clockwise, if the contour at infinity is traversed clock-wise.
\begin{figure}[!ht]
\begin{center}
\end{center}
\includegraphics[width=16cm]{RL_l_r2_v31.eps}
\caption[]{Root Loci of the polynomial $\lambda(z;{\bf c})$, Eq. (\ref{eqn:lambdazdef}).
The figures are for $(l,r)$ - values (starting from the top left and going
clockwise) $(4,2)$, $(8,2)$, $(8,3)$ and $(8,4)$ .
Plotted in each figure are the roots associated with the possible equivalence classes.
For $l=4$ these are ${\bf c} = 000$ (+), ${\bf c} = 001$ (*),
${\bf c} = 010$ (diamonds) and ${\bf c} = 111$ (triangles), while for the $l=8$ cases
they are ${\bf c} = 0000000 $ (+), ${\bf c} = 0000001 $ (*),
${\bf c} = 0000010 $ (diamonds), ${\bf c} = 0000011 $ (triangles) and we have shown
the roots associated with the remaining classes as small dots.
The dashed circles correspond to $\| z \| =1$ and $\| z \| =r$ and have been
inserted as a guide to the eye. All classes have a root near
$z = (1,0)$. The remaining roots cluster around and beyond the circle $\| z \| =r $. }
\label{rootplot}
\end{figure}
Considering the matching probability, we find
\begin{equation}
p(m,{\bf c}) = - \sum_{i=0}^{{\cal N}(\lambda)} \frac{1}{2\pi i} \oint_{\partial D_i} dz \frac{1}{z^{m+1}} \; \; \frac{1}{(1-z)\lambda(z;{\bf c})},
\end{equation}
where $\partial D_i$ is a clock-wise contour around the $i^{\rm th}$ zero of
$(1-z)\lambda(z;{\bf c})$, ${\cal N}(\lambda)$ is the number of zeroes, and we assume
that the zeroes are ordered such that
$z_0 = 1 < z_1 < \| z_2 \| < \ldots < \| z_{{\cal N}} \|$.
Evaluating explicitly the residues for the first two poles we have,
\begin{eqnarray}
p(m,{\bf c}) &=& 1 - A_1 \left ( \frac{1}{z_1} \right )^{m+1} \nonumber \\
&-& \sum_{i=2}^{{\cal N}(\lambda)}
\frac{1}{2\pi i} \oint_{\partial D_i} dz \frac{1}{z^{m+1}} \; \; \frac{1}{(1-z)\lambda(z;{\bf c})},
\nonumber \\
\end{eqnarray}
with the residue $A_1$ given by
\begin{equation}
A_1 = \frac{1}{\lambda^{\prime} (z_1;{\bf c})( 1 - z_1)}.
\label{eqn:A1def}
\end{equation}
The remaining zeroes $z_2, z_3, \ldots z_l$ of $\lambda(z;{\bf c})$
are located near and beyond $\| z \| \approx r $, so that in the limit of large $m$, their
relative contributions are smaller. We thus arrive at the asymptotic form
\begin{equation}
p(m,{\bf c}) \rightarrow 1 - A_1 \left ( \frac{1}{z_1} \right )^{m+1}
\label{eqn:pmcasyexact}
\end{equation}
for large $m$.
We can obtain approximate expressions for $z_1$ and thus an approximation of the
asymptotic behavior as follows. With
\begin{equation}
\lambda(z;{\bf c}) = z^l + r^l(1-z) \left [ 1 + c(z/r) \right ]
\end{equation}
we see that when $\| z \| \sim 1$, the second term in the above equation is
a large term, $r^l$, multiplied with a term that will be small due to the
$z-1$ prefactor. The product of these two terms can be made of order $1$, if
$z-1 \sim 1/r^l$, which then can be made to cancel the first term $z^l$ if $z>1$.
Using the Lagrange Inversion Formula, $z-1$ can be expanded in a power series in $1/r^l$
\cite{wilf}: Letting $u = z-1$ and $t = 1/r^l [1 + c(1/r)]^{-1}$, the equation
$\lambda(z;{\bf c}) = 0$ can be written in the form
\begin{equation}
u = t\; \phi(u),
\end{equation}
where
\begin{equation}
\phi(u) = (1+u)^l \frac{1 + c \left( \frac{1}{r} \right ) }{1 + c \left(\frac{1+u}{r} \right )}.
\end{equation}
is a formal power series in u. Thus
\begin{equation}
z_1 = 1 + u(t) = 1 + \sum_{i=1}^{\infty} u_i t^i,
\label{eqn:z1expan}
\end{equation}
with
\begin{equation}
u_i = \frac{1}{i!} \left. \frac{{\rm d}^{i-1} \phi^i}{{\rm d}u^{i-1}}\right |_{u=0}.
\end{equation}
One finds to leading and sub-leading order
\begin{equation}
u_1 = 1,
\label{eqn:zeta1def}
\end{equation}
and
\begin{equation}
u_2 = l - \frac{1}{r} \; \frac{ c^\prime(1/r)} { 1 + c(1/r)},
\end{equation}
with
\begin{equation}
c^\prime(1/r) = \sum_{i=1}^{l-1} ic_i \left ( \frac{1}{r} \right )^{i-1}
\end{equation}
so that to leading order we have
\begin{equation}
z_1 = 1 + \frac{1}{1 + c(1/r) }\; \; \frac{1}{r^l}
\label{eqn:z1leadingorder}
\end{equation}
The residue $A_1$ can be evaluated similarly,
and we find to order $1/r^l$ that
\begin{equation}
A_1 = 1 - \frac{1}{r^l} \frac{ \frac{1}{r}c^\prime(1/r)} {\left [ 1 + c(1/r) \right ]^2} .
\label{eqn:A1expansion}
\end{equation}
Note that $A_1$ is of order one.
The asymptotic behavior of $h(b;{\bf c})$ and $d(b;{\bf c})$ can be worked out
in a similar manner. For large $b$ we find,
\begin{equation}
h(b) \rightarrow h_{asy}(b) = \frac{1}{r^l}\; \frac{A_1}{z_1} \left [ r^l \left ( z_1 - 1 \right ) \right ]^2 \;
\left ( \frac{r}{z_1} \right )^b
\label{eqn:hasy}
\end{equation}
and
\begin{equation}
d(b) \rightarrow d_{asy}(b) = \frac{A_1}{z_1} \left [ r^l \left ( z_1 - 1 \right ) \right ]
\; \left ( \frac{r}{z_1} \right )^b.
\label{eqn:dasy}
\end{equation}
where $A_1$ is given by Eq.~(\ref{eqn:A1def}), $z_1$ is the smallest root of
$\lambda(z;{\bf c})$ and from the expansion of $z_1$, Eq.~(\ref{eqn:z1expan}), we see that
the terms in square brackets are of order one
Taking the asymptotic forms of $h$ and $d$ to calculate the $n$-match
distribution one finds
\begin{equation}
p^{(1)}(n;m,{\bf c}) =
A_1\left ( \begin{array}{c} m + n \\ n \end{array} \right )
\;
\left [ A_1 r^l \left ( 1 - \frac{1}{z_1} \right )^2 \right ]^n \left ( \frac{1}{z_1} \right )^{m+1-n}.
\label{eqn:p1nm}
\end{equation}
\vspace*{1cm}
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=10cm]{pkl_asy.eps}
\caption[]{Comparison of the asymptotic form, Eq.~(\ref{eqn:pmcasyexact}), with the
exact matching probabilities $p(m;\bf c)$. The figure shows
the matching probability for $l=4$, $r=2$, and $6 < k = m+l < 88$. The open circles
correspond to the numerically obtained matching probabilities for
${\bf c} = 000$, $001$, $010$ and $111$ (from top to bottom). The lines
correspond to the asymptotic form, Eq.~(\ref{eqn:pmcasyexact}), with $A_1$ and
$z_1$ calculated numerically. Inset: Plot of
$p(m;\bf c)$ for intermediate values of $m$. The symbols are as in the main figure.
The equivalence classes are (from top to bottom): $000$, $001$, $010$ and $111$.
}
\label{pklplot}
\end{center}
\end{figure}
Figure \ref{pklplot} shows a comparison of the asymptotic form, Eq.~(\ref{eqn:pmcasyexact}),
with the exact matching probabilities $p(m;\bf c)$. The figure shows
the matching probability for $l=4$, $r=2$, and $6 < k = m+l < 104$. For $l=2$
there are $4$ equivalence classes: $000$, $001$, $010$, and $111$ with $6$, $6$,
$2$ and $2$ members, respectively ({\em cf.} Tables \ref{bitv1} and \ref{bitv2}
for the case $r=2$). The open circles
correspond to the numerically obtained matching probabilities for
${\bf c} = 000$, $001$, $010$ and $111$ (from top to bottom). For $m \le 20$ $(k \le 24)$,
$p(m;{\bf c})$ was obtained by direct enumeration of all possible strings and checking
for matches. For $m > 20$ a sampling algorithm was used: for each value of $k$, $10^6$
strings of length $k$ were generated randomly and the matching probability was obtained
by counting the matching strings of the sample. The solid lines
correspond to the asymptotic form, Eq.~(\ref{eqn:pmcasyexact}), with $z_1$ and
$A_1$ calculated numerically, from Eqs.~(\ref{eqn:lambdazdef}) and (\ref{eqn:A1def}) for
each of the equivalence classes. The inset shows $p(m;\bf c)$ for intermediate values of $m$,
where we do not expect the asymptotic form to be very good.
The discrepancies become much more severe when we consider the $n$-match distribution.
Fig. \ref{ndist_plot} shows the $n$-match distributions for a $4$ letter binary string
inside a random string of length $k=256$ for the four possible equivalence classes
${\bf c} = 000$ (top left), ${\bf c} = 001$ (top right), ${\bf c} = 010$ (bottom left),
and ${\bf c} = 111$ (bottom right). The solid circles are the exact matching probabilities
that were obtained numerically using the algorithm described above.
The dotted line corresponds to the approximation Eq.~(\ref{eqn:p1nm}), normalized by an
overall constant. The dashed line corresponds to the gaussian approximation of Kleffe
and Borodovsky \cite{kleffe}, while the dot-dashed line is the compound poisson
approximation of Chrysaphinou and Papastavridis \cite{chrys}, Geske {\it et al.} \cite{geske},
and Schbath \cite{schbath}.
Note that while the
approximation Eq.~(\ref{eqn:p1nm}) performs very poorly, the gaussian and compound-poisson
distributions approximate well the true distribution only for some equivalence
classes ${\bf c}$, but fail for others, as was noted by Robin and Schbath
\cite{robschbath}. The solid line on the other hand, is the single analytical result of this
article and agrees well with the actual distributions.
We now turn to the description of the $n$-match probability in terms
of the (configurational) partition function of a 1d lattice gas.
\vspace*{1cm}
\begin{figure}[!ht]
\begin{center}
\end{center}
\includegraphics[width=16cm]{ndist_0256kl04r02.eps}
\caption[]{The $n$-match distribution for matching a $l=4$ letter binary
string $x$ inside a random string of length $k=256$, for $x=0001$ (top left),
$x=1001$ (top right), $x=1010$ (bottom left) and $x=1111$ (bottom right).
The circles are the exact probabilities, the dotted line corresponds to
the approximation Eq.~(\ref{eqn:p1nm}) (normalized by an overall constant) and
the dashed and dashed-dotted lines correspond to the Gaussian and compound
poisson approximation (see text for details). The solid line is the analytical
result of this paper.
}
\label{ndist_plot}
\end{figure}
\section{The $n$-match probability as the partition function of a 1d lattice gas}
In this section we present the statistical mechanics approach to calculating the $n$-match
distribution function. We first map the problem into one of calculating the
(configurational) partition sum of a $1d$-lattice gas. We next analyze the interaction
emerging in such a description, then set up a virial expansion leading to an
approximate evaluation of the partition function and finally discuss asymptotic limits.
\subsection{The $n$-particle partition function}
Our starting point is Eq.(\ref{eqn:pnmaspartition}), which we reproduce below for
convenience,
\begin{equation}
p(n;m,{\bf c})
= \sum_{a_1<a_2<\cdots<a_n} \frac{1}{r^{m+l}} d(a_1)
\left [ \prod_{i=1}^{n-1}h(a_{i+1}-a_{i}) \right ] d(m-a_n).
\end{equation}
with $d$ and $h$ as defined in Eqs.~(\ref{eqn:ddef}) and (\ref{eqn:gdef}).
The expression above for $p(n;m,{\bf c})$ already resembles
the partition function of a gas of $n$ particles with particle boundary
interactions proportional to $-\ln d$ and nearest neighbor particle-particle
interactions proportional to $- \ln h$.
In order to make this analogy work, we need to consider what we mean by the
free-particle, i.e. no interaction limit.
Note that $d(b)$ and $h(b)$ are
conditional matching weights. For example,
$h(b)$ is the weight of the compound event: given a match at position $a$
what is the likelihood that the next
match is at $a+b$. The asymptotic behavior of $d(b)$ and $h(b)$,
Eqs.~(\ref{eqn:dasy}) and (\ref{eqn:hasy}), can be interpreted
to correspond to the approximation when the correlations inherent in
the compound events are ignored. Thus the ratios
$d(b)/d_{asy}(b)$ and $h(b)/h_{asy}(b)$ measure the strength of
the correlations in such events.
It is therefore natural to define the
particle-boundary and particle-particle interactions,
$U^{boun}(b)$ and $U(b)$, respectively as
\begin{eqnarray}
e^{-\beta U^{boun}(b)} &=& \frac{d(b)}{d_{asy}(b)} \label{eqn:Ubdef} \\
e^{-\beta U(b)} &=& \frac{ h(b)}{h_{asy}(b)}. \label{eqn:Udef}
\end{eqnarray}
We thereby obtain meaningful physical interactions that
vanish as $ b \rightarrow \infty $.
Note that since the potentials do not have any characteristic
scale, a temperature by itself is meaningless and we will write
"energies" always with the pre-factor $\beta$, {\em i.e.} in dimension-less units.
The (configurational) partition function, Eq.~(\ref{eqn:pnmaspartition})
can now be written in terms of these interactions as
\begin{equation}
p(n;m,{\bf c}) = \frac{A_1}{z_1^{m+1}} e^{\beta \mu n} \;
\sum_{a_1<a_2<\cdots<a_n}e^{-\beta \mathcal{H}_n(a_1,\ldots , a_n) },
\label{eqn:pzmmdef_good}
\end{equation}
with
\begin{equation}
e^{\beta \mu } = A_1 \frac{r^l}{z_1}\;\left (z_1 -1 \right )^2
\label{eqn:ebmudef}
\end{equation}
and the Hamiltonian given by
\begin{equation}
\mathcal{H}_n(a_1,\ldots , a_n) = U^{boun}(a_1) + U^{boun}(m-a_n) + \sum_{i=1}^{n-1} U(a_{i+1} - a_i)
\label{eqn:Hamiltonian}
\end{equation}
Eq.~(\ref{eqn:p1nm}) corresponds to the free-particle limit ($U=U_b = 0$),
which in the probability language is the limit of all correlations suppressed.
Before proceeding, it is instructive to study these interactions in more detail.
\subsection{Interactions}
\label{interac}
Consider the particle-particle interaction first. From Eqs.~(\ref{eqn:hasy}) and
(\ref{eqn:Udef}) we find that
\begin{equation}
e^{-\beta U(b)} = \left [ \frac{z_1}{A_1} \; \frac{1}{r^{2l} (z_1-1)^2} \right ] h(b) r^{l-b} z_1^b ,
\label{eqn:ppint}
\end{equation}
where the term in square brackets is of order one, with respect to the small parameter
$1/r^l$, cf. Eqs.~(\ref{eqn:A1expansion}) and (\ref{eqn:z1expan}).
Figure \ref{ueffl0406} shows the particle particle interactions
for words of length $l=4$ and $l=6$ as parameterized
by their associated equivalence classes ${\bf c}$.
The potentials are plotted against distance measured in
units of the word length $l$ and have been vertically
offset for clarity with the dashed lines representing
$U=0$. The crosses on the dashed line indicate
that the associated potential at that value is $+\infty$.
We see that the potential have infinite values only
for $b \le l$. Also, the values of the potential in the regime $b \le l$
are generally much bigger than in the regime $b >l $, meaning that
the potential is stronger in the former region. We will refer to
the region $b \le l$ and $b > l$ as the core and tail of the
interaction, respectively.
For a given length $l$ and depending on ${\bf c}$, we also see that
the interactions have different features. For
${\bf c} = 0\cdots0$ the interaction has a hard-core of size
$l$ followed by a repulsive tail, while for ${\bf c} = 1\cdots1$
the interaction has a strongly attractive compontent at $b=1$, followed
by a hard-core region for $1 \le b \le l$, that goes over into
an oscillatory but decaying
tail. The potentials for the other values of ${\bf c}$ seem to be a
mixture of these two types of behavior.
Figure \ref{ueff0011} shows the behavior of the potentials associated
with the equivalence classes ${\bf c} = 0\cdots0$ (left) and
${\bf c} = 1\cdots1$ (right) in their dependence on the word length $l$.
For both equivalence classes we see
that the tail of the interaction becomes weaker as $l$ increases.
When ${\bf c} = 0\cdots0$, the core is hard-core and only
the core-size $l$ changes. The situation is different for
${\bf c} = 1\cdots1$. For the ${\bf c} = 1\cdots1$ family of interactions we
see that the attractive part of the core actually becomes stronger with
increasing $l$. It turns out that the same is also true
for the other equivalence classes, namely with increasing $l$,
the cores of the interactions become stronger, while the tails
become weaker.
\vspace*{1cm}
\begin{figure}[!ht]
\begin{center}
\end{center}
\includegraphics[width=16cm]{ueff_l04-l06r02.eps}
\caption[]{Plot of the effective potentials $\beta U(b)$,
Eq.~(\ref{eqn:ppint}), associated with the equivalence classes
${\bf c}$ of strings of lengths $l$. The potentials are
plotted against distance measured in units of the word length $l$.
Note that the potentials have been vertically
offset for clarity. The dashed lines represent the
$U=0$ lines for each potential. The crosses on the line U=0 indicate
that the associated potential at that point is $+\infty$.
Left: Interparticle potentials associated with words of length $l=4$, for
which the possible equivalence classes are ${\bf c} = 000, 010$ and
$111$, as indicated in the figure. Right: same as left but for $l=6$.
Notice how the attractive part of the interaction emerges and grows
stronger as the fundamental period of the string decreases to $1$ (
${\bf c} = 1\cdots1$). The tail of the interaction corresponds to
the regime $b/l >1$.
}
\label{ueffl0406}
\end{figure}
In summary, Figs. \ref{ueffl0406} and \ref{ueff0011} suggest
the following generic features of the interactions:
(i) a strong core $b \le l$, followed by a weak tail for $b > l$,
and, (ii) for a given family of interactions, as $l$ increases the core
of the interaction tends to become stronger, while the tail of the
interaction becomes weaker.
\vspace*{1cm}
\begin{figure}[!ht]
\begin{center}
\end{center}
\includegraphics[width=16cm]{ueff_0000_1111.eps}
\caption[]{Plot of the effective potentials $\beta U(b)$,
Eq.~(\ref{eqn:ppint}), associated with the equivalence classes
${\bf c} = 0\cdots0$ and $1\cdots1$ and their dependence on
the lengths $l$. The potentials are
plotted against distance measured in units of the word length $l$.
Note that the potentials have been vertically
offset for clarity. The dashed linerepresent the
$U=0$ line for each potential. The crosses on the a line indicate
that the associated potential at that value is $+\infty$.
Left: Interparticle potentials associated with the equivalence
class ${\bf c} = 0\cdots0$ for words of length $l=3,4,6$ and $8$.
Note that the interactions have a hard-core of size $b/l =1$ followed
by a repulsive tail. The strength of the tail weakens with increasing
$l$. Right: Interparticle potentials associated with the equivalence
class ${\bf c} = 1\cdots1$ for words of length $l=3,4,6$ and $8$.
Note that the interactions have an attractive part at $b=1$, followed
by a hard-core for $b/l < 1$, and
a weak, oscillatory decaying tail. Also note the opposite behavior
of the strength of the core and the tail: With increasing $l$, the strength
of the attractive part of the core is seen to increase,
while the strength of the tail
decreases.
}
\label{ueff0011}
\end{figure}
These observations can be readily proven from the small $b$ behavior
of $h(b)$, which in turn can be extracted from the recursion for $h(b)$,
Eqs.~(\ref{eqn:grec}) and (\ref{eqn:Concemore}}). Thus we find
for $b < l$
\begin{equation}
h(b) = \left \{ \begin{array}{ll}
c_b, & \mbox {if $\chi$ does not divide $b$,} \\
1, & \mbox{if $b = \chi $,} \\
0, & \mbox{otherwise,} \\
\end{array} \right.
\label{eqn:hcore}
\end{equation}
where $\chi$ is the fundamental period associated with ${\bf c}$ that was
defined at the end of Section \ref{bit-sect}. Recall that by definition,
$h(0) = 0$.
Thus the interaction in the core region can be written as
\begin{equation}
\beta U(b) = -\ln h(b) \; + b \ln \left ( \frac{r}{ z_1} \right ) \; -l \ln r \; + \beta U_0, \; \; \; \mbox{$b \le l$}
\label{eqn:Uppcore}
\end{equation}
where
\begin{equation}
\beta U_0 = \ln \left ( \frac{z_1}{A_1 \left [r^l (z_1-1) \right ]^2} \right )
\end{equation}
is a constant that is of order $1/r^l$, since the argument of the logarithm
is of order 1 to the same order.
We see that the interaction becomes $+\infty$, whenever $h(b) = 0$. This is
certainly the case for $b < \chi$. Furthermore, since $r/z_1 > 1$, in the core
region finite values of $U(b)$ increase with increasing $b$, as clearly seen
in Fig. \ref{ueffl0406}.
The first finite value of $U(b)$ occurs at $b = \chi$.
From Eq.~(\ref{eqn:Uppcore}) we obtain for $U(\chi)$
\begin{equation}
\beta U(\chi) = \chi \ln \left ( \frac{r}{z_1} \right ) - l \ln r + \mathcal{O} \left (\frac{1}{r^l} \right ).
\label{eqn:Uxhi}
\end{equation}
Thus it is apparent that for fixed $\chi$, $\beta U(\chi)$ becomes more negative as either
$l$ or $r$ increase. In fact we see that to leading order, the dependence of
$\beta U(\chi)$ on $l$ is linear, while its dependence on $r$ is logarithmic.
Also for $\chi < l$ we see that $\beta U(\chi)$ is negative.
The case when $\chi = l$, corresponding to ${\bf c} = 00\ldots0$,
is a little more complicated. In that case the $\ln r$ terms in
Eq.~(\ref{eqn:Uxhi}) cancel and we are left with a term $l \ln z_1$ which is of order
$1/r^l$ as well and thus $\beta U_0$ cannot be neglected anymore. However this means
that the potential is of order $1/r^l$, which turns out to be the correct scale of the
strength of the tail and indeed decreases as $r$ or $l$ increase (see Fig.\ref{ueff0011}).
To be specific, for $\chi = l$, it can be readily checked
that $h(b) = r^{b-l}$ for $l \le b < 2l$ and the potential in this regime thus becomes
\begin{equation}
\beta U(b) = \frac{2l + 1 - b}{r^l} + \mathcal{O} \left (\frac{1}{r^{2l}} \right ),
\end{equation}
where we have substituted the expansions of $A_1$ and $z_1$, Eqs.~(\ref{eqn:A1expansion})
and (\ref{eqn:z1expan}), respectively, to lowest non-trivial order. Thus we see that
for $\chi = l$ the characteristic energy scale of the tail of the
interaction scales like $\sim l/r^l$, and
decreases as $l$ or $r$ increase.
The case of general $\chi$ and ${\bf c}$ is similar, but the calculation are tedious
yet straightforward. Rather than doing this, we will motivate the result by
considering the value of the interaction at $b = l + \chi$, which is readily worked out from
\begin{equation}
h(l+\chi) = \left \{ \begin{array}{ll}
r^\chi - h(l) - 1, & \mbox {for $\chi < l/2$,} \\
r^\chi - h(l) - 1, - \sum_{\tau = \chi + 1}^{l-1} h(\tau) c(l+\chi-1),
& \mbox{for $l/2 \le \chi < l $,} \\
r^\chi - h(l), & \mbox{for $\chi = l $.}
\end{array} \right.
\end{equation}
This means that
\begin{equation}
h(l+\chi) = r^\chi \left ( 1 - \epsilon \right ),
\end{equation}
where $\epsilon r^\chi$ is at most $l-\chi+1$ and hence of order $l$.
Substituting this result into
the expression for $U(b)$ along with the expansions for $z_1$ and $A_1$, one finds that
the result is of the form
\begin{equation}
-\beta U(l+\chi) = \frac{\alpha}{r^l} + \mathcal{O} \left (\frac{1}{r^{2l}} \right ),
\end{equation}
where $\alpha$ is a ${\bf c}$ and $l$ dependent constant of order one.
To conclude, we find that the characteristic energy of the core of the interaction
scales like $-(l-\chi) \ln r$ ($\chi < l$), while the energy of the tail goes to leading
order like $1/r^l$. These results are consistent with the behavior observed in
Figs.~\ref{ueffl0406} and \ref{ueff0011}
Turning to the particle boundary interactions, note that Eq.~(\ref{eqn:drec}),
which can be conveniently written as
\begin{equation}
\frac{d(b)}{r^b} = 1 - \sum_{a=1}^{b} \frac{h(a)}{r^a}
\end{equation}
relates the properties of $h$ to those of $d$. We thus see that
analogous results can be obtained for the boundary interaction $U^{boun}(b)$ and
we leave the details to the interested reader.
\subsection{The Hamiltonian}
The results of the previous section allow us to obtain approximate expressions for the
probability $p(n;m,{\bf c})$, by first approximating the effective Hamiltonian
$H_n$ and then carrying out the configurational sums. This is most easily done
using generating functions.
Define the generating functions associated with Eqs.~(\ref{eqn:Ubdef})
and (\ref{eqn:Udef}) as
\begin{eqnarray}
D(z) &=& \sum_{b=0}^{\infty} z^b e^{-\beta U_b(b)}, \\
H(z) &=& \sum_{b=0}^{\infty} z^b e^{-\beta U(b)} .
\end{eqnarray}
It is not difficult to show that in terms of the generating functions of $d(b)$ and
$h(b)$, $D(z)$ and $H(z)$ are given by
\begin{eqnarray}
D(z) &=& e^{-\beta \mu} (z_1-1) d\left (\frac{zz_1}{r};{\bf c} \right ) \label{eqn:Ddef} \\
H(z) &=& e^{-\beta \mu} h\left (\frac{zz_1}{r};{\bf c} \right ) \label{eqn:Hdef}.
\end{eqnarray}
Using the convolution property, Eq.~(\ref{eqn:pzmmdef_good}) can be
written in terms of the generating functions $D(z)$ and $H(z)$ as
\begin{equation}
p(n;m,{\bf c}) = \frac{A_1}{z_1^{m+1}} e^{\beta \mu n}
\frac{1}{2\pi i} \oint_{\partial D} dz \frac{1}{z^{m+1}} \; \; D^2(z) H^{n-1}(z),
\label{eqn:pnmc_contour}
\end{equation}
where the contour is again the boundary of a domain enclosing the origin inside of
which $D^2(z)H^{n-1}(z)$ is analytic. Eq.~(\ref{eqn:pnmc_contour}) is the lattice
analog of the partition function of a 1d gas with pairwise nearest neighbor interactions.
The 1d continuum case has been treated in detail by G\"ursey \cite{gursey}
({\em see also} Fisher \cite{fisher}).
Next, define the truncated generated functions $D_{\Lambda}(z)$ and $H_{\Lambda}(z)$
as
\begin{eqnarray}
D_{\Lambda}(z) &=& \sum_{b=0}^{\Lambda -1} z^b e^{-\beta U_b(b)}, \label{eqn:Ubzdef} \\
H_{\Lambda}(z) &=& \sum_{b=0}^{\Lambda -1} z^b e^{-\beta U(b)} \label{eqn:Uzdef}.
\end{eqnarray}
It is readily seen that these generating functions are associated with the Boltzmann
factor of an interaction that has been cut-off at $b \ge \Lambda$. The idea is that
since, by construction, the interactions decay to zero at large distances, introducing
a finite cut-off $\Lambda$ will introduce only a small and controllable error in the
overall calculation. In what follows, we will use this to set up a perturbation expansion
of the probability distribution.
We need to note however that since the result has to be a normalized distribution, setting
the potential to zero beyond the cut-off will destroy the normalization of the distribution.
Indeed there are at least two ways to handle the interaction beyond the cut-off:
(i) we can either set the interaction to a constant $U_\Lambda$ for $b \ge \Lambda$ and
eventually choose $U_\Lambda$ such that the distribution is normalized, or (ii) we take the
interaction beyond $\Lambda$ to be rapidly decaying. It turns out that the calculation
can be done for either of the cases. The approximation by a constant potential beyond the
cut-off lends itself readily for obtaining error bounds, as we will sketch below.
On the other hand, it turns out that the tail of the actual interactions
{\em does} asymptotically decay exponentially.
Thus letting the interaction decay exponentially beyond the cut-off turns out to be a
very good approximation and we will calculate the probability distributions in this way.
Consider the case of a constant potential beyond the cut-off first and
define the approximate interaction $\hat{U}(b)$ as
\begin{equation}
\hat{U}_\Lambda(b) = \left \{ \begin{array}{ll} U(b), & b < \Lambda \ \\
U_\Lambda, & b \ge \Lambda, \end{array} \right.
\end{equation}
with the corresponding generating function given by
\begin{equation}
\hat{H}_\Lambda(z) = \sum_{b=0}^{\infty} z^b e^{-\beta \hat{U}_\Lambda(b)} = H_{\Lambda}(z)
+ e^{-\beta U_\Lambda} \; \frac{z^\Lambda}{1-z}.
\end{equation}
Since $d(z)$ is related to $h(z)$ via Eq.~(\ref{eqn:dzexplicit}), this implies
a corresponding boundary interaction which can be worked out as
\begin{equation}
\hat{D}_\Lambda(z) = \sum_{b=0}^{\infty} z^b e^{-\beta \hat{U}_\Lambda(b)} = D_{\Lambda}(z)
+ e^{-\beta U_\Lambda} \; \frac{z^\Lambda}{1-z}.
\end{equation}
Define the approximation to $p(n;m,{\bf c})$, Eq.~(\ref{eqn:pnmc_contour}), as
\begin{equation}
\hat{p}(n;U_\Lambda,m,{\bf c}) = \frac{A_1 e^{\beta \mu n}}{z_1^{m+1}}
\frac{1}{2\pi i} \oint_{\partial D} dz \frac{1}{z^{m+1}} \; \;
\hat{D}_\Lambda^2(z) \hat{H}_\Lambda^{n-1}(z),
\end{equation}
It is clear that as $\Lambda \rightarrow \infty$ we must have $U_\Lambda \rightarrow 0$,
since an increasingly larger part of the true interactions is kept.
By using the definition of $U_\Lambda(b)$ and writing
$\hat{p}(n;U_\Lambda,m,{\bf c})$ in the partition sum form of
Eq.~(\ref{eqn:pnmaspartition}), it can readily be verified that if
\begin{equation}
U_{-} \le U_\Lambda \le U_{+}
\end{equation}
this implies that
\begin{equation}
\hat{U}_{-}(b) \le \hat{U}_\Lambda(b) \le \hat{U}_{+}(b)
\end{equation}
for all values of $b$, which in turn implies that
\begin{equation}
\hat{p}(n;U_{+},m,{\bf c}) \le \hat{p}(n;U_\Lambda,m,{\bf c}) \le \hat{p}(n;U_{-},m,{\bf c}).
\end{equation}
Thus by choosing $U_{+}$ and $U_{-}$ as
\begin{eqnarray}
U_{+} &=& \max_{b \ge \Lambda} \left \{ U(b), U^{boun}(b) \right \}, \\
U_{-} &=& \min_{b \ge \Lambda} \left \{ U(b), U^{boun}(b) \right \}
\end{eqnarray}
one could in principle obtain error bounds on the approximate distribution, which will become
tighter as $\Lambda \rightarrow \infty$. We will not pursue this any further in the present
article, but instead perform the calculation with an exponentially decaying interaction beyond
the cut-off $\Lambda$.
Recall that the tail of the true interaction is due to the other
zeroes of $\lambda(z;{\bf c})$, which
are located a distance $\sim r$ from the origin, ({\em see} Fig.\ref{rootplot}).
Thus superposed on the asymptotic behavior
of $h(b)$, which we have shown to fall-off like $z_1^{-b}$, there will be terms that
decay more rapidly and roughly as $r^{-b}$, since $z_1 < r$.
In fact it is the latter that are responsible for the asymptotic behavior
of the interactions. For $b$ large, we therefore take approximately
\begin{equation}
h(b) \approx e^{\beta \mu} z_1^{-b} + \gamma e^{\beta \mu} r^{-b}
\end{equation}
which upon taking logarithms and factoring out the first terms implies that
asymptotically
\begin{equation}
\beta U(b) \approx \beta \mu b \ln z_1 - \gamma \left ( \frac{z_1}{r} \right )^{b},
\end{equation}
where we have neglected higher order terms $\gamma^k (z_1/r)^{kb}$. Of course,
with increasing cut-off $\Lambda$, the residual tail will be less important.
This suggest taking the following approximate interactions:
\begin{equation}
\hat{U}_\Lambda(b) = \left \{ \begin{array}{ll} U(b), & b < \Lambda \ \\
-\gamma \left ( \frac{z_1}{r} \right )^{b} , & b \ge \Lambda, \end{array} \right.
\end{equation}
with the corresponding approximate generating function given by
\begin{equation}
\hat{H}_\Lambda(z) = \sum_{b=0}^{\infty} z^b e^{-\beta \hat{U}_\Lambda(b)} = H_{\Lambda}(z)
+ \gamma \left ( \frac{z_1}{r} \right )^{\Lambda}\; \frac{z^\Lambda}{1-\frac{z_1}{r}z}
+ \frac{z^\Lambda}{1-z}.
\label{eqn:hhat}
\end{equation}
Since $d(z)$ is related to $h(z)$ via Eq.~(\ref{eqn:dzexplicit}), this implies
a corresponding approximate interaction for the boundary interaction,
which can be worked out,
\begin{equation}
\hat{D}_\Lambda(z) = \sum_{b=0}^{\infty} z^b e^{-\beta \hat{U}_\Lambda(b)} = D_{\Lambda}(z)
+ \gamma \left ( \frac{z_1}{r} \right )^{\Lambda}\; \frac{z^\Lambda}{1-\frac{z_1}{r}z}
+ \frac{z^\Lambda}{1-z}.
\label{eqn:dhat}
\end{equation}
Denoting the generating function of the approximate tail of the interaction as
\begin{equation}
\Gamma(z) = \gamma \left ( \frac{z_1}{r} \right )^{\Lambda}\;
\frac{z^\Lambda}{1-\frac{z_1}{r}z},
\end{equation}
$\hat{p}(n;m,{\bf c})$ becomes
\begin{equation}
\hat{p}(n;\gamma,m,{\bf c}) = \frac{A_1 e^{\beta \mu n}}{z_1^{m+1}}
\frac{1}{2\pi i} \oint_{\partial D} dz \frac{1}{z^{m+1}} \; \;
\hat{D}_\Lambda^2(z) \hat{H}_\Lambda^{n-1}(z).
\label{eqn:pnmc_contour_app}
\end{equation}
What therefore remains to be done is to evaluate the contour integral,
Eq.~(\ref{eqn:pnmc_contour_app}), which can be carried out by the method of
stationary phase, which in the context of generating functions is also known as
Hayman's method \cite{wilf}:
\subsection{Distributions}
Write the integral in Eq.~(\ref{eqn:pnmc_contour_app}) as
\begin{equation}
I = \frac{1}{2\pi i} \oint_{\partial D} dz \frac{1}{z^{m+1}} \; \; f(z).
\end{equation}
Then for large $m$, the value of the integral is given approximately by
\begin{equation}
I \approx \left (\frac{1}{u_m}\right )^m \frac{f(u_m)}{\sqrt{2\pi b_m}},
\label{eqn:Iapp}
\end{equation}
where $u_m$ is the smallest positive real root of the equation
\begin{equation}
m = u \frac{\rm d}{\rm du}\ln f(u)
\end{equation}
and $b_m$ is given by
\begin{equation}
b_m = u \frac{\rm d} {\rm du} \ln f(u)+ u^2 \frac{{\rm d}^2} {{\rm du}^2} \ln f(u).
\end{equation}
Applying Hayman's method to the integral, Eq.~(\ref{eqn:pnmc_contour_app}), we let
\begin{equation}
f(u) = \hat{D}_\Lambda^2(u) \hat{H}_\Lambda^{n-1}(u)
\label{eqn:fudef}
\end{equation}
and find after a little bit of algebra
\begin{eqnarray}
m &=& u \frac{{\rm d} }{{\rm du}} \ln f(u) \nonumber \\ &=& \frac{2}{x}
\frac{1 + \Lambda x + x^2 (1+x)^{\Lambda-2} \left [ \hat{D}^\prime_\Lambda\left (\frac{1}{1+x} \right )
+ \Gamma^\prime\left (\frac{1}{1+x} \right ) \right ]}
{ 1 + x (1+x)^{\Lambda-1} \left [ \hat{D}_\Lambda\left (\frac{1}{1+x} \right )
+ \Gamma\left (\frac{1}{1+x} \right )\right ] } \nonumber \\
&+& \frac{n-1}{x}
\frac{1 + \Lambda x + x^2 (1+x)^{\Lambda-2} \left [ \hat{H}^\prime_\Lambda\left (\frac{1}{1+x} \right )
+ \Gamma^\prime\left (\frac{1}{1+x} \right ) \right ]}
{ 1 + x (1+x)^{\Lambda-1} \left [ \hat{H}_\Lambda\left (\frac{1}{1+x} \right )
+ \Gamma\left (\frac{1}{1+x} \right )\right ], }
\label{eqn:mofx}
\end{eqnarray}
where we have parameterized $u$ as
\begin{equation}
u = \frac{1}{1+x}.
\label{eqn:xu}
\end{equation}
Since we are interested in solutions for large $m$, it is clear from the above that to leading
order $x \propto 1/m$. Multiplying both sides of the above equation by $x$ and expanding
the fractions in a power series around $x=0$, we obtain
\begin{equation}
mx = (n+1) \left \{ 1 + \epsilon_1 x + \epsilon_2 x^2 + \ldots \right \}
\label{eqn:mxeq}
\end{equation}
The first two orders can be readily worked out, yielding
\begin{equation}
\epsilon_1 = \Lambda - \tilde{\gamma}\xi -
\frac{2\hat{D}_\Lambda(1) + (n-1)\hat{H}_\Lambda(1) } {n+1}
\label{eqn:eps1}
\end{equation}
and
\begin{eqnarray}
\epsilon_2 &=& \tilde{\gamma}\xi \left [ 2 \xi + \tilde{\gamma}\xi - 1 \right ]
+ \frac{1}{n+1} \left \{ 2\hat{D}^2_\Lambda(1) + (n-1) \hat{H}^2_\Lambda(1)
+ 2 \left [ 2 \hat{D}^\prime_\Lambda(1) + (n-1) \hat{H}^\prime_\Lambda(1) \right ] \right. \nonumber \\
&-& \left [ 2(\Lambda - \tilde{\gamma}\xi) -1 \right ]
\left. \left [ 2 \hat{D}_\Lambda(1) - (n-1) \hat{H}_\Lambda(1) \right ] \right \},
\end{eqnarray}
where
\begin{equation}
\xi = \frac{1}{1-\frac{z_1}{r}}
\end{equation}
and $\tilde{\gamma} = \gamma (1-1/\xi)^\Lambda$.
Rewriting Eq.~(\ref{eqn:mxeq}) in a form suitable for Lagrange's Inversion Formula,
\begin{equation}
x = \frac{n+1}{m - (n+1)\epsilon_1} \left \{ 1 + \epsilon_2 x^2 + \ldots \right \},
\end{equation}
We obtain an expansion of $x$ in terms of $(n+1)/[m-(n-1)\epsilon_1]$ and the coefficients $\epsilon_i$ as
\begin{equation}
x = \frac{n+1}{m-(n-1)\epsilon_1} + \left [ \frac{n+1}{m-(n-1)\epsilon_1} \right ]^3 \epsilon_2 \ldots \; .
\label{eqn:virialx}
\end{equation}
The term $b_m$ can be worked out in a similar manner and we find
\begin{equation}
b_m = m + \frac{n+1}{x^2} - (n+1)(\epsilon_1 + \epsilon_2) + \ldots \; ,
\label{eqn:bmexp}
\end{equation}
where the omitted terms are of order $x$ and higher.
Combining Eqs.~(\ref{eqn:pnmc_contour_app}), (\ref{eqn:Iapp}), (\ref{eqn:fudef}), with
Eqs.~(\ref{eqn:virialx}) and (\ref{eqn:bmexp}) we finally obtain
\begin{equation}
\hat{p}(n;\gamma,m,{\bf c}) \approx \frac{A_1 e^{\beta \mu n}}{z_1^{m+1}}
\left ( 1 + x \right )^m \hat{D}_\Lambda^2 \left ( \frac{1}{1+x} \right )
\hat{H}_\Lambda^{n-1} \left ( \frac{1}{1+x} \right ) \frac{1}{\sqrt{2\pi b_m}}.
\label{eqn:phat}
\end{equation}
The strength of the tail, $\tilde{\gamma} $, is still undertermined and we will determine
it by fitting the approximate tail to the actual interaction in the interval
$b \in [ \Lambda, \Lambda + l-1 ] $. Note that this way there are no adjustable
parameters and since the tail is only approximate, the normalization is not
perfect and is found to vary by a few percent. Alternatively, one can choose
$\tilde{\gamma}$ such that normalization is achieved. In either of the cases
the distributions do not vary significantly, meaning that for a certain range of
$\tilde{\gamma}$ values, the shape of the distribution is robust.
The solid lines in Fig.~\ref{ndist_plot} show the approximate distribution,
Eq.~(\ref{eqn:phat}), for the four equivalence classes associated with words
of length $l=4$ and with $r=2$, $k=256$. We will refer to this approximation
as the liquid theory approximation. In this and all the other
results that we will present, the cut-off $\Lambda$ was chosen as $\Lambda = 3l$
and $x$ was expanded to 2nd order. The dashed lines in Fig.~\ref{ndist_plot}
are the Gaussian approximation of Kleffe and Borodovsky (KB) \cite{kleffe}
with the distribution mean and variance given by Eqs.~(\ref{eqn:nave})
and (\ref{eqn:sigman}). The dot-dashed lines are the compound poisson (CP)
approximation of Chrysaphinou and Papastavridis \cite{chrys},
Geske {\it et al.} \cite{geske} and Schbath \cite{schbath}.
The variation between actual and approximate
distributions, $p(n)$ and $\hat{p}(n)$, can be quantified by the
{\it total variational distance} \cite{barbour}
between the two distributions and is defined as
\begin{equation}
d_{TV}(p,\hat{p}) = \frac{1}{2}\sum_n \| \hat{p}(n) - p(n) \| .
\end{equation}
Table 3 shows the variational distances between the actual and approximate
distributions depicted in Fig. \ref{ndist_plot}, $(l=4)$ and $k=256$.
\begin{table}
\begin{center}
\vskip 0.25cm
\begin{tabular}{||r|l|l|l|l||}
\hline
${\bf c}$ & $d^{L}_{TV}$ & $d^{NL}_{TV}$ & $d^{CP}_{TV}$ & $d^{KB}_{TV}$ \\
\hline
000 & 0.052 & 0.053 & 0.189 & 0.052 \\
001 & 0.035 & 0.031 & 0.079 & 0.075 \\
010 & 0.011 & 0.003 & 0.108 & 0.071 \\
111 & 0.032 & 0.021 & 0.047 & 0.148 \\
\hline
\end{tabular}
\caption{Total variational distance between the actual distribution and the various
approximate distribution for the case $r=2$, $k=256$:
liquid theory approximation (L), Eq.~(\ref{eqn:phat}),
the liquid theory approximation normalized by an overall constant (NL), the compound
poisson approximation (CP) and the gaussian approximation (KB). }
\end{center}
\label{dtvl4_r2}
\end{table}
We see that the (un-normalized) liquid theory approximation, Eq.~(\ref{eqn:phat}) (L),
as well as the liquid theory approximation normalized by an overall constant (NL)
perform better then the compound poisson (CP) and gaussian approximation (KB). Note
that for ${\bf c} = 000$, none of the approximations captures the
height of the peak of the distribution accurately and we will remark on this shortly.
Tables 4 and 5 show the total variational distances
between the actual and approximate distributions for word lengths $l=3,4,5,6,7$ and
$l=8$ and string lengths $k$ chosen such that $k/r^l = 16$, {\it. i.e.} the
distributions have approximately the same mean. Overall, the
liquid theory approximation, Eq.~(\ref{eqn:phat}) (L) , as well as the liquid theory
approximation normalized by an overall constant (NL) perform better then or as well as
the compound poisson (CP) and gaussian approximation (KB) taken by themselves.
The CP approximation gives a better approximation for
${\bf c} = 11\cdots 1$ and for some of the low and high $\chi$
equivalence classes associated with $l=7$ and $l=8$. Also note that
for $l \ge 6$ the CP approximation performs generally better than the KB
approximation, as was noted before by Robin and Schbath \cite{robschbath}.
The poor performance of the liquid theory approximation for the
case $l=3$ and ${\bf c} = 00$ turns out to be due to the fact
that the expansion of $x$ and $b_m$
to second order is not adequate. Upon calculating $x$ (and $b_m$) more accurately,
the agreement with the actual distributions turns out to be nearly perfect.
\begin{table}
\begin{center}
\vskip 0.25cm
\begin{tabular}{||r|l|l|l|l||}
\hline
${\bf c}$ & $d^{L}_{TV}$ & $d^{NL}_{TV}$ & $d^{CP}_{TV}$ & $d^{KB}_{TV}$ \\
\hline
00 & ***** & (0.933) & 0.227 & 0.006 \\
01 & 0.027 & 0.008 & 0.156 & 0.084 \\
11 & 0.018 & 0.016 & 0.121 & 0.131 \\
\hline
000 & 0.052 & 0.053 & 0.189 & 0.052 \\
001 & 0.035 & 0.031 & 0.079 & 0.075 \\
010 & 0.011 & 0.003 & 0.108 & 0.071 \\
111 & 0.032 & 0.021 & 0.047 & 0.148 \\
\hline
0000 & 0.009 & 0.010 & 0.090 & 0.018 \\
0001 & 0.018 & 0.016 & 0.056 & 0.043 \\
0010 & 0.010 & 0.008 & 0.061 & 0.050 \\
0011 & 0.040 & 0.036 & 0.034 & 0.089 \\
0101 & 0.021 & 0.024 & 0.075 & 0.056 \\
1111 & 0.044 & 0.026 & 0.012 & 0.154 \\
\hline
00000 & 0.013 & 0.011 & 0.034 & 0.028 \\
00001 & 0.006 & 0.004 & 0.040 & 0.030 \\
00010 & 0.009 & 0.011 & 0.053 & 0.028 \\
00011 & 0.018 & 0.019 & 0.061 & 0.028 \\
00100 & 0.013 & 0.011 & 0.032 & 0.053 \\
00101 & 0.010 & 0.006 & 0.037 & 0.055 \\
01010 & 0.019 & 0.011 & 0.042 & 0.066 \\
11111 & 0.049 & 0.027 & 0.011 & 0.152 \\
\hline
\end{tabular}
\caption{Total variational distance between the actual distribution and the various
approximate distribution for the case $r=2$ and $(l,k)$
$(3,128)$, $(4,256)$, $(5,512)$, $(6,1024)$, $(7,2048)$ and $(8,4096)$:
liquid theory approximation (L), Eq.~(\ref{eqn:phat}),
the liquid theory approximation normalized by an overall constant (NL), the compound
poisson approximation (CP) and the gaussian approximation (KB). }
\end{center}
\label{dtvl3_6_r2}
\end{table}
\begin{table}
\vskip 0.25cm
\begin{center}
\begin{tabular}{||r|l|l|l|l||}
\hline
${\bf c}$ & $d^{L}_{TV}$ & $d^{NL}_{TV}$ & $d^{CP}_{TV}$ & $d^{KB}_{TV}$ \\
\hline
000000 & 0.025 & 0.024 & 0.004 & 0.037 \\
000001 & 0.003 & 0.003 & 0.028 & 0.028 \\
000010 & 0.004 & 0.002 & 0.023 & 0.031 \\
000011 & 0.005 & 0.006 & 0.031 & 0.031 \\
000100 & 0.004 & 0.002 & 0.023 & 0.038 \\
000101 & 0.004 & 0.003 & 0.029 & 0.037 \\
000111 & 0.004 & 0.003 & 0.029 & 0.042 \\
001001 & 0.011 & 0.012 & 0.033 & 0.041 \\
010101 & 0.023 & 0.013 & 0.026 & 0.067 \\
111111 & 0.052 & 0.022 & 0.015 & 0.146 \\
\hline
0000000 & 0.023 & 0.022 & 0.009 & 0.040 \\
0000001 & 0.022 & 0.021 & 0.006 & 0.040 \\
0000010 & 0.004 & 0.002 & 0.013 & 0.031 \\
0000011 & 0.018 & 0.017 & 0.004 & 0.041 \\
0000100 & 0.003 & 0.002 & 0.014 & 0.034 \\
0000101 & 0.010 & 0.008 & 0.007 & 0.039 \\
0000111 & 0.003 & 0.002 & 0.014 & 0.037 \\
0001000 & 0.003 & 0.003 & 0.017 & 0.036 \\
0001001 & 0.005 & 0.003 & 0.012 & 0.040 \\
0010010 & 0.005 & 0.005 & 0.017 & 0.046 \\
0010011 & 0.010 & 0.007 & 0.007 & 0.055 \\
0101010 & 0.025 & 0.009 & 0.012 & 0.072 \\
1111111 & 0.054 & 0.020 & 0.015 & 0.140 \\
\hline
\end{tabular}
\caption{Total variational distance between the actual distribution and the various
approximate distribution for the case $r=2$ and $(l,k)$
values of $(7,2048)$ and $(8,4096)$:
liquid theory approximation (L), Eq.~(\ref{eqn:phat}),
the liquid theory approximation normalized by an overall constant (NL), the compound
poisson approximation (CP) and the gaussian approximation (KB). }
\end{center}
\label{dtvl7_8_r2}
\end{table}
Regarding the robustness of the liquid theory approximations (L) and (NL), we have
checked that going to a higher cut-off does not improve the distributions
very much. Also, it turns out that for large $\chi$ and $l$, the first order expression
for $x$ is often sufficient, however it is almost always insufficient for small
$\chi$ and in particular when $\chi = 1$, {\it i.e.} $x$ belongs to the
equivalence class ${\bf c} = 11\ldots1$.
Fig.~\ref{ndist_plot_k4096} shows the $n$ match distributions for $l=4$ and
with a string length that has been increased to $k=4096$. Comparing with
the case $k=256$, Fig.~\ref{ndist_plot}, the distributions for small $\chi$
are more symmetric around their mean.
\vspace*{1cm}
\begin{figure}[!ht]
\includegraphics[width=16cm]{ndist_4096kl04r02_norm.eps}
\caption[]{The $n$-match distribution for matching a $l=4$ letter binary
string $x$ inside a random string of length $k=4096$, for $x=0001$ (top left),
$x=1001$ (top right), $x=1010$ (bottom left) and $x=1111$ (bottom right).
The circles are the exact probabilities, the dashed and dashed-dotted lines
correspond to the Gaussian and compound poisson approximation (see text for details).
The solid line is the analytical result, Eq.~(\ref{eqn:phat}) normalized by an
overall constant.
}
\label{ndist_plot_k4096}
\end{figure}
The total variatonal distances are given in the table below. Note that they are comparable
with the values that we obtained for $k=256$, Table 3.
\begin{table}
\vskip 0.25cm
\begin{center}
\begin{tabular}{||r|l|l|l|l||}
\hline
${\bf c}$ & $d^{L}_{TV}$& $\bar{d}^{NL}_{TV}$ & $d^{CP}_{TV}$ & $d^{KB}_{TV}$ \\
\hline
000 & 0.061 & 0.060 & 0.197 & 0.060 \\
001 & 0.035 & 0.035 & 0.076 & 0.075 \\
010 & 0.011 & 0.004 & 0.108 & 0.065 \\
111 & 0.045 & 0.023 & 0.038 & 0.140 \\
\hline
\end{tabular}
\caption{Total variational distance between the actual distribution and the various
approximate distribution for the case $r=2$, $l=4$ and $k=4096$:
liquid theory approximation (L), Eq.~(\ref{eqn:phat}),
the liquid theory approximation normalized by an overall constant (NL), the compound
poisson approximation (CP) and the gaussian approximation (KB). }
\end{center}
\label{dtv_k4096_l4_r2}
\end{table}
The discrepancy between actual and approximate distributions for
${\bf c} = 000$ is persistent: it does not improve with
increasing $\Lambda$, or going to third order
in the expansion of $x$, or by taking the stationary phase
approximation to higher order (which turns out to be a $1/n$ expansion).
The discrepancy for ${\bf c} = 000$ does not seem to be a finite-size effect
either as can be seen by comparing Figs.~\ref{ndist_plot_k4096} and \ref{ndist_plot}
On the other hand, increasing $r$, does reduce the total variations.
Fig.~\ref{ndist_plot_k4096r04} shows the $n$-match distribution for
$l=4$, $m=4092$ and strings whose letters come from a $4$ letter alphabet.
Notice that the total variation of the approximate distributions, are overall
much smaller and all three approximations yield similar results.
In particular the deviations for ${\bf c} = 000$ have disappeared now.
Table 7 gives the corresponding variational distances:
\begin{table}
\vskip 0.25cm
\begin{center}
\begin{tabular}{||r|l|l|l|l||}
\hline
${\bf c}$ & $d^{L}_{TV}$& $\bar{d}^{NL}_{TV}$ & $d^{CP}_{TV}$ & $d^{KB}_{TV}$ \\
\hline
000 & 0.008 & 0.008 & 0.016 & 0.030 \\
001 & 0.005 & 0.003 & 0.004 & 0.034 \\
010 & 0.006 & 0.007 & 0.014 & 0.036 \\
111 & 0.028 & 0.013 & 0.005 & 0.080 \\
\hline
\end{tabular}
\caption{Total variational distance between the actual distribution and the various
approximate distribution for the case $r=4$, $l=4$ and $k=4096$:
liquid theory approximation (L), Eq.~(\ref{eqn:phat}),
the liquid theory approximation normalized by an overall constant (NL), the compound
poisson approximation (CP) and the gaussian approximation (KB). }
\end{center}
\label{dtv_l4_r4}
\end{table}
Comparing with Table 3, we see indeed that for
$r=4$ the total variational distances are overall smaller.
It seems that for the case ${\bf c} = 000$ and $r=2$, the stationary phase approximation
around the single point $u \approx 1$ is not capturing all the contributions to the
probability distribution.
\vspace*{1cm}
\begin{figure}[!ht]
\includegraphics[width=16cm]{ndist_4096kl04r04_norm.eps}
\caption[]{The $n$-match distribution for matching a $l=4$ letter 4-ary
string $x$ inside a random string of length $k=4096$, for $x=0001$ (top left),
$x=1001$ (top right), $x=1010$ (bottom left) and $x=1111$ (bottom right).
The circles are the exact probabilities, the dashed and dashed-dotted lines
correspond to the Gaussian and compound poisson approximation (see text for details).
The solid line is the analytical Eq.~(\ref{eqn:phat}).
}
\label{ndist_plot_k4096r04}
\end{figure}
Finally, we would like to remark that the expansion of $x$, Eq.~(\ref{eqn:virialx})
is in fact the virial expansion of the equation of state for the (discrete) lattice
gas. The parameter $x$ is related to $z$ as $x=1/z - 1$, Eq.~(\ref{eqn:xu}). In the
continuous 1d gas of $n$ particles in a "volume" $L$ and nearest-neighbor interactions,
the partition function can be written as \cite{gursey,fisher}
\begin{equation}
Q(n,L) = \frac{1}{2\pi i} \oint ds e^{s L} \; \;
D^2(s) H^{n-1}(s)
\label{eqn:partcont}
\end{equation}
where $D(s)$ and $H(s)$ are the Laplace transforms of the Boltzmann factor
for the particle-boundary and particle-particle interactions, and Eq.~(\ref{eqn:partcont})
is the inverse Laplace transform with an appropriately chosen contour.
For physical interactions and in the thermodynamical
limit, it turns out that the integral in the above equation
can be evaluated by a saddle point expansion around the point $s_0$ \cite{fisher} and
as a result, it turns out that $s_0 = \beta P$, where
$\beta$ is the Boltzmann factor and $P$ is the pressure \cite{gursey}, \cite{fisher}.
Comparing with Eq.~(\ref{eqn:pnmc_contour_app}) we see that upon discretizing
the length of the container by letting $L = m \Delta$,
and assuming that the interactions vary slowly with
respect to $\Delta$, Eq.~((\ref{eqn:pnmc_contour_app}) can be recovered under the identification
\begin{equation}
e^{-s_0\Delta} = u = \frac{1}{1+x},
\end{equation}
which for small $\Delta$ implies that x = $s_0 \Delta = \beta P \Delta$. We thus see
that the virial expansion Eq.~(\ref{eqn:virialx}) leads to a van der Waals type
equation of state \cite{uhlenbeck}. Indeed as can be seen from Eq.~(\ref{eqn:eps1}),
$\epsilon_1$ is the {\em effective} hard-core size and the term $(n-1)\epsilon_1$ is
the total excluded "volume" due to the interaction (core + tail).
Fig.~\ref{eq_state} shows the "$P-V$ isotherms" of the lattice gas with $l=4$, $r=2$ and
fixed particle number $n = 15$ for the four equivalence classes ${\bf c} = 000,001,010$
and $111$ (from top to bottom). The thick solid line is the "ideal gas" law $x=n/m$.
The data points have been obtained from numerically solving Eq.~(\ref{eqn:mofx}).
Using the approximate equation of state, Eq.~(\ref{eqn:mxeq}) give similar results
but with increasing deviations at high densities.
\vspace*{1cm}
\begin{figure}[!ht]
\includegraphics[width=16cm]{eq_state_nb15l4.eps}
\caption[]{The "P-V diagram" of the lattice gas with $l=4$, $r=2$ and fixed
particle number $n=15$ for the four possible interactions .
${\bf c} = 000,001,010$ and $111$ (from top to bottom).
The thick solid line corresponds to the "ideal gas" law $x=n/m$
(refer to text for details).
}
\label{eq_state}
\end{figure}
\subsection{Asymptotics}
We now consider the asymptotic form of the $n$-match distributions in the limit that
the length $k=m+l$ of the random string is large. It turns out that this is
most readily done using generating functions. We define the generating
function $p(\zeta,z;{\bf c})$ of $p(n,m;{\bf c})$ as
\begin{equation}
p(\zeta;m,{\bf c}) = \sum_{n=0}^{\infty} p(n;m,{\bf c}) \zeta^n
\end{equation}
From Eq.~(\ref{eqn:pnmc_contour}) we thus find that
\begin{equation}
p(\zeta;m,{\bf c}) = \frac{A_1}{z_1^{m+1}}
+ \frac{A_1}{z_1^{m+1}} \sum_{n=1}^{\infty} \left ( \zeta e^{\beta \mu} \right )^n
\frac{1}{2\pi i} \oint_{\partial D} dz \frac{1}{z^{m+1}} \; \; D^2(z) H^{n-1}(z),
\label{eqn:pzetamc_contour}
\end{equation}
where we have used the asymptotic form $p(0;m,{\bf c}) = A_1/z_1^{m+1}$ for the $n=0$
term, since $m$ is assumed to be large. The order of summation and integration
can be exchanged if the integrand is uniformly converging in the region of
integration. It is not hard to show that this can be achieved for example by a
circular path $\| z \| = R$, with a suitably chosen $R < 1$. Thus carrying out
the sum first, we obtain
\begin{equation}
p(\zeta;m,{\bf c}) = \frac{A_1}{z_1^{m+1}}
+ \frac{A_1}{z_1^{m+1}} \zeta e^{\beta \mu}
\frac{1}{2\pi i} \oint_{\partial D} dz \frac{1}{z^{m+1}} \; \;
\frac{D^2(z)}{1 - \zeta e^{\beta \mu} H(z)}.
\label{eqn:pzetamc_contour2}
\end{equation}
Substituting the approximate forms for $D(z)$ and $H(z)$, Eqs.~(\ref{eqn:dhat}) and
(\ref{eqn:dhat}), we find
\begin{eqnarray}
&\hat{p}~(\zeta;\gamma,m,{\bf c})& = \frac{A_1}{z_1^{m+1}} \nonumber \\
&+& \frac{A_1}{z_1^{m+1}}
\frac{\zeta e^{\beta \mu}}{2\pi i} \oint_{\partial D} \frac{dz}{z^{m+1}} \; \frac{1}{1-z} \; \;
\frac{ \left [ z^\Lambda+ (1-z) \left ( D_\Lambda(z) + \Gamma(z) \right ) \right ]^2}
{ (1-z) \left [ 1 - \zeta e^{\beta \mu} \left ( H_\Lambda(z) + \Gamma(z) \right ) \right ] - \zeta e^{\beta \mu} z^\Lambda } . \nonumber \\
\label{eqn:pzetamc_contour3}
\end{eqnarray}
Denote the expression in the denominator by $\bar{\lambda}(z;\zeta,{\bf c})$,
\begin{equation}
\bar{\lambda}(z;\zeta,{\bf c}) = (1-z) \left [ 1 - \zeta e^{\beta \mu} \left ( H_\Lambda(z) + \Gamma(z) \right ) \right ] - \zeta e^{\beta \mu} z^\Lambda .
\label{eqn:lambdabar}
\end{equation}
Since $\exp(\beta \mu)$, is of order $1/r^l$ it follows that $\bar{\lambda}(z;\zeta,{\bf c})$
has a root near $z = 1$. It turns out again that this is the root closest to the origin and
that all other roots are of order $\| z \|^\Lambda \zeta \exp(\beta \mu) \sim 1$.
Denoting the root of smallest magnitude by $\bar{z}_1$, and using the method of
Section \ref{asySec}, a series expansion of $\bar{z}_1$ can be made. One
finds to lowest order that
\begin{equation}
\bar{z}_1 = 1- \frac{\zeta e^{\beta \mu} }
{1- \zeta e^{\beta \mu} H_\Lambda(1) - \zeta e^{\beta \mu} \Gamma(1)}.
\label{eqn:z1bar}
\end{equation}
The integrand in Eq.~(\ref{eqn:pzetamc_contour3}) has therefore two dominant poles
at $z=1$ and $z=\bar{z}_1$. For large $m$, the contour integral can again be evaluated
approximately by pushing the countour out to infinity and keeping only the residues from
the dominant poles (which are traversed counter-clockwise), as explained in Section \ref{asySec}.
We find
\begin{equation}
\hat{p}(\zeta;\gamma,m,{\bf c}) =
\frac{A_1}{\left (z_1 \bar{z}_1 \right )^{m+1}}
\frac{\zeta e^{\beta \mu}}{1-\bar{z}_1}
\left ( - \frac{1}{\bar{\lambda}^\prime ((z;\zeta,{\bf c}) } \right )
\left [ (1-\bar{z}_1)\left (D_\Lambda(\bar{z}_1) + \Gamma(\bar{z}_1) \right ) + \bar{z}_1^\Lambda \right ]^2.
\label{eqn:phat_largem}
\end{equation}
Notice that the $m$ dependence is entirely confined to the
term $1/(z_1 \bar{z}_1)^{m+1}$. Thus this
term alone is responsible for the large $m$ behavior.
The term in the square brackets is the effect due to the
boundaries of the string. When $m$ is
large boundary effects should not matter and we will set
this term to $1$. Alternatively, we
can assume that the random string is circular and
in this case the boundary term will not arise.
Apart from the cut-off assumption on the behavior of the tails, and the assumption of
large $m$ leading to the $m$-asymptotic expression, Eq.~(\ref{eqn:phat_largem}),
we have not made any assumptions on $r$ or $l$ so far.
To proceed further, we will assume that $1/r^l \ll 1$ so that the lowest order expressions
for $\bar{z}_1$ and $z_1$ will provide the leading order approximation
to Eq.~(\ref{eqn:phat_largem}).
Substituting the lowest order expression for $\bar{z}_1$, Eq.~(\ref{eqn:z1bar})
and noting that
to this order
$-\bar{\lambda}^\prime ((z;\zeta,{\bf c}) = 1 - \zeta \exp(\beta \mu) H_\Lambda(1)
- \zeta \exp(\beta \mu) \Gamma(1) $,
the result simplifies to
\begin{equation}
\hat{p}(\zeta;\gamma,m,{\bf c}) = \frac{A_1}{\left (z_1 \bar{z}_1 \right )^{m+1}}
\end{equation}
The compound poisson distribution arises in the limit when $m \rightarrow \infty$ and
$\langle n \rangle $ is finite. From Eq.~(\ref{eqn:nave}) this implies that the word
length $l$ scales as $l \sim \log_r (m+1)$. From the properties of the interactions
that were derived in Section \ref{interac}, we see that the tails are very
weak and of order $1/m$, while the core is relatively strong and of order $\log m$.
Thus it is permissible to set $\Lambda = l$ and ignore the tails ($\Gamma = 0)$.
Note that in this limit $1/r^l \sim 1/m$ and thus to lowest order $A_1 =1 $,
and
\begin{equation}
e^{\beta \mu} = \frac{1}{r^l} \; \frac{1}{\left [ 1 + c(1/r) \right ]^2}
\end{equation}
We thus obtain
\begin{equation}
\hat{p}(\zeta;\gamma,m,{\bf c}) = \left [
\left ( 1 + \frac{1}{1 + c(1/r) }\; \; \frac{1}{r^l} \right )
\left ( 1- \frac{\zeta}{r^l} \; \frac{1}{\left [ 1 + c(1/r) \right ]^2}
\frac{1}{1-\zeta e^{\beta \mu} H_l(1) } \right ) \right ]^{-(m+1)},
\label{eqn:almost}
\end{equation}
Further simplifications occur, noting
that from Eq.~(\ref{eqn:hexpl0}) to order $1/r^l$ we have
\begin{equation}
\frac{1}{1 + c(1/r) } = 1 - h\left ( \frac{1}{r};{\bf c} \right ),
\label{eqn:hcrelation}
\end{equation}
while from Eq.~(\ref{eqn:Hdef}) we find that
\begin{equation}
e^{\beta \mu} H_l(1) = h\left (\frac{1}{r};{\bf c} \right ).
\end{equation}
Multiplying out the product in Eq.~(\ref{eqn:almost}) and keeping only
terms to order $1/r^l \sim 1/m$, we thus obtain
\begin{equation}
\hat{p}(\zeta;\gamma,m,{\bf c}) = \left [
1 + \frac{1}{r^l} \left ( 1- h\left (\frac{1}{r};{\bf c} \right ) \right )^2
\left ( \frac{1}{1 - h\left (\frac{1}{r};{\bf c} \right )} -
- \frac{\zeta}
{1-\zeta h \left ( \frac{1}{r};{\bf c} \right ) } \right ) \right ]^{-(m+1)},
\end{equation}
Taking now the limit $m \rightarrow \infty$
such that $(m+1)/r^l = \langle n \rangle $ is finite,
the expression is readily brought to the form
\begin{equation}
\hat{p}(\zeta;\gamma,m,{\bf c}) = e^{- \sum_{j=1}^{\infty}
\left ( 1- \zeta^j \right ) \bar{\lambda}_j }
\label{eqn:cppoisson}
\end{equation}
with
\begin{equation}
\bar{\lambda}_j = \langle n \rangle \left [ 1 - h\left (\frac{1}{r};{\bf c} \right )
\right ]^2 h\left (\frac{1}{r};{\bf c} \right )^j .
\end{equation}
Eq.~(\ref{eqn:cppoisson}) is the generating function of a
compound poisson distribution \cite{feller}
and precisely the result derived by various other methods by
Chrysaphinou and Papastavridis \cite{chrys},
Geske {\it et al.} \cite{geske}, and Schbath \cite{schbath}
in the special case of uniformly i.i.d letters.
Also note that the CP distribution is normalized, $\hat{p}(1;\gamma,m,{\bf c}) =1 $.
Note that setting the tails ($b \ge l$) of the interactions to zero means that given the
next match is a distance at least $l$ away, it can occur with
equal probability at any $b\ge l$.
Since nearest neighbor match separations $b<l$ define an overlapping cluster, this means
that the location of the clusters themselves, $b \ge l$, are distributed like the arrivals
of a poisson process \cite{chrys,reinertetal,waterman}. We therefore see that the
liquid theory description in terms of interactions along with the separation of
cores and tails provides an alternative and very simple explanation of this property.
Conversely, strong tails mean that the positions of the clusters themselves are correlated
and deviate from a poisson process (meaning that the probability of initiating a new cluster
depends on the distance from the last cluster).
We now consider the limit $m \rightarrow \infty$ and
$n \rightarrow \infty$ such that in this limit
the number density $n/(m+1) = 1/r^l$ remains constant and is small.
In this limit the tails of the
interaction are also small, and we obtain (to lowest order in $1/r^l$)
\begin{eqnarray}
\hat{p}(\zeta;\gamma,m,{\bf c}) &=& \left [
\left ( 1 + \frac{1}{1 + c(1/r) }\; \; \frac{1}{r^l} \right ) \right ]^{-(m+1)} \nonumber \\
&\times& \left [ \left ( 1- \frac{\zeta}{r^l} \; \frac{1}{\left [ 1 + c(1/r) \right ]^2}
\frac{1}{1-\zeta e^{\beta \mu} H_l(1) - \zeta e^{\beta \mu} \Gamma(1)} \right ) \right ]^{-(m+1)},
\label{eqn:almost-asy}
\end{eqnarray}
Notice that if $\Lambda = \infty$, there would be nothing left for the remaining tail and thus
$\Gamma$ would be zero and we would obtain
\begin{equation}
\hat{p}(\zeta;0,m,{\bf c}) = \left [
\left ( 1 + \frac{1}{1 + c(1/r) }\; \; \frac{1}{r^l} \right )
\left ( 1- \frac{\zeta}{r^l} \; \frac{1}{\left [ 1 + c(1/r) \right ]^2}
\frac{1}{1-\zeta h(1/r;{\bf c})} \right ) \right ]^{-(m+1)},
\label{eqn:almost-asy2}
\end{equation}
The normalization is given by $\hat{p}(1;0,m,{\bf c}) =1$, and using the relation
Eq.~(\ref{eqn:hcrelation}) it is readily seen that the distribution is normalized to order
$1/r^l$. This observation immediately gives us a way to estimate $\Gamma(1)$, which must be
chosen such that the distribution is normalized to that order. We have
\begin{equation}
e^{\beta \mu} \Gamma(1) = e^{\beta \mu} H_l(1) - h\left ( \frac{1}{r}; {\bf c} \right )
\end{equation}
and the normalized distribution becomes
\begin{equation}
\hat{p}(\zeta;m,{\bf c}) = \left [
\left ( 1 + \left [ 1 - h\left (\frac{1}{r};{\bf c} \right ) \right ]\frac{1}{r^l} \right )
\left ( 1- \frac{\zeta}{r^l} \;
\frac{\left [ 1 - h\left (\frac{1}{r};{\bf c} \right ) \right ]^2} {1-\zeta h(1/r;{\bf c})} \right )
\right ]^{-(m+1)}.
\label{eqn:almost-asy3}
\end{equation}
The large $n$ limit can again be obtained using Hayman's method introduced in the previous sub-section.
Choosing $\zeta_0$ such that
\begin{equation}
n = \left. \left ( \zeta \frac{\rm d}{{\rm d}\zeta } \ln \hat{p}(\zeta;m,{\bf c}) \right )
\right |_{\zeta = \zeta_0}
\end{equation}
we find to order $1 - \langle n \rangle /n$
\begin{equation}
\zeta_0 = 1 + \frac{1}{2} \; \frac{1-h(1/r;{\bf c})}{ 1+ h(1/r;{\bf c})}
\left ( 1 - \frac{\langle n \rangle}{n} \right ),
\end{equation}
where $ \langle n \rangle$ is as defined in
Eq.~(\ref{eqn:nave}). Using this approximation for
$\zeta_0$, we find after a little bit of algebra that
the distribution of $n$ around its mean is Gaussian distributed,
\begin{equation}
\hat{p}(\zeta;m,{\bf c}) \frac{1}{\sqrt{2\pi \hat{\sigma}^2_n}}
\exp \left (- \frac{(n - \langle n \rangle )^2}{2\hat{\sigma}^2_n} \right ),
\end{equation}
with
\begin{equation}
\hat{\sigma}^2_n = \langle n \rangle \; \frac{1-h(1/r;{\bf c})}{ 1+ h(1/r;{\bf c})}.
\end{equation}
In concluding this section we would like to point out that our
derivation of the CP and Gaussian asymptotic forms rests
on determing the dominant root of $\bar{\lambda}(z;\zeta,{\bf c})$, Eq.~(\ref{eqn:lambdabar}),
which in turn emerges as a result of introducing a cut-off $\Lambda$ and approximating the
interactions beyond $\Lambda$. In a sense, it is the presence of the cut-off that
simplifies the analytical treatment of the problem, since it makes explicit the separation
of small and therefore negligible terms from the dominant ones.
\section{The case of general random letter strings}
All the calculations and results presented so far, have been worked out for the case of
uniformly and i.i.d letters of the random string. However for many applications
this requirement is too restrictive. Letter distributions that have been considered
in the literature are non-uniform i.i.d letters and letter sequences generated by
a Markov process. For either of the cases asymptotic results in the form of large
deviations, Gaussian and compound poisson distributions exist
\cite{chrys,geske,fudos,regszpan,goldwater,schbath,prum,reinertetal,waterman}.
In this section we show that the $n$-match probability associated with a broader class
of letter distributions can be worked out using the lattice gas description introduced
in the previous section. The essential insights gained from this approach are not
changed by this generalization. The problem to be solved is still that of calculating
the partition function of a 1d lattice gas of $n$ particles
with nearest-neighbor interactions among themselves and the boundaries. The only
difference is that the interactions and hence the calculations become more involved.
The required generating functions have been already derived by
R\'egnier and Szpankowski \cite{regszpan} and we will adopt their results
to our notation. Let again ${\bf y} = (y_1,y_2,\ldots,y_k)$ be the letters of the
random string and let ${\bf x} = (x_1,x_2,\ldots,x_l)$ be the word to be matched.
R\'egnier and Szpankowski consider the case of i.i.d letters with arbitrary
letter distribution (Bernoulli Model) and letter sequences
generated by a one-step Markov process with transition matrix ${\bf P}$, such that
$P_{ij}$ is the transition probability $P\{y_{a+1} = i | y_a = j\}$,
${\bf \pi} = (\pi_1,\pi_2,\ldots,\pi_r)$ is the stationary letter distribution satisfying
${\bf \pi} {\bf P} = {\bf \pi}$,
and the stationary matrix $\Pi$ is the matrix whose $r$ rows
are ${\bf \pi}$ (Markov Model).
Given any subsequence of letters $y_{a+1}, y_{a+2},\ldots, y_{a+l}$,
denote by $p({\bf y}_{a,l})$ the probability of encountering
${\bf y}_{a,l}$, without any conditions on the letters preceeding or following it.
Likewise, denote by $p({\bf x})$ the probability of generating
the word ${\bf x}$. The generating function of the $n$-match probability
is given by \cite{regszpan}:
\begin{equation}
p(n;z,{\bf c}) = p({\bf x}) \tilde{d}^2(z;{\bf c}) \tilde{h}^{n-1}(z;{\bf c}),
\label{eqn:pnztilde}
\end{equation}
with
\begin{equation}
\tilde{d}(z;{\bf c}) = \frac{1 - \tilde{h}(z;{\bf c})}{1-z}
= \frac{1}{p({\bf x})} \; \frac{1}{\lambda(z;{\bf c})},
\label{eqn:dztilde}
\end{equation}
\begin{equation}
\tilde{h}(z;{\bf c}) = 1 - \frac{1}{p({\bf x})} \; \frac{1 - z}{\lambda(z;{\bf c})},
\label{eqn:hztilde}
\end{equation}
and
\begin{equation}
\lambda(z;{\bf c}) = z^l + \frac{1}{p({\bf x})} (1-z) \left [ 1 + \tilde{c}(z)
+ \frac{p({\bf x})}{\pi(x_1)} T(z) z^l \right ],
\label{eqn:lz}
\end{equation}
In the last equation $\pi(x_1)$ is the steady state
probability of encountering the letter $x_1$ and $T(z)$ is the generating
function for the steady-state transition probability from the end of one
word match to the beginning of the next word match as a function of the gap
length between the two words (for the Bernoulli Model $T(z) = 0$).
The generating function
$\tilde{c}(z)$ is defined as
\begin{equation}
\tilde{c}(z) = \sum_{b=1}^{l-1} c_b p({\bf x}_{1,b}) z^b,
\label{eqn:cztilde}
\end{equation}
where $c_b({\bf x})$ are the bit-vectors associated with the word ${\bf x}$.
Note that an overall factor of $z^l$ in the definition of $p(n;z,{\bf c})$
in \cite{regszpan} is absent, since the generating function
$p(n;z,{\bf c})$, as defined above,
corresponds in our case to $p(n;m,{\bf c})$, where $m = k-l$ is the effective
length of the string.
Comparing with the corresponding equations of the uniformly distributed
random letter case, Eqs.~(\ref{eqn:pnzdef}), (\ref{eqn:hzexplicit}),
(\ref{eqn:dzexplicit}) and (\ref{eqn:lambdazdef}), we see that the form
of the equations as well as the relationships between the generating
functions are identical.
In particular, all recursions can be recovered by making the replacements
$h_a/r^a \rightarrow \tilde{h}_a$, $d_a/r^a \rightarrow \tilde{d}_a$,
and $c_a/r^a \rightarrow \tilde{c}_a$ so that
$h(z/r) \rightarrow \tilde{h}(z)$ etc.
The Markov property introduces the additional complication that
one has to propagate the end of one word match at $a_i$ to the
beginning of the next match at $a_{i+1}$ through the $(a_{i+1} - a_i -l)$-step
steady-state transition probability.
R\'egnier and Szpankowski have also proven that the polynomial
$\lambda(z;{\bf c})$ has at least one real root and that all roots
have $ \| z \| \ge 1$, as in the
case of uniform letter distributions.
The asymptotic behavior of $\tilde{h}$ and $\tilde{d}$ is again due to
the root closest to $z=1$.
As can be seen from Eq.~(\ref{eqn:lz}), for $p({\bf x})$ small, the root closest
to $z=1$ is located at roughly
\begin{equation}
z_1 \approx 1 + \frac{p({\bf x})}{ 1 + \tilde{c}(1)
+ \frac{p({\bf x})}{\pi(x_1)} T(1) }
\end{equation}
and all other roots are roughly located at $\| z \| \sim 1/p({\bf x})$.
Recall that in the case of uniformly distributed letters,
$p({\bf x}) = 1/r^l \le 1/2$. For the general letter distributions, both the
distribution as well as the word ${\bf x}$ can be chosen arbitrarily and
thus there is no constraint on the values that $0 \le p({\bf x}) \le 1$ can take.
This means in particular that there is a broader class of possible interactions.
Defining again the effective particle-particle interaction as
\begin{equation}
e^{-\beta U(b)} = \frac{\tilde{h}(b)}{\tilde{h}_{asy}(b)},
\end{equation}
${\tilde{h}_{asy}(b)}$ is readily worked out as
\begin{equation}
\tilde{h}_{asy}(b) = p({\bf x}) \frac{A_1}{z_1} \;
\left [ \frac{(z_1 - 1)^2}{p({\bf x})} \right ] \; \left ( \frac{1}{z_1} \right )^b
\equiv e^{\beta \mu} \left ( \frac{1}{z_1} \right )^b
\end{equation}
where $A_1(z_1 - 1) = -1/\lambda^\prime(z_1)$, {\it cf.} Eq.~(\ref{eqn:A1def}).
We thus obtain for the particle-particle interaction
\begin{equation}
\beta U(b) = - \ln \tilde{h}(b) - b \ln z_1 + \ln p({\bf x}) + \beta U_0,
\label{eqn:Uppcore_gen}
\end{equation}
where
\begin{equation}
\beta U_0 = \ln \left [ \frac{A_1}{z_1} \; \left ( \frac{z_1 - 1}{p({\bf x})} \right )^2 \right ] .
\end{equation}
If $p({\bf x}) \ll 1$, $\beta U_0$ is a constant of order
$p({\bf x})$, since the argument of the logarithm
is of order 1 to the same order.
In the core-region $b<l$, the non-zero values of $\tilde{h}$ are still determined by the bit-vector
${\bf c}$ associated with ${\bf x}$ and we find analogous to Eq.~(\ref{eqn:hcore}) that
\begin{equation}
\tilde{h}(b) = \left \{ \begin{array}{ll}
c_b p({\bf x}_{1,b}), & \mbox {if $\chi$ does not divide $b$,} \\
p({\bf x}_{1,\chi}), & \mbox{if $b = \chi $,} \\
0, & \mbox{otherwise,} \\
\end{array} \right.
\label{eqn:hcore_gen}
\end{equation}
where $\chi$ is the fundamental period associated with ${\bf c}$ that was
defined at the end of Section \ref{bit-sect} and by definition,
$\tilde{h}(0) = 0$.
We see that the interaction is $+\infty$, whenever $c_b = 0$. This is
certainly the case for $b < \chi$. The interaction in the core-region is given
by
\begin{equation}
\beta U(b) = - \ln c_b - b \ln \left [ z_1 p^{1/b}({\bf x}_{1,b} ) \right ] + \ln p({\bf x}) + \beta U_0,
\label{eqn:Uppcore_gen2}
\end{equation}
Comparing the above with the uniform letter distribution case, Eq.~(\ref{eqn:Uppcore}),
we see similarities as well as differences: the argument of the logarithm in square
brackets is no longer necessarily smaller than one, but can depend on the subtle
interplay of the overall word matching probability $p({\bf x})$ (which determines $z_1$)
with the (smaller) probabilities of matching the subwords $p({\bf x}_{1,b})$.
Thus such cases
require more care. Assuming that $p({\bf x})$ is sufficiently small so that the expression
in square brackets is smaller than one, it is again the case that the energy of the core
region is of order $\ln p({\bf x})$ and increases with $b$. Under the same assumptions, the
characteristic energy of the tail region can be worked out and one finds that it goes like
$p({\bf x})$ similarly to the line of reasoning in Section \ref{interac}.
Thus we see that by suitably choosing the set of probabilities $ p({\bf x}_{1,b})$, for
$b=1,2\ldots,l$ the strengths of the core and tail of the interactions can be varied and
can possibly move the distribution functions into a regime where approximations
ignoring the contributions from the tails (such as the compound poisson approximation)
are inappropriate.
Also note that by letting $p({\bf x})$ to be arbitrarily close to one, the difference of
magnitudes between the root closest to the origin $z_1$ and the other roots can be made to
vanish. Since the attenuation length of the tail of the interactions depends on the
separation of these roots, we see that in this limit $z_1$ ceases to be the dominant
root. For the interactions this means an increasingly more slowly decaying tail as the
two roots approach each other. In such a regime the tails of the interactions
should become very important and thus cannot be neglected.
It is possible that for certain distributions and choices of words, this can
cause a break-down of the liquid theory approach, which essentially
is a perturbation theory and it would be interesting
to find out if and how this can happen. Strong tails will certainly affect the quality
of approximations such as the compound poisson distribution, which was based on the
assumption that tails can be ignored, which turned out to be equivalent to
assuming a poissonian distribution of cluster locations,
as explained in Section \ref{asySec}.
We will further discuss these points in the Discussion section below.
\section{Discussion}
We have presented a new approach to calculating the probability distribution for
the number of matches of a given word inside a random string of letters.
Our approach rests on the observation that the exact expression for such a distribution
can be interpreted as the partition function of an $n$-particle system on a
linear lattice, with pairwise nearest neighbor interactions. By exploiting this
analogy and focusing on the generic properties of the interaction, we have been
able to set up a virial expansion for the equation of state of this lattice
gas and thereby obtained an analytical expression for the $n$-match probability
distribution, which besides extrapolating between the known asymptotic forms,
also provides a good approximation in the intermediate regimes.
The identification and subsequent analysis of the effective interactions in the
lattice gas description turns out to be key in our solution of this problem.
The interactions are characterized by a strong core-region of the size of the
word-length followed by a relatively weak and exponentially decaying tail.
Although we have carried out the detailed analysis for the special case of
uniform letter distributions, we showed in Section IV, that our
method is readily extended to the broader class of distributions, such as
non-uniform letter distributions and random letter sequences generated by
a Markov process. Regardless of the underlying stochastic process for the
random string, the generic feature of the interactions are still the same,
namely a relatively strong core and a weaker tail and our approach should be
readily applicable to these types of problems as well.
We should also point out that our method of approach bears some
similarity with the work of R\'egnier and Szpankowski \cite{regszpan}, who
also use generating functions in their approach to this problem. Our
approach is however distinct in at least one crucial point: In the cited
work, upon deriving the generating functions
for the $n$-match distribution, Eqs.~(\ref{eqn:pnztilde}), (\ref{eqn:dztilde}),
(\ref{eqn:hztilde}) and (\ref{eqn:lz}), the authors perform a
Laurent expansion of the generating function around its dominant $n+1$ order
pole at $z_1$. Such an expansion is asymptotic in the interactions
and runs the risk of capturing more accurately the tail of the
interaction rather than its core (or at least many terms must be kept in order
to capture the core part to a sufficient degree of accuracy \cite{wilf}).
The approximation scheme presented here precisely avoids this
by introducing a cut-off distance $\Lambda$ and keeping
the {\em exact} interaction upto $\Lambda$, while approximating
the interactions only beyond $\Lambda$. As we have shown, this is
easily done, since the structure of the core of the interaction ($b < l$)
directly follows from the overlap properties of the string to be matched.
Our analysis also shows that since the core part of the interaction
is typically stronger than the exponentially decaying tail, keeping the core
is crucial in determining the global properties of the distribution.
Moreover, our approach allows us to understand approximations such
as the compound distribution as being applicable in a regime where
the tails of the interaction can be neglected and only the core is kept.
This also highlights the relative importance of the core part of the
interaction with respect to its tail.
Lastly, we would like to remark that our treatment of interactions,
by separating out its strong and short-ranged core from its
weak tail, is actually not new. Interactions with a strong core
and an exponentially decaying tail are known as Kac potentials,
named after M. Kac, who along with co-workers studied one-dimensional
particle systems with such interactions (continuum and lattice
version) in considerable detail,
as part of an effort to understand the liquid-gas
transition in the context of the van der Waals equation of
state \cite{kac,kacetal,hemleb} (for an overview, see
the review article by Hemmer and Lebowitz \cite{hemleb}).
Such systems are interesting, since they lead to phase transitions
in the limit when the characteristic decay length of the interaction
tends to infinity \cite{kac,kacetal,hemleb} (for an overview of phase transitions in one
dimensions see, the review article by Griffiths \cite{griffiths}).
The similarity of such systems with the
string matching problem is at hand, since one can make the
interactions to decay as slowly as one wishes by
choosing a suitable random letter distribution and string ${\bf x}$ to
be matched such that the dominant poles of the generating function
of the $n$-match distribution function become arbitrarily
close to each other. It would therefore be of interest
to see whether the distribution functions in this regime
can be calculated using the more sophisticated techniques,
such as integral equations and operator methods,
which have been introduced particularly for the purpose
of dealing with such types of interactions \cite{hemleb}.
\vskip 1cm
{\bf Acknowledgments}
I would like to thank Ay\c{s}e Erzan for both initially bringing to my attention
the string matching problem and later pointing out the connection with Kac
potentials. This work was supported in part by the Nahide and Mustafa
Saydan Foundation and T\"ubitak, the Turkish Science and Technology
Research Council.
\vskip 2cm
|
{
"timestamp": "2005-08-25T22:49:20",
"yymm": "0411",
"arxiv_id": "cond-mat/0411706",
"language": "en",
"url": "https://arxiv.org/abs/cond-mat/0411706"
}
|
\section{\label{sec:introduction}Introduction}
The LOCKSS\footnote{LOCKSS is a trademark of Stanford University.}
(Lots Of Copies Keep Stuff Safe) program has developed and deployed
test versions of a system for preserving access to academic journals
published on the Web. The fundamental problem for any digital
preservation system is that it must be affordable for the long term.
To reduce the cost of ownership, the LOCKSS system uses generic PC
hardware, open source software, and peer-to-peer technology. It is
packaged as a ``network appliance,'' a single-function box that can be
connected to the Internet, configured and left alone to do its job
with minimal monitoring or administration. The system has been under
test at about 50 libraries worldwide since 2000.
Like other Internet hosts, these appliances are continually subject to
attack. Although measures~\cite{Rosenthal2003b} have been taken to
render the operating system platform resistant to attack, its
compromise must be anticipated. The appliances cooperate with each
other to detect and repair damage in a peer-to-peer network. The
first version~\cite{Rosenthal2000} of this protocol turned out to be
vulnerable to various attacks. We recently redesigned the
protocol~\cite{Maniatis2003lockssSOSP} to make it more resistant to attack.
The redesign needed as input an assessment of the capabilities and
strategies of the potential adversaries, but we were unable to find
this information off-the-shelf. We present our assessment, and the
implications we drew from it, as a contribution to an eventual
reusable adversary specification.
\section{\label{sec:adversaryAssessment}Adversary Assessment}
Military intelligence seeks to develop so-called ``appreciations'' of a
potential adversary's ``capabilities'' (what the adversary \emph{could}
do) and ``intentions'' (what the adversary is \emph{expected} to attempt
with the capabilities) as a basis for
planning~\cite{Army1985}. Similarly, plans and techniques for defending
distributed systems exposed to the Internet need an appreciation of the
capabilities and intentions of the adversary they may encounter when
deployed.
Our assessment identified the following probable adversary capabilities:
\begin{itemize}
\item Unlimited Power
\item Unlimited Identities
\item Conspiracy
\item Eavesdropping and Spoofing
\item Exploiting Common Vulnerabilities
\item Uncovering Secrets
\end{itemize}
\subsection{\label{sec:unlimitedPower}Unlimited Power}
Techniques~\cite{Staniford2002} have been described by which a worm
could compromise a large proportion of vulnerable Internet hosts
in a short time.
In practice,
even much less sophisticated techniques~\cite{SlammerAnalysis} have proven
capable of compromising large numbers of hosts quickly,
despite widespread knowledge of both the vulnerabilities themselves
and their cures for
six months prior to the attack~\cite{SlammerVulnerability}.
Further:
\begin{itemize}
\item Experience with Code Red~\cite{CodeRedAnalysis} shows that at least 1/3
of the compromised hosts remain compromised a month after the start of the
attack.
Two years after the attack a pool of 20,000 infected hosts was still
available~\cite{Krebs2003}.
\item Experience with Slapper~\cite{Rescorla2003} shows that 1/3 of
vulnerable hosts were still vulnerable 3 months after the vulnerability
was announced and 1 month after the start of the attack.
\item Experience with a BIND vulnerability~\cite{BindVulnerability}
shows that a significant proportion of professionally maintained systems
are still vulnerable two months after the vulnerability was made public.
\item Advertisements are rumored to be appearing that invite spam senders
to rent access to a network of 450K compromised hosts they
can use to disguise the origin of e-mails.
\end{itemize}
So far, these networks of compromised hosts have been used to mount
crude but effective~\cite{MiMailAttack} network-level denial of
service attacks.
However,
it would be a simple matter for the payload of such a worm to be
an application-level attack targeted at a particular victim system.
If the worm were based on a vulnerability
as widespread (350K+ hosts) as the ones Code Red~\cite{CodeRedAnalysis}
or Blaster (385K+ hosts)~\cite{Krebs2003} exploited,
the attacker could expect on the order of 10K machine-years of
computation to be available for the attack on the victim system
(30\% of systems compromised for 1 month, 10\% for 3 months).
This is,
for example,
about 35 times the effort used to win the RSA DES Challenge III in
1999~\cite{DESCrack}.
There is a practical difficulty for the adversary hoping to use these
pools of compromised hosts as a resource for attacking a given system.
Many other adversaries with other targets are in competition for the
resource, which is not infinite although it may be large. This
difficulty, however, is not a comfort to the designer of system defenses,
whose worst-case analysis must assume that all available resources may
be used for a single-minded attack against his system.
\subsection{\label{sec:unlimitedIdentities}Unlimited Identities}
Given the relative ease by which an adversary can compromise and
control a large number of hosts across the Internet, we must assume
that the adversary can pose as an unlimited number of identities,
e.g., IP addresses. The adversary can either directly use the
compromised host's IP address or make the compromised host spoof other
IP addresses on the same subnet. Even if ingress filtering
\cite{ingress} were turned on in all routers across the Internet, the
cost for a host to spoof an IP address on the same subnet is
negligible.
There is a practical difficulty for the adversary in that he can only
steal identities on subnets in which he maintains a presence, either
legitimately or through compromise. This difficulty is not a comfort to
the designer of system defenses who must assume that the adversary can
have a presence in thousands of subnets spread across the Internet.
The assessment above is not unique to IP addresses. Email addresses,
identity certificates, DNS domains are just as easy for an adversary to
hoard or spoof or both. Techniques for making this more difficult or
time-consuming for an adversary include client puzzles and reverse
Turing tests~\cite{captcha}, but the adversaries are adapting to them.
For example,
it is now rumored that reverse Turing tests can be forwarded to a
service run by porn sites,
which exploit their customers to solve them and return their
responses.
\subsection{\label{sec:conspiracy}Conspiracy}
The Fizzer worm uses IRC~\cite{FizzerComms} to communicate with a central
control site.
It would be possible for a worm to use peer-to-peer communication techniques
instead,
avoiding the difficulties the Fizzer worm suffered
when its IRC channel was subverted by its enemies~\cite{FizzerCounter}.
It has to be assumed,
therefore,
that all the adversary's identities mask a single distributed adversary
with instantaneous self-awareness. Any state, such as messages
sent, received, or observed by one identity acting on behalf of
the adversary is immediately made available to all other identities.
In addition,
it must be assumed that some apparently benign identities are
conspiring with the adversary.
Anything known to these ``spies,''
including supposed secrets such as session keys,
is known to the adversary.
It is practically difficult for the adversary
to distribute information rapidly and completely among the
components of a distributed system with as many nodes as there are
compromised hosts. This difficulty is not a comfort for the designer of
system defenses, who must assume that the adversary can succeed in
getting the critical information to the nodes that need it.
\subsection{\label{sec:eavesdropping}Eavesdropping and Spoofing}
A single compromised host on a subnet can eavesdrop on traffic to
and from all hosts on the same subnet. It can also send spoofed
messages on behalf of the co-located hosts,
as well as send messages with spoofed source addresses from
anywhere in the Internet to
co-located hosts. By doing so it can often abuse trust relationships
mediated by IP addresses. This behavior is very difficult to detect
and prevent when compromised hosts are not regularly monitored and
maintained.
\subsection{\label{sec:commonVulnerabilities}Common Vulnerabilities}
Even if the design of the system's defenses is perfect, the designer
cannot assume that their implementation is as perfect. It is likely
that, at some point, an exploitable implementation vulnerability will be
discovered. A well-designed flash worm exploiting it can compromise
the vast majority of the vulnerable hosts in a very short time.
In different contexts including traditional Byzantine Fault
Tolerance~\cite{Castro1999}, Distributed Hash Tables~\cite{Castro2002}
and sampled voting~\cite{Maniatis2003lockssSOSP} it has been shown that
systems with more than about 1/3 faulty or malign peers cannot survive
for long. Given this, even in fault-tolerant systems, peers need to be
assigned at random one of at least four independent implementations if
the system is to survive the discovery of an implementation
vulnerability. Rodrigues et al.~\cite{Rodrigues2001} describe a
framework within which independent implementations can be accommodated
in a fault-tolerant system.
It is important to note that even a perfectly designed and implemented
system cannot avoid vulnerabilities brought about by human operators who
are coerced to misbehave. An invulnerable computer system, though
unimaginably hard to build, is certainly easier to imagine than an
incorruptible human.
\subsection{\label{sec:uncoveringSecrets}Uncovering Secrets}
Most systems rely on secret-based encryption systems to preserve
system integrity. The assumption is that the adversary does not
know and cannot in a timely fashion obtain any of the secrets.
This is not a robust assumption. A recent survey~\cite{BallPointPasswords}
purported to show that the vast majority of commuters at
a London station would reveal their passwords if offered a ball-point
pen. The adversary may conspire with an insider, he may be the beneficiary
of lax security by insiders such as poor password choice~\cite{Klein1990},
he may steal authentication tokens, and, given the resources we assume, he may
even use brute-force techniques to break the encryption.
System designers should not treat encryption as a
panacea~\cite{RosenthalSeiden}. An individual analysis is needed of the
consequences of compromise of each key in the system, if only to assess
the precautions appropriate for its protection.
\section{\label{sec:intentions}Intentions}
We have presented an assessment of some of the putative adversary's
capabilities. We must now assess his possible intentions. What
might the adversary be intending to achieve by exploiting these
capabilities?
Our initial attempt classifies possible adversary intentions into
five classes: Stealth, Nuisance, Attrition, Thief, and Spy.
\subsection{\label{sec:Stealth}Stealth}
The \emph{Stealth} adversary's goal is to damage the system by affecting
its state. A necessary sub-goal is to avoid detection before the damage
is complete, for example to dodge an intrusion detection system.
\subsection{\label{sec:nuisance}Nuisance}
The \emph{Nuisance} adversary's goal is to discredit the system by
continually raising intrusion alarms. There is no intention to
cause any actual damage to the system or prevent it from functioning.
An attack from the Nuisance adversary might, for example, be
intended to get the victim's system administrators to disable or
ignore the intrusion alarms as a prelude to other forms of attack.
\subsection{\label{sec:Attrition}Attrition}
The \emph{Attrition} adversary's goal is to prevent the system from functioning
for long enough to inflict damage on the organization it supports.
Some forms of the adversary are referred to as ``Denial of Service,''
but this has come to mean a technique rather than a goal.
The Blaster worm was an Attrition attack, attempting to mount a flooding
attack on a Microsoft website from its 385K infected hosts. The MiMail
virus is an Attrition attack against a set of anti-spam
services~\cite{MiMailAttack}.
\subsection{\label{sec:thief}Thief}
The goal of the \emph{Thief} adversary is to steal services provided by
the system (possibly over long time periods) or steal valuable
information protected by the system. The Thief is different from the
Stealth adversary in that he does not necessarily want to alter the
state of the system, nor does he want to bring the system down or
subvert it. The Thief of services wants unauthorized access to
resources for as long as possible without being detected. The Thief of
information hopes that his intrusion remains undetected for as long as
possible.
The Sobig series of viruses~\cite{Sullivan2003} is believed to be a
Thief who steals services from victim machines by using them as a
spam-sending network. It is also thought to be used to mount
Attrition attacks on anti-spam services~\cite{SobigSpam}.
\subsection{\label{sec:spy}Spy}
The \emph{Spy} adversary's goal is to observe as much about the system
as possible: who participates, where users are located, and what
transactions take place. The Spy could be a powerful corporation
wanting to harass or prosecute users. The Spy could also be a
government collecting information on the on-line activities of its
citizens.
\section{\label{sec:rulesOfThumb}Rules Of Thumb}
We summarize these assessments with some conservative ``rules of thumb.''
The assumptions underlying them are a worm infecting three times
as many hosts as Code Red,
with the bulk of the infection lasting four days,
and 10\% still infected after three months.
The adversary can:
\begin{itemize}
\item exert bursts of computational effort lasting 100 hours and
using 1,000,000 hosts,
\item sustain computational effort over 100 days using 100,000 hosts,
\item masquerade behind 1,000,000 IP addresses,
\item eavesdrop on and spoof traffic from 10\% of the hosts in the
victim system for 100 days.
\item break 100 well-chosen DES keys.
\end{itemize}
\section{\label{sec:implications}Implications}
Our adversary is very powerful, posing a number of important
implications. First, it is economically infeasible to test,
or even simulate,
attacks of this scale.
Assurance that a system does not fail under expected attacks is not likely to
be available or credible.
Design should focus on:
\begin{itemize}
\item Graceful, or at least survivable, failure.
\item Assisting diagnosis, perhaps by using bimodal
behaviors~\cite{Birman1999} to raise alarms.
\item Assisting recovery.
\end{itemize}
Second, the adversary can mount a full-scale attack with no warning.
Rate-limiting techniques~\cite{Rosenthal2000,Forrest2000,Williamson2002}
are important in slowing the rate of failure enough to allow for
human intervention before failure is total.
Third,
the adversary can appear as huge numbers of new peers or clients.
Limiting the rate at which the system accepts new peers or clients
using techniques such as ``newcomer pays''~\cite{Friedman2001} may
help slow the failure.
\section{\label{sec:relatedWork}Related Work}
Researchers in many different fields have tackled the task of
characterizing malicious adversaries. In this section, we outline only
a few of the approaches we have identified in the literature.
\begin{itemize}
\item \emph{Cryptography} typically uses game-theoretic analyses to
construct sets of ``games'' resulting in the adversary behavior observed
by benign protocol participants, and investigate whether those sets
contain games with malign participants.
\item \emph{Protocol design} typically uses exhaustive search of the
transitive closure of the state space of the protocol without explicitly
modeling an adversary's capabilities or intentions. Finite-state
analysis takes the same approach in an automated fashion, with some
notable successes (see, for example, an automated analysis of
authentication protocols~\cite{Mitchell1997}).
\item \emph{Distributed systems theory} typically works backwards from a
bad state of the system (e.g., a state in which an exploit has been used
to damage the system) to identify the sequence of events that must have
happened to arrive at that state. The system has to be specified in a
suitable formalism (e.g., Lamport's TLA+~\cite{Lamport2002} or Lynch and
Tuttle's Input/Output Automata~\cite{Lynch1989,Lynch1996}), but in some
cases it is possible to conduct an invariant analysis without a full
system specification.
\item \emph{Fault tolerance} typically places broad limits on the
adversary (e.g., ``no more than 1/3 of the nodes can be malign'' in the
case of byzantine fault tolerance~\cite{Lamport1982}). In other cases,
nodes with similar failure modes can be grouped together into distinct
equivalence classes with respect to failures (e.g., in Malkhi and
Reiter's work on quorum systems~\cite{Malkhi1998}). These can be
loosely considered an adversary model.
\end{itemize}
Previous work on defending systems against attack classifies adversaries
as either ``computationally bounded or unbounded'' and considers the
time interval over which the adversary collects or modifies state
\cite{Dingledine2000b}. Although the pool of vulnerable machines on
which an adversary can draw is in fact limited, it is large enough and
the repair rates low enough that the adversary may be considered
effectively unbounded in effort and time.
RFC3607~\cite{rfc3607} describes how a worm payload can be used for
cryptanalysis,
and identifies the first such payload observed in the wild.
\section{\label{sec:conclusion}Conclusion}
We have presented what we believe is a conservative assessment of
the putative adversary the designers of defenses for an Internet
system must take into account.
This adversary is based on reasonable extrapolations from
the observed behavior of worms exploiting vulnerabilities in
applications and systems that are widely deployed on the Internet,
and on the assumption that the payload of such worms might be
targeted at the system under consideration. We believe that
discussion of this and alternative adversary assessments
leading to some consensus as a basis for future designs would
be valuable.
Our adversary is powerful enough to pose design, implementation
and testing problems well beyond those current technology can solve.
It appears that designing systems to survive attacks of this
magnitude unimpaired is unlikely to succeed.
Further,
even if the design appeared to succeed,
testing implementations to assure that success was manifest in practice
is unlikely to be affordable. A more reasonable goal may be to
slow and delay the process of failure under attack to allow for
human intervention.
\section{\label{sec:acknowledgements}Acknowledgments}
This material is based upon work supported
by the National Science Foundation under Grant No.\ 9907296, however any
opinions, findings, and conclusions or recommendations expressed in this
material are those of the authors and do not necessarily reflect the
views of the National Science Foundation.
The LOCKSS program is grateful for support from the National Science
Foundation, the Andrew W. Mellon Foundation, Sun Microsystems
Laboratories, and Stanford Libraries.
Vicky Reich has made the LOCKSS program possible.
\bibliographystyle{plain}
|
{
"timestamp": "2004-11-22T01:22:24",
"yymm": "0411",
"arxiv_id": "cs/0411078",
"language": "en",
"url": "https://arxiv.org/abs/cs/0411078"
}
|
\section{Introduction}
\label{sec:1}
Since the fundamental work by Kolmogorov in $1941$ \cite{frish},\cite{K41} one of the key points for many theoretical achievements in turbulence research is the (statistical) restoration of homogeneity and isotropy of fluctuations at small scales \cite{sree},\cite{arneo}.
It is however impossible to provide a proper and consistent description of a great variety of systems where ``non idealized'' fluctuations are still alive. The characterization of the emission of a tracer from point-like sources, the study of scalar concentrations along channels with non-homogeneous boundaries, or in those systems whose large scales are driven by strong shears, are remarkable examples in which strong anisotropies \cite{Kur},\cite{Shen} and inhomogeneities \cite{Dhr} must be taken into account in order to obtain a correct picture of the statistical properties of those systems.\\
In the last decade a consistent progress in the development of a systematic analysis to separate isotropic fluctuations from the anisotropic ones in real turbulent flows and turbulent transport \cite{procaccia},\cite{bif1},\cite{bif2} has been carried out. It has been understood how to treat and face systems whose rotational symmetries are broken by the presence of an external forcing inducing anisotropic contributions. In particular, the study of a simplified model for passive scalar advection, known as Kraichnan model \cite{kraic1},\cite{kraic2}, has provided a clear understanding of the statistical properties in all anisotropic sectors of the scalar fluctuations. Indeed, closed equations for the equal-time correlation functions can be obtained: these are linear partial differential equations whose unforced solutions (also called zero modes \cite{gaw}) generally exhibit anomalous scaling. This is in contrast with the forced solutions that possess (non-anomalous) dimensional scaling. This has given the insight to explain the universality in the statistical framework. Indeed, the anomalous properties of small scale statistics result from a decoupling between the zero mode scaling and dimensional scaling, and the universality of these properties naturally emerges because the zero-mode scaling properties are independent on the forcing mechanism (see \cite{falk} for an exhaustive review).\\
In the present paper we formulate the concept of the possible small-scale homogeneity restoration by focusing on the two-point equal-time scalar correlation function for the Kraichnan advection model. The advecting velocity is still homogeneous and isotropic but this is not for the scalar injection mechanism which is supposed here to be neither isotropic nor homogeneous. As we will show, the inhomogeneous forcing induces a new lengthscale $\ell_q$ into the scalar dynamics, in terms of which quantitative conclusions on the persistence of small-scale inhomogeneity will be given. The aim of this investigation is twofold. Firstly, we want to show how the correct homogeneous limit can be restored going at separations (in the two-point scalar correlation function) smaller than $\ell_q$. Secondly, we want to give some analytical insights about the opposite physical situation represented by the presence of inhomogeneous fluctuations on scales of the same order of our separation. In the latter regime the pure power-law behaviour (homogeneous limit) is replaced by the ``beating'' in superposition of different power laws originating from the scalar inhomogeneities.\\
The paper is organized as follows: in section~\ref{sec:2} we formulate the general problem of anisotropic and inhomogeneous correlations. This problem will be studied analytically for the Kraichnan ensembles. In section~\ref{sec:3} the statistical description of the isotropic sector is provided with particular emphasis on some kinds of physically meaningful forcings. Conclusions follow in section~\ref{sec:4}.
\section{The two-point non homogeneous, non isotropic scalar correlation functions}
\label{sec:2}
\subsection{Basic equations}
Let us start with the equation for a passive scalar field transported by the velocity field ${\bi v}$:
\begin{equation} \label{partenza}
\partial_t\theta({\bi x}_1,t)+{\bi v}({\bi x}_1,t)\cdot\bnabla\theta({\bi x}_1,t)=\kappa\Delta\theta({\bi x}_1,t)+f({\bi x}_1,t)\;.
\end{equation}
The velocity field ${\bi v}$ is assumed incompressible, statistically homogeneous and
isotropic, whereas the source term $f$ is a large-scale random forcing
which is not invariant under translations. Let us now specialize to the case of the Kraichnan ensemble \cite{kraic1},\cite{kraic2} where the velocity field is Gaussian, white in time, zero-mean and with two-point correlation function
\[\langle v_{\alpha}({\bi x}_1,t_1)v_{\beta}({\bi x}_2,t_2)\rangle=D_{\alpha\beta}({\bi x}_1-{\bi x}_2)\delta(t_1-t_2)\;.\]
The spatial behaviour is described by
\[D_{\alpha\beta}({\bi r})=D_0\delta_{\alpha\beta}-d_{\alpha\beta}({\bi r})\;,\]
with
\[d_{\alpha\beta}({\bi r})=D_1r^{\xi}\left[(d+\xi-1)\delta_{\alpha\beta}-\xi\frac{r_{\alpha}r_{\beta}}{r^2}\right]\]
for $r=|{\bi r}|$ smaller than the integral scale of the velocity field ($L_v$), above which $d_{\alpha\beta}({\bi r})$ saturates to an almost constant value whose order of magnitude is $D_1L_v^{\xi}$; consequently, since the correlation $D_{\alpha\beta}({\bi r})$ has to vanish for $r \to \infty$, the relation $D_0 \sim D_1L_v^{\xi}$ holds. $d$ is the space dimension ($\ge 2$) and $\xi$ is the scaling exponent, describing the degree of roughness present in the velocity field, lying in the open interval $(0,2)$. The forcing is assumed to be Gaussian, white in time, zero-mean and with two-point correlation function $\langle f({\bi x}_1,t_1)f({\bi x}_2,t_2)\rangle=F({\bi x}_1,{\bi x}_2)\delta(t_1-t_2)$. The steady-state equation for the two-point equal-time scalar correlation function $C({\bi x}_1,{\bi x}_2) \equiv \langle\theta({\bi x}_1,t)\theta({\bi x}_2,t)\rangle$ reads
\begin{equation} \label{equazione}
\fl d_{\alpha\beta}({\bi r})\frac{\partial^2 C}{\partial r_\alpha\partial r_\beta}\!+\!\frac{1}{4}[D_{\alpha\beta}({\bi r})\!+\!D_{\alpha\beta}({\bf 0})]\frac{\partial^2 C}{\partial z_{\alpha}\partial z_{\beta}}\!+\!2\kappa\frac{\partial^2 C}{\partial r_{\alpha}\partial r_{\alpha}}\!+\!\frac{\kappa}{2}\frac{\partial^2 C}{\partial z_{\alpha}\partial z_{\alpha}}\!+\!F({\bi r},{\bi z})=0\;,
\end{equation}
where ${\bi z}=({\bi x}_1+{\bi x}_2)/2$ is the centre of mass and ${\bi r}={\bi x}_1-{\bi x}_2$ is the relative separation. To obtain (\ref{equazione}) we have multiplied (\ref{partenza}) by $\theta({\bi x}_2,t)$, averaged over the stochastic fields ${\bi v}$ and $f$ (exploiting Furutsu-Novikov's rule on integration by parts \cite{Novikov}), symmetrized the resulting expression and eventually dropped the temporal derivative term. Fourier transforming in ${\bi z}$ and defining
\[\hat{C} \equiv \hat{C}({\bi r},{\bi q})=\int\rme^{\rmi{\bi q}\cdot{\bi z}}C({\bi r},{\bi z})\rmd{\bi z}\;,\]
\[\hat{F} \equiv \hat{F}({\bi r},{\bi q})=\int\rme^{\rmi{\bi q}\cdot{\bi z}}F({\bi r},{\bi z})\rmd{\bi z}\;,\]
we obtain, in the limit $\kappa \to 0$, the corresponding equation for $\hat{C}$:
\[d_{\alpha\beta}({\bi r})\frac{\partial^2\hat{C}}{\partial r_{\alpha}\partial r_{\beta}}-
\frac{1}{4}[D_{\alpha\beta}({\bi r})+D_{\alpha\beta}({\bf 0})]\,q_{\alpha}q_{\beta}\hat{C}
+\hat{F}({\bi r},{\bi q})=0\;.\]
At small separations $r \ll L_v$, since $D_1 \sim D_0L_v^{-\xi}$, $d_{\alpha\beta}({\bi r}) \sim D_0(r/L_v)^{\xi}$ is negligible with respect to $D_0$ and $D_{\alpha\beta} \simeq D_0\delta_{\alpha\beta}$. In the limit $r \ll L_v$ we can thus consider the simpler equation
\[d_{\alpha\beta}({\bi r})\frac{\partial^2\hat{C}}{\partial r_{\alpha}\partial r_{\beta}}-
\frac{1}{2}D_0q^2\hat{C}+\hat{F}({\bi r},{\bi q})=0\;.\]
Generally, all the anisotropic (in the separation $r$) components of $\hat{F}({\bi r},{\bi q})$ may induce effects that are not globally invariant under rotations. It is thus better to consider the following decomposition \cite{ARAD}:
\[\hat{C}({\bi r},{\bi q})=\sum_{l,m}\hat{C}_{l,m}(r,{\bi q})Y_{l,m}(\Omega)\;,\]
\[\hat{F}({\bi r},{\bi q})=\sum_{l,m}\hat{F}_{l,m}(r,{\bi q})Y_{l,m}(\Omega)\;,\]
with $\Omega$ denoting the solid angle associated with ${\bi r}$, and study the behaviour of the correlation function at separations $r$ smaller than the forcing correlation length $L$, where $\hat{F}_{l,m}(r,{\bi q}) \simeq \hat{F}_{l,m}(0,{\bi q})$. The equation in each anisotropic sector $(l,m)$ for the correlation function $\hat{C}_{l,m}=\hat{C}_{l,m}(r,{\bi q})$ reads
\begin{equation} \label{eq:1}
\fl r^{-(d-1)}\partial_rr^{d+\xi-1}\partial_r\hat{C}_l-\frac{(d+\xi-1)}{d-1}l(d-2+l)r^{-2}\hat{C}_l-\ell_q^{-(2-\xi)}\hat{C}_l+\phi_{{\bi q},l}=0
\end{equation}
where, because of the degeneration, we have dropped the dependence on the subscript $m$ and where we have introduced a rescaled forcing in the form $\phi_{{\bi q},l}=\hat{F}_l(0,{\bi q})/(d-1)D_1$. A new scale $\ell_q$, associated to the strength of the scalar inhomogeneities, has also been introduced. It measures the separation above which inhomogeneities become important (e.g. it can be derived through a dimensional-analysis balance between the first and the third term in (\ref{eq:1})) and it is defined as
\[\ell_q=\left[\frac{q^2D_0}{2(d-1)D_1}\right]^{-1/(2-\xi)}\;.\]
The Fourier scale $q^{-1}$ of the forcing inhomogeneities can be larger or of the same order of $L$. In the latter case we have $\ell_q/L \sim (L/L_v)^{\xi/(2-\xi)} \ll 1$ and the lengthscale $\ell_q$ lies inside the inertial range. The homogeneous case is recovered when $q^{-1} \gg L$ so as to insure $\ell_q/L \gg 1$. An important exception to the previous limitation on $q^{-1}$ is provided, in way of example, by the emission of a tracer from a point source, in which all $q$'s are excited with the same strength. This case is however quite peculiar because $L$ would be infinitesimal and $\ell_q$ would fall in the diffusive range, thus preventing us from taking the limit $\kappa \to 0$. In what follows we will always consider $\kappa \to 0$ and $L_v \to \infty$.
\subsection{General solution}
It is easy to verify that the general solution of (\ref{eq:1}), as a function of $r$ and $\ell_q$, is
\begin{equation} \label{eq:2}
\hat{C}_{l}(r;\ell_q)=\hat{C}_{l;{\rm part}}(r;\ell_q) +r^{-(d+\xi-2)/2}\left[B_1K_{\nu_l}(w)+B_2I_{\nu_l}(w)\right]\;,
\end{equation}
where $w=2(2-\xi)^{-1}(r/\ell_q)^{(2-\xi)/2}$ and $\nu_l=[(d+\xi-2)^2+4(d+\xi-1)l(d-2+l)/(d-1)]^{1/2}/(2-\xi)$. The constants $B_1$,$B_2$ are fixed by the boundary conditions and $K$,$I$ represent the Bessel functions \cite{GRAD} of complex argument whose behaviours for small arguments are $K_{\nu_l}(w) \sim w^{-\nu_l}$ and $I_{\nu_l}(w) \sim w^{\nu_l}$. The particular solution $\hat{C}_{l;{\rm part}}(r;\ell_q)$ of (\ref{eq:1}) can be found, for instance, exploiting the method of variation of constants \cite{couhil} which leads to
\[\fl\hat{C}_{l;{\rm part}}(r;\ell_q)=\phi_{{\bi q},l}Ar^{(2-d-\xi)/2}\left[K_{\nu_l}(w)\int_0^w\rho^{\nu_0+1}I_{\nu_l}(\rho)\rmd\rho+I_{\nu_l}(w)\int_w^{\infty}\rho^{\nu_0+1}K_{\nu_l}(\rho)\rmd\rho\right]\;,\]
where $A=\ell_q^{(2+d-\xi)/2}((2-\xi)/2)^{\nu_0}$ and $\nu_0=(d+\xi-2)/(2-\xi)$. Regularity at $r=0$ imposes $B_1=0$ since $K_{\nu_l}(w) \sim w^{-\nu_l}$, while the term with $B_2$ is regular as $r \to 0$ since $I_{\nu_l}(w) \sim w^{\nu_l}$.\\
An exact solution can also be found for the values $r \gg L$. Indeed, in this case, we have $\hat{F}_l(r,{\bi q}) \simeq 0$, as the forcing correlation rapidly decreases for separations much greater than $L$, and an unforced equation arises:
\begin{equation} \label{eq:5}
r^{-(d-1)}\partial_rr^{d+\xi-1}\partial_r\hat{C}_l-\frac{(d+\xi-1)}{d-1}l(d-2+l)r^{-2}\hat{C}_l-\ell_q^{-(2-\xi)}\hat{C}_l=0\;.
\end{equation}
The solution of (\ref{eq:5}) reduces to the zero mode,
\begin{equation} \label{eq:6}
\hat{C}_l(r;\ell_q)=r^{-(d+\xi-2)/2}\left[B_3K_{\nu_l}(w)+B_4I_{\nu_l}(w)\right]\;,
\end{equation}
where regularity at $\infty$ imposes $B_4=0$ as $I_{\nu_{l}}(w) \sim w^{-1/2}\rme^w$ for $r \to \infty$, while the term with $B_3$ is regular because $K_{\nu_{l}}(w) \sim w^{-1/2}\rme^{-w}$ for $r \to \infty$.\\
The correlation of a passive scalar field in the presence of inhomogeneous fluctuations, whose characteristic length is $\ell_q$, can be thus computed in the limits of small ($r \ll L$) and large ($r \gg L$) separations with respect to the integral scalar scale $L$. We find a dependence on some unknown constants ($B_2$,$B_3$) that can be fixed upon boundary conditions. To provide a clear and instructive example we can assume a forcing whose correlation function is a step function in $r$, i.e. $\hat{F}_l(r,{\bi q})=\hat{F}_l({\bi q})\Theta(L-r)$. In this case we can perform an exact matching in $r=L$ comparing the limits ($r \to L^-$, $r \to L^+$) of both $\hat{C}_l$ and $\hat{C}'_l$ (prime means derivative with respect to the variable $r$) deriving from the two expressions (\ref{eq:2}) and (\ref{eq:6}) (see Appendix). The final result is
\begin{equation} \label{eq:8}
\fl\hat{C}_l(r;\ell_q)=\cases{
\phi_{{\bi q},l}\ell_q^{2-\xi}w^{-\nu_0}\left[I_{\nu_l}(w)\displaystyle\int_w^W\rho^{\nu_0+1}K_{\nu_l}(\rho)\rmd\rho\right.&\\\ns
\hspace{2.2cm}\left.+K_{\nu_l}(w)\displaystyle\int_0^w\rho^{\nu_0+1}I_{\nu_l}(\rho)\rmd\rho\right]&for $0<r<L$\\
\phi_{{\bi q},l}\ell_q^{2-\xi}w^{-\nu_0}K_{\nu_l}(w)\displaystyle\int_0^W\rho^{\nu_0+1}I_{\nu_l}(\rho)\rmd\rho&for $L<r<\infty$\;.}
\end{equation}
where $W \equiv w|_{r=L}=2(2-\xi)^{-1}(L/\ell_q)^{(2-\xi)/2}$.\\
Once we assume a small-scale description with respect to the integral scale of the velocity field (i.e. $L_v \to \infty$), the general solution thus depends on three fundamental scales $r$, $L$ and $\ell_q$. These scales represent the separation, the forcing correlation scale and the characteristic length of the inhomogeneities, respectively. From the general solution (\ref{eq:8}) it is important to note (see Appendix) that taking the limit $\ell_q \to \infty$ for fixed $L$ and $r$, (\ref{eq:8}) reduces to the well-known solution for the homogeneous case. Indeed, the latter solution satisfies the equation:
\[r^{-(d-1)}\partial_rr^{d+\xi-1}\partial_r\hat{C}_l-\frac{(d+\xi-1)}{d-1}l(d-2+l)r^{-2}\hat{C}_l+\phi_{{\bi q},l}=0\;.\]
Anyway, in a small scale description with respect to the forcing correlation ($r \ll L$), the presence of a finite $\ell_q$ can reduce the range of pure power-law behaviour because of the presence of the Bessel functions in our solution. This scenario is clearly opposite to the one considered in the homogeneous limit where a pure power-law behaviour is found.\\
In order to get a deeper insight about these two different regimes, in the next section we concentrate ourselves on a purely isotropic situation where the forcing correlation function depends only on $q=|{\bi q}|$ and the correlation of the scalar field coincides with its isotropic sector ($l=0$). This is the simplest, physically relevant assumption with a physical meaning that one can consider to obtain a clear and systematic description of the influence of inhomogeneous contributions.
\section{Analysis for the isotropic case}
\label{sec:3}
We focus here on the isotropic sector ($l=0$) for the two-point equal-time scalar correlation function (the notation $\hat{C}$ is used to indicate $\hat{C}_{l=0}$). The technical advantage now is that alternatively to the method of variation of constants, we can choose the particular solution of (\ref{eq:1}) to be a constant, since the second term in (\ref{eq:1}) vanishes for $l=0$ and the coefficient of the function $\hat{C}(r;\ell_q)$ reduces to a constant. We can repeat the same arguments given for the general case to obtain
\begin{equation} \label{eq:12}
\hat{C}(r;\ell_q)=\cases{
\phi_{\bi q}\ell_q^{2-\xi}+A_2r^{-(d+\xi-2)/2}I_\nu(w)&for $0<r<L$\\
A_3r^{-(d+\xi-2)/2}K_\nu(w)&for $L<r<\infty$\;,}
\end{equation}
where $A_2$,$A_3$ are known functions of $L$ and $\ell_q$ (see Appendix). As discussed in the previous section, the presence of the Bessel functions (the fingerpoint of the scalar inhomogeneities) makes it impossible to see a clear power-law behaviour in the inertial range when $\ell_q/L \sim 1$. This is clearly seen by calculating local-slopes (figure 1) of the difference
\[\hat{S}(r;\ell_q) \equiv \hat{C}(0;\ell_q)-\hat{C}{(r;\ell_q)}\;,\]
strictly related to the second-order structure function, where $\hat{C}(r;\ell_q)$ is the general solution (\ref{eq:12}). For a fixed $L$ and $\xi$ (say, $\xi=4/3$, corresponding to the KOC51 scaling \cite{monyag}) we can change the ratio $\ell_q/L$ and examine the local slope behaviours as a function of $r/L$: the homogeneous case (single power-law behaviour in the inertial range) is recovered only when $\ell_q \gg L$, while for $\ell_q$ of the same order of $L$ a coexistence of power laws spoils the pure scaling of the homogeneous case.
\begin{figure}[h]
\begin{picture}(450,250)
\put(75,0){\includegraphics{fig.eps}}
\end{picture}
\caption{Local slopes of the correlation function in the inertial range. We plot $\zeta(r)=\frac{\rmd \ln \hat{S} (r;\ell_q)}{\rmd \ln r}$ as a function of $r/L$ for different values of the lenghtscale $\ell_q$. We plot $\ell_q/L=0.1$ ($+$), $\ell_q/L=1$ ($\times$), $\ell_q/L=10$ ($\star$). For comparison, the homogeneous case is also plotted (dotted plot).}
\end{figure}
\\To be more precise, let us focus on the asymptotic properties of (\ref{eq:12}). First of all we can perform the limit $r/\ell_q \to 0$ to obtain
\begin{equation} \label{eq:13}
\hat{C}(r;\ell_q) \approx \hat{C}_{\rm hom}(r;\ell_q) \equiv \cases{
a(\ell_q)+b_2(\ell_q)r^{2-\xi}&for $0<r<L$\\
b_3(\ell_q)r^{2-d-\xi}&for $L<r<\infty$\;,}
\end{equation}
where $a(\ell_q),b_2(\ell_q),b_3(\ell_q)$ can be obtained from the expansion of Bessel functions. In the limit $L \ll \ell_q$, $a$, $b_2$ and $b_3$ reduce to the well-known coefficients of the homogeneous isotropic case $\alpha$,$\beta_2$,$\beta_3$ (see Appendix).\\
In the opposite situation ($\ell_q \ll L$) the function $\hat{C}(r;\ell_q)$ approximates the step function ${\cal C}\Theta(L-r)$, where ${\cal C} \simeq \phi_{\bi q}\ell_q^{2-\xi}$. Thus, if $\phi_{\bi q}=F_0$ is a constant (we shall call it ``forcing of the first kind''), the plot of $\hat{C}(r;\ell_q)$ collapses on the axis of the abscissas when $\ell_q \to 0$. On the contrary, if one wanted to keep ${\cal C}$ finite, a scaling $\phi_{\bi q}=F_0\ell_q^{-(2-\xi)} \propto q^2$ could be assumed: the collapse would now take place for $\ell_q \to \infty$. This simply tells us that some kinds of forcing are not allowed (e.g. we also rule out $\phi_{\bi q}$'s giving unbounded ${\cal C}$'s or $a$'s, as for example $\phi_{\bi q} \sim \ell_q^{\gamma}$ with $\gamma>0$ or $<-(2-\xi)$). The finiteness of both $a$ and ${\cal C}$ may thus be guaranteed assuming e.g. $\phi_{\bi q}=F_{\ell}\ell_q^{-(2-\xi)}+F_LL^{-(2-\xi)}$ (``forcing of the second kind''), with $F_{\ell},F_L$ constants. Of course there would be infinite kinds of allowed forcing, but we will focus on these two because of their physical relevance. In what follows we will consider forcings of the first kind with $F_0=1$ and forcings of the second kind with $F_{\ell}=F_L=1$. The value chosen for $\xi$ is its Kolmogorov value $\xi=4/3$ and we will focus on the three-dimensional case ($d=3$).
\begin{figure}[htbp]
\begin{picture}(450,250)
\put(60,0){\includegraphics{coeff.eps}}
\put(20,225){$a(\ell_q)/\alpha$}
\put(40,210){\&}
\put(20,195){$b_2(\ell_q)/\beta_2$}
\put(420,10){$\ell_q/L$}
\end{picture}
\caption{Ratios between ``actual'' (functions of $\ell_q$) and ``homogeneous'' (limit values for $\ell_q \to \infty$) coefficients: dashed lines represent $a(\ell_q)/\alpha$, solid ones $b_2(\ell_q)/\beta$. Thin lines are related to the forcing of the first kind and thick lines are related to the forcing of the second kind.}
\end{figure}
\\Figure 2 represents the ratios of the additive and multiplicative coefficients given by $a(\ell_q)$ and $b_2(\ell_q)$ to the corresponding homogeneous ones $\alpha$ and $\beta_2$, as functions of $\ell_q/L$, for both kinds of forcing. The ``actual'' values attain the ``homogeneous'' ones only for large $\ell_q$'s.
\begin{figure}[htbp]
\begin{picture}(450,250)
\put(60,0){\includegraphics{graf.eps}}
\put(35,225){$\hat{S}(r;\ell_q)$}
\put(50,210){\&}
\put(30,195){$b_2(\ell_q)r^{2-\xi}$}
\put(420,10){$r/L$}
\end{picture}
\caption{Plots of $\hat{S}(r;\ell_q)=\hat{C}(0;\ell_q)-\hat{C}(r;\ell_q)$ and of the respective power-law approximations (dashed lines) for $\ell_q/L=10^2$ and forcing of the first kind (upper plot) and $=10^{-2}$ and both kind of forcings (lower plots). Thin lines are related to the forcing of the first kind and thick lines are related to the forcing of the second kind.}
\end{figure}
\\In figure 3 we show the plots of the difference $\hat{C}(0;\ell_q)-\hat{C}(r;\ell_q)$, together with the respective power-law approximations, for two different values of $\ell_q/L$, $10^2$ and $10^{-2}$: in the former case the two kinds of forcing substantially give the same result (only the first kind is thus represented) and the agreement is perfect all over the inertial range, while in the latter the separation takes place for $r<\ell_q$ for both kinds of forcing. One should also notice that by decreasing $\ell_q$, besides the slower convergence to power-law behaviour (as remarked in figure 1), the value $\hat{C}(L;\ell_q)$ tends to decrease with the ``first'' kind of forcing and to increase with the ``second'' kind.
\begin{figure}[htbp]
\begin{picture}(450,250)
\put(60,0){\includegraphics{min.eps}}
\put(20,225){$\Delta\hat{C}(r_L;\ell_q)$}
\put(420,10){$\ell_q/L$}
\end{picture}
\caption{Difference between approximated (power-law behaviour with ``actual'' coefficients $a(\ell_q)$ and $b_2(\ell_q)$) and actual expressions of $\hat{C}(r)$ for $r={\rm golden\ section\ of\ }L$. The thin line is related to the forcing of the first kind while the thick one to the forcing of the second kind.}
\end{figure}
\\Figure 4 shows
\[\Delta\hat{C}(r_L;\ell_q) \equiv \hat{C}_{\rm hom}(r_L;\ell_q)-\hat{C}(r_L;\ell_q)\;,\]
i.e. the difference between the approximated and the actual expressions of $\hat{C}(r;\ell_q)$ as functions of $\ell_q/L$, calculated for $r=r_L$ lying in the inertial range (in this case $r_L$ has been chosen as the golden section of $L$ , but similar plots exist $\forall r<L$). The presence of a maximum is quite intuitive for the first kind of forcing, as both expressions vanish for infinitesimal $\ell_q$, but is remarkable for the second kind, which means that the ``error'' of the approximation becomes negligible not only for large but also for small $\ell_q$'s.\\
The previous discussion has been carried out in the pseudo-Fourier space $({\bi r},{\bi q})$, but the final results must be expressed in the physical space $({\bi r},{\bi z})$ and a superposition is then needed. It is thus useful to analyze some instructive cases of superpositions.
\begin{figure}[htbp]
\begin{picture}(450,250)
\put(60,0){\includegraphics{ant.eps}}
\put(15,225){$\displaystyle\sum_{i=1}^2\hat{S}(r;\ell_{q_i})$}
\put(40,205){\&}
\put(5,185){$\displaystyle\sum_{i=1}^2b_2(\ell_{q_i})r^{2-\xi}$}
\put(420,10){$r/L$}
\end{picture}
\caption{Plots of the superpositions of two $\hat{C}(0;\ell_q)-\hat{C}(r;\ell_q)$ and of the respective power-law approximations (dashed lines) for $\ell_{q_1}/L=10^3$ and $\ell_{q_2}/L=1$ (upper two plots) or $=10^{-3}$ (lower two plots). Thin lines are related to the forcing of the first kind and thick lines are related to the forcing of the second kind.}
\end{figure}
\\Figure 5 represents the simplest case of superposition, i.e. the excitation of two $q$'s, e.g. $q_1$ and $q_2$, with constant or $\ell_q$-dependent amplitude. In particular the upper two plots are the (weighted) sum of the modes $\ell_{q_1}/L=10^3$ and $\ell_{q_2}/L=1$ and show very similar behaviours between each other (departure from the $r^{2-\xi}$ straight line at $r$'s about one order of magnitude smaller than the smaller $\ell_q$), while in the lower two $\ell_{q_1}$ is kept fixed but $\ell_{q_2}$ is reduced to $10^{-3}L$. In the former case since the correlation function collapses towards the step function in the limit $\ell_q \to 0 $, the structure function does not ``feel'' the smallest $\ell_q$ in the inertial range. Obviously the restoration of the correct power-law behaviour depends on the degree of convergence of $\hat{C}(r;\ell_q)$ towards the step function (in this case the first kind of forcing converges more rapidly than the second one).
More realistic cases are connected to the excitation of a finite set of discrete modes: in this case the correctness of the power-law approximation is guaranteed (at least) for $r$'s sufficiently smaller than the minimum $\ell_q$. On the contrary, if the forcing has a continuum spectrum, one has to compute the continuum Fourier antitransformed ($\int\rmd^dq\,\rme^{\rmi{\bi q}\cdot{\bi z}} \sim \int\rmd\ell_q\,\ell_q^{-(4-\xi)/2}\ell_q^{-(d-2)(2-\xi)/2}$) which is well defined for the forcings we have considered.
\section{Conclusions}
\label{sec:4}
The properties of the two-point equal-time correlation function for the Kraichnan model of advection have been studied in presence of anisotropies and inhomogeneities. The system can be described by the following three different scales: the separation ($r$), the forcing correlation length ($L$) and, finally, the lengthscale of the inhomogeneities ($\ell_q$). The model can be treated analytically and the properties of both small scales and large scales can be related to the typical lengthscale $\ell_q$. This offers the possibility to analyze the breaking of translationally invariant properties by means of an external forcing term and to check if the small scale statistics can be regarded as universal in the sense that it does not depend on the details of the inhomogeneous contribution. This somehow universal property is strictly connected to the restoration of a homogeneous limit in an inhomogeneous situation for scales smaller that the typical inhomogeneous one. The homogeneous limit ($r \ll \ell_q$, $L \ll \ell_q $) has been studied and it has been shown how the solution reproduces exactly the one that can be obtained starting from homogeneous equations. On the other side, the homogeneous power-law behaviour is completely spoiled when $\ell_q$ is of the order of the separation $r$ and it can be seen as a ``beating'' of different power laws originating from the scalar inhomogeneities.\\
Summarizing, a pure power-law behaviour exists $\forall\ell_q$ going at sufficiently small $r$'s and this is a clear indication of the fact that the statistical description can be seen as the same of the homogeneous case but with a reduced range of pure scaling law behaviour. When we pass to the physical space, and if more inhomogeneous modes are excited, the restoration of an inertial range is guaranteed if the excitation takes place only at large scales or at large scales together with very small scales ($\ell_q \to 0$). The calculations are carried out with the overall exception for those $\ell_q$'s falling in the diffusive range: these have been excluded from our analysis.
\ack
We thank L. Biferale, A. Celani and A. Mazzino for useful discussions and suggestions.
|
{
"timestamp": "2004-11-03T12:45:35",
"yymm": "0411",
"arxiv_id": "nlin/0411008",
"language": "en",
"url": "https://arxiv.org/abs/nlin/0411008"
}
|
\section*{Acknowledgments}
The work of T.H. was supported in part
by the US DOE under contract No. DE-FG02-95ER40896, in part by the
Wisconsin Alumni Research Foundation, and in part by
National Natural Science Foundation of China.
The work of G.V. and Y. W. was supported
in part by DOE under contact number DE-FG02-01ER41155.
|
{
"timestamp": "2004-11-03T20:12:25",
"yymm": "0411",
"arxiv_id": "hep-ph/0411055",
"language": "en",
"url": "https://arxiv.org/abs/hep-ph/0411055"
}
|
\section{Goal}
In order to investigate the properties of the thick disk and its interface
with the thin disk we have compiled a catalogue of elemental abundances
of O, Na, Mg, Al, Si, Ca, Ti, Ni, Fe including 830 stars (Girard and Soubiran 2004).
The classification of
thin disk and thick disks stars has been performed on the basis of their
(U,V,W) velocities. The two populations overlap greatly in metallicity but
at a given [Fe/H] the thick disk shows on average an enhancement of 0.07 dex in
[$\alpha$/Fe] (Fig.1). In order to go further in this investigation we want to be able
to measure [Fe/H] and [$\alpha$/Fe] from a large collection of spectra with
an automatic procedure.
\begin{figure}[h]
\begin{center}
\leavevmode
\centerline{\epsfig{file=girardF2.ps,angle=90, width=8.0cm}}
\end{center}
\caption{[$\alpha$/Fe] vs [Fe/H]}
\label{}
\end{figure}
\begin{figure}[h!]
\begin{center}
\leavevmode
\centerline{\epsfig{file=girardF1.ps,angle=0, width=7.0cm}}
\end{center}
\caption{Comparison of [Fe/H] from the TGMET code with
[Fe/H] from the catalogue of abundances. Rms = 0.11.}
\label{}
\end{figure}
\section{Tools and material}
In this section we summerize the libraries and the codes
used for this investigation :
\begin{itemize}
\item The ELODIE library of 1962 spectra ($\lambda \lambda$390.6-681.1nm, R=42000)
of 1388 stars with measured Lick indices
(Prugniel \& Soubiran 2004) and its intersection with
the abundance catalogue : 449 spectra of 308 stars.
\item The grid of synthetic spectra with 3 values of [$\alpha$/Fe]
(Barbuy et al. 2003).
\item The TGMET code : a minimun
distance algorithm to measure (Teff, logg, [Fe/H]) (Katz et al. 1998).
\item The ETOILE code : a modified version of TGMET
with determination of [$\alpha$/Fe] (D.Katz, priv. com.).
\\
\\
\end{itemize}
\begin{figure}[h]
\begin{center}
\leavevmode
\centerline{\epsfig{file=girardF4.ps,angle=90, width=8.0cm}}
\end{center}
\caption{[Fe/H] from ETOILE vs [Fe/H] from the catalogue.
The modification of Teff in the input of the code provides a variation of [Fe/H].}
\label{}
\end{figure}
\begin{figure}[h!]
\begin{center}
\leavevmode
\centerline{\epsfig{file=girardF3.ps,angle=90, width=8.0cm}}
\end{center}
\caption{[$\alpha$/Fe] from ETOILE vs [$\alpha$/Fe] from the catalogue.}
\label{}
\end{figure}
TGMET relies on the least-square comparison of an ELODIE
spectrum of a target star to a library of ELODIE spectra of reference
stars with well determined atmospheric
parameters.\\
ETOILE is a minimum distance algorithm based on the perturbation
method described in Cayrel et al. (1991). With this method,
the reference library must sample the parameter space with
regular steps.
That is why synthetic spectra are used instead of empirical
spectra.\\
We use the grid of synthetic spectra computed by Barbuy et al.(2003) :
$\lambda \lambda$460-560nm, 4000 $\leq$ Teff $\leq$ 7000 K in steps of 250 K,
0.5 $\leq$ log g $\leq$ 5.0 in steps of 0.5,
[Fe/H] : -3.0, -2.5, -2.0, -1.5, -1.0, -0.5, -0.3, -0.2, -0.1, 0.0 and +0.3 and
[$\alpha$/Fe] : 0.0, +0.2 and +0.4. \\
A first step is to validate the grid,
that is verify that
computed spectra and observed ones with same parameters match
on the whole wavelength interval.
\section{Results}
A bootstrap method is used to test the performances of TGMET.
Based on 449 spectra, TGMET is able to
retrieve the atmospheric parameters
with a typical accuracy of 134K in Teff and 0.11 in
[Fe/H] (Fig.2).
The main limitation of TGMET is its empirical reference library
which does not sample perfectly the parameter space. A limitation
overcome with the use of ETOILE and a grid of synthetic spectra.
As a starting point ETOILE uses the TGMET solution.
Preliminary results from ETOILE
suggest that the catalogue of abundances and the grid are not on
the same temperature scale : metallicities are correctly recovered
if a hotter temperature is given in input (Fig.3). [$\alpha$/Fe] is not yet
correctly estimated (Fig.4). Possible causes are currently investigated.
|
{
"timestamp": "2004-11-10T11:53:00",
"yymm": "0411",
"arxiv_id": "astro-ph/0411263",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411263"
}
|
\section{Introduction}
The early discussions of realistic vacua in heterotic superstring
theory were within the context of the ``standard embedding''
\cite{GSW}
of the spin connection into the gauge connection. Said differently,
these vacua always involve a holomorphic $E_8$ vector bundle, $V$,
which is induced by the tangent bundle $TX$ of the smooth Calabi-Yau
threefold $X$.
Although leading to interesting low
energy physics, this approach suffers from the fact that it is highly
constrained, the tangent bundle being only one out of an enormous
number of possible holomorphic bundles $V$. One consequence of this
constraint is the fact that all heterotic vacua based on the standard
embedding require the spontaneous breaking of $E_8$ to $E_6$, which is
then further broken by Wilson lines. Although $E_6$ is a possible
grand unified group, other groups, such as $SU(5)$ or $Spin(10)$, are
simple and more compelling given recent experimental data.
Equally significant is that, in the standard embedding, the low energy
spectrum and couplings are completely determined by the cohomology of
the tangent bundle $TX$. This seriously constrains these quantities,
and it has been difficult to find realistic models in this context.
A technical breakthrough in this regard was presented in
\cite{FMW1,D,FMW2},
where it was shown how to construct a large class of
stable, holomorphic vector bundles on simply connected elliptically
fibered Calabi-Yau threefolds where $V \ne TX$. Such bundles admit
connections satisfying the hermitian Yang-Mills equations. This work
was extended in \cite{Donagi:1998xe}-\cite{Donagi:1999gc},
and it was shown that these bundles can lead
to heterotic string vacua with a wide range of low energy gauge
groups, including $SU(5)$ and $Spin(10)$. Many of the physical
properties of these vacua have been studied, including supersymmetry
breaking \cite{Lukas:1999kt,Lalak1},
the moduli space of the vector bundle
\cite{Buchbinder:2002pr}-\cite{moduli2}, and,
in the strongly coupled case, the associated M5-brane moduli space
\cite{Donagi:1999jp},
small instanton phase transitions
\cite{Ovrut:2000qi}-\cite{Douglas:2004yv},
non-perturbative superpotentials
\cite{Buchbinder:2002pr,Buchbinder:2002ic,Lima:2001nh,Lima:2001jc},
and fluxes \cite{Krause:2000gp}-\cite{Cardoso:2002hd}.
More recently, it was
shown how to compute the sheaf cohomology of $V$ and its tensor
products, thus determining the complete particle physics spectrum
\cite{spec1,spec}. An important conclusion of these papers is that
the spectrum depends on the region of vector bundle moduli space in
which it is evaluated. Although constant for generic moduli, the
spectrum can jump dramatically on subspaces of co-dimension one or
higher always containing, however, three families of quarks and
leptons. These vacua also underlie the theory of brane universes
\cite{Lukas:1997fg}-\cite{Lukas:1998yy} and ekpyrotic and Big Crunch/Big Bang
cosmology \cite{Khoury:2001zk}-\cite{Khoury:2001wf}.
The major drawback of these vacua is that the compactification
manifold is simply connected. It follows that these are all GUT
theories which cannot be broken to the standard model with Wilson
lines \cite{Sen:1985eb}-\cite{Andreas}.
Although many of these vacua contain Higgs multiplets whose
vacuum expectation values could induce symmetry breaking,
it would be simpler and more natural if Wilson lines could be introduced.
This was accomplished in \cite{Donagi:2000fw}-\cite{Donagi:1999ez},
where stable holomorphic vector
bundles with structure group $SU(5)$ were constructed over
torus-fibered Calabi-Yau threefolds with fundamental group $\pi_1(X) =
\mathbb{Z}_2$. These heterotic vacua lead, using a $\mathbb{Z}_2$ Wilson line, to
low energy theories that are anomaly free, have three families of
quarks/leptons and the gauge group $(SU(3)_C \times SU(2)_L \times U(1)_Y)/\IZ_6$.
This work was extended
to vector bundles with
structure group $SU(4)$ on torus-fibered Calabi-Yau threefolds with
$\pi_1(X) = \mathbb{Z}_2 \times \mathbb{Z}_2$ in
\cite{Donagi:2003tb}-\cite{Ovrut:2002jk} and $\pi_1(X) = \mathbb{Z}_3 \times
\mathbb{Z}_3$ in \cite{Braun:2004xv}. Although very promising, it is
essential that one now compute the exact spectrum and couplings in
these standard model vacua. In this paper, we take a major step
in this direction by computing the particle spectrum for the
vacua in \cite{Donagi:2000fw}-\cite{Donagi:1999ez}.
This is accomplished as follows. In
\cite{Donagi:2000fw}-\cite{Donagi:1999ez}, $X$ is the quotient $X =
\tilde{X} / \mathbb{Z}_2$, where $\tilde{X}$ is a simply connected Calabi-Yau
threefold. Denote by $\tilde{V}$ the pull-back of $V$ to $\tilde{V}$. To find the
particle spectrum, one must first compute the sheaf cohomology of
$\tilde{V}$ and its tensor products. This is a non-trivial task involving
various techniques in cohomological algebra and algebraic geometry. In
this paper, we present a systematic approach to such computations, and
determine all relevant cohomology groups in our theory. The next step
is to find the explicit representations of $\mathbb{Z}_2$ in each of these
spaces. We give a precise methodology for accomplishing this. This
approach is then used to determine each of the requisite $\mathbb{Z}_2$
representations. The above information, in conjunction with the action
of the $\mathbb{Z}_2$ Wilson line, can be utilized to find all group
multiplets that are invariant under $\mathbb{Z}_2$, as well as their
multiplicities. When constructing the quotient Calabi-Yau threefold $X
= \tilde{X} / \mathbb{Z}_2$, these invariant multiplets descend to $X$ and form the
massless particle physics spectrum.
Using these techniques, we compute and tabulate the spectrum.
Specifically, we do the following. In Section 2, we present a general
formalism for describing $G( \subset E_8)$-bundles, Wilson lines and
the massless spectrum associated with non-simply connected Calabi-Yau
threefolds $X$ with $\pi_1(X) = F$. It is shown that determining this
spectrum requires the computation of specific sheaf cohomologies on
the covering Calabi-Yau threefold $\tilde{X}$, as well as the action of $F$
on these groups. This formalism is illustrated for several values of
$F$, including $F = \mathbb{Z}_2$. Section 3 is devoted to a brief review of
the results in \cite{Donagi:2000fw}-\cite{Donagi:1999ez}.
Specifically, we discuss the construction of
torus-fibered Calabi-Yau threefolds $X$ with fundamental group $F =
\mathbb{Z}_2$. It is shown how to construct stable, holomorphic bundles $V$
with structure group $SU(5)$ on $X$. These arise from $\mathbb{Z}_2$
invariant bundles $\tilde{V}$ on $\tilde{X}$ and satisfy the basic
phenomenological constraints of particle physics. Computing the
massless spectrum of this theory requires determining the sheaf
cohomology of $\tilde{V}$ and its tensor products. A general method for
doing this is presented in Section 4 and used to compute the relevant
cohomology groups in our theory. Section 5 is devoted to finding the
explicit representations of $\mathbb{Z}_2$ on these cohomology
groups. Combining the results of Section 5 with the $F = \mathbb{Z}_2$
example in Section 2, the massless spectrum of our theory is computed,
tabulated and discussed in Section 5. Finally, in the Appendix
we present some useful mathematical facts used throughout the paper.
\section{The Spectra of Heterotic Compactifications \\
with Wilson Lines}\label{s:1}
A vacuum in weakly coupled heterotic string theory is specified by a
pair $(X,\overline{{\mathcal V}})$, where $X$ is a Calabi-Yau threefold and
$\overline{{\mathcal V}}$ is a stable $E_8 \times E_8$ holomorphic principal bundle
on $X$ satisfying the Green-Schwarz anomaly cancellation condition
\cite{GS}
\begin{equation}
c_2(\overline{{\mathcal V}}) = c_2(TX) \ .
\end{equation}
Note that
specifying the $E_8 \times E_8$ bundle $\overline{{\mathcal V}}$ is
the same as giving two $E_8$ bundles ${\mathcal V}$ and ${\mathcal V}''$.
The anomaly cancellation condition can be written as
\begin{equation}\label{anom1}
c_2({\mathcal V}) + c_2({\mathcal V}'') = c_2(TX) \ .
\end{equation}
In this
work, we will always take ${\mathcal V}''$ to be trivial.
Then, condition \eref{anom1} becomes
\begin{equation}\label{anom}
c_2({\mathcal V}) = c_2(TX) \ .
\end{equation}
In heterotic M-theory compactifications, this condition is relaxed to
\begin{equation}\label{anom2}
c_2({\mathcal V}) + [C] = c_2(TX) \ ,
\end{equation}
where ${\mathcal V}$ is a stable holomorphic $E_8$ principal bundle in the
observable sector and
$[C]$ is the class of some effective curve $C \subset X$ on which
M5-branes wrap.
The particle
spectrum of this compactification consists \cite{GSW} of
zero-modes of the ten-dimensional Dirac operator
\begin{equation}
\slash{\! \! \! \! D} : \Gamma({\mathop {\rm ad}} {\mathcal V} \otimes S^+_{10}) \rightarrow
\Gamma({\mathop {\rm ad}} {\mathcal V} \otimes S^-_{10}) \ .
\end{equation}
Here ${\mathop {\rm ad}}{\mathcal V}$ is the rank-248 vector bundle associated to ${\mathcal V}$ by the
adjoint representation of $E_8$, $S^{\pm}_{10}$ are the bundles of
positive and negative chirality spinors in 10-dimensions, and
$\Gamma$ denotes global sections of a bundle over the 10-dimensional
space $\mathbb{R}^4 \times X$.
(Note that we can consider ${\mathop {\rm ad}} {\mathcal V}$ to be a bundle on $\mathbb{R}^4 \times
X$ by simply pulling it back from $X$).
The 10-dimensional spinors decompose in terms of their (Minkowski)
$\mathbb{R}^4$ and (internal) $X$ components as
\begin{equation}
S^+_{10} = (S^+_4 \otimes S^+_6) \oplus
(S^-_4 \otimes S^-_6).
\end{equation}
The internal spinors, on the Calabi-Yau threefold $X$, can be
identified with the $(0,q)$ forms ${\cal A}^{0,q}$ on $X$, with even/odd $q$
corresponding to positive/negative chirality:
\begin{equation}
S^+_6 \simeq {\cal A}^{0,0} \oplus {\cal A}^{0,2}, \qquad
S^-_6 \simeq {\cal A}^{0,1} \oplus {\cal A}^{0,3}.
\end{equation}
In terms of this identification, the Dirac operator becomes
$\slash{\! \! \! \! D} = \overline{\partial} + \overline{\partial}^* + \slash{\! \! \! \! D}_4$
coupled to ${\mathop {\rm ad}} {\mathcal V}$, where
$\overline{\partial}$ is the Dolbeault operator on $X$, and $\slash{\! \! \! \! D}_4$ is
the Dirac operator on flat $\mathbb{R}^4$.
Putting these facts together, we find that the spectrum is
\begin{equation}
\ker(\slash{\! \! \! \! D}) \simeq
\left( \bigoplus_{q=0,2} H^q(X, {\mathop {\rm ad}} {\mathcal V}) \otimes {\bf S}^+_4
\right)
\oplus
\left( \bigoplus_{q=1,3} H^q(X, {\mathop {\rm ad}} {\mathcal V}) \otimes {\bf S}^-_4
\right),
\end{equation}
where ${\bf S}^{\pm}_4$ denote the constant sections of the bundle
$S^{\pm}_4$ on $\mathbb{R}^4$.
The positive chirality particles are those which multiply
${\bf S}^+_4$, so they are given by (a basis of)
\begin{equation}\label{poschiral}
\bigoplus_{q=0,2} H^q(X, {\mathop {\rm ad}} {\mathcal V}).
\end{equation}
Their negative chirality anti-particles are similarly given by a basis
of
\begin{equation}\label{negchiral}
\bigoplus_{q=1,3} H^q(X, {\mathop {\rm ad}} {\mathcal V}).
\end{equation}
By Serre duality, this is the dual space to \eref{poschiral}, as it
should be by CPT.
Recall that, for each charged particle, CPT predicts the existence of an
anti-particle of opposite charge.
The annihilation of a particle with its
anti-particle can be interpreted as a natural pairing.
Hence, we can
interpret the space of anti-particles as the dual of the space of
particles.
In order to describe the complete spectrum, we will
in this work calculate
\begin{equation}
Spec := \bigoplus_{q=0,1} H^q(X, {\mathop {\rm ad}} {\mathcal V}).
\end{equation}
Then, $\ker(\slash{\! \! \! \! D})$ is obtained by adding the duals to $Spec$.
In practice, the $E_8$ bundle ${\mathcal V}$ is often associated to some
stable $G$-bundle $V$ on $X$,
where $G \subset E_8$ is some subgroup, e.g.,
$G = SU(n)$ for $n=3,4$ or $5$~\footnote{
Since all of our bundles are holomorphic, the relevant
structure groups are actually $G = SL(n, \mathbb{C})$. However, to
conform to the usual physics notation, we will throughout this
paper refer to these groups as $G=SU(n)$.
}:
\begin{equation}\label{def-cV}
{\mathcal V} = V \uptimes{G} E_8 \ .
\end{equation}
The resulting compactification then has a low energy gauge group
\begin{equation}\label{GH}
H = Z_{E_8}(G) \ ,
\end{equation}
the commutant of $G$ in $E_8$. The decomposition of the
248-dimensional representation ${\mathop {\rm ad}} E_8$ under the product $G \times
H$ then gives an associated decomposition for ${\mathop {\rm ad}} {\mathcal V}$ and the
Dirac-operator zero-modes. For example, we can take $V$ to be an
$SU(3)$ bundle, or equivalently, a rank 3 vector bundle with trivial
determinant.
The usual embedding of $G = SU(3)$ into $E_8$ has commutant $H=E_6$. The
decompostion of ${\mathop {\rm ad}} E_8$ into irreducible representations
of $SU(3) \times E_6$ involves four terms
\begin{equation}\label{248-3}
248 =
(1,78) \oplus (8,1) \oplus (3, 27) \oplus (\overline{3}, \overline{27}) \ .
\end{equation}
Here, 8 and 78 are the adjoints of $SU(3)$ and $E_6$ respectively,
3 is the
fundamental representation of $SU(3)$, and 27, $\overline{27}$ are the
smallest representations of $E_6$. For the zero-modes we get:
\begin{equation}\label{3decomp}
Spec =
\left( H^0(X, {\mathcal O}_X) \otimes 78 \right) \oplus
\left( H^1(X, {\mathop {\rm ad}} V) \otimes {\rlap{1} \hskip 1.6pt \hbox{1}} \right) \oplus
\left( H^1(X, V) \otimes 27 \right) \oplus
\left( H^1(X, V^*) \otimes \overline{27} \right) \ .
\end{equation}
Here we think of $V$ as a rank 3 vector bundle on $X$ associated to
the principal $SU(3)$ bundle by the fundamental representation,
$V^*$ is its dual vector bundle, ${\mathop {\rm ad}} V$ is the rank-8 vector
bundle of traceless endomorphisms of $V$, and ${\mathcal O}_X$ is the trivial
rank-1 bundle on $X$. Note that the stability of $V$ and the
Calabi-Yau property of $X$ guarantee that for each of the associated
bundles $({\mathcal O}_X, {\mathop {\rm ad}} V, V, V^*)$, the cohomology can be non-zero
for either $q=0$ or $q=1$ but not both, as indicated in
\eref{3decomp}.
As another example, we consider the usual embedding of $G=SU(5)$ into
$E_8$. The commutant is $H=SU(5)$ and
the $SU(5)_G \times SU(5)_H$-decomposition is
\begin{equation}\label{E8-su5}
248 =
(1,24) \oplus (24,1)
\oplus (10,5) \oplus (\overline{10},\overline{5})
\oplus (5, \overline{10}) \oplus (\overline{5},10) \ .
\end{equation}
The zero-modes are
\begin{eqnarray}
Spec &=&
\left( H^0(X, {\mathcal O}_X) \otimes 24 \right) \oplus
\left( H^1(X, {\mathop {\rm ad}} V) \otimes {\rlap{1} \hskip 1.6pt \hbox{1}} \right) \oplus
\left( H^1(X, \wedge^2 V) \otimes 5 \right) \oplus
\left( H^1(X, \wedge^2 V^*) \otimes \overline{5} \right) \nonumber \\
&& \oplus \left( H^1(X, V) \otimes \overline{10} \right) \oplus
\left( H^1(X, V^*) \otimes 10 \right) \ .
\end{eqnarray}
More generally, for $G \subset E_8$ with commutant $H$, we write
\begin{equation}\label{adE8}
{\mathop {\rm ad}} E_8 = \bigoplus_i U_i \otimes R_i \ ,
\end{equation}
where $U_i$ runs over irreducible representations of $G$, and $R_i$ are
corresponding representations of $H$.
Using this decomposition of the representation ${\mathop {\rm ad}} E_8$ on each fiber
of the $E_8$ bundle defined in \eref{def-cV},
we find the decomposition
\begin{equation}\label{decomp}
{\mathop {\rm ad}} {\mathcal V} = \bigoplus_i U_i(V) \otimes R_i \ ,
\end{equation}
where $U_i(V)$ are the vector bundles associated to the $G$-bundle
$V$ via the representations $U_i$ of $G$.
Next we want to see how these results are modified by Wilson
lines. Let $F \subset H$ be a finite subgroup which acts on a
Calabi-Yau threefold $\tilde{X}$ freely with a Calabi-Yau quotient
$X = \tilde{X} / F$.
The $G$-bundle $V$ and the $E_8$-bundle
${\mathcal V} = V \uptimes{G} E_8$ on $X$ pull back to a $G$-bundle $\tilde{V}
= p^* V$ and an $E_8$-bundle
$\tilde{{\mathcal V}} = p^*{\mathcal V} = \tilde{V} \uptimes{G} E_8$
on $\tilde{X}$, where $p : \tilde{X} \rightarrow X$ is the covering map.
The action of $F$ on $\tilde{X}$ lifts to actions, denoted $\rho$,
on $\tilde{V}$, $\tilde{{\mathcal V}}$, hence on their cohomologies.
The cohomology group computed on $X$ is precisely the
$\rho(F)$-invariant part of the cohomology on $\tilde{X}$
\begin{equation}\label{invcoho}
H^q(X, {\mathop {\rm ad}} {\mathcal V}) = H^q(\tilde{X}, {\mathop {\rm ad}} \tilde{{\mathcal V}})^{\rho(F)} \ .
\end{equation}
The Wilson line $W$ is the flat $H$-bundle on $X$ induced from the
$F$-cover $p: \tilde{X} \rightarrow X$ via the given embedding of $F$ in $H$:
\begin{equation}\label{wilson}
W := \tilde{X} \uptimes{F} H \ .
\end{equation}
The $(G \times H)$-bundle $V \oplus W$ induces another $E_8$-bundle
on $X$:
\begin{equation}\label{VpGH}
{\mathcal V}' = (V \oplus W) \uptimes{(G \times H)} E_8 \ .
\end{equation}
Our goal in this work is to study the particle spectrum and other
properties of the heterotic vacuum given by compactification on
$(X,{\mathcal V}')$. Since the structure group of ${\mathcal V}'$ can be reduced to $G
\times F$ (but not to $G$), we see in analogy with \eref{GH} that this
vacuum has low energy gauge group
\begin{equation}\label{S}
S := Z_H(F) = Z_{E_8} (G \times F) \ .
\end{equation}
We will work primarily with a particular class of geometric examples
which is reviewed in Section 2. In the remainder of the present
section we will describe the general approach. This is based on the
observation that, when pulled backed to $\tilde{X}$, the two bundles
${\mathcal V}$, ${\mathcal V}'$ coincide:
\begin{equation}
p^* {\mathcal V}' \simeq p^* {\mathcal V} = \tilde{{\mathcal V}} \ .
\end{equation}
This is because
the finite structure group $F$ of the Wilson line $W$ is killed
in the passage from $X$ to $\tilde{X}$. Another way to describe this
is to note that there are two actions $\rho$, $\rho'$ of $F$ on
$\tilde{{\mathcal V}}$, both lifting the given $F$ action on $\tilde{X}$.
The quotient by $\rho$ gives ${\mathcal V}$, and the quotient by $\rho'$ gives
${\mathcal V}'$.
The analogue of \eref{invcoho} is:
\begin{equation}\label{invcoho2}
H^q(X, {\mathop {\rm ad}} {\mathcal V}') = H^q(\tilde{X}, {\mathop {\rm ad}} \tilde{{\mathcal V}})^{\rho'(F)} \ .
\end{equation}
We can write the decomposition \eref{decomp} on $\tilde{X}$:
\begin{equation}
{\mathop {\rm ad}} \tilde{{\mathcal V}} = \bigoplus_i U_i(\tilde{V}) \otimes R_i
\end{equation}
and use formulas \eref{invcoho}, \eref{invcoho2} to descend to
$X$. The $\rho$ action of $F$ acts only on the associated vector
bundles $U_i(\tilde{V})$, hence on their cohomology, so:
\begin{equation}\label{Hdecomp}
H^q(X, {\mathop {\rm ad}} {\mathcal V}) = \bigoplus_i
H^q(\tilde{X}, U_i(\tilde{V}))^{\rho(F)}
\otimes R_i
\ .
\end{equation}
The $\rho'$ action of $F$ is a combination of the
$\rho$ action on the $U_i(\tilde{V})$ with the action of $F \subset H$
on the $R_i$:
\begin{equation}\label{Hdecomp2}
H^q(X, {\mathop {\rm ad}} {\mathcal V}') = \bigoplus_i
\left( H^q(\tilde{X}, U_i(\tilde{V})) \otimes R_i \right)^{\rho'(F)} \ .
\end{equation}
Recall that $H^q(X, {\mathop {\rm ad}} {\mathcal V})$ and its decomposition \eref{Hdecomp}
carry an action of $H$ (which is the natural action on $R_i$ in
\eref{Hdecomp}), but only the subgroup $S \subset H$ acts
on $H^q(X, {\mathop {\rm ad}} {\mathcal V}')$ and its decomposition \eref{Hdecomp2}. To make
the latter more explicit, we decompose each $H$-representation $R_i$
in terms of the irreducible $F$-representations $A_j$:
\begin{equation}\label{Ri}
R_i = \bigoplus_j (A_j \otimes B_{ij}), \quad
B_{ij} := {\rm Hom}_F(A_j, R_i) \ .
\end{equation}
Our formula \eref{Hdecomp2} for the particle spectrum then becomes
\begin{equation}\label{spec}
H^q(X, {\mathop {\rm ad}} {\mathcal V}') = \bigoplus_{i,j}
(H^q(\tilde{X}, U_i(\tilde{V})) \otimes A_j)^{\rho'(F)}
\otimes B_{ij} \ .
\end{equation}
Here each $B_{ij}$ carries a representation of the low energy gauge
group $S$, which occurs in $H^q(X, {\mathop {\rm ad}} {\mathcal V}')$ with multiplicity
$m_{ij}$ equal to the dimension of the space of $F$-invariants in
$H^q(\tilde{X}, U_i(\tilde{V})) \otimes A_j$. Note that the
$S$-representation $B_{ij}$ is often not irreducible. Rather, we
should think of $B_{ij}$ as a block of irreducible
$S$-representations, each of which corresponds to some particle. All
the particles in a given block $B_{ij}$ occur in the spectrum with the
same multiplicity $m_{ij}$.
We can summarize our procedure so far as follows. The input involves
\begin{itemize}
\item a structure group $G \subset E_8$,
\item a finite subgroup $F$ of the commutant $H = Z_{E_8}(G)$,
\item a free action of $F$ on a Calabi-Yau threefold $\tilde{X}$ with
Calabi-Yau quotient $X = \tilde{X} / F$, and
\item a $G$-bundle $V$ on $X$ satisfying the anomaly cancellation
condition \eref{anom2}.
\end{itemize}
These data determine a Wilson line $W$ on $X$ (as in \eref{wilson}) and
a heterotic vacuum $(X,{\mathcal V}')$ where ${\mathcal V}'$ combines the $G$-bundle $V$
with the Wilson line $W$, as in \eref{VpGH}. The low energy gauge
group of this vacuum is the subgroup $S \subset H$ given in
\eref{S}. The particle spectrum is determined as follows:
\begin{itemize}
\item Decompose ${\mathop {\rm ad}} E_8$ as in \eref{adE8} in terms of irreducible
$G$-representations $U_i$ and corresponding
$H$-representations $R_i$.
\item Decompose each $R_i$ as in \eref{Ri} in terms of irreducible
$F$-representations $A_j$ and corresponding blocks of
irreducible $S$-representations $B_{ij}$.
\item Most of the work then goes into computing the cohomology groups
$H^q(\tilde{X}, U_i(\tilde{V}))$ of the associated vector
bundles on $\tilde{X}$, and the action of $F$ on these
cohomologies. The multiplicity $m_{ij}$ of the irreducible
$F$-representation $A_j^*$ in $H^q(\tilde{X}, U_i(\tilde{V}))$
is the multiplicity of all particles
from block $B_{ij}$ in the particle spectrum of $(X,{\mathcal V}')$.
\end{itemize}
We illustrate the general procedure in two cases. First consider
$G = SU(3)$, $H = E_6$. As we saw in \eref{248-3}, the $U_i$ are 1, 8,
3 and $\overline{3}$, and the corresponding $R_i$ are 78, 1, 27 and
$\overline{27}$. Now $H=E_6$ has a maximal subgroup
\begin{equation}
H_0 = SU(3)_C \times SU(3)_L \times SU(3)_R \ ,
\end{equation}
where we can think of $C$, $L$, $R$ as standing for color, left,
right.
We can, for example, take $F = F(n,\hat{n}) = \mathbb{Z}_n \times
\mathbb{Z}_{\hat{n}}$ whose two generators are mapped to $H_0$ as
\begin{equation}
{\rlap{1} \hskip 1.6pt \hbox{1}}_C \times
\mat{\alpha & & \cr &\alpha& \cr &&\alpha^{-2}}_L \times {\rlap{1} \hskip 1.6pt \hbox{1}}_R \ ,
\qquad \qquad
{\rlap{1} \hskip 1.6pt \hbox{1}}_C \times {\rlap{1} \hskip 1.6pt \hbox{1}}_L \times
\mat{\hat{\alpha} & & \cr &\hat{\alpha}& \cr &&\hat{\alpha}^{-2}}_R
\ ,
\end{equation}
where $\alpha$ and $\hat{\alpha}$ are roots of unity of orders $n$ and
$\hat{n}$ respectively. Another possibility is to work with $F_0$, the
diagonal subgroup $\mathbb{Z}_n$ in $F(n,n)$, with generator
\begin{equation}
{\rlap{1} \hskip 1.6pt \hbox{1}}_C \times
\mat{\alpha & & \cr &\alpha& \cr &&\alpha^{-2}}_L \times
\mat{\alpha & & \cr &\alpha& \cr &&\alpha^{-2}}_R \ .
\end{equation}
Either $F$ (with $n, \hat{n} > 1$) or $F_0$ (with $n > 1$) break $E_6$ to
\begin{equation}
S = SU(3)_C \times \left(\frac{SU(2) \times U(1)}{\mathbb{Z}_2}\right)_L
\times \left(\frac{SU(2) \times U(1)}{\mathbb{Z}_2}\right)_R \ .
\end{equation}
In this case, it is easier to first decompose each $R_i$ under $H_0$,
and then to further decompose each $H_0$ component under $F$ and
$S$. Under $H_0$ we have:
\begin{eqnarray}
78 &=& (8,1,1) \oplus (1,8,1) \oplus (1,1,8) \oplus (3,3,3) \oplus
(\overline{3},\overline{3},\overline{3}) \nonumber \\
1 &=& (1,1,1) \nonumber \\
27 &=& (3,\overline{3},1) \oplus (1,3,\overline{3}) \oplus (\overline{3},1,3) \nonumber \\
\overline{27} &=& (\overline{3},3,1) \oplus (1,\overline{3},3) \oplus (3,1,\overline{3}) \ ,
\end{eqnarray}
where $(a,b,c)$ is shorthand for the $H_0$-representation $a_C \otimes
b_L \otimes c_R$. When we further decompose under $S$, the color
representations are unchanged, while the 3 of $L$ or $R$ breaks as
$2_1 \oplus 1_{-2}$, and the adjoint 8 breaks as $3_0 \oplus 1_0 \oplus
2_3 \oplus 2_{-3}$. (Here $b_w$ denotes the $b$-dimensional
representation of $SU(2)$, on which $U(1)$ acts with weight $w$. This
representation of $SU(2) \times U(1)$ factors through $(SU(2) \times
U(1))/\mathbb{Z}_2$ if and only if the integers $b$ and $w$ have opposite
parity.) So the $(8,1,1)$ of $H_0$ becomes $(8,1,1)_{0,0}$ of $S$,
while the $(1,8,1)$ becomes $(1,3,1)_{0,0} \oplus (1,1,1)_{0,0} \oplus
(1,2,1)_{3,0} \oplus (1,2,1)_{-3,0}$. The two subscripts give the
weights of the two $U(1)$'s in $S$. The same subscripts taken modulo
$n$ and $\hat{n}$ give the weights of $F(n,\hat{n})$, so they
determine the representation $A_j$.
We tabulate the results in \tref{tab:eg1}.
In that table, the only reducible block is $B_{00}$. However, if
we replace $F$ by its subgroup $F_0$, many of the $A_j$ coalesce,
resulting in many reducible $B_{ij}$'s.
\begin{table}
\[
\begin{array}{||c|c|c|c|c||}\hline\hline
U_i & H^q(\tilde{X}, U_i(\tilde{V})) & R_i & A_j & B_{ij} \\ \hline
\hline
1 & H^0(\tilde{X}, {\mathcal O}_{\tilde{X}}) & 78 & 0,0 &
(8,1,1)\oplus(1,3,1)\oplus(1,1,3)\oplus2 \times (1,1,1) \\
\hline
& & & 3,0 & (1,2,1) \\ \hline
& & & -3,0 & (1,2,1) \\ \hline
& & & 0,3 & (1,1,2) \\ \hline
& & & 0,-3 & (1,1,2) \\ \hline
& & & 1,1 & (3,2,2) \\ \hline
& & & -2,-2 & (3,1,1) \\ \hline
& & & -1,-1 & (\overline{3},2,2) \\ \hline
& & & 2,2 & (\overline{3},1,1) \\ \hline
& & & 1,-2 & (3,2,1) \\ \hline
& & & -2,1 & (3,1,2) \\ \hline
& & & -1,2 & (\overline{3},2,1) \\ \hline
& & & 2,-1 & (\overline{3},1,2) \\ \hline
8 & H^1(\tilde{X}, {\mathop {\rm ad}} \tilde{V}) & 1 & 0,0 & (1,1,1) \\ \hline
3 & H^1(\tilde{X},\tilde{V}) & 27 & -1,0 & (3,2,1) \\ \hline
& & & 2,0 & (3,1,1) \\ \hline
& & & 1,-1 & (1,2,2) \\ \hline
& & & -2,-1 & (1,1,2) \\ \hline
& & & 1,2 & (1,2,1) \\ \hline
& & & -2,2 & (1,1,1) \\ \hline
& & & 0,1 & (\overline{3},1,2) \\ \hline
& & & 0,-2 & (\overline 3,1,1) \\ \hline
\overline 3 & H^1(\tilde{X},\tilde{V^*}) &
\overline{27} & -1,0 & (\overline 3,2,1) \\ \hline
& & & 2,0 & (\overline 3,1,1) \\ \hline
& & & 1,-1 & (1,2,2) \\ \hline
& & & -2,-1 & (1,1,2) \\ \hline
& & & 1,2 & (1,2,1) \\ \hline
& & & -2,2 & (1,1,1) \\ \hline
& & & 0,1 & (3,1,2) \\ \hline
& & & 0,-2 & (3,1,1) \\ \hline \hline
\end{array}
\]
\caption{The decomposition of $H^q(X, {\mathop {\rm ad}} {\mathcal V}')$ where $G=SU(3)$
and $F=\mathbb{Z}_n \times \mathbb{Z}_{\hat{n}}$.}\label{tab:eg1}
\end{table}
For our second example we consider $G=SU(5)$, so $H=SU(5)$ and the
decomposition of ${\mathop {\rm ad}} E_8$ is given in \eref{E8-su5}. The finite group
$F$ is $\mathbb{Z}_2$, where the generator is embedded in $H=SU(5)$
diagonally with eigenvalues $(1,1,1,-1,-1)$. This breaks $H$ to the
standard model group $S = (SU(3)_C \times SU(2)_L \times U(1)_Y)/\IZ_6$. In
\tref{tab:eg2}, we use $(a,b)_w$ to denote the product of an
$a$-dimensional representation of $SU(3)$ with a $b$-dimensional
representation of $SU(2)$, where $U(1)$ acts with weight $w=3Y$. The
corresponding representation $A_j$ of $F$ depends only on the parity
of $w$.
\begin{table}[h]
\[
\begin{array}{||c|c|c|c|c||}\hline\hline
U_i & H^q(\tilde{X}, U_i(\tilde{V})) & R_i & A_j & B_{ij} \\ \hline
\hline
1 & H^0(\tilde{X}, {\mathcal O}_{\tilde{X}}) & 24 & 0 &
(8,1)_0\oplus(1,3)_0\oplus(1,1)_0 \\ \hline
& & & 1 & (3,2)_{-5}\oplus(\overline 3,2)_{5} \\ \hline
24 & H^1(\tilde{X}, {\mathop {\rm ad}} \tilde{V}) & 1 & 0 & (1,1)_0 \\ \hline
10 & H^1(\tilde{X}, \wedge^2 \tilde{V}) & 5 & 0 & (3,1)_{-2} \\ \hline
& & & 1 & (1,2)_{3} \\ \hline
\overline{10} & H^1(\tilde{X}, \wedge^2 \tilde{V}^*) & \overline 5 & 0 & (\overline 3,1)_{2} \\ \hline
& & & 1 & (1,2)_{-3} \\ \hline
5 & H^1(\tilde{X}, \tilde{V}) & \overline{10} & 0 & (3,1)_{4} \oplus (1,1)_{-6}
\\ \hline
& & & 1 & (\overline 3,2)_{-1} \\ \hline
\overline 5 & H^1(\tilde{X}, \tilde{V}^*) &
10 & 0 & (\overline 3,1)_{-4} \oplus (1,1)_{6} \\ \hline
& & & 1 & (3,2)_{1} \\ \hline \hline
\end{array}
\]
\caption{The decomposition of $H^q(X, {\mathop {\rm ad}} {\mathcal V}')$ where $G=SU(5)$
and $F=\mathbb{Z}_2$.
The $A_j$ correspond to characters of the $\mathbb{Z}_2$ action on $R_i$.
The $a,b$ in $(a,b)_w$ are the representations of $SU(3)_C$ and
$SU(2)_L$ respectively, whereas $w = 3Y$.}\label{tab:eg2}
\end{table}
\section{Standard Model Bundles}
In this section we recall the standard model bundles constructed in
\cite{Donagi:2000fw,Donagi:2000zs,Donagi:2000zf}.
We need a quadruple $(\tilde{X}, A, \tau, \tilde{V})$ satisfying:
\begin{eqnarray}\label{cond}
\bullet&({\bf \mathbb{Z}_2})& \mbox{$\tilde{X}$ is a smooth
Calabi-Yau 3-fold and
$\tau : \tilde{X} \rightarrow \tilde{X}$ is a freely acting involution.}\nonumber\\
&&\mbox{$A$ is a fixed
ample line bundle (K\"{a}hler structure) on $\tilde{X}$.}\nonumber\\
\bullet&{\bf (S)}& \mbox{$\tilde{V}$ is an $A$-stable vector bundle of rank
five on $\tilde{X}$ with structure group $G=SU(5)$.}\nonumber\\
\bullet&{\bf (I)}& \mbox{$\tilde{V}$ is $\tau$-invariant.}\nonumber\\
\bullet&{\bf (C1)}& \mbox{$c_{1}(\tilde{V}) = 0$.}\nonumber\\
\bullet&{\bf (C2)}& \mbox{$c_{2}(\tilde{X}) - c_{2}(\tilde{V})$ is effective.}\nonumber\\
\bullet&{\bf (C3)}& \mbox{$c_{3}(\tilde{V}) = 12$.}
\end{eqnarray}
The involution $\tau$ generates a subgroup
$\mathbb{Z}_2 = F \subset H = Z_{E_8}(SU(5)) = SU(5)$.
The quotient $X := \tilde{X} / F$ is another
Calabi-Yau threefold, and invariance of $\tilde{V}$ allows us to identify it
with the pullback of a stable $SU(5)$ bundle $V$ on $X$, as in
\sref{s:1}.
This produces a heterotic M-theory vacuum $(X,{\mathcal V}')$ with
particle spectrum as given in \tref{tab:eg2} of \sref{s:1}.
\subsection{Rational Elliptic Surfaces and Their Products}
The simply connected threefold $\tilde{X}$ is a complete intersection in
$\mathbb{P}^1 \times \mathbb{P}^2 \times (\mathbb{P}^2)'$ of two hypersurfaces of
multidegrees $(1,3,0)$ and $(1,0,3)$ respectively. This is a
Calabi-Yau, by adjunction, and it has two
elliptic fibrations. These threefolds were first studied by Schoen
\cite{schoen}. Choose projective coordinates: $[t_0 : t_1]$ on $\mathbb{P}^1$;
$z = [z_0 : z_1 : z_2]$ on $\mathbb{P}^2$; and $z' = [z_0' : z_1' : z_2']$ on
$(\mathbb{P}^2)'$. The two hypersurfaces can be written:
\begin{eqnarray}
t_0 f_0(z) - t_1 f_1(z) &=& 0 \label{h1} \\
t_0 f_0'(z') - t_1 f_1'(z') &=& 0 \label{h2} \ ,
\end{eqnarray}
where $f_0, f_1, f_0', f_1'$ are homogeneous cubic polynomials. Since
equation \eref{h1} does not involve $z'$, it defines a hypersurface $B
\subset \mathbb{P}^1 \times \mathbb{P}^2$. Similarly equation \eref{h2} defines a
hypersurface $B' \subset \mathbb{P}^1 \times (\mathbb{P}^2)'$. The surfaces $B$,
$B'$ are called rational elliptic surfaces, or
(inaccurately) $dP_9$'s.
The projections of these surfaces to $\mathbb{P}^1$ are elliptic fibrations:
\begin{equation}
\beta : B \rightarrow \mathbb{P}^1, \qquad \beta' : B' \rightarrow \mathbb{P}^1 \ .
\end{equation}
The original threefold $\tilde{X}$ comes with the two projections
\begin{equation}\label{pipip}
\pi : \tilde{X} \rightarrow B', \qquad \pi' : \tilde{X} \rightarrow B
\end{equation}
which are again elliptic fibrations. In fact, $\tilde{X}$ is the fiber
product
\begin{equation}\label{fiberprod}
\tilde{X} = B \times_{\mathbb{P}^1} B' \ ,
\end{equation}
meaning that a point of $\tilde{X}$ is uniquely specified by a pair of
points $b \in B$, $b' \in B'$ with $\beta(b) = \beta'(b') \in
\mathbb{P}^1$.
The opposite projection $\nu : B \rightarrow \mathbb{P}^2$ is birational,
exhibiting $B$ as the blowup of $\mathbb{P}^2$ at the 9 points $A_i$, $i = 1,
\ldots, 9$ where $f_0(z) = f_1(z)=0$, and similarly for $B'$.
(This is the origin of the ``$dP_9$'' name -- but these surfaces are
not del Pezzos.)
It follows that $H^2(B, \mathbb{Z}) = Pic(B)$ has rank 10. An orthogonal
basis consists of the class $\ell := \nu^* {\mathcal O}_{\mathbb{P}^2}(1)$ together
with the 9 exceptional classes $e_1, \ldots, e_9$. The only non-zero
intersection numbers on $B$ are
$\ell^2 = 1, \quad e_i^2 = -1$, $i = 1, \ldots, 9$.
The class $f := \beta^{-1}(\mbox{point})$ of an elliptic fiber is given
by $f = 3 \ell - \sum\limits_{i=1}^9 e_i$.
There is an analogous basis
$\ell', e_1', \ldots, e_9'$ on $B'$.
The rank of $H^2(\tilde{X}, \mathbb{Z})$ is
therefore 19, with basis $\pi^*\ell' = (\pi')^*\ell, \pi^*e_1', \ldots,
\pi^*e_9', (\pi')^*e_1, \ldots, (\pi')^*e_9$.
\subsection{Special Rational Elliptic Surfaces}\label{s:B}
In order to obtain the involution $\tau$ on $\tilde{X}$, and also in order
to have invariant bundles $\tilde{V}$ on $\tilde{X}$ satisfying the required
conditions, the rational elliptic surfaces $B$, $B'$ need to be
specialized to a particular subfamily. This can be specified as
follows.
Let $\Gamma_{1} \subset \mathbb{P}^{2}$ be a nodal cubic with a node
$A_{8}$. Choose four generic points on $\Gamma_{1}$ and label them
$A_{1}, A_{2}, A_{3}, A_{7}$. Let $\Gamma \subset \mathbb{P}^{2}$ be the
unique smooth cubic which passes through $A_{1}, A_{2}, A_{3}, A_{7},
A_{8}$ and is tangent to the lines $\langle A_{7}A_{i} \rangle$ for $i
= 1, 2, 3$ and $8$. Consider the pencil of cubics spanned by
$\Gamma_{1}$ and $\Gamma$. All cubics in this pencil pass through
$A_{1}, A_{2}, A_{3}, A_{7}, A_{8}$ and are tangent to $\Gamma$ at
$A_{8}$. Let $A_{4}, A_{5}, A_{6}$ be the remaining three base points,
and let $B$ denote the blow-up of $\mathbb{P}^{2}$ at the points $A_{i}$, $i =
1,2, \ldots, 8$ and the point $A_{9}$ which is infinitesimally near
$A_{8}$ and corresponds to the line $\langle A_{7}A_{8} \rangle$.
The pencil becomes the anti-canonical map $\beta : B \rightarrow \mathbb{P}^{1}$ which
is an elliptic fibration with a section. The map $\beta$ has two
reducible fibers $f_{i} = n_{i}\cup o_{i}$, $i = 1,2$ of type
$I_{2}$. We denote by $e_{i}$, $i = 1, \ldots, 7$ and $e_{9}$ the
exceptional divisors corresponding to $A_{i}$, $i = 1, \ldots, 7$ and
$A_{9}$, and by $e_{8}$ the reducible divisor $e_{9} + n_{1}$. The
divisors $e_{i}$ together with the pullback $\ell$ of a class of a line
from $\mathbb{P}^{2}$ form a standard basis in $H^{2}(B,{\mathbb Z})$.
The surface $B$ has an involution $\alpha_{B}$ which is uniquely
characterized by the properties: $\beta \circ \alpha_{B} = \tau_{\mathbb{P}^1}
\circ \beta$, where $\tau_{\mathbb{P}^1}$ is the involution
$t_0 \rightarrow t_0, t_1 \rightarrow -t_1$ on $\mathbb{P}^{1}$,
and $\alpha_{B}$ fixes
the proper transform of $\Gamma$ pointwise. Note that $\tau_{\mathbb{P}^1}$
leaves two points in $\mathbb{P}^1$ fixed, which we call $0$ and
$\infty$. Furthermore, $\alpha_B$ acts as $(-1)_B$ when
restricted to the fiber $f_\infty = \beta^{-1}(\infty)$ and, hence,
leaves four points fixed in $f_\infty$.
Choosing $e_{9} := e$ as the zero section of $\beta$, we can interpret
any other section $\xi$ as an automorphism $t_{\xi} : B \rightarrow B$
which acts along the fibers of $\beta$. The automorphism $\tau_{B} =
t_{e_1}\circ \alpha_{B}$ is again an involution of $B$ which
commutes with $\beta$, induces the same involution on $\mathbb{P}^{1}$ as
$\alpha_{B}$, and has four isolated fixed points sitting on the
same fiber $f_\infty$ of $\beta$.
The special rational elliptic surfaces form a four dimensional
irreducible family. Their geometry was the subject of \cite{Donagi:2000fw}.
The structure of a special rational elliptic surface
$B$ is shown in \fref{fig:B} and the action of $\tau_B$ on
$H^2(B,\mathbb{Z})$ is summarized in \tref{tab:tB}.
\begin{table}
\[
\begin{array}{||l||l||}
\hline\hline
& \tau_{B}^{*} \\
\hline
e_1 & e_9 \\
e_j (j=2,3) & f - e_j + e_1 + e_9 \\
e_i (i=4,5,6) & f - l + e_i + e_1 + e_7 + e_9 \\
e_7 & l - e_2 - e_3 \\
e_8 & f - l + e_1 + e_7 + e_8 + e_9 \\
e_9=e & e_1 \\
l & 2f + 2(e_1+e_9) - (e_2+e_3) + e_7 \\
f = 3l - \sum\limits_{i=1}^9 e_i & f \\
\hline \hline
\end{array}
\]
\caption{The action of $\tau_B$ on $H^2(B, \mathbb{Z})$.}
\label{tab:tB}
\end{table}
\begin{figure}
\parbox{\textwidth}{
\begin{center}
\psfig{file=specialB.eps,height=2.5in}
\end{center}
\caption{A special rational elliptic surface $B$. It has 8 $I_1$
singular fibers.
In addition, there are 2 $I_2$ fibers
$f_1 = n_1 \cup o_1$ and $f_2 = n_2 \cup o_2$.
Under the involution $\tau_B = t_\xi \circ \alpha_B$, there are 4
fixed points, which we have marked, on the fiber $f_\infty$.}
\label{fig:B}
}
\end{figure}
\begin{figure}
\parbox{\textwidth}{
\begin{center}
\psfig{file=X.eps,height=2.5in}
\end{center}
\caption{The Calabi-Yau threefold $\tilde{X}$ is constructed as the fiber
product over $\mathbb{P}^1$
of two non-generic $dP_9$ surfaces $B$ and
$B'$. We have matched the fibers $f_0$ and $f_\infty$ of $B$ with
the fibers $f'_\infty$ and $f'_0$ of $B'$ respectively. The image points
in $\mathbb{P}^1$ of these fibers, namely $0$ and $\infty$ for $B$ and
$0'$ and $\infty'$ for $B'$, are identified as
$0 = \infty'$ and $\infty = 0'$.}
\label{fig:X}
}
\end{figure}
\subsection{Building $\tilde{X},\tau$ and $A$}\label{s:tau}
Choose two special rational elliptic surfaces $\beta : B \rightarrow \mathbb{P}^{1}$
and $\beta' : B' \rightarrow \mathbb{P}^{1}$ in $\tilde{X}$
so that the discriminant loci of $\beta$
and $\beta'$ in $\mathbb{P}^{1}$ are disjoint, $\alpha_{B}$ and $\alpha_{B'}$
induce the same involution on $\mathbb{P}^{1}$, and the fixed loci of $\tau_{B}$
and $\tau_{B'}$ sit over different points 0 and $\infty$
in $\mathbb{P}^{1}$. The fiber
product $\tilde{X} = B\times_{\mathbb{P}^{1}} B'$ is a smooth Calabi-Yau threefold
which is elliptic and has a freely acting involution
$
\tau := \tau_{B}\times_{\mathbb{P}^1}\tau_{B'}
$
and another (non-free) involution $\alpha_{X} := \alpha_{B}\times_{\mathbb{P}^1}
\alpha_{B'}$. For concreteness we fix the elliptic
fibration of $\tilde{X}$ to
be the projection
$
\pi : \tilde{X} \rightarrow B'
$
to $B'$.
The structure of $\tilde{X}$ is shown in \fref{fig:X}.
The stability of the bundle $\tilde{V}$ which we describe below is with
respect to a particular choice of K\"ahler class $A$. If $A_0$ is any
K\"ahler class on $\tilde{X}$, $h'$ a K\"ahler class on $B'$, and $n \gg 0$,
the class of $A = A_0 + n \pi^* h'$ will be K\"ahler on $\tilde{X}$. The
specific value that was found in \cite{Donagi:2000zs} to satisfy all
the requirements was given by
$h' = 193 f' + 144 e_1' + 168(e_9' + e_4' - e_5')$.
\subsection{The Construction of $V$}\label{s:V}
The construction of the $SU(5)$ bundle $V$ on $X := \tilde{X}/F$ is
equivalent to the construction of an $SU(5)$ bundle $\tilde{V}$ on $\tilde{X}$
together with an action of the involution $\tau$ on $\tilde{V}$. The
construction of $\tilde{V}$ in \cite{Donagi:2000zs} employs a combination of two
techniques: extensions and the spectral construction.
The rank 5 bundle $\tilde{V}$ is constructed as an extension
\begin{equation}\label{v23}
0 \rightarrow V_2 \rightarrow \tilde{V} \rightarrow V_3 \rightarrow 0
\end{equation}
involving two simpler bundles $V_2$, $V_3$, of ranks 2 and 3
respectively. Given the $V_i$, we can immediately construct their
direct sum $\tilde{V}_0 = V_2 \oplus V_3$, which is the trivial
extension. In terms of an open cover $\{U_{\alpha}\}$ and $i \times i$
transition matrices $\{g_{i\alpha\beta}\}$ for each $V_i$, the
transition matrices for $\tilde{V}_0$ are
\begin{equation}
g_{0 \alpha \beta} =
\mat{g_{2 \alpha \beta} & 0 \cr 0 & g_{3 \alpha \beta}} \ .
\end{equation}
A general extension $\tilde{V}$ is a rank 5 bundle containing $V_2$ as a
subbundle with quotient $V_3$, but $V_3$ cannot be realized as a
subbundle of $\tilde{V}$ unless $\tilde{V}$ is the trivial extension $\tilde{V}_0$. The
transition matrices for such an extension must be of the form:
\begin{equation}
g_{\alpha \beta} =
\mat{g_{2 \alpha \beta} & h_{\alpha \beta} \cr
0 & g_{3 \alpha \beta}} \ .
\end{equation}
In order for these $g_{\alpha \beta}$ to define a vector bundle, the
upper right corner $h_{\alpha \beta}$ must satisfy a cocycle
condition. Working this out shows that the set of isomorphism classes
of extensions is described by the sheaf cohomology group:
\begin{equation}
{\rm Ext}^1_{\tilde{X}}(V_3, V_2):= H^1(\tilde{X}, V_3^* \otimes V_2) \ .
\end{equation}
The direct sum $\tilde{V}_0 = V_2 \oplus V_3$ corresponds to the 0 element
of this extension group. Our standard model bundle $\tilde{V}$ turns out to
correspond to a non-trivial extension $[\tilde{V}] \in {\rm Ext}_{\tilde{X}}^1(V_3, V_2)$. In
order for $\tilde{V}$ to be $\tau$-invariant, we require first that $V_2$
and $V_3$ be $\tau$-invariant, so we can choose an action of $\tau$ on
$V_2$ and $V_3$. This induces an action of $\tau$ on
${\rm Ext}_{\tilde{X}}^1(V_3, V_2)$.
In order for $\tilde{V}$ to be $\tau$-invariant, we
require that the extension class $[\tilde{V}]$ be $\tau$-invariant.
\subsection{The Construction of the $V_i$}
The construction of the bundles $V_i$, $i = 2,3$, involves the
{\bf spectral construction} or {\bf Fourier-Mukai transform}
\cite{FMW1,D,FMW2}. The Fourier-Mukai transform is a self-equivalence
of the derived category $D^b(\tilde{X})$ of coherent sheaves on $\tilde{X}$
\begin{eqnarray}
FM : && D^b(\tilde{X}) \rightarrow D^b(\tilde{X}) \nonumber \\
&& {\mathcal F} \rightarrow Rp_{1*}(p_2^* {\mathcal F} \stackrel{L}{\otimes} {\mathcal P}) \ .
\end{eqnarray}
Here, $p_1$, $p_2$ are the projections of the fiber product $\tilde{X}
\times_{B'} \tilde{X}$ to the two $\tilde{X}$ factors, $Rp_{1*}$ is the right
derived functor of $p_{1*}$, ${\mathcal P}$ is the Poincar\'e
sheaf on $\tilde{X} \times_{B'} \tilde{X}$, and $\stackrel{L}{\otimes}$ is the left
derived functor of $\otimes$.
If $V_i$ is a rank $i$ vector bundle
on $\tilde{X}$ which is semistable and of degree 0 on each elliptic fiber
$f$ of $\pi:\tilde{X} \rightarrow B'$, then $FM^{-1}(V_i)$ is a rank 1 sheaf
$N_{\Sigma_i}$ supported on a divisor $\Sigma_i \subset \tilde{X}$ which is
finite of
degree $i$ over the base $B'$. In other words, $\Sigma_i$ intersects
each elliptic fiber $f$ in $i$ points. In case $\Sigma_i$ is smooth,
$N_{\Sigma_i}$ is actually a line bundle on $\Sigma_i$. The spectral
construction starts with $(\Sigma_i, N_{\Sigma_i})$ and recovers the
bundle $V_i$ as the Fourier-Mukai transform.
When $\Sigma_i$ is irreducible, the
resulting bundle $V_i$ is automatically stable.
In our case we do not need the full spectral construction on the
threefold $\tilde{X}$. The map $\beta : B \rightarrow \mathbb{P}^1$ is an elliptic
fibration, so there is a Fourier-Mukai transform $FM_B$ on
$D^b(B)$. We will describe below certain curves $C_i \subset B$ and
line bundles $N_i \in Pic(C_i)$ for $i =2,3$. These determine two
bundles $W_i := FM_B(C_i, N_i)$ with $\mbox{rk}(W_i) = i$.
Our desired bundles $V_i$
are then recovered as
\begin{equation}\label{vi}
V_i = \pi'^* W_i \otimes \pi^* L_i
\end{equation}
for appropriate line bundles $L_i \in Pic(B')$. The spectral data on
$B$ and on $\tilde{X}$ are related by
\begin{equation}
\Sigma_i = (\pi')^{-1} C_i = C_i \times_{\mathbb{P}^1} B', \qquad
N_{\Sigma_i} = (\pi')^* N_i \otimes \pi^* L_i \ .
\end{equation}
This is summarized in \fref{f:comm}.
\begin{figure}
\begin{diagram}
&&& & V_i && \\
&&& & \dTo && \\
&&& & \tilde{X} && \\
&&&\ldTo^{\pi'} & & \rdTo{\pi}& \\
W_i &\rTo &B& & && B' &\lTo &L_i \ . &\\
&&&\rdTo_{\beta} & & \ldTo_{\beta'} &\\
&&& & \mathbb{P}^1 &&
\end{diagram}
\caption{The structure of the vector bundles $V_i$, $i=2,3$.}
\label{f:comm}
\end{figure}
The specific values we take for the $C_i$, $N_i$ and $L_i$ are as
follows. Choose curves $\overline{C}_{2}, C_{3} \subset B$, so that
\[
\begin{array}{lcl}
\bullet&& \overline{C}_{2} \in |{\mathcal O}_{B}(2e_{9} + 2f)|, \quad
C_{3} \in |{\mathcal O}_{B}(3e_{9} + 6 f)|, \nonumber \\
\bullet&& \mbox{$\overline{C}_{2}$ and $C_{3}$ are
$\alpha_{B}$-invariant,} \nonumber\\
\bullet&& \mbox{$\overline{C}_{2}$ and $C_{3}$ are smooth and irreducible.}
\end{array}
\]
Set $C_{2} =
\overline{C}_{2} + f_{\infty}$ where $f_{\infty}$ is the smooth fiber
of $\beta$ containing the four fixed points of $\tau_{B}$.
We choose the line bundles $N_2 \in Pic^{3,1}(C_2)$, $N_3 \in Pic^{16}(C_3)$
to transform correctly under the involution $\alpha_B|_{C_i}$:
\begin{equation}\label{Ni}
N_i \simeq
(\alpha_B |_{C_i})^* N_i \otimes {\mathcal O}_{C_i}(e_1 -e_9+f), \qquad
i = 2,3 \ .
\end{equation}
Here $Pic^{3,1}(C_2)$ denotes line bundles of degree 3 on
$\overline{C}_2$ and degree 1 on $f_\infty$ \cite{Donagi:2000zs}.
(It is shown in \cite{Donagi:2000zs} that such $N_i$ do exist.)
A useful quantity associated with the bundle $W_2$
is the degree $-1$ line bundle $G \in Pic^{-1}(f_\infty)$ on
the elliptic curve $f_\infty$, defined as
\begin{equation}\label{defG}
G = N_2 |_{f_\infty}(-D),
\end{equation}
where $D$ is the divisor $D = \overline{C}_2 \cap f_\infty$.
This fits into an exact sequence
\begin{equation}\label{seqW2}
0 \rightarrow W_2 \rightarrow \overline{W}_2 \rightarrow i_{f_\infty *}(G^*) \rightarrow 0 \ ,
\end{equation}
where $\overline{W}_2$ is the rank 2 vector bundle associated
with the spectral cover $\overline{C}_2$ and spectral line bundle
$\overline{N}_2 = N_2 \otimes {\mathcal O}_{\overline{C}_2}$.
The Chern characters can be read from Lemma 5.1 of \cite{Donagi:2000zs}:
\begin{equation}\label{chernWi}
\begin{array}{cc}
\mbox{ch}(W_2) = 2 - f - 3 {\rm pt}, &
\mbox{ch}(\overline{W}_2) = 2 - 2 {\rm pt}, \nonumber \\
\mbox{ch}(W_3) = 3 + f - 6 {\rm pt}, &
\mbox{ch}(G^*) = f + {\rm pt}.
\end{array}
\end{equation}
Finally, the line bundles $L_i$ on $B'$ are given by
\begin{eqnarray}\label{L23}
L_2 &=& {\mathcal O}_{B'}(3 r') \nonumber \\
L_3 &=& {\mathcal O}_{B'}(-2 r')
\end{eqnarray}
where
\begin{equation}\label{defr}
r' = e_1' + e_4' - e_5' + e_9' + f'
= 3\ell' - 2e_4' - (e_2'+e_3'+e_6'+e_7'+e_8').
\end{equation}
Formula \eref{L23} holds with the specific choices $N_2 \in Pic^{3,1}(C_2)$,
$N_3 \in Pic^{16}(C_3)$ which we made above, and only with those
choices. This is why we specify the general solution in \cite{Donagi:2000zs}
to these values.
This completes the specification of the bundles
$V_i$ for $i=2,3$.
It was seen in \cite{Donagi:2000zs} that $\tau$-invariant extensions
$[\tilde{V}] \in {\rm Ext}_{\tilde{X}}^1(V_3, V_2)_{(+)}$ exist,
and that the bundle $\tilde{V}$ corresponding
to a general such $[\tilde{V}]$ has structure group $G = SU(5)$, is stable, is
$\tau$-invariant, and satisfies the requirements $(\mathbb{Z}_2, S, I, C1, C2,
C3)$ in \eref{cond}.
\subsection{Comments}
The reason we did not build $\tilde{V}$ directly by a spectral construction
applied to the surface $\Sigma = \Sigma_2 \cup \Sigma_3$ in $\tilde{X}$ (or
to the curve $C = C_2 \cup C_3$ in $B$) is that on singular spectral
covers (such as $\Sigma$, $C$), the rank 1 sheaf ($\overline{N}$ or $N$) can
fail to be a line bundle, leading to technical complications. A
closely related problem is that it is harder to check the stability of
$\tilde{V}$ when the spectral cover is reducible.
Another subtlety is that our $C_2$ is not finite over $\mathbb{P}^1$. It
intersects the generic elliptic fiber in 2 points, but it contains the
entire fiber $f_\infty$.
We chose $N_2$ carefully so that our $W_2$ is still the Fourier-Mukai
transform of $(C_2,N_2)$.
But in practice it is often easier to work with $\overline{C}_2$, $\overline{N}_2$
and $\overline{W}_2$, and to relate $W_2$ and $\overline{W}_2$ via \eref{seqW2}.
The construction in \cite{Donagi:2000zs} involves additional degrees of
freedom in the form of Hecke transforms applied to the $\tilde{V}$. Later
checks, motivated by questions of Mike Douglas,
suggest that most or
all of these extra degrees of freedom
may be illusory.
At any rate, we do not use them in the present work.
\section{Cohomologies of $U_i(\tilde{V})$}\label{s:coh}
In order to compute the relevant cohomologies on a rational elliptic
surface such as $B'$, we need some basic facts about the line bundle
${\mathcal O}_{B'}(r')$ of \eref{defr}.
We claim that the direct image is:
\begin{equation}\label{br}
\beta'_* {\mathcal O}_{B'}(r') \simeq {\mathcal O}_{\IP^1} \oplus {\mathcal O}_{\IP^1},
\end{equation}
or equivalently that
\begin{equation}\label{b(r-f)}
\beta'_* {\mathcal O}_{B'}(r'-f') \simeq {\mathcal O}_{\IP^1}(-1) \oplus {\mathcal O}_{\IP^1}(-1).
\end{equation}
Indeed, the left hand side of \eref{b(r-f)} is a rank 2 bundle on
$\mathbb{P}^1$, since $(r'-f') \cdot f' = 2$, so it must be of the form
${\mathcal O}_{\IP^1}(a) \oplus {\mathcal O}_{\IP^1}(b)$ for some integers $a$, $b$.
Now $r'-f' = e_1' + e_9' + e_4' - e_5'$ cannot be effective
(any effective representative has negative intersection with $e_1'$,
$e_4'$, $e_9'$, so must contain all of them), and therefore our
integers $a$, $b$ must be negative.
To conclude that $a=b=-1$ as claimed in \eref{b(r-f)}, it suffices to
note that $a+b$ is the degree of $\beta'_* {\mathcal O}_{B'}(r'-f')$, which by
{Groethendieck-Riemann-Roch}~(GRR) equals $-2$.
Instead of GRR, we can obtain the same result using a bit of geometry.
We saw in \eref{defr} that $r' = 3\ell' -(e_2'+e_3'+e_6'+e_7'+e_8') -
2e_4'$, so we can identify sections of ${\mathcal O}_{B'}(r')$
with cubic polynomials on $\mathbb{P}^2$ vanishing at $A_i$ for
$i=2,3,6,7,8$, and vanishing to second order at $A_4$.
The space $H^0({\mathcal O}_{\mathbb{P}^2}(3 \ell))$ of cubics is
10 dimensional, the vanishing at each of the five $A_i$ imposes one
linear condition, and vanishing to second order at $A_4$ imposes 3
more conditions, for a total of 8 conditions.
Therefore $2= 10-8 \le h^0({\mathcal O}_{B'}(r')) =
h^0(\mathbb{P}^1,\beta'_* {\mathcal O}_{B'}(r')) =
h^0({\mathcal O}_{\IP^1}(a+1)) + h^0({\mathcal O}_{\IP^1}(b+1))$.
Recalling that $a$, $b$ are negative, this is possible only for
$a=b=-1$; so we have found another argument for \eref{br},
\eref{b(r-f)}.
It follows from \eref{br} that $H^0({\mathcal O}_{B'}(r'))$ is 2 dimensional.
We let $x_0$ and $x_1$ be a basis.
We claim that the quotient $x_1 / x_0$ is everywhere defined, so it
gives a map
\begin{equation}
\chi : B' \rightarrow \mathbb{P}^1_x,
\end{equation}
and the $x_i$ can be interpreted as homogeneous coordinates on the
target $\mathbb{P}^1_x$.
Checking that $\chi$ is everywhere defined is equivalent to verifying
that $x_0$ and $x_1$ cannot vanish at the same point.
Since $r'^2 = 0$, two divisors in the linear system $|r'|$ cannot
intersect each other unless they have a common component.
So to conclude, it suffices to check that some (and hence almost
all) of these divisors are irreducible.
This follows immediately from the geometric model: in fact, the fibers
of $\chi$, identified as the pencil of cubics vanishing at the five
$e_i'$ and doubly at $e_4'$, include precisely 8 reducible curves,
namely:
\begin{equation}\label{Kij}
\begin{array}{lll}
K_i^1 \cup K_i^2, &
K_i^1 = \ell' - e_5' - e_i', &
K_i^2 = 2\ell'-(e_2'+e_3'+e_6'+e_7'+e_8')-e_5'+e_i',
\quad i=2,3,6,7,8 \\
K_j^0 \cup K_j^3, &
K_j^0 = e_j', &
K_j^3 = 3\ell'-(e_2'+e_3'+e_6'+e_7'+e_8')-2e_5'-e_j',
\quad j=1,4,9.
\end{array}\end{equation}
The first five curves occur as reducible cubics in $\mathbb{P}^2$, consisting
of the line joining $A_5$ to $A_i$ and the conic through $A_5$ and the
remaining 4 points.
The last three consist of cubics which happen to pass through one of
the $A_j$, so their inverse image in $B'$ contains the corresponding
$e_j'$.
All other cubics in our system are singular (at $A_5$) but
irreducible.
We conclude that $\chi$ is indeed everywhere defined, its generic
fiber is a $\mathbb{P}^1$, and precisely the 8 fibers listed in \eref{Kij}
split into a pair of $\mathbb{P}^1$'s meeting at one point.
Clearly, the target space $\mathbb{P}^1_x$ of the map $\chi$ defined by the
line bundle ${\mathcal O}_{B'}(r')$ has nothing to
do with the target space $\mathbb{P}^1_t$ of the map $\beta'$ defined by the
line bundle ${\mathcal O}_{B'}(f')$.
In fact, we can put these two maps together, to get a map
\begin{equation}
\Delta = (\beta', \chi) : B' \rightarrow {\sf Q} := \mathbb{P}^1_t \times \mathbb{P}^1_x
\end{equation}
given by the two pairs of homogeneous coordinates $(t_0, t_1)$,
$(x_0, x_1)$.
The product surface ${\sf Q} $ could be identified with a smooth quardric in
$\mathbb{P}^3$ via the embedding $(t_0x_0, t_1x_0, t_0x_1, t_1x_1)$, but we
will not use this.
The product map $\Delta$ is onto ${\sf Q} $, and is of degree $f' \cdot r' =
2$; in other words, we have realized the rational elliptic surface
$B'$ as a double cover of the quadric surface ${\sf Q} $.
The fibers of $\beta'$ are of course the elliptic curves $f'$ which now
appear as double covers of $\mathbb{P}^1_x$ branched at 4 points.
The general fiber of $\chi$, on the other hand, is isomorphic to a
$\mathbb{P}^1$, as is seen by adjunction.
It appears as a double cover of $\mathbb{P}^1_t$ branched at 2 points.
The branch locus $Br_\Delta$ of $\Delta$ is therefore a divisor of
bidegree $(4,2)$ in {\sf Q} .
Line bundles on {\sf Q} are of the form ${\mathcal O}_{{\sf Q} }(k,l) :=
pr_t^*{\mathcal O}_{\mathbb{P}^1_t}(l) \otimes pr_x^*{\mathcal O}_{\mathbb{P}^1_x}(k)$, with integers
$k$, $l$, where $pr_t$, $pr_x$ are the projections to $\mathbb{P}^1_t$,
$\mathbb{P}^1_x$ respectively: $\beta' = pr_t \circ \Delta$, $\chi = pr_x
\circ \Delta$.
Let us introduce the abbreviation
\begin{equation}
{\mathcal O}_{B'}(k,l) := \Delta^* {\mathcal O}_{{\sf Q} }(k,l) = {\mathcal O}_{B'}(k r' + l f')
\end{equation}
for the corresponding line bundles on $B'$.
So for example ${\mathcal O}_{B'}(0,1)$ is the anticanonical bundle $K_{B'}^{-1}
\simeq {\mathcal O}_{B'}(f')$, ${\mathcal O}_{B'}(1,0)$ is ${\mathcal O}_{B'}(r')$, ${\mathcal O}_{B'}(3,0)$ is our $L_2$,
and ${\mathcal O}_{B'}(-2,0)$ is $L_3$.
On $B'$ there is a unique involution $\iota$ which exchanges the two
sheets of $B'$ over {\sf Q} .
Its fixed locus is the ramification divisor $Ram_\Delta \subset B'$.
The image $\Delta(Ram_\Delta)$ is of course $Br_\Delta$.
Since
\begin{equation}
\Delta^*{\mathcal O}_{{\sf Q} }(Br_\Delta) = {\mathcal O}_{B'}(2Ram_\Delta)
\end{equation}
and the Picard group of $B'$ has no torsion, we find that:
\begin{equation}\label{ob-ram}
{\mathcal O}_{B'}(Ram_\Delta) \simeq \Delta^*{\mathcal O}_{{\sf Q} }\left(\frac12 Br_\Delta\right) =
\Delta^*{\mathcal O}_{{\sf Q} }(2,1) = {\mathcal O}_{B'}(2,1).
\end{equation}
For any double cover such as $\Delta$, sections of ${\mathcal O}_{B'}$ can be
decomposed into $\iota$-invariants and anti-invariants.
This can be written formally as a decomposition of the direct image:
\begin{equation}
\Delta_* {\mathcal O}_{B'} \simeq 1\cdot {\mathcal O}_{{\sf Q} } \oplus y \cdot
{\mathcal O}_{{\sf Q} }\left(-\frac12 Br_\Delta\right),
\end{equation}
where $y \in H^0({\mathcal O}_{B'}(Ram_\Delta))$ is the
$\iota$-anti-invariant
section characterized up to scalars by its vanishing precisely on
$Ram_\Delta$. (This is another special case of GRR).
In our case, \eref{ob-ram} shows that
\begin{equation}\label{def-y}
y \in H^0({\mathcal O}_{B'}(2,1)), \quad \iota y = -y
\end{equation}
and
\begin{equation}
\Delta_* {\mathcal O}_{B'} = {\mathcal O}_{{\sf Q} } \oplus y {\mathcal O}_{{\sf Q} }(-2,-1).
\end{equation}
This can be tensored with the pullback of ${\mathcal O}_{{\sf Q} }(k,l)$, giving the
decomposition
\begin{equation}\label{deltak,l}
\Delta_* {\mathcal O}_{B'}(k,l) = {\mathcal O}_{{\sf Q} }(k,l) \oplus y {\mathcal O}_{{\sf Q} }(k-2,l-1)
\end{equation}
which will be the foundation for our cohomological calculations.
Let $S_x^k := H^0({\mathcal O}_{\IP^1_x}(k))$ denote the $(k+1)$-dimensional
vector space of homogeneous polynomials of degree $k \ge 0$ in
$x_0,x_1$, with basis consisting of the monomials $x_0^k,
x_0^{k-1}x_1, \ldots, x_1^k$.
We set $S_x^k = 0$ for $k < 0$, and let $(S_x^k)^*$ denote the dual
vector space.
The cohomology of $\mathbb{P}^1_x$ is given by:
\begin{equation}
H^0({\mathcal O}_{\IP^1_x}(k)) = S_x^k, \quad
H^1({\mathcal O}_{\IP^1_x}(k)) \simeq (S_x^{-2-k})^*,
\end{equation}
where the second formula involves Serre duality and therefore depends
on choosing, once and for all, an isomorphism of $K_{\mathbb{P}^1_x}$ with
${\mathcal O}_{\mathbb{P}^1_x}(-2)$.
This formula can be applied to the product surface ${\sf Q} = \mathbb{P}^1_t
\times \mathbb{P}^1_x$, yielding a formula for the direct images
(for a general definition of direct image sheaves we refer the reader
to the Appendix)
\begin{equation}
R^i pr_{t*} {\mathcal O}_{{\sf Q} }(k,l) =
H^i({\mathcal O}_{\IP^1_x}(k)) \otimes {\mathcal O}_{\IP^1_t}(l) \simeq
\left\{\begin{array}{ll}
S_x^k \otimes {\mathcal O}_{\IP^1_t}(l), & i = 0 \\
(S_x^{-2-k})^* \otimes {\mathcal O}_{\IP^1_t}(l), & i = 1
\end{array}\right. \ ,
\end{equation}
and therefore for the cohomology:
\begin{eqnarray}\label{Hn-qkl}
H^n({\mathcal O}_{{\sf Q} }(k,l)) &=&
\bigoplus_{i+j=n} H^i({\mathcal O}_{\IP^1_x}(k)) \otimes
H^j({\mathcal O}_{\IP^1_t}(l))\nonumber\\
&\simeq&
\left\{\begin{array}{ll}
S_x^k \otimes S_t^l, & n = 0 \\
(S_x^{-2-k})^* \otimes S_t^l \oplus S_x^k \otimes (S_t^{-2-l})^*,
& n = 1 \\
(S_x^{-2-k})^* \otimes (S_t^{-2-l})^* & n=2.
\end{array}\right.
\end{eqnarray}
The power of formula \eref{deltak,l} is that it allows us to write
down analogous formulas for the much more complicated surface $B'$:
\begin{equation}\label{bRb-ob}
\begin{array}{ccccc}
\beta'_* {\mathcal O}_{B'}(k,l) &=& S_x^k \otimes {\mathcal O}_{\IP^1_t}(l) &\oplus&
y S_x^{k-2} \otimes {\mathcal O}_{\IP^1_t}(l-1) \nonumber \\
R^1\beta'_* {\mathcal O}_{B'}(k,l) &\simeq& (S_x^{-2-k})^* \otimes {\mathcal O}_{\IP^1_t}(l) &\oplus&
y (S_x^{-k})^* \otimes {\mathcal O}_{\IP^1_t}(l-1).
\end{array}\end{equation}
Note that for $k>0$ only the $\beta'_*$ term is non-zero, while for $k<0$
only the $R^1\beta'_*$ term is non-zero.
The cohomology on $B'$ can be obtained from \eref{bRb-ob}, or directly
from \eref{deltak,l}:
\begin{equation}
H^n({\mathcal O}_{B'}(k,l)) = H^n({\mathcal O}_{{\sf Q} }(k,l)) \oplus
y H^n({\mathcal O}_{{\sf Q} }(k-2,l-1)),
\end{equation}
where the individual terms are given in \eref{Hn-qkl}.
Explicitly, this formula gives bases for the various cohomology groups
on $B'$ consisting of monomials in $t_0, t_1, x_0, x_1, y$.
For example:
\begin{equation}
\begin{array}{ll}
H^0({\mathcal O}_{B'}(0,1)): & t_0, t_1 \\
H^0({\mathcal O}_{B'}(1,0)): & x_0, x_1 \\
H^0({\mathcal O}_{B'}(3,0)): & x_0^3, x_0^2x_1, x_0x_1^2, x_1^3 \\
H^0({\mathcal O}_{B'}(2,1)): & t_0x_0^2, t_0x_0x_1, t_0x_1^2,
t_1x_0^2, t_1x_0x_1, t_1x_1^2, y.
\end{array}
\end{equation}
Now, we are ready to calculate the cohomology groups which we need on
$\tilde{X}$.
\paragraph{$\bullet$ \fbox{$V_2$}}
We have
\begin{equation}\label{b-w2=0}
\beta_* \overline{W}_2 = \beta_* W_2 = 0
\end{equation}
since these sheaves are
torsion-free and vanish at a generic point.
We also have $R^1\beta_* \overline{W}_2 = 0$
because it is supported on $\overline{C}_2 \cap e_9$, which is empty.
The long exact sequence induced from \eref{seqW2} therefore gives:
\begin{equation}
0 = \beta_*\overline{W}_2 \rightarrow \beta_* i_{f_\infty *}(G^*) \rightarrow R^1 \beta_* W_2
\rightarrow R^1 \beta_* \overline{W}_2 \rightarrow 0,
\end{equation}
so $R^1 \beta_* W_2 = \beta_* i_{f_\infty *}(G^*)$.
The Leray spectral sequence for $\pi: \tilde{X} \rightarrow B'$ therefore gives:
\begin{eqnarray}\label{H1V2-1}
H^1(\tilde{X}, V_2)
&=& H^1(\tilde{X}, \pi'^*W_2 \otimes \pi^*L_2)
= H^0(B', R^1\pi_*\pi'^*W_2 \otimes L_2) \nonumber \\
&=& H^0(B', \beta'^*R^1\beta_*W_2 \otimes L_2)
= H^0(f_\infty, G^*) \otimes H^0(f_0',L_2).
\end{eqnarray}
Note that $h^0(f_\infty,G^*) = 1$, $h^0(f_0', L_2) = 6$, hence
$h^1(\tilde{X}, V_2) = 6$.
\paragraph{$\bullet$ \fbox{$V_3$}}
We again have $\beta_*W_3=0$, so for $i=0,1$:
\begin{equation}
H^i(\tilde{X}, V_3) = H^0(B', \beta'^*R^i\beta_*W_3 \otimes L_3)
= H^0(\mathbb{P}^1, R^i\beta_*W_3 \otimes \beta_*' L_3) = 0,
\end{equation}
where we have used that $\beta'_*L_3 = 0$, which holds since $L_3 \cdot
f' = -4 < 0$.
\paragraph{$\bullet$ \fbox{$\tilde{V}$}}
The long exact sequence induced from \eref{v23} gives:
\begin{equation}\label{H1tv}
0 = H^0(\tilde{X}, V_3) \rightarrow H^1(\tilde{X}, V_2) \rightarrow H^1(\tilde{X}, \tilde{V})
\rightarrow H^1(\tilde{X}, V_3) = 0,
\end{equation}
so
$H^1(\tilde{X}, \tilde{V}) = H^1(\tilde{X}, V_2) = H^0(f_\infty, G^*) \otimes
H^0(f_0',L_2)$ by \eref{H1V2-1}.
\paragraph{$\bullet$ \fbox{$\wedge^2 V_2$}}
From \eref{chernWi} we know that $\wedge^2W_2 = c_1(W_2) = -f$.
But $\pi'^*{\mathcal O}_B(-f) \simeq \pi^* {\mathcal O}_{B'}(-f')$, since both pull back
from the same sheaf ${\mathcal O}_{\IP^1}(-1)$ on $\mathbb{P}^1$.
Therefore,
\begin{equation}
\wedge^2 V_2 = \pi'^* \wedge^2 W_2 \otimes \pi^* (L_2 ^{\otimes 2})
\simeq \pi^* {\mathcal O}_{B'}(6,-1).
\end{equation}
Combining this with:
\begin{equation}
\pi_* {\mathcal O}_{\tilde{X}} = {\mathcal O}_{B'}, \quad
R^1 \pi_* {\mathcal O}_{\tilde{X}} = {\mathcal O}_{B'}(-f')
\end{equation}
gives us formulas for the direct images of $\wedge^2 V_2$:
\begin{equation}
\pi_* \wedge^2 V_2 \simeq {\mathcal O}_{B'}(6,-1), \quad
R^1 \pi_* \wedge^2 V_2 \simeq {\mathcal O}_{B'}(6,-2).
\end{equation}
We then push on to $\mathbb{P}^1$ as in \eref{bRb-ob}, and since $R^1 \beta'_* =
0$ for $k=6$, we find:
\begin{eqnarray}
(\beta' \circ \pi)_* \wedge^2 V_2 &=& \beta'_*(\pi_* \wedge^2 V_2) = \beta'_* {\mathcal O}_{B'}(6,-1)
= S_x^6 \otimes {\mathcal O}_{\IP^1_t}(-1) \oplus y S_x^{4} \otimes {\mathcal O}_{\IP^1_t}(-2)
\nonumber \\
R^1 (\beta' \circ \pi)_* \wedge^2 V_2 &=& \beta'_*(R^1 \pi_* \wedge^2 V_2) =
\beta'_* {\mathcal O}_{B'}(6,-2) =
S_x^6 \otimes {\mathcal O}_{\IP^1_t}(-2) \oplus y S_x^{4} \otimes {\mathcal O}_{\IP^1_t}(-3)
\nonumber \\
R^2 (\beta' \circ \pi)_* \wedge^2 V_2 &=& 0.
\end{eqnarray}
Since none of these sheaves have any global sections, we find the
cohomology on $\tilde{X}$ by taking $H^1$ of the images on $\mathbb{P}^1_t$:
\begin{equation}\label{h1-av2}\begin{array}{ll}
H^0(\tilde{X}, \wedge^2 V_2) = 0, & h^0(\tilde{X}, \wedge^2 V_2) = 0, \\
H^1(\tilde{X}, \wedge^2 V_2) = y S_x^4, & h^1(\tilde{X}, \wedge^2 V_2) = 5, \\
H^2(\tilde{X}, \wedge^2 V_2) = S_x^6 \oplus y S_x^4 \otimes (S_t^1)^*,
& h^2(\tilde{X}, \wedge^2 V_2) = 7+2\times 5 = 17, \\
H^3(\tilde{X}, \wedge^2 V_2) = 0, & h^3(\tilde{X}, \wedge^2 V_2) = 0.
\end{array}\end{equation}
\paragraph{$\bullet$ \fbox{$\wedge^2 V_2^*$}}
The cohomology of $\wedge^2 V_2^*$ can be obtained from that of $\wedge^2 V_2$ by
Serre duality.
Equivalently, we can apply the above procedure to $\wedge^2 V_2^* =
\pi^*{\mathcal O}_{B'}(-6,1)$, noting that for $k=-6$ all the $\beta'_*$ terms
in \eref{bRb-ob} vanish:
\begin{equation}
\pi_* \wedge^2 V_2^* = {\mathcal O}_{B'}(-6,1), \quad
R^1 \pi_* \wedge^2 V_2^* = {\mathcal O}_{B'}(-6,0).
\end{equation}
\begin{eqnarray}
(\beta' \circ \pi)_* \wedge^2 V_2^* &=& 0, \nonumber \\
R^1 (\beta' \circ \pi)_* \wedge^2 V_2^* &=& R^1 \beta'_*(\pi_* \wedge^2 V_2^*) =
R^1 \beta'_* {\mathcal O}_{B'}(-6,1) \nonumber \\
&=&
S_x^{4*} \otimes {\mathcal O}_{\IP^1_t}(1) \oplus y S_x^{6*} \otimes {\mathcal O}_{\IP^1_t},
\nonumber \\
R^2 (\beta' \circ \pi)_* \wedge^2 V_2^* &=& R^1\beta'_*(R^1\pi_* \wedge^2 V_2^*)
= R^1\beta'_* {\mathcal O}_{B'}(-6,0) \nonumber \\
&=& S_x^{4*} \otimes {\mathcal O}_{\IP^1_t} \oplus y S_x^{6*} \otimes {\mathcal O}_{\IP^1_t}(-1),
\end{eqnarray}
\begin{equation}\label{H-av2*}\begin{array}{llll}
H^0(\tilde{X}, \wedge^2 V_2^*) &=& 0, & h^0(\tilde{X}, \wedge^2 V_2^*) = 0, \\
H^1(\tilde{X}, \wedge^2 V_2^*) &=&
H^0(\mathbb{P}^1_t, S_x^{4*} \otimes {\mathcal O}_{\IP^1_t}(1) \oplus
y S_x^{6*} \otimes {\mathcal O}_{\IP^1_t}) & \\
&=& S_x^{4*} \otimes S^1_t \oplus y S_x^{6*},
& h^1(\tilde{X}, \wedge^2 V_2^*) = 5 \times 2 + 7 = 17, \\
H^2(\tilde{X}, \wedge^2 V_2^*) &=&
H^0(\mathbb{P}^1_t, S_x^{4*} \otimes {\mathcal O}_{\IP^1_t} \oplus y S_x^{6*} \otimes
{\mathcal O}_{\IP^1_t}(-1)) & \\
&=& S_x^{4*}, & h^2(\tilde{X}, \wedge^2 V_2^*) = 5, \\
H^3(\tilde{X}, \wedge^2 V_2^*) &=& 0, & h^3(\tilde{X}, \wedge^2 V_2^*) = 0.
\end{array}\end{equation}
\paragraph{$\bullet$ \fbox{$V_2 \otimes V_3^*$}}
We recall that $C_2 = \overline{C}_2 \cup f_\infty$, and $W_2$ is related to
$\overline{W}_2$ by sequence \eref{seqW2}.
If we tensor \eref{seqW2} by $W_3^*$ and push to $\mathbb{P}^1$ with $\beta_*$,
we find
\begin{equation}\label{seqW2xW3*}
0 \rightarrow \beta_*( i_{f_\infty *}G^* \otimes W_3^*) \rightarrow {\mathcal F} \rightarrow \overline{{\mathcal F}} \rightarrow
0,
\end{equation}
where
\begin{equation}
{\mathcal F} := R^1 \beta_*(W_2 \otimes W_3^*), \quad
\overline{{\mathcal F}} := R^1 \beta_*(\overline{W}_2 \otimes W_3^*),
\end{equation}
and the last term in \eref{seqW2xW3*} is 0 because $G^*$ has degree $+1$
on $f_\infty$.
All the sheaves in \eref{seqW2xW3*} have finite support:
\begin{itemize}
\item[$-$] $\overline{{\mathcal F}}$ is supported on $\beta(\overline{C}_2 \cap C_3)$.
If we choose things generically, $\overline{C}_2 \cap C_3$ will
consist of 12 points $p_j$ in $B'$, the image
$\beta(\overline{C}_2 \cap C_3)$ will consist of 12 distinct points $\hat{p}_j
:= \beta(p_j) \in \mathbb{P}^1$, $j=1, \ldots, 12$, and $\overline{{\mathcal F}}$ will decompose
as the sum of 12 rank 1 skyscraper sheaves ${\mathcal F}_j$ near each
$\hat{p}_j$: $\overline{{\mathcal F}} = \bigoplus_{j=1}^{12} {\mathcal F}_j$.
\item[$-$] $\beta_*(i_{f_\infty *}G^* \otimes W_3^*)$ is supported
at $\infty \in \mathbb{P}_t^1$, and has rank 3 there.
It can therefore be decomposed (non-canonically) as
$\bigoplus_{j=13}^{15} {\mathcal F}_j$, with each ${\mathcal F}_j$ a rank 1 skyscraper
sheaf supported at $\infty$.
For $j=13,14,15$ we use $\hat{p}_j$ as another notation for the point
$\infty \in \mathbb{P}_t^1$, the support of ${\mathcal F}_j$.
\item[$-$]
The sequence \eref{seqW2xW3*} splits, so
${\mathcal F} = \bigoplus_{j=1}^{15} {\mathcal F}_j$.
\end{itemize}
We can now combine this with formula \eref{bRb-ob} applied to $L_2
\otimes L_3^* = {\mathcal O}_{B'}(5,0)$, to compute $H^1(\tilde{X}, V_2 \otimes V_3^*)$:
\begin{eqnarray}\label{h1v2v3*}
H^1(\tilde{X}, V_2 \otimes V_3^*)
&=& H^0(\mathbb{P}^1_t, R^1\beta_*(W_2\otimes W_3^*) \otimes \beta_*'(L_2 \otimes
L_3^*) ) \nonumber \\
&=& H^0(\mathbb{P}^1_t, {\mathcal F} \otimes [ S_x^5 \otimes {\mathcal O}_{\IP^1_t} \oplus
y S_x^3 \otimes {\mathcal O}_{\IP^1_t}(-1) ]) \nonumber \\
&=& \bigoplus_{j=1}^{15} H^0(\mathbb{P}^1_t, {\mathcal F}_j) \otimes
[S_x^5 \oplus y S_x^3 \otimes \{\hat{p}_j\mathbb{C}\} ].
\end{eqnarray}
Here, we use the notation $ \{\hat{p}_j\mathbb{C}\} \subset S_t^{1*}$ for the
line inside the 2-dimensional plane $S_t^{1*}$ consisting of all
points proportional to $\hat{p}_j \in \mathbb{P}^1_t = \mathbb{P}(S_t^{1*})$.
This line is the fiber at $\hat{p}_j$ of the line bundle ${\mathcal O}_{\IP^1_t}(-1)$.
In particular, the dimension is
\begin{equation}
h^1(\tilde{X}, V_2 \otimes V_3^*) = 150 = 15 \times [6+4].
\end{equation}
\paragraph{$\bullet$ \fbox{$V_2^* \otimes V_3^*$}}
From the Chern character formula \eref{chernWi} we know that $W_2^*
\simeq W_2 \otimes {\mathcal O}_{B'}(f)$, and therefore
\begin{equation}
R^1\beta_*(W_2^*\otimes W_3^*) \simeq R^1\beta_*(\beta^* {\mathcal O}_{\IP^1_t}(1) \otimes W_2
\otimes W_3^*)
= {\mathcal F} \otimes {\mathcal O}_{\IP^1_t}(1).
\end{equation}
In analogy with \eref{h1v2v3*} we therefore get
\begin{eqnarray}\label{H2-v2*v3*}
H^2(\tilde{X}, V_2^* \otimes V_3^*) &=&
H^0(\mathbb{P}^1_t, R^1\beta_*(W_2^*\otimes W_3^*) \otimes
R^1\beta'_*(L_2^* \otimes L_3^*)) \nonumber \\
&=& H^0(\mathbb{P}^1_t, {\mathcal F} \otimes [y S_x^{1*}]) \nonumber \\
&=& \bigoplus_{j=1}^{15}
H^0(\mathbb{P}^1_t, {\mathcal F}_j) \otimes y S_x^{1*},
\end{eqnarray}
and the dimension is
\begin{equation}\label{h2v2*v3*}
h^2(\tilde{X}, V_2^* \otimes V_3^*) = 30 = 15 \times 2.
\end{equation}
\paragraph{$\bullet$ \fbox{$\wedge^2 \tilde{V}$}}
We note that the short exact sequence \eref{v23}
which defines $\tilde{V}$ implies the exact sequence
\begin{equation}\label{defQ}
0 \rightarrow \wedge^2 V_2 \rightarrow \wedge^2 \tilde{V} \rightarrow Q \rightarrow 0 \ ,
\end{equation}
where $Q$ is defined by the quotient of the map $\wedge^2 V_2 \rightarrow \wedge^2 \tilde{V}$.
However, the natural map $\wedge^2 \tilde{V} \rightarrow \wedge^2 V_3$ factors through $Q$ with the
kernel $V_2 \otimes V_3$. A simple consistency
check for this statement is by
dimension counting. Recall that $V_2$, $V_3$ and $\tilde{V}$ have rank 2, 3
and 5 respectively. Then,
$Q$ has dimension $\frac{5\cdot 4}{2} -
\frac{2\cdot 1}{2} = 9$ from \eref{defQ}, $\wedge^2 V_3$ has dimension
$\frac{3\cdot 2}{2} = 3$, so the kernel should have dimension
$9-3=6$. This is indeed the dimension of
$V_2 \otimes V_3$, which is $2 \cdot 3 = 6$.
In summary, we have an intertwined pair of short exact sequences as
follows.
\begin{equation}
\begin{array}{cccccccccc}
&&&&&&0&&&\\
&&&&&&\uparrow&&&\\
&&&&&&\wedge^2 V_3&&&\\
&&&&&&\uparrow&&&\\
0 &\rightarrow& \wedge^2 V_2 &\rightarrow& \wedge^2 \tilde{V} &\rightarrow& Q &\rightarrow& 0 \ . \\
&&&&&&\uparrow&&&\\
&&&&&&V_2 \otimes V_3&&&\\
&&&&&&\uparrow&&&\\
&&&&&&0&&&
\end{array}
\end{equation}
This then induces the following two long exact sequences in
cohomology,
\begin{equation}\label{seqavt1}
\begin{array}{ccccccccc}
0 & \rightarrow & H^0(\tilde{X}, \wedge^2 V_2) & \rightarrow & H^0(\tilde{X}, \wedge^2 \tilde{V}) & \rightarrow & H^0(\tilde{X}, Q) &
\rightarrow & \\
& \rightarrow & H^1(\tilde{X}, \wedge^2 V_2) & \rightarrow & \fbox{$H^1(\tilde{X}, \wedge^2 \tilde{V})$} &
\rightarrow & H^1(\tilde{X}, Q) & \rightarrow & \\
& \rightarrow & H^2(\tilde{X}, \wedge^2 V_2) & \rightarrow & H^2(\tilde{X}, \wedge^2 \tilde{V}) & \rightarrow & H^2(\tilde{X}, Q) &
\rightarrow & \\
& \rightarrow & H^3(\tilde{X}, \wedge^2 V_2) & \rightarrow & H^3(\tilde{X}, \wedge^2 \tilde{V}) & \rightarrow & H^3(\tilde{X}, Q) &
\rightarrow & 0 \ ,
\end{array}
\end{equation}
and
\begin{equation}\label{seqQ}
\begin{array}{ccccccccc}
0 & \rightarrow & H^0(\tilde{X}, V_2 \otimes V_3) & \rightarrow & H^0(\tilde{X}, Q) &
\rightarrow & H^0(\tilde{X}, \wedge^2 V_3) & \rightarrow & \\
& \rightarrow & H^1(\tilde{X}, V_2 \otimes V_3) & \rightarrow & H^1(\tilde{X}, Q) &
\rightarrow & H^1(\tilde{X}, \wedge^2 V_3) & \rightarrow & \\
& \rightarrow & H^2(\tilde{X}, V_2 \otimes V_3) & \rightarrow & H^2(\tilde{X}, Q) &
\rightarrow & H^2(\tilde{X}, \wedge^2 V_3) & \rightarrow & \\
& \rightarrow & H^3(\tilde{X}, V_2 \otimes V_3) & \rightarrow & H^3(\tilde{X}, Q) &
\rightarrow & H^3(\tilde{X}, \wedge^2 V_3) & \rightarrow & 0 \ .
\end{array}
\end{equation}
We have boxed $H^1(\tilde{X}, \wedge^2 \tilde{V})$ since it is the term we wish
to compute.
First consider the second sequence \eref{seqQ}.
By the same arguments as \eref{b-w2=0}, we have that
\begin{equation}\label{H03v23}
H^0(\tilde{X}, V_2 \otimes V_3) =
H^3(\tilde{X}, V_2 \otimes V_3) =
H^0(\tilde{X}, \wedge^2 V_3) =
H^3(\tilde{X}, \wedge^2 V_3) = 0 \ .
\end{equation}
It then follows from \eref{seqQ} that
\begin{equation}\label{h03Q0}
H^0(\tilde{X}, Q) = H^3(\tilde{X}, Q) = 0.
\end{equation}
Furthermore, using the Leray spectral sequence and the fact that
$\pi_* \wedge^2 V_3 = 0$ implies
\begin{equation}\label{H1av3=H0}
H^1(\tilde{X}, \wedge^2 V_3) \simeq H^0(B', R^1\pi_* \wedge^2 V_3).
\end{equation}
Now,
\begin{equation}\label{R1piwV3}
R^1\pi_* \wedge^2 V_3 = \beta^{'*}(R^1 \beta_* \wedge^2 W_3) \otimes L_3^{\otimes
2}.
\end{equation}
Therefore, pushing \eref{R1piwV3} down to $\mathbb{P}^1$, \eref{H1av3=H0}
becomes
\begin{equation}
H^0(B', R^1\pi_* \wedge^2 V_3) = H^0(\mathbb{P}^1, (R^1 \beta_* \wedge^2 W_3) \otimes
\beta'_* L_3^{\otimes 2}).
\end{equation}
Using \eref{L23}, we see that $L_3^{\otimes 2}$
has negative degree along a generic fiber. Therefore, assuming that
the support of $R^1\beta_* \wedge^2 W_3$ is on irreducible fibers,
$\beta'_* L_3^{\otimes 2}$ vanishes and
\begin{equation}\label{H1av3}
H^1(\tilde{X}, \wedge^2 V_3) = 0 \ .
\end{equation}
Substituting \eref{H03v23} and \eref{H1av3} into \eref{seqQ} implies
\begin{equation}\label{H1Q}
H^1(\tilde{X},Q) \simeq H^1(\tilde{X}, V_2\otimes V_3) \ ,
\end{equation}
and that $H^2(\tilde{X},Q)$ fits into the short exact sequence
\begin{equation}
0 \rightarrow H^2(\tilde{X}, V_2 \otimes V_3) \rightarrow H^2(\tilde{X}, Q)
\rightarrow H^2(\tilde{X}, \wedge^2 V_3) \rightarrow 0.
\end{equation}
Having established these results, let us now consider the first sequence
\eref{seqavt1}.
Substituting \eref{h03Q0} into \eref{seqavt1} gives
\begin{equation}
H^0(\tilde{X}, \wedge^2 \tilde{V}) \simeq H^0(\tilde{X}, \wedge^2 V_2) \ ,
\end{equation}
and
\begin{equation}\label{seqavt2}
0 \rightarrow H^1(\tilde{X},\wedge^2 V_2) \rightarrow \fbox{$H^1(\tilde{X}, \wedge^2 \tilde{V})$}
\rightarrow H^1(\tilde{X}, Q) \rightarrow H^2(\tilde{X},\wedge^2 V_2) \rightarrow
\ldots
\end{equation}
Putting \eref{H1Q} into \eref{seqavt2} then leads to the exact
sequence
\begin{equation}\label{seqavt}
0 \rightarrow H^1(\tilde{X},\wedge^2 V_2) \rightarrow
\fbox{$H^1(\tilde{X}, \wedge^2 \tilde{V})$} \rightarrow H^1(\tilde{X}, V_2\otimes V_3)
\stackrel{M^T}{\longrightarrow}
H^2(\tilde{X},\wedge^2 V_2) \rightarrow \ldots
\end{equation}
with which we will determine the desired boxed term.
In \eref{seqavt}, we have explicitly labeled a map $M^T$,
namely the coboundary map
\begin{equation}\label{defMt}
M^T : H^1(\tilde{X}, V_2 \otimes V_3) \rightarrow H^2(\tilde{X},\wedge^2 V_2) \ .
\end{equation}
It is given by cup product with
\begin{equation}\label{defMt2}
[\tilde{V}] \in H^1(\tilde{X}, V_3^* \otimes V_2) =
{\rm Ext}^1_{\tilde{X}}(V_3,V_2) \ ,
\end{equation}
the extension class of $\tilde{V}$, via the pairing
\begin{equation}
\begin{array}{cccccc}
{\mathcal M}^T: & H^1(\tilde{X}, V_2 \otimes V_3) &\times&
H^1(\tilde{X}, V_3^* \otimes V_2)
&\rightarrow& H^2(\tilde{X},\wedge^2 V_2) \\
& A & \times & B & \rightarrow & C \ .
\end{array}
\end{equation}
This can be dualized to
\begin{equation}\label{dualM}
\begin{array}{cccccc}
{\mathcal M}:&H^1(\tilde{X},\wedge^2 V_2^*) &\times&
H^1(\tilde{X}, V_3^* \otimes V_2)
&\rightarrow&
H^2(\tilde{X}, V_2^* \otimes V_3^*) \\
& C^* & \times & B & \rightarrow & A^*
\end{array} \ .
\end{equation}
In formulas \eref{H-av2*}, \eref{h1v2v3*} and \eref{H2-v2*v3*} we have
expressed the three cohomology groups in \eref{dualM} as $H^0$ on
$\mathbb{P}^1_t$ of appropriate sheaves.
The naturality of our construction implies that the multiplication map
${\mathcal M}$ on cohomologies is itself induced from the natural multiplication
map of the underlying sheaves on $\mathbb{P}^1_t$, namely:
\begin{equation}
\left(S_x^{4*} \otimes {\mathcal O}_{\IP^1_t}(1) \oplus y S_x^{6*} \otimes {\mathcal O}_{\IP^1_t}\right)
\otimes
\left({\mathcal F} \otimes [ S_x^5 \otimes {\mathcal O}_{\IP^1_t} \oplus
y S_x^3 \otimes {\mathcal O}_{\IP^1_t}(-1)]\right)
\rightarrow
{\mathcal F} \otimes y S_x^{1*}.
\end{equation}
By taking global sections, we find that ${\mathcal M}$ is the product:
\begin{equation}\label{M-sec}
{\mathcal M}:
\left(S_x^{4*} \otimes S^1_t \oplus y S_x^{6*}\right)
\otimes
\left(\bigoplus_{j=1}^{15} H^0(\mathbb{P}^1_t, {\mathcal F}_j) \otimes
[S_x^5 \oplus y S_x^3 \otimes \{\hat{p}_j\mathbb{C}\}]\right)
\rightarrow
\bigoplus_{j=1}^{15} H^0(\mathbb{P}^1_t, {\mathcal F}_j) \otimes y S_x^{1*}.
\end{equation}
In particular, our ${\mathcal M}$ breaks into blocks.
The three spaces involved in ${\mathcal M}$ have dimensions 17, 150 and 30
respectively.
This breaks into 15 blocks $(j=1,\ldots,15)$, each sending a $17
\times 10$ dimensional space to a 2-dimensional space.
Each block breaks further into a $10 \times 4 \rightarrow 2$ sub-block and a $7
\times 6 \rightarrow 2$ sub-block, corresponding to the products
\begin{equation}
(S_x^{4*} \otimes S_t^1)
\otimes
(S_x^{3} \otimes \{\hat{p}_j\mathbb{C}\})
\rightarrow
S_x^{1*}
\end{equation}
and
\begin{equation}
(S_x^{6*}) \otimes (S_x^{5}) \rightarrow S_x^{1*},
\end{equation}
respectively. (We have suppressed a $y H^0({\mathcal F}_j)$ factor on each side).
The transpose $M : C^* \rightarrow A^*$ of our map $M^T$ is obtained from
\eref{M-sec} by evaluating at the extension class $[\tilde{V}] \in B$.
We can express this $[\tilde{V}]$ in terms of its coefficients
$a_{i,j}$, $i=0,\ldots,5$, $j=1,\ldots,15$ and $b_{k,j}$,
$k=0,\ldots,3$, $j=1,\ldots,15$, in the $S_x^5$ and $S_x^3$ factors
respectively.
Now the map $S_x^{6*} \rightarrow S_x^{1*}$ given by the $a_{i,j}$ is
represented by the $2 \times 6$ matrix
\begin{equation}
M_{I, j} =
\left(
\begin{array}{ccc}
a_{0,j} & \ldots~~a_{5,j} & 0 \\
0 & a_{0,j}~~\ldots & a_{5,j}
\end{array}
\right),
\end{equation}
while the map $S_x^{4*} \otimes S_t^1 \rightarrow S_x^{1*}$ given by the
$b_{k,j}$ is represented by the $2\times 10$ matrix
\begin{equation}
M_{II,j} = \left(\begin{array}{c|c}
\begin{array}{ccc}
b_{0,j}t_0(\hat{p}_j) & \ldots ~~ b_{3,j}t_0(\hat{p}_j)& 0 \\
0 & b_{0,j}t_0(\hat{p}_j) ~~\ldots & b_{3,j}t_0(\hat{p}_j)
\end{array}
&
\begin{array}{ccc}
b_{0,j}t_1(\hat{p}_j) & \ldots ~~b_{3,j}t_1(\hat{p}_j) & 0 \\
0 & b_{0,j}t_1(\hat{p}_j) ~~\ldots & b_{3,j}t_1(\hat{p}_j)
\end{array}
\end{array}\right).
\end{equation}
So the full $30 \times 17$ matrix $M$ is then
\begin{equation}\label{M}
M = \left(\begin{array}{ccc}
M_{I,1} & & M_{II,1} \\
\vdots & & \vdots \\
M_{I,15} & & M_{II,15}
\end{array}\right).
\end{equation}
For a generic choice of the $a_{i,j}$ and $b_{k,j}$, the rank of $M$
is 17 and $M$ is surjective.
It is easy to see that this remains true also for generic
$\tau$-invariant extension $[\tilde{V}]$.
Plugging this, along with formulas \eref{H-av2*} and \eref{h2v2*v3*},
into \eref{seqavt}, we find:
\begin{equation}\label{avt=18}
h^1(\tilde{X}, \wedge^2 \tilde{V}) = 5+30 - 17 = 18.
\end{equation}
Using Serre duality on $\tilde{X}$ and the fact that $\mbox{ind}(\tilde
V)=\mbox{ind}(\wedge^2 \tilde{V})=6$ \cite{spec},
it is now straightforward to determine the
remaining cohomology groups of $\tilde{V}$, $\tilde{V}^*$, $\wedge^2
\tilde V$ and $\wedge^2 \tilde V^*$.
\section{The $\mathbb{Z}_2$ Action}\label{s:z2}
In subsection \ref{s:tau} we described the involutions $\tau_B$,
$\tau_{B'}$, $\tau$ acting compatibly on $B$, $B'$ and $\tilde{X}$.
The action of $\tau_{B'}$ on line bundles on $B'$ is specified in
\tref{tab:tB}.
In particular, the line bundles ${\mathcal O}_{B'}(0,1)$ and ${\mathcal O}_{B'}(1,0)$ are
$\tau$-invariant.
It follows that there are induced involutions $\tau_{\mathbb{P}^1_t}$,
$\tau_{\mathbb{P}^1_x}$ that commute with the corresponding maps $\beta':B' \rightarrow
\mathbb{P}^1_t$, $\chi : B' \rightarrow \mathbb{P}^1_x$.
We have already encountered the involution $\tau_{\mathbb{P}^1_t}$ in
subsection \ref{s:B}, where we denoted it simply $\tau_{\mathbb{P}^1}$.
It sends $t_0 \mapsto t_0, \quad t_1 \mapsto -t_1$.
We claim that $\tau_{\mathbb{P}^1_x}$ is also a non-trivial involution, so
with an appropriate choice of the coordinates $x_0$, $x_1$ on
$\mathbb{P}^1_x$ (note that we never fixed these coordinates up till now!) it
acts as $x_0 \mapsto x_0, \quad x_1 \mapsto -x_1$.
For this, we must determine the action of $\tau$
on the $\mathbb{P}^1$ family of rational curves $r'$.
For a general, non-singular member of this family, all we learn from
\tref{tab:tB} is that it goes to another such.
But the table also tells us the image under $\tau_{B'}$
of each of the line
bundles ${\mathcal O}_{B'}(K_i^d)$, as $K_i^d$ runs over the 16 components of the 8
reducible curves in the system $|r'|$, specified in \eref{Kij}.
Each of these has the property that $K_i^d$ is the only effective
curve in its class: $h^0(B', K_i^d) = 1$.
So we can deduce from \tref{tab:tB}
not only the cohomology class of the
image, but the actual physical image:
\begin{equation}
K_2^d \leftrightarrow K_3^d, \quad
K_1^d \leftrightarrow K_9^d, \quad
K_4^d \leftrightarrow K_7^{3-d}, \quad
K_6^d \leftrightarrow K_8^{3-d}.
\end{equation}
At any rate, this clearly demonstrates that $\tau_{\mathbb{P}^1_x}$ is not
the identity, as claimed.
Via the map $\Delta$, our surface $B'$ is a double cover of ${\sf Q} =
\mathbb{P}^1_t \times \mathbb{P}^1_x$.
Its equation can be written explicitly as
\begin{equation}
y^2 = F_{4,2}(x,t),
\end{equation}
with $F_{4,2}(x,t)$ a bihomogeneous polynomial, of degree 4 in
$x_0,x_1$ and of degree 2 in $t_0, t_1$.
By \eref{def-y}, $y$ is a section of ${\mathcal O}_{B'}(2,1)$ which vanishes on the
ramification locus $Ram_\Delta$.
Since $Ram_\Delta$ goes to itself under $\tau_{B'}$, $y$ must go to a
multiple of itself.
Since $\tau_{B'}$ is an involution, this multiple is $\pm 1$, so in
particular $F_{4,2}$ must be invariant (rather than anti-invariant).
From \eref{def-y}, it follows that either $\tau_{B'} y = y$ or
$\iota \tau_{B'} y = y$.
Both involutions $\tau_{B'}$, $\iota \tau_{B'}$ have the same properties.
So by relabelling $\iota \tau_{B'}$ as $\tau_{B'}$
if necessary, we may as well
assume that the action of $\tau_{B'}$ is given explicitly by:
\begin{equation}\label{tx-trans}
t_0 \mapsto t_0, \quad t_1 \mapsto -t_1, \quad
x_0 \mapsto x_0, \quad x_1 \mapsto -x_1, \quad
y \mapsto y.
\end{equation}
In subsection \ref{s:V} we chose compatible actions of $\tau$ on
$V_2$, $V_3$ and $\tilde{V}$.
It turns out that the particle spectrum on $X$ is independent of these
choices and is precisely half the spectrum on $\tilde{X}$ which we computed
above.
We compute it as follows.
\paragraph{$\bullet$ \fbox{$H^1(\tilde{X}, \tilde{V})$}}
We have identified $H^1(\tilde{X}, \tilde{V})$ with $H^0(f_\infty, G^*) \otimes
H^0(f_0',L_2)$ in \eref{H1V2-1}, \eref{H1tv}.
We plug $k=3$, $l=0$ into formula \eref{deltak,l}, and
restrict the double cover $\Delta : B' \rightarrow {\sf Q} $ to $\chi: f_0' \rightarrow
\mathbb{P}^1_x$, finding:
\begin{equation}
\chi_* {\mathcal O}_{f_0'}(3r') = {\mathcal O}_{\IP^1_x}(3) \oplus y {\mathcal O}_{\IP^1_x}(1).
\end{equation}
We get a natural identification of $H^0(f_0', L_2) = H^0(f_0', 3r')$
with $S_x^3 \oplus y S_x^1$.
From \eref{tx-trans} we see that the $\tau$ action on this
6-dimensional space has a 3-dimensional invariant subspace and
3-dimensional anti-invariant subspace.
There is also a $\tau$-action on the 1-dimensional $H^0(f_\infty,
G^*)$, which must be either invariant or anti-invariant.
Either way, we find:
\begin{equation}\label{res-tv}
h^1(\tilde{X}, \tilde{V})_+ = 3, \quad
h^1(\tilde{X}, \tilde{V})_- = 3.
\end{equation}
\paragraph{$\bullet$ \fbox{$H^1(\tilde{X}, \wedge^2 \tilde{V})$}}
From the identification of $H^1(\tilde{X}, \wedge^2 V_2)$ with $y S_x^4$ in
\eref{h1-av2}, we see that
\begin{equation}
h^1(\tilde{X}, \wedge^2 V_2)_+ = 3, \quad
h^1(\tilde{X}, \wedge^2 V_2)_- = 2,
\end{equation}
while the identification of $H^2(\tilde{X},\wedge^2 V_2)$ with
$S_x^6 \oplus y S_x^4 \otimes (S_t^1)^*$ gives
\begin{equation}
h^2(\tilde{X}, \wedge^2 V_2)_+ = 4+5 = 9, \quad
h^2(\tilde{X}, \wedge^2 V_2)_- = 3+5 = 8.
\end{equation}
On the other hand, we saw in \eref{H2-v2*v3*} that $H^1(\tilde{X}, V_2 \otimes V_3)$ is
dual to $\bigoplus_{j=1}^{15} H^0(\mathbb{P}^1_t, {\mathcal F}_j) \otimes
(y S_x^{1*})$.
Again, the action of $\tau$ on the 2-dimensional space $y S_x^{1*}$
has 1-dimensional invariants and 1-dimensional anti-invariants, so
regardless of its action on the 15 1-dimensional spaces $H^0(\mathbb{P}^1_t,
{\mathcal F}_j)$, we get:
\begin{equation}
h^1(\tilde{X}, V_2 \otimes V_3)_+ = 15, \quad
h^1(\tilde{X}, V_2 \otimes V_3)_- = 15.
\end{equation}
Combining the last three formulae with \eref{seqavt} and recalling
that $M^T$ is $\tau$-equivariant (since it is cup product with the
class $[\tilde{V}]$, which was taken in subsection \ref{s:V} to be
$\tau$-invariant), we see that for those generic choices to which
\eref{avt=18} applies we have:
\begin{equation}\label{res-avt}
h^1(\tilde{X}, \wedge^2 \tilde{V})_+ = 3 + 15 - 9 = 9, \quad
h^1(\tilde{X}, \wedge^2 \tilde{V})_- = 2 + 15 - 8 = 9.
\end{equation}
\paragraph{$\bullet$ \fbox{$H^1(\tilde{X},\tilde{V}^*)$ and $H^1(\tilde{X},\wedge^2 \tilde{V}^*)$}}
The spectrum also requires the terms $H^1(\tilde{X}, \tilde{V}^*)$ and
$H^1(\tilde{X}, \wedge^2 \tilde{V}^*)$.
These can be obtained from the three-family condition (C3) in
\eref{cond}, in conjunction with the index theorem \eref{donagi4},
as well as Serre duality \eref{Serre} presented in the Appendix.
Together with the fact that $H^0(\tilde{X}, \tilde{V})$, $H^0(\tilde{X}, \tilde{V}^*) =
H^3(\tilde{X}, \tilde{V})^*$, $H^0(\tilde{X}, \wedge^2 \tilde{V})$, and $H^0(\tilde{X}, \wedge^2 \tilde{V}^*) = H^3(\tilde{X},
\wedge^2 \tilde{V})^*$ all vanish, we have that
\begin{equation}
-h^1(\tilde{X}, U_i(\tilde{V})) + h^1(\tilde{X}, U_i(\tilde{V^*}))
= 6, \qquad
U_i(\tilde{V}) = \tilde{V},~\wedge^2 \tilde{V} \ .
\end{equation}
In fact, a $\mathbb{Z}_2$-graded version of the index theorem implies the
stronger result that
\begin{equation}\label{z2index}
-h^1(\tilde{X}, U_i(\tilde{V}))_{(\pm)} + h^1(\tilde{X}, U_i(\tilde{V^*}))_{(\pm)}
= 3, \qquad
U_i(\tilde{V}) = \tilde{V},~\wedge^2 \tilde{V} \ .
\end{equation}
Alternatively, we can think of it as
the index theorem applied to each of
the $\tau$-invariant and anti-invariant pieces of the cohomology.
Therefore, combining \eref{z2index} with \eref{res-tv}, we have that
\begin{equation}
h^1(\tilde{X}, \tilde{V}^*)_+ = 6, \quad
h^1(\tilde{X}, \tilde{V}^*)_- = 6.
\end{equation}
Similarly, combining \eref{z2index} with \eref{res-avt}, we have that
\begin{equation}
h^1(\tilde{X}, \wedge^2 \tilde{V})_+ = 12, \quad
h^1(\tilde{X}, \wedge^2 \tilde{V})_- = 12.
\end{equation}
Let us summarize the conclusions of the last two sections.
It is convenient to introduce the following notation.
Consider, for example, the cohomology group $H^1(\tilde{X}, \tilde{V})$.
We showed in Section \ref{s:coh} and Section \ref{s:z2} that
$h^1(\tilde{X},\tilde{V})=6$ and $h^1(\tilde{X},\tilde{V})_{(+)}=h^1(\tilde{X},\tilde{V})_{(-)}=3$
respectively.
Henceforth, we will express both of these facts by writing
\begin{equation}
H^1(\tilde{X},\tilde{V}) = \mathbb{C}^3_{(+)} \oplus \mathbb{C}^3_{(-)}.
\end{equation}
Using this notation, we encapsulate the results of Section \ref{s:coh}
and Section \ref{s:z2} in \tref{t:sum}.
\begin{table}[h]
\[
\begin{array}{||c|c|c|c|c|c||}\hline\hline
U_i & H^q(\tilde{X}, U_i(\tilde{V})) & R_i & h^q(\tilde{X}, U_i(\tilde{V}))
& A_j & \mathbb{C}_{(+)}^{r} \oplus \mathbb{C}_{(-)}^s \\ \hline \hline
1 & H^0(\tilde{X}, {\mathcal O}_{\tilde{X}}) & 24 & 1 & 0 &
\mathbb{C}_{(+)}^{1}
\\ \hline
10 & H^1(\tilde{X}, \wedge^2 \tilde{V}) & 5 & 18 & 0 & \mathbb{C}_{(+)}^{9}
\\ \hline
& & & & 1 & \mathbb{C}_{(-)}^{9}
\\ \hline
\overline{10} & H^1(\tilde{X}, \wedge^2 \tilde{V}^*) & \overline 5 & 24 &
0 & \mathbb{C}_{(+)}^{12}
\\ \hline
& & & & 1 & \mathbb{C}_{(-)}^{12}
\\ \hline
5 & H^1(\tilde{X}, \tilde{V}) & \overline{10} & 6 & 0 & \mathbb{C}_{(+)}^{3}
\\ \hline
& & & & 1 & \mathbb{C}_{(-)}^{3}
\\ \hline
\overline 5 & H^1(\tilde{X}, \tilde{V}^*) & 10 & 12 & 0 & \mathbb{C}_{(+)}^{6}
\\ \hline
& & & & 1 & \mathbb{C}_{(-)}^{6}
\\ \hline \hline
\end{array}
\]
\caption{The dimensions and $\mathbb{Z}_2$ actions on the cohomology spaces
$H^q(\tilde{X}, U_i(\tilde{V}))$.
}\label{t:sum}
\end{table}
\section{Low Energy Spectrum}
We know from the discussion in Section 2, and specifically from
equation \eref{spec}, that the multiplicities of the representations
$B_{ij}$ of the low energy
gauge group are determined by
$(H^q(\tilde{X}, U_i(\tilde{V})) \otimes A_j)^{\rho'(F)}$, the
invariant part of $H^q(\tilde{X}, U_i(\tilde{V})) \otimes A_j$ under
the joint action of $\mathbb{Z}_2$ on $H^q(\tilde{X}, U_i(\tilde{V}))$ and
$A_j$.
By combining the results in \tref{tab:eg2} with the $\mathbb{Z}_2$ action on
the cohomology groups listed in \tref{t:sum},
we can now compute the complete low energy spectrum of
our theory.
The associated multiplets descend to $X = \tilde{X} / \mathbb{Z}_2$ to form the
$(SU(3)_C \times SU(2)_L \times U(1)_Y)/\IZ_6$ particle physics spectrum.
The results are listed in \tref{t:final}. The representation $R_i =
1$, corresponding to the moduli $H^0(\tilde{X}, {\mathop {\rm ad}} \tilde{V})$, is not presented.
\begin{table}[h]
\[
\begin{array}{||c|c|c|c||}\hline\hline
R_i & A_j &
(H^q(\tilde{X}, U_i(\tilde{V})) \otimes A_j)^{\rho'(F)} & B_{ij}
\\ \hline \hline
24 & 0 & \mathbb{C}_{(+)}^{1} &
(8,1)_0 \oplus (1,3)_0 \oplus (1,1)_0
\\ \hline
5 & 0 & \mathbb{C}_{(+)}^{9} & (3,1)_{-2}
\\ \hline
& 1 & \mathbb{C}_{(-)}^{9} & (1,2)_{3}
\\ \hline
\overline 5 & 0 & \mathbb{C}_{(+)}^{12} & (\overline 3,1)_{2}
\\ \hline
& 1 & \mathbb{C}_{(-)}^{12} & (1, 2)_{-3}
\\ \hline
\overline{10} & 0 & \mathbb{C}_{(+)}^{3} & (3,1)_{4} \oplus (1,1)_{-6}
\\ \hline
& 1 & \mathbb{C}_{(-)}^{3} & (\overline 3, 2)_{-1}
\\ \hline
10 & 0 & \mathbb{C}_{(+)}^{6} & (\overline 3,1)_{-4} \oplus (1,1)_{6}
\\ \hline
& 1 & \mathbb{C}_{(-)}^{6} & (3, 2)_{1}
\\ \hline
\hline
\end{array}
\]
\caption{The particle spectrum of the low-energy $(SU(3)_C \times SU(2)_L \times U(1)_Y)/\IZ_6$ theory.
The $A_j$ correspond to
characters of the $\mathbb{Z}_2$ representations on $R_i$.
The $U(1)$ charges listed are $w=3Y$.
}
\label{t:final}
\end{table}
To begin with, the spectrum contains one copy of vector supermultiplets
transforming under $(SU(3)_C \times SU(2)_L \times U(1)_Y)/\IZ_6$ as
\begin{equation}
(8,1)_0 \oplus (1,3)_0 \oplus (1,1)_0.
\end{equation}
Contained in these multiplets are the gauge connections for $SU(3)_C$,
$SU(2)_L$ and $U(1)_Y$ respectively.
Furthermore, it contains three families of quarks and lepton
superfields, each family transforming as
\begin{equation}\label{ql}
(3,2)_{1}, \quad (\overline{3},1)_{-4}, \quad (\overline{3},1)_{2}
\end{equation}
and
\begin{equation}
(1,2)_{-3}, \quad (1,1)_{6}
\end{equation}
respectively.
Each of these multiplets is a chiral superfield, none of which has a
conjugate partner.
However, there are additional chiral superfields in the spectrum.
It follows from \tref{t:final} that these occur as conjugate pairs
of the $(SU(3)_C \times SU(2)_L \times U(1)_Y)/\IZ_6$ representations
\begin{equation}\label{5}
(3,1)_{-2}, \quad (1,2)_{3}
\end{equation}
and
\begin{equation}\label{10bar}
(3,1)_{4} \oplus (1,1)_{-6}, \quad (\overline{3},2)_{-1}.
\end{equation}
These multiplets represent extra matter in the spectrum, such as Higgs
and other exotic fields.
Let us explain how the quark/lepton fermions and conjugate pairs arise.
Consider, for example, the $B_{ij}$ representations $(\overline{3},2)_{-1}$
and $(3,2)_{1}$, corresponding to the $\overline{10}$ and 10 representations
respectively.
From \tref{t:final}, we see that there are 3 copies of
$(\overline{3},2)_{-1}$ and 6 copies of $(3,2)_{1}$.
Note that $6-3=3$
copies of $(3,2)_{1}$ are unpaired, as a consequence of
the index theorem.
Each unpaired $(3,2)_{1}$ multiplet contributes to a single
quark/lepton generation, as in \eref{ql}.
This leaves 3 conjugate pairs of $(\overline{3},2)_{-1}$ and $(3,2)_{1}$
superfields.
Being non-chiral pairs, these do not contribute to a quark/lepton
family but, rather, are additional supermultiplets
as listed in \eref{5} and \eref{10bar}.
It remains to enumerate the number of additional superfields.
From \tref{t:final}, we see that the spectrum has
\begin{equation}
n_{(3,1)_{-2}} = 9, \quad
n_{(1,2)_{3}} = 9
\end{equation}
and
\begin{equation}
n_{(3,1)_{4} \oplus (1,1)_{-6}} = 3, \quad
n_{(\overline 3,2)_{-1}} = 3
\end{equation}
copies of \eref{5} and \eref{10bar} respectively.
The multiplicity $n_{(1,2)_{3}}$ corresponds to the number of Higgs
doublet conjugate pairs in the low energy spectrum.
The remaining multiplets in \eref{5} and \eref{10bar} are exotic.
We conclude that the low energy spectrum of the simple,
representative model discussed in this paper includes the requisite
three chiral
families of quarks and leptons. Additionally, it naturally has
Higgs doublet supermultiplet pairs.
Unfortunately, the spectrum contains extra, exotic
chiral supermultiplets which, potentially, are phenomenologically
unacceptable.
However, these conjugate pairs of exotic multiplets may couple to the
moduli fields coming from $H^1(X, V \otimes V^*)$ to form mass terms.
If the moduli can
acquire a sufficiently high vacuum expectation value, then the exotics
multiplets will decouple at low energy and be compatible with
phenomenology.
These couplings will be discussed
elsewhere.
Armed with the technology developed in this paper, one can
now compute the spectra of standard-like models based on arbitrary
stable vector bundles on a wide range of elliptically fibered
Calabi-Yau threefolds. These spectra can be constrained to always
contain three families of quarks and leptons. We are presently
searching for such vacua with a phenomenologically acceptable number
of Higgs doublets and, hopefully, no exotic matter.
\paragraph{Acknowledgements}
We are grateful to Volker Braun and Tony Pantev for enlightening
discussions.
R.~D.~would like to acknowledge conversations with Jacques Distler.
This Research was supported in part by
the Department of Physics and the Maths/Physics Research Group
at the University of Pennsylvania
under cooperative research agreement DE-FG02-95ER40893
with the U.~S.~Department of Energy and an NSF Focused Research Grant
DMS0139799 for ``The Geometry of Superstrings.'' R.~D.~is partially
supported by an NSF grant DMS 0104354.
R.~R.~is also supported
by the Department of Physics and Astronomy of Rutgers University under
grant DOE-DE-FG02-96ER40959.
|
{
"timestamp": "2005-12-05T16:15:46",
"yymm": "0411",
"arxiv_id": "hep-th/0411156",
"language": "en",
"url": "https://arxiv.org/abs/hep-th/0411156"
}
|
\section{Introduction}
Blazars are a peculiar class of active galactic nuclei, consisting
of optically violently variable (OVV), gamma-ray loud quasars and
BL Lac objects. They have been observed at all wavelengths, from
radio through very-high energy (VHE) $\gamma$-rays. Six
blazars (Mrk~421: \cite{punch92}; Mrk~501: \cite{quinn96};
PKS 2155-314: \cite{chadwick99}; 1ES~2344+514: \cite{catanese98};
1H~1426+428: \cite{horan02}; 1ES~1959+650: \cite{kajino99,holder03})
have now been detected at VHE $\gamma$-rays ($> 350$~GeV) by
ground-based air \v Cerenkov telescopes. Blazars exhibit variability
at all wavelengths on various time scales. Radio interferometry
often reveals one-sided kpc-scale jets with apparent superluminal
motion.
The high inferred isotropic luminosities, short variability
time scales, and superluminal motion provide conclusive evidence
that
blazars are sources of relativistic jets pointing at a small
angle with respect to our line of sight.
One of the key unresolved questions in the field of blazar research
to date is the nature of relativistic particles in blazar jets.
In the framework of relativistic jet models, the low-frequency
(radio
-- optical/UV) emission from blazars is interpreted as
synchrotron
emission from nonthermal relativistic electrons in the
jet. The
high-frequency (X-ray -- $\gamma$-ray) emission could either
be produced via Compton upscattering of low frequency radiation by the
same electrons responsible for the synchrotron emission \citep[for
a recent review see, e.g.,][]{boettcher02}, or due to hadronic processes
initiated by relativistic protons co-accelerated with the electrons
\citep[for a recent discussion see, e.g.][]{muecke03}. The lack of
knowledge of the primary jet launching mechanism and the difficulty
in constraining the jet composition from general energetics
considerations
currently leave both leptonic and hadronic models
open as viable possibilities. In many cases, both types of models
can provide acceptable fits to the observed broadband spectral energy
distributions (SEDs) of BL~Lac objects, in particular the TeV blazars
(see, e.g., \cite{mk97,pian98,petry00,kraw02} for leptonic and
\cite{muecke03} for hadronic models).
In the framework of leptonic jet models, TeV blazars are successfully
modelled by SSC models in which the high-energy emission is produced
by Compton scattering of electron-synchrotron radiation off the same
ultrarelativistic electrons producing the synchrotron emission
\citep[e.g.][]{mk97,pian98,petry00,kraw02}. Such models have been
successful in modeling not only the SEDs, but also the detailed
spectral variability, including spectral hysteresis at X-ray energies, of
several TeV blazars \citep[e.g.,][]{krm98,gm98,kataoka00,kusunose00,lk00}.
An inevitable prediction of the SSC model is that any flaring activity
at TeV energies should be accompanied by a quasi-simultaneous flare
in the synchrotron component. Even if the synchrotron flare does not
necessarily have to be very pronounced at X-ray energies, since the
TeV photons might be produced by Compton upscattering of seed photons
that are observed predominantly in the radio -- optical regime, there
should always be a significant imprint of the TeV flare in the optical
and X-ray light curves.
This prediction is in striking contrast to the recent observation of
\cite{kraw04} of an ``orphan'' TeV flare seen in the Whipple light
curve of the TeV blazar 1ES~1959+650 during a multiwavelength campaign
in the late spring and summer of 2002. The object displayed first a
quasi-simultaneous TeV and X-ray (RXTE) flare, followed by a well
sampled, smooth decline of the X-ray flux over the following
$\sim 1$~month. However, during this smooth decline, a second
TeV flare, $\sim 20$~days after the initial one, was observed,
which was only accompanied by very moderate $\lesssim 0.1^{\rm mag}$
flaring activity in the R and V bands. This behavior is clearly
unexpected in a purely leptonic SSC blazar jet model.
In light of their great success to model both the broadband SEDs
and spectral variability of TeV blazars in great detail, leptonic
models might still be a very reasonable starting point for further
investigations of this peculiar flaring behavior of 1ES~1959+650.
However, even if one assumes that the high-energy emission is
usually dominated by leptonic processes in blazar jets in general
and in 1ES~1959+650 in particular, one would naturally expect
that the emitting plasma in blazar jets is not a pure $e^+ e^-$
pair plasma, but contains a non-negligible admixture of protons.
For example, based on X-ray luminosity constraints from observations,
\cite{sm00} find that even if $e^+ e^-$ pairs outnumber protons
by a large margin (factor of $\sim 50$), blazar jets might still
be dynamically dominated by their baryon content. Similar conclusions
have been reached by \cite{kt04}, ruling out a pure electron-proton
plasma in energy equilibrium between electrons and protons or a
pure electron-positron pair plasma. These conclusions are also
supported by energy requirements in large-scale extragalactic
X-ray jets observed by {\it Chandra} which seem to remain
relativistic out to kpc and even Mpc distances from the central
engine \citep[see, e.g.][]{gc01,sambruna03}).
Detailed simulations of particle acceleration at relativistic
shocks or shear layers show that a wide variety of particle
spectra may result in such scenarios \citep[e.g.,][]{ob02,so03,ed04},
greatly differing from the standard spectral
index of 2.2 -- 2.3
previously believed to be a universal
value in relativistic shock
acceleration \citep[e.g.,][]{gallant99,achterberg01}. Thus, both
the nature of the matter in blazar jets and the energy spectra of
ultrarelativistic particles injected into the emission regions in
blazar jets are difficult to constrain from first principles.
Consequently, also their kinetic luminosity is hard to constrain.
However, if Fermi acceleration plays a major role in the energization
of electrons (pairs) in leptonic jets, then one would naturally expect
that also protons are accelerated to relativistic energies, though
conceivably not exceeding the energy threshold to boost the bulk of
the available soft photons up to the energy of the $\Delta$ resonance at
1232~MeV in the proton's rest frame to initiate pion production processes.
While the size-scale constraint would allow the acceleration of protons
up to Lorentz factors of $\gamma_{\rm p, max} \sim 3 \times 10^8 \,
B_{-1} \, R_{16}$ (where $B = 0.1 \, B_{-1}$~G is the co-moving
magnetic field and $R = 10^{16} \, R_{16}$~cm is the size of the
emitting region), factors related to, e.g., the incomplete development
of plasma wave turbulences and superluminal magnetic-field configurations
at oblique shocks \citep{ob02,ed04} may severely limit the maximum energies
of protons by several orders of magnitude.
It has previously been suggested \citep[e.g.,][]{ad03} that the presence
of external photon fields may substantially lower the effective proton
energy
threshold for $p\gamma$ pion production compared to the
standard hadronic-jet scenario based on synchrotron target photons.
They have also pointed out that the conversion of protons to neutrons
via charged pion production ($p\gamma \to n\pi^+$) may facilitate
the transport of kinetic energy in baryons out to kpc scales. For
$\gamma_{\rm p, max} = 10^4 \, \gamma_4$, photon energies of ${E'}_{\rm ph}
\sim E_{\Delta} / {\gamma'}_{\rm p,max} \sim 30 \, \gamma_4^{-1}$~keV
in the co-moving frame of the emission region would be required in
order to initiate $p\gamma$ processes. Such photon energies are unlikely
to be achieved by intrinsic (electron synchrotron) photons, but they
may occasionally be provided by external photon sources due to the
Doppler blue shift into the emission region. For example, quasi-isotropic
radiation fields from re-processed accretion-disk photons \citep{sbr94,dss97}
or reflected jet synchrotron emission \citep{gm96,bd98} are good candidates
for external soft photon sources to occasionally exceed the $p\gamma$ pion
production threshold for relativistic protons of ${\gamma'}_p \sim 10^3$
-- $10^4$.
This paper presents a discussion of the idea that the ``orphan'' TeV
flare in 1ES~1959+650 resulted from $\pi^0$ decay following $\gamma$p
pion production on an external photon field dominated by photons from
the first, simultaneous synchrotron + TeV
flare.
In \S \ref{model}, the basic model geometry and parameter choices,
guided by the observations of 1ES~1959+650, are outlined. Analytic
estimates constraining model parameters, in particular the hadron
number density and energy content in the jet are presented in
\S \ref{results}. \S \ref{summary} contains a summary and brief
discussion.
\section{\label{model}Model setup and parameter estimates}
The basic model geometry is sketched in Fig. \ref{geometry}. A blob filled
with ultrarelativistic electrons and relativistic protons is traveling along
the relativistic jet, defining the positive $z$ axis. Particles are accelerated
very close to the central engine (F1) in an explosive event which is producing
the initial synchrotron + TeV flare via the leptonic SSC mechanism. Synchrotron
emission from this flare is reflected off a gas cloud (the mirror M) located
at a distance $R_m$ from the central engine. The cloud has a reprocessing
optical depth $\tau_m = 10^{-1} \, \tau_{-1}$ and a radius $R_c = 10^{17} \,
R_{c,17}$~cm, implying an average density of $n_c = 10^6
\, n_6$~cm$^{-3}$
with $n_6 \sim 1.5$.
The characteristic synchrotron photon energy during the primary
flare of 1ES~1959+650 was ${E'}_{\rm sy}
\sim 1 \, \Gamma_1^{-1}
\, E_{\rm sy, 1}$~keV in the co-moving frame, implying a
characteristic
photon energy of the reflected synchrotron radiation
of ${E'}_{\rm Rsy}
\sim 100 \, \Gamma_1 \, E_{\rm sy, 1}$~keV, where
$\Gamma = 10 \, \Gamma_1$ is the bulk
Lorentz factor of the emission
region, and it is assumed that the
Doppler boosting factor $D \approx
\Gamma$. Here, the observed peak of the synchrotron spectrum has been
parametrized as $E_{\rm sy} = 10 \, E_{\rm sy,1}$~keV. Relativistic
electrons with
${\gamma'}_e \gtrsim 10$ will be very inefficient in
Compton upscattering this radiation field due to the rapid decline of
the Klein-Nishina cross section. Here and in the remainder of this paper,
quantities in the frame of the emission region (``blob'') are denoted
by primed symbols, while unprimed symbols refer to quantities in the
stationary system of the AGN. Considering VHE photon production from
the decay of neutral pions with co-moving Lorentz factors
${\gamma'}_{\pi^0}$, we may assume that ${\gamma'}_{\pi^0} \approx
{\gamma'}_{\Delta} \approx {\gamma'}_p$. The observable spectrum of
$\pi^0$ decay photons will then extend out to $E_{\pi^0 \to 2 \gamma}
\sim 7 \, \gamma_4 \, \Gamma_1$~TeV.
The observed time delay between the primary synchrotron flare and
the
secondary flare due to interactions of the blob with the first
reflected
synchrotron flare photons to arrive back at the blob was
$\Delta t_{\rm obs} = 20 \, \Delta t_{20}$~days, and is related to
the distance of the reflector by
\begin{equation}
\Delta t_{\rm obs} \approx {R_m \over 2 \, \Gamma^2 c}.
\label{Delta_t}
\end{equation}
Thus, $R_m \approx 3 \Gamma_1^2 \, \Delta t_{20}$~pc. A cloud of
reflecting
gas with the characteristics specified above, at this
distance from a central source of the ionizing continuum radiation
from a central accretion disk with luminosity $L_D = 10^{44} \,
L_{44}$~ergs~s$^{-1}$ will remain largely neutral (ionization
parameter $\xi = L_D / (4 \pi R_m^2 n_c) \sim 8 \times 10^{-2} L_{44} \,
(R_m / 3 {\rm pc})^{-2} \, n_6^{-1}$). Its optical emission line luminosity
will be limited by $L_{\rm line} < L_D \, (R_c/R_m)^2 = 10^{40} \,
L_{44} \, R_ {c,17}^2 \, (R_m/3 \, {\rm pc})^{-2}$~ergs~s$^{-1}$,
corresponding to a line flux of $F_{\rm line} < 2 \times 10^{-15} \,
L_{44} \, R_ {c,17}^2 \, (R_m/3 \, {\rm pc})^{-2}$~ergs~cm$^{-2}$~s$^{-1}$
which is negligible compared to the jet synchrotron continuum, consistent
with the classification of 1ES~1959+650 as a BL~Lac object.
The duration of the flare, $w_{\rm fl}^{\rm obs}$ will then be
determined by the time it takes for the blob to travel from the
location $z_0$ of the onset of the secondary flare to the mirror
at $R_m$:
\begin{equation}
w_{\rm fl}^{\rm obs} = {(R_m - z_0) \, (1 - \beta) \over
\beta \, c} \approx {R_m / 8 \, \Gamma^4 \, c} \approx
1.2 \, \Gamma_1^{-2} \; {\rm hr},
\label{flare_duration}
\end{equation}
where $\beta \, c = \sqrt{1 - 1/\Gamma^2} \, c$ is the speed of
the
blob which is assumed to remain constant throughout the period
considered
here. From the observed $\nu F_{\nu}$ fluxes of the
primary synchrotron
flare and the secondary TeV flare, $\nu F_{\nu}
({\rm sy}) \sim 5 \times
10^{-10}$~ergs~s$^{-1}$~cm$^{-2}$ and
$\nu F_{\nu} (600 \, {\rm GeV}) \sim 3 \times 10^{-10}$~ergs~s$^{-1}$~cm$^{-2}$
\citep[see Fig. \ref{bbspectrum} and][]{kraw04}, we find the co-moving
luminosities, ${L'}_{\rm sy} \sim 2.5 \times
10^{41} \,
\Gamma_1^{-4}$~ergs~s$^{-1}$ and ${L'}_{\rm VHE} \sim 1.5
\times 10^{41}
\, \Gamma_1^{-4}$~ergs~s$^{-1}$. Here, an $\Omega_{\Lambda} = 0.7$,
$\Omega_{\rm m} = 0.3$ cosmology
with $H_0 = 70$~km~s$^{-1}$~Mpc$^{-1}$
was used. With these parameters, 1ES~1959+650 with $z = 0.047$ is
located at a luminosity distance of $d_L = 210$~Mpc.
If the reflecting cloud is located in
a direction close
to our line of sight, the energy density of jet
synchrotron
photons impinging onto the mirror is
\begin{equation}
u_{\rm sy} (R_m) \sim {d_L^2 \over R_m^2 c} \, \nu F_{\nu} ({\rm sy})
\sim 1.3 \times 10^5 \, \Gamma_1^{-4} \, \Delta t_{20}^{-2} \; {\rm ergs
\, cm}^{-3}.
\label{u_sy}
\end{equation}
The reflected synchrotron flux will be received by the blob very close
to (and within) the mirror, so that its photon energy density, in the
co-moving frame of the blob, is given by
\begin{equation}
{u'}_{\rm Rsy} \sim {\tau_m \, \Gamma^2 \, u_{\rm sy} (R_m) \over 4 \pi}
\sim 1.0 \times 10^5 \, \Gamma_1^{-2} \, \Delta t_{20}^{-2} \, \tau_{-1}
\; {\rm ergs \, cm}^{-3}.
\label{u_Rsy}
\end{equation}
This reflected synchrotron photon field can now be used to estimate
the energy and density of relativistic protons needed in the jet to
produce the ``orphan'' TeV flare in 1ES~1959+650 via $\gamma$p pion
production and subsequent $\pi^0$ decay, and to estimate the expected
signatures of such a scenario at lower (optical -- X-ray) frequencies.
\section{\label{results}Results}
The co-moving luminosity from $p\gamma \to \Delta \to p+\pi^0 \to
p +
2 \gamma$ produced by protons of a given energy ${\gamma'}_p$
is
given by
\begin{equation}
{L'}_{\rm VHE} \sim {8 \over 3} \, c \, \sigma_{\Delta} \, {u'}_{\rm Rsy}
\gamma'_p {70 \, {\rm MeV} \over {E'}_{\rm Rsy}} \, N_p ({\gamma'}_p).
\label{LRsy}
\end{equation}
where $\sigma_{\Delta} \approx 300 \, \mu$b is the $\Delta$ resonance cross
section and $N_p ({\gamma'}_p)$ is the number of protons at energy
${\gamma'}_p \approx (300 {\rm MeV})/E'_{\rm Rsy} \approx 3 \times 10^3
\, \Gamma_1^{-1} \, E_{\rm sy,1}^{-1}$. With this, the observable
$\nu F_{\nu}$ peak flux in the TeV flare can be estimated as
\begin{equation}
\nu F_{\nu} ({\rm VHE}) \sim {{L'}_{\rm VHE} \, \Gamma^4 \over 4 \pi \,
d_L^2} \sim 1.0 \times 10^{-56} \, N_p ({\gamma'}_p) \,
\Delta t_{20}^{-2}
\, \tau_{-1} \, E_{\rm sy,1}^{-2} \; {\rm ergs \, cm}^{-2}
\, {\rm s}^{-1}.
\label{nFn_VHE}
\end{equation}
Setting this equal to the observed VHE peak flux yields
\begin{equation}
N_p ({\gamma'}_p) \sim 3.0 \times 10^{46} \, \Delta t_{20}^2 \,
\tau_{-1}^{-1} \, E_{\rm sy,1}^2.
\label{Np_gp}
\end{equation}
The spectrum of non-thermal protons in the blob may be expected
to have a low-energy cut-off at relativistic energies. For example,
if the non-thermal protons are injected into the jet as pick-up
ions from a relativistic shock wave traveling along the jet
\cite[see, e.g.][]{ps00}, this low-energy cutoff is expected
at $\gamma_{\rm p, min} \sim \Gamma = 10 \, \Gamma_1$.
Assuming that the relativistic proton spectrum is a straight
power-law with index $s$, the estimate (\ref{Np_gp}) corresponds
to a total relativistic proton number of
\begin{equation}
N_p \sim
{(3,000)^{1 + s} \cdot 10^{44 - s} \over s - 1} \,
\Gamma_1^{1 - 2 s} \, \Delta t_{20}^2 \, \tau_{-1}^{-1} \,
E_{\rm sy,1}^{2-s}.
\label{Np_total}
\end{equation}
For a typical index $s = 2$, this corresponds to $N_p \sim
2.7 \cdot 10^{52} \, \Gamma_1^{-3} \, \Delta t_{20}^2 \,
\tau_{-1}^{-1}$ and a relativistic proton number density of
\begin{equation}
{n'}_p \sim 6.4 \times 10^3 \, \Gamma_1^{-3} \, \Delta t_{20}^2
\, \tau_{-1}^{-1} \, R_{16}^{-3} \; {\rm cm}^{-3}.
\label{np}
\end{equation}
Note the strong dependence on the bulk Lorentz factor. With
values of $\Gamma_1 \sim 2$, the required proton density can be
substantially less than the typical electron densities found
in spectral
modeling of blazars (${n'}_e \sim 10^3$~cm$^{-3}$),
which is perfectly consistent with the pair/proton
number density
ratios inferred by \cite{sm00}. With $s = 2$ and a maximum Lorentz
factor of relativistic protons of $\gamma_{\rm p, max} \sim 10^4$,
the total co-moving kinetic energy in
relativistic protons in the
blob is then
\begin{equation}
{E'}_{b,p} \sim 2.8 \times
10^{51} \, \Gamma_1^{-2} \,
\Delta t_{20}^2 \, \tau_{-1}^{-1} \; {\rm erg}.
\label{Ep}
\end{equation}
The kinetic luminosity carried by relativistic protons in the
jet can then be estimated as
\begin{equation}
L_p^{\rm kin.} \sim
7.3 \times 10^{44} \, R_{16}^{-1} \,
\Delta t_{20}^2
\, \tau_{-1}^{-1} \, f_{-3} \; {\rm ergs \; s}^{-1}
\label{Lp}
\end{equation}
where $f = 10^{-3} \, f_{-3}$ is a filling factor accounting
for the likely case that the relativistic proton
plasma is
concentrated only in individual blobs along the jet rather
than being continuously distributed throughout the jet.
The radiative output from the $\Delta^+$ decay channel
$p\gamma \to \Delta^+ \to n\pi^+$, followed by $\pi^+ \to
\mu^+ + \nu_{\mu}$ and $\mu^+ \to \overline{\nu_{\mu}} +
e^+ + \nu_e$ will primarily consist of positron synchrotron
radiation. Considering the kinematics of the pion and muon
decay processes, one finds that the positron will carry away
$\sim 1/3$ of the total pion energy. Consequently, we have
$\gamma_{e^+} \sim (1/3) \, (m_{\pi}/m_e) \, \gamma'_p \sim
2.1 \times 10^5 \, \Gamma_1^{-1} \, E_{\rm sy,1}^{-1}$. The
synchrotron emission from the secondary positrons will peak
at
\begin{equation}
E_{\rm sy, e^+} \sim 500 \, B_{-1} \, E_{\rm sy,1}^{-2}
\; {\rm eV},
\label{E_psyn}
\end{equation}
i.e. typically in the UV or soft X-ray regime. Note that
unlike the case of a proton blazar (with higher magnetic
fields and much higher proton and positron Lorentz factors),
the charged-pion decay channel will {\it not} initiate an
electromagnetic cascade.
An estimate of the expected $\nu F_{\nu}$ flux in the $e^+$
synchrotron emission can be found in the following way. First of
all, considering the co-moving dynamical time scale, $t'_{\rm dyn}
\sim R/c \sim 3.3 \times 10^5 \, R_{16}$~s and the synchrotron
cooling time scale,
\begin{equation}
t'_{e^+ \rm sy} \sim 3.7 \times 10^5 \, \Gamma_1 \,
B_{-1}^{-2} \, E_{\rm sy,1} \; {\rm s},
\label{t_psyn}
\end{equation}
we find that those are comparable, implying that the secondary
positrons might lose a substantial fraction of their kinetic
energy to radiation before potentially leaking out of the emission
region. Second, we realize that the synchrotron cooling time
scale will set the natural duration of the secondary $e^+$
synchrotron flare, which will be (in the observer's frame)
$w_{e^+ \rm sy}^{\rm obs} \sim 3.7 \times 10^4 \, B_{-1}^{-2}
\, E_{\rm sy,1}$~s. Consequently, the duration of the $\pi^0$
decay VHE $\gamma$-ray flare is a factor of $f_w \equiv
w_{\pi^0}^{\rm obs} / w_{e^+ \rm sy}^{\rm obs} \sim 0.12
\, B_{-2}^2 \, E_{\rm sy,1} \, \Gamma_1^{-2}$ shorter than
the secondary synchrotron flare, so the observed $\nu F_{\nu}$
peak flux of the $e^+$ synchrotron flare should have been a
factor of $f_w / 3 \sim 0.04$ lower than that of the $\pi^0$
decay flare (note that 2/3 of the $\pi^+$ energy will go
into neutrino emission), which yields
\begin{equation}
\nu F_{\nu}^{e^+ \rm sy} \sim 1.2 \times 10^{-11} \, \Gamma_1^{-2}
\, B_{-2}^2 \, E_{\rm sy,1} \; {\rm ergs \, cm}^{-2} \, {\rm s}^{-1}
\label{nuFnu_psyn}
\end{equation}
if the observed secondary VHE flare was due to $\pi^0$
decay photons. The expected $e^+$ synchrotron peak flux
would thus have been only a few \% of the observed RXTE
$\nu F_{\nu}$ flux level during the secondary VHE flare,
and have peaked at energies well below the RXTE energy
range, leaving no observable trace in the X-ray light
curve of the 1ES~1959+650 campaign of 2002. The expected
level and spectral shape of the secondary $e^+$ synchrotron
emission is represented by the dot-dashed line in Fig.
\ref{bbspectrum}.
An estimate of a possible optical flare can be obtained
by realizing that positrons emitting synchrotron radiation
in the optical regime are expected to be slow-cooling and
thus basically reproduce the spectrum of the primary
relativistic protons ($\sim \gamma^{-2}$), resulting in
a synchrotron spectrum $\nu F_{\nu} \propto \nu^{1/2}$,
which yields an R band flux from the secondary positron
synchrotron emission of
\begin{equation}
\nu F_{\nu}^{e^+ \rm sy} (R) \sim 7.0 \times 10^{-13}
\, \Gamma_1^{-1.5} \, B_{-1}^{1.5} \, E_{\rm sy,1}^2
\; {\rm ergs \, cm}^{-2} \, {\rm s}^{-1}.
\label{nuFnu_R}
\end{equation}
Comparing this to the average R-band flux around the
time of the secondary (``orphan'') TeV flare, results
in a predicted optical flare of
\begin{equation}
\Delta m \sim 0.05^{\rm mag},
\label{Delta_m}
\end{equation}
which is perfectly consistent with the observed very
small optical activity of $\Delta m_{\rm obs} \lesssim
0.1^{\rm mag}$ of 1ES~1959+650 at that time \citep{kraw04}.
\section{\label{summary}Summary and Discussion}
In this paper, I have suggested a model to explain the
``orphan'' TeV flare of 1ES~1959+650 in 2002, which followed
a correlated X-ray + TeV flare by about 20~days. In this
model, the secondary TeV flare resulted from $\pi^0$ decay
following p$\gamma$ pion production by relativistic protons
($\gamma_p \sim 10^3$ -- $10^4$) on the primary synchrotron
flare photons, reflected off a mirror cloud at a distance
of a few pc from the central engine. Using the observational
data from the 1ES~1959+650 observations in 2002, I have
estimated the required parameters pertaining to the relativistic
proton population in the jet in order to produce the secondary
TeV flare with this mechanism. The main results of this investigation
are:
\begin{itemize}
\item The required model setup is consistent with the BL~Lac
classification of 1ES~1959+650.
\item The required density of relativistic protons in the jet
is very well consistent with earlier findings that blazar jets
might be dynamically dominated by the kinetic energy of relativistic
protons, even if they are by far outnumbered by electron/positron
pairs, which may dominate the radiative output of 1ES~1959+650
most of the time.
\item The secondary $e^+$ synchrotron emission resulting from
$\pi^+$ decay in this scenario is too weak and peaks at too low
energies to leave an observable imprint in the RXTE light curve
at the time of the secondary TeV flare, consistent with its
non-detection (and, thus, with the appearance of the TeV flare
as an ``orphan'' flare).
\item The optical flare produced by secondary $e^+$ synchrotron
emission is expected to produce only a very mild bump of
$\Delta m \sim 0.05^{\rm mag}$ in the R and V bands, which
is perfectly consistent with the very moderate activity of
the source during the secondary TeV flare.
\end{itemize}
A detailed investigation of the spectral and light curve
features resulting in this scenario is currently underway
and will be published in a forthcoming paper (Postnikov \&
B\"ottcher 2004, in preparation). This will also include
the characteristics of the expected neutrino emission
resulting from $\pi^+$ decay.
Another signature of relativistic protons in the framework
of the model suggested here might arise from photo-pair
production, $p\gamma \to p e^+ e^-$. The threshold proton
energy for this process, in our parametrization is $\gamma_{\rm
thr, pair} \sim 5 \, \Gamma^{-1} \, E_{\rm sy, 1}^{-1}$. The
bulk of pairs injected into the emission region from this process
would thus have only mildly relativistic energies and would not
leave significant non-thermal radiation signatures. However, it
has been demonstrated by \cite{km99} \cite[see also][for the
application of this process to gamma-ray bursts]{kgm02,kgm04}
that the photo-pair production process can exceed a critical
threshold beyond which a pair avalanche on synchrotron radiation
of secondary pairs develops. The threshold proton energy to
initiate such an avalanche has been evaluated by \cite{km99}
to be $\gamma_{\rm p, crit} \sim 10^4 \, B_{\rm G}^{-1/3}
\, \Gamma_1^{-2/3}$. Thus, if the emitting volume contains
protons with energies $\gamma_p \gg 10^4$, this supercritical
pair avalanche can lead to a strong synchrotron signal, extending
far into the X-ray regime, which would naturally be expected to
produce a corresponding SSC signature at $\gamma$-ray energies.
Because of the strong synchrotron component of this scenario,
these signatures are easily distinguishable. It is very well
conceivable that the ``supercritical pile'' scenario \citep{km99},
indicative of protons with Lorentz factors of $\gamma_p \gg 10^4$,
is responsible for simultaneous X-ray + TeV $\gamma$-ray flares,
while the pion production scenario discussed here, indicative of
protons with Lorentz factors of $10^3 \lesssim \gamma_p \lesssim
10^4$ produces orphan TeV flares. Protons with yet lower Lorentz
factors may ultimately be probed by the radiation signatures of
mildly relativistic or even thermal pairs injected through the
$p\gamma$ pair production process of protons near threshold.
\acknowledgments
The author wishes to thank H. Krawczynski for providing the
broadband spectral data of 1ES~1959+650, and the anonymous
referee for pointing out the potential importance of the
$p\gamma$ process in the framework of this model. This work
was partially supported by NASA through INTEGRAL GO grant
no. NAG~5-13684 and XMM-Newton GO grant no. NNG~04GF70G.
|
{
"timestamp": "2004-11-09T23:20:20",
"yymm": "0411",
"arxiv_id": "astro-ph/0411248",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411248"
}
|
\section{Introduction}
Let $G$ be a Lie group acting properly on a configuration manifold
$Q$.
Consider the cotangent lifted $G$-action on $T^*Q$.
This action is
Hamiltonian
with respect to the
standard exact symplectic form on $T^*Q$ and
with equivariant momentum map\xspace
denoted by $\mu: T^*Q\to\mathfrak{g}^*$.
Assume $\mbox{$\mathcal{O}$}$ is a
coadjoint orbit contained in the image of $\mu$.
The first part of this paper
is concerned with the study of the singular symplectic
quotient
\[
\mu^{-1}(\mbox{$\mathcal{O}$})/G =: T^{*}Q\mbox{$/\negmedspace/_{\mathcal{O}}$} G.
\]
Indeed, this quotient cannot be a smooth manifold, in
general, since we do not assume the $G$-action on $Q$ to be free.
However, we can apply the theory of
singular symplectic
reduction as developed by Sjamaar and Lerman \cite{SL91}, Bates and
Lerman \cite{BL97}, and Ortega and Ratiu \cite{OR04} (see also Theorem
\ref{thm:sing_spr}), and this exhibits $T^*Q\mbox{$/\negmedspace/_{\mathcal{O}}$} G$ to be a Whitney
stratified space with strata of the form
\[
(\mu^{-1}(\mbox{$\mathcal{O}$})\cap(T^{*}Q)_{(L)})/G
=:
(T^{*}Q\mbox{$/\negmedspace/_{\mathcal{O}}$} G)_{(L)}
\]
where $(L)$ is an element of the isotropy lattice of the $G$-action on
$T^{*}Q$.
One of the aims of this paper is to develop a bundle picture for the
reduced phase space $T^*Q\mbox{$/\negmedspace/_{\mathcal{O}}$} G$, i.e., to obtain a fiber bundle
$T^*Q\mbox{$/\negmedspace/_{\mathcal{O}}$} G\to T^*(Q/G)$ in a suitable (singular) sense. One can
hope to construct such a fiber bundle in the presence of an
additionally chosen generalized connection form (see
Section~\ref{sec:gauged_red}) on $Q\twoheadrightarrow Q/G$.
However, for reasons explained
in Remark~\ref{rem:stratQ} there cannot exist a surjection
$T^*Q\mbox{$/\negmedspace/_{\mathcal{O}}$} G\to T^*(Q/G)$ with locally constant fiber type
for general proper $G$-actions.
In Remark~\ref{rem:stratQ}(\ref{equ:F}) we state a weak substitute of
a bundle picture for $T^*Q\mbox{$/\negmedspace/_{\mathcal{O}}$} G$.
To obtain a bundle picture and a useful description of the reduced
phase space $T^*Q\mbox{$/\negmedspace/_{\mathcal{O}}$} G$
we have to assume that the base manifold $Q$ is of
single orbit type, that is, $Q=Q_{(H)}$ for a subgroup $H$ of
$G$. Assuming this we get a first result that says that, locally,
\[
\xymatrix{
{\mbox{$\mathcal{O}$}\spr{0}H}
\ar @{^{(}->}[r]&
{T^{*}Q\mbox{$/\negmedspace/_{\mathcal{O}}$} G}
\ar[r]&
{T^{*}(Q/G)}
}
\]
is a symplectic fiber bundle (Theorem~\ref{thm:bun_pic}).
This result is obtained by applying the Palais
Slice Theorem to the $G$-action on the base space $Q$, and then using the
Singular Commuting Reduction Theorem of Section~\ref{s:sg_com_red}.
This is an inroad that was also taken by
Schmah~\cite{Sch04} to get a local description of $T^{*}Q\mbox{$/\negmedspace/_{\mathcal{O}}$} G$.
However, one can also give a global symplectic description of the
reduced phase space and this is done in Section \ref{sec:gauged_red}.
This follows an approach that is generally called Weinstein
construction (\cite{Wei78}).
(We do not consider the closely related construction of Sternberg~\cite{Ste77}.)
In the case that the $G$-action on
the configuration space $Q$ is free this global description was first given
by Marsden and Perlmutter~\cite{MP00}. Their result says that
the choice of a principal bundle connection on $Q\twoheadrightarrow Q/G$ yields a
realization of
the symplectic
quotient $T^{*}Q\mbox{$/\negmedspace/_{\mathcal{O}}$} G$ as a fibered product
\[
T^{*}(Q/G)\times_{Q/G}(Q\times_{G}\mbox{$\mathcal{O}$}),
\]
and they compute the reduced symplectic structure in terms of data
intrinsic to this realization -- \cite[Theorem 4.3]{MP00}.
In the presence of a single non-trivial isotropy $(H)$
on the configuration space one
obtains a non-trivial isotropy lattice on $T^{*}Q$
whence the symplectic reduction of $T^*Q$ is to be carried out in the
singular context of \cite{SL91,BL97,OR04}.
The result is the following:
The choice of a generalized connection form
(see Section~\ref{sec:gauged_red})
on $Q\twoheadrightarrow Q/G$ yields a realization of
each symplectic stratum $(T^{*}Q\mbox{$/\negmedspace/_{\mathcal{O}}$} G)_{(L)}$ of the reduced space
as a fibered product
\[
(\mbox{$\mathcal{W}$}\mbox{$/\negmedspace/_{\mathcal{O}}$} G)_{(L)}
=
T^{*}(Q/G)\times_{Q/G}(\mbox{$\bsc_{q\in Q}\mathcal{O}\cap\ann\gu_{q}$})_{(L)}/G
\]
where
\[
\mbox{$\mathcal{W}$} :=
(Q\times_{Q/G}T^{*}(Q/G))\times_{Q}\mbox{$\bigsqcup$}_{q\in Q}\mbox{$\textup{Ann}\,$}{\mathfrak{g}_{q}}
\cong
T^{*}Q
\]
as symplectic manifolds with a Hamiltonian $G$-action.
Moreover, we compute the reduced symplectic structure in terms
intrinsic to this realization. This is the content of
Theorem~\ref{thm:WWred}.
Even though we have to restrict to single orbit type
manifolds this result is quite general
in the sense that it is valid for arbitrary
coadjoint orbits $\mbox{$\mathcal{O}$}$.
The first to have studied symplectic reduction of cotangent bundles
for non-free actions seems to have been Montgomery~\cite{Mon83}.
Using the point reduction approach this paper gives conditions under
which the reduced phase space carries a smooth manifold structure.
The
first to study this subject
in the context of singular symplectic reduction as developed by \cite{SL91,BL97,OR04}
is Schmah~\cite{Sch04}
who proves a cotangent bundle specific slice theorem at points whose
momentum values are fully isotropic.
The other
important paper on singular cotangent bundle reduction is by
Perlmutter, Rodriguez-Olmos and Sousa-Diaz~\cite{PRS03}.
By restricting to do
reduction at fully isotropic values of the momentum map\xspace $\mu:
T^{*}Q\to\mathfrak{g}^{*}$ they are able to drop all assumptions on the
isotropy lattice of the $G$-action on $Q$, and give a very complete
description of the reduced symplectic space.
As an application of the bundle picture found in
Theorem~\ref{thm:WWred} we
consider Calogero-Moser systems with spin in Section \ref{sec:cms}.
In fact, it was an idea of Alekseevsky, Kriegl, Losik, Michor\xspace \cite{AKLM03} to consider polar
representations of compact Lie groups $G$ on a Euclidean vector
space $V$ to obtain new versions of
Calogero-Moser models. We make these ideas precise by using the
singular cotangent bundle reduction machinery. Thus let $\Sigma$ be a
section for the $G$-action in $V$, let $C$ be a Weyl chamber in
this section, and put $M:=Z_{G}(\Sigma)$. Under a strong but not
impossible condition on a chosen coadjoint orbit in $\mathfrak{g}^{*}$ we
get
\[
T^{*}V\mbox{$/\negmedspace/_{\mathcal{O}}$} G
=
T^{*}C_{r}\times\mbox{$\mathcal{O}$}\spr{0}M
\]
from the general theory (Theorem~\ref{thm:WWred}),
where $C_{r}$ denotes the sub-manifold of
regular elements in $C$.
This is the effective phase space of the spin
Calogero-Moser system. The corresponding Calogero-Moser function is obtained as a
reduced Hamiltonian from the free Hamiltonian on $T^{*}V$. The
resulting formula is
\[
\hcm(q,p,[Z])
=
\by{1}{2}\sum_{i=1}^{l}p_{i}^{2}
+
\by{1}{2}\sum_{\lambda\in R}
\by{\sum_{i=1}^{k_{\lambda}}z_{\lambda}^{i}z_{\lambda}^{i}}{\lambda(q)^{2}}.
\]
This is made precise with the necessary notation in Section
\ref{sec:cms}.
It was first observed by Kazhdan, Kostant and Sternberg
\cite{KKS78} that one can obtain
Calogero-Moser models via Hamiltonian reduction of $T^*\mathfrak{g}$ or
$T^*G$.
(See also Subsections~\ref{sub:cms-constr} and \ref{sub:kks}.)
In fact, \cite{KKS78}
chose their example such that the reduction procedure yields a smooth
symplectic manifold, and moreover there appear no spin variables, i.e.,
$\mbox{$\mathcal{O}$}\spr{0}H=\set{\textup{point}}$.
This reduction approach was further pursued in \cite{AKLM03}
as alluded to above.
However, these spin models are Hamiltonian systems on non-smooth
spaces, and it was not clear in which sense one should regard
the reduced systems as being Hamiltonian systems.
Moreover, the precise form of the
singularities was not clear. These questions are answered by
Theorem~\ref{thm:WWred}: The reduced system is a stratified
Hamiltonian system, the strata of the reduced phase space are
described, and when the system is restricted to a stratum it is a
Hamiltonian system in the usual sense on this stratum. Moreover, the
stratification is Whitney whence by \cite{SL91}
the singularities which appear are
of conic form.
As an interesting side product (Remark~\ref{rem:r+A})
our (singular) cotangent bundle
reduction approach yields a connection to the $r$-matrix theoretic
construction of Calogero-Moser models of Li and
Xu~\cite{LX00,LX02}. This connection is new and
interesting since it explains in geometric terms
why solutions of the classical dynamical
Yang-Baxter equation lead to Calogero-Moser systems.
Finally, we use a result on non-commutative integrability from Zung
\cite[Theorem 2.3]{Zung02} to show that these Calogero-Moser systems are
integrable in the non-commutative sense.
\textbf{Thanks.}
This paper is part of my PhD thesis written under
the supervision of Peter Michor. I am grateful to him for introducing me
to symplectic geometry and proposing the subject of Calogero-Moser
systems associated to polar representations of compact Lie
groups. Further, I wish to thank the Centre Bernoulli for their hospitality
in September 2004. This stay contributed a lot toward the
finishing of this paper. I am also thankful to Stefan Haller and Armin
Rainer for helpful remarks and comments.
Finally, I want to thank the referees for their detailed report.
\section{Preliminaries and notation}\label{sec:prel}
All manifolds to be considered are Hausdorff, para-compact,
finite dimensional, and smooth in the $\mbox{$C^{\infty}$}$-sense.
We do not assume that manifolds are connected but allow for finitely
many connected components of varying (finite) dimension. Thus the
dimension of a manifold is only locally constant.
A proper
$G$-space $M$ is a manifold $M$ acted upon properly by a Lie group
$G$.
For proper $G$-spaces the
Slice Theorem~(\cite{Pal61,PT88,DK99}) holds,
and we shall make frequent use of this theorem.
Let $M$ be a proper $G$-space. An \caps{isotropy class} $(H)$ of the
$G$-action on $M$
is a conjugacy class of an isotropy subgroup $G_x$ of a point
$x\in M$,
that is, $(H) = \set{gG_xg^{-1}: g\in G}$.
If we want to explicit that $(H)$ is the conjugacy class of $H$ with
respect to $G$ we shall write $(H)^G$.
The
\caps{isotropy lattice} $\mbox{$\mathcal{IL}$}(M)$ is defined to be the lattice consisting of all
isotropy classes $(H)$ of the $G$-space $M$.
For $(H)\in\mbox{$\mathcal{IL}$}(M)$ the \caps{orbit type} sub-manifold is
\begin{align}
M_{(H)} :=
M_{(H)^G} :=
\set{x\in M: G_x \textup{ is conjugate to } H \textup{ within } G},
\end{align}
the \caps{symmetry type} sub-manifold is
$M_H := \set{x\in M: G_x = H}$, and the \caps{fixed point} sub-manifold is
$M^H := \set{x\in M: H\subset G_x}$.
More generally,
let $N$ be a closed topological
subspace of $M$ and let $K$ be a compact subgroup of $G$ which acts
(continuously)
on $N$.
We may view $M$ also as a proper $K$-space, and thus obtain an orbit type stratification of
$M$
with respect to the $K$-action.
Since $N$ is $K$-invariant this induces a decomposition
of $N$ according to orbit types.
Let $N_{(L_0)} = N_{(L_0)^K} = N\cap M_{(L_0)^K}$
be such an orbit type stratum, that is $L_0 = K_x$ for some $x\in N$.
With regard to $L_0\subset K$
we will be concerned with the \caps{generalized isotropy class}
\begin{align}
(L_0)_K^G := \set{L\subset K: \textup{ there is } g\in G \textup{ s.t.\ } gL_0g^{-1} = L}
\end{align}
and the corresponding generalized orbit type space
\begin{align}
N_{(L_0)_K^G} := \set{z\in N: K_z\in(L_0)_K^G}\label{def:weirdo-type}.
\end{align}
Clearly, $N_{(L_0)_K^G}$ itself
decomposes into orbit type strata with respect to the induced $K$-action.
\begin{comment}
NONSENSE:
We will make use of the following lemma in the proof of Lemma~\ref{lm:WWorb}.
\begin{lemma}
Suppose $(M,\omega)$ is a proper Hamiltonian $G$-space with equivariant momentum map\xspace $J: M\to\mathfrak{g}^*$,
let $K$ be a compact subgroup of $G$, and let $\iota: \mathfrak{k}\hookrightarrow\mathfrak{g}$ denote the inclusion. Then
the induced $K$-action on $(M,\omega)$ is also Hamiltonian with momentum map\xspace $J_K = \iota^*\circ J$ and the
level set $J_K^{-1}(0)=:N$ is stratified according to isotropy classes $(L_0)^K$
with respect to the $K$-action.
Moreover, the space $N_{(L_0)_K^G}$ is a smooth
disconnected manifold with finitely many connected components which are the connected
components of the orbit type strata $N_{(L')^K}$ where $L'\in(L_0)_K^G$.
\end{lemma}
\begin{proof}
This lemma follows entirely from \cite[Theorem~2.1]{SL91}. In particular, we have
to check that, for $L_0,L_1\in(L_0)_K^G$, the relation
$\ov{N_{(L_0)^K}} \cap N_{(L_1)^K} \neq \emptyset$ implies $N_{(L_0)^K} = N_{(L_1)^K}$. Indeed,
by the condition of the frontier (\cite{SL91,Mat70,GorMac88}) we have
$N_{(L_1)^K} \subset \ov{N_{(L_0)^K}}$ whence it follows that $L_0$ is conjugate within $K$ to a
subgroup of $L_1$ (\cite[Theorem~2.1]{SL91}). By assumption this means that there are
elements $k\in K$ and $g\in G$ such that $L_0\subset k^{-1}L_1k = k^{-1}gL_0g^{-1}k=L_0$.
That is, $(L_0)^K = (L_1)^K$.
\end{proof}
\end{comment}
\begin{comment}
***********************SUPERFLUOUS:*******************************************************
Using the
condition of the frontier (\cite{Mat70,GorMac88})
in conjunction with the partial ordering of $K$-isotropy classes given by reverse sub-conjugacy
(see~\cite{Bre72,DK99,SL91})
it follows that $N_{(L_0)_K^G}$
is a finite disjoint union of orbit type strata. That is, we view $N_{(L_0)_K^G}$ as a
smooth manifold with finitely many connected components.
for a subgroup $K\subset G$ we define
\begin{align}
M_{(H)^K}
:=
\set{x\in M: G_x \textup{ is conjugate to } H \textup{ within } K} \label{STRANGO}
\end{align}
whence
$M_{(H)} = M_{(H)^G}$.
Further, let
$H$ be a subgroup of $G$ and let
$(L)\in\mbox{$\mathcal{IL}$}(M)$
such that $L$ is conjugate within $G$ to a subgroup $L_0\subset H$. Then
we define
\begin{align}
M_{(L_0)_H^G}
:=
\set{x\in M: H_x = H\cap G_x
\text{ is conjugate to }L_0\text{ within }G}. \label{WEIRDO}
\end{align}
This subset can be seen to be a manifold consisting of finitely many
connected components as follows: Consider the restricted $H$-action on
$M$ and note that, by compactness of $H$,
$M$ decomposes into finitely many orbit type strata with respect to the $H$-action.
In
particular, the $H$-invariant subset
$M_{(L_0)_H^G}$ decomposes into finitely many orbit type
strata with respect to the $H$-action,
and since any two of these strata are diffeomorphic to each
other (by virtue of the $G$-action) none can intersect the closure of
the other. Therefore, we can view $M_{(L_0)_H^G}$ as a finite co-product
modeled on an $(L_0)^H$-orbit type sub-manifold $M_{(L_0)^H}$.
*******************************************************************************************
\end{comment}
Using the Slice Theorem one can show (see
\cite{DK99}) that the stratification of $M$ into orbit type
sub-manifolds forms a Whitney stratification. Likewise the
stratification of the topological space $M/G$ into strata of the type
$M_{(H)}/G$ where $(H)\in\mbox{$\mathcal{IL}$}(M)$
forms a Whitney stratification. Thus $M/G$ becomes a stratified
space, i.e., a stratified space such that the stratification satisfies
the Whitney conditions. See \cite{Mat70,GorMac88,Pfl01,Pfl01a} for
more on stratified spaces. If $X$ and $Y$ are stratified spaces a
\caps{stratified map} $\phi: X\to Y$ is a continuous map which respects the
stratifications, that is, the pre-image under $\phi$ of every stratum of $Y$
decomposes into a union of strata of $X$.
For example, the orbit projection map $\pi: M\twoheadrightarrow M/G$ is a stratified
map.
If the action is written as $l: G\times M\to M$, $(k,x)\mapsto
l(k,x)=l_{k}(x)=l^{x}(k)=k.x$
we can tangent bundle lift it via
$k.(x,v):=(k.x,k.v):=Tl_{k}.(x,v)=(l_{k}(x),T_{x}l_{k}.v)$
for $(x,v)\in TM$ to an action on $TM$.
As the
action consists of transformations by diffeomorphisms
it may also be lifted to
the cotangent bundle. This is the cotangent lifted action which is
defined by
$k.(x,p):=(k.x,k.p):=T^{*}l_{k}.(x,p)=
(k.x,T_{k.x}^{*}l_{k^{-1}}.p)$ where $(x,p)\in T^{*}M$.
Our notation for the fundamental vector field is
$
\zeta_{X}(x)
:= \zeta(x)(X)
:= \dd{t}{}|_{0}l(\exp(+tX),x)
= T_{e}l^{x}(X)
$
where $X\in\mathfrak{g}$.
\section{Singular symplectic reduction}\label{s:sg_com_red}
Let $(M,\omega)$ be a connected symplectic manifold\xspace, and $G$ a Lie group that
acts on $(M,\omega)$ in a proper and Hamiltonian fashion such that there is an
equivariant momentum map\xspace $J: M\to\mathfrak{g}^{*}$.
The very strong machinery of singular symplectic reduction is
(for the case of compact $G$) due to
Sjamaar and Lerman~\cite{SL91} who prove that the singular symplectic
quotient is a Whitney stratified space that has symplectic manifolds
as its strata. This result which is the Singular Reduction Theorem was
then generalized to the case of proper actions by Bates and
Lerman~\cite{BL97},
Ortega and Ratiu~\cite{OR04}, and others.
\begin{theorem}[Singular symplectic reduction]\label{thm:sing_spr}
Let $(H)$ be in the isotropy lattice of the $G$-action on $M$, and
suppose that $J^{-1}(\mbox{$\mathcal{O}$})\cap M_{(H)}\neq\emptyset$ for a coadjoint
orbit $\mbox{$\mathcal{O}$}\subseteq\mathfrak{g}^{*}$.
Then the following are true.
\begin{itemize}
\item
The subset $J^{-1}(\mbox{$\mathcal{O}$})\cap M_{(H)}$ is an initial sub-manifold of
$M$.
\item
The topological quotient $(J^{-1}(\mbox{$\mathcal{O}$})\cap M_{(H)})/G$ has a unique
smooth structure such that the projection map
\[
\xymatrix{
{J^{-1}(\mbox{$\mathcal{O}$})\cap M_{(H)}}
\ar @{->>}[r]^-{{\pi}}&
{(J^{-1}(\mbox{$\mathcal{O}$})\cap M_{(H)})/G}
}
\]
is a smooth surjective submersion.
\item
Let $\iota: J^{-1}(\mbox{$\mathcal{O}$})\cap M_{(H)}\hookrightarrow M$ denote the inclusion
mapping. Then $(J^{-1}(\mbox{$\mathcal{O}$})\cap M_{(H)})/G$ carries a symplectic
structure $\omega_{0}$ which is uniquely characterized by the formula
\[
\pi^{*}\omega_{0}
=
\iota^{*}\omega
- (J|(J^{-1}(\mbox{$\mathcal{O}$})\cap M_{(H)}))^{*}\Omega^{\mathcal{O}}
\]
where $\Omega^{\mathcal{O}}$ is the canonical (positive Kirillov-Kostant-Souriau)
symplectic form on $\mbox{$\mathcal{O}$}$.
\item
Consider a $G$-invariant function $H\in\mbox{$C^{\infty}$}(M)^{G}$. Then the flow of
the Hamiltonian vector field $\ham{H}$ leaves the connected components
of $J^{-1}(\mbox{$\mathcal{O}$})\cap M_{(H)}$ invariant. Moreover, $H$ factors to a
smooth function $h$ on the quotient
$(J^{-1}(\mbox{$\mathcal{O}$})\cap M_{(H)})/G$.
Finally, $\ham{H}$ and the Hamiltonian vector field to $h$
are related via the canonical projection $\pi$, whence the flow of the former
projects to the flow of the latter.
\item
The collection of all strata of the form $(J^{-1}(\mbox{$\mathcal{O}$})\cap M_{(H)})/G$
constitutes a Whitney stratification of the topological space
$J^{-1}(\mbox{$\mathcal{O}$})/G$.
\end{itemize}
\end{theorem}
\begin{proof}
This theorem is contained in \cite[Section 8]{OR04}. See also
\cite[Corollary 14]{BL97} and \cite{SL91}.
\end{proof}
In fact, \cite{OR04} state the above theorem only for connected
components of strata. This is so because they do not require
the momentum map\xspace $J$ to be equivariant with respect to the co-coadjoint action
on $\mathfrak{g}^*$.
As a matter of convention we write shorthand $M\mbox{$/\negmedspace/_{\mathcal{O}}$} G :=
J^{-1}(\mbox{$\mathcal{O}$})/G$ for the reduced space of $M$ with respect to the
Hamiltonian action by $G$. If $\mbox{$\mathcal{O}$}$ is the coadjoint orbit passing
through $\alpha$ then we shall also abbreviate
$J^{-1}(\alpha)/G_{\alpha} = M\spr{\alpha}G = M\mbox{$/\negmedspace/_{\mathcal{O}}$} G$.
\begin{theorem}[Singular commuting reduction]\label{thm:sg_comm_red}
Let $G$ and $H$ be Lie groups that act properly and by symplectomorphisms
on $(M,\omega)$ with momentum maps $J_{G}$ and $J_{H}$
respectively. Assume that the actions commute, that $J_{G}$ is
$H$-invariant, and that $J_{H}$ is $G$-invariant. Let $\alpha\in\mathfrak{g}^{*}$ be in
the image of $J_{G}$ and $\beta\in\mathfrak{h}^{*}$ in the image of $J_{H}$.
Then the $G$ action drops to a Poisson action on $M\spr{\beta}H$ and
$J_{G}$ factors to a momentum map\xspace $j_{G}$ for the induced action. Likewise,
the $H$ action drops to a Poisson action on $M\spr{\alpha}G$ and
$J_{H}$ factors to a momentum map\xspace $j_{H}$ for the induced
action. Furthermore, we have
\[
(M\spr{\alpha}G)\spr{\beta}H
\cong
M\spr{(\alpha,\beta)}(G\times H)
\cong
(M\spr{\beta}H)\spr{\alpha}G
\]
as symplectic stratified spaces.
\end{theorem}
\begin{proof}
An outline of a proof of this result is given in \cite[Section~4]{SL91} for the case that
$G$ and $H$ are compact. Using the machinery of singular symplectic reduction
for proper Hamiltonian actions as described in \cite{OR04} the proof
of \cite{SL91} extends to the more general setting.
\end{proof}
\section{The bundle picture}\label{sec:bun-pic}
From now on let $G$ be a Lie group acting properly from the left on a
manifold $Q$.
The $G$ action then induces a Hamiltonian action
on the cotangent bundle $T^{*}Q$ by cotangent lifts. This means that
the lifted action respects the canonical symplectic form
$\Omega=-d\theta$ on $T^{*}Q$ where $\theta$ is the Liouville form on
$T^{*}Q$, and,
moreover, there is an
equivariant momentum map\xspace
$\mu:
T^{*}Q\to\mathfrak{g}^{*}$ given by
$\vv<\mu(q,p),X>
= \theta(\zeta^{T^{*}Q}_{X})(q,p)
= \vv<p,\zeta_{X}(q)>$
where
$(q,p)\in T^{*}Q$, $X\in\mathfrak{g}$, $\zeta_{X}$ is the fundamental
vector field associated to the $G$-action on $Q$,
and $\zeta^{T^{*}Q}_{X}\in\mathfrak{X}(T^{*}Q)$ is the
fundamental vector field associated to the cotangent lifted action.
In this section we want to apply the Slice Theorem (\cite{PT88,DK99,OR04})
to the action of $G$ on $Q$ to get a local model of
the singular symplectic reduced space $T^{*}Q\mbox{$/\negmedspace/_{\mathcal{O}}$} G =
\mu^{-1}(\mbox{$\mathcal{O}$})/G$ where $\mbox{$\mathcal{O}$}$ is a coadjoint orbit in the image of
$\mu$.
Thus we consider a tube $U$ in $Q$ around an orbit $G.q$ with
$G_{q}=H$. And we denote the slice at $q$ by $S$ such that
\[
U\cong G\times_{H}S
\]
as $G$-spaces. Here the action on $U$ is given by the restriction of
the $G$-action on $Q$ to the invariant neighborhood $U$ of $G.q$.
On the other hand, the
action on $G\times_H S$ is given by $g.[(k,s)]_H = [(gk,s)]_H$ where
$g\in G$ and $[(k,s)]_H$ denotes the class of $(k,s)\in G\times S$ in
$G\times_H S$. Moreover, note that the $H$-action on $G\times S$ is
given by $h.(k,s) = (kh^{-1},h.s)$ where $h\in H$.
(See \cite{PT88,DK99,OR04}.)
In particular, it follows that $U/G\cong S/H$ as stratified spaces
with smooth structure. (See \cite{Pfl01,Pfl01a}.)
Assume for a moment that the action by $G$ on $U$ is free, whence
$U\cong G\times S$. Let $\mu: T^{*}U\to\mathfrak{g}^{*}$ be the canonical
momentum mapping, $\lambda\in\mathfrak{g}^{*}$ a regular value in the image of
$\mu$, and $\mbox{$\mathcal{O}$}$ the coadjoint orbit passing through $\lambda$. Then we
have
\begin{align*}
(T^{*}U)\spr{\mathcal{O}}G
&=
(T^{*}U)\spr{\lambda}G
=
(T^{*}G\times T^{*}S)\spr{\lambda}G
=
(T^{*}G)\spr{\lambda}G\times T^{*}S\\
&=
\mbox{$\mathcal{O}$}\times T^{*}(U/G)
\end{align*}
as symplectic spaces; since $T^{*}G\spr{\lambda}G = \mbox{$\mathcal{O}$}$.
The aim of this section is to drop the
freeness assumption. To do so we will take the same approach as Schmah
\cite{Sch04} and use singular commuting reduction.
Now we return to the case where $U=G\times_{H}S$ as introduced
above. On $G\times S$ we will be concerned with two commuting
actions. These are
\begin{align}
\lambda: G\times G\times S&\longrightarrow G\times S, \quad\lambda_{g}(k,s)=(gk,s)\\
\tau: H\times G\times S&\longrightarrow G\times S, \quad\tau_{h}(k,s)=(kh^{-1},h.s).
\end{align}
These actions obviously commute. The latter, i.e., $\tau$ is called the
twisted action by $H$ on $G\times S$. We can cotangent lift $\lambda$ and
$\tau$ to give Hamiltonian transformations on $T^{*}(G\times S)$ with
momentum mappings $J^{\lambda}$ and $J^{\tau}$, respectively. By left
translation we trivialize
$T^{*}(G\times S)
=
(G\times\mathfrak{g}^{*})\times T^{*}S$.
To facilitate the notation we will denote the cotangent lifted action
of $\lambda$, $\tau$ again by $\lambda$, $\tau$ respectively.
\begin{lemma}\label{lem:actions}
Let $(k,\eta;s,p)\in G\times\mathfrak{g}^{*}\times T^{*}S$. Then we have the
following formulas.
\begin{align}
J^{\lambda}(k,\eta;s,p)
&= \mbox{$\text{\upshape{Ad}}$}(k^{-1})^{*}.\eta\label{eq:J-lam}
=: \mbox{$\text{\upshape{Ad}}$}^{*}(k).\eta\in\mathfrak{g}^{*},\\
J^{\tau}(k,\eta;s,p)
&= -\eta|\mathfrak{h}+\mu(s,p)\in\mathfrak{h}^{*}\label{eq:J-tau}
\end{align}
where $\mu$ is the canonical momentum map on $T^{*}S$. Moreover,
the actions $\lambda$ and $\tau$ commute, and
$J^{\lambda}$ is $H$-invariant and $J^{\tau}$ is $G$-invariant.
\end{lemma}
Since the formula of the canonical momentum map on $T^{*}S$
with regard to the $H$-action
is the same as that on
$T^{*}U$
with regard to the $G$-action
we use the same symbol $\mu$ for both
these maps. It will be clear from the context whether $\mu$
denotes the $H$- or the $G$-momentum map whence this will not cause any
confusion.
\begin{proof}
We denote the left action by $G$ on itself by $L$, the right action by
$R$, and the conjugate action by $\mbox{$\text{\up{conj}}$}$. In this notation we then
have $\mbox{$\text{\upshape{Ad}}$}(k).X=T_{e}\mbox{$\text{\up{conj}}$}_{k}.X$, and
$\mbox{$\text{\up{conj}}$}_{k}=L_{k}\circ R^{k^{-1}}=R^{k^{-1}}\circ L_{k}$.
It is straightforward to verify that the
cotangent lifted actions of $L$ and $R$ on $T^{*}G=G\times\mathfrak{g}^{*}$
are given by
\begin{align*}
T^{*}L_{g}(k,\eta)&=(gk,\eta)=(gk,-\eta\circ\zeta^{R^{-1}}(g))\\
T^{*}R^{g}(k,\eta)&=(kg,\mbox{$\text{\upshape{Ad}}$}^*(g^{-1}).\eta)=(kg,\eta\circ\zeta^{L}(g^{-1}))
\end{align*}
where $\zeta^{L}$ and $\zeta^{R}$ denote the fundamental vector field
mappings associated to $L$ and $R$ respectively.
Using the left trivialization $TG = G\times\mathfrak{g}$ we thus find that
\begin{align*}
\vv<J^{\lambda}(k,\eta;s,p),X>
&= \vv<\eta,\zeta^{L}_{X}(k)>
= \vv<\eta,T_kL_{k^{-1}}\dd{t}{}|_0\exp{(tX)}k>\\
&= \vv<\eta,T_e(L_{k^{-1}}\circ R^k).X>
= \vv<\mbox{$\text{\upshape{Ad}}$}^{*}(k).\eta,X>
\end{align*}
for all $X\in \mathfrak{g}$ which shows the first claim.
Likewise, it furthermore follows that $\vv<J^{\tau}(k,\eta;s,p),Z> = \vv<-\eta,Z> +
\vv<p,\zeta_{X}(s)>$ for all $Z\in\mathfrak{h}$.
The invariance of $J^{\lambda}$ and $J^{\tau}$ is
immediate from the formulas of the trivialized cotangent lifted
actions.
\end{proof}
\begin{corollary}
Let $\alpha\in\mathfrak{g}^{*}$ and $\beta\in\mathfrak{h}^{*}$ such that $\alpha$,
$\beta$ is in the image of $J^{\lambda}$, $J^{\tau}$ respectively. Then
the following are true.
\begin{enumerate}[\upshape (1)]
\item The action $\lambda$ descends to a Hamiltonian action on the
Marsden-Weinstein reduced space $T^{*}(G\times
S)\spr{\beta}H$. Moreover, $J^{\lambda}$ factors to a momentum map\xspace
$j_{\lambda}: T^{*}(G\times S)\spr{\beta}H\to\mathfrak{g}^{*}$
for this action.
\item The action $\tau$ descends to a Hamiltonian action on the
Marsden-Weinstein reduced space $T^{*}(G\times
S)\spr{\alpha}G$. Moreover, $J^{\tau}$ factors to a momentum map\xspace
$j_{\tau}: T^{*}(G\times S)\spr{\alpha}G\to\mathfrak{h}^{*}$
for this action.
\item The product action $G\times H\times T^{*}(G\times S)\to
T^{*}(G\times S)$, $(k,h,u)\mapsto \lambda_{k}.\tau_{h}.u$ is
Hamiltonian with momentum map\xspace $(J^{\lambda},J^{\tau})$. Moreover,
\begin{align*}
(T^{*}(G\times S)\spr{\alpha}G)\spr{\beta}H
=
T^{*}(G\times S)\spr{(\alpha,\beta)}(G\times H)
=
(T^{*}(G\times S)\spr{\beta}H)\spr{\alpha}G
\end{align*}
as singular symplectic spaces.
\end{enumerate}
\end{corollary}
\begin{proof}
Since the actions by $\lambda$ and $\tau$ are free the first two
assertions can be deduced from the regular commuting reduction theorem
(\cite{MMOPR03})
with the necessary conditions being verified in the above
lemma. Clearly, the product action by $G\times H$ is well-defined and
Hamiltonian with asserted momentum map\xspace. However, the product action will not
be free in general. Thus the last point is a consequence of the
singular commuting reduction theorem of Section \ref{s:sg_com_red}.
\end{proof}
We will only be interested in the case where $\beta=0$.
Moreover,
on $T^*G$ we shall only be concerned with the lifted $\lambda$-action.
Thus the expression
$T^{*}G\spr{\alpha}G$ will throughout stand for
$(J^{\lambda})^{-1}(\alpha)/G_{\alpha}$.
\begin{proposition}\label{prop:bun_pic}
Clearly, $0$ is in the image of $J^{\tau}$. Therefore,
\begin{align*}
T^{*}U\spr{\alpha}G
&\cong
T^{*}(G\times_{H}S)\spr{\alpha}G
=
T^{*}(G\times S)\spr{0}H\spr{\alpha}G
=
T^*(G\times S)\spr{\alpha}G\spr{0}H\\
&=
(T^{*}G\spr{\alpha}G\times T^{*}S)\spr{0}H
=
(\mbox{$\mathcal{O}$}\times T^{*}S)\spr{0}H
\end{align*}
as stratified symplectic spaces, and
where $\mbox{$\mathcal{O}$}=\mbox{$\text{\upshape{Ad}}$}^{*}(G).\alpha$.
\end{proposition}
\begin{proof}
Since the isomorphism\xspace $T^{*}U\xrighto{\simeq}T^{*}(G\times_{H}S)$ comes
from an equivariant diffeomorphism\xspace $U\xrighto{\simeq}G\times_{H}S$ on the base it is
an equivariant symplectomorphism\xspace that intertwines the respective momentum
maps. Now the regular reduction theorem
for cotangent bundles at zero
momentum says that $T^{*}(G\times_{H}S)$ and $T^{*}(G\times
S)\spr{0}H$ are symplectomorphic.
(See \cite[Theorem~4.3.3]{AM78} and the remark immediately below \cite[Theorem~4.3.3]{AM78}.)
Further it is well-known
(and immediate from Lemma~\ref{lem:actions}(\ref{eq:J-lam}))
that
$T^{*}G\spr{\alpha}G=\mbox{$\mathcal{O}$}$. The rest is a direct consequence of
Theorem \ref{thm:sg_comm_red} on singular commuting reduction.
\end{proof}
From now on we make the assumption that
\[
Q=Q_{(H)},
\]
i.e., all isotropy subgroups of points $q\in Q$ are conjugate within
$G$ to $H$.
Obviously, this assumption imposes a rather strong restriction on the
generality of the subsequent. However, in applications such as in the
Calogero-Moser system of Section~\ref{sec:cms} this is the generic case
in a certain sense.
See also Remark~\ref{rem:stratQ}.
\begin{theorem}[Bundle picture]\label{thm:bun_pic}
Let $Q=Q_{(H)}$ and
let $\mbox{$\mathcal{O}$}\subseteq\mathfrak{g}^{*}$ be a coadjoint orbit in the image
of the momentum map\xspace
$\mu: T^{*}Q\to\mathfrak{g}^{*}$.
Then, locally, we have a singular symplectic fiber bundle
\[
\xymatrix{
{\mbox{$\mathcal{O}$}\spr{0}H}\ar @{^{(}->}[r]
&{T^{*}Q\spr{\mathcal{O}}G} \ar[r]
&{T^{*}(Q/G)}
}
\]
with typical fiber the singular symplectic space $\mbox{$\mathcal{O}$}\spr{0}H$ and smooth base
$T^{*}(Q/G)$.
\end{theorem}
The fiber bundle in this theorem is singular in the
sense that it is a topological fiber bundle and the transition
functions act by strata preserving transformations on $\mbox{$\mathcal{O}$}\spr{0}H$
which are smooth in the sense that they preserve the algebra
$\mbox{$C^{\infty}$}(\mbox{$\mathcal{O}$}\spr{0}H) := W^{\infty}(\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h})^H$
where $W^{\infty}(\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h})$ denotes the Whitney $\mbox{$C^{\infty}$}$
functions on $\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h}\subset\mbox{$\mathcal{O}$}$.
See
\cite{ACG91,Pfl01,Pfl01a} for more on smooth structures on singular
(symplectic) spaces.
\begin{proof}
Consider a tube $U$ of the $G$-action on $Q$. By virtue of the Slice
Theorem (\cite{PT88,DK99,OR04})
there thus exists a slice $S$ such that there is a
$G$-equivariant diffeomorphism\xspace
\[
U\cong G\times_{H}S = G/H\times S.
\]
Indeed, this is true
since all points of $Q$ are regular by assumption
whence the slice
representation is trivial. We can lift this diffeomorphism to a
symplectomorphism of cotangent bundles to get
\[
T^{*}U\spr{\mathcal{O}}G
\cong \mbox{$\mathcal{O}$}\spr{0}H\times T^{*}S
\]
as in Proposition \ref{prop:bun_pic} above. Since $T^{*}S$ is a typical
neighborhood in $T^{*}(Q/G)$ the result follows.
\end{proof}
\begin{rem}[On fully singular reduction]\label{rem:stratQ}
For the purpose of this remark assume that the isotropy lattice of the
$G$-action on $Q$ consists of more than one isotropy class. Let $(H)$
be an isotropy class on $Q$, and let $(L)$ be an isotropy class of the
lifted $G$-action on $T^*Q$.
Let $\mbox{$\textup{Ann}\,$} Q_{(H)}\to Q_{(H)}$ denote the sub-bundle of $(T^*Q)|Q_{(H)}$
consisting of those co-vectors which vanish upon insertion of a vector
tangent to $Q_{(H)}$. Clearly, we have
\[
(T^*Q)_{(L)}|Q_{(H)}
=
(T^*Q_{(H)}\times_{Q_{(H)}}\mbox{$\textup{Ann}\,$} Q_{(H)})_{(L)},
\]
and note that the momentum map\xspace $\mu: T^*Q\to\mathfrak{g}^*$ vanishes on
$\mbox{$\textup{Ann}\,$} Q_{(H)}$.
Therefore,
for an orbit $\mbox{$\mathcal{O}$}$ in the image of $\mu$ we have that
\[
\mu^{-1}(\mbox{$\mathcal{O}$})|Q_{(H)}
=
\mu_{(H)}^{-1}(\mbox{$\mathcal{O}$})\times_{Q_{(H)}}\mbox{$\textup{Ann}\,$} Q_{(H)}
\]
where $\mu_{(H)}$
denotes the momentum map\xspace of the cotangent lifted
$G$-action on $T^*Q_{(H)}$.
The $G$-equivariant projection
$\mu_{(H)}^{-1}(\mbox{$\mathcal{O}$})\times_{Q_{(H)}}\mbox{$\textup{Ann}\,$} Q_{(H)}\to\mu_{(H)}^{-1}(\mbox{$\mathcal{O}$})$
gives rise to a
mapping
\[
\tag{F}\label{equ:F}
\eta_{(L)}:
(\mu_{(H)}^{-1}(\mbox{$\mathcal{O}$})\times_{Q_{(H)}}\mbox{$\textup{Ann}\,$} Q_{(H)})_{(L)}/G
\longrightarrow\mu_{(H)}^{-1}(\mbox{$\mathcal{O}$})/G
= T^*(Q_{(H)})\mbox{$/\negmedspace/_{\mathcal{O}}$} G
\]
the base of which is
described by Theorem~\ref{thm:WWred} in the presence of a generalized
connection form on $Q_{(H)}\twoheadrightarrow Q_{(H)}/G$.
The map $\eta_{(L)}$ is, in general, neither surjective nor does it have locally constant fiber type.
The fiber over a point $[x]\in\mu_{(H)}^{-1}(\mbox{$\mathcal{O}$})/G$
(such that $G_{\tau(x)}=H$)
is of the form
\[
\eta_{(L)}^{-1}([x])
=
\set{w\in\mbox{$\textup{Ann}\,$}_{\tau(x)}Q_{(H)}: H_w\cap G_x\text{ is conjugate to } L \text{ within } G}/G_x
\]
where $\tau: T^*Q_{(H)}\to Q_{(H)}$
is the cotangent projection.
Note that $gLg^{-1}\subset G_x\subset G_{\tau(x)}$
for some $g\in G$
by equivariance of projections.
The image of $\eta_{(L)}$ clearly is a union of orbit type strata. Moreover, using the
notation of Duistermaat and Kolk~\cite[Definition~2.6.1]{DK99} it is evident
that
\[
\mbox{$\text{\up{im}}\,$}\eta_{(L)}\subset(\mu_{(H)}^{-1}(\mbox{$\mathcal{O}$}))_{x}^{\lesssim}/G
= G.(\mu_{(H)}^{-1}(\mbox{$\mathcal{O}$}))^L/G
\]
where
$x\in(\mu_{(H)}^{-1}(\mbox{$\mathcal{O}$}))_{(L)} = (\mu_{(H)}^{-1}(\mbox{$\mathcal{O}$}))_x^{\sim}$,
whence it follows that
$(\mu_{(H)}^{-1}(\mbox{$\mathcal{O}$}))_{(L)}/G\subset\mbox{$\text{\up{im}}\,$}\eta_{(L)}$ is open and dense
since it is the regular stratum of $\mbox{$\text{\up{im}}\,$}\eta_{(L)}$.
Let
$M_0
:= (T^*(Q_{(H)})\mbox{$/\negmedspace/_{\mathcal{O}}$} G)_{(L)}
:= (\mu_{(H)}^{-1}(\mbox{$\mathcal{O}$}))_{(L)}/G$
(see Theorem~\ref{thm:WWred})
and consider the restriction
$\eta_0 := \eta_{(L)}|\eta_{(L)}^{-1}(M_0)$ which yields a bundle like object
\[
\eta_0:
\eta_{(L)}^{-1}(M_0)\longrightarrow M_0,
\]
that is, $\eta_0$ is surjective and the fiber over a point $[x]\in
M_0$ such that $G_x=L$ is $(\mbox{$\textup{Ann}\,$}_{\tau(x)}Q_{(H)})^L$.
We believe that this object can be shown to constitute a smooth fiber bundle.
However, the employability of this `bundle'
is quite limited by the fact that
we cannot give a satisfactorily useful description of
$\eta_{(L)}^{-1}(M_0)$.
Further problems are deciding what a generalized connection form on
$Q\twoheadrightarrow Q/G$ should be and determining how the `secondary
strata'
$(\mu_{(H)}^{-1}(\mbox{$\mathcal{O}$})\times_{Q_{(H)}}\mbox{$\textup{Ann}\,$} Q_{(H)})_{(L)}/G$ fit
together to yield the `primary stratum'
$(\mu^{-1}(\mbox{$\mathcal{O}$})\cap(T^*Q)_{(L)})/G$. The latter problem was solved in
\cite{PRS03} for reduction at trivial orbits $\mbox{$\mathcal{O}$}=\set{\textup{point}}$.
If $Q_{(H)}=Q_{\textup{reg}}$ is the regular stratum which is open dense in
$Q$ then $\mbox{$\textup{Ann}\,$} Q_{(H)}$ is trivial. In this (generic) case
Theorems~\ref{thm:bun_pic} and \ref{thm:WWred} thus provide a full answer to the
reduction problem. In the more general situation these results clearly
provide only a partial answer to the reduction problem. However, it is
expected that any solution to this problem will rely on these single
orbit type results.
\end{rem}
\section{Gauged cotangent bundle reduction}\label{sec:gauged_red}
Continue
to assume that we are in the situation of Section~\ref{sec:bun-pic}.
In particular, we suppose that $Q=Q_{(H)}$ is of single orbit type.
However, as an additional input datum we assume from now on a
generalized
principal bundle connection form $A\in\Omega^1(Q;\mathfrak{g})$ on $Q\twoheadrightarrow Q/G$
given. The term generalized is to be understood in the context of
Alekseevsky and Michor~\cite[Section~3.1]{AM95}. This means that $A: TQ\to\mathfrak{g}$ is
$G$-equivariant and that $\zeta = \zeta\circ A\circ\zeta$.
In particular, the connection form $A$ induces a right inverse to the
projection $\mathfrak{g}\twoheadrightarrow\mathfrak{g}/\mathfrak{g}_q$ depending smoothly on $q\in Q$.
According to
\cite[Section 4.6]{AM95} the curvature form associated to $A$ is
defined by
\[
\mbox{$\textup{Curv}$}^{A}
:=
dA-\by{1}{2}[A,A]^{\wedge}
\]
where
\[
[\varphi,\psi]^{\wedge}(v_1,\dots,v_{l+k})
:=
\by{1}{k!l!}
\sum_{\sigma}\sign\sigma
[\varphi(v_{\sigma1},\dots,v_{\sigma l}),\psi(v_{\sigma(l+1)},\dots,v_{\sigma(l+k)})]
\]
is the graded Lie bracket on
$
\Omega(Q;\mathfrak{g})
:= \bigoplus_{k=0}^{\infty}\Gamma(\Lambda^{k}T^{*}Q\otimes\mathfrak{g})
$,
and $\varphi\in\Omega^{l}(Q;\mathfrak{g})$
and $\psi\in\Omega^{k}(Q;\mathfrak{g})$. The sign in our definition of $\mbox{$\textup{Curv}$}^A$
differs from that in \cite{AM95} because we are concerned with left
$G$-actions as opposed to right actions.
Since the $G$-action on $Q$ is of single orbit type the orbit
space $Q/G$ is a smooth manifold, and the projection $\pi: Q\to Q/G$
is a fiber bundle with typical fiber $G/H$.
However, the isotropy lattice of the lifted action by $G$ on
$T^{*}Q$ is, in general (for $H\neq\set{e}$), non-trivial whence
the quotient space $(T^{*}Q)/G$
is a stratified space. Its strata are of the form
$(T^{*}Q)_{(L)}/G$ where $(L)$ is in the isotropy lattice of $T^{*}Q$.
The vertical sub-bundle of $TQ$ with respect to $\pi: Q\to Q/G$ is
$\mbox{$\textup{Ver}$} := \ker T\pi$. Via the connection $A$ we can also define the
horizontal sub-bundle $\mbox{$\textup{Hor}$} := \ker A$. We define the dual
horizontal sub-bundle of $T^{*}Q$ as the sub-bundle $\mbox{$\textup{Hor}$}^{*}$
consisting of those co-vectors that vanish on all vertical
vectors. Likewise, we define the dual
vertical sub-bundle of $T^{*}Q$ as the sub-bundle $\mbox{$\textup{Ver}$}^{*}$
consisting of those co-vectors that vanish on all horizontal
vectors. As usual, the connection $A$ provides a trivialization of the
vertical sub-bundle, i.e., $\mbox{$\textup{Ver}$} \cong_{A} \mbox{$\bigsqcup$}_{q\in Q}\mathfrak{g}/\mathfrak{g}_q$.
In particular, $\mbox{$\bigsqcup$}_{q\in Q}\mathfrak{g}/\mathfrak{g}_q$ and $\mbox{$\bigsqcup$}_{q\in Q}\mbox{$\textup{Ann}\,$}\mathfrak{g}_q$
are smooth vector bundles.
\subsection{Mechanical connection}\label{sub:mech-con}
If $(Q,\vv<.,.>)$ is a Riemannian manifold and $G$ acts on $Q$ by isometries
there is a certain connection which is particularly well adapted to
mechanical systems on $Q$. This is the so-called mechanical connection
which is defined as follows.
For $X,Y\in\mathfrak{g}$ and $q\in Q$ we define
$\mbox{$\mathbb{I}$}_{q}(X,Y) := \vv<\zeta_{X}(q),\zeta_{Y}(q)>$ and call this
the \caps{locked inertia tensor}.
This defines a
non-degenerate pairing on $\mathfrak{g}/\mathfrak{g}_q$ whence
it provides an identification $\check{\mbox{$\mathbb{I}$}_{q}}:
\mathfrak{g}/\mathfrak{g}_{q}\to(\mathfrak{g}/\mathfrak{g}_{q})^{*} = \mbox{$\textup{Ann}\,$}\mathfrak{g}_{q}$. We use this isomorphism\xspace
to define a one-form $\tilde{A}$ on $Q$ with values in the bundle
$\mbox{$\bigsqcup$}_{q\in Q}\mathfrak{g}/\mathfrak{g}_{q}$ by the following diagram.
\[
\xymatrix{
{T_{q}^{*}Q} \ar[r]^-{\mu_{q}}
&
{\mbox{$\textup{Ann}\,$}\mathfrak{g}_{q}}
\ar[d]^{(\check{\mathbb{I}_{q}})^{-1}}
\\
{T_{q}Q} \ar[u]^{\simeq}
\ar @{-->}[r]^{\tilde{A}_{q}}&
{\mathfrak{g}/\mathfrak{g}_{q}}
}
\]
Notice that $\mbox{$\text{\up{im}}\,$}\mu_q = \mbox{$\textup{Ann}\,$}\mathfrak{g}_q$ by reason of dimension.
Thus we have a trivialization
$\mbox{$\textup{Ver}$}\cong_{\tilde{A}}\mbox{$\bigsqcup$}_{q\in Q}\mathfrak{g}/\mathfrak{g}_q$
of the vertical sub-bundle. However, to obtain a
generalized connection form on $Q\twoheadrightarrow Q/G$ from this trivialization
we have to assume one additional object: namely, let $r_q: \mathfrak{g}/\mathfrak{g}_q$
be a right inverse to the projection $\mathfrak{g}\twoheadrightarrow \mathfrak{g}/\mathfrak{g}_q$ depending
smoothly on $q\in Q$.
The (generalized) \caps{mechanical connection} $A: TQ\to\mathfrak{g}$
on $Q\twoheadrightarrow
Q/G$ is thus defined as the composition
$A_q = r_q\circ\tilde{A}_q:
T_qQ\to\mathfrak{g}/\mathfrak{g}_q\to\mathfrak{g}$.
One obvious way to obtain such a right inverse
is to choose a $G$-invariant non-degenerate bilinear
form on $\mathfrak{g}$ such that $\mathfrak{g}/\mathfrak{g}_q\cong\mathfrak{g}_q^{\bot}\hookrightarrow\mathfrak{g}$ with respect to
this form. In many examples such a non-degenerate form is given
canonically.
The mechanical connection was first defined by Smale \cite{Sma70}
for the case of Abelian group actions. See also
Marsden, Montgomery, and Ratiu \cite[Section 2]{MMR90}.
To verify that the mechanical connection is indeed a generalized
principal connection form one checks that
$A: TQ\to\mathfrak{g}$
is equivariant
and
$\zeta(q)(A_{q}(\zeta_{X}(q)))=\zeta_X(q)$ for all $X\in\mathfrak{g}$.
\subsection{Weinstein realization of $T^*Q$}
Let $A$ continue to denote a generalized
principal connection form on $Q$, and let $\vv<.,.>$ denote the dual pairing.
We define a point-wise dual
$A_{q}^{*}:
\mbox{$\textup{Ann}\,$}\mathfrak{g}_{q}\to
\mbox{$\textup{Ver}$}_{q}^{*}\subseteq T_{q}^{*}Q$
by the formula
$
\vv<A_{q}^{*}(\lambda),v>
=
\vv<\lambda,A_{q}(v)>
$
where $\lambda\in\mbox{$\textup{Ann}\,$}\mathfrak{g}_{q}$ and $v\in T_{q}Q$. Notice that
$
A_{q}^{*}(\mu_{q}(p)) = p
$
for all $p\in\mbox{$\textup{Ver}$}_{q}^{*}$
and
$
\mu_{q}(A_{q}^{*}(\lambda)) = \lambda
$
for all $\lambda\in\mbox{$\textup{Ann}\,$}\mathfrak{g}_{q}$ since $A$ is a connection form.
Let $\pi: Q\twoheadrightarrow Q/G$ and $\tau_Q: TQ\to Q$ denote the
projections. From the connection form $A$ we obtain the horizontal
lift mapping which we denote by
\[
C := ((\tau_Q,T\pi)|\mbox{$\textup{Hor}$})^{-1}:
Q\times_{Q/G}T(Q/G)\to \mbox{$\textup{Hor}$}\hookrightarrow TQ.
\]
Its fiber restriction shall be
denoted by $C_q: \set{q}\times T_{\pi(q)}(Q/G)\to\mbox{$\textup{Hor}$}_q\hookrightarrow T_qQ$.
Using the horizontal lift
$C$
on the one hand and the connection
$A$ on the other hand
we obtain a $G$-equivariant isomorphism\xspace
\[
TQ = \mbox{$\textup{Hor}$}\oplus\mbox{$\textup{Ver}$}
\longrightarrow
(Q\times_{Q/G}T(Q/G))\times_{Q}\mbox{$\bigsqcup$}_{q\in Q}\mathfrak{g}/\mathfrak{g}_{q}
\]
of bundles over $Q$.
There is a dual
version to this isomorphism\xspace, and we will abbreviate
\[
\mbox{$\mathcal{W}$} :=
(Q\times_{Q/G}T^{*}(Q/G))\times_{Q}\mbox{$\bigsqcup$}_{q\in Q}\mbox{$\textup{Ann}\,$}{\mathfrak{g}_{q}}
\cong
\mbox{$\textup{Hor}$}^{*}\oplus\mbox{$\textup{Ver}$}^{*}.
\]
The explicit form of the isomorphism\xspace $\mbox{$\mathcal{W}$}\cong T^*Q$ is stated in
Proposition~\ref{prop:WW} below.
To set up some notation for the upcoming proposition,
and clarify the picture consider the following
stacking of pull-back diagrams.
\[
\xymatrix{
{\mbox{$\mathcal{W}$}} \ar[r]^-{\rho^{*}\widetilde{\tau}=\widetilde{\widetilde{\tau}}}
\ar[d]_{\widetilde{\tau}^{*}\rho=\widetilde{\rho}} &
{\mbox{$\bigsqcup$}_{q}\mbox{$\textup{Ann}\,$}\mathfrak{g}_{q}}
\ar[d]^{\rho} \\
{Q\times_{Q/G}T^{*}(Q/G)}
\ar[r]^-{\pi^{*}\tau=\widetilde{\tau}}
\ar[d]_{\tau^{*}\pi=\widetilde{\pi}} &
Q
\ar[d]^{\pi} \\
T^{*}(Q/G)
\ar[r]^-{\tau} &
Q/G
}
\]
The upper stars in this diagram are, of course, not pull-back
stars. By slight abuse of notation we shall denote elements
$(q;\pi(q),\eta;q,\lambda)\in\mbox{$\mathcal{W}$}$ simply by $(q,\eta,\lambda)$.
Further, let $\tau_{\mathcal{W}}: T\mbox{$\mathcal{W}$} \to \mbox{$\mathcal{W}$}$ denote the tangent
projection.
\begin{proposition}[Symplectic structure on \mbox{$\mathcal{W}$}]\label{prop:WW}
The chosen connection form $A$ induces an isomorphism\xspace
\begin{align*}
\psi=\psi(A):
(&Q\times_{Q/G}T^{*}(Q/G))\times_{Q}\mbox{$\bigsqcup$}_{q\in Q}\mbox{$\textup{Ann}\,$}\mathfrak{g}_{q} = \mbox{$\mathcal{W}$}
\longrightarrow T^{*}Q, \\
(&q,\eta,\lambda)
\longmapsto
(q,(T_q\pi)^*\eta+A_q^{*}(\lambda))
\end{align*}
of bundles over $Q$,
and the following are true.
\begin{enumerate}[\upshape (1)]
\item
There is an induced $G$-action on $(\mbox{$\mathcal{W}$},\sigma=\psi^*\Omega)$ by
symplectomorphisms.
Here $\Omega=-d\theta$ is the canonical symplectic form on $T^*Q$.
Moreover, this action is Hamiltonian with momentum
map
\[
\mu_{A} = \mu\circ\psi: \mbox{$\mathcal{W}$}\longrightarrow\mathfrak{g}^{*}, (q,\eta,\lambda)\longmapsto\lambda,
\]
where $\mu$ is the
momentum map $T^{*}Q\to\mathfrak{g}^{*}$,
and $\psi$ is equivariant.
\item
The induced symplectic form $\sigma$ on the connection
dependent realization $\mbox{$\mathcal{W}$}$ of $T^{*}Q$
is given by the formula
\[
\sigma =
(\wt{\pi}\circ\wt{\rho})^{*}\Omega^{Q/G} - dB.
\]
Here
$\Omega^{Q/G}=-d\theta^{Q/G}$ is the canonical symplectic form on $T^{*}(Q/G)$,
and
$B\in\Omega^{1}(\mbox{$\mathcal{W}$})$
is given by
\[
B
=
\vv<\tau_{\mathcal{W}}^*\mu_A,(\wt{\tau}\circ\wt{\rho})^*A>.
\]
Moreover,
\begin{align*}
dB
&=
\vv<\tau_{\mathcal{W}}^*d\mu_A\stackrel{\wedge}{,}(\wt{\tau}\circ\wt{\rho})^*A>
+ \vv<\tau_{\mathcal{W}}^*\mu_A , (\wt{\tau}\circ\wt{\rho})^*\mbox{$\textup{Curv}$}^A>\\
&\phantom{=}
+ \vv<\tau_{\mathcal{W}}^*\mu_A , \by{1}{2}[(\wt{\tau}\circ\wt{\rho})^*A,(\wt{\tau}\circ\wt{\rho})^*A]^{\wedge}>.
\end{align*}
Here $\vv<.\stackrel{\wedge}{,}.>$ denotes the exterior multiplication
of a $\mathfrak{g}^*$-valued form with a $\mathfrak{g}$-valued form.
In local coordinates where we may use a splitting of tangent vectors
$\xi_1,\xi_2\in T_{(q,\eta,\lambda)}\mbox{$\mathcal{W}$}$ as
$\xi_i
= (q'_i,\eta'_i,\lambda'_i)
$
and
$q'_i = v_{i}^{\textup{hor}}+\zeta_{Z_{i}}(q)$
for $i=1,2$
this means that
\[
dB_{(q,\eta,\lambda)}(\xi_1,\xi_2) \\
=
\vv<\lambda'_1,Z_2> - \vv<\lambda'_{2},Z_1>
+
\vv<\lambda,\mbox{$\textup{Curv}$}^{A}_{q}(q'_1,q'_2)>
+ \vv<\lambda,[Z_{1},Z_{2}]>.
\]
\end{enumerate}
\end{proposition}
\begin{proof}
Obviously, $\psi$ is a bundle map. Its inverse is given by
the bundle map
$(q,p)\mapsto(q,C_q^*(p),\mu_q(p))$ where $C_q: \set{q}\times
T_{\pi(q)}(Q/G)\to T_qQ$ is the horizontal lift mapping and $\mu_q :=
\mu|T_q^*Q$ is the fiber restriction of the momentum map\xspace.
Assertion (1) is clear from the construction.
To see assertion (2) we work locally in $Q$. That is, let $U$ be a
trivializing patch for $\mbox{$\mathcal{W}$}\to Q$ as well as for $T^*Q\to Q$ and
consider $\psi|U: \mbox{$\mathcal{W}$}|U\to T^*Q|U$. Let $(q,\eta,\lambda)\in\mbox{$\mathcal{W}$}|U$ and
$\xi\in T_{(q,\eta,\lambda)}\mbox{$\mathcal{W}$}$. By locality we may split $\xi$ as $\xi
= (q',\eta',\lambda')$ where $q'\in T_qQ$, $\eta'\in T_{\eta}(T^*(Q/G))$,
and $\lambda'\in T_{\lambda}(\mbox{$\bigsqcup$}_{q\in Q}\mbox{$\textup{Ann}\,$}\mathfrak{g}_q)$.
However, in order to make the notation not too cumbersome we do not invent new
symbols (like $\xi^U$ or $A^U$) for the local versions of global
objects (like $\xi$ or $A$).
To find the desired
formula for $\sigma = \psi^*\Omega = -d\psi^*\theta$, note that
\begin{align*}
(\psi^*\theta)_{(q,\eta,\lambda)}(\xi)
&=
\theta_{(q,(T_q\pi)^*\eta+A_q^*(\lambda))}(T_{(q,\eta,\lambda)}\psi.\xi)\\
&=
\theta_{(q,(T_q\pi)^*\eta+A_q^*(\lambda))}(T_{(q,\eta,\lambda)}\psi^1.\xi,T_{(q,\eta,\lambda)}\psi^2.\xi)\\
&=
\vv<(T_q\pi)^*\eta+A_q^*(\lambda),T_{(q,(T_q\pi)^*\eta+A_q^*(\lambda))}\tau_Q.T_{(q,\eta,\lambda)}\psi^1.\xi>\\
&=
\vv<(T_q\pi)^*\eta,q'> + \vv<A_q^*(\lambda),q'>\\
&=
\vv<\eta,T_q\pi.q'> + \vv<\lambda,A_q(q')>\\
&=
((\wt{\pi}\circ\wt{\rho})^*\theta^{Q/G})_{(q,\eta,\lambda)}(\xi)
+
(\vv<\tau_{\mathcal{W}}^*\mu_A,(\wt{\tau}\circ\wt{\rho})^*A>)_{(q,\eta,\lambda)}(\xi)\\
&=
(((\wt{\pi}\circ\wt{\rho})^*\theta^{Q/G})+B)_{(q,\eta,\lambda)}(\xi)
\end{align*}
where
$\psi|U = (\psi^1,\psi^2): \mbox{$\mathcal{W}$}|U\to U\times V = T^*Q|U$
and $V$ is the standard fiber of
$\tau_Q: T^*Q\to Q$.
Since the first
and the last expressions in this computation are global objects it is
true that
\begin{align*}
\sigma
&=
(\wt{\pi}\circ\wt{\rho})^*\Omega^{Q/G}
-
dB\\
&=
(\wt{\pi}\circ\wt{\rho})^*\Omega^{Q/G}
-
\vv<\tau_{\mathcal{W}}^*d\mu_A\stackrel{\wedge}{,}(\wt{\tau}\circ\wt{\rho})^*A>\\
&\phantom{=i}
- \vv<\tau_{\mathcal{W}}^*\mu_A , (\wt{\tau}\circ\wt{\rho})^*\mbox{$\textup{Curv}$}^A>
+ \vv<\tau_{\mathcal{W}}^*\mu_A , \by{1}{2}[(\wt{\tau}\circ\wt{\rho})^*A,(\wt{\tau}\circ\wt{\rho})^*A]^{\wedge}>.
\end{align*}
Indeed, this is because
$\mbox{$\textup{Curv}$}^A = dA-\by{1}{2}[A,A]^{\wedge}$.
\end{proof}
The $G$-action on $\mbox{$\mathcal{W}$}$ is, of course, given by $g.(q,\eta,\lambda) = (g.q,\eta,\mbox{$\text{\upshape{Ad}}$}^*(g).\lambda)$.
Similarly there is an induced $G$-action on
$\mbox{$\bigsqcup$}_{q\in Q}\mathfrak{g}/\mathfrak{g}_q \cong \mbox{$\textup{Ver}$}$
which is given by $g.(q,X+\mathfrak{g}_q) = (g.q,\mbox{$\text{\upshape{Ad}}$}(g).X+\mathfrak{g}_{g.q})$.
The notion of a stratified map which appears in the following theorem
is defined in Section~\ref{sec:prel}.
\begin{theorem}[Weinstein space]
There are stratified isomorphisms of stratified bundles over $Q/G$:
\begin{align*}
\alpha=\alpha(A):
\mbox{$\bigsqcup$}_{(L)}(TQ)_{(L)}/G &\longrightarrow
T(Q/G)\times_{Q/G}\mbox{$\bigsqcup$}_{(L)}(\mbox{$\bigsqcup$}_{q\in Q}\mathfrak{g}/\mathfrak{g}_{q})_{(L)}/G,\\
[(q,v)] &\longmapsto(T\pi(q,v),[(q,A_{q}(v))])
\end{align*}
where $(L)$ runs through the isotropy lattice of $TQ$.
The dual isomorphism\xspace is given by
\begin{align*}
\beta=(\alpha^{-1})^{*}:
(T^{*}Q)/G &\longrightarrow T^{*}(Q/G)\times_{Q/G}(\mbox{$\bigsqcup$}_{q\in
Q}\mbox{$\textup{Ann}\,$}\mathfrak{g}_{q})/G =: W,\\
[(q,p)] &\longmapsto(C^{*}_q(p),[(q,\mu(q,p))])
\end{align*}
where the stratification was suppressed.
Here
\[
\smash{
C^{*}_q:
T^{*}_qQ\overset{\iota_q^*}{\to}\mbox{$\textup{Hor}$}_q^{*}\to T_{\pi(q)}^{*}(Q/G)
}
\]
is
the point wise dual to the horizontal lift mapping
\[
\smash{
C:
Q\times_{Q/G}T(Q/G)\longrightarrow\mbox{$\textup{Hor}$}\overset{\iota_q}{\hookrightarrow} TQ},
\text{ }
([q],v;q)\longmapsto C_{q}(v).
\]
Moreover, $\beta$ is an isomorphism\xspace of Poisson spaces as follows: we can
naturally identify
\begin{align*}
\mbox{$\mathcal{W}$}/G \overset{=}{\longrightarrow} W, \;
[(q;[q],\eta;q,\lambda)] \longmapsto ([q],\eta;[(q,\lambda)])
\end{align*}
thus obtaining a quotient Poisson bracket on
$\mbox{$C^{\infty}$}(W)=\mbox{$C^{\infty}$}(\mbox{$\mathcal{W}$})^{G}$ as the quotient Poisson bracket.
\end{theorem}
In the case that $G$ acts on $Q$ freely the first assertion of the
above theorem can also be
found in Cendra, Holm, Marsden, Ratiu \cite{CHMR98}. Following Ortega
and Ratiu \cite[Section 6.6.12]{OR04} the above constructed
interpretation $W$ of $(T^{*}Q)/G$ is called \caps{Weinstein space}
referring to Weinstein \cite{Wei78} where this universal
construction first appeared.
In fact, the original construction of \cite{Wei78} was the following:
Let $Q$ be a left free and proper $G$-space such that $Q\twoheadrightarrow Q/G$ is
endowed with a principal bundle connection form $A$ and let $F$ be a
right Hamiltonian $G$-space with equivariant momentum map\xspace $\Phi: F\to\mathfrak{g}^*$.
To make $F$ into a left Hamiltonian $G$-space we use the inversion in
the group. The momentum map\xspace with respect to the thus obtained $G$-action is
given by $-\Phi$.
Under these assumptions \cite{Wei78} proves
that the smooth symplectic quotient
$
(T^*Q\times F)\spr{0}G
=
(\mu-\Phi)^{-1}(0)/G
\cong_A
T^*(Q/G)\times_{Q/G}(Q\times_G F)
$
is symplectomorphic to the Sternberg space
$(Q\times_{Q/G}T^*(Q/G))\times_G F$ of \cite{Ste77}. Taking $F$ to be
the Hamiltonian $G$-space
$\mbox{$\mathcal{O}$}$ acted upon by $\mbox{$\text{\upshape{Ad}}$}^*(G)$, and employing the shifting trick
$T^*Q\mbox{$/\negmedspace/_{\mathcal{O}}$} G \cong (T^*Q\times\mbox{$\mathcal{O}$})\spr{0}G$, this construction
yields the realization
$T^*Q\mbox{$/\negmedspace/_{\mathcal{O}}$} G
\cong_A
T^*(Q/G)\times_{Q/G}(Q\times_G \mbox{$\mathcal{O}$})$. In particular, one thus obtains
a fiber bundle $\mbox{$\mathcal{O}$}\hookrightarrow T^*Q\mbox{$/\negmedspace/_{\mathcal{O}}$} G \to T^*(Q/G)$.
It thus makes sense to refer to the realization $W$ of $(T^*Q)/G$ which is
constructed along similar lines as a Weinstein space. The induced Poisson
structure on $W$ is explicitly described in \cite{HR05}.
\begin{proof}
As already noted above $(TQ)/G$ is a stratified space. Since the base
$Q$ is stratified as consisting only of a single stratum, the
equivariant foot
point projection map $\tau: TQ\to Q$ is trivially a stratified
map. Thus, we really get a stratified bundle $(TQ)/G\to Q/G$. In the
same spirit $(\mbox{$\bigsqcup$}_{q\in Q}\mathfrak{g}/\mathfrak{g}_{q})/G$ is stratified into
orbit types, and the projection onto $Q/G$ is a stratified bundle
map. According to Davis \cite{Dav78} pullbacks are well defined in
the category of stratified spaces and stratified maps
and thus it makes sense to define
$T(Q/G)\times_{Q/G}(\mbox{$\bigsqcup$}_{q\in Q}\mathfrak{g}/\mathfrak{g}_{q})/G$.
The map $\alpha$ is well defined: indeed, for $(q,v)\in TQ$ and
$k\in G$
we have
$T\pi(k.q,k.v)
= (\pi(k.q),T_{k.q}\pi(T_{q}l_{k}(v)))
= (\pi(q),T_{q}(\pi\circ l_{k})(v))
= T\pi(q,v)$,
and
$[(k.q,A_{k.q}(k.v))] = [(q,A_q(v))]$
by equivariance of $A$.
It is clearly continuous
as a composition of continuous maps.
We claim that $\alpha$ maps strata onto strata, and moreover we
have the formula
\[
\alpha((TQ)_{(L)}/G)
=
T(Q/G)\times_{Q/G}(\mbox{$\bigsqcup$}_{q\in Q}\mathfrak{g}/\mathfrak{g}_{q})_{(L)}/G.
\]
Indeed, this follows immediately since $\alpha$ lifts to a smooth equivariant
isomorphism\xspace
$\wt{\alpha}: TQ\to(Q\times_{Q/G}T(Q/G))\times_Q\mbox{$\bigsqcup$}_{q\in Q}\mathfrak{g}/\mathfrak{g}_q$,
$(q,v)\mapsto(q;T\pi(q,v);q,A_q(v))$
of vector bundles over $Q$,
and we have clearly that
$\wt{\alpha}((TQ)_{(L)}) =
(Q\times_{Q/G}T(Q/G))\times_Q(\mbox{$\bigsqcup$}_{q\in Q}\mathfrak{g}/\mathfrak{g}_q)_{(L)}$.
The restriction of $\alpha$ to any stratum is smooth
as a composition of smooth maps.
Since $\zeta(q)(A_{q}(\zeta_{X}(q))) = \zeta_X(q)$ for
$X\in\mathfrak{g}$ we can write down an inverse
\[
\alpha^{-1}:
([q],v;[(q,X)])
\to
[(q,C_{q}(v)+\zeta_{X}(q))]
\]
and again it is an easy matter to notice that this map is well
defined, continuous, and smooth on each stratum.
In fact, we have used here that the connection $A$ by definition
provides a right inverse to $\mathfrak{g}\twoheadrightarrow\mathfrak{g}/\mathfrak{g}_q$ whence by slight
abuse of notation we may consider elements $X\in\mathfrak{g}/\mathfrak{g}_q$ as elements
in $\mathfrak{g}$.
It makes sense to define the dual $\beta$ of the inverse map
$\alpha^{-1}$ in a point wise manner, and it only remains to compute
this map.
\begin{align*}
\vv<\beta[(q,p)],([q],v;[(q,X)])>
&=
\vv<[(q,p)],[(q,C_{q}(v)+\zeta_{X}(q))]> \\
&=
\vv<p,C_{q}(v)> + \vv<p,\zeta_{X}(q)> \\
&=
\vv<C^{*}_q(p),v> + \vv<\mu(q,p),X> \\
&=
\vv<(C^{*}_q(p),[(q,\mu(q,p))]),([q],v;[(q,X)])>
\end{align*}
where we used the $G$-invariance of the dual pairing over $Q$.
Finally, $\beta$ is an isomorphism\xspace of Poisson spaces:
note first that the identifying map
$\mbox{$\mathcal{W}$}/G\to W$,
$[(q;[q],\eta;q,\lambda)]\mapsto ([q],\eta;[(q,\lambda)])$
is well-defined because $G_{q}$ acts
trivially on $\mbox{$\textup{Hor}$}_{q}^{*} \cong T^{*}_{[q]}(Q/G)\ni\eta$ which in turn is due
to the fact that all points of $Q$ are regular. The quotient Poisson
bracket is well-defined since $\mbox{$C^{\infty}$}(\mbox{$\mathcal{W}$})^{G}\subseteq\mbox{$C^{\infty}$}(\mbox{$\mathcal{W}$})$ is a
Poisson sub-algebra. The statement now follows because the diagram
\[
\xymatrix{
T^{*}Q \ar[r]^-{\psi^{-1}}\ar @{->>}[d] & {\mbox{$\mathcal{W}$}}\ar @{->>}[dr] \\
(T^{*}Q)/G\ar[r]^-{\beta} & W\ar @{=}[r] & {\mbox{$\mathcal{W}$}}/G
}
\]
is commutative, and composition of top and down-right arrow is
Poisson and the left vertical arrow is surjective.
\end{proof}
\subsection{The reduced phase space}
The following lemmas are key to the subsequent. They
guarantee, in particular,
that every non-empty pre-image of a coadjoint orbit
$\mbox{$\mathcal{O}$}\subset\mathfrak{g}^*$ under $\mu$ fibrates surjectively over $\mbox{$\textup{Hor}$}^*$.
In Theorem~\ref{thm:WWred} we use this to show that every non-empty
symplectic stratum of the
reduced phase space fibrates over $T^*(Q/G)$.
The meta principle motivating these results is that $\mu$
(which is defined by means of the universal connection $\theta$)
can be thought of
as a kind of
universal connection form on $Q\twoheadrightarrow Q/G$.
\begin{lemma}
Let $\mbox{$\mathcal{O}$}\subseteq\mathfrak{g}^{*}$ be a coadjoint orbit, $\mu: T^{*}Q\to\mathfrak{g}^{*}$
the canonical momentum mapping, and $\mu_{q}:=\mu|(T_{q}^{*}Q)$. Then
either $\mu_{q}^{-1}(\mbox{$\mathcal{O}$})=\emptyset$ for all $q\in Q$ or
$\mu_{q}^{-1}(\mbox{$\mathcal{O}$})\neq\emptyset$ for all $q\in Q$. In the latter case
we have
\[
\mu_{q}^{-1}(\mbox{$\mathcal{O}$})
=
\mbox{$\textup{Ann}\,$}_{q}(T_{q}(G.q))\times\set{A_{q}^{*}(\lambda):
\lambda\in\mbox{$\textup{Ann}\,$}\mathfrak{g}_{q}\cap\mbox{$\mathcal{O}$}}
\]
which is an equality of topological spaces and
where $A_{q}^{*}: \mbox{$\textup{Ann}\,$}\mathfrak{g}_q\to\mbox{$\textup{Ver}$}_q^*$
is the adjoint of $A_{q}: T_{q}Q\to\mathfrak{g}/\mathfrak{g}_{q}$.
\end{lemma}
\begin{proof}
Assume firstly that $q_1,q_2\in Q$ lie in the same $G$-orbit. Then it
is obviously true that $\mu_{q_1}^{-1}(\mbox{$\mathcal{O}$})$ is empty if and only if
$\mu_{q_2}^{-1}(\mbox{$\mathcal{O}$})$ is empty. Thus for the purpose of this proof we
can assume that $G_{q_1}=G_{q_2}=H$:
all isotropy subgroups are conjugate to each other, and $q_{2}$ can be
moved around in its orbit without loss of generality.
Now assume that
$\mu_{q_1}^{-1}(\mbox{$\mathcal{O}$})$ is non-empty, i.e., there is
$\lambda = \mu_{q_1}(p_1)\in\mbox{$\textup{Ann}\,$}\mathfrak{h}\cap\mbox{$\mathcal{O}$}$. Using the
connection $A$ we may then define
$p_2 := A_{q_2}^*(\lambda)\in\mbox{$\textup{Ver}$}_{q_2}^*\cap\mu_{q_2}^{-1}(\mbox{$\mathcal{O}$})$
whence $\mu_{q_2}^{-1}(\mbox{$\mathcal{O}$})$ is non-empty as well.
In fact, this construction also proves the last claim of the lemma.
\end{proof}
\begin{lemma}\label{lm:WWorb}
Let $\mbox{$\mathcal{O}$}$ be a coadjoint orbit in the image of the momentum map\xspace $\mu_{A}:
\mbox{$\mathcal{W}$}\to\mathfrak{g}^{*}$. Further, let $(L)$ be in the isotropy lattice of the
$G$-action on $\mbox{$\mathcal{W}$}$ such that
$\mu_{A}^{-1}(\mbox{$\mathcal{O}$})\cap\mbox{$\mathcal{W}$}_{(L)}\neq\emptyset$. Then
\[
\mbox{$\mathcal{W}$}_{(L)}
=
(Q\times_{Q/G}T^{*}(Q/G))\times_{Q}(\mbox{$\bigsqcup$}_{q\in Q}\mbox{$\textup{Ann}\,$}\mathfrak{g}_{q})_{(L)}
\]
and
\[
\mbox{$\mathcal{W}$}_{(L)}\cap\mu_{A}^{-1}(\mbox{$\mathcal{O}$})
=
(Q\times_{Q/G}T^{*}(Q/G))\times_{Q}(\mbox{$\bigsqcup$}_{q\in Q}\mbox{$\textup{Ann}\,$}\mathfrak{g}_{q}\cap\mbox{$\mathcal{O}$})_{(L)}
\]
are smooth manifolds.
Moreover,
\[
\xymatrix{
{\mbox{$\mathcal{O}$}_{(L_0)_H^G}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h}}
\ar @{^{(}->}[r]&
{(\mbox{$\bigsqcup$}_{q\in Q}\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{g}_{q})_{(L)}}
\ar[r]
&Q
}
\]
is a smooth fiber bundle where $L_{0}$ is a subgroup of $H$ such that
$L_{0}$ is conjugate to $L$ within $G$.
\end{lemma}
Notice that we do not assume $\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h}$ to be smooth.
The notation $\mbox{$\mathcal{O}$}_{(L_0)_H^G}$ is explained in Section~\ref{sec:prel}(\ref{def:weirdo-type}).
The $G$-action on $\mbox{$\bigsqcup$}_{q\in Q}\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{g}_q$ is induced from the $G$-action on $\mbox{$\mathcal{W}$}$
from Proposition~\ref{prop:WW}, and is given by $g.(q,\lambda) = (g.q,\mbox{$\text{\upshape{Ad}}$}^*(g).\lambda)$.
\begin{proof}
The statement about $\mbox{$\mathcal{W}$}_{(L)}$ is clear. Thus also the description
of $\mbox{$\mathcal{W}$}_{(L)}\cap\mu_{A}^{-1}(\mbox{$\mathcal{O}$})$ follows from the previous lemma
together with Theorem~\ref{thm:sing_spr}.
Concerning the second assertion let $q_{0}\in Q$ with $G_{q_{0}}=H$. Then
\[
(q_{0},\lambda)\in(\mbox{$\bigsqcup$}_{q\in Q}\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{g}_{q})_{(L)}
\]
if and only if
\[
\lambda\in\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h}
\textup{ and }
H\cap G_{\lambda}=H_{\lambda}=L_{0}\textup{ is conjugate to } L
\textup{ in }G
\]
which is true if and only if
\[
\lambda\in(\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h})_{(L_0)_H^G}
\]
where $L_{0}$ is a subgroup of $H$ conjugate to $L$ within $G$,
and we view $\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h}\subset\mbox{$\mathcal{O}$}$ as an $H$-space by
virtue of the restricted $\mbox{$\text{\upshape{Ad}}$}^*(H)$-action.
By Lemma~\ref{lem:weirdo-type} the space
$(\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h})_{(L_0)_H^G}$ is a smooth manifold.
\begin{comment}
***********************SUPERFLUOUS***********************************
Consider the $\mbox{$\text{\upshape{Ad}}$}^{*}(H)$ action on $\mbox{$\mathcal{O}$}$. This action is a
Hamiltonian one with momentum map given by $\rho: \mbox{$\mathcal{O}$}\to\mathfrak{h}^{*}$,
$\lambda\mapsto\lambda|\mathfrak{h}$, i.e., by restriction. Thus
the level set
$\rho^{-1}(0) =
\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h}$
is a stratified space by Theorem~\ref{thm:sing_spr};
typical smooth strata of this space are of the
form $\mbox{$\mathcal{O}$}_{(L_0)}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h}$ with $L_{0}$ a subgroup of $H$,
and $\mbox{$\mathcal{O}$}_{(L_0)_H^G}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h}$ is a finite disjoint union of
such strata (see Section~\ref{sec:prel}).
********************************************************************
\end{comment}
To see smooth local triviality we proceed as follows. Let again
$q_{0}\in Q$ with $G_{q_{0}}=H$, and let $S$ be a slice at $q_{0}$ and
$U$ a tube around $G.q_{0}$. That is, $G/H\times S\cong U$,
$(kH,s)\mapsto k.s$
as proper $G$-spaces
by virtue of the Slice Theorem (\cite{PT88,DK99}).
Then we consider the smooth trivializing map
\begin{align*}
S\times G\times_{H}(\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h})_{(L_0)_H^G}
&\longrightarrow
(\mbox{$\bigsqcup$}_{q\in Q}\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{g}_{q})_{(L)}|U,\\
(s,[(k,\lambda_0)]_H)
&\longmapsto
(k.s,\mbox{$\text{\upshape{Ad}}$}^{*}(k).\lambda_0)
\end{align*}
which is well defined since
$[(k,\lambda_0)]_H\in G\times_H(\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h})_{(L_0)_H^G}$ implies that
$H_{\lambda_0} = gL_0g^{-1}\in (L)$ for some $g\in G$ and this yields
$G_{(k.s,\textup{Ad}^*(k).\lambda_0)} = kG_{(s,\lambda_0)}k^{-1}
= kH_{\lambda_0}k^{-1} = kgL_0g^{-1}k^{-1}$.
Hereby we use, firstly, that
the diagonal $H$-action cancels
out, i.e., $\mbox{$\text{\upshape{Ad}}$}^*(kh^{-1}).\mbox{$\text{\upshape{Ad}}$}^*(h).\lambda_0 = \mbox{$\text{\upshape{Ad}}$}^*(k).\lambda_0$ for all
$h\in H$;
secondly, we use that $\mathfrak{g}_s = \mathfrak{g}_{q_0} = \mathfrak{h}$ for all $s\in S$ since $S$ is a
slice at $q_0$ -- see, e.g., \cite[Corollary~5.1.13(2)]{PT88}.
Clearly, this map is smooth with smooth inverse
$
(q,\lambda)=(k.s,\mbox{$\text{\upshape{Ad}}$}^{*}(k).\lambda_{0})
\longmapsto
(s,[(k,\lambda_{0})]_H).
$
Therefore, this construction provides smooth bundle charts of the total space
$(\mbox{$\bigsqcup$}_{q\in Q}\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{g}_{q})_{(L)}$.
\end{proof}
\begin{lemma}\label{lem:weirdo-type}
Let $G$, $\mbox{$\mathcal{O}$}$, $H$ and $L_0$ be as in Lemma~\ref{lm:WWorb}. Then
$(\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h})_{(L_0)_H^G}$ is a smooth (possibly disconnected) pre-symplectic
manifold.
\end{lemma}
This fact is an instance of the general theory of
singular commuting reduction of \cite{MMOPR03}. However,
the argument is rather explicit.
\begin{proof}
As in Lemma~\ref{lem:actions} consider the $G$-action on
$T^*G=G\times\mathfrak{g}^*$ given by $g.(k,\lambda) = (gk,\lambda)$ and the
$H$-action given by $h.(k,\lambda) = (kh^{-1},\mbox{$\text{\upshape{Ad}}$}^*(h).\lambda)$. These
actions are Hamiltonian.
By Lemma~\ref{lem:actions}(\ref{eq:J-tau})
symplectic reduction of
$G\times\mathfrak{g}^*$ with respect to $H$ at $0\in\mathfrak{h}^*$ yields
$(G\times\mathfrak{g}^*)\spr{0}H = G\times_{H}\mbox{$\textup{Ann}\,$}\mathfrak{h}$.
Let $\pi_H: G\times\mbox{$\textup{Ann}\,$}\mathfrak{h}\twoheadrightarrow G\times_H\mbox{$\textup{Ann}\,$}\mathfrak{h}$ denote the orbit projection.
The corollary of
Lemma~\ref{lem:actions} (or straightforward computation) implies that there
is an induced $G$-action on $G\times_H\mbox{$\textup{Ann}\,$}\mathfrak{h}$
given by $g.[(k,\lambda)]_H = [(gk,\lambda)]_H$
with momentum map\xspace $j$ given
by $j[(k,\lambda)]_H = \mbox{$\text{\upshape{Ad}}$}^*(k).\lambda$.
(The formula for $j$ follows from
Lemma~\ref{lem:actions}(\ref{eq:J-lam}).)
Now the theory of singular symplectic reduction
(see Theorem~\ref{thm:sing_spr})
implies that
\begin{multline*}
(j^{-1}(\mbox{$\mathcal{O}$}))_{(L_0)^G}
= j^{-1}(\mbox{$\mathcal{O}$})\cap(G\times_H\mbox{$\textup{Ann}\,$}\mathfrak{h})_{(L_0)^G} = \\
\set{[(k,\lambda)]_H\in G\times_H(\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h}):
G_{[(k,\lambda)]_H} = kH_{\lambda}k^{-1}
\textup{ is conjugate to } L_0 \textup{ in } G}
\end{multline*}
is a sub-manifold of
$G\times_H\mbox{$\textup{Ann}\,$}\mathfrak{h}$. Therefore, the pre-image
$\pi_H^{-1}((j^{-1}(\mbox{$\mathcal{O}$}))_{(L_0)^G})
= G\times(\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h})_{(L_0)_H^G}$ is a
sub-manifold of $G\times\mbox{$\textup{Ann}\,$}\mathfrak{h}$, and
$(\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h})_{(L_0)_H^G}$ is a sub-manifold of $\mbox{$\textup{Ann}\,$}\mathfrak{h}$.
Finally, the manifold in question is pre-symplectic since
$(\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h})_{(L_0)_H^G}/H
\cong (G\times_H(\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h})_{(L_0)^G_H})/G
= (j^{-1}(\mbox{$\mathcal{O}$}))_{(L_0)^G}/G$ is a
symplectic manifold.
\end{proof}
\begin{comment}
\begin{rem}
To see that the fiber of the bundle $(\mbox{$\bigsqcup$}_{q\in Q}\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{g}_q)_{(L)}\to Q$
in Lemma~\ref{lm:WWorb}
cannot, in general,
be of the form $(\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h})_{(L_0)^H}$,
as claimed in a previous version of this paper, consider the following example.
Let $G$ be a semisimple Lie group
acting on itself by conjugation,
let $H\subset G$ a Cartan subgroup,
and consider $Q:=G_{(H)}$ and $q_0\in Q$ with $G_{q_0}=H$.
Suppose $\mbox{$\mathcal{O}$}$ is a coadjoint orbit in $\mathfrak{g}^*$
and $\lambda\in\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h}$ is such that $H_{\lambda}=:L_0$ is neither $\set{e}$ nor $H$.
Assume further that there is a group element
$n\in N_G(H)\setminus H$
satisfying $nL_0n^{-1}\neq L_0$.
(E.g., in $G=\mbox{$\textup{SO}$}(5)$ one can find such data explicitly.)
By construction we
thus have $\mbox{$\text{\upshape{Ad}}$}^*(n).\lambda\in\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h}$ and
$L_1 := H_{\textup{Ad}^*(n).\lambda} = nL_0n^{-1}$.
Therefore, $(q_0,\lambda)$ and $(q_0,\mbox{$\text{\upshape{Ad}}$}^*(n).\lambda)$
both are elements of $(\mbox{$\bigsqcup$}_{q\in Q}\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{g}_q)_{(L)}$
whence $\lambda$ and $\mbox{$\text{\upshape{Ad}}$}^*(n).\lambda$ both sit in the fiber over $q_0$.
However, since $H$ is Abelian, $L_0$ cannot be conjugate to
$L_1$ within $H$, that is,
$\mbox{$\text{\upshape{Ad}}$}^*(n).\lambda\notin(\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h})_{(L_0)^H}$.
\end{rem}
\end{comment}
The singular reduction diagram of Ortega and Ratiu~\cite[Theorem 8.4.4]{OR04}
adjoined to the universal reduction procedure of Arms, Cushman, and
Gotay~\cite{ACG91} (see also \cite[Section 10.3.2]{OR04}) applied to
the Weinstein space has the following form.
\[
\xymatrix{
{\mu_{A}^{-1}(\mbox{$\mathcal{O}$})}
\ar @{<-^{)}}[r]
\ar @{->>}[d]&
{\mu_{A}^{-1}(\lambda)}
\ar @{^{(}->}[r]
\ar @{->>}[d]&
{\mbox{$\mathcal{W}$}}
\ar @{->>}[d]\\
{\mu_{A}^{-1}(\mbox{$\mathcal{O}$})/G}
\ar @{<-}[r]^-{\simeq}&
{\mu_{A}^{-1}(\lambda)}/G
\ar @{^{(}->}[r]&
{\mbox{$\mathcal{W}$}/G}
\ar @{=}[r]&
W
}
\]
where $\lambda\in\mu_{A}(\mbox{$\mathcal{W}$})$ and $\mbox{$\mathcal{O}$}$ is the coadjoint orbit passing
through $\lambda$. Therefore it is a sensible generalization of the smooth
case to interpret the reduced space
$\mu_{A}^{-1}(\mbox{$\mathcal{O}$})/G = \mbox{$\mathcal{W}$}\spr{\mathcal{O}}G$ as a typical
stratified symplectic leaf of the stratified Poisson space $W$.
The following thus generalizes the result of Marsden and Perlmutter
\cite[Theorem 4.3]{MP00} to the case of a non-free but single orbit
type action of $G$ on $Q$.
Let $\mbox{$\mathcal{O}$}$ be a
coadjoint orbit in the image of the momentum map\xspace $\mu_{A}: \mbox{$\mathcal{W}$}\to\mathfrak{g}^{*}$,
and let $(L)$ be in the isotropy lattice of the $G$-action on
$\mbox{$\mathcal{W}$}$ such that
$\lorb{\mbox{$\mathcal{W}$}}:=\mu_{A}^{-1}(\mbox{$\mathcal{O}$})\cap\mbox{$\mathcal{W}$}_{(L)}\neq\emptyset$.
Then we define
\[
\lorb{\iota}: \lorb{\mbox{$\mathcal{W}$}}\hookrightarrow\mbox{$\mathcal{W}$},
\]
the canonical embedding, and the orbit projection mapping
\[
\lorb{\pi}: \lorb{\mbox{$\mathcal{W}$}}\twoheadrightarrow\lorb{\mbox{$\mathcal{W}$}}/G=:(\mbox{$\mathcal{W}$}\mbox{$/\negmedspace/_{\mathcal{O}}$} G)_{(L)}.
\]
Further, we denote the Kirillov-Kostant-Souriau symplectic form on
$\mbox{$\mathcal{O}$}$ by $\mbox{$\Om^{\mathcal{O}}$}$, that is
$\mbox{$\Om^{\mathcal{O}}$}(\lambda)(\mbox{$\text{\upshape{ad}}$}^{*}(X).\lambda,\mbox{$\text{\upshape{ad}}$}^{*}(Y).\lambda) = \vv<\lambda,[X,Y]>$.
Remember from Proposition \ref{prop:WW} that the symplectic structure
on $\mbox{$\mathcal{W}$}=\mbox{$\mathcal{W}$}(A)$ is denoted by $\sigma = \psi^*\Omega$.
\begin{theorem}[Gauged symplectic reduction]\label{thm:WWred}
Let $Q=Q_{(H)}$
and let $A$ be a generalized connection form on $\pi: Q\twoheadrightarrow Q/G$.
Let $\mbox{$\mathcal{O}$}$ be a
coadjoint orbit in the image of the momentum map\xspace $\mu_{A}: \mbox{$\mathcal{W}$}\to\mathfrak{g}^{*}$,
and let $(L)$ be in the isotropy lattice of the $G$-action on
$\mbox{$\mathcal{W}$}$ such that
$\lorb{\mbox{$\mathcal{W}$}}:=\mu_{A}^{-1}(\mbox{$\mathcal{O}$})\cap\mbox{$\mathcal{W}$}_{(L)}\neq\emptyset$. Then the
following are true.
\begin{enumerate}[\upshape (1)]
\item
The smooth manifold $(\mbox{$\mathcal{W}$}\mbox{$/\negmedspace/_{\mathcal{O}}$} G)_{(L)}$ is a typical symplectic
stratum of the singular symplectic space $\mbox{$\mathcal{W}$}\mbox{$/\negmedspace/_{\mathcal{O}}$} G$. The smooth
symplectic manifold
\[
(\mbox{$\mathcal{O}$}\spr{0}H)_{(L_0)_H^G} := (\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h})_{(L_0)_H^G}/H,
\]
where $L_0$ is an isotropy subgroup of
the $H$-action on $\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h}$,
is a disjoint union of
typical smooth symplectic strata of the singular symplectic space
$\mbox{$\mathcal{O}$}\spr{0}H$.
\item
The symplectic stratum $(\mbox{$\mathcal{W}$}\mbox{$/\negmedspace/_{\mathcal{O}}$} G)_{(L)}$ can be globally
described as
\[
(\mbox{$\mathcal{W}$}\mbox{$/\negmedspace/_{\mathcal{O}}$} G)_{(L)}
=
T^{*}(Q/G)\times_{Q/G}(\mbox{$\bsc_{q\in Q}\mathcal{O}\cap\ann\gu_{q}$})_{(L)}/G
\]
whence it is the total space of the smooth symplectic fiber bundle
\[
\xymatrix{
{(\mbox{$\mathcal{O}$}\spr{0}H)_{(L_0)_H^G}}
\ar @{^{(}->}[r]&
{(\mbox{$\mathcal{W}$}\mbox{$/\negmedspace/_{\mathcal{O}}$} G)_{(L)}}
\ar[r]^{\chi}
&{T^{*}(Q/G).}
}
\]
Hereby $L_{0}\subset H$ is an
isotropy subgroup of the induced $H$-action on $\mbox{$\mathcal{O}$}$ which is
conjugate to $L$ within $G$.
\item
The symplectic structure $\lorb{\sigma}$ on $(\mbox{$\mathcal{W}$}\mbox{$/\negmedspace/_{\mathcal{O}}$} G)_{(L)}$
is uniquely determined and given by the formula
\[
(\lorb{\pi})^{*}\lorb{\sigma}
=
(\lorb{\iota})^{*}\sigma - (\mu_{A}|\lorb{\mbox{$\mathcal{W}$}})^{*}\mbox{$\Om^{\mathcal{O}}$}.
\]
Therefore,
\[
\lorb{\sigma} = \chi^*\Omega^{Q/G} - \lorb{\beta}
\]
where $\lorb{\beta}\in\Omega^{2}((\mbox{$\mathcal{W}$}\mbox{$/\negmedspace/_{\mathcal{O}}$} G)_{(L)})$ is defined by
\[
(\lorb{\pi})^{*}\lorb{\beta}
=
(\lorb{\iota})^{*}dB + (\mu_A|\lorb{\mbox{$\mathcal{W}$}})^{*}\mbox{$\Om^{\mathcal{O}}$}.
\]
Finally, $B$ is the form that was introduced in Proposition
\ref{prop:WW}.
\item
The stratified symplectic space can be globally described as
\[
\mbox{$\mathcal{W}$}\mbox{$/\negmedspace/_{\mathcal{O}}$} G
=
T^{*}(Q/G)\times_{Q/G}\mbox{$\bsc_{q\in Q}\mathcal{O}\cap\ann\gu_{q}$}/G
\]
whence it is the total space of
\[
\xymatrix{
{\mbox{$\mathcal{O}$}\spr{0}H}
\ar @{^{(}->}[r]&
{\mbox{$\mathcal{W}$}\mbox{$/\negmedspace/_{\mathcal{O}}$} G}
\ar[r]
&{T^{*}(Q/G)}
}
\]
which
is a stratified symplectic fiber bundle with singularities
confined to the fiber direction.
\end{enumerate}
\end{theorem}
\begin{proof}
Assertion (1).
This is an implication of the general theory of stratified
symplectic reduction -- see Ortega and Ratiu~\cite[Section 8.4]{OR04}
or Section~\ref{s:sg_com_red} for a statement of the relevant theorem
and Section~\ref{sec:prel} for the notation.
To see that
$
(\mbox{$\mathcal{O}$}\spr{0}H)_{(L_0)_H^G}
$
is a union of
typical smooth symplectic strata of the singular symplectic space
$\mbox{$\mathcal{O}$}\spr{0}H$ note firstly that
$
(\mbox{$\mathcal{O}$}\spr{0}H)_{(L_0)_H^G}
$
is,
according to the proof of Lemma~\ref{lem:weirdo-type},
a typical stratum of $(T^*G)\spr{0}H\mbox{$/\negmedspace/_{\mathcal{O}}$} G$.
By the corollary of
Lemma~\ref{lem:actions} (with $S=\set{\textup{point}}$ and $\beta = 0$)
there is an isomorphism
$
(T^*G)\spr{0}H\mbox{$/\negmedspace/_{\mathcal{O}}$} G
\cong (T^*G)\mbox{$/\negmedspace/_{\mathcal{O}}$} G\spr{0}H
=
\mbox{$\mathcal{O}$}\spr{0}H
$,
$
[[(g,\lambda)]_H]_G
\mapsto
[[(g,\lambda)]_G]_H
=
[\lambda]_H$
of singular symplectic spaces, whence strata are mapped
symplectomorphically onto unions of strata.
Assertion (2).
The description of the stratum $(\mbox{$\mathcal{W}$}\mbox{$/\negmedspace/_{\mathcal{O}}$} G)_{(L)}$ follows from
Proposition~\ref{prop:WW}.
We know from (1) that all spaces involved in the
diagram really are smooth. As in the proof of Lemma \ref{lm:WWorb}
let $q_{0}\in Q$ with $G_{q_{0}}=H$, $S$ a slice at $q_{0}$, and
$U\cong G/H\times S$ a tube around the orbit $G.q_{0}$. Then we get
the local description
\begin{align*}
(\mbox{$\mathcal{W}$}\mbox{$/\negmedspace/_{\mathcal{O}}$} G)_{(L)}|U
&=
T^{*}S\times_{S}(\mbox{$\bigsqcup$}_{q\in U}\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{g}_{q})_{(L)}/G\\
&\cong
T^{*}S\times_{S}S\times(\mbox{$\mathcal{O}$}\cap\mbox{$\textup{Ann}\,$}\mathfrak{h})_{(L_0)_H^G}/H\\
&=
T^{*}S\times(\mbox{$\mathcal{O}$}\spr{0}H)_{(L_0)_H^G}
\end{align*}
as claimed.
The bundle is symplectic by Theorem~\ref{thm:bun_pic}.
Assertion (3). The defining property of the reduced symplectic form
$\lorb{\sigma}$, namely,
\[
(\lorb{\pi})^{*}\lorb{\sigma}
=
(\lorb{\iota})^{*}\sigma - (\mu_{A}|\lorb{\mbox{$\mathcal{W}$}})^{*}\mbox{$\Om^{\mathcal{O}}$}
\]
is a well-established fact,
see e.g.\ Bates and Lerman~\cite[Proposition~11]{BL97}.
Thus it is clear from Proposition~\ref{prop:WW} that
\[
\lorb{\sigma}
=
\chi^*\Omega^{Q/G}-\lorb{\beta}
\]
and it remains to check that
$\lorb{\beta}$ is a well defined two-form on $(\mbox{$\mathcal{W}$}\mbox{$/\negmedspace/_{\mathcal{O}}$} G)_{(L)}$.
To see this notice firstly that
\[
\tilde{\beta}
:=
(\lorb{\pi})^{*}\lorb{\beta}
=
(\lorb{\iota})^{*}dB + (\mu_A|\lorb{\mbox{$\mathcal{W}$}})^{*}\mbox{$\Om^{\mathcal{O}}$}
\in \Omega^2(\lorb{\mbox{$\mathcal{W}$}})
\]
is
$G$-invariant. Furthermore, we claim that $\tilde{\beta}$ is
horizontal, i.e., vanishes upon insertion of a vertical
vector field. Indeed, let
$
(q,\eta,\lambda)\in\lorb{\mbox{$\mathcal{W}$}}
$
and consider $\zeta_{Z_1}(q,\eta,\lambda),\xi_2\in
T_{(q,\eta,\lambda)}\lorb{\mbox{$\mathcal{W}$}}$.
We proceed locally as in the proof of Proposition~\ref{prop:WW} so
that there is a splitting of tangent vectors as
$\zeta_{Z_1}(q,\eta,\lambda) = (\zeta_{Z_1}(q),0,\mbox{$\text{\upshape{ad}}$}^*(Z_1).\lambda)$ and
$\xi_2 = (q_2',\eta_2',\mbox{$\text{\upshape{ad}}$}^*(Y).\lambda)$.
Therefore,
\begin{align*}
\tilde{\beta}_{(q,\lambda)}(&\zeta_{Z_1}(q,\eta,\lambda),\xi_2)\\
&=
\vv<\mbox{$\text{\upshape{ad}}$}^{*}(Z_{1}).\lambda,Z_{2}>
- \vv<\mbox{$\text{\upshape{ad}}$}^{*}(Y).\lambda,Z_{1}>
+ 0
+ \vv<\lambda,[Z_{1},Z_{2}]>
+ \vv<\lambda,[Z_{1},Y]>\\
&=
- \vv<\lambda,[Z_{1},Z_{2}]>
- \vv<\lambda,[Z_{1},Y]>
+ \vv<\lambda,[Z_{1},Z_{2}]>
+ \vv<\lambda,[Z_{1},Y]>
= 0.
\end{align*}
That is, $\tilde{\beta}$ is a basic form and thus descends to a form
$\lorb{\beta}$.
Assertion (4) is a pasting together of the results in (2).
\end{proof}
\begin{corollary}\label{cor:WWred}
Let $\mbox{$\mathcal{O}$}$ be a
coadjoint orbit in the image of the momentum map\xspace $\mu_{A}: \mbox{$\mathcal{W}$}\to\mathfrak{g}^{*}$,
and let $(L)$ be in the isotropy lattice of the $G$-action on
$\mbox{$\mathcal{W}$}$ such that
$\lorb{\mbox{$\mathcal{W}$}}:=\mu_{A}^{-1}(\mbox{$\mathcal{O}$})\cap\mbox{$\mathcal{W}$}_{(L)}\neq\emptyset$. Assume
further that there is a global slice $S$ such that $Q\cong G/H\times
S$. Then we have the global description
\[
\mbox{$\mathcal{W}$}\mbox{$/\negmedspace/_{\mathcal{O}}$} G =
T^{*}S\times\mbox{$\mathcal{O}$}\spr{0}H.
\]
Moreover, the reduced symplectic form
$\lorb{\sigma}$ on a symplectic stratum
$(\mbox{$\mathcal{W}$}\mbox{$/\negmedspace/_{\mathcal{O}}$} G)_{(L)} = T^{*}S\times(\mbox{$\mathcal{O}$}\spr{0}H)_{(L_0)_H^G}$
is given by the formula
\[
\lorb{\sigma} = \Omega^{Q/G}-\Omega^{\mathcal{O}}_{(L_0)_H^G}
\]
where $\Omega^{\mathcal{O}}_{(L_0)_H^G}$ is the canonically reduced
symplectic form on
$(\mbox{$\mathcal{O}$}\spr{0}H)_{(L_0)_H^G}$, and $L_{0}$ is a subgroup of $H$ which
is conjugate to $L$ within $G$.
\end{corollary}
\begin{proof}
This is an immediate consequence of Theorems \ref{thm:bun_pic} and
\ref{thm:WWred}.
\end{proof}
\section{Spin Calogero-Moser systems}\label{sec:cms}
In this section we give a mechanical application of
Theorem~\ref{thm:WWred} to obtain spin Calogero-Moser models.
This approach follows, in essence, the idea of Kazhdan, Kostant and
Sternberg~\cite{KKS78} that such models may be obtained via
projection of geodesic systems on Lie groups or Lie algebras.
This construction can be
carried out in various guises and at different levels of generality.
For the simplest case
we describe the way Theorem~\ref{thm:WWred} allows to understand this
projection procedure in the following subsection. The emphasis is here
on the use of the mechanical connection.
\subsection{The construction based on cotangent bundle reduction}\label{sub:cms-constr}
Let $G$ be a (real or complex)
simple Lie group, $\mathfrak{g}$ its Lie algebra, $\mathfrak{h}$ a Cartan sub-algebra,
and $H$ a corresponding Cartan subgroup.
Then we consider either $Q = G_{(H)}$ acted upon by $G$ via
conjugation or $Q = \mathfrak{g}_{(H)}$ acted upon by $\mbox{$\text{\upshape{Ad}}$}(G)$.
Note that $G_{(H)}$ is open dense in $G$ and $\mathfrak{g}_{(H)}$ is open dense
in $\mathfrak{g}$.
We will see below that choosing $Q=\mathfrak{g}_{(H)}$ leads to Calogero-Moser
systems with rational potential while choosing $Q=G_{(H)}$ leads to
Calogero-Moser systems with trigonometric potential.
As in the
construction of the previous sections we may then consider the lifted
$G$-action on $T^*Q$ which is Hamiltonian with equivariant momentum map\xspace
$\mu: T^*Q\to\mathfrak{g}^*$. Since $G$ is assumed simple we can use the
Killing form $B$
to identify $T^*Q\cong_B TQ$ and $\mathfrak{g}^*\cong_B\mathfrak{g}$ whence $\mu$ becomes a
mapping $\mu: TQ\to\mathfrak{g}$. Let $\mbox{$\mathcal{O}$}$ be an adjoint orbit in the image
of $\mu$.
Via the Killing form we may thus define a mechanical connection $A$ on
$Q\twoheadrightarrow Q/G$ as in Subsection~\ref{sub:mech-con}.
By Theorem~\ref{thm:WWred} and its corollary the singular symplectic
quotient of $TQ$ at $\mbox{$\mathcal{O}$}$ is therefore given by
\[
\mu^{-1}(\mbox{$\mathcal{O}$})/G
=
TQ\mbox{$/\negmedspace/_{\mathcal{O}}$} G
\cong_{A}
T(Q/G)\times\mbox{$\mathcal{O}$}\spr{0}H,
\]
and this space is called a \caps{spin Calogero-Moser space}.
(Recall that $\mbox{$\mathcal{O}$}\spr{0}H = (\mbox{$\mathcal{O}$}\cap\mathfrak{h}^{\bot})/H$.)
This terminology is justified as follows. Use the left multiplication
in the group to trivialize the tangent bundle as $TQ =
Q\times\mathfrak{g}$. Let $\mathcal{H}: Q\times\mathfrak{g}\to\mbox{$\bb{R}$}$,
$(q,X)\mapsto\by{1}{2}B(X,X)$ denote the free Hamiltonian. In the
notation of Proposition~\ref{prop:WW} the reduced Hamiltonian system
on $T(Q/G)\times\mbox{$\mathcal{O}$}\spr{0}H$ is thus given by Hamiltonian reduction
of $(\mbox{$\mathcal{W}$},\sigma,\psi^*\mathcal{H})$ at $\mbox{$\mathcal{O}$}$.
Since the Hamiltonian in this
picture is given by
\[
(\psi^*\mathcal{H})(q,\eta,\lambda)
=
\by{1}{2}B(\eta,\eta) + \by{1}{2}B(A_q^*(\lambda),A_q^*(\lambda))
\]
for $(q,\eta,\lambda)\in\mbox{$\mathcal{W}$}$ one thus needs to compute $A_q^*(\lambda)$. In
fact, it obviously suffices to compute $A_q^*(\lambda)$ for $q\in Q_H$.
The crucial point now is that with respect to the identifications
$T^*Q = TQ$ and $\mathfrak{g}^* = \mathfrak{g}$ we have
\[
A_q^*(\lambda) = \zeta(q)(\check{\mbox{$\mathbb{I}$}}_q^{-1}(\lambda))
\]
where $\check{\mbox{$\mathbb{I}$}}_q: \mathfrak{g}_q^{\bot}\to\mathfrak{g}_q^{\bot}$
is the $G_q$-equivariant isomorphism\xspace obtained from the locked inertia
tensor $\mbox{$\mathbb{I}$}$.
This object can be computed using structure theory, and we shall do so
in the next paragraph.
\subsubsection{The rational case}
Assume that $\mathfrak{g}$ is a complex simple Lie algebra.
(The case of real simple Lie algebras works analogously.)
Let $Q=\mathfrak{g}_{(H)}$, let $\Delta\subset\mathfrak{h}^*$ be a root system,
$\Delta^+$ a system of positive roots, and $\mathfrak{g}
= \mathfrak{h}\oplus\oplus_{\alpha\in\Delta}\mathfrak{g}_{\alpha}$ the corresponding
root space decomposition. For each $\alpha\in\Delta$ we choose a
vector $E_{\alpha}\in\mathfrak{g}_{\alpha}$ such that $B(E_{\alpha},E_{-\alpha})
= 1$.
Assume that $q\in Q_H = \mathfrak{h}_{\textup{reg}}$ where $\mathfrak{h}_{\textup{reg}}$
denotes the set of regular elements.
Then we may write
$\lambda\in\mbox{$\mathcal{O}$}\cap\mathfrak{h}^{\bot}$ as
$\lambda =
\sum_{\alpha\in\Delta}\lambda_{\alpha}E_{\alpha}$
and
$
\smash{
Z :=
\check{\mbox{$\mathbb{I}$}}_q^{-1}(\lambda)\in\mathfrak{h}^{\bot}
}
$
as
$Z = \sum_{\alpha\in\Delta}z_{\alpha}E_{\alpha}$.
Since $\zeta_Z(q) = \mbox{$\text{\upshape{ad}}$}(Z).q$
it follows that
\begin{align*}
\lambda_{\alpha}
&=
B(\lambda,E_{-\alpha})
=
\mbox{$\mathbb{I}$}_q(Z,E_{-\alpha})
=
B(\zeta_Z(q),\zeta_{E_{-\alpha}}(q))
=
B(Z,-\alpha(q)^2 E_{-\alpha})\\
&=
-z_{\alpha}\alpha(q)^2.
\end{align*}
Therefore, we find that
$
\smash{
Z
=
-\sum_{\alpha\in\Delta}\lambda_{\alpha}\alpha(q)^{-2}E_{\alpha}
}
$,
and
\[
\tag{R}\label{equ:R}
A_q^*(\lambda)
=
\zeta_Z(q)
=
+\mbox{$\text{\upshape{ad}}$}(q)\sum_{\alpha\in\Delta}\by{\lambda_{\alpha}}{\alpha(q)^2}E_{\alpha}
=
\sum_{\alpha\in\Delta}\by{\lambda_{\alpha}}{\alpha(q)}E_{\alpha}.
\]
This in turn implies that the reduced Hamiltonian
$(\psi^*\mathcal{H})_0$ is given by
\begin{align*}
(\psi^*\mathcal{H})_0(q,\eta,[\lambda]_H)
&=
\by{1}{2}B(\eta,\eta) +
\by{1}{2}B(\sum_{\alpha\in\Delta}\by{\lambda_{\alpha}}{\alpha(q)}E_{\alpha}
,\sum_{\alpha\in\Delta}\by{\lambda_{\alpha}}{\alpha(q)}E_{\alpha})\\
&=
\by{1}{2}B(\eta,\eta)
-
\sum_{\alpha\in\Delta^+}\by{\lambda_{\alpha}\lambda_{-\alpha}}{\alpha(q)^2}
\end{align*}
where $(q,\eta,[\lambda]_H)\in T(Q/G)\times\mbox{$\mathcal{O}$}\spr{0}H =
C_{\textup{reg}}\times\mathfrak{h}\times\mbox{$\mathcal{O}$}\spr{0}H$ and where $C_{\textup{reg}}$ denotes
the interior of a Weyl chamber.
This function is the Hamiltonian of the rational
Calogero-Moser system with spin.
\subsubsection{The trigonometric case}
Let $Q=G_{(H)}$ and continue the notation regarding the structure
theoretic objects of the previous paragraph. Assume $q = \exp{a}\in Q_H =
H_{\textup{reg}}$ whence $a\in\mathfrak{h}_{\textup{reg}}$. Consider
$\lambda\in\mbox{$\mathcal{O}$}\cap\mathfrak{h}^{\bot}$ and $Z$ as above. Since $\zeta_Z(q) =
\mbox{$\text{\upshape{Ad}}$}(q^{-1}).Z-Z = (e^{-\textup{ad}(a)}-1).Z$ we may compute
\[
\lambda_{\alpha}
=
B(\lambda,E_{-\alpha})
=
\mbox{$\mathbb{I}$}_q(Z,E_{-\alpha})
=
(e^{-\alpha(a)}-1)(e^{\alpha(a)}-1)z_{\alpha}
=
(2-2\cosh \alpha(a))z_{\alpha}
\]
and thus obtain the adjoint to the mechanical connection
\[
\tag{T}\label{equ:T}
A_q^*(\lambda)
=
\zeta_Z(q)
=
\by{1}{2}\sum_{\alpha\in\Delta}\by{2}{e^{-\alpha(a)}-1}\lambda_{\alpha}E_{\alpha}
=
\by{1}{2}\sum_{\alpha\in\Delta}\lambda_{\alpha}E_{\alpha}
+
\by{1}{2}\sum_{\alpha\in\Delta}\coth(\by{-\alpha(a)}{2})\lambda_{\alpha}E_{\alpha}
\]
in the same way as above.
Therefore, the reduced Hamiltonian
$(\psi^*\mathcal{H})_0$ is given by
\[
(\psi^*\mathcal{H})_0(q,\eta,[\lambda]_H)
=
\by{1}{2}B(\eta,\eta) + \by{1}{2}B(\zeta_Z(q),\zeta_Z(q))
=
\by{1}{2}B(\eta,\eta)
-
\by{1}{4}\sum_{\alpha\in\Delta^+}\by{\lambda_{\alpha}\lambda_{-\alpha}}{\sinh^2\alpha(a)}
\]
where $(q,\eta)\in T(Q_{(H)}/G) = T(H_{\textup{reg}}/W)$ and
$[\lambda]_H\in\mbox{$\mathcal{O}$}\spr{0}H$, and where $W=N(H)/H$ is the Weyl group.
This function is the Hamiltonian of the trigonometric
Calogero-Moser system with spin.
\begin{rem}[Mechanical connection \& classical dynamical
$r$-matrix]\label{rem:r+A}
It is noted in the introduction that the idea of obtaining
Calogero-Moser systems through Hamiltonian reduction is originally due
to Kazhdan, Kostant and Sternberg~\cite{KKS78}.
However, a completely different approach to obtain such systems was
taken by Li and Xu~\cite{LX00,LX02}. They used
an analysis based on
classical dynamical
$r$-matrices (associated to complex simple Lie algebras)
to directly write down
the Hamiltonian of spin Calogero-Moser systems (associated to complex simple
Lie algebras).
(See also Feh\'{e}r and Pusztai~\cite[Section~2]{FP06} for an outline of
this construction.)
This approach is based on the classification of classical
dynamical $r$-matrices of Etingof and Varchenko~\cite{EV98}.
It was noticed by Feh\'{e}r and Pusztai~\cite{FP06} that
one can obtain
the same
Calogero-Moser models which appear in \cite{LX00,LX02} through
Hamiltonian reduction of cotangent bundles.
Moreover,
constructing certain new spin Calogero-Moser models
it was observed by \cite[Proof of Prop.~3]{FP06} that
dynamical $r$-matrices appear in the process of Hamiltonian reduction
of
$T^*G$ where $G$ is a complex or real simple Lie group acting on itself
by twisted conjugation.
However, the relationship
of the $r$-matrix and the reduction approach was still mysterious
in the sense that there was no explanation for it other than the
computations obviously yielding the correct results.
We claim that this
relationship can be further
explained in a geometric framework using the
mechanical connection.\footnote{This point of view essentially evolved
during discussions with Laszlo Feh\'{e}r.}
Indeed,
in \cite{EV98} a classical dynamical $r$-matrix
associated to a complex simple Lie algebra $\mathfrak{g}$
is defined as a
meromorphic function $r: \mathfrak{h}^*=_B\mathfrak{h}\to\mathfrak{g}\otimes\mathfrak{g}$ which satisfies
the classical dynamical Yang-Baxter equation
(CDYBe) and certain other conditions (\cite[Subsection~3.2]{EV98})
in the completed tensor product
$\mathfrak{g}\hat{\otimes}\mathfrak{g}$. Via the isomorphism\xspace $\mathfrak{g}\otimes\mathfrak{g} \cong
\hom(\mathfrak{g}^*,\mathfrak{g}) =_B \hom(\mathfrak{g},\mathfrak{g})$ we may think of a classical
dynamical $r$-matrix as a meromorphic function
$R:
\mathfrak{h}\to\hom(\mathfrak{g},\mathfrak{g})$ subject to the appropriate equations.
For the \emph{rational case} let us use Equation~(\ref{equ:R}) to define
a holomorphic function $R: \mathfrak{h}_{\textup{reg}}\to\hom(\mathfrak{g},\mathfrak{g})$ by
\[
R(q)(\lambda)
:=
A_q^*(\lambda)
=
\sum_{\alpha\in\Delta}\by{\lambda_{\alpha}}{\alpha(q)}E_{\alpha}
\]
By \cite[Theorem~3.2]{EV98} any classical dynamical $r$-matrix
associated to a complex simple Lie algebra $\mathfrak{g}$ with
coupling constant $\epsilon=0$ is of this form
(where $X=\Delta$, $C=0$ and
$\nu=0$ in the
notation of \cite[Theorem~3.2]{EV98}).
For the \emph{trigonometric case} let us use Equation~(\ref{equ:T})
to define
a holomorphic function $R: \mathfrak{h}_{\textup{reg}}\to\hom(\mathfrak{g},\mathfrak{g})$ by
$R(a)(\lambda)=\by{1}{2}\lambda$ for $\lambda\in\mathfrak{h}$, and
\begin{align*}
R(a)(\lambda)
&:=
A_{\exp a}^*(\lambda)
=
\by{1}{2}\sum_{\alpha\in\Delta}\lambda_{\alpha}E_{\alpha}
+
\by{1}{2}\sum_{\alpha\in\Delta}\coth(\by{-\alpha(a)}{2})\lambda_{\alpha}E_{\alpha}\\
&=
i_{\lambda}
(\by{1}{2}\Omega
+
\by{1}{2}\sum_{\alpha\in\Delta}\coth(\by{\alpha(a)}{2})E_{\alpha}\otimes E_{-\alpha})
\end{align*}
for $\lambda\in\mathfrak{h}^{\bot}$.
Here
$\Omega
= \sum_{i=1}^{l}x_i\check{B}(x_i)
+
\sum_{\alpha\in\Delta}E_{\alpha}\check{B}(E_{-\alpha})$
is the Casimir element of $(\mathfrak{g},B)$
where $x_1,\ldots,x_l$ is an orthonormal basis of $\mathfrak{h}$,
and
$X=\Delta_+$,
$C_{i,j}=0$,
$\epsilon = 1$ and $\nu=0$
in the notation of \cite[Theorem~3.10]{EV98}.
By \cite[Theorem~3.10]{EV98} any classical dynamical $r$-matrix
associated to a simple Lie algebra with
coupling constant $\epsilon=1$ is of this form.
We view this as a new geometric explanation of why it is
possible to associate Calogero-Moser systems to classical dynamical
$r$-matrices. It remains a goal for future work
to find a general
relationship between the condition (i.e., CDYBe) defining classical
dynamical $r$-matrices and the properties
(such as $\mbox{$\textup{Curv}$}^A = 0$)
of the mechanical connection.
\end{rem}
\subsection{$\mbox{$\textup{SL}$}(m,\mbox{$\bb{C}$})$ by hand}\label{sub:kks}
As an example consider $G=\mbox{$\textup{SL}$}(m,\mbox{$\bb{C}$})$. Here we work along the lines of Kazdhan, Kostant, Sternberg\xspace
\cite[Section 2]{KKS78} who considered the case $G=\mbox{$\textup{SU}$}(m,\mbox{$\bb{C}$})$. See
also Alekseevsky, Kriegl, Losik, Michor\xspace \cite[Section 5.7]{AKLM03}.
The point to this example is that we try to say as much as possible
about the reduced phase space by using an \emph{ad hoc} approach.
Let $\mbox{$\mathcal{O}$}=\mbox{$\text{\upshape{Ad}}$}(G)Z_{0}$ be an
orbit passing through a semi-simple element $Z_{0}$. Consider $(a,\alpha)\in
G_{r}\times\mathfrak{g}$ with $\alpha-a\alpha a^{-1} = \mu(a,\alpha) = Z$.
Note that $\mu: G_r\times\mathfrak{g}\to\mathfrak{g}$ is the equivariant momentum map\xspace of the
action which is obtained by lifting the $G$-action on $G_r$ by
conjugation
to the (co-)tangent bundle $G_r\times\mathfrak{g}$.
As usual $G_{r}$ denotes the set of regular elements, that is, $G_{r}$
consists of those matrices that have $m$ different
eigenvalues. Moreover, we let $H$ denote the subgroup of diagonal
matrices, and $H_{r} := H\cap G_{r}$.
Via the $\mbox{$\text{\upshape{Ad}}$}(G)$-action we can bring $a$ in diagonal form with entries
$a_{i}\neq a_{j}$ for $i\neq j$.
Since $Z_{ij} =
\alpha_{ij}-\by{a_{i}}{a_{j}}\alpha_{ij}$ the following are
coordinates on
$(\mu^{-1}(\mbox{$\mathcal{O}$})\cap(G_{r}\times\mathfrak{g}))/\mbox{$\text{\upshape{Ad}}$}(G)
= T^*G_r\mbox{$/\negmedspace/_{\mathcal{O}}$} G$:
\begin{itemize}
\item $a_{i}$ for $i=1,\ldots,m$.
\item $\alpha_{i}:=\alpha_{ii}$ for $i=1,\ldots,m$.
\item $\alpha_{ij}=(1-\by{a_{i}}{a_{j}})^{-1}Z_{ij}$ for $i\neq j$.
\end{itemize}
These coordinates give an identification
\[
(\mu^{-1}(\mbox{$\mathcal{O}$})\cap(G_{r}\times\mathfrak{g}))/\mbox{$\text{\upshape{Ad}}$}(G) =
(T^{*}H_{r}\times(\mbox{$\mathcal{O}$}\cap\mathfrak{h}^{\bot})/\mbox{$\text{\upshape{Ad}}$}(H))/W
\]
where $W=N(H)/H$ is the Weyl group.
\caps{Claim:} \emph{If $\mbox{$\mathcal{O}$}$ is an orbit which is of minimal non-zero
dimension then we have that $\mbox{$\mathcal{O}$}\cap\mathfrak{h}^{\bot}/\mbox{$\text{\upshape{Ad}}$}(H) =
\set{\textup{point}}$. Moreover, the reduced phase space can be
described as
$(\mu^{-1}(\mbox{$\mathcal{O}$})\cap(G_{d}\times\mathfrak{g}))/\mbox{$\text{\upshape{Ad}}$}(G) \cong T^{*}H_{r}/W$.}
Here $G_{d}$ denotes the open and dense subset of all diagonable elements in
$\mbox{$\textup{SL}$}(m,\mbox{$\bb{C}$})$. Indeed, let $\mu(a,\alpha) = Z\in\mbox{$\mathcal{O}$}\cap\mathfrak{h}^{\bot}$
with $a$ in diagonal form. Thus
$Z=vw^{t}-cI$ where $c:=\by{1}{m}\vv<v,w>\neq0$, $v,w\in\mbox{$\bb{C}$}^{m}$,
and $w^{t}$ is the transposed to the column vector $w$. Since
$Z\in\mathfrak{h}^{\bot}$ we infer that $v_{i}w_{i}=c$. Hence
\[
\mbox{$\mathcal{O}$}\cap\mathfrak{h}^{\bot} =
\set{(\by{c}{v_{1}}v,\dots,\by{c}{v_{m}}v)-cI:
v_{i}\in\mbox{$\bb{C}$}\setminus\set{0}}.
\]
Take such an
$(\by{c}{v_{1}}v,\dots,\by{c}{v_{m}}v)-cI=:Z_{1}$. Let
$h=\prod_{i=1}^{m}v_{i}\cdot\diag{v_{1}^{-1},\dots,v_{m}^{-1}}$. Then we
can bring $Z_{1}$ into the normal form
$\mbox{$\text{\upshape{Ad}}$}(h)Z_{1} = c(1)_{ij}-cI$ where $(1)_{ij}$ denotes the $m\times
m$-matrix with all entries equal to $1$. Finally note that
$\alpha_{ij}-\by{a_{i}}{a_{j}}\alpha_{ij} = \by{c}{v_{j}}v_{i} \neq 0$
implies that $a=\diag{a_{1},\dots,a_{m}}$ is actually regular.
The coordinates for $T^*G_r\mbox{$/\negmedspace/_{\mathcal{O}}$} G$ found above by evaluating the momentum constraint
equation and factoring out the $G$-action have been the motivating
point for the formulation of the general Theorem~\ref{thm:WWred}.
\subsection{Application: Hermitian matrices}
Consider $V$ the space of complex Hermitian $n\times n$ matrices as
the configuration space to start from. This space shall be endowed
with the inner product $V\times V\to\mbox{$\bb{R}$}$,
$(a,b)\mapsto\mbox{$\textup{Tr}$}(ab)$. Moreover, we let $G=\mbox{$\textup{SU}$}(n,\mbox{$\bb{C}$})$ act on $V$ by
conjugation. Clearly this action leaves the trace form invariant. Via
the inner product we can trivialize the cotangent bundle as $T^{*}V =
V\times V^{*} = V\times V$, and the cotangent lifted action of $G$ is
simply given by the diagonal action.
The canonical symplectic form on $T^{*}V$ is given by
\[
\Omega_{(a,\alpha)}((a_{1},\alpha_{1}),(a_{2},\alpha_{2})) =
\mbox{$\textup{Tr}$}(\alpha_{2}a_{1})-\mbox{$\textup{Tr}$}(\alpha_{1}a_{2}).
\]
The free Hamiltonian on $T^{*}V=V\times V$ is given by
\[
\mbox{$H_{\textup{free}}$}: (a,\alpha)\longmapsto\by{1}{2}\mbox{$\textup{Tr}$}(\alpha\alpha).
\]
Trajectories of this Hamiltonian are given by straight lines of the
form $t\mapsto a+t\alpha$ in the configuration space $V$.
Let us further identify
$\mbox{$\mathfrak{su}$}(n)^{*}=\mbox{$\mathfrak{su}$}(n)$ via the Killing form. The momentum mapping is then
given by
\[
\mu: (a,\alpha)\longmapsto[a,\alpha] = \mbox{$\text{\upshape{ad}}$}(a).\alpha.
\]
Consider also an orbit
$\mbox{$\mathcal{O}$}$ together with its canonically induced symplectic structure in
the image of the momentum mapping.
\textbf{Assumption:} The orbit $\mbox{$\mathcal{O}$}$ is such that
$\mu^{-1}(\mbox{$\mathcal{O}$})\subseteq V_{r}\times V$. Here $V_{r}$ denotes the set of
regular elements in $V$ with respect to the $G$ action.
This assumption is, for example, fulfilled if the projection from $\mbox{$\mathcal{O}$}$
to any root space is non-trivial.
On the other hand, if the assumption is not satisfied for a particular
orbit $\mbox{$\mathcal{O}$}$ one can also consider the restricted $G$-action on $V_r$
and proceed with reduction of the Hamiltonian system
$(T^*V_r,\Omega,\mbox{$H_{\textup{free}}$})$ at the orbit level $\mbox{$\mathcal{O}$}$. This has, however,
the disadvantage that the Hamiltonian flow lines may leave
$T^*V_r\mbox{$/\negmedspace/_{\mathcal{O}}$} G$ in finite time.
Let $\Sigma$ denote the subspace of $V$ consisting of diagonal
matrices. Then $\Sigma$ is a section of the $G$-action on $V$, see
Section \ref{sec:pol_rep}. Further, we define
$\Sigma_{r}:=V_{r}\cap\Sigma$. Within $\Sigma$ we choose the positive Weyl
chamber $C:=\set{\diag{q_{1},\dots,q_{n}}: q_{1}>\ldots>q_{n}}$ so
that $C = \Sigma/W = V/G$
where
$W = W(\Sigma) = N_{G}(\Sigma)/Z_{G}(\Sigma)$.
Thus $C_{r}:=\Sigma_{r}\cap C$
may be considered as
a global slice for the $G$-action on $V_{r}$ so that
$G/M\times C_{r}\cong V_{r}$, $(gM,a)\mapsto g.a$
where $M:=Z_{G}(\Sigma_{r})=Z_{G}(\Sigma)$. That
is, $M$ is the subgroup of $\mbox{$\textup{SU}$}(n)$ consisting of diagonal matrices
only. Now we may apply Corollary \ref{cor:WWred} to get
\[
T^{*}V\mbox{$/\negmedspace/_{\mathcal{O}}$} G
=
T^{*}C_{r}\times\mbox{$\mathcal{O}$}\spr{0}M
\]
as symplectic stratified spaces. The strata are of the form
\[
(T^{*}V\mbox{$/\negmedspace/_{\mathcal{O}}$} G)_{(L)}
=
T^{*}C_{r}\times(\mbox{$\mathcal{O}$}\spr{0}M)_{(L_0)_M^G}
\]
where $L_{0}$ is a subgroup of $M$ conjugate to $L$ within $G$.
Moreover, the reduced symplectic structure $\lorb{\sigma}$ on
$(T^{*}V\mbox{$/\negmedspace/_{\mathcal{O}}$} G)_{(L)}$ is of product form, i.e.,
\[
\lorb{\sigma} =
\Omega^{C_{r}}-\Omega^{\mathcal{O}}_{(L_0)_M^G}
\]
where $\Omega^{\mathcal{O}}_{(L_0)_M^G}$ is the canonically reduced
symplectic form on $(\mbox{$\mathcal{O}$}\spr{0}M)_{(L_0)_M^G}$.
From the general theory (Theorem~\ref{thm:sing_spr})
we know that the Hamiltonian $\mbox{$H_{\textup{free}}$}$ reduces
to a Hamiltonian $\hcml$ on the stratum $(T^ {*}V\mbox{$/\negmedspace/_{\mathcal{O}}$} G)_{(L)}$, and
that integral curves of $\mbox{$H_{\textup{free}}$}$ project to integral curves of
$\hcml$.
In particular, the dynamics remain confined to the symplectic stratum.
The reduced Hamiltonian is thus given by
\[
\hcml(q,p,[\lambda]) = \mbox{$H_{\textup{free}}$}(q,p+A_{q}^{*}(\lambda))
\]
where $[\lambda]$ is the class of $\lambda$ in $(\mbox{$\mathcal{O}$}\spr{0}M)_{(L_0)_M^G}$ and
$A_{q}^{*}: \mathfrak{g}_{q}^{\bot}=\mathfrak{m}^{\bot}\to T_{q}(G.q)=\Sigma^{\bot}$
is the point wise dual to the mechanical connection as
introduced in Section \ref{sec:gauged_red}. Assume that
$q=\diag{q_{1}>\ldots>q_{n}}$ and that
$\lambda=(\lambda_{ij})_{ij}\in(\mbox{$\mathcal{O}$}\cap\mathfrak{m}^{\bot})_{(L_0)_M^G}$. Then
\[
A_{q}^{*}(\lambda)_{ij} = \by{\lambda_{ij}}{q_{i}-q_{j}}
\textup{ for } i\neq j,
\textup{ and }
A_{q}^{*}(\lambda)_{ii} = 0.
\]
Therefore, for $p=\diag{p_{1},\ldots,p_{n}}\in\Sigma$ and $q,[\lambda]$
as introduced we obtain
\begin{align*}
\hcml(q,p,[\lambda])
&=
\by{1}{2}\mbox{$\textup{Tr}$}(p)^{2}+\by{1}{2}\mbox{$\textup{Tr}$}(A_{q}^{*}(\lambda))^{2}
=
\by{1}{2}\sum_{i=1}^{n}p_{i}^{2}
+ \by{1}{2}\sum_{i\neq
j}\by{\lambda_{ij}\lambda_{ji}}{(q_{i}-q_{j})(q_{j}-q_{i})}\\
&=
\by{1}{2}\sum_{i=1}^{n}p_{i}^{2}
+ \sum_{i>j}\by{|\lambda_{ij}|^{2}}{(q_{i}-q_{j})^{2}}
\end{align*}
since $\lambda_{ji}=-\overline{\lambda_{ij}}$ and
$\mbox{$\textup{Tr}$}(pA_{q}^{*}(\lambda))=\mbox{$\textup{Tr}$}(A_{q}^{*}(\lambda)p)=0$.
This is the Hamiltonian
function of the Calogero-Moser system with spin. Integrability of this
system in the non-commutative sense is proved in the next section in a
more general context.
\subsection{Application: Polar representations of compact Lie groups}\label{sub:pol-cm}
The idea of considering polar representations of compact Lie groups to
obtain new versions of Spin Calogero-Moser systems is due to Alekseevsky, Kriegl, Losik, Michor\xspace
\cite{AKLM03}.
As in Section \ref{sec:pol_rep} let $V$ be a real Euclidean
vector space and $G$ a connected compact Lie group that acts on $V$ via
a polar representation. Via the inner product we consider the
cotangent bundle of $V$ as a product $T^{*}V=V\times V$. The canonical
symplectic form $\Omega$ is thus given by
\[
\Omega_{(a,\alpha)}((a_{1},\alpha_{1}),(a_{2},\alpha_{2})) =
\vv<\alpha_{2},a_{1}>-\vv<\alpha_{1},a_{2}>
\]
where $\vv<\phantom{a},\phantom{a}>$ is the inner product on $V$.
The free Hamiltonian on $T^{*}V=V\times V$ is given by
\[
\mbox{$H_{\textup{free}}$}: (a,\alpha)\longmapsto\by{1}{2}\vv<\alpha,\alpha>.
\]
Trajectories of this Hamiltonian are given by straight lines of the
form $t\mapsto a+t\alpha$ in the configuration space $V$.
Of course, the
cotangent lifted action of $G$ is just the diagonal action
of $G$ on $V\times V$. By Section \ref{sec:pol_rep} we may
think of the action by $G$ on $V$ as a symmetric space representation
and thus consider
$\mathfrak{g}\oplus V =: \mathfrak{l}$ as a real semi-simple Lie algebra with Cartan decomposition
into $\mathfrak{g}$ and $V$, and with bracket relations
$[\mathfrak{g},\mathfrak{g}]\subseteq\mathfrak{g}$, $[\mathfrak{g},V]\subseteq V$, and $[V,V]\subseteq V$. The
momentum mapping corresponding to the $G$-action on $T^{*}V=V\times V$
is now given by
\[
\mu: V\times V\longrightarrow\mathfrak{g}^{*}=\mathfrak{g},
\quad
(a,\alpha)\longmapsto[a,\alpha]=\mbox{$\text{\upshape{ad}}$}(a).\alpha
\]
where we identify $\mathfrak{g}=\mathfrak{g}^{*}$ via an $\mbox{$\text{\upshape{Ad}}$}(G)$-invariant inner product.
Consider also an orbit
$\mbox{$\mathcal{O}$}$ together with its canonically induced symplectic structure in
the image of the momentum mapping.
\textbf{Assumption:} The orbit $\mbox{$\mathcal{O}$}$ is such that
$\mu^{-1}(\mbox{$\mathcal{O}$})\subseteq V_{r}\times V$. Here $V_{r}$ denotes the set of
regular elements in $V$ with respect to the $G$ action.
We proceed as above, and let $\Sigma$ denote a fixed section of the
$G$-action on $V$, consider $C$ a Weyl chamber in $\Sigma$, and put
$M:=Z_{G}(\Sigma)$.
We may apply Corollary \ref{cor:WWred} to get
\[
T^{*}V\mbox{$/\negmedspace/_{\mathcal{O}}$} G
=
T^{*}C_{r}\times\mbox{$\mathcal{O}$}\spr{0}M
\]
as symplectic stratified spaces. The strata are of the form
\[
(T^{*}V\mbox{$/\negmedspace/_{\mathcal{O}}$} G)_{(L)}
=
T^{*}C_{r}\times(\mbox{$\mathcal{O}$}\spr{0}M)_{(L_0)_M^G}
\]
where $L_{0}$ is a subgroup of $M$ conjugate to $L$ within $G$.
Moreover, the reduced symplectic structure $\lorb{\sigma}$ on
$(T^{*}V\mbox{$/\negmedspace/_{\mathcal{O}}$} G)_{(L)}$ is of product form, i.e.,
\[
\lorb{\sigma} =
\Omega^{C_{r}}-\Omega^{\mathcal{O}}_{(L_0)_M^G}
\]
where $\Omega^{\mathcal{O}}_{(L_0)_M^G}$ is the canonically reduced
symplectic form on $(\mbox{$\mathcal{O}$}\spr{0}M)_{(L_0)_M^G}$.
From the general theory we know that the Hamiltonian $\mbox{$H_{\textup{free}}$}$ reduces
to a Hamiltonian $\hcml$ on the stratum $(T^ {*}V\mbox{$/\negmedspace/_{\mathcal{O}}$} G)_{(L)}$, and
that integral curves of $\mbox{$H_{\textup{free}}$}$ project to integral curves of
$\hcml$.
In particular the dynamics remain confined to the symplectic stratum.
The reduced Hamiltonian is thus given by
\[
\hcml(q,p,[Z]) = \mbox{$H_{\textup{free}}$}(q,p+A_{q}^{*}(\lambda))
\]
where $[Z]$ is the class of $Z$ in $(\mbox{$\mathcal{O}$}\spr{0}M)_{(L_0)_M^G}$ and
$A_{q}^{*}: \mathfrak{g}_{q}^{\bot}=\mathfrak{m}^{\bot}\to T_{q}(G.q)=\Sigma^{\bot}$
is the point wise dual to the mechanical connection as
introduced in Section \ref{sec:gauged_red}.
Let $q\in C_{r}$, $p=\sum_{i=1}^{l}p_{i}B_{0}^{i}$, and
$Z=\sum_{\lambda\in
R}\sum_{i=1}^{k_{\lambda}}z_{\lambda}^{i}E_{\lambda}^{i} \in
(\mbox{$\mathcal{O}$}\cap\mathfrak{m}^{\bot})_{(L_0)_M^G}$ where $l=\dim\Sigma$ and
$k_{\lambda}=\by{1}{2}\dim\mathfrak{l}_{\lambda}$. The notation here is as in
Section~\ref{sec:pol_rep},
and $R=R(\mathfrak{l},\Sigma)\subseteq\Sigma^{*}$ denotes the
set of restricted roots, in particular. With these definitions the
dual mapping to the mechanical connection is given by
\[
A_{q}^{*}(Z)
=
\sum_{\lambda\in R}\sum_{i=1}^{k_{\lambda}}
\by{z_{\lambda}^{i}}{\lambda(q)}B_{\lambda}^{i}.
\]
Note that $\lambda(q)\neq0$ for all $\lambda\in R$ since $q\in C_{r}$ is
regular.
The reduced Hamiltonian thus computes to
\[
\hcml(q,p,[Z])
=
\by{1}{2}\vv<p+A_{q}^{*}(Z),p+A_{q}^{*}(Z)
=
\by{1}{2}\sum_{i=1}^{l}p_{i}^{2}
+
\by{1}{2}\sum_{\lambda\in R}
\by{\sum_{i=1}^{k_{\lambda}}z_{\lambda}^{i}z_{\lambda}^{i}}{\lambda(q)^{2}}.
\]
The reduced Hamiltonian system $(T^{*}V\mbox{$/\negmedspace/_{\mathcal{O}}$}
G,\sigma^{\mathcal{O}},\hcm)$ is thus a new version of a
Calogero-Moser system with
spin. It is, in fact, a stratified Hamiltonian system in the sense that
it is a Hamiltonian system on each symplectic stratum $(T^{*}V\mbox{$/\negmedspace/_{\mathcal{O}}$}
G)_{(L)}$, and the dynamics remain confined to these strata.
We now show that the thus obtained Calogero-Moser system is
integrable in the non-commutative sense.
To do so we will use Theorem \ref{thm:zung}. We start by choosing
coordinates $q_{1},\dots,q_{n},p_{1},\dots,p_{n}$ on $T^{*}V=V\times
V$ such that the Poisson bracket of functions $f,g\in\mbox{$C^{\infty}$}(V\times V)$
is given by the usual equation
$\bra{f,g}
=
\sum_{i=1}^{n}
(\by{\del f}{\del p_{i}}\by{\del g}{\del q_{i}}
- \by{\del f}{\del q_{i}}\by{\del g}{\del q_{i}})$.
Moreover we assume that $q_{1}\dots,q_{l},p_{1},\dots,p_{l}$ are
coordinates on $\Sigma\times\Sigma\hookrightarrow V\times V$. Let us now
consider the map
\[
\Phi: V\times V\longrightarrow \Sigma^{\bot}\times V
\]
given by projection, and endow $\Sigma^{\bot}\times V$ with the
inherited Poisson structure. Clearly, $\mbox{$C^{\infty}$}(\Sigma^{\bot}\times V)$
has a center and this is just generated by $p_{1},\dots,p_{l}$. Thus
we may identify $Z(\mbox{$C^{\infty}$}(\Sigma^{\bot}\times V)) = \mbox{$C^{\infty}$}(\Sigma)$.
Now the set of all first integrals of $\mbox{$H_{\textup{free}}$}$,
i.e.,
\[
\mbox{$\mathcal{F}$}_{H_{\textup{free}}}
=
\set{F\in\mbox{$C^{\infty}$}(V\times V): \bra{F,H}=0}
\]
can be identified with $\mbox{$C^{\infty}$}(\Sigma^{\bot}\times V)$ via
$\Phi$ since $\mbox{$H_{\textup{free}}$}$ factors over the projection onto the second
factor and is $G$-invariant, and thus can be considered as a function
on $\Sigma$.
Therefore,
\begin{align*}
\dim V\times V
=
\dim\Sigma^{\bot}\times V + \dim\Sigma
=
\mbox{$\text{ddim}\,$}\mbox{$\mathcal{F}$}_{H_{\textup{free}}}+\mbox{$\text{ddim}\,$} Z(\mbox{$C^{\infty}$}(\Sigma^{\bot}\times V)),
\end{align*}
and we are exactly in the situation of the following theorem to
conclude non-commutative integrability of the reduced system.
\begin{theorem}\label{thm:zung}
Assume the Hamiltonian system $(M,\omega,H)$ is invariant under a
Hamiltonian action of a compact Lie group $G$.
If $(M,\omega,H)$ is non-commutatively integrable
(Definition~\ref{def:int}) then the
reduced system is integrable as well:
\begin{itemize}
\item The singular Poisson reduced system
is non-commutatively integrable.
\item The singular symplectic reduced system is non-commutatively
integrable.
\end{itemize}
\end{theorem}
\begin{proof}
This theorem is proved by Zung \cite[Theorem 2.3]{Zung02}. For
material on singular reduction we refer to Ortega and Ratiu
\cite{OR04} and Section~\ref{s:sg_com_red}.
\end{proof}
The idea of non-commutative integrability under the name of degenerate
integrability is due to Nehoro\v{s}ev \cite{Neh72}
who also introduced the appropriate concept of action-angle variables.
This section follows mainly the approach of Zung
\cite{Zung02,Zung03}. See also Mishchenko and Fomenko \cite{MF78}. The
following definition is less general than that given in the above
cited references but better suited for the applications in this paper.
\begin{definition}\label{def:int}
Let $(M,\bra{\phantom{f},\phantom{f}})$
be a Poisson manifold, and consider a Hamiltonian
function $H: M\to\mbox{$\bb{R}$}$.
We denote the Poisson sub-algebra of all first integrals of $H$ by
$\mbox{$\mathcal{F}$}_{H}$, that is
\[
\mbox{$\mathcal{F}$}_{H}
:=
\set{F\in\mbox{$C^{\infty}$}(M): \bra{F,H}=0}.
\]
The Hamiltonian system is called
\caps{non-commutatively integrable} if there is a finite dimensional
Poisson vector space $W$ and a generalized momentum map\xspace $\Phi: M\to W$ which is a Poisson
morphism with respect to the Poisson structure on $W$
such that the following are satisfied.
\begin{itemize}
\item $\Phi^{*}: \mbox{$C^{\infty}$}(W)\to\mbox{$\mathcal{F}$}_{H}$ is an isomorphism\xspace of Lie-Poisson algebras.
\item $\dim M = \mbox{$\text{ddim}\,$}\mbox{$C^{\infty}$}(W)+\mbox{$\text{ddim}\,$} Z(\mbox{$C^{\infty}$}(W))$ where
$Z(\mbox{$C^{\infty}$}(W))$ denotes the commutative sub-algebra of Casimir
functions on $W$, and $\mbox{$\text{ddim}\,$}\mbox{$C^{\infty}$}(W)=\dim W$ is the functional
dimension of $\mbox{$C^{\infty}$}(W)$.
\end{itemize}
\end{definition}
It is crucial in the formulation of the above theorem that $\dim M =
\mbox{$\text{ddim}\,$}\mbox{$\mathcal{F}$}_{H} + \mbox{$\text{ddim}\,$} Z(\mbox{$\mathcal{F}$}_{H})$, and $\mbox{$\mathcal{F}$}_{H}$ is the set of
\emph{all} first integrals of $H$.
\section{Appendix: Polar representations}\label{sec:pol_rep}
Let $V$ be a real Euclidean vector space, and $G$ be a connected compact Lie
group. Further, let $\rho: G\to\mbox{$\textup{SO}$}(V,\vv<\_\,,\_>)$ be a \caps{polar
representation} of $G$ on $V$. That is, there is subspace
$\Sigma\subseteq V$ (a \caps{section}) such that $\Sigma$ meets all
$G$-orbits, and does so orthogonally.
The following is due to Dadok \cite{Dad85} and is a consequence of his
classification of polar actions.
\begin{proposition}
There exists a connected Lie group $\tilde{G}$ together with a
representation $\tilde{\rho}:\tilde{G}\to\mbox{$\textup{SO}$}(V)$ such that the
following hold.
There is a real semi-simple Lie algebra $\mathfrak{l}$ with a Cartan
decomposition $\mathfrak{l} = \mathfrak{g}\oplus\mathfrak{p}$. Moreover, there is a Lie algebra
isomorphism $A: \lie{\tilde{G}}=\tilde{\mathfrak{g}}\to\mathfrak{g}$ and a linear
isomorphism
$B: V\to\mathfrak{p}$ such that
$B(\tilde{\rho}^{\prime}(X).v) = [A(X),B(v)]$ for all $X\in\tilde{\mathfrak{g}}$ and $v\in
V$. Finally, the $G$-orbits coincide with the $\tilde{G}$-orbits, that
is $V/G=V/\tilde{G}$.
\end{proposition}
\begin{proof}
See Dadok \cite[Proposition 6]{Dad85}.
\end{proof}
Thus, for the purpose of this paper, it suffices to assume that
the representation of $G$ on $V$ is a symmetric space representation
whence
$\mathfrak{l}=\mathfrak{g}\oplus V$ is a Cartan decomposition, and hence
$[\mathfrak{g},\mathfrak{g}]\subseteq\mathfrak{g}$, $[\mathfrak{g},V]\subseteq V$, and $[V,V]\subseteq\mathfrak{g}$.
Therefore, $G\times V\cong L$, $(g,v)\mapsto
g\exp(v)$ is a global Cartan decomposition with compact $G$ where $\lie{L}=\mathfrak{l}$.
An element $v\in V$ is said to be \caps{regular} (with respect to the
$G$-action) if the orbit $\mbox{$\mathcal{O}$}(v)=\rho(G).v=G.v$ is of maximal
possible dimension. The set of regular elements will be denoted by
$V_{r}$. The following assertions which are easy to verify are used in
Subsection~\ref{sub:pol-cm}.
(See also Knapp \cite[Chapter VI]{Knapp96}.)
Let $v\in V$. Then, by reason of dimension,
$\mbox{$\text{\upshape{ad}}$}(v)|Z_{\mathfrak{g}}(v)^{\bot}: Z_{\mathfrak{g}}(v)^{\bot}\to Z_{V}(v)^{\bot}$
and $\mbox{$\text{\upshape{ad}}$}(v)|Z_{V}(v)^{\bot}: Z_{V}(v)^{\bot}\to Z_{\mathfrak{g}}(v)^{\bot}$
both are linear isomorphisms.
The set $V_{r}$ of regular elements is open dense in $V$. Moreover,
$v\in V_{r}$ if and only if $Z_{V}(v)=:\Sigma$ is a section in $V$. This is the
case if and only if $\Sigma$ is maximally Abelian.
Let $\Sigma\subset V$ be a section, and put $\mathfrak{m}:=Z_{\mathfrak{g}}(\Sigma)$. The
set $R=R(\mathfrak{l},\Sigma)\subseteq\Sigma^{*}$ shall denote the set of restricted
roots. This gives rise to the restricted root space decomposition
\[
\mathfrak{l}
= \mathfrak{m}\oplus\Sigma\oplus\oplus_{\lambda\in R}\mathfrak{l}_{\lambda}.
\]
Any
Cartan subalgebra\xspace $\mathfrak{h}\subseteq\mathfrak{l}$ of $\mathfrak{l}$ is of the form
$
\mathfrak{h}=\mathfrak{t}\oplus\Sigma
$
where $\mathfrak{t}\subseteq\mathfrak{m}$ is a Cartan subalgebra\xspace (Lie algebra to a maximal torus) of $\mathfrak{g}$.
Each restricted root space
$\mathfrak{l}_{\lambda}$ has an orthonormal basis
$
E_{\lambda}^{i}\in\mathfrak{g},
B_{\lambda}^{i}\in V
$
where $i=1,\ldots,k_{\lambda}=\by{1}{2}\dim\mathfrak{l}_{\lambda}$, and which is such
that
$\mbox{$\text{\upshape{ad}}$}(v)E_{\lambda}^{i} = \lambda(v)B_{\lambda}^{i}$ and
$\mbox{$\text{\upshape{ad}}$}(v)B_{\lambda}^{i} = \lambda(v)E_{\lambda}^{i}$ for all
$v\in\Sigma$.
The vectors
$
E_{0}^{i},
$
where $i=1,\dots,\dim\mathfrak{m}$,
and
$
B_{0}^{j},
$
where
$
j=1,\dots,\dim\Sigma
$
will denote an orthonormal basis of $\mathfrak{m}$ and $\Sigma$ respectively.
|
{
"timestamp": "2008-10-30T10:07:36",
"yymm": "0411",
"arxiv_id": "math/0411068",
"language": "en",
"url": "https://arxiv.org/abs/math/0411068"
}
|
\section{Introduction}
Recently, new experimental data on leptonic widths in heavy quarkonia
(HQ) has been presented \cite{ref.1,ref.2,ref.3}. In the BaBar
experiment the mass and the total and electronic widths of the $\Upsilon
(10580)$ resonance have been measured with great accuracy \cite{ref.1},
while the CLEO Collaboration has observed significantly larger muonic
branching ratios of the $\Upsilon (nS)$ resonances $(n=1,2,3)$
compared to the values adopted till now \cite{ref.4}. Besides, existing
experimental data on the total cross section for hadron production in
$e^+e^-$ annihilation (in the region $\sqrt{s} = 3.8 \div 4.8$ GeV)
have been reanalysed \cite{ref.3}, and the total and electronic widths of
the $\psi(4040), \psi(4160), \psi(4415)$ resonances are shown to be
larger by 20\% for the $\psi(4040)$ and by 70\% for the $\psi(4415)$
resonance than their values from the Particle Data Group (PDG) \cite{ref.4}.
Accurate knowledge of the leptonic widths of high meson
excitations is of special importance for the theory, because the
wave functions (w.f.) at the origin of vector $nS$
resonances, $|R_n(0)|^2$, proportional to $\Gamma_{e^+e^-}$,
directly provides information about the static $Q\bar Q$
interaction at all distances including large $r$. In HQ they can
be expressed through the matrix element (m.e) of the static
force. Unfortunately, the true behavior of the static potential
$V(r)$ at $r\mathrel{\mathpalette\fun >} 1$~fm is still undefined in QCD and from lattice
measurements there is only an indication that the confining linear
potential $\sigma_0 r$ is becoming more flat at large $r$
\cite{ref.5}.
The origin of this phenomenon has been discussed in \cite{ref.6} where
it was shown that flattening occurs due to the creation of virtual
$q\bar q$ pairs in the Wilson loop before the string breaking takes
place \cite{ref.6}. Due to the presence of virtual loop(s) the surface
of $\langle W(C)\rangle$, and therefore the effective string tension, is
becoming smaller: the string tension $\sigma(r)$ depends on $r$ and
its derivative $\sigma'(r)<0$. For light mesons this phenomenon gives
rise to a correlated shift down of radial excitations which increases
with $n$ (for $\rho(3S)$ this shift is about 150 MeV \cite{ref.6}).
Our chereent alculations show that a similar {\em correlated} shift
down takes place for high excitations in HQ, being about 60 MeV for
$\psi(4415)$ and about 30 MeV for $\Upsilon (6S)$ \cite{ref.7}, while
the leptonic widths of high excitations provide an additional
opportunity to test the confining potential at large $r$.
In this paper we concentrate on the leptonic widths of HQ. For
low-lying resonances they have been calculated in many papers [8-10],
where it has been observed that agreement with the experimental values
of $\Gamma_{e^+e^-}$ can be obtained only if the asymptotic freedom
(AF) behaviour of the vector coupling $\alpha_V(r)$ in the
gluon-exchange (GE) interaction is taken into account (this effect is
about 50\%). For the Coulomb interaction $(\alpha_V=const)$ the
leptonic widths of both low- and high-lying resonances in the
$\Upsilon$- and $\psi$-families appear to be 50-100\% higher than
their experimental values.
However, even if the AF behavior of $\alpha_V(r)$ is taken into account
and for low-lying resonances (like $J/\psi, \psi(2S), \Upsilon(nS)
(n\leq 3)$ the leptonic widths are in agreement with experiment, still
for very high excitations (like the $\psi(4040),$ $ \psi(4415),$ $
\Upsilon(11019)$) the calculated $\Gamma_{e^+e^-}$ appear to be
significantly larger than the experimental values. The characteristic
feature of these resonances is that they have very large sizes (their
r.m.s. radii $r_n\mathrel{\mathpalette\fun >} 1.2$ fm) and therefore their w.f. at the origin
are very sensitive to details of the $Q\bar Q$ interaction at all
distances. It does not seem accidental that better agreement with
experiment for high resonances is obtained in \cite{ref.11} where a
mild (logarithmic) confining potential (instead of linear $\sigma_0 r$
potential) has been used.
In this paper we study two effects which give rise to a decrease of the
leptonic widths of high excitations in HQ. The first one is the
flattening of the confining potential at large $r$. The second effect
occurs if the GE interaction is very much suppressed or even switched
off due to a screening at distances $r\mathrel{\mathpalette\fun >} 1.0$ fm. The reason of such a
total screening needs a special analysis \cite{ref.7}, but the
dynamics of a resonance with large radius, defined by the confining
potential only, appears to be rather simple.
\section{Leptonic widths as probes of the gluon exchange interaction}
The electronic width of the vector meson $V(nS)$ is given by the
Van Royen--Weisskopf formula \cite{ref.12} with the QCD correction
taken into account \cite{ref.13}. It contains the w.f. at the origin
and some known quantities:
\begin{equation}
\Gamma_{e^+e^-} (V(nS)) = \frac{4e^2_Q\alpha^2}{M_n^2(V)}
|R_n(0)|^2\left(1-\frac{16}{3\pi} \alpha_s (2m_Q)\right),
\label{eq.1}
\end{equation}
Here for $\alpha_s(2m_c)$ and $\alpha_s(2m_b)$ we use the conventional
values: $\alpha_s(2m_c)=0.253, \,\alpha_s(2m_b)=0.177$ (e.g. see
\cite{ref.10}). The w.f. at the origin is proportional to the leptonic
width and on the other hand it can be expressed through the m.e. over
the static force $F(r)=\frac{dV}{dr}$.
In the nonrelativistic (NR) approximation the relation is \cite{ref.14}
\begin{equation}
|R_n^{NR} (0)|^2= m_Q\langle F(r)\rangle_{nS}.
\label{eq.2}
\end{equation}
Here for $m_Q$ the heavy quark pole mass entering NR Hamiltonian must
be used \cite{ref.15}. For relativistic kinematics and a relativistic
Hamiltonian with the use of the ``einbein approximation'' for the
spinless Salpeter equation instead of Eq.~(\ref{eq.2}) the following
relation can be obtained \cite{ref.16}:
\begin{equation}
|R_n(0)|^2=\omega_Q\langle F(r)\rangle_{nS},
\label{eq.3}
\end{equation}
where
\begin{equation}
\omega_Q(nS) = \langle \sqrt{\mbox{\boldmath${\rm p}$}^2+m^2_Q}\rangle_{nS}
\label{eq.4}
\end{equation}
is the average kinetic energy of a heavy quark, or the quark
constituent mass. For $c$ and $b$ quarks in HQ the difference between
$\omega_Q$ and the pole mass $m_Q$ is about 200 MeV for low-lying
states and about $250\div 300$ MeV for high excitations and this
difference gives about 20\% (5\%) corrections to $|R_n(0)|^2$ in
charmonium (bottomonium).
In the general case the static potential can be presented in the form
\begin{equation}
V(r) =r\sigma(r) -\frac43 \frac{\alpha_V(r)}{r} f_{\rm scr} (r).
\label{eq.5}
\end{equation}
To describe low-lying states (below the open-flavor threshold) it is
sufficient to take a linear confining potential with
$\sigma(r)=const=\sigma_0$ and to put the screening function
$f_{sc}(r)=1.$ For high-lying resonances both effects--the flattening
of the confining potential and the screening of GE interaction--are
becoming important. We shall consider the effects coming from screening
in detail in our next paper \cite{ref.7}, while here we take the
screening function.
\begin{equation}
f_{\rm scr} = \left\{\begin{array}{ll}
1,&r<R_{\rm scr},\\
f_0\exp(-(\sqrt{\sigma }\,r)^{4/3}),& r\geq R_{\rm scr}.\end{array}
\right.
\label{eq.6a}
\end{equation}
The choice of this function with $R_{\rm scr}\approx 0.6$ fm is
motivated by the analysis of the screening effects in \cite{ref.17}.
Here we take a larger value for the screenig radius: $R_{\rm scr}
\approx 1.0$ fm.
Then for low and high excitations one can use different
approximations in Eq.~(\ref{eq.3}). For low excitations $\sigma'(r)$ is
negligible, $\langle \sigma (r) \rangle\approx\sigma_0$, but the
contribution from the derivative $\alpha'_V(r)$ in Eq.~(\ref{eq.3}) is
important (it reflects the influence of the AF behavior of the
coupling $\alpha_V(r)$) and one obtains
\begin{equation}
|R_n(0)|^2= \omega_Q(n) \sigma_0 +\frac43 \omega_Q(n)
\left\{ \langle r^{-2} \alpha_V (r)\rangle_{nS} -
\langle r^{-1} \alpha'_V(r) \rangle_{nS}\right\}\quad ({\rm small~} n).
\label{eq.6}
\end{equation}
For high-lying excitations, on the contrary, the derivative
$\alpha'_V(r)$ is small and, moreover, in bottomonium the negative term
$\langle r \sigma'(r) \rangle_{n}$ also remains much smaller than $\langle
\sigma (r)\rangle$. For such resonances effects from the screening of GE
interaction is becoming important:
\begin{eqnarray}
|R_n(0)|^2 & = &
\omega_Q(n) \left\{\langle \sigma(r)\rangle_{nS}-\langle r\sigma'(r)\rangle_{nS} \right.
\nonumber \\
& & \quad\quad
+ \left. \frac43 \langle r^{-2} \alpha_V(r) f_{\rm scr}(r)\rangle_{nS}
-\frac43 \langle r^{-1} \alpha_V(r) f'_{s\rm cr}(r)\rangle_{nS}\right\}.
\label{eq.7}
\end{eqnarray}
In Tables \ref{Table.1} and \ref{Table.2} we present the HQ leptonic
widths calculated for three potentials still neglecting the screening
effects:
1. For the Cornell potential with the parameters taken from \cite{ref.8}
($\alpha_V(r)=const=0.52$) the leptonic widths are very large, being
50$\div$ 70\% larger for all $\Upsilon (nS)$ resonances $(n\leq 6)$
than the experimental values.
2. For the potential taken from \cite{ref.15},
\begin{equation}
V_B(r)=\sigma_0 r-\frac43 \frac{\alpha_B(r)}{r}, \label{eq.8}
\end{equation}
the vector coupling $\alpha_B(r)$ is defined in background
perturbation theory. This vector coupling $\alpha_B(r)$ has the correct
perturbative limit at small distances and also possesses the property
of freezing at large $r$. As seen from Tables 1 and 2 this potential gives
a good description of the electronic widths for many HQ states:
$J/\psi, \psi(3686)$ in charmonium and for all $\Upsilon(nS) $
resonances with exception of the $\Upsilon (6S)$ resonance (the mass
$M_{\exp}(6S) =11019$ MeV).
3. To demonstrate the sensitivity of the leptonic widths to the
behavior of the confining potential at large distances we consider the
``modified'' potential,
\begin{equation} V_M(r) =r\sigma (r) -\frac43
\frac{\alpha_B(r)}{r},
\label{eq.9}
\end{equation}
where the flattening effect is taken into account and the string
tension $\sigma(r)$ is taken as for light mesons \cite{ref.6} while
the vector coupling $\alpha_B(r)$ is the same as in the potential
Eq.~(\ref{eq.8}).
\begin{table}[ht]
\caption{\label{Table.1} The leptonic widths (in keV) of the $\Upsilon
(nS)$ resonances for the Cornell potential and the potentials given by
Eqs.~(\ref{eq.8}) and (\ref{eq.9}).}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Potential &1S & 2S & 3S & 4S & 5S & 6S \\
\hline
$\sigma_0r-\frac{\kappa}{r}$ $^{a)}$ & 2.60 & 0.94& 0.66 & 0.54 &0.47 & 0.42 \\
\hline
$ \sigma _0r-\frac43 \frac{\alpha_B(r)}{r} ~^{b)}$& 1.21 & 0.56 & 0.41 & 0.34 & 0.30 & 0.27 \\
\hline
$\sigma (r) r-\frac43 \frac{\alpha_B(r)}{r} ~^{c)}$ & 1.14 & 0.54 & 0.40 & 0.32 & 0.27 & 0.24 \\
\hline
experiment \cite{ref.4} &
1.32(7) & 0.52(8) & 0.48(11) & 0.248(31) & 0.31(7) & 0.130(30) \\
\quad\quad \cite{ref.2} & & & & 0.321(46) & & \\
\hline
\end{tabular}
$^{a)}$ From \cite{ref.8} where $\sigma_0=0.1826$ GeV$^2$,
$\kappa=0.52$; $m_b=5.17$ GeV is in fact the constituent mass
$\omega_b$ Eq.~(\ref{eq.4}). \\
$^{b)}$ Here $\sigma_0=0.18 $ GeV$^2$,
$\alpha_B(r)$ is taken from \cite{ref.15}, where
$\Lambda_{\overline{MS}}(2-{\rm loop}) =242$ MeV $(n_f = 5)$, and the pole mass
$m_b(2-{\rm loop})=4.83$ GeV.\\
$^{c)}$ Here $\sigma(r)=\sigma_0g(r)$ is taken from \cite{ref.6} with
$\sigma_0=0.18$ GeV$^2$, $(g(0)=1)$, $\alpha_B(r)$ is taken as
in footnote ${}^{b)}$.
\end{table}
\begin{table}[ht]
\caption{\label{Table.2} The leptonic widths (in keV) of the $\psi (nS)$
resonances for the same potentials as in Table~\ref{Table.1}.}
\begin{tabular}{|c|c|c|c|c|}
\hline
Potential &1S & 2S & 3S & 4S \\
\hline
$\sigma_0 r-\frac{\kappa}{r}$ $^{a)}$ & 8.18 & 3.68& 2.62 & 2.01 \\
\hline
$\sigma_0 r-\frac43 \frac{\alpha_B(r)}{r}^{b)}$
& 5.13 & 2.48 & 1.80 & 1.39 \\
\hline
$\sigma(r) r-\frac43 \frac{\alpha_B(r)}{r} ~^{c)}$ & 5.10 & 2.42 & 1.70 & 1.18
\\
\hline
experiment \cite{ref.4} & 5.26(37) & 2.19(15) & 0.75(15) & 0.47(10) \\
\quad\quad \cite{ref.3} & & & 0.89(8) & 0.71(10) \\
\hline
\end{tabular}
$^{a)}$ The parameters of the Cornell potential are the same as
in footnote ${}^{a)}$ in Table \ref{Table.1} and $m_c=1.84$ GeV.\\
$^{b)}$ See footnote ${}^{b)}$ in Table \ref{Table.1}; the pole mass $m_c=1.44$
GeV, $\Lambda_{\overline{MS}} = 260$ MeV $(n_f = 4)$.\\
$^{c)}$ See footnote ${}^{c)}$ in Table \ref{Table.1}; the pole mass $m_c=1.44$
GeV, $\Lambda_{\overline{MS}} = 260$ MeV $(n_f = 4)$.\\
\end{table}
From Tables \ref{Table.1} and \ref{Table.2} one can see that for the
modified potential Eq.~(\ref{eq.9}) the leptonic widths of the
$J/\psi,$ $\psi(2S)$ and $\Upsilon(nS) (n\leq 5)$ are practically the
same as for the linear potential $\sigma_0r$ Eq.~(\ref{eq.8}) while
for the higher resonances $(\psi(4040)$, $\psi(4415)$, and
$\Upsilon(11019))$ they are smaller by only $\sim 10\%$ and still
exceed $\Gamma_{e^+e^-} (\exp)$. The characteristic feature of these
three resonances is their large sizes (even in single-channel
approximation) $r_3(\psi(3S))\approx 1.2$ fm; $r_4(\psi(4S))\approx
1.4$ fm, $ r_6(\Upsilon(6S)\approx 1.4$ fm. There can be two possible
reasons for a further decrease of their leptonic widths. First, one may
think of the coupling of the considered $Q\bar Q$ resonance to an open
meson-meson channel. Comparison of the experimental data with our
calculations show that the $4S$ state, $\Upsilon (10580)$, has a
hadronic shift of about 50 MeV due to coupling to the $B\bar B$ channel
\cite{ref.7}, nevertheless, the calculated electronic width (see
Table~\ref{Table.1}) appears to be in very good agreement with the new
precision measurements of $\Gamma_{e^+e^-}(\Upsilon(10580))$
\cite{ref.1}. Also for the $\Upsilon(10865)$, the $5S\; b\bar b$ state,
for which the mass is close to the $B_s^*\bar B_s^*$ threshold,
agreement between the calculated and experimental value of
$\Gamma_{e^+e^-}$ is obtained. So, one can assume that open channels do
not drastically change the leptonic width of a resonance considered and
cannot explain the $\sim 70\%$ difference between the theoretical and
experimental leptonic widths for $\psi(4415)$ and $\Upsilon(11019)$. (A
small mixing of the $Q\bar Q$ and meson-meson channels was also
observed in Lattice QCD (second ref. \cite{ref.5})).
Therefore we assume here that a significant reduction of the leptonic
widths of the $\psi(4415)$ and $\Upsilon (11019)$ resonances occurs
due to a change in the static potential: a screening of the GE
interaction at large $r$ and flattening of the confining potential.
\section{Leptonic widths of highly excited resonances}
If the screening of the GE interaction with $f_{\rm scr}(r)$
Eq.~(\ref{eq.6}) is taken into account, then the w.f. at the origin is
defined by the relation Eq.~(\ref{eq.8}) where the contribution from
the GE term appears to be small ($<10\%)$ for high excitations, so
that
\begin{equation}
|R_n(0)|^2 =\omega_Q(n)\{\langle\sigma(r)\rangle_n -
\langle r\sigma'(r)\rangle_n\}\equiv\omega_Q(n)\bar\sigma_n
\label{eq.10}
\end{equation}
Here we shall use the function $\sigma (r)$ in the form and with the
parameters suggested in \cite{ref.6}. Its characteristic values are
following:
\begin{eqnarray}
\sigma(r) \approx \sigma_0 \;\; & {\rm for} & r\mathrel{\mathpalette\fun <} 1 \; {\rm fm},
\nonumber \\
\sigma(r=1.3 \; {\rm fm}) & \approx & 0.94\,\sigma_0,
\nonumber \\
\sigma(r=2.5 \; {\rm fm}) & \approx & 0.78\,\sigma_0,
\nonumber \\
\sigma(r\mathrel{\mathpalette\fun >} 4 \; {\rm fm}~) & = & 0.6 \,\sigma_0.
\label{eq.9a}
\end{eqnarray}
i.e. this string tension is slowly decreasing for larger $Q\bar Q$
separations $r$ and has asymptotic value $\sigma_{\rm asym}=
0.6\sigma_0(\approx 0.11$ GeV$^2$ for $\sigma_0=0.18$ GeV$^2$). Our
flattening confining potential continues to grow (with a smaller
slope), and it significantly differs from the one suggested in
\cite{ref.18}, where the confining potential is taken as a constant
equal to $R_{SC}\,\sigma (R_{SC})$ for $r\geq R_{SC}\approx 1.6$ fm. One
may notice that with such an assumption the quarks in a meson are not
confined and can be liberated.
For this simple asymptotic potential $V_{\rm asym}({\rm large~}r)
=r\sigma(r)$ the constituent masses $\omega_n(Q\bar Q)$ and $\langle
\sigma(r)\rangle_{nS}$ can be calculated easily from the solutions of the
spinless Salpeter equation \cite{ref.15}. For the $\Upsilon (6S)$ and
$\Upsilon (7S)$, using $(\sigma_0=0.18$ GeV$^2$ and $m_b\approx 4.83$
GeV) the following numbers are obtained,
\begin{eqnarray}
\omega_7(b\bar b) & \approx & \omega_6(b\bar b) = 5.1~{\rm GeV},
\nonumber \\
\langle \sigma (b\bar b, r )\rangle_{6S}& = & 0.171~{\rm GeV}^2
\nonumber \\
\langle \sigma (b\bar b, r )\rangle_{7S} & = & 0.167~{\rm GeV}^2.
\label{eq.10a}
\end{eqnarray}
and the term $\langle r\sigma'(r) \rangle$ is relatively small. Then
from Eq.~(\ref{eq.10})
\begin{eqnarray}
|R_6(b\bar b, 0)|^2 & = & 0.87 ~{\rm GeV}^3,
\nonumber \\
|R_7(b\bar b, 0)|^2 & = & 0.85
~{\rm GeV}^3.
\label{eq.11}
\end{eqnarray}
and from Eq.~(\ref{eq.1}) one obtains
\begin{equation}
\Gamma_{e^+e^-}(\Upsilon(11.019))=0.12~{\rm keV}.
\label{eq.14a}
\end{equation}
This value is in good agreement with the experimental value
$\Gamma_{e^+e^-}(\Upsilon(6S))=0.130\pm 0.030$ keV [4]. With the
use of Eq.~(\ref{eq.13}) the electronic width of the still unobserved
$\Upsilon(7S)$ can also be predicted:
\begin{equation}
\Gamma_{e^+e^-}(\Upsilon(7S))=0.11~{\rm keV},
\label{eq.15a}
\end{equation}
where the value of the mass, $M_7=11.25$ GeV (obtained in
single-channel approximation) has been used. Note that
the mass difference $M(7S) - M(6S)$ is not small, about 230 MeV.
In charmonium for better accuracy, the negative correction in
$\overline{\sigma}_n$ Eq.~(\ref{eq.9}) coming from the derivative $\langle
r \sigma'(r)\rangle$, is becoming larger and gives a contribution of
$\sim 15\%$. The value of $\bar \sigma$ for the $\psi (4S)$ is $ \langle
\sigma(r)-r\sigma'(r)\rangle_{4S}=0.14$ GeV$^2$ and $\langle
\sigma(r)-r\sigma'(r)\rangle_{5S}=0.13$ GeV$^2$ for $\psi(5S)$, while the
constituent masses are: $\omega_4(c\bar c) =1.71$ GeV and
$\omega_5(c\bar c)=1.67$ GeV. Then from Eq.~(\ref{eq.10}) it follows
that
\begin{equation}
|R_4(c\bar c,
0)|^2=0.24 ~{\rm GeV}^3,\\ |R_5(c\bar c, 0)|^2=0.22 ~{\rm
GeV}^3,\label{eq.12}
\end{equation}
and correspondingly, the electronic widths are
\begin{equation}
\Gamma_{e^+e^-} (\psi(4415))=0.66 ~{\rm keV},~~\Gamma_{e^+e^-}
(\psi(5S))=0.54 ~{\rm keV}.
\label{eq.13}
\end{equation}
For the $\psi(4415)$ resonance our theoretical prediction in
Eq.~(\ref{eq.13}) agrees very well with that from the analysis of
Seth~\cite{ref.3} ($\Gamma_{e^+e^-}(\psi(4415))_{\exp}=0.71\pm0.10$
keV), while both numbers significantly differ from PDG's
$\Gamma_{e^+e^-}(\psi(4415))=0.47\pm 0.10$ keV.
Our treatment above was done in single-channel approximation when the
possibility of string breaking is neglected, while the creation of
virtual $q\bar q$ pairs is taken into account through the dependence of
$\sigma (r)$ on $r$. Since at present there is no fundamental
string-breaking theory in QCD, we neither do know what the probability
of string breaking and is nor do we know the probability of the
existence of very high $Q\bar Q$ excitations. Therefore we do not know
what is the upper limit, or the admittable size $R_{max}$ of a
high-lying resonance (the $Q\bar Q$ string) above which a resonance
cannot exist.
Still, the resonances $\Upsilon (7S)$ and $\psi(5S)$ do not have large
sizes, the splitting $\Upsilon (7S)-\Upsilon(6S)$ is not small,
($\Delta M\sim 230$ MeV), and therefore one may expect them to exist.
In our calculations $\bar r_7(b\bar b)=1.6$ fm and $\bar r_5 (c\bar
c)=1.8$ fm (in single-channel approximation) and their masses (without
a hadronic shift) are $M(\Upsilon(7S)) \approx 11.24$ GeV,
$M(\psi(5S))\approx 4.63$ GeV. In Eqs.~(\ref{eq.15a}) and
(\ref{eq.13}) their electronic widths,
$\Gamma_{e^+e^-}(\Upsilon(7S))=0.11~{\rm keV},
\Gamma_{e^+e^-}(\psi(5S))=0.54~{\rm keV}$ are given (see
Table~\ref{Table.3}).
\begin{table}
\caption{\label{Table.3} The leptonic widths (in keV) of highly excited
states in charmonium and bottomonium for the flattening potential
$\sigma(r) r$, taken from \cite{ref.6} $(\sigma_0=0.18$ GeV$^2$).}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
& $\psi(4040)$ & $\psi(4415)$ & $\psi(5S)$ & $\Upsilon(11019) $& $\Upsilon(7S)$ \\
& & & $M\approx 4630$ MeV & & $M\approx 11250$ MeV \\
\hline
~this paper~ & 0.94 & 0.66 & 0.54 & 0.12 & 0.11 \\
\hline
Exper.~\cite{ref.4} & 0.75(15) & 0.47(10) & & 0.13(3) & \\
\quad\quad\quad\cite{ref.3} &0.89(8)& 0.71(10)& &&\\
\hline
\end{tabular}
\end{table}
\section{Conclusions}
The electronic widths of high-lying resonances in HQ are of special
interest for the theory because they provide important information
about the QCD confining potential at large distances.
Our calculations, performed with a relativistic Hamiltonian, show
that three effects give rise to a decrease of the electronic
widths of vector mesons:\\
(i) The asymptotic-freedom behavior of the vector coupling, which
determines the GE potential, gives a decrease of the leptonic widths of
about 70\% for the $\Upsilon(nS)$ resonances $(n\leq 6)$ and about
50\% for the $\psi(nS)$ $(n\leq 4)$ resonances.\\
(ii) The flattening of the confining potential at large distances gives
an additional drop ($\sim 15\%)$ in the leptonic widths of HQ but only
for high excitations: $n\geq 4$ for the $\Upsilon(nS)$ family and
$n\geq 3$ for the $\psi(nS)$ states. In this case good agreement with
experiment is obtained for all $\Gamma_{e^+e^-}(\Upsilon(nS))(n\leq 5)$
and for $\Gamma_{e^+e^-}(J/\psi)$, $\Gamma_{e^+e^-}(\psi(3686))$.\\
(iii) If for some reason the GE interaction is totally switched off for
resonances of large size, then the leptonic widths of very high
excitations, like $\Upsilon (11019)$, $ \psi(4040)$, and $\psi(4415)$,
strongly decrease and appear to be in good agreement with the experimental
data. These three resonances have large r.m.s. radii, $r_n(Q\bar Q)\mathrel{\mathpalette\fun >} 1.2$ fm
and their purely nonperturbative dynamics turns out to be
rather simple. It is essential that here the string tension
$\sigma(r)$ is taken just the same as in the light meson analysis of
radial excitations \cite{ref.6}.\\
(iv) The electronic widths and masses of the still unobserved
resonances: $\Gamma_{e^+e^-} (\Upsilon(7S))=0.11$~ keV
$(M_7(b\bar{b})\approx 11250)$ and $\Gamma_{e^+e^-}(\psi(5S))\approx 0.54 $
keV $( M_5(c\bar c)\approx 4630)$ are predicted.\\
\vspace{1cm}
\acknowledgments
We thank Yu.A. Simonov for fruitful discussions. This work was partly supported by the PRF Grant for leading scientific
schools, Nr. 1774.2003.2.
|
{
"timestamp": "2004-11-22T14:02:04",
"yymm": "0411",
"arxiv_id": "hep-ph/0411291",
"language": "en",
"url": "https://arxiv.org/abs/hep-ph/0411291"
}
|
\section{Introduction}
The theory of input-to-state stable (ISS) systems plays a central
role in modern non-linear control analysis and controller design
(see \citep{MRS04,MS04,S98,S01,SW95}). The ISS property was
introduced by Sontag in \citep{S89} and an ISS Lyapunov
characterization was obtained by Sontag and Wang in \citep{SW95}.
The ISS Lyapunov characterization provides necessary and
sufficient conditions for time-invariant systems to be ISS, in
terms of the existence of so-called strict ISS Lyapunov functions;
see Section \ref{sec2} below for the relevant definitions and
\citep{ELW00} for an extension to time-varying systems. Strict
Lyapunov functions have been used to design stabilizing feedback
laws that render asymptotically controllable systems ISS to
actuator errors and small observation noise; see \citep{MS04,S01}.
Such control laws are expressed in terms of gradients of Lyapunov
functions and therefore require explicit strict Lyapunov functions
in order to be implemented. This has motivated a great deal of
research devoted to constructing explicit strict Lyapunov functions.
One obstacle to these constructions is that the known strict
Lyapunov functions from the existence theory are optimal control
value functions, involving a supremum of a cost criterion over
infinitely many possible solution paths (see \citep{BR01,ELW00,SW95,TP00}),
and therefore are not explicit. Although value
functions can often be expressed as unique solutions of
Hamilton-Jacobi (HJ) equations subject to appropriate side
conditions, the usual techniques for computing value functions in
terms of HJ equation solutions can be difficult to implement. For
certain special kinds of systems, strict ISS Lyapunov functions
can be explicitly constructed by ad hoc means. On the other hand,
there are numerous important cases where it is relatively
straightforward to use backstepping or other known methods to
construct explicit {\em non}-strict ISS Lyapunov functions (see our
definitions of ISS and non-strict ISS Lyapunov functions in Section
\ref{sec2} and Section \ref{sec5} for an explicit example). For
instance, applying the methods of \citep{JN97} to tracking problems
for nonholonomic systems in chained form results in non-strict
Lyapunov functions. The constructions in \citep{MazPrat} also
frequently give rise to non-strict Lyapunov functions.
This motivates the search for techniques for constructing strict
ISS Lyapunov functions for time-varying systems, in terms of known
non-strict ISS Lyapunov functions. This search is the focus of
this note. For time-varying systems with no controls, the paper
\citep{M03} constructed strict globally asymptotically stable
(GAS) Lyapunov functions in terms of given non-strict GAS Lyapunov
functions. Here we further develop the approach in \citep{M03}.
We provide the necessary background on ISS systems and Lyapunov
functions in Section \ref{sec2}. We then introduce a non-strict
generalization of ISS in which the dissipation rate depends on a
non-negative time-dependent decay parameter. The parameter can be
zero along intervals of positive length. However, when the
parameter is identically one, our non-strict ISS property agrees
with the usual ISS condition. Under a mild non-degeneracy
assumption on this parameter, which is of persistency of
excitation type (see for instance \citep{udel} and \citep{lopa}
for definitions and discussions of the concept of persistency of
excitation), we show that our non-strict ISS property is
equivalent to the existence of a strict ISS Lyapunov function and
is therefore also equivalent to the standard ISS condition. We
prove these equivalences in Section \ref{sec3}. They are proved by
explicitly constructing strict ISS Lyapunov functions. In Section
\ref{sec5}, we illustrate our constructions using a tracking
example. Concluding remarks in Section \ref{secc} end the paper.
\section{Preliminaries}
\label{sec2} Let $\mathcal{K}_\infty$ denote the set of all
continuous functions $\rho:[0,\infty)\to[0,\infty)$ for which (i)
$\rho(0) = 0$ and (ii) $\rho$ is increasing and unbounded. Let
$\mathcal{K}\mathcal{L}$ denote the set of all continuous
functions $\beta:[0,\infty) \times [0,\infty)\to[0,\infty)$ for
which (1) for each $t\ge 0$, $\beta(\cdot, t)$ is strictly
increasing and $\beta(0,t) = 0$ (2) $\beta(s,\cdot)$ is
non-increasing for each $s\ge 0$, and (3) $\beta(s,t)\to 0$ as
$t\to +\infty$ for each $s\ge 0$.
We study the stability properties of the fully nonlinear
nonautonomous system
\begin{equation}
\label{sys}
\dot x = f(t,x,u),\; \; \; t\ge 0, \; x\in {\mathbb R}^n,
\; u\in {\mathbb R}^m
\end{equation}
where we always assume $f$ is locally Lipschitz in $(t,x,u)$.
Following \citep{M03}, we also assume $f$ is periodic in $t$,
which means there exists a constant $T>0$ such that
$f(t+T,x,u)=f(t,x,u)$ for all $t\ge 0$, $x\in {\mathbb R}^n$, and
$u\in {\mathbb R}^m$. However, most of our arguments remain valid
if this periodicity assumption is weakened to requiring $f$ to be
uniformly locally bounded in $t$, meaning,
\begin{equation}
\label{boundedness}
\sup\{|f(t,x,u)|: (x,u)\in K, t \ge 0\} < + \infty
\end{equation}
where $|\cdot|$ is the usual Euclidean norm.
The control functions for
our system (\ref{sys}) comprise the set of all measurable locally
essentially bounded functions $\alpha:[0,\infty)\to{\mathbb R}^m$;
we denote this set by $\mathcal{U}$. We let $|\alpha|_I$ denote
the essential supremum of any control $\alpha\in \mathcal{U}$
restricted to any interval $I\subseteq [0,\infty)$. For each
$t_o\ge 0$, $x_o\in {\mathbb R}^n$, and $\alpha\in \mathcal{U}$,
we let $I\ni t\mapsto \phi(t;x_o, t_o, \alpha)$ denote the unique
trajectory of (\ref{sys}) for the input $\alpha$ satisfying
$x(t_o) = x_o$ and defined on its maximal interval $I\subseteq [t_o,\infty)$.
This trajectory will be denoted by $\phi$ when this would not lead to confusion.
We say that $f$ is {\em forward complete} provided each such trajectory $\phi$ is
defined on all of $[t_o,\infty)$.
A $C^1$ function $V:[0,\infty) \times {\mathbb R}^n \to [0,\infty)$
is said to be of class ${\rm UPPD}$ (written $V \in {\rm UPPD}$)
provided it is uniformly proper and positive definite, which means
there exist $\alpha_1, \alpha_2, \alpha_3 \in \mathcal{K}_\infty$ such
that, for all $t \ge 0, x \in {\mathbb R}^n$,
\begin{equation}
\label{uppd} \alpha_1(|x|) \le V(t,x)\le \alpha_2(|x|), \; \;
|\nabla V(t,x)|\le \alpha_3(|x|).
\end{equation}
We say that $V$ has period $\tau$ in $t$ provided there exists a
constant $\tau>0$ such that $V(t+\tau,x)=V(t,x)$ for all $t\ge 0$
and $x\in {\mathbb R}^n$; in this case, the bound on $\nabla V$ in
(\ref{uppd}) is redundant. We assume $\alpha_1$ and $\alpha_2$
in (\ref{uppd}) are $C^1$, e.g., by taking
$\alpha_2(s)=\int_o^{\scriptscriptstyle s}\alpha_3(r)dr$ and
minorizing $\alpha_1$ by a $C^{\scriptscriptstyle 1}$ function of
class $\mathcal{K}_\infty$.
Given $V\in {\rm UPPD}$, we set
\[\dot V(t,x,u):=\frac{\partial V}{\partial t}(t,x)+
\frac{\partial V}{\partial x}(t,x) f(t,x,u).\] Notice that
$s\mapsto\sup\{|\dot V(t,x,u)|: t\ge 0, |x|\le \chi(s),|u|\le s\}
+s$ is of class $\mathcal{K}_\infty$ for each $\chi\in
\mathcal{K}_\infty$ (by (\ref{boundedness})-(\ref{uppd})). We let
$\mathcal{P}$ denote the set of all continuous functions
$p:{\mathbb R} \to [0,\infty)$ that admit constants
$\tau,\varepsilon,\bar p>0$ for which
\begin{equation}
\label{calp}
\begin{array}{rcl}
\int_{t-\tau}^tp(s)ds\; \ge \; \varepsilon\; {\rm \and\ } \; p(t)\le \bar p\; , \; \; \forall t\ge 0.
\end{array}
\end{equation}
We write $p\in \mathcal{P}(\tau,\varepsilon,\bar p)$ to indicate that
(i) $p\in \mathcal{P}$ and (ii) $\tau,\varepsilon,\bar p>0$ are
constants such that (\ref{calp}) holds. In particular, any
continuous periodic function $p:{\mathbb R}\to[0,\infty)$ that is
not identically zero admits constants $\tau,\varepsilon,\bar p>0$
satisfying (\ref{calp}). On the other hand, (\ref{calp}) also
allows non-periodic $p$ with arbitrarily large null sets, e.g., for
fixed $r > 0$, set $p_r(t) = (1 + e^{-t})\max\{0, \sin^3(\frac{t}{r})\}$.
The elements of $\mathcal{P}$
serve as the decay rates for our non-strict Lyapunov functions as
follows:
\begin{defn}
Let $p\in \mathcal{P}$. A function $V\in {\rm UPPD}$ is called
an {\rm ISS(p) Lyapunov function} for (\ref{sys}), provided there exist
$\chi\in \mathcal{K}_\infty$ and $\mu\in \mathcal{K}_\infty\cap C^1$
such that
\begin{equation}
\label{nonstr} |x| \ge \chi(|u|) \Rightarrow \dot V(t,x,u) \le -
p(t)\mu(|x|)\; \; \; \forall t\ge 0.
\end{equation}
An ISS(p) Lyapunov function for (\ref{sys}) and $p(t)\equiv 1$
is also called a {\rm strict ISS Lyapunov function}.
\end{defn}
Notice that (\ref{nonstr}) allows $\dot V(t,x,u)=0$ for those $t$
where $p(t) = 0$. This corresponds to allowing $V$ to non-strictly
decrease along the solutions $\phi$ of (\ref{sys}).
\begin{defn}
Let $p\in \mathcal{P}$. We say that (\ref{sys}) is {\rm ISS(p)},
or that it is {\rm input-to-state stable (ISS) with decay rate}
$p$, provided there exist $\beta\in \mathcal{K}\mathcal{L}$ and
$\gamma \in \mathcal{K}_\infty$ such that for all $t_o\ge 0$,
$x_o\in {\mathbb R}^n$, $u_o\in \mathcal{U}$ and $h\ge 0$,
\begin{equation}
\label{siss}
\begin{array}{rcl}
|\phi(t_o+h; x_o, t_o, u_o)| & \le & \beta\left(|x_o|, \int_{t_o}^{t_o+h} p(s)ds\right)
\\
& & + \gamma\left(|u_o|_{[t_o,t_o+h]}\right).
\end{array}
\end{equation}
If (\ref{sys}) is ISS(p) with $p \equiv 1$, then we say that (\ref{sys}) is {\rm ISS}.
\end{defn}
Notice that ISS(p) systems are automatically forward complete. We
also study dissipation-type decay conditions as follows:
\begin{defn}
\label{zpb} Let $p\in \mathcal{P}$. A function $V\in {\rm UPPD}$
is called a {\rm non-strict dissipative Lyapunov function} for
(\ref{sys}) and $p$, or a {\rm DIS(p) Lyapunov function}, provided
there exist $\Omega\in \mathcal{K}_\infty$ and $\mu\in
\mathcal{K}_\infty\cap C^1$ such that, for all $t \ge 0, x \in
{\mathbb R}^n, u \in {\mathbb R}^m$
\begin{equation}
\label{i3}\dot V(t,x,u) \; \le - p(t)\mu(|x|) + \Omega(|u|) \; \; .
\end{equation}
A DIS(p) Lyapunov function for (\ref{sys}) and $p(t)\equiv 1$ is
also called a {\rm strict DIS Lyapunov function}.
\end{defn}
\begin{rem}
\label{rte} Definition \ref{zpb} is a nonlinear version of the
property used in \citep{lopa} to ensure the global uniform
exponential stability of time-varying linear systems belonging to
a specific family of systems. Thus, the explicit construction of a
strict DIS Lyapunov function in terms of a given DIS(p) Lyapunov
function we present in the next section, extends \citep{lopa}
where only linear systems are studied and no strict Lyapunov
function is constructed.
\end{rem}
We use the following elementary observations:
\begin{lem}
\label{plem}
Let $\tau,\varepsilon, \bar p > 0$ be constants and
$p\in \mathcal{P}(\tau,\varepsilon,\bar p)$ be given. Then: \newline
\noindent
(i) $0 \leq \int_{t-\tau}^t\left(\int_s^tp(r)dr\right)ds \le \frac{\tau^2\bar p}{2}$
for all $t\ge 0$ and\newline
\noindent
(ii) $[0,\infty)\ni h\mapsto {\underline
p}(h)=\inf\left\{\int_t^{t+h}p(r)dr: t\ge 0\right\}$ is
continuous, non-decreasing, and unbounded.
\end{lem}
We leave the proof of this lemma to the reader as a simple
exercise.
\section{Equivalent Characterizations of Non-Strict ISS}
\label{sec3}
We next relate the Lyapunov functions and stability notions we
introduced in the last section. We show that ISS(p)
is equivalent to the existence of an ISS(p) Lyapunov function
and the existence of a strict ISS Lyapunov function. Our proof
explicitly constructs a strict ISS
Lyapunov function for (\ref{sys}) in terms of a given DIS(p)
Lyapunov function. Moreover, if $p\in \mathcal{P}(\tau,\varepsilon,
\bar p)$ and our given DIS(p) Lyapunov function both have period
$\tau$, then the strict ISS Lyapunov function we construct also
has period $\tau$. We next prove:
\begin{thm}
\label{mainthm}
Let $p\in \mathcal{P}$ and $f$ be as above. The following are equivalent: \newline
\noindent
$(C_1)$ $f$ admits an ISS(p) Lyapunov function.\newline
\noindent
$(C_2)$ $f$ admits a strict ISS Lyapunov function.\newline
\noindent
$(C_3)$ $f$ admits a DIS(p) Lyapunov function.\newline
\noindent
$(C_4)$ $f$ admits a strict DIS Lyapunov function.\newline
\noindent
$(C_5)$ $f$ is ISS(p). \newline
\noindent
$(C_6)$ $f$ is ISS.
\end{thm}
\noindent We prove the following implications: $(C_1)
\Rightarrow (C_2) \Rightarrow (C_4)\Rightarrow (C_1)$,
$(C_3)\Leftrightarrow(C_4)$, $(C_2)\Leftrightarrow(C_6)$, and
$(C_5)\Leftrightarrow(C_6)$.
We fix $\tau,\varepsilon, \bar p>0$ such that $p\in
\mathcal{P}(\tau,\varepsilon,\bar p)$.\newline \noindent {\bf Step
1:} $(C_1) \Rightarrow (C_2)$. If $(C_1)$ holds, then we can find
an ISS(p) Lyapunov function $V$ for $f$, and therefore
$\alpha_{1}, \alpha_{2}\in \mathcal{K}_{\infty}\cap C^1$
satisfying (\ref{uppd}) and $\chi\in \mathcal{K}_\infty$ and $\mu
\in \mathcal{K}_{\infty}\cap C^1$ satisfying (\ref{nonstr}). Set
\begin{equation}
\label{choices}
\begin{array}{rcl}
\tilde\alpha_2(s) & := & \max\left\{\frac{\tau\bar
p}{2},1\right\}(\alpha_2(s)+\mu(s)+s),
\\
w(s) & := & \frac{1}{4\tau}\mu(\tilde \alpha^{-1}_2(s)).
\end{array}
\end{equation}
Then $\tilde \alpha_2,\tilde \alpha^{-1}_2 \in
\mathcal{K}_\infty\cap C^1$. Since $V(t,x)\le \tilde
\alpha_2(|x|)$ for all $t\ge 0$ and $x\in {\mathbb R}^n$, the
following holds for all $t\ge 0$:
\begin{equation}
\label{g1} |x| \ge \chi(|u|) \Rightarrow \dot V(t,x,u) \le -
p(t)\mu(\tilde \alpha_2^{-1}(V(t,x))) .
\end{equation}
Note too that $w\in \mathcal{K}_{\infty}\cap C^1$. We later use
the fact that
\begin{equation}
\label{ig7} \begin{array}{lll} \displaystyle 0\; \le\;
w'(s)&\le&\displaystyle \frac{\mu'(\tilde\alpha^{-1}_2(s))}{4\tau
\max\{\frac{\tau\bar p}{2},1\}(\mu'(\tilde \alpha^{-1}_2(s))+1)}\\
&\le& \displaystyle\frac{1}{2\tau^2\bar p}\end{array}
\end{equation}
for all $s\ge 0$. Consider the UPPD function
\begin{equation}
\label{vs}
V^\sharp(t,x) = V(t,x) + \xi(t) w(V(t,x))
\end{equation}
with $\xi(t) = \int_{t-\tau}^{t}\left(\int_s^t p(r)\, dr\,\right)
ds$. Then
\[
\begin{array}{rcl}
\dot{V}^\sharp(t,x,u) & = & [1 + \xi(t) w'(V(t,x))]\dot{V}(t,x,u)
\\
& & + \left[\tau p(t) - \int_{t - \tau}^{t} p(r)\, dr\right]
w(V(t,x))
\end{array}\]
follows from a simple calculation. When $|x| \ge \chi(|u|)$,
condition (\ref{g1}) gives ${\scriptstyle \dot V}(t,x,u)\le 0$ and
therefore also
\[
\begin{array}{rcl}
\dot{V}^\sharp(t,x,u) & \leq & - p(t)\mu(\tilde
\alpha^{-1}_2(V(t,x)))
\\
& +& \left[\tau p(t) - \int_{t - \tau}^{t} p(r)\, dr\right]
\frac{1}{4\tau}\mu(\tilde \alpha_2^{-1}(V(t,x)))
\\
& \leq & - \frac{3}{4}p(t)\mu(\tilde \alpha^{-1}_2(V(t,x)))
\\
& -& \left(\int_{t - \tau}^{t} p(r)\, dr\right)
\frac{1}{4\tau}\mu(\tilde \alpha_2^{-1}(V(t,x)))
\\
& \leq & -
\frac{\varepsilon}{4\tau}\mu(\tilde\alpha_2^{-1}(\alpha_1(|x|)))\;
\; \forall t\ge 0.
\end{array}
\]
Since $\mu\circ\tilde \alpha_2^{-1}\circ\alpha_1\in C^1\cap
\mathcal{K}_\infty$, it follows that $V^\sharp$ is a strict ISS
Lyapunov function for (\ref{sys}).\newline \noindent {\bf Step 2:}
$(C_2) \Rightarrow (C_4)$. Assume $(C_2)$, so $f$ admits a strict
ISS Lyapunov function $V$. Let $\mu$ and $\chi$ satisfy condition
(\ref{nonstr}) with $p\equiv 1$. Then the strict dissipative
condition (\ref{i3}) with $p\equiv 1$ follows by choosing any
$\Omega \in \mathcal{K}_\infty$ satisfying
\[
\Omega(s) \ge \displaystyle\max_{\{t\ge 0, |x|\le \chi(s), |u|\le
s\}}\{\dot V(t,x,u) + \mu(|x|)\} \; \; \forall s\ge 0.
\]
Such an $\Omega$ exists by our assumptions
(\ref{boundedness})-(\ref{uppd}). Therefore, $V$ is itself a
strict DIS Lyapunov function for $f$. \newline \noindent {\bf Step
3:} $(C_4) \Rightarrow (C_1)$. Assume $(C_4)$, so $f$ admits a
strict DIS Lyapunov function $V$. Let $\mu,\Omega\in
\mathcal{K}_\infty$ satisfy (\ref{i3}) with $p\equiv 1$; then if
$|x| \geq \chi(|u|):=\mu^{-1}(2\Omega(|u|))$, then
\[\dot V(t,x,u) \le - \frac{1}{2} \mu(|x|),\; \; {\rm so}\; \;
\dot V(t,x,u) \le - \frac{p(t)}{2\bar p} \mu(|x|)\] for all $t\ge
0$.
Therefore, $V$ is also an ISS(p) Lyapunov function for $f$, so
$(C_1)$ is satisfied.\newline \noindent {\bf Step 4:}
$(C_3)\Leftrightarrow(C_4)$. Since $p\in \mathcal{P}$ is bounded,
we easily conclude that $(C_4)$ implies $(C_3)$. Conversely,
assume $V\in {\rm UPPD}$ is a DIS(p) Lyapunov function for $f$ and
$\alpha_1,\alpha_2,\mu,\Omega\in \mathcal{K}_\infty$ satisfy
(\ref{uppd}) and the DIS(p) requirements. Define $\tilde \alpha_2,
w\in \mathcal{K}_\infty\cap C^1$ and $V^\sharp$ by (\ref{choices})
and (\ref{vs}). As before, when $\tilde \mu=\mu\circ \tilde
\alpha^{\scriptscriptstyle -1}_2$, we have ${\scriptstyle \dot
V}(t,x,u) \le - p(t)\tilde \mu(V(t,x)) + \Omega(|u|)$ for all $t
\ge 0, x \in {\mathbb R}^n, u \in {\mathbb R}^m$. It follows from
Lemma \ref{plem}(i) and (\ref{ig7}) that
\begin{equation}\label{bounds}
1 + \xi(t) w'(V(t,x)) \in \left[1,\frac{5}{4}\right]\; , \; \;
\forall t\ge 0, x\in {\mathbb R}^n.
\end{equation}
Since $w = \frac{1}{4\tau}\tilde \mu$, we deduce that
\begin{equation}
\begin{array}{rcl}
\dot{V}^\sharp & \leq & - p(t)\tilde{\mu}(V(t,x)) +
\frac{5}{4}\Omega(|u|)\nonumber
\\
& +& \tau p(t) w(V(t,x))\nonumber - \left(\int_{t - \tau}^{t}
p(r)dr\right) w(V(t,x))\nonumber
\\
& \leq & - \varepsilon w(\alpha_1(|x|)) + \frac{5}{4}\Omega(|u|). \nonumber
\end{array}
\end{equation}
Since $w\circ\alpha_1\in C^1\cap \mathcal{K}_{\infty}$, it follows
that $V^\sharp$ is the desired strict DIS Lyapunov
function.\newline \noindent {\bf Step 5:} $(C_2)\Leftrightarrow
(C_6)$. The implication $(C_2)\Rightarrow (C_6)$ follows from
\citep[Theorem 4.19, p.176]{K02}. (In \citep{K02}, the controls
are bounded piecewise continuous functions $\alpha:[0,\infty)\to
{\mathbb R}^m$, but the result from \citep{K02} can be extended to
our general control set $\mathcal{U}$ using a standard denseness
argument (see e.g. Remark C.1.2 and the proof of Theorem 1 in
\citep{S98a}).) The converse was announced in \citep[Theorem
1]{ELW00} and can be deduced from \citep{BR01} as follows. If $f$
is ISS, then \citep{SW95} provides $\chi\in \mathcal{K}_\infty$
such that the constrained input system $\dot x =
f_\chi(t,x,d):=f(t,x,d\chi^{-1}(|x|))$, $|d|\le 1$ is uniformly
globally asymptotically stable (UGAS); i.e., there exists $\beta
\in \mathcal{K}\mathcal{L}$ such that for each $t_o\ge 0$ and
$x_o\in {\mathbb R}^n$ and each trajectory $y$ of $f_\chi$
satisfying $y(t_o)=x_o$, we have $|y(t_o+h)|\le \beta(|x_o|,h)$
for all $h\ge 0$. By minorizing $\chi^{-1}$, we can assume it is
$C^1$. This means the locally Lipschitz set-valued dynamics
$F(t,x)=\{f(t,x,u): \chi(|u|)\le |x|\}$ is UGAS, as is its
convexification $\overline{\rm co}(F)$, namely $(t,x)\mapsto
\overline{\rm co}\{F(t,x)\}$ where $\overline{\rm co}$ denotes the
closed convex hull (cf. \citep[Proposition 4.2]{BR01}). Since
$\overline{\rm co}(F)$ is continuous and compact and convex
valued, and since we are assuming $f$ is periodic in $t$,
\citep[Theorem 4.5]{BR01} provides a time-periodic $V\in {\rm
UPPD}$ such that, for all $x\in {\mathbb R}^n, \; t\ge 0,\; w \in
F(t,x)$,
\[
\frac{d}{dt}V(t,x)+\frac{d}{dx} V(t,x)w\le - V(t,x).
\]
Recalling the definition of $F$ and assuming (without loss of
generality) that $V$ satisfies (\ref{uppd}) with $\alpha_1\in
\mathcal{K}_\infty\cap C^1$,
\[
\begin{array}{l}
|x|\ge \chi(|u|)\; \Rightarrow\; f(t,x,u)\in F(t,x)\\
\Rightarrow\; \dot V(t,x,u)\le -V(t,x)\le
-\alpha_1(|x|)\end{array}\] for all $t\ge 0$, so $V$ is the
desired strict ISS Lyapunov function for $f$. This establishes
$(C_6)\Rightarrow (C_2)$.\newline \noindent {\bf Step 6:}
$(C_5)\Leftrightarrow (C_6)$. Assuming $(C_6)$, there are
$\beta\in \mathcal{K}\mathcal{L}$ such that for all $t_o\ge 0$,
$x_o\in {\mathbb R}^n$, $u_o\in \mathcal{U}$, and $h\ge 0$,
\[
\begin{array}{lll}
|\phi(t_o+h; x_o, t_o, u_o)|& \le & \beta(|x_o|, \bar p h)
+\gamma(|u_o|_{[t_o,t_o+h]})
\\
& \le & \beta(|x_o|,\int_{t_o}^{t_o+h} p(s)ds ) \\&+&
\gamma(|u_o|_{[t_o,t_o+h]}),\end{array}
\]
where $\phi$ is the trajectory of $f$ we defined
in Section \ref{sec2}. Therefore, $f$ is ISS(p) so $(C_6)\Rightarrow (C_5)$.
Conversely, if $f$ is ISS(p), then we can find
$\beta\in \mathcal{K}\mathcal{L}$ such that for all
$t_o \ge 0$, $x_o \in {\mathbb R}^n$, $u_o \in \mathcal{U}$, and $h \ge 0$,
\[
\begin{array}{lll}
|\phi(t_o+h; x_o, t_o, u_o)|& \le & \beta\left( |x_o|,
\int_{t_o}^{t_o+h} p(s)ds\right)
\\
& & + \gamma(|u_o|_{[t_o,t_o+h]})
\\
& \le & \beta\left(|x_o|,\underline{p}(h)\right)+\gamma(|u_o|_{[t_o,t_o+h]}).
\end{array}
\]
By Lemma \ref{plem}(ii) , $\hat \beta(s,t):=
\beta(s,\underline{p}(t))\in \mathcal{K}\mathcal{L}$, so $(C_5)\Rightarrow
(C_6)$, as desired. This proves Theorem \ref{mainthm}.
\medskip
\begin{rem}
Observe that if the functions $V$, $\alpha_2$, $\mu$, $p$ are of
class $C^k$, where $k$ is a positive integer or $\infty$, then the
particular function $\tilde{\alpha}_2$ in (\ref{choices}) we have
chosen implies that the function $V^\sharp(t,x)$ is of class
$C^k$.
\end{rem}
\begin{rem}
Our proof of Theorem \ref{mainthm} shows that if $V$ is a strict
ISS Lyapunov function for $f$, then $V$ is also a strict DIS
Lyapunov function for $f$. The preceding implication is no longer
true if our boundedness requirement (\ref{boundedness}) on $f$ is
dropped, as illustrated by the following example from
\citep{ELW00}: Take the one-dimensional single input system $\dot
x=f(t,x,u):=-x+(1+t)q(u-|x|)$, where $q:{\mathbb R}\to{\mathbb R}$
is any $C^1$ function for which $q(r)\equiv 0$ for $r\le 0$ and
$q(r)>0$ otherwise.
Then $V(x)=x^2$ is a strict ISS Lyapunov function for the system
since $|x|\ge |u| \Rightarrow {\scriptstyle \dot V}\le -x^2$ but
$V$ does
not satisfy the strict DIS condition (\ref{i3}) for any
choices of $\mu$ and $\Omega$. This does not contradict our
results because (\ref{boundedness}) is not satisfied. This
contrasts with the time-invariant case where strict ISS Lyapunov
functions are automatically strict DIS Lyapunov functions.
\end{rem}
\section{Illustration}
\label{sec5} We next use our results to construct a strict ISS
Lyapunov function for a tracking problem for a rotating rigid body
(see \citep{C84,SSPJ95,SS97} for the background and motivation for
this problem). Following Lefeber \citep[p.31]{L00}, we only
consider the dynamics of the velocities, which, after a change of
feedback, are
\begin{equation}
\label{transformed}
\dot{\omega}_1 = \delta_1 + u_1\; ,\; \;
\dot{\omega}_2 = \delta_2 + u_2\; ,\; \; \dot{\omega}_3 = \omega_1\omega_2.
\end{equation}
where $\delta_1$ and $\delta_2$ are the inputs and $u_1$ and $u_2$ are
the disturbances.
We consider the reference state trajectory
\begin{equation}
\label{zl1}
\omega_{1r}(t) = \sin(t) \; ,\; \; \omega_{2r}(t)= \omega_{3r}(t) = 0
\end{equation}
but our method applies to more general reference trajectories as
well; see Remark \ref{lasr} below. The substitution
$\tilde{\omega}_i(t) = \omega_i(t) - \omega_{ir}(t)$ transforms
(\ref{transformed}) into the error equations
\begin{equation}
\label{cnh}
\begin{array}{rcl}
\dot{\tilde{\omega}}_1 & = & \delta_1 + u_1 - \cos(t) \; ,
\\
\dot{\tilde{\omega}}_2 & = & \delta_2 + u_2 \; ,
\\
\dot{\tilde{\omega}}_3 & = & (\tilde{\omega}_1 + \sin(t))\tilde{\omega}_2\; .
\end{array}
\end{equation}
By applying the backstepping approach as it is applied in
\citep{JN97}, or through direct calculations, one shows that the
derivative of the class UPPD function
\begin{equation}
\label{tjb}
V(t,\tilde{\omega}) = \frac{1}{2}\left[\tilde{\omega}_1^2
+ (\tilde{\omega}_2 + \sin(t)\tilde{\omega}_3)^2 + \tilde{\omega}_3^2\right]
\end{equation}
with $\tilde{\omega} = (\tilde{\omega}_1, \tilde{\omega}_2, \tilde{\omega}_3)^\top$
along the trajectories of (\ref{cnh}) in closed-loop with the control laws
\begin{equation}
\label{poc}
\begin{array}{rcl}
\delta_1(t,\tilde{\omega}) & = & - \tilde{\omega}_1 - \tilde{\omega}_2\tilde{\omega}_3 + \cos(t)
\\
\delta_2(t,\tilde{\omega}) & = & - [1 + \sin(t)\tilde{\omega}_1 + \sin^2(t)]\tilde{\omega}_2
\\
& & - (2\sin(t) + \cos(t))\tilde{\omega}_3
\end{array}
\end{equation}
satisfies
\begin{equation}
\label{ufg}
\begin{array}{rcl}
\dot{V} & = & - \tilde{\omega}_1^2 - (\tilde{\omega}_2 + \sin(t)\tilde{\omega}_3)^2
- \sin^2(t)\tilde{\omega}_3^2
\\
& & + \tilde{\omega}_1 u_1 + (\tilde{\omega}_2 + \sin(t)\tilde{\omega}_3) u_2
\\
& \leq & - \frac{1}{2}\tilde{\omega}_1^2 - \frac{1}{2}(\tilde{\omega}_2 + \sin(t)\tilde{\omega}_3)^2
- \sin^2(t)\tilde{\omega}_3^2
\\
& & + \frac{1}{2}(u_1^2 + u_2^2)
\\
& \leq & - p(t)\tilde\mu(V(\tilde{\omega}))
+ \Omega(|u|)
\end{array}
\end{equation}
with $u = (u_1,u_2)^\top \in {\mathbb R}^2$, $p(t) = \sin^2(t)$,
$\tilde\mu(s) = s$ and $\Omega(s) = \frac{1}{2}s^2$. Therefore $V$
is a DIS(p) Lyapunov function for (\ref{cnh}) in closed-loop with
the control laws (\ref{poc}). Observe that, in this case, $p\in
\mathcal{P}(\pi,\pi/2,1)$. Setting $\tau = \pi$ and $w(s) =
\frac{1}{8\tau}\tilde\mu(s) = \frac{s}{8\pi}$, it follows that
(\ref{bounds}) also holds. Therefore, Steps 3-4 from our proof of
Theorem \ref{mainthm} show
\[\begin{array}{ccc}
\! \! & V^\sharp(t,\tilde{\omega}) = V(t,\tilde{\omega}) +
\left[\int_{t-\tau}^t\left(\int_s^t p(r) dr\right) ds\right]
w(V(t,\tilde{\omega}))
\\
& = \left[1 + \frac{\pi}{32} - \frac{1}{32}
\sin(2t)\right]V(t,\tilde{\omega})
\end{array}
\]
is a strict DIS Lyapunov function and also a strict ISS Lyapunov
function for the system (\ref{cnh}) in closed-loop with
the control laws (\ref{poc}). \medbreak\medbreak \noindent
\begin{rem}
\label{lasr} We chose to work with the reference trajectory
(\ref{zl1}) because it leads to the simple error equations
(\ref{cnh}). However, one can easily check that a strict ISS
Lyapunov function can be constructed for any reference state
trajectory $(\omega_{1r}(t),\omega_{2r}(t), \omega_{3r}(t))$ such
that \[\begin{array}{l}\sup_t
\left|\int_{0}^{t}\omega_{1r}(s)\omega_{2r}(s) ds\right| <
\infty\; \; {\rm and}\\ \int_{t - \tau}^{t}[\omega^2_{1r}(s) +
\omega^2_{2r}(s)] ds \geq \varepsilon \; , \; \forall t\ge
\tau\end{array}\]
for some constants $\tau,\varepsilon>0$.
\end{rem}
\section{Conclusion}
\label{secc}
For ISS time-varying systems, we provided explicit strict
Lyapunov function {\em constructions} that can easily be performed
in practice. The knowledge of these Lyapunov functions allows us
to extend the well-known and useful theory of ISS systems to a
broad class of time-varying nonlinear dynamics. We conjecture
that a discrete-time version of our main result can be proved.
\thebibliography{xx}
\harvarditem[Bacciotti \& Rosier]{Bacciotti \& Rosier}{2001}{BR01}
Bacciotti, A. \& Rosier, L. (2001). {\em Liapunov Functions and
Stability in Control Theory.} Lecture Notes in Control and Inform.
Sci. Vol. 267, Springer-Verlag London, Ltd., London.
\harvarditem[Crouch]{Crouch}{1984}{C84} Crouch, P. (1984).
Spacecraft attitude control and stabilization: applications of
geometric control theory to rigid body models. {\em IEEE Trans.
Automat. Control}, 29(4), 321-331.
\harvarditem[Edwards {\em et al.}]{Edwards, Lin \&
Wang}{2000}{ELW00} Edwards, H., Lin, Y. \& Wang, Y. (2000). On
input-to-state stability for time-varying nonlinear systems. {\em
Proceedings of the 39th IEEE Conference on Decision and Control,
Sydney, Australia.}
\harvarditem[Jiang \& Nijmeijer]{Jiang \& Nijmeijer}{1997}{JN97}
Jiang, Z.-P. \& Nijmeijer, H. (1997). Tracking control of mobile
robots: a case study in backstepping. {\em Automatica}, 33(7),
1393-1399.
\harvarditem[Khalil]{Khalil}{2002}{K02}
Khalil, H. (2002).
{\em Nonlinear Systems, Third Edition.} Prentice Hall, 2002.
\harvarditem[Lefeber]{Lefeber}{2000}{L00}
Lefeber, E., (2000).
{\em Tracking Control of Nonlinear Mechanical Systems.}
PhD Thesis, University of Twente, Enschede, The Netherlands, April 2000.
(On-line at $\mathtt{http://se.wtb.tue.nl/\sim lefeber/}$.)
\harvarditem[Loria \& Panteley]{Loria \& Panteley}{2002}{lopa}
Loria, A. \& Panteley, E. (2002). Uniform exponential stability of
linear time-varying systems: revisited. {\em Systems \& Control
Letters}, 47(1), 13-24.
\harvarditem[Loria {\em et al.}]{Loria, Panteley, Popovic \&
Teel}{2002}{udel} Loria, A., Panteley, E., Popovic, D. \& Teel, A.
(2002). $\delta$-persistency of excitation: a necessary and
sufficient condition for uniform attractivity. {\em Proceedings of
the 41st IEEE Conference on Decision and Control, Las Vegas, NV.}
\harvarditem[Malisoff {\em et al.}]{Malisoff, Rifford \&
Sontag}{2004}{MRS04} Malisoff, M., Rifford L. \& Sontag E. (2004).
Global asymptotic controllability implies input-to-state
stabilization. {\em SIAM Journal on Control and Optimization,}
42(6), 2221-2238.
\harvarditem[Malisoff \& Sontag]{Malisoff, Rifford \&
Sontag}{2004}{MS04} Malisoff, M. \& Sontag, E. (2004). {\em
Asymptotic controllability and input-to-state stabilization: The
effect of actuator errors.} In: Optimal Control, Stabilization,
and Nonsmooth Analysis, Lecture Notes in Control and Inform.
Sci., Springer-Verlag, Heidelberg. Vol. 301, 155-171.
\harvarditem[Mazenc]{Mazenc}{2003}{M03} Mazenc, F. (2003). Strict
Lyapunov functions for time-varying systems. {\em Automatica},
39(2), 349-353.
\harvarditem[Mazenc \& Praly]{Mazenc \& Praly}{2000}{MazPrat}
Mazenc, F. \& Praly, L. (2000). Asymptotic Tracking of a State
Reference for Systems with a Feedforward Structure. {\em
Automatica}, 36(2), 179-187.
\harvarditem[Morin \& Samson]{Morin \& Samson}{1997}{SS97} Morin,
P. \& Samson, C. (1997). Time-varying exponential stabilization of
a rigid spacecraft with two controls. {\em IEEE Trans. Automatic
Control}, 42(4), 528-534.
\harvarditem[Morin {\em et al.}]{Morin, Samson, Pomet \&
Jiang}{1995}{SSPJ95} Morin, P., Samson, C., Pomet, J.-B. \& Jiang,
Z.-P. (1995). Time-varying feedback stabilization of the attitude
of a rigid spacecraft with two controls. {\em Systems \& Control
Letters}, 25(5), 375-385.
\harvarditem[Sontag]{Sontag}{1989}{S89} Sontag, E. (1989). Smooth
stabilization implies coprime factorization. {\em IEEE Trans.
Automatic Control}, 34(4), 435-443.
\harvarditem[Sontag]{Sontag}{1998}{S98} Sontag, E. (1998).
Comments on integral variants of ISS. {\em Systems \& Control
Letters}, 34(1-2), 93-100.
\harvarditem[Sontag]{Sontag}{1998}{S98a}
Sontag, E. (1998).
{\em Mathematical Control Theory. Deterministic
Finite-Dimensional Systems. Second Edition.} Texts in Applied
Mathematics 6. Springer-Verlag, New York, 1998.
\harvarditem[Sontag]{Sontag}{2001}{S01} Sontag, E. (2001). {\em
The ISS philosophy as a unifying framework for stability-like
behavior}. In: Nonlinear Control in the Year 2000, Vol. 2,
Lecture Notes in Control and Inform. Sci., Springer, London. Vol.
259, 443--467.
\harvarditem[Sontag \& Wang]{Sontag \& Wang}{1995}{SW95} Sontag,
E. \& Wang, Y. (1995). On characterizations of the input-to-state
stability property. {\em Systems \& Control Letters}, 24(5),
351-359.
\harvarditem[Teel \& Praly]{Teel \& Praly}{2000}{TP00} Teel, A. \&
Praly, L. (2000). A smooth Lyapunov function from a
class-$\mathcal{K}\mathcal{L}$ estimate involving two positive
semidefinite functions. {\em ESAIM Control Optim. Calc. Var.}, 5,
313-367.
\endthebibliography
\end{document}
|
{
"timestamp": "2004-11-07T18:22:19",
"yymm": "0411",
"arxiv_id": "math/0411150",
"language": "en",
"url": "https://arxiv.org/abs/math/0411150"
}
|
\section{INTRODUCTION}
After H and He, C, N, O are, together with Ne, the most abundant elements in
the Universe. As such, they are key ingredients in a large number of
astrophysical issues. Their abundances in metal-poor stars are tracers of the
nucleosynthetic sites that contributed to the different phases of galactic
evolution. Moreover, they are important contributors to the opacity in stellar
interiors and act as catalysts in the CNO-cycle of H-burning.
Stars of globular clusters (GCs) offer an ideal diagnostic in order to
understand
stellar evolution for low and intermediate stellar masses.
However, since the pioneering study of Osborn (1971) it is known that a spread
in the light elements (C, N, O, but also among heavier species such as Na, Al and
Mg) is present among cluster stars of similar evolutionary phase, unlike their
analogs in the galactic field (e.g. Gratton et al. 2000 and references
therein). Among these last, C and N abundances follow well defined
evolutionary paths, with two episodes of mixing, the first one related to the
first dredge-up and the second one after the red giant branch (RGB) bump, when
the molecular weight barrier created by the maximum inward penetration of the
outer convective envelope is canceled by the outward expansion of the
H-burning shell. The first episode was theoretically predicted by Iben (1964);
while the second one is not present in canonical non-rotating models, it
may be easily accomodated in models where some type of circulation is
activated e.g. by core rotation (Sweigart \& Gross 1978; Charbonnel 1994).
Gratton et al. (2000) showed that among field stars no variations
corresponding to these mixing episodes are observed for the remaining
elements (namely, O and Na): again, this agrees with models, that do not allow
deep enough mixing along the RGB. Clearly this pointed toward a peculiarity of
globular cluster stars.
As shown by Denisenkov \& Denisenkova (1989), and later in a more
quantitative way by Langer et al. (1993), the observed star-to-star scatter
in cluster stars may be explained by the CNO-cycle and the accompanying
proton-capture reactions at high temperature. More uncertain is where these
reactions occurred, whether in the observed star themselves, prior to an
internal (extra- or enhanced-) very deep mixing episode, or elsewhere, perhaps
in some form of H-burning at high temperature taking place e.g. in now extinct
intermediate-mass AGB stars (IM-AGB), followed by ejection of polluting matter
(see Gratton, Sneden and Carretta 2004 for an updated review and references on
the huge literature on this subject).
Evidences from red giants are ambiguous, since both mixing (causing a decrease
of [C/Fe] as a function of luminosity, see Bellman et al. 2001 and references
therein) and pollution/accretion of processed matter (e.g. Yong et al. 2003,
Sneden et al. 2004) might be invoked to explain observations.
Cleaner conclusions can be drawn from unevolved or slightly evolved stars,
where no mixing is expected and inner temperatures are not high enough to
permit the $p-$capture reactions in the NeNa and AlMg cycles. However, up to
a short time ago, only low-dispersion observations of molecular bands of
hydrides such as CH and NH or bi-metallic molecules such as CN were available to study
the chemical composition of faint GC turn-off (TO) dwarfs or subgiants (SGB).
More important, no O indicator was accessible, since the atmospheric cutoff
and the low throughput of existing spectrographs severely hampered the use of
OH bands in the UV regions, and the remaining O features are only observable
on high dispersion spectra.
In spite of these limitations, a number of studies (see e.g. Briley et al.
2004b and references therein) uncovered that large spreads in the CH and CN
band strengths, anticorrelated with each other, do exist in unevolved stars in
several GCs. The only explanation must necessarily rest on an event that
polluted the material forming these stars, likely early in the cluster
lifetime.
Previous results from the present program (Gratton et al. 2001, Carretta et al.
2004) provided further strong evidences favouring primordial abundance
variations. Deriving the first reliable O abundances in cluster dwarfs, we
found a clear anticorrelation between Na and O among TO and SGB stars in NGC
6752 and 47 Tuc. In NGC 6397, early results were not conclusive, but they were
hampered by small number statistics. Similar anticorrelations were found also
for Mg and Al. These facts require some non internal mechanism.
However, after initial successes (see e.g. Ventura et al. 2001), more recent
models of metal-poor IM-AGB stars met serious problems in reproducing the O-Na
anticorrelation and related phenomenology (Denissenkov \& Herwig 2003; Fenner
et al. 2004; Herwig 2004), and unveiled that not only Hot Bottom Burning
occurs (HBB), but also vigorous H-burning at somewhat cooler temperatures
during the interpulse phases. To overcome these problems, Denissenkov \& Weiss
(2004) recently proposed that the site for the $p-$capture reactions is the
interior of RGB stars slightly more massive than those currently observed in
globular clusters, and that they exchanged mass with the currently unevolved
stars where the anomalous abundances are observed.
In the present paper we complete the analysis of the spectra presented in
Gratton et al. (2001) and Carretta et al. (2004) by including detailed
abundances of C, N and O. From these abundances and
from isotopic $^{12}$C/$^{13}$C ratios, measured for the
first time in such unevolved stars, we suggest that both
triple-$\alpha$ captures in He-burning, to form fresh $^{12}$C, and typical
H-burning processing at high temperatures are required to reproduce the
observed pattern of abundances in these stars. If confirmed, this would
exclude the possibility that mass-exchange with RGB stars might be responsible
for the observed abundances. In the discussion, we will also comment on other
possible shortcomings of this hypothesis.
Furthermore, we also present results for Na and O abundances in a more
extended sample of subgiants in NGC 6397, showing that large variations,
anticorrelated with each other, in these two elements do exist also in this
metal-poor cluster.
\section{OBSERVATIONS}
Details of observations are given in Gratton et al. (2001; Paper I) and
Carretta et al. (2004). Briefly, spectra were acquired using the
Ultraviolet-Visual Echelle Spectrograph (UVES) mounted at the ESO VLT-UT2
within several runs (June and September 2000; August and October 2001, July
2002) of the ESO Large Program 165.L-0263 (P.I. R. Gratton). On the whole, we
have observational material for 6 dwarfs and 9 subgiant stars in NGC 6397, 9
dwarfs and 9 subgiants in NGC 6752 and 3 dwarfs and 9 subgiants in 47 Tuc.
Relevant data for the observed stars are given in Gratton et al. (2001) and
Carretta et al. (2004). Those for the additional subgiants in NGC 6397 are
the same as given for the other subgiants in Gratton et al. (2001).
Data were acquired using the dichroic beamsplitter \#2. In the blue arm we
used the CD2, centered at 420 nm, to cover both the CH G-band at $\sim
4300$~\AA\ and the CN UV system at $\sim 3880$~\AA. The spectral coverage is
about $\lambda\lambda$ 356-484 nm. The CD4, centered at 750 nm (covering
$\lambda\lambda$ 555-946 nm), was adopted in the red arm. Observations in the
run of June 2000 (mostly for NGC 6397) were made with a slightly different
setup, resulting in a spectral coverage $\lambda\lambda$ 338-465 nm in the
blue and $\lambda\lambda$ 517-891 nm in the red. Slit length was always 8
arcsec, while the slit width was mostly set at 1 arcsec (corresponding to a
resolution of 43000). In a few cases, according to the seeing conditions, this
value was slightly modified downward or upward.
In NGC 6397 and NGC 6752, typical exposure times were $\sim 1$ hour for
subgiants, while in 47 Tuc we doubled this time. Each turn-off star was
observed for a total of about 4 hours, split into several exposures.
At $\lambda \sim 4300$ we reached a typical value of the $S/N$ per
pixel of $\sim 30$, increasing to $\sim 70$ for stars in NGC 6397.
\section{ATMOSPHERIC PARAMETERS}
The adopted values of the atmospheric parameters are discussed in detail in
Gratton et al. (2001) and Carretta et al. (2004). Here we only recall the main
features of the analysis, that was also applied to the 6 newly observed SGB
stars in NGC 6397.
We compared effective temperatures from observed colours (both Johnson $B-V$
and Str\"omgren $b-y$ were used) with spectroscopic temperatures derived from
fitting Balmer lines (namely H$\alpha$). This approach was devised to derive
precise values of the reddenings on the same scale for both cluster stars and
field stars. These were used in the estimate of accurate distances to these
clusters (see Gratton et al. 2003).
Average values of temperatures were finally used for stars of NGC 6752 and NGC
6397. However, in 47 Tuc we found that the adoption of individual T$_{\rm
eff}$'s for the subgiant stars provided best agreement with the values given
by line excitation. Adopted values for 47 Tuc are given in Table 2 of Carretta
et al. (2004).
Values of the surface gravity were derived from the location of stars in the
colour magnitude diagram; an age of 14 Gyr and corresponding masses were
assumed.
Estimates of the microturbulent velocity $v_t$ for each star were derived, as
usual, by eliminating trends of abundances with expected line strengths. Again,
average values for each group of stars in similar evolutionary phases were
adopted in NGC 6397 and NGC 6752.
Finally, the overall model metallicities [A/H] were chosen as equal to the Fe
abundances that best reproduce the measured equivalent widths ($EW$), using the
model atmospheres from the Kurucz (1995) grid with the overshooting option
switched off.
\section{ANALYSIS}
\subsection{Carbon and isotopic ratios}
Carbon abundances for the program stars were obtained from a comparison of
observed and synthetic spectra in the wavelength region from 4300~\AA\ to
4340~\AA. This spectral region includes the band head of the (0-0),(1-1) and
(2-2) bands of the A$^2$$\Delta$-X$^2$$\Pi$ transitions of CH. We used newly
derived line lists from Lucatello et al. (2003). Briefly, the starting line list was
extracted from Kurucz's database (Kurucz CD-ROM 23, 1995), including
atomic species and molecular lines of C$_2$, CN, CH, NH and OH. A few lines,
missing in the Kurucz database, were added from the solar tables (Moore,
Minnaert \& Houtgast 1966); when unidentified, we arbitrarily attributed these
lines to Fe I, with an excitation potential of EP=3.5 eV.
The dissociation potential of CH has been determined with high accuracy at
$D^0_0 = 3.464$ eV (Brzozowoski et al. 1976), while band oscillator strenghts
were modified in order to reproduce the observed solar spectrum (Kurucz et al.
1984), using the solar carbon abundance of Anders and Grevesse (1989). We
found that, in order to have a good match, a corrective factor of $-$0.3 dex in
the $\log gf$ values of the electronic transitions and a shift of $-$0.05~\AA\ in
wavelength were required, with respect to the values given by Kurucz. As found
by many authors (see e.g. Grevesse \& Sauval 1998), high excitation CH lines
listed by Kurucz are missing in the spectra of the Sun and other stars, due to
pre-dissociation. We omitted from our line list those lines rising from levels
with excitation potential over 1.5 eV.
The excellent match of the synthetic spectrum with the observed solar spectrum
in part of the G-band region is shown in Figure~\ref{f:solargband}.
\begin{figure}
\psfig{figure=solargband.eps,width=8.8cm,clip=}
\caption[]{The observed solar spectrum (Kurucz et al. 1984), shown as a
continuous line, with overimposed the synthetic spectrum obtained using the
adopted line list and the solar model (dotted line), in the G-band region at
$\sim$ 4309-18~\AA. Notice that the spectral region shown in this Figure
includes less than 1/4 of that used in our comparison with synthetic spectra.}
\label{f:solargband}
\end{figure}
Using the appropriate atmospheric parameters, synthetic spectra in the spectral
region 4300-4340~\AA\ were computed varying [C/Fe] in steps of 0.2 dex, in the
range from 0.3 to $-$0.7 dex. A constant oxygen abundance [O/Fe] = 0 was adopted
in all these computations. The exact values affect only negligibly the
derived C abundances, since for stars warmer than $\sim 4500$ K the coupling
of C and O is not relevant.
After the synthesis computations, the generated spectra were convolved with
Gaussians of appropriate FWHM to match the broadening mechanisms (in
particular that due to the instrumental response) of the observed spectra.
Carbon abundances were then derived from a set of 15-17 CH features within the
region under scrutiny, inspecting by eye all features, and computing an
average value for each star.
Results are summarized in Table~\ref{t:cno}, while Figure~\ref{f:figch} shows
two examples of the synthetic spectrum fits to the observed CH features for a
subgiant (star 478) and a turn-off star (star 1012) in 47 Tuc. The average
$rms$ deviations of the abundances from the individual observed features in
[C/Fe] are 0.10-0.12 dex.
\begin{figure}
\psfig{figure=figch.eps,width=8.8cm,clip=}
\caption[]{Left panel: spectrum synthesis of some features of CH band in a
subgiant star of 47 Tuc. The heavy solid line is the observed spectrum, while
dashed, solid and dotted lines are the synthetic spectra computed for three
values of the C abundances (listed on top of figure). Right panel: the same, for
a dwarf star of 47 Tuc. Note that the synthetic spectra are now computed with
different C values. All synthetic spectra were convolved with a Gaussian
to take into account the instrumental profile of observed spectra.}
\label{f:figch}
\end{figure}
Isotopic ratios $^{12}$C/$^{13}$C were estimated from spectrum synthesis in
the two regions 4228-4240~\AA\ and 4360-4372~\AA\ containing various clean
features of $^{13}$CH (see e.g. Sneden, Pilachowski and Vandenberg 1986;
Gratton et al. 2000). Synthetic spectra were computed using the C abundances
for each star derived from the G-band synthesis and the appropriate
atmospheric parameters, and a range for the $^{12}$C/$^{13}$C values. The
adopted isotopic ratios were derived as the averages from several features in
both regions.
\subsection{Nitrogen}
Since the violet CN band at 4200~\AA\ is vanishingly weak in warm metal-poor
stars (see Cannon et al. 1998, but see below the case of 47 Tuc), we used the
UV CN band and derived N abundances from a number of CN features in the
wavelength range 3876-3890~\AA, where the bandhead of the $\Delta v=0$ bands of
the UV system lies, again using line lists optimized by Lucatello et al. (2003).
These lists use a CN dissociation potential of 7.66 eV from Engleman and Rouse
(1975); a corrective factor of -0.3 dex in the $\log gf$ of the electronic
transitions was applied also in this case, with respect to the values listed by
Kurucz. In Figure~\ref{f:solarcn} the comparison between the observed solar
spectrum (Kurucz et al. 1984) and the synthetic spectrum computed with the
optimized line list is shown.
\begin{figure}
\psfig{figure=solarcn.eps,width=8.8cm,clip=}
\caption[]{The observed solar spectrum (Kurucz et al. 1984), shown as a
continuous line, with overimposed the synthetic spectrum obtained using the
adopted line list and the solar model (dotted line), in the region at
$\sim$ 3880~\AA\ including the $\Delta v=0$ bandheads of UV CN transition.}
\label{f:solarcn}
\end{figure}
Carbon abundances derived above from synthesis of the CH bands were adopted in
the computation of synthetic spectra, relevant for individual stars, together
with the appropriate atmospheric parameters (from Gratton et al. 2001 and
Carretta et al. 2004).
In the case of 47 Tuc, which is about 1.3 dex and 0.7 dex more metal-rich than
NGC 6397 and NGC 6752, respectively, we were able to use also the violet CN
band strengths at $\sim 4215$~\AA\ in order to estimate the N abundances, at
least in the subgiant stars. A procedure similar to that described above was
used to compute synthetic spectra in the region from 4202~\AA\ to 4226~\AA,
with the proper C abundance for each star. The N abundances resulted to be in
very good agreement with those derived from the synthesis of the 3880~\AA\
region, so for the subgiants in 47 Tuc the [N/Fe] values are those obtained as
the average of N abundances in the two regions.
No observations for the NH band were available for our program stars, apart from
the very first run (June 2000), where the setup covered the region from 3376
to 3560~\AA, missed in the following observing runs. In this run stars in both
NGC 6397 and NGC 6752, but not in 47 Tuc were observed. This choice of the
setup was driven by the consideration that the expected $S/N$ of the spectra
of the (fainter) stars in NGC 6752 and 47 Tuc was so low that likely no
meaningful abundances could be obtained.
For these spectra of NGC 6397 stars, we then prepared a line list in the
spectral range 3400-3410~\AA, where some NH lines lie, using again the solar
spectrum as a starting point; however, in order to obtain a good match, the
$\log gf$ values of NH lines in Kurucz's list had to be lowered by about 0.5 dex.
Results of the NH synthesis in stars of NGC 6397 are given in the next Sect.
\subsection{Oxygen and Sodium}
Oxygen abundances in these warm stars were derived almost exclusively from the
permitted near-IR triplet at 7771-75~\AA, as discussed at length in Gratton et
al. (2001) and Carretta et al. (2004). Only for one subgiant in 47 Tuc
could we measure the forbidden [O I] lines. For the other stars, the
very weak [O I] line was masked by much stronger telluric features, so that no
reliable abundance could be derived. Final abundances and upper limits are
given in Table~\ref{t:cno}, corrected for non-LTE effects as described in
Gratton et al. (1999), from statistical equilibrium calculations based on
empirically calibrated collisional H I cross sections. The appropriate
corrections were also applied to the Na abundances, derived from the strong
doublet at 8183-94~\AA.
\section{RESULTS}
Derived abundances for C, N, O and isotopic ratios $^{12}$C/$^{13}$C for stars
in NGC 6397, NGC 6752 and 47 Tuc are listed in Table~\ref{t:cno}.
Carbon isotopic ratios could not be reliably
derived for stars in NGC 6397 and dwarfs in NGC 6752; only upper limits for C
and N abundances were obtained for dwarf stars in NGC 6397, due to the weakness
of the features and the low $S/N$ ratio in the blue region of the spectra.
Table~\ref{t:cno} also lists, for an easier comparison of the relevant elements
involved in H-burning at high temperatures, the abundances of Na taken
from the previous papers of this series (Gratton et al. 2001, Carretta et al.
2004). For the 6 subgiants in NGC 6397 observed in July 2002, newly derived
Na and O abundances are also shown in this Table, where stars are ordered
according to increasing Na abundances.
{\tiny
\begin{table*}
\caption[]{Abundances of C, N, O, Na, and isotopic ratios $^{12}$C/$^{13}$C
in stars of 47 Tuc, NGC 6752 and NGC 6397 }
\begin{tabular}{rccrcr}
\hline
Star & [C/Fe] & [N/Fe]& [O/Fe] & [Na/Fe]& $^{12}$C/$^{13}$C \\
\hline
\\
\multicolumn{6}{c}{NGC 6397 - dwarfs} \\
\\
1905 &$<$+0.50& $<$2.0& +0.24 & +0.09 & \\
202765 &$<$+0.50& $<$1.5& +0.33 & +0.13 & \\
201432 &$<$+0.50& $<$1.5& +0.21 & +0.15 & \\
1543 &$<$+0.50& $<$1.5& +0.28 & +0.28 & \\
1622 &$<$+0.50& $<$2.0& +0.23 & +0.35 & \\
\\
\multicolumn{6}{c}{NGC 6397 - subgiants} \\
\\
706 & +0.10& $-$0.5& +0.54 &$-$0.48 & \\
760 & +0.10& $-$0.5& +0.39 &$-$0.43 & \\
777 & +0.15& +0.2& +0.39 &$-$0.25 & \\
737 & +0.15& +1.4& $<$0.06 & +0.19 & \\
793 & $-$0.10& +1.2& $<$0.06 & +0.23 & \\
703 & +0.00& +1.5& $<$0.06 & +0.30 & \\
206810 & $-$0.07& +1.3& $<$0.31 & +0.32 & \\
729 & $-$0.10& +1.4& $<$0.06 & +0.48 & \\
669 & +0.01& +1.3& +0.31 & +0.53 & \\
\\
\multicolumn{6}{c}{NGC 6752 - dwarfs} \\
\\
4428 & +0.09& +1.1& +0.33 &$-$0.29 & \\
4383 & +0.12& +1.2& +0.57 &$-$0.18 & \\
202316 & +0.12& +1.5& +0.27 &$-$0.06 & \\
4341 & +0.21& +1.4& +0.20 & +0.20 & \\
4661 & $-$0.20& +1.4& $-$0.35 & +0.29 & \\
4458 & $-$0.20& +1.5& +0.02 & +0.31 & \\
5048 & $-$0.20& +1.5& $-$0.30 & +0.37 & \\
4907 & $-$0.20& +1.5& $-$0.25 & +0.58 & \\
200613 & $-$0.20& +1.7& & +0.62 & \\
\\
\multicolumn{6}{c}{NGC 6752 - subgiants} \\
\\
1406 & $-$0.13& +0.0& +0.44 & +0.02 & 9 \\
1665 & $-$0.28& +1.0& & +0.10 & 11 \\
1445 & $-$0.42& +1.2& & +0.14 & 5 \\
1400 & $-$0.25& +1.0& +0.38 & +0.20 & 9 \\
1563 & $-$0.33& +1.3& +0.42 & +0.26 & 5 \\
1461 & $-$0.32& +1.3& & +0.28 & 5 \\
202063 & $-$0.37& +1.2& +0.53 & +0.31 & 3 \\
1460 & $-$0.49& +1.4& $<$0.29 & +0.42 & 5 \\
1481 & $-$0.51& +1.3& & +0.51 & 4 \\
\\
\multicolumn{6}{c}{NGC 104 - dwarfs} \\
\\
1081 &$-$0.13 &$-$0.50& +0.57& $-$0.34& $>$10\\
1012 &$-$0.10 &$-$0.30& +0.48& $-$0.14& $>$10\\
975 &$-$0.13 &$-$0.30& +0.40& +0.22& $>$10\\
\\
\multicolumn{6}{c}{NGC 104 - subgiants} \\
\\
482 &$-$0.16 & +0.10& +0.61& +0.06 & 12 \\
206415 &$-$0.11 &$-$0.25& +0.52& +0.10 & 10 \\
201075 &$-$0.12 &$-$0.30& +0.41& +0.11 & 9 \\
433 &$-$0.32 & +1.10& & +0.24 & 10 \\
456 &$-$0.28 & +1.00& $<+0.19$& +0.28 & 12 \\
201600 &$-$0.50 & +1.10& $<+0.09$& +0.30 & 10 \\
435 &$-$0.31 & +0.70& $<-0.19$& +0.31 & 9 \\
429 &$-$0.35 & +0.50& $-$0.01& +0.31 & 6 \\
478 &$-$0.30 & +0.90& & +0.37 & 9 \\
\hline
\end{tabular}
\begin{list}{}{}
\item[] Values of [N/Fe] in the subgiants of 47 Tuc are the average of the
abundances estimated from the synthesis of the 3883~\AA\ and the 4215~\AA\
regions.
\end{list}
\label{t:cno}
\end{table*}
}
In the following, some features of the analysis of individual clusters are
discussed.
\paragraph{47 Tuc}
Being much more metal-rich than the other two clusters, 47 Tuc is the only one
for which we were able to obtain meaningful lower limits of the isotopic
ratios $^{12}$C/$^{13}$C for dwarf stars. The values found are listed in
last column of Table~\ref{t:cno}.
For the three dwarfs, the rather low quality of the spectra in the blue
hampered a precise determination of the isotopic ratio. Hence, we choose to
smooth somewhat the spectra, degrading the resolution to enhance the $S/N$.
However, even in this case, the best result we could secure is that the ratio
$^{12}$C/$^{13}$C is $>10$ in these turn-off stars.
It should be noticed that these are, to our knowledge, {\it the first
determinations}, of the $^{12}$C/$^{13}$C isotopic ratios in stars less
evolved than the RGB-bump in GCs.
\paragraph{NGC 6752}
For dwarfs in NGC 6752, the quality of spectra does not allow a clearcut
determination of the C abundances in the O-poor dwarfs, which are very rich in
N but with low C abundances. Hence, in these warm stars, the features of CH are
rather weak, and we adopted the following procedure, in order to obtain a more
reliable estimate.
The spectra of individual dwarf stars with low or not detected oxygen were
summed up and this coadded spectrum was then used to derive an abundance of C
through comparison with synthetic spectra. Our best estimate is [C/Fe]$=-0.2
\pm 0.1$. Analogously, the N abundances were then derived from the region
3876-3890~\AA\ using [C/Fe]$=-0.2$ dex and synthetic spectra computed with
different values of the [N/Fe] ratio. Notice that apparently there are no
N-poor dwarfs, and only one N-poor subgiant, in our sample.
\paragraph{NGC 6397}
In some stars of NGC 6397, acquired with the bluest setup during our first run,
we were able to investigate the NH molecular bands. The great advantage of
using the hydrides bands in metal-poor clusters like NGC 6397 is that bands of
bi-metallic molecules like CN become vanishingly weak at low metallicity,
due to their quadratic dependence on the metal abundance. Figure~\ref{f:nh1}
shows the observed spectrum of the subgiant star 206810 as compared to five
synthetic spectra computed with different [N/Fe] ratios. The lines of NH are
clearly observed and the best match is obtained with the synthetic spectrum
computed with [N/Fe]$\simeq 1.3$ dex, in very good agreement with the value
that was derived from the CN bands. This supports our derivation of the N
abundance for the subgiants in this cluster.
\begin{figure}
\psfig{figure=nh1.eps,width=8.8cm,clip=}
\caption[]{The observed spectrum of the subgiant 206810 in NGC 6397 (heavy
solid line) in the spectral region 3400-3410~\AA. The thin solid lines are
synthetic spectra computed by using values of [N/Fe]= 1.0, 1.25, 1.50, 1.75 and
2.0 from top to bottom, respectively. The NH lines are clearly observed.}
\label{f:nh1}
\end{figure}
In Figure~\ref{f:nh2} the same comparison is made with the average spectrum
obtained from the three dwarfs in NGC 6397 having the best spectra in the UV,
namely stars 202765, 201432 and 1543. The resulting average spectrum was
decontaminated for a relevant (about 20\% of the total value) contribution of
scattered light due to the sky, not properly taken into account by our spectrum
extraction procedure (note that these lines lie at the extreme UV edge of the
observed spectrum).
\begin{figure}
\psfig{figure=nh2.eps,width=8.8cm,clip=}
\caption[]{The same as in previous Fig. but for the average spectrum of 3
dwarfs in NGC 6397, namely stars 202765, 201432 and 1543.}
\label{f:nh2}
\end{figure}
In this average spectrum we cannot firmly conclude that NH lines are actually
observed: only some lines, but not all, are detected, and even these are very
close to the noise level. The comparison with synthetic spectra in this region
shows that a reasonable fit might be achieved at [N/Fe]$\leq 1.5$ dex: a value
greater than 2.0 dex is clearly excluded. This upper limit is however more
stringent that that derived from the CN lines.
\section{DISCUSSION}
\subsection{The light elements: C and N }
Variations of the abundances of C and N in the examined stars, as derived from
the features of the G-band and UV CN band, respectively, are shown in
Figure~\ref{f:chcn}. Typical error bars are also shown; they are conservative
estimates, including both the scatter from the observed individual lines of CH
and CN and the effect (almost negligible) of errors in the adopted atmospheric
parameters (see Gratton et al. 2001 and Carretta et al. 2004 for the estimates
of these uncertainties).
\begin{figure}
\psfig{figure=chcn.eps,width=8.8cm,clip=}
\caption[]{Run of the [C/Fe] ratio as a function of [N/Fe] for stars in 47 Tuc
(red triangles), NGC 6752 (green circles) and NGC 6397 (blue squares). Open
symbols represent dwarfs and filled symbols are subgiant stars, for all three
clusters. Arrows represent upper limits in N, C abundances. Typical error bars
are also shown.}
\label{f:chcn}
\end{figure}
In 47 Tuc the dwarfs are clustered around [C/Fe]$\sim -0.15$ dex, [N/Fe]$\sim
-0.4$ dex, while the subgiants seem to be divided into two distinct groups, one
with low N-high C and the other with low C-high N. Abundances of N and C seem
to be anticorrelated in the other two clusters tt, even if for NGC 6397 we
derived only upper limits for dwarfs, and the evidence of anticorrelation is
somewhat weaker in the subgiants, with respect to stars in NGC 6752 and 47
Tuc\footnote{Note that at [N/Fe]$=-0.5$ and [C/Fe]$=+0.10$ there are $two$
subgiant stars in NGC 6397, where in Figure~\ref{f:chcn} only one point is
displayed.}.
Even taking into account the small number statistics, it does not seem
premature to
conclude that in all 3 clusters there are a few subgiants with very low N
abundances, well separated from high-N/low-C subgiants. The average C
abundance of the 3 low-N subgiants in 47 Tuc is [C/Fe]$=-0.13$ dex ($rms=0.03$
dex, 3 stars), whereas the 6 subgiants with high N abundances have an average
of [C/Fe]$=-0.34$ ($rms$=0.08). This difference in [C/Fe] is very similar to
the one between the N-poor subgiant in NGC 6752 ([C/Fe]$=-0.13$) and the
average obtained from the other (N-rich) subgiants: [C/Fe]$=-0.37$
($rms=0.09$, 8 stars). The spread in C abundances is smaller (about 0.15 dex)
for subgiants in NGC 6397: in this case we obtain [C/Fe]$=+0.12$ dex
($rms=$0.12 dex, 3 stars) and [C/Fe]$=-0.02$ dex ($rms=$0.10 dex, 6 stars)
respectively for N-poor and N-rich stars.
From these numbers we note what is immediately apparent in Figure~\ref{f:chcn}:
while the spread in [N/Fe] is well above 1 dex, in each cluster there is a
rather small variation in C abundances. Since the C/N ratio in C-rich, N-poor
stars is roughly solar ($\sim 0.6$ dex), N in N-rich stars cannot be produced
only by transformation of C into N. Furthermore, even if carbon is a minority
species in these stars, the residual C observed in N-rich stars is much more
than that expected for material processed by CN-cycle at high temperature
([C/Fe]$\;\lower.6ex\hbox{$\sim$}\kern-7.75pt\raise.65ex\hbox{$<$}\; -0.8$; see Langer et al. 1993)
\begin{figure}
\psfig{figure=lowr.eps,width=8.8cm,clip=}
\caption[]{Run of the [C/Fe] ratio as a function of [N/Fe]. For our stars in
47 Tuc, NGC 6397 and NGC 6752 symbols are as in Fig. 6, with typical error
bars shown. For literature data (all smaller symbols), filled yellow circles
are SGB stars in M 5 from Cohen, Briley and Stetson (2002), black crosses are
main sequence turn-off stars in M 13 from Briley et al. (2004a), magenta empty
triangles are M 71 turn-off stars from Briley and Cohen (2001) and magenta
empty exploded stars are 47 Tuc MS stars from Briley et al. (2004b).}
\label{f:lowr}
\end{figure}
\subsection{Comparison with previous works}
The anti-correlation between C and N abundances, already known from low
resolution spectra, is confirmed by our high dispersion spectra. To show this,
we reproduced in Figure~\ref{f:lowr} a similar plot shown by Briley et al.
(2004a) with abundances of C and N for unevolved or slightly evolved stars in a
number of clusters. While zero-point offsets are likely present between our
data set and the [C/Fe] and [N/Fe] values derived by Briley et al. (as shown
by mean ridge lines for 47 Tuc), the behaviour is essentially the same. In all
clusters examined so far, variations in C and N are anti-correlated with each
other, with N showing large spreads, with respect to the more modest scatter
in C abundances. Only among the SGB stars in M5 studied by Cohen et al. (2002)
does the spread in C seem to equal the spread in N, and the most C-poor stars have
a C depletion close to that expected by complete CN-cycling.
Is this C-N anticorrelation tied to evolutionary phenomena occurring within the
stars themselves or are we seeing the outcome of an already established
nucleosynthesis implanted early in the material? To answer this question, we
have to look into the evolutionary status of our program stars and seek for
relationships with luminosity.
\begin{figure}
\psfig{figure=fieldcmv1.eps,width=8.8cm,clip=}
\caption[]{Run of the [C/Fe] ratio as a function of absolute magnitude for
program stars in NGC 6397, NGC 6752 and 47 Tuc. Symbols are as in Fig. 6 .
Small (magenta) crosses are the field stars of the Gratton et al. (2000)
sample with metallicity in the range $-2.0 \leq$ [Fe/H] $\leq -1$.}
\label{f:fieldcmv1}
\end{figure}
In Figure~\ref{f:fieldcmv1} we plotted the [C/Fe] values of all program stars
in the three clusters as a function of the absolute magnitude of the stars.
Distance moduli are those corrected for the effect of binarity in Gratton et
al. (2003). As a reference, we also plotted halo field stars from Gratton et
al. (2000), in the same luminosity range; we only considered field stars in the
range $-2.0 \leq$ [Fe/H] $\leq -1$, that closely matches the metallicity range
of our three globular clusters. This figure shows that the C abundance drops moderately
(less than a factor of 2) both in field and cluster stars at the expected
luminosity for the first dredge-up, in agreement with the standard stellar
evolution models.
Admittedly, the magnitude range is rather limited, but we can extend it by
using literature data available for bright giants in the studied clusters. In
Figure~\ref{f:fieldcmv2} we added to our data also C measurements for red
giants in NGC 6397 (Briley et al. 1990), NGC 6752 (Suntzeff and Smith 1991)
and 47 Tuc (Brown et al. 1990). In all cases, the absolute magnitude scale is
that defined by Gratton et al. (2003).
\begin{figure}
\psfig{figure=fieldcmv2.eps,width=8.8cm,clip=}
\caption[]{Run of the [C/Fe] ratio as a function of absolute magnitude for
program stars in NGC 6397, NGC 6752 and 47 Tuc. Symbols are as in Fig. 6.
Small (magenta) crosses are the field stars of the Gratton et al. (2000)
sample with metallicity in the range $-2.0 \leq$ [Fe/H] $\leq -1$. Open (blue)
diamonds are the bright giants in NGC 6397 from Briley et al. (1990); open
(green) diamonds are red giants in NGC 6752 from Suntzeff and Smith (1991);
open (red) diamonds are red giants in 47 Tuc from Brown et al. (1990).}
\label{f:fieldcmv2}
\end{figure}
Despite the heterogeneity of data sources and methods used to obtain the C
abundances (low-dispersion spectroscopic measurements of the G-band in stars of
NGC 6397 by Briley et al. 1990; infrared spectra of the first overtone CO bands
for stars studied by Suntzeff and Smith 1991 in NGC 6752; moderately high
resolution spectra of the CN red system and of the G-band for stars in 47 Tuc
by Brown et al. 1990; synthesis of high resolution spectra of the UV CN system
and of the G-band in our program stars), and offsets between the bright giants
and the unevolved stars might be present, the overall pattern among cluster
stars seems to follow rather well the one defined by "undisturbed" field
stars. Namely, the second step-like decrease in [C/Fe] ratios around the red
giant branch bump ($M_V\sim 0.5$) seems to be present also for cluster stars.
This is explained (e.g. Sweigart and Mengel 1979; Charbonnel 1995; Gratton et
al. 2000) as the onset of a second mixing episode during the red giant
evolution of a population II star, once the molecular weight barrier
established by the retreating convective envelope is wiped out by the
advancing shell of H-burning. From now on, CN-processed material is able to
reach the surface layer, where a further decrease of C is visible, as shown by
Figure~\ref{f:fieldcmv2}.
We conclude that in spite of a larger (intrinsic) scatter, the same mixing
episodes observed among field stars can be traced also among cluster stars.
This result is not new. Very recently, Smith and Martell (2003) used the same
field sample by Gratton et al. (2000) and literature data for C abundances in
red giants in M 92, NGC 6397, M 3 and M 13 to show that the rate of decline of
[C/Fe] on the RGB as a function of $M_V$ is very similar between cluster and
halo field giants. Our study, however, has the advantage of sampling regions
along the RGB well below the so called bump in the RGB luminosity function,
where standard theories for extra-mixing (see e.g. Sweigart and Mengel 1979)
fix the threshold in magnitude for the onset of additional mixing. In globular
clusters the chemical anomalies can be traced down to very faint magnitudes
and we clearly detect a steady increase in the average C abundance going from
red giants to subgiants and to dwarf stars.
Figure~\ref{f:fieldcmv2} also shows another well known feature: cluster
stars seems to reach more extreme C depletions than those experienced by field
analogs, as clearly indicated by red giants in NGC 6752 and, partly, by
giants in NGC 6397.
\begin{figure}
\psfig{figure=fieldnmv1.eps,width=8.8cm,clip=}
\caption[]{Run of the [N/Fe] ratio as a function of absolute magnitude for
program stars in NGC 6397, NGC 6752 and 47 Tuc, field stars (Gratton et al.
2000) and cluster red
giants from literature (Brown et al. 1990). Symbols and color codes are as in
the previous Figure.}
\label{f:fieldnmv1}
\end{figure}
The analogous plot for N abundances, in Figure~\ref{f:fieldnmv1}, also reveals
an extreme behaviour of cluster stars. Spreads in [N/Fe] are very large in
cluster dwarfs and subgiants with respect to field stars of similar
evolutionary status. Moreover, when coupled with literature data (from Brown
et al. 1990), a hint for increasing [N/Fe] at increasing luminosity seems to
appear for stars in 47 Tuc. On the other hand, no clear indication of such
increase in NGC 6752 is present.
\begin{figure}
\psfig{figure=fieldccmv1.eps,width=8.8cm,clip=}
\caption[]{Run of the isotopic ratio $^{12}$C/$^{13}$C as a function of $M_V$ for
program stars in NGC 6397, NGC 6752 and 47 Tuc. Symbols are as in the previous
Figures. Small (magenta) crosses and upper limits are the field stars of the
Gratton et al.
(2000) sample with metallicity in the range $-2.0 \leq$ [Fe/H] $\leq -1$.
Open (green) diamonds are red giants in NGC 6752 from Suntzeff and Smith
(1991); open (red) diamonds are red giants in 47 Tuc from Brown et al. (1990)
and Shetrone (2003).}
\label{f:fieldccmv1}
\end{figure}
Finally, Figure~\ref{f:fieldccmv1} shows the isotopic ratios $^{12}$C/$^{13}$C
measured in program stars as compared to the field database of Gratton et al.
(2000), as well as the literature data available for these clusters.
Also in this case, cluster stars show extremely low values of the isotopic
ratios, when compared to the field stars of similar magnitude. However, the
interesting feature here is that the $^{12}$C/$^{13}$C values are not at the
CN-cycle equilibrium value, not even for stars that are extremely N-rich. For
these stars one would expect a value of $\sim 3$, at odds with our findings.
Even considering literature data for bright giants, their isotopic ratios seem
to be somewhat lower than those in field stars but always at a level slightly
higher than the equilibrium value.
\subsubsection{First conclusions from light elements}
In summary, the analyses of C and N and the relative abundances of the carbon
isotopes in slightly evolved globular cluster stars show that:
\begin{itemize}
\item there are moderately small variations in C abundances, anticorrelated to
(large) variations in N abundances. However, except for a few stars in M5, the C
is generally not as low as expected for CN-cycling material. In no dwarf or
early subgiant do we find C abundances as low as those observed in (all) highly
evolved RGB stars, i.e. stars brighter than the RGB-bump.
\item N shows a large spread and, with the cautionary warning of a rather
limited range in sampled magnitudes, does not seem to have a particular
increase at the first dredge-up position. Note that a large
fraction of dwarfs and early subgiants have N abundances as high as those
observed in highly evolved RGB stars (Figure~\ref{f:fieldnmv1}).
\item the isotopic ratios $^{12}$C/$^{13}$C are low but do not reach the
equilibrium value of $\sim 3$.
\end{itemize}
From these observations, it is already clear that
no unevolved star has the same surface composition
as highly RGB stars, as required by some
recent scenarios advocated to explain the O-Na anticorrelation (Denissenkov and
Weiss 2004). Rather, the observed pattern resembles much more that predicted by
the evolutionary models of the most massive intermediate-mass, low
metallicity AGB stars (e.g., Ventura et al. 2002, 2004). According to these
models, these stars have complete CNO cycling at the base of the convective
envelope and the coupling of nucleosynthesis plus release of processed matter
into the protocluster clouds or onto second-generation stars is able to
produce a surface composition where C is depleted, but not to values as low as
those expected from CN-cycling, because some fresh $^{12}$C produced by
triple$-\alpha$ reactions is dredged up to the surface, N is enhanced and the
carbon isotopic ratio approaches the equilibrium value, due to the large
enhancement of $^{13}$C. Apparently, this is almost exactly the chemical
composition of the unevolved cluster stars under scrutiny.
At this point of our discussion, however, this assertion is not yet proven,
because dilution with material not contaminated by CNO burning could be
invoked to explain the observed trends for C and N abundances. Observations of
heavier elements involved in high temperature $p-$capture reactions may give a
deeper insight.
\subsection{Oxygen and Sodium}
Looking now at heavier elements (O and Na), we proceed along a path of
stronger Coulomb barriers. The temperatures involved are much higher and we are
considering deeper regions in the H-burning shell.
The well known Na-O anticorrelation (see Gratton, Sneden and Carretta 2004 for
a recent review) is summarized in Figure~\ref{f:ona2} for our program clusters.
In this Figure we also added the available literature
data, which is mostly for bright red giant stars,
even if, apart from a few cases, no systematic studies have been
performed for these 3 clusters, often used as calibrators. Abundances of Na
and O in NGC 6397 include 2 stars studied by Norris and Da Costa (1995) and 2
stars from Castilho et al. (2000). For 47 Tuc, additional data are from Norris
and Da Costa (1995) and Carretta (1994). For NGC 6752, we added 6 stars from
Norris and Da Costa (1995), 4 stars from Carretta (1994) and the bright red
giants studied by Yong et al. (2003). Individual values of Na and O for the
bump stars in NGC 6752 analyzed by Grundahl et al. (2002) have not been published
anywhere, therefore they are not used. Whenever possible, as in the Yong et
al. sample, we started from original $\log n(O)$ and $\log n(Na)$ values,
bringing them onto a homogeneous scale by using the average [Fe/H] values and
the solar values adopted in the present study.
Note that values for our dwarfs and subgiants do include corrections for
departures from LTE. For literature data, this was possible only for stars
analyzed in Carretta (1994). However, since O abundances are usually derived in
red giants from the forbidden [O I] doublet, these corrections are negligible.
NLTE corrections for Na abundance might be of more concern in giants, depending
on what lines were used in the various analyses, but the overall appearence of
Figure~\ref{f:ona2} shows that if there are some offsets, they are rather
small.
\begin{figure}
\psfig{figure=ona2.eps,width=8.8cm,clip=}
\caption[]{Run of the [Na/Fe] ratio as a function of [O/Fe], for stars in
47 Tuc, NGC 6752 and NGC 6397. Symbols for our program stars are as in Fig. 6.
Literature data are as follows: (green) diamonds with crosses
inside are bright red giants from the extensive study by Yong et al. (2003),
open (blue, green and red) diamonds are stars of NGC 6397, NGC 6752 and 47 Tuc,
respectively, from Norris and Da Costa (1995), Carretta (1994) and Castilho et
al. (2000), as described in the text.}
\label{f:ona2}
\end{figure}
The O-Na anticorrelation is in fact very well defined for all clusters; there
seem not to be large differences among the different clusters over
the metallicity range sampled, nor among stars of different evolutionary status
within a given cluster. For the first time, the existence of a Na-O
anticorrelation also among stars in NGC 6397 is clearly shown. This Figure
shows among slightly evolved cluster stars the same trends that have previously
been observed among the red giant stars of several globular clusters.
The difference with respect to field stars is striking. With their highly
homogeneous sample, Gratton et al. (2000) convincingly showed that field stars
have constant Na and O abundances, not anticorrelated with each other. The
obvious implication is that in cluster stars there is something else able to
alter simultaneously the abundances of these two elements.
To get a deeper insight, we show in Figure~\ref{f:cna} and Figure~\ref{f:nna}
the derived abundances of C and N, respectively, for program stars as a
function of the [Na/Fe] ratio.
\begin{figure}
\psfig{figure=cna.eps,width=8.8cm,clip=}
\caption[]{Run of the [C/Fe] ratio as a function of [Na/Fe], for stars in
47 Tuc, NGC 6397 and NGC 6752. Symbols are as in Fig. 6 .}
\label{f:cna}
\end{figure}
\begin{figure}
\psfig{figure=nna.eps,width=8.8cm,clip=}
\caption[]{Run of the [N/Fe] ratio as a function of [Na/Fe], for stars in
47 Tuc, NGC 6752 and NGC 6397. Symbols are as in Fig. 6 .}
\label{f:nna}
\end{figure}
In all three clusters, Figure~\ref{f:cna} and Figure~\ref{f:nna} clearly show a
trend for C and N abundances to decrease and increase respectively with the
increase of Na abundances. In particular, the C-Na anticorrelation closely
mimics the well-known O-Na anticorrelation, summarized in Figure~\ref{f:ona2}.
Furthermore, while turn-off stars show a large range in Na abundances (at
almost constant C), carbon abundances are anticorrelated with Na abundances for
SGB stars. On the other hand, N abundance correlates well with sodium among
subgiant stars, even if the anticorrelation is less evident among TO stars.
Finally, O is anticorrelated with N, as shown in Figure~\ref{f:no}.
\begin{figure}
\psfig{figure=no.eps,width=8.8cm,clip=}
\caption[]{Run of the [N/Fe] ratio as a function of [O/Fe], for stars in
47 Tuc, NGC 6752 and NGC 6397. Symbols are as in Fig. 6 .}
\label{f:no}
\end{figure}
The overall distribution of light elements seems to point out that processes of
proton-capture are at work. In the atmospheres of the stars studied we are just
seeing the products of these reactions. In this case, the line of thought is
the same as in Gratton et al. (2001): turn-off stars do not reach the
temperature regime where the ON and NeNa cycles required to produce the Na-O
anticorrelation are active, and moreover these stars have convective
envelopes that are too small to have efficiently mixed the ashes of these
nuclear processes up to
the surface. The same conclusion holds also for subgiants.
The bottom line is that what we are seeing are the by-products of nuclear
burning and dredge-up in $other$ stars, that are now not observable, but that
have returned their elements to the intracluster medium or directly to the
surface of the presently observed stars.
\subsection{CNO arithmetic}
Having now also O abundances at hand for program stars, it is possible to test
in another way if the observed pattern of C, N, O abundances can be explained
by the CNO-cycle alone. In fact, in this hypothesis it is only the relative
content of C, N and O that may change, as a consequence of different reaction
rates; their sum has to be constant.
In Figure~\ref{f:lowr3} the C abundances are plotted as a function of the sum
C$+$N for our program stars and, as a reference, for stars with abundances from
low dispersion studies, as in Figure~\ref{f:lowr}.
\begin{figure}
\psfig{figure=lowr3.eps,width=8.8cm,clip=}
\caption[]{Run of the [C/Fe] ratio as a function of [(C+N)/Fe]. For our stars
in 47 Tuc, NGC 6397 and NGC 6752 symbols are as in Fig. 6. For literature data
(all smaller symbols), filled yellow circles are SGB stars in M 5 from Cohen
et al. (2002), black crosses are main sequence turn-off stars in M 13 from
Briley et al. (2004a), magenta empty triangles are M 71 turn-off stars from
Briley and Cohen (2001) and magenta empty exploded stars are 47 Tuc MS stars
from Briley et al. (2004b).}
\label{f:lowr3}
\end{figure}
From this Fig. one has the impression that the sum C$+$N increases with
decreasing C abundance; this is confirmed by computing Kendall's $\tau$, which
implies that this anti-correlation is highly significative, at the 99.3\%
level. Hence the observed pattern cannot be due simply to a
transformation of C into N by the incomplete CN-cycle of H-burning.
The possible alternatives are then either that the ON-cycle is also involved,
adding N freshly produced from O, or that some variable amount of already
existing N is superimposed on the effects of C$\rightarrow$N reconversion.
In this regard, clearcut evidence is provided by Figure~\ref{f:ncno}, where
the run of [N/Fe] as a function of the total sum C$+$N$+$O is shown.
Over a spread in N of almost 2 dex, the sum remains almost (but not exactly,
see below) constant; this by itself implies that the complete CNO-cycle
has been at
work to produce the observed pattern. Once more, neither subgiants nor, in
particular, unevolved turn-off stars are able to forge elements (such as Na)
that require high-temperature proton-capture reactions. Moreover, they are
unable to mix in their convective envelopes such products to the atmospheric
layers; hence, we are forced to conclude that the CNO cycle was at work in
stars other than those presently observed.
\begin{figure}
\psfig{figure=ncno.eps,width=8.8cm,clip=}
\caption[]{Run of the [N/Fe] ratio as a function of [(C+N+O)/Fe] for our stars
in 47 Tuc, NGC 6397 and NGC 6752; symbols are as in Fig. 6.}
\label{f:ncno}
\end{figure}
The best candidates at hand are the intermediate-mass AGB stars.
Another class of possible polluters was recently suggested by Denissenkov and
Weiss (2004). According to their computations, as well as those previously
reported by Denissenkov and Herwig (2003), nucleosynthesis in IM-AGB stars
with strong O-depletion is not accompanied by large Na production (hence, the
matter is not Na-enhanced as required by the Na-O anticorrelation); instead,
strong Mg depletions are expected, and this has never been observed in
globular cluster stars. Similar results has been recently obtained by Herwig
(2004) and Fenner et al. (2004). As a way out, Denissenkov and Weiss (2004)
suggested that peculiar CNO abundances, as observed in unevolved cluster
stars, might be a result of the H-burning shell in upper RGB stars of mass
slightly larger than those presently observed in GCs, provided that they have
experienced some degree of extra-mixing (see Denissenkov and Herwig 2003),
followed by mass transfer onto less evolved stars.
\begin{figure}
\psfig{figure=cnon2.eps,width=8.8cm,clip=}
\caption[]{Run of the [C/N] ratio as a function of [O/N] for our stars
in 47 Tuc, NGC 6397 and NGC 6752; symbols are as in Fig. 6. Superimposed on the
data are the three models outlined in the text: a simple dilution with
material processed through the
complete CNO cycle (solid, black line), contamination from N-poor RGB stars
(dashed blue line) with composition typical of field RGB stars, and
contamination from N-rich upper-RGB stars experiencing very deep mixing (dotted
magenta line).}
\label{f:cnon2}
\end{figure}
\subsubsection{A simple dilution model}
So far, it has been shown (i) that globular cluster stars exhibit the same
mixing episodes observed for field stars; (ii) that no slightly evolved
cluster star has an abundance pattern the same as that observed among stars
close to
tip of the red giant branch; and (iii) that there may be an excess in the sum
of C+N+O in N-rich stars, that can possibly be attributed to some $^{12}$C
in excess with respect to the predictions of the complete CNO cycle.
To test how various schemes work to explain these observational facts, we
will consider here simple models for the dilution of the products of CNO burning at
various temperatures, and will compare their predictions with the run of the
[C/N] ratio against [O/N] ratio shown in Figure~\ref{f:cnon2}, as well as
with the other pieces of information we have gathered.
When constructing our dilution models, we started noticing that in the complete
CNO-cycle at high temperature ($40 \times 10^6$ K), at equilibrium the O
abundance is decreased much more than the C abundance (factors of about 50 and
6, respectively: Langer et al. 1993). This is quite different from the case of
the incomplete CN-cycle at low temperature ($\sim 10^6$~K), where the C
abundance is decreased by a factor of 6, as before, but the O abundance is not
modified.
\medskip
\noindent
{\bf Testing the scenario of pollution by the complete CNO-cycle}
\medskip
We started by considering the standard scenario of pollution by the complete
CNO-cycle. Let $a$ be the fraction of gas forming unevolved stars in globular
clusters that has been burnt through the complete CNO-cycle and let us assume
the remaining gas to be of primordial composition. It follows that the
abundance of C and O in the mixed gas with respect to the initial composition
is: $f= a \cdot b(C,O) + (1-a)$, where $b(C,O)$ is the abundance of C and O at
the equilibrium of the CNO-cycle (0.17 and 0.02 for C and O, respectively). The
N abundance can be derived by the constraint that the total CNO abundance is
constant.
If we assume that initially [C/N]=0.5 and [O/N]=1.0 (which is approximatively
the composition of N-poor stars in the three clusters: see
Figure~\ref{f:cnon2}), we may then predict the values of [C/N] and [O/N] for
different fractions $a$ of the gas consumed in the complete CNO-cycle, with
$a=0$ being the original composition. This trend is shown as a solid line in
Figure~\ref{f:cnon2}. This line reproduces fairly well the location of
observational points in Figure~\ref{f:cnon2}, albeit it predicts somewhat too
low C abundances for N-rich dwarfs. Dilution factors $a$\ adequate to
reproduce N-rich stars are $0.5<a<0.8$. Such values also allow reproducing
the isotopic ratio $^{12}$C/$^{13}$C$\sim 7$ observed in unevolved stars.
Moreover, with these factors we might quite easily obtain a Li abundance
roughly similar to the original one, provided that the diluting material be
Li-rich (as expected from some of the IM-AGB stars: Ventura et al. 2001).
On the other hand, with such large dilution factors, the models by Langer et
al. (1993) would predict too much Na, by more than a factor of about 10; this
is because in these models Na is produced also by $^{20}$Ne. In order to
reproduce the observations we would need that Na be forged only from $^{22}$Ne.
\medskip
\noindent
{\bf Testing the scenario of polluting RGB stars}
\medskip
Let us now use a similar model to test the scenario envisioned by Denissenkov
and Weiss (2004) of pollution by RGB stars. In this case, mixing occurs in a
fraction of upper-RGB stars and afterward a transfer of material onto the
dwarfs occurs.
Let us assume that the upper-RGB stars might have either one of the following
2 compositions: (i) a chemical composition typical of field upper-RGB stars
(N-poor, i.e. N only coming from incomplete CN-cycle) or (ii) a composition
from very deep mixing, where complete-CNO cycle and Na-enrichment are
involved. For these stars we will use the most extreme case observed (i.e.
N-rich). Note that in both groups of stars all Li is destroyed.
The starting compositions ($\log n({\rm C})/({\rm N})/({\rm O})/({\rm Na})/
({\rm C+N+O})$) assumed are then:
8.60/8.00/8.90/6.30/9.11,
8.60/7.50/9.30/5.80/9.38,
7.50/9.33/8.40/6.80/9.38
and 8.00/8.50/9.30/5.80/9.38 for the solar, original, N-rich and N-poor cases,
respectively.
By varying the dilution factor $a$, we obtain the abundance pattern for the two
cases original+$a \times$ (N-rich) and original+$a \times$ (N-poor). Results
are overplotted as a dotted (magenta) line and a dashed (blue) line,
respectively, over our data in Figure~\ref{f:cnon2}. The first case is very
similar to the case made above for the complete CNO cycle, differences being
only due to the slightly different assumptions made about the compositions.
If this scenario is correct, it would be expected that the observed points
should lie between the two lines. Actually, the line representing pollution by
N-poor stars does not reproduce the observations; on the other hand, the line
representing N-rich stars fits the data reasonably well (although not as well
the N-rich dwarfs), requiring values $0.5<a<0.8$\ similar to those obtained in
the previous subsection.
The inadequacy of models with pollution by N-poor stars is evident when
noticing that within this scheme we should expect to find C-poor, Na-poor
stars. However, these stars are not observed at all (see Figure~\ref{f:cna}).
The inference is that in globular clusters there are no dwarfs polluted by RGB
stars with a chemical composition typical of field RGB stars. Within this
scheme it should then be assumed that only stars experiencing very deep mixing
polluted unevolved stars. One would then be forced to conclude that only a
fraction of RGB stars, and only those in clusters, lose a great amount of mass,
and that these very same stars do experience also very deep mixing, likely due
to the same physical mechanism (rotation? binarity?).
An additional problem with this scheme is that N-rich giants have generally
no Li at their surface. We would then expect that Li be depleted by a factor
of 2 to 5 in main sequence stars of globular clusters like NGC 6397, in
contrast with observations (Bonifacio et al. 2002).
There are further additional concerns in a mechanism involving pollution by RGB
stars. The lost material ends up polluting other stars. It cannot be a simple
surface pollution: in fact, in this case there should be also noticeable
differences between dwarfs and subgiants (due to different masses of the
convective envelopes) which is not observed. Since most of the unevolved stars
observed in clusters like NGC 6397 and NGC 6752 are N-rich, the total amount of
mass lost by these RGB stars should be large, about 80$\%$ of the cluster
mass. This seems unlikely, since an RGB star cannot lose more than $\sim 50\%$
of its mass, the remaining being locked in the degenerate core. Another
problem concerns the epoch when this pollution occurred. In fact, if the mass
was lost in recent times, one would expect a large numbers of young stars,
obviously not observed.
On the other hand, IM-AGB stars may eject almost 80\% of their mass
(see e.g. Marigo et al. 1998), hence the mass requirement in this case would be
met if the original initial mass function (IMF) of the cluster stars is not too
steep, allowing to form many AGB stars. Evidences for a flatter local IMF are
discussed, in this context, by e.g. D'Antona (2004) and Briley et al. (2001).
\subsubsection{The triple-$\alpha$ scenario}
As pointed out in the previous discussion, another problem is evident from
Figure~\ref{f:cnon2}. The dilution models predict that stars having [O/N]$\sim
-1.5$\ should have [C/N]$\sim -1.9$, whereas our observations show [C/N]$\sim
-1.5$. This suggests the presence of an additional source of $^{12}$C, very
likely through the triple$-\alpha$ process. A similar excess of $^{12}$C is
also suggested by the $^{12}$C/$^{13}$C isotopic ratio. Let us in fact assume
that $c$ is the fraction of $^{12}$C produced by triple$-\alpha$ and $(1-c)$
the fraction of $^{12}$C resulting by CNO-processing. Hence, we may write for
$^{13}$C (which is produced only by the CNO-cycle) $^{13}$C =
$\frac{(1-c)}{R_e}$, where $R_e \sim 3.5$ is the equilibrium value of the
isotopic ratio $^{12}$C/$^{13}$C. We can now re-write the fraction of $^{12}$C
from CNO by subtracting the contribution of $^{13}$C and of $^{12}$C by
triple$-\alpha$, i.e. as
$^{12}$C = $(1-c) -$ $^{13}$C = $1-c$ -$\frac{(1-c)}{R_e}$, hence
as $(1-\frac{1}{R_e})\times (1-c)$.
The observed C isotopic ratio is
then $R_o$ = $(\frac{^{12}C}{^{13}C})_o = \frac{(1-\frac{1}{R_e}) \cdot(1-c) +
c}{\frac{(1-c)}{R_e}}$ from which, with simple algebra, the fraction of
$^{12}$C due to triple$-\alpha$ is $$ c = 1 - \frac{R_e}{1+R_o} .$$ By using an
observed ratio (see Table 1) of about 8, we derive that about 60\% of the C
observed is likely to come from triple$-\alpha$ burning.
This estimate compares well with what is known from models of IM-AGB stars.
In fact, typical estimates of the C abundance in NGC 6752, [C/Fe]$\sim
-0.3$\ corresponds to a mass fraction of $\sim 2\times 10^{-5}$ of
$^{12}$C, a value entirely consistent with model predictions for a
5~M$_\odot$ star of similar metallicity (Ventura et al. 2004).
Up to now, our conclusions are based only on the observation that there is too
much C, with respect to the very large N-enhancements, and carbon
isotopic ratios too high to be explained purely with a re-arrangement of elements
involved in the CNO-cycle. However, other additional evidence comes from the O
abundances. In the low-O, low-C region of Figure~\ref{f:cnon2} the existence
of a sort of plateau also suggests that the products of triple$-\alpha$ are
involved. This is not very clear in subgiants, but quite evident in dwarfs,
whose C abundances have not been modified by the first dredge-up.
\section{CONCLUSIONS}
Summarizing, we can consider various mass ranges of likely polluters that
contributed to the chemical composition in stars presently observed:
\begin{itemize}
\item metal-poor stars in the intermediate mass range, say $1.2 \leq M \leq
3-5 M_\odot$, are the classical donors considered for CH-stars (McClure and
Woodsworth 1990). This is likely not the correct mass range for typical stars
polluting globular clusters, since they also produce consistent amounts of
$s-$process elements, that are not seen to vary in cluster stars.
\item stars less massive than this range (the range considered by Denissenkov
and Weiss 2004) are also unlikely. Apart from the possible excess of $^{12}$C
discussed above, either (i) they release a large mass to be successively
accreted, and this is unlikely, since the polluted stars would have then a
large mass range, the cluster TO would be smeared out and we would end up with
a cluster mostly composed of blue stragglers\footnote{Actually, we think that
the mechanism proposed by Denissenkov and Weiss (2004) is indeed active in a
minority of globular cluster stars belonging to binary systems, producing the
blue straggler stars with the classical McCrea (1964) mechanism}; or (ii) they
provide a small amout of polluting mass. This second possibility is also
unpalatable because a thin layer accreted on the surface of a main sequence
star would be diluted when the stars climb toward the SGB phase by the
extension of the convective envelope. The evidence of large spread in C, N, O,
Na abundances at all luminosities (see Briley, Cohen and Stetson 2002)
strongly argues against a simple contamination of stellar surface.
\item finally the stars more massive than the above range are just the IM-AGB
stars that, at the present, are in our opinion the best candidate polluters.
\end{itemize}
Finally, we must recall that at least half or even 2/3 of stars observed in
globular clusters seem to be heavily affected by large alterations in their
chemical composition. In order to explain this with mass exchange from a RGB
star onto a companion in a binary system, we would probably end up with a huge
number of low-mass X-ray binaries in GCs, at odds with observations.
In summary, our findings and discussion strongly suggest that the polluters
were intermediate mass AGB stars, and $not$ upper RGB stars.
On the other hand, the scenario in which IM-AGB stars are the primary
contributors in shaping the chemical mix of the early cluster environment
still has to face problems (Denissenkov and Herwig 2003, Denissenkov and Weiss
2004, Herwig 2004, Fenner et al. 2004): current models do not seem to
reproduce the required observational pattern. It must be however reminded
that, as noticed by Denissenkov and Weiss, the yields computed from models of
these stars strongly depend on two poorly known physical inputs, namely the
treatment of mass loss and the efficiency of convective transport (see also
Ventura et al. 2002). Further progress in stellar modeling is strongly urged.
\begin{acknowledgements}
{ This research has made use of the SIMBAD data base, operated at CDS,
Strasbourg, France. We thank the ESO staff at Paranal (Chile) for their help
during observing runs, Elena Sabbi for useful discussion on LMXB's in GCs, and
the referee for very careful reading of the manuscript.}
\end{acknowledgements}
|
{
"timestamp": "2004-11-09T22:03:50",
"yymm": "0411",
"arxiv_id": "astro-ph/0411241",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411241"
}
|
\section{Introduction}
\label{sect:intro}
Soft gamma-ray repeaters (SGRs) are a small group of high-energy
sources that were discovered through the observation of short
bursts in the hard X/soft $\gamma$-ray range (for a review see
Hurley 2000). The bursts have typical durations of the order of a
few hundred milliseconds and are emitted during sporadic
``active'' periods that can last from weeks to months. Their
spectra above $\sim$20 keV have been well described by optically
thin thermal bremsstrahlung with temperatures $\sim$30-40 keV (but
see Feroci et al. 2004 and Olive et al. 2004 for the spectra of
bursts at other energies).
For several years after their discovery, SGRs could not be
unambiguously associated with persistent sources at other
wavelengths, with the exception of SGR 0525--66. The precise
localization of the latter source placed it in the Large
Magellanic Cloud supernova remnant N49, hinting at a neutron star
origin (Cline et al. 1982). More recent observations of the SGR
burst locations with X-ray imaging satellites led to the
identification of their persistent counterparts in the 0.5--10 keV
X-ray range. This proved to be crucial in confirming the neutron
star hypothesis through the discovery of coherent pulsations
(periods of 7.5 and 5.2 s) and secular spin-down in the range
10$^{-11}$--10$^{-10}$ s s$^{-1}$ in two of these sources (SGR~1806--20\
and SGR 1900+14 ; Kouveliotou et al. 1998; Hurley et al. 1999a).
Furthermore, the association with clusters of massive stars
(Mirabel et al. 2000) indicates that SGRs are relatively young
objects. The most successful model advanced to explain the SGRs
involves neutron stars with a very high magnetic field, or
``magnetars'' (Duncan \& Thompson 1992, Thompson \& Duncan 1995).
In magnetars, which are assumed to have internal fields much
higher than the quantum critical value $B_{c} =
\frac{m_{e}^{2}c^{3}}{e\hbar}=4.4\times10^{13}$~G, the dominant
source of free energy is the magnetic field, rather than rotation
as in the ordinary radio pulsars. This energy is enough to power
both the bursts and the persistent X-ray emission
(L$\sim$10$^{34}$-10$^{35}$ erg s$^{-1}$).
The study of the persistent emission from SGRs has been carried
out at energies below $\sim$10-20 keV to date. The only
observations of SGRs at higher energies, besides those of the
bursts, were limited to a small number of exceptionally bright
transient events, the so called ``giant'' and ``intermediate"
flares, which occurred in SGR 0525--66 (Mazets et al. 1979) and in
SGR 1900+14 (Hurley et al. 1999b, Guidorzi et al. 2004). Here we
report the discovery of emission extending up to 150 keV from the
quiescent counterpart of the soft gamma-ray repeater SGR~1806--20. Some
evidence for this high-energy component was reported in a
preliminary analysis of a subset of the data presented here
(G\"{o}tz et al. 2004b, Bird et al. 2004). In a companion paper
Molkov et al. (2005) report independent evidence from different
INTEGRAL observations of persistent hard X-ray ($>$20 keV)
emission from this source.
\begin{table*}[ht!]
\caption{{\it INTEGRAL Core Program Observations of SGR~1806--20\ }}
\begin{center}
\begin{tabular}{lccc}
\hline
Observing & Net exposure & Flux 20-60 keV & Flux 60-100 keV \\
Period & (ksec) & (counts s$^{-1}$) & (counts s$^{-1}$) \\
\hline
2003 March 12 - April 23 & 233 & 0.286$\pm$0.043 & 0.10$\pm$0.03 \\
2003 September 27 - October 15 & 278 & 0.344$\pm$0.041 & 0.09$\pm$0.03 \\
2004 February 17 - April 19 & 285 & 0.341$\pm$0.043 & 0.05$\pm$0.03 \\
2004 September 21 - October 14 & 213 & 0.511$\pm$0.050 & 0.19$\pm$0.03 \\
\hline
2003 March 12 - 2004 Oct. 14 & 1033 & 0.375$\pm$0.022 & 0.102$\pm$0.015 \\
\hline
\end{tabular}
\end{center}
\end{table*}
\begin{figure*}[th!]
\begin{center}
\psfig{figure=newimagebw.ps,width=15cm}
\caption{Images of a 4$^{\circ}\times$3$^{\circ}$ region
containing the position of SGR~1806--20\ obtained with the IBIS/ISGRI
instrument in the 20-60 keV (left) and 60-100 keV energy range
(right). The other detected sources are (1) GX 9+1, (2) IGR
J18027--2016 (Revnivtsev et al. 2004), and (3) IGR J17597--2201
(Lutovinov et al. 2003).}
\end{center}
\end{figure*}
\section{Data Analysis and Results}
The location of SGR~1806--20\ has been repeatedly observed by the INTEGRAL
gamma-ray satellite in 2003-2004 (Winkler et al. 2003) in the
course of Core Program (consisting of regular scans along the
Galactic plane and bulge), deep exposures of the Galactic Center
region, and Target of Opportunity observations. Here we
concentrate on the Core Program data obtained with the IBIS
instrument (Ubertini et al. 2003), which provides imaging above 15
keV with an unprecedented combination of sensitivity
($\sim$milliCrab) and angular resolution (12$'$) using the coded
mask imaging technique.
The available data consist of more than 500 individual pointings
in which SGR~1806--20\ was observed at different off-axis angles. The
typical integration time of most pointings is $\sim$1800 s (those
obtained after May 2004 lasted $\sim$2200 s each). We restrict our
analysis to the pointings in which SGR~1806--20\ was in the central part
of the field of view (radius $<$8$^{\circ}$), where the
sensitivity is higher and the instrument calibration is better
known. This results in a total exposure time of one million
seconds (corrected for the vignetting effect). The observations,
spanning almost two years, were concentrated in four periods of
1-2 months each due to the visibility constraints of the
satellite, as indicated in Table~1. We used the data obtained
with the IBIS lower energy detector ISGRI (Lebrun et al. 2003), an
array of 128$\times$128 CdTe pixels providing a geometric area of
2600 cm$^{2}$ in the nominal energy range 15 keV - 1 MeV.
The detector images in the 20-60 and 60-100 keV energy ranges of
the individual pointings were deconvolved, cleaned and co-added
using version 4.1 of the Offline Scientific Analysis software
(Goldwurm et al. 2003) provided by the INTEGRAL Science Data
Center (Courvoisier et al. 2003). A
4.5$^{\circ}\times$4.5$^{\circ}$ region of the resulting mosaics
is shown in Fig.~1. A source with a significance of 17.2$\sigma$
is detected in the lowest energy range at the Chandra position of
SGR~1806--20\ (Kaplan et al. 2002). Several other sources are visible in
the image (see figure caption for details). SGR~1806--20\ is also detected
in the 60-100 keV energy range, with a significance of
6.9$\sigma$. The source count rates of 0.375$\pm$0.022 counts
s$^{-1}$ and 0.102$\pm$0.015 counts s$^{-1}$ measured in the soft
and hard energy ranges correspond to a flux of $\sim$3 mCrabs.
Except for the period March-April 2003, the source was in a
bursting state during the reported observations, leading to the
detection of several bursts in the IBIS data\footnote{the analysis
of those observed in September-October 2003, showing an
anticorrelation between hardness and intensity of the bursts, has
been reported in G\"{o}tz et al. (2004a)}.
A few pointings
containing some strong bursts (Mereghetti et al. 2004a, Golenetski
et al. 2004) were removed from the analysis described above. The
effect of the remaining ones is negligible: their cumulative
contribution of $\sim$16,000 counts in about 50 bursts,
corresponds to 4\% of the total number of source counts in the
integrated images.
Analyzing the data for the four observing periods separately in
the same way, we derived the fluxes reported in Table 1, which
give some evidence for an increase in the source hard X-ray
luminosity with time (see Fig.~3b). A fit of the 20-60 keV count
rate values with a constant flux gives a $\chi^2$=12.3,
corresponding to a 6.5$\times$10$^{-3}$ probability.
For the spectral analysis we produced images in 9 energy intervals
in the range 15-300 keV. The count rates extracted from the images
were fit using the latest available response matrices (RMF v11,
ARF v5). Since the first three observing periods yielded
consistent spectral parameters, we analyzed them together,
obtaining a best fit power law photon index $\Gamma$=1.9$\pm$0.2
and a 20-100 keV flux F$_{20-100}$=(4.7$\pm$0.5)$\times$10$^{-11}$
erg cm$^{-2}$ s$^{-1}$ (all errors are 1$\sigma$). During the last
observing campaign (Sept-Oct 2004) the corresponding parameters
were $\Gamma$=1.5$\pm$0.3 and
F$_{20-100}$=(8.0$\pm$0.9)$\times$10$^{-11}$ erg cm$^{-2}$
s$^{-1}$. Thus there is some evidence for a harder spectrum
coupled with a flux increase. The two spectra are shown in Fig.~2a
and 2b. Note that the higher flux and harder spectrum of the
Sept.-Oct. 2004 period lead to a significant detection of SGR~1806--20\ up
to the 100-150 keV energy bin. For comparison, Fig.~2c shows the
typical spectrum of one of the bursts (note that the intensity has
been scaled down by a factor 10$^4$). It can be seen that the
bursts have a much softer spectrum than the persistent emission.
\section{Discussion}
Observations of SGR~1806--20\ below 10 keV carried out with BeppoSAX and
ASCA indicated a power law spectrum with photon index $\sim$2 and
unabsorbed 2-10 keV flux of $\sim$1.6$\times$10$^{-11}$ erg
cm$^{-2}$ s$^{-1}$ (Mereghetti et al. 2000). Although the
detection of pulsations in the 10-20 keV range with RossiXTE
(G\"{o}\v{g}\"{u}\c{s} et al. 2002) showed that the emission from
SGR~1806--20\ extends above the soft X-ray range, no spectral information
was available above 10 keV and the BeppoSAX spectrum could be
fitted equally well by a thermal bremsstrahlung model with
temperature kT=11 keV. The extrapolation of the BeppoSAX power law
spectrum lies below the average value obtained with INTEGRAL. On
the other hand, a good agreement is found between the INTEGRAL
spectra and those measured below 10 keV with XMM-Newton in
2003-2004 (Mereghetti et al. 2004b). This is shown by the dashed
lines in Fig. 2, which indicate the extrapolations of the
XMM-Newton power law spectra measured at dates consistent with the
INTEGRAL observations (Fig.2a: 7 October 2003; Fig.2b: 6 October
2004).
In Fig.~3 we compare the hard X-ray flux measured with INTEGRAL
with the level of activity from SGR~1806--20\ as a function of time. This
is indicated by the number of bursts detected by the
Interplanetary Network (IPN). The source was quiescent during the
first INTEGRAL observing period and moderately active in the two
following periods. The last INTEGRAL observations, in which a
harder and brighter persistent emission was detected, took place
after the strong reactivation of the Summer 2004. SGR~1806--20\ was still
particularly active in September-October 2004 (Mereghetti et al.
2004a, Golenetski et al. 2004).
The detection of pulsed hard X-rays (20-150 keV) from the
Anomalous X-ray Pulsar 1E 1841--045 has been recently reported by
Kuiper et al. (2004). Anomalous X-ray Pulsars (AXPs) share many
similarities with the SGRs (Mereghetti et al. 2002) and are also
thought to be magnetars (see Woods \& Thompson 2004 for a recent
review). It is therefore interesting to compare our results with
those obtained for this source. The high-energy emission from
1E~1841--045, with a power law photon index
$\Gamma$=1.47$\pm$0.05, is definitely harder than that displayed
by SGR~1806--20\ before September 2004. This value of the photon index,
obtained with the RXTE/HEXTE instrument, refers to the total
(pulsed plus unpulsed) flux and is consistent with the results of
the INTEGRAL flux measurements in the 18-60 and 60-120 keV ranges
(Molkov et al. 2004). The spectrum of the pulsed component in
1E~1841--045 is even harder ($\Gamma$=0.94$\pm$0.16; Kuiper et al.
2004). When compared to lower energy measurements, these results
imply that the spectrum of the AXP has a significant hardening
above $\sim$10 keV. This contrasts with the situation seen in
Figs.~2(a) and 2(b) for SGR~1806--20.
The hard X-ray emission from magnetars has been discussed in the
context of a model involving currents in a globally twisted
magnetosphere by Thompson, Lyutikov \& Kulkarni (2002). A
non-thermal tail in the X-ray spectrum is predicted as a result of
multiple resonant cyclotron scattering. According to these
authors, the differences between SGRs and AXPs, i.e. the emission
of bursts in SGRs, as well as their harder spectra and faster
spin down rate, are explained by a larger degree of twist in the
external magnetic field. Our observation of a harder persistent
spectrum when the bursting activity of SGR~1806--20\ was at its highest
level indicates that this correlation, previously seen when
comparing different sources, also holds within the same object.
\begin{figure}[t]
\psfig{figure=spectranew.ps,width=6cm,angle=90}
\caption{IBIS/ISGRI spectra of SGR~1806--20: (a) persistent emission
March 2003-Apr. 2004, (b) persistent emission
Sept.-Oct. 2004, (c) one burst (scaled down by a factor 10$^{4}$).
The solid lines are the best fits (power laws in (a) and (b),
thermal bremsstrahlung in (c)). The dashed lines indicate the
extrapolation of power-law spectra measured in the 1-10 keV
range with XMM-Newton (Mereghetti et al. 2004b).}
\end{figure}
\begin{figure}[t]
\psfig{figure=2plot.ps,width=6cm,angle=-90}
\caption{(a) histogram of number of bursts per day detected by the
third Interplanetary Network (IPN). This histogram includes mostly
those bursts with fluences larger than 10$^{-7}$ erg cm$^{-2}$ and
is not corrected for experiment duty cycles and source visibility.
It therefore represents only the trend of the burst rate and not
its true value. (b) count rate in the 20-60 keV range. The
horizontal error bars indicate the time spanned by the (non
continuous) observations.}
\end{figure}
\section{Conclusions}
Thanks to the high sensitivity of the INTEGRAL IBIS instrument we
have discovered a hard X-ray component extending to 150 keV in the
persistent emission from SGR~1806--20. The imaging capabilities of IBIS
have been crucial for the observation of this crowded region of
the Galactic plane: they allow us to unambiguously associate the
observed hard X-ray emission with SGR~1806--20.
The hard X-rays
correlate in intensity and spectral hardness with the level of
bursting activity, and, contrary to what is observed in the AXP
1E~1841--045, a comparison with lower energy data does not
indicate evidence for a spectral hardening at $\sim$10-20 keV. The
INTEGRAL data provide the first detection of persistent emission
in this energy range for a SGR and open a new important diagnostic
to study the physics of magnetars (see, e.g., Thompson \&
Beloborodov 2004).
\begin{acknowledgements}
This work has been partially supported by the Italian Space
Agency. KH is grateful for support under NASA's INTEGRAL U.S.
Guest Investigator program, grant NAG5-13738.
\end{acknowledgements}
|
{
"timestamp": "2005-02-16T16:11:28",
"yymm": "0411",
"arxiv_id": "astro-ph/0411695",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411695"
}
|
\section{Introduction}
The origin of the stellar initial mass function (IMF) is a fundamental
problem in astrophysics because the stellar IMF determines photometric
properties of galaxies and the dynamical and chemical evolutions of their
interstellar medium. In this contribution we address the relation between
statistical properties of turbulence and the origin of the stellar IMF
as discussed in Padoan \& Nordlund (2002, 2004).
The main result of these works is that the power law slope, $s$, of the stellar
IMF measured by \cite{Salpeter55} is the consequence of the turbulent
nature of the star--forming gas and is directly related to the turbulent
power spectrum slope, $\beta$, $s=3/(4-\beta)$. From this point of view it
is not surprising that the origin of Salpeter's result has remained
mysterious for half a century, as our understanding of turbulence
has not improved much since the seminal work by \cite{Kolmogorov41}.
The situation has changed during the last decade, because
ever increasing computer resources have recently allowed significant
progress in both fields of turbulence and star formation.
Since Salpeter's work, the stellar IMF has been measured successfully
in many systems. We now know that the IMF in young clusters
(e.g. Chabrier 2003) reaches a maximum at a fraction of a solar
mass, and then turns around with a relative abundance of brown dwarfs
(BDs) that may vary from cluster to cluster (e.g. Luhman et al. 2000).
The work we present here explains also this feature of the IMF as a
consequence of the supersonic turbulence in the star--forming gas.
Using the properties of supersonic turbulence we have derived from
recent numerical simulations, we predict the stellar IMF essentially
without free parameters. This predicted IMF is shown to depend on the
rms Mach number, mean density and temperature of the turbulent flow.
It is also shown to agree well with Salpeter's IMF for large stellar
masses and with the low mass IMF derived for young stellar clusters.
Our view of the IMF as a natural property of supersonic turbulence
in the magnetized and isothermal star--forming gas provides an
explanation for the origin of BDs as well. According to
this picture, BDs may be formed in the same way as
hydrogen--burning stars.
\section{Statistics of Supersonic MHD Turbulence}
The velocity power spectrum in the inertial range of turbulence (between
the scales of energy injection and viscous dissipation) is a power law,
$E_v(k)\propto k^{-\beta}$, where $k$ is the wave--number. The spectral
index is $\beta\approx5/3$ for incompressible turbulence (Kolmogorov 1941),
and $\beta\approx2$ for pressureless turbulence (Burgers 1974; Gotoh \& Kraichnan1993).
In recent numerical simulations of isothermal, super--Alfv\'{e}nic and highly
supersonic magneto--hydrodynamic (MHD) turbulence we have obtained a power
spectrum intermediate between the Burgers and the Kolmogorov power spectra
(Boldyrev, Nordlund \& Padoan 2002) and consistent with the prediction by
\cite{Boldyrev02}, $E(k)\propto k^{-1.74}$.
The probability density function (PDF) of gas density in isothermal
turbulent flows is well approximated by a Log--Normal distribution
with moments depending on the rms Mach number of the flow
(Nordlund \& Padoan 1999; Ostriker, Gammie \& Stone 1999). The density
structure is characterized by a complex system of interacting shocks
resulting in a fractal network of dense cores, filaments, sheets and
low density ''voids'', with a large density contrast. Most of the mass
concentrates in a small volume fraction (according to the Log--Normal PDF),
a manifestation of the intermittent nature of the turbulence.
An example of a projected turbulent density field is shown in Figure~\ref{fig0}.
\begin{figure}[ht]
\centerline{
\epsfxsize=10cm \epsfbox{ppadoan_fig1.eps}
}
\caption[]{Projected density field from a numerical simulation of isothermal,
supersonic hydrodynamic turbulence with an effective resolution of $1024^3$
computational zones (Kritsuk, Padoan \& Norman, in preparation). The contrast
has been reduced in order to show details of the low density regions.}
\label{fig0}
\end{figure}
\section{From Kolmogorov to Salpeter: The Mass Distribution of Collapsing Cores}
A simple model of the expected mass distribution of dense cores
generated by supersonic turbulence has been proposed in
\cite{Padoan+Nordlund02imf}. The model is based on two assumptions:
i) The power spectrum of the turbulence is a power law; ii) the
typical size of a dense core scales as the thickness of the postshock
gas. The first assumption is a basic result for turbulent flows and
holds also in the supersonic regime (Boldyrev et al. 2002).
The second assumption is suggested by the fact that postshock condensations
are assembled by the turbulent flow in a dynamical time.
Condensations of virtually any size can therefore be formed,
independent of their Jeans' mass.
With these assumptions, together with the jump conditions for MHD
shocks, the mass distribution of dense cores can be related to the
power spectrum of turbulent velocity, $E_v(k)\propto k^{-\beta}$:
\begin{equation}
N(m)\,{\rm d}\ln m\propto m^{-3/(4-\beta)}{\rm d}\ln m ~.
\label{imf}
\end{equation}
If the turbulence spectral index $\beta$ is taken from the
analytical prediction by \cite{Boldyrev02}, which
is consistent with the observed velocity dispersion--size Larson
relation (Larson 1979, 1981) and with our numerical results
(Boldyrev et al. 2002), then $\beta \approx 1.74$ and the
mass distribution is $N(m)\,{\rm d}\ln m\propto m^{-1.33}{\rm d}\ln m$,
almost identical to Salpeter's stellar IMF (Salpeter 1955).
The exponent of the mass distribution is rather well constrained,
because the value of $\beta$ for supersonic turbulence cannot be
smaller than the incompressible value, $\beta= 1.67$ (slightly larger
with intermittency corrections), and the Burgers case, $\beta=2.0$.
As a result, the exponent of the mass distribution is predicted to be
within the range of values of 1.3 and 1.5.
While massive cores are usually larger than their critical Bonnor--Ebert mass,
$m_{\rm BE}$, the probability that small cores are dense enough to collapse is
determined by the statistical distribution of core density.
In order to compute this collapse probability for small cores, we assume
i) the distribution of core density can be approximated by the
\begin{figure}[ht]
\centerline{
\epsfxsize=6cm \epsfbox{ppadoan_fig2a.eps}
\epsfxsize=6cm \epsfbox{ppadoan_fig2b.eps}
}
\caption[]{Left panel: Analytical mass distributions computed
for $\langle n\rangle=10^4$~cm$^{-3}$, $T=10$~K and
for three values of the sonic rms Mach number, $M_{\rm S}=5$, 10 and 20
(solid lines). The dotted lines show the mass distribution for
$T=10$~K, $M_{\rm S}=10$ and $\langle n\rangle=5\times10^3$~cm$^{-3}$
(lower plot) and $\langle n\rangle=2\times10^4$~cm$^{-3}$ (upper plot).
Right panel: IMF of the cluster IC 348 in Perseus obtained by
\cite{Luhman+2003} (solid line histogram) and theoretical IMF computed for
$\langle n\rangle=5\times 10^4$~cm$^{-3}$, $T=10$~K and
$M_{\rm S}=7$ (dashed line).}
\label{fig1}
\end{figure}
Log--Normal PDF of gas density and ii) the core density and mass are
statistically independent. Because of the intermittent
nature of the Log-Normal PDF, even very small (sub--stellar) cores
have a finite chance to be dense enough to collapse.
Based on the first assumption, we can compute the distribution of the
critical mass, $p(m_{\rm BE})\,d m_{\rm BE}$, from the Log--Normal
PDF of gas density assuming constant temperature (Padoan, Nordlund \& Jones 1997).
The fraction of cores of mass $m$ larger than their critical
mass is given by the integral of $p(m_{\rm BE})$ from 0 to
$m$. Using the second assumption of statistical independence of core density
and mass, the mass distribution of collapsing cores is
\begin{equation}
N(m)\, {\rm d}\ln m\propto m^{-3/(4-\beta)}\left[\int_0^m{p(m_{\rm BE}){\rm d}m_{\rm BE}}\right]\,{\rm d}\ln m ~.
\label{imfpdf}
\end{equation}
This mass distribution is a function of the rms Mach number, mean density and
temperature of the turbulent flow, as these parameters enter the PDF of gas
density and thus $p(m_{\rm BE})$. Figure~\ref{fig1} (left panel) shows five
mass distributions computed from equation (\ref{imfpdf}) with three different
values of the sonic rms Mach number and two different values of density.
\cite{Padoan+Nordlund04bd} have suggested that BDs may originate from
the process of turbulent fragmentation like hydrogen--burning stars.
This is illustrated by the analytical IMF in the left panel of Figure~\ref{fig1},
which shows a relatively large abundance of BDs is predicted for sufficiently
large values of the mean density or rms Mach number.
The IMF of the cluster IC 348 in Perseus, obtained by \cite{Luhman+2003}, is
plotted in the right panel of Figure~\ref{fig1} (solid line histogram).
The IMF of this cluster has been chosen for the comparison with the theoretical
model because it is probably the most reliable observational IMF including both
BDs and hydrogen burning stars. In Figure~\ref{fig1} we have also plotted the
theoretical mass distribution computed for $\langle n\rangle=5\times 10^4$~cm$^{-3}$,
$T=10$~K and $M_{\rm S}=7$. These parameters are appropriate
for the central $5\times 5$~arcmin of the cluster ($0.35\times 0.35$~pc),
where the stellar density corresponds to approximately $2\times 10^4$~cm$^{-3}$.
The figure shows that the theoretical distribution of collapsing
cores, computed with parameters inferred from the observational data,
is roughly consistent with the observed stellar IMF in the cluster IC 348.
Similar IMFs have been obtained for several other young clusters (Chabrier 2003).
\section{Conclusions}
We have proposed to explain the stellar IMF as a property of supersonic
turbulence. This scenario is very different from previous theories of star
formation (see Shu, Adams \& Lizano 1987), where it is assumed that stars of small
and intermediate mass are formed from sub--critical cores evolving
quasi--statically, on the time--scale of ambipolar drift. These theories
do not account for the ubiquity of turbulence in star--forming clouds
and therefore ignore the effect of turbulence in the fragmentation process.
A naive interpretation of our results may lead to the conclusion that
the stellar IMF at large masses should be a universal power law, with
a slope very close to Salpeter's value. The statistics of turbulence
controlling the origin of the stellar IMF are indeed universal (they
depend on {\it flow} properties such as the rms Mach number, not
{\it fluid} properties, and are insensitive to initial conditions
that are soon ``forgotten'' due to the chaotic nature of the turbulence).
However, such statistics are derived from ensemble averages.
According to our derivation, massive stars originate from shocks
on relatively large scales. In any given system (for example a molecular cloud)
the number of large scale compressions (thus the number of massive stars)
is relatively small. Because of the small size of the statistical sample
and because of the intermittent nature of the turbulence, large deviations
from Salpeter's IMF are possible in individual systems. Salpeter's IMF is
therefore a universal property of supersonic turbulence, but it is
not necessarily reproduced precisely in every star--forming region.
\begin{chapthebibliography}{}
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\bibitem[{{Boldyrev} (2002)}]{Boldyrev02}
{Boldyrev}, S. 2002, ApJ, 569, 841
\bibitem[{{Boldyrev} {et~al.} (2002)}]{Boldyrev+02scaling}
{Boldyrev}, S., {Nordlund}, {\AA}., \& {Padoan}, P. 2002, ApJ, 573, 678
\bibitem[{Burgers (1974)}]{Burgers74}
Burgers, J.~M. 1974, in The Nonlinear Diffusion Equation (Reidel, Dordrecht)
\bibitem[{{Chabrier} (2003)}]{Chabrier03}
{Chabrier}, G. 2003, PASP, 115, 763
\bibitem[{Gotoh \& Kraichnan (1993)}]{Gotoh+Kraichnan93}
Gotoh, T. \& Kraichnan, R.~H. 1993, Phys. Fluids, A, 5, 445
\bibitem[{Kolmogorov (1941)}]{Kolmogorov41}
Kolmogorov, A.~N. 1941, Dokl. Akad. Nauk. SSSR, 30, 301
\bibitem[{{Larson} (1979)}]{Larson79}
{Larson}, R.~B. 1979, MNRAS, 186, 479
\bibitem[{{Larson} (1981)}]{Larson81}
---. 1981, MNRAS, 194, 809
\bibitem[{{Luhman} {et~al.} (2000)}]{Luhman+2000}
{Luhman}, K.~L., {Rieke}, G.~H., {Young}, E.~T., {Cotera}, A.~S., {Chen}, H.,
{Rieke}, M.~J., {Schneider}, G., \& {Thompson}, R.~I. 2000, ApJ, 540, 1016
\bibitem[{{Luhman} {et~al.} (2003)}]{Luhman+2003}
{Luhman}, K.~L., {Stauffer}, J.~R., {Muench}, A.~A., {Rieke}, G.~H., {Lada},
E.~A., {Bouvier}, J., \& {Lada}, C.~J. 2003, ApJ, 593, 1093
\bibitem[{{Nordlund} \& {Padoan} (1999)}]{Nordlund+Padoan99}
{Nordlund}, {\AA}.~K. \& {Padoan}, P. 1999, in Interstellar Turbulence, 218
\bibitem[{{Ostriker} {et~al.} (1999)}]{Ostriker+99}
{Ostriker}, E.~C., {Gammie}, C.~F., \& {Stone}, J.~M. 1999, ApJ, 513, 259
\bibitem[{{Padoan} \& {Nordlund} (2002)}]{Padoan+Nordlund02imf}
{Padoan}, P. \& {Nordlund}, {\AA}. 2002, ApJ, 576, 870
\bibitem[{{Padoan} \& {Nordlund} (2004)}]{Padoan+Nordlund04bd}
---. 2004, ApJ, in press (see also astro-ph/0205019)
\bibitem[{Padoan {et~al.} (1997)}]{Padoan+97imf}
Padoan, P., Nordlund, {\AA}., \& Jones, B. 1997, MNRAS, 288, 145
\bibitem[{{Salpeter} (1955)}]{Salpeter55}
{Salpeter}, E.~E. 1955, ApJ, 121, 161
\bibitem[{Shu {et~al.} (1987)}]{Shu+87}
{Shu}, F.~H., {Adams}, F.~C. \& {Lizano}, S., 1987, ARA\&A, 25, 23
\end{chapthebibliography}
\end{document}
|
{
"timestamp": "2004-11-18T18:04:26",
"yymm": "0411",
"arxiv_id": "astro-ph/0411474",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411474"
}
|
\section{Introduction}
Oriented knots $K_0$ and $K_1$ in the 3-sphere are called
(topologically) \emph{concordant} if there is a topologically
locally flat proper embedding of $S^1\times [0,1]$ into $S^3\times
[0,1]$ that restricts to the $K_i$ on $S^3\times \{i\}$.
Equivalently, $K_0$ is concordant to $K_1$ if $K_0\#(-K_1)$ is
(topologically) slice, that is, if $K_0\#(-K_1)$ bounds a
topologically locally flat disk in the 4-ball. Here $`\#'$ denotes
the connected sum, and $-K_1$ is the mirror image of $K_1$ with
reversed string orientation. Concordance is an equivalence
relation on the set of oriented knots, and the set of equivalence
classes (\emph{concordance classes}) forms an abelian group,
$\mathcal C$, under the operation of connected sum. This group is called
the \emph{(topological) knot concordance group}. The
classification of the knot concordance group is still open, and it
has been one of the central problems in knot theory. A common
strategy of investigating the knot concordance group is to extract
information from abelian covers or metabelian covers of the
exterior of a knot (see \cite{L,CS,CG,G,KL,Le,Fr}). On the other
hand, requiring smooth embedding instead of topologically locally
flat embedding, we can define the \emph{smooth knot concordance
group}, which is denoted by $\mathcal C^{smooth}$. In smooth knot
concordance theory, there have been recent and very interesting
developments using knot Floer homology \cite{OS} and Khovanov
homology \cite{Ra}. In this paper we concern ourselves mainly with
topological knot concordance. Hence we work in the topologically
locally flat category unless mentioned otherwise.
Cochran, Orr, and Teichner (henceforth COT) recently made
significant progress in the study of $\mathcal C$ using \emph{derived}
covers of the exterior of a knot \cite{COT1}. (A \emph{derived
cover} is a covering space corresponding to a derived subgroup of
the fundamental group of the space.) In particular, they defined a
filtration $\{\mathcal F_{(n)}\}_{n\in \frac12 {\mathbb N}_0}$ of the knot
concordance group:
$$
0\subset\cdots\subset\mathcal F_{(n.5)}\subset\mathcal F_{(n)}\subset\cdots
\subset\mathcal F_{(1.5)}\subset\mathcal F_{(1.0)}\subset\mathcal F_{(0.5)}
\subset\mathcal F_{(0)}\subset\mathcal C.
$$
Here $\mathcal F_{(n)}$ is a subgroup of $\mathcal C$ consisting of
\emph{$n$-solvable} knots. (A knot $K$ is said to be
\emph{$n$-solvable} if the zero surgery on the knot in the
3-sphere bounds a spin 4-manifold $W$ which satisfies certain
conditions on integral homology groups and intersection form on
the $n$-th derived cover of $W$. In this case, we say $K$ is
\emph{$n$-solvable via $W$}, and $W$ is called an
\emph{$n$-solution} for $K$. Refer to \cite[Section 8]{COT1}.) COT
showed that the previously known abelian and metabelian
concordance invariants are reflected at the first stage of their
filtration \cite{COT1}, implying that the abelian group
$\mathcal F_{(1.0)}/\mathcal F_{(1.5)}$ has infinite rank. They also established
that $\mathcal F_{(2.0)}/\mathcal F_{(2.5)}$ has infinite rank \cite{COT2}.
Moreover, Cochran and Teichner showed that $\mathcal F_{(n)}/\mathcal F_{(n.5)}$
has positive rank for every integer $n \ge 2$ \cite{CT}. It is
still unknown whether or not $\mathcal F_{(n)}/\mathcal F_{(n.5)}$ is infinitely
generated for $n>2$, or whether or not $\mathcal F_{(n.5)}/\mathcal F_{(n+1)}$
is non-trivial. Thus, although $\mathcal F$ has been shown to be highly
non-trivial, many questions about its structure remain open.
These non-triviality results have been refined in the case $n=1$.
C. Livingston recently asked if one can always find non-concordant
knots which share a given (classical) Seifert form. That is, given
$K$, can one always find other knots, distinct up to concordance,
which share the same (classical) Seifert form as $K$? As he
observed, if $K$ has the Seifert form of an unknot (has Alexander
polynomial $1$) then the answer is certainly ``No'', since all
knots with Alexander polynomial $1$ are topologically concordant
to the trivial knot and hence concordant to each other (by M.
Freedman's work \cite{F,FQ}). Livingston gave a partial answer in
the positive to his question using Casson-Gordon invariants
\cite{Li}. This work was completed by the second author, who
showed that there always exist such knots for a given (classical)
Seifert form if and only if the Alexander polynomial of the
Seifert form is not trivial. More precisely, in \cite{K} he showed
that for a given knot $K$ whose Alexander polynomial has degree at
least $2$, there are infinitely many knots $K_i$ (with $K_0 = K$)
such that $K_i-K$ is $1$-solvable, but $K_i - K_j$ ($i\ne j)$ is
not $(1.5)$-solvable, and $K_i$ shares the same (classical)
Seifert form (hence the same (classical) Alexander module) as the
knot $K$. We view these results as non-triviality results for
$\mathcal F$ that are finer (for $n=1$) than those of
\cite{COT1}\cite{COT2}\cite{CT}. They suggest further questions.
Is the Livingston-Kim result true for $n>1$? Moreover, is there an
even finer result that constructs such examples while fixing not
only the (classical) Seifert form (hence the (classical) Alexander
module) but also the higher-order Seifert forms and higher-order
Alexander modules developed in \cite{C}? This paper answers these
questions in the positive as long as the degree of the Alexander
polynomial is at least $4$ (see below). To do so we found the need
to extend the technology of \cite{COT1} to a ``relative'' setting,
and to greatly generalize the key technical result of Cochran and
Teichner \cite[Theorem 4.3]{CT}. We expect that this extended
technology will be of independent interest, enabling further work
in this area.
In this paper, we introduce the notion of an \emph{$n$-cylinder}
(Definition~\ref{defn:n-cylinder}), generalizing the notion of an
$n$-solution of COT . The difference is that an $n$-solution
allows only one boundary component, whereas an $n$-cylinder can
have multiple boundary components. A basic (trivial) example of an
$n$-cylinder (with two boundary components) is a (spin) homology
cobordism between the zero surgeries on two knots. If two knots
are concordant then one can easily find such an $n$-cylinder
(which in this case is a homology cobordism), by doing surgery on
$S^3\times [0,1]$ along the annulus cobounded by the knots
(Remark~\ref{rem:n-cylinder}(4)). Using this, we define a family
of new equivalence relations on the knot concordance group, that
we call \emph{$n$-solvequivalence} (see
Definition~\ref{defn:n-solvequivalence}) : $K_0$ is
\emph{$n$-solvequivalent} to $K_1$ if the zero-framed surgery on
$K_0$ and the zero-framed surgery on $K_1$ cobound an
$n$-cylinder. From the basic example above, it is clear that
concordant knots are $n$-solvequivalent for all $n$.
On the other hand, the filtration $\mathcal F$ of COT immediately
suggests another family of equivalence relations on $\mathcal C$, given
as follows. We say $K_0$ and $K_1$ are \emph{concordant modulo
$n$-solvability} if $K_0\#(-K_1)$ is $n$-solvable. There is a
close relationship between $n$-solvability and
$n$-solvequivalence. It is not difficult to show that if two knots
are concordant modulo $n$-solvability, then they are
$n$-solvequivalent (Proposition~\ref{prop:old and new}), but we
have not been able to establish the converse. Hence
$n$-solvequivalence is a possibly weaker obstruction to
concordance than $n$-solvability.
Note that $n$-solvequivalence reflects information on the zero
surgeries on \emph{each} $K_i$ separately, but \emph{concordance
modulo $n$-solvability} reflects only information on the zero
surgery on $K_0\#(-K_1)$ (see Proposition~\ref{prop:P and P
perb}).
As an application of $n$-solvequivalence and of our other
extensions of the technology of \cite{COT1}\cite{COT2} \cite{CT},
we obtain the following theorem, generalizing the nontriviality
results cited above. The condition that the degree of the
Alexander polynomial be at least $4$ may seem ad hoc. However,
very recent work of S. Friedl and Teichner \cite{FT} complements
our theorem and we use it to show that our theorem is ``best
possible", in the sense that it is false for certain knots whose
Alexander polynomial has degree $2$.
The reader is referred to \cite[Section 3]{CT} for the definition
of symmetric Gropes. Let $G$ be a group. Then the \emph{$n$-th
derived group of $G$} is inductively defined by $G^{(0)}\equiv G$
and $G\ensuremath{^{(n+1)}} \equiv [G^{(n)},G^{(n)}]$.
\newtheorem*{main}{Theorem~\ref{thm:main}}
\begin{main}[{\bf Main Theorem}]
Let $n$ be a positive integer. Let $K$ be a knot whose Alexander
polynomial has degree greater than 2 (if $n=1$ then degree equal
to $2$ is allowed). Then there is an infinite family of knots
$\{K_i \,\,|\,\, i=0,1,2,\ldots\}$ with $K_0 = K$ that satisfies
the following :
\begin{itemize}
\item [(1)] For each $i$, $K_i - K$ is $n$-solvable. In
particular, $K_i$ is $n$-solvequivalent to $K$. Moreover, $K_i$
and $K$ cobound, in $S^3\times\[0,1\]$, a smoothly embedded symmetric
Grope of height $n+2$.
\item [(2)] If $i\ne j$, then $K_i$ is not $(n.5)$-solvequivalent
to $K_j$. In particular, $K_i-K_j$ is not $(n.5)$-solvable, and
$K_i$ and $K_j$ do not cobound, in $S^3\times\[0,1\]$, an embedded
symmetric Grope of height $(n+2.5)$.
\item [(3)] For each $i$, $K_i$ has the same order $m$ integral
higher-order Alexander module as $K$ for $m=0,1,\ldots, n-1$.
Indeed, if $G_i$ and $G$ denote the knot groups of $K_i$ and $K$
respectively, then there is an isomorphism $G_i/(G_i)^{(n+1)}\to
G/G^{(n+1)}$ that preserves the peripheral structures.
\item [(4)] For each $i$, $K_i$ has the same $m^{th}$-higher-order
Seifert presentation as $K$ for $m=0,1,\ldots, n-1$. In
particular, all of the knots admit the same classical Seifert
matrix.
\item [(5)] If $i>j$, $K_i - K_j$ is of infinite order in
$\mathcal F_{(n)}/\mathcal F_{(n.5)}$.
\item [(6)] If $i,j>0$, $s(K_i)=s(K_j)$ and $\tau(K_i)=\tau(K_j)$,
where $s$ is Rasmussen's smooth concordance invariant and $\tau$
is the smooth concordance invariant of P.~Ozsv{\'a}th and
Z.~Szab{\'o} \cite{Ra}\cite{OS}.
\end{itemize}
\end{main}
The general term \emph{higher-order Alexander modules} was first
used in \cite{COT1}. The \emph{higher-order Alexander modules} and
\emph{higher-order Seifert presentations} of a knot that we use
here were introduced and developed by Cochran \cite{C}. These are
defined in terms of the homology of the derived covers of the knot
exterior. Definitions are given in Section~\ref{sec:main}. But in
particular, these generalize (the case $n=0$) the classical
Alexander module and Seifert matrix. Therefore, the case $n=1$ of
our theorem recovers the previously mentioned results of
Livingston and Kim. If the degree of the Alexander polynomial is
precisely $2$ then the Livingston-Kim result says that it is
possible to fix the \emph{classical} Seifert matrix ($m=0$), but
we show, using recent work of Friedl and Teichner, that there are
at least \emph{some} such knots for which it is \emph{not
possible} to fix any higher-order Seifert presentation matrices.
Specifically, the main theorem fails for $n>1$ if we allow knots
whose Alexander polynomials have degree $2$ (see
Proposition~\ref{prop:failure}). More details are within.
As another obvious
corollary we recover the aforementioned result of Cochran and Teichner
\cite{CT}.
\newtheorem*{cor of main}{Corollary~\ref{cor:cot filtration}}
\begin{cor of main}\cite{CT}
For every positive integer $n$, $\mathcal F_{(n)}/\mathcal F_{(n.5)}$ has
positive rank.
\end{cor of main}
\noindent Indeed, this corollary is essentially equivalent to parts (1) and
(5) in Theorem~\ref{thm:main}, and so our new contribution lies
in being able to impose (3) and (4).
Now consider the filtration of $\mathcal C$ given by the subgroups
$\mathcal G_n$ of knots that bound topologically embedded symmetric
Gropes of height $n$ in $B^4$ and the filtration of the smooth
knot concordance group $\mathcal C^{smooth}$ given by the subgroups
$\mathcal G_n^{smooth}$ of knots that bound smoothly embedded symmetric
Gropes of height $n$ in $B^4$. It was shown in \cite[Theorem
1.3]{CT} that $\mathcal G_{n+2}/\mathcal G_{n+2.5}$ has positive rank if $n\geq
0$. It follows from their work (although not explicitly stated)
that $\mathcal G_{n+2}^{smooth}/\mathcal G_{n+2.5}^{smooth}$ has positive rank.
Our work recovers these results, together with the added control
of the Alexander and Seifert data, and allows for the following
additional observations concerning the smooth concordance
invariants of Rasmussen and Ozsv{\'a}th-Szab{\'o}.
\newtheorem*{gropecor}{Corollary~\ref{cor:gropefiltrations}}
\begin{gropecor} Let $n$ be a positive integer.
\begin{itemize}
\item [1.] $\mathcal G_{n+2}^{smooth}/\mathcal G_{n+2.5}^{smooth}$ has
elements of infinite order represented by knots $K$ with $s(K)=0$
and $\tau(K)=0$. \item [2.] The kernel of the homomorphism
$ST:\mathcal G_{n+2}^{smooth}\to \mathbb{Z}\times \mathbb{Z}$ given by
$ST(K)=(s(K),\tau(K))$ is infinite.
\end{itemize}
\end{gropecor}
Our Proposition~\ref{prop:geomalg} and
Theorem~\ref{thm:paininbutt} represent a significant strengthening
of the crucial technical results of Proposition $6.2$ and Theorem
$6.3$ in \cite{CT}. In the process, we introduce the notion of an
\emph{algebraic $n$-solution} which should be considered as an
algebraic abstraction of our notion of an $n$-cylinder
(Definition~\ref{defn:algebraic} and
Proposition~\ref{prop:geomalg}). In fact, an algebraic
$n$-solution was introduced by Cochran and Teichner in \cite{CT},
but our notion is much more general.
For an obstruction to $n$-solvequivalence, as in
\cite{COT1}\cite{COT2}\cite{CT}, we use the Cheeger-Gromov
\emph{von Neumann $\rho$-invariant} (Theorem~\ref{thm:rho=0} and
Corollary~\ref{thm:rho=0}), \cite{ChG}. For this purpose, we
analyze the twisted homology groups of an $n$-cylinder and its
boundary in Section~\ref{sec:homology}.
This paper is organized as follows. In
Section~\ref{sec:n-cylinder}, we define an $n$-cylinder and
$n$-solvequivalence. We also investigate the relationship between
$n$-solvability and $n$-solvequivalence. In
Section~\ref{sec:homology}, the twisted homology groups of an
$n$-cylinder and its boundary are analyzed. In
Section~\ref{sec:obstruction}, we explain how to realize von
Neumann $\rho$-invariants as obstructions to $n$-solvequivalence.
All of these sections generalize \cite{COT1} by extending to ``the
relative case''. The main theorem is proved in
Section~\ref{sec:main}. In the last section, we define and
investigate an algebraic $n$-solution. Here, we not only
generalize \cite{CT} to the relative case but also significantly
strengthen a primary technical tool. In a strictly logical order,
Section~\ref{sec:algebraic} would precede Section~\ref{sec:main},
but we have chosen to place Section~\ref{sec:algebraic} after
Section~\ref{sec:main}, since the arguments in
Section~\ref{sec:algebraic} are very technical.
\section{$n$-cylinders and $n$-solvequivalence}
\label{sec:n-cylinder} In this section, we define an $n$-cylinder
and $n$-solvequivalence. Throughout this
paper integer coefficients are understood for homology groups
unless specified otherwise.
Recall that a group $G$ is called \emph{$n$-solvable} if $G\ensuremath{^{(n+1)}} =
1$. If $X$ is a topological space, $X^{(n)}$ denotes the covering
space of $X$ corresponding to the $n$-th derived group of $\ensuremath{\pi_1}(X)$.
For a compact spin 4-manifold $W$ there is an equivariant
intersection form
$$
\lambda_n : H_2(W^{(n)}) \times H_2(W^{(n)}) \longrightarrow
{\mathbb Z}[\ensuremath{\pi_1}(W)/\ensuremath{\pi_1}(W)^{(n)}]
$$
and a self-intersection form $\mu_n$ \cite[Chapter
5]{Wa}\cite[Section 7]{COT1}. (Here $H_2(W^{(n)})$ is considered
to be a right ${\mathbb Z}[\ensuremath{\pi_1}(W)/\ensuremath{\pi_1}(W)^{(n)}]$-module.) In general, these
intersection forms are singular.
Let us denote the boundary components of $W$ by $M_i$
($i = 1,2,\ldots, \ell$). That is, $\partial W =
\coprod^\ell_{i=1}M_i$ where each $M_i$ is a connected 3-manifold.
Let $I \equiv \operatorname{Image} \{\operatorname{inc}_* : H_2(\partial W) \to H_2(W)\}$
where $\operatorname{inc}_*$ is the homomorphism induced by the inclusion from
$\partial W$ to $W$. Then the usual intersection form factors
through
$$
\overline{\lambda_0} : H_2(W)/I \times H_2(W)/I \longrightarrow {\mathbb Z}.
$$
We define an \emph{$n$-Lagrangian} to be a
${\mathbb Z}[\ensuremath{\pi_1}(W)/\ensuremath{\pi_1}(W)^{(n)}]$-submodule of $H_2(W^{(n)})$ on which
$\lambda_n$ and $\mu_n$ vanish and which maps onto a $\frac12$-rank
direct summand of $H_2(W)/I$ under the covering map. An
\emph{$n$-surface} is defined to be a based and immersed surface
in $W$ that can be lifted to $W^{(n)}$. Observe that any class in
$H_2(W^{(n)})$ can be represented by an $n$-surface and that
$\lambda_n$ can be calculated by counting intersection points in
$W$ among representative $n$-surfaces weighted appropriately by
signs and by elements of $\pi_1(W)/\pi_1(W)^{(n)}$. We say an
$n$-Lagrangian $L$ admits \emph{$m$-duals} (for $m\le n$) if $L$
is generated by (lifts of) $n$-surfaces $\ell_1,\ell_2,\ldots,\ell_g$ and
there exist $m$-surfaces $d_1,d_2,\ldots, d_g$ such that $H_2(W)/I$
has rank $2g$ and $\lambda_m(\ell_i,d_j)=\delta_{i,j}$. Similarly we
can define a \emph{rational $n$-Lagrangian} and \emph{rational
$m$-duals}. Here we do not require that $W$ be spin and $\mu_n$ is
not discussed. A \emph{rational $n$-Lagrangian} $L$ is a
${\mathbb Q}[\ensuremath{\pi_1}(W)/\ensuremath{\pi_1}(W)^{(n)}]$-submodule of $H_2(W^{(n)};{\mathbb Q})$ on
which $\lambda_n$ (with ${\mathbb Q} [\pi_1(W)/\pi_1(W)^{(n)}]$ coefficients)
vanishes and which maps onto a $\frac12$-rank direct summand of
$H_2(W;{\mathbb Q})/I_{\mathbb Q}$ under the covering map. Here $I_{\mathbb Q} \equiv
\operatorname{Image} \{\operatorname{inc}_* : H_2(\partial W;{\mathbb Q}) \to H_2(W;{\mathbb Q})\}$. We say
a rational $n$-Lagrangian $L$ admits \emph{rational $m$-duals}
(for $m\le n$) if $L$ is generated by $n$-surfaces
$\ell_1,\ell_2,\ldots,\ell_g$ and there are $m$-surfaces
$d_1,d_2,\ldots, d_g$ such that $H_2(W;{\mathbb Q})/I_{\mathbb Q}$ has rank $2g$
and $\lambda_m(\ell_i,d_j)=\delta_{i,j}$.
\begin{defn}
\label{defn:n-cylinder} Let $n$ be a nonnegative integer.
\begin{enumerate}
\item A compact, connected, spin 4-manifold $W$ with
$\partial W = \coprod^\ell_{i=1}M_i$ where each $M_i$ is a
connected component of $\partial W$ with $H_1(M_i) \cong {\mathbb Z}$ is
an \emph{$n$-cylinder} if the inclusion from $M_i$ to $W$ induces
an isomorphism on $H_1(M_i)$ for each $i$ and $W$ admits an
$n$-Lagrangian with $n$-duals. In addition to this, if the
$n$-Lagrangian is the image of an $(n+1)$-Lagrangian under the
covering map, then $W$ is called an \emph{$(n.5)$-cylinder}.
\item A compact, connected 4-manifold $W$ with
$H_1(W;\mathbb{Q})\cong \mathbb{Q}$ and with $\partial W =
\coprod^\ell_{i=1}M_i$ where each $M_i$ is a connected component
of $\partial W$ with $H_1(M_i) \cong {\mathbb Q}$ is a \emph{rational
$n$-cylinder of multiplicity $\{m_1,m_2,\ldots m_\ell\}$}
($m_i\in{\mathbb Z}$) if the inclusion from $M_i$ to $W$ induces an
isomorphism on $H_1(M_i;{\mathbb Q})$ such that a generator 1 in
$H_1(M_i)/torsion$ is sent to $m_i$ in $H_1(W)/torsion$ and $W$
admits a rational $n$-Lagrangian with rational $n$-duals. In
addition to this, if the rational $n$-Lagrangian is the image of a
rational $(n+1)$-Lagrangian under the covering map, then $W$ is
called a \emph{rational $(n.5)$-cylinder of multiplicity
$\{m_1,m_2,\ldots,m_\ell\}$}.
\end{enumerate}
\end{defn}
\begin{rem}
\label{rem:n-cylinder}
\begin{enumerate}
\item By naturality of intersection forms and commutativity of the
diagram below, the image of an $n$-Lagrangian in $H_2(W)/I$
becomes a metabolizer for the $\overline{\lambda_0}$ and it
follows that the signature of $W$ is zero. The same is true for a
rational $n$-Lagrangian.
$$
\begin{diagram}\dgARROWLENGTH=1.0em
\node{H_2(W^{(n)})} \arrow{s} \node{\times} \node{H_2(W^{(n)})}
\arrow{s} \node{\longrightarrow} \node[2]{{\mathbb Z}[\ensuremath{\pi_1}(W)/\ensuremath{\pi_1}(W)^{(n)}]} \arrow{s}\\
\node{H_2(W)/I} \node{\times} \node{H_2(W)/I} \node{\longrightarrow}
\node[2]{{\mathbb Z}}
\end{diagram}
$$
\item Using twisted local coefficients, $H_2(W^{(n)})$ is
identified with $H_2(W;{\mathbb Z}[\ensuremath{\pi_1}(W)/\ensuremath{\pi_1}(W)^{(n)}])$ as (right)
${\mathbb Z}[\ensuremath{\pi_1}(W)/\ensuremath{\pi_1}(W)^{(n)}]$-modules.
\item An $n$-cylinder is, in particular, a rational $n$-cylinder
of multiplicity $\{1,1,\ldots,1\}$.
\item A spin homology cobordism, say $W$, between two closed,
connected, oriented 3-manifolds is an $n$-cylinder since
$H_2(W)/I=0$.
\end{enumerate}
\end{rem}
The notion of $n$-cylinder extends the notion of $n$-solution in
\cite[Definitions 1.2,8.5,8.7]{COT1}. This is clear from the
following proposition.
\begin{prop}
\label{prop:n-cylider and n-solution} A 4-manifold $W$ with one
boundary component is an $n$-cylinder if and only if it is an
$n$-solution.
\end{prop}
\begin{proof}
Suppose $V$ is a compact, connected, spin 4-manifold with
$\partial V = N$ where $N$ is connected, $H_1(N) \cong {\mathbb Z}$, and
the inclusion from $N$ to $V$ induces an isomorphism on $H_1(N)$.
To prove the proposition, we only need to show $J \equiv
\operatorname{Image}\{\operatorname{inc}_* : H_2(N) \longrightarrow H_2(V)\} = 0$ where $\operatorname{inc}_*$ is the
homomorphism induced by the inclusion. Since the inclusion induces
an isomorphism on the first homology, $\operatorname{inc}^* : H^1(V) \longrightarrow
H^1(N)$ is an isomorphism. By Poincar$\acute{\text{e}}$ duality,
the boundary map $H_3(V,N) \longrightarrow H_2(N)$ is an isomorphism. Hence
$\operatorname{inc}_* : H_2(N) \longrightarrow H_2(V)$ is a zero homomorphism in the
homology long exact sequence of the pair $(V,N)$.
\end{proof}
\begin{rem}
\label{rem:rational n-cylinder and rational n-solution} In a
similar fashion, one can easily see that a 4-manifold $W$ with one
boundary component is a rational $n$-cylinder if and only if it is
a rational $n$-solution (see \cite[Definition 4.1]{COT1}).
\end{rem}
\noindent Using $n$-cylinders, we define $n$-solvequivalence
between 3-manifolds and between knots.
\begin{defn}
\label{defn:n-solvequivalence}Let $n\in \frac12{\mathbb N}_0$. Let $M_1$
and $M_2$ be closed, connected, oriented 3-manifolds.
\begin{enumerate}
\item If there
exists an $n$-cylinder $W$ such that $\partial W = M_1\coprod
-M_2$, then $M_1$ is \emph{$n$-solvequivalent to $M_2$ via $W$}.
\item If there exists a rational $n$-cylinder (of multiplicity
$\{m_1,-m_2\}$) $W$ such that $\partial W = M_1\coprod -M_2$, then
$M_1$ is \emph{rationally $n$-solvequivalent to $M_2$ via $W$ (of
multiplicity $\{m_1,m_2\}$)}.
\item For given two knots $K_i$ $(i=1,2)$, if the zero framed
surgery on $S^3$ along $K_1$ is (rationally) $n$-solvequivalent to
the zero framed surgery on $S^3$ along $K_2$ via $W$, then we say
$K_1$ is \emph{(rationally) $n$-solvequivalent to $K_2$ via $W$}.
\end{enumerate}
\end{defn}
\noindent It is not hard to show that $n$-solvequivalence is an
equivalence relation. Here we give a proof only for transitivity.
\begin{prop}
\label{prop:transitivity} Let $L$, $M$, and $N$ be closed,
connected, oriented 3-manifolds with $H_1(L) \cong H_1(M) \cong
H_1(N) \cong {\mathbb Z}$. Let $n\in \frac12 {\mathbb N}_0$. Suppose $L$ is
$n$-solvequivalent to $M$ via $V$ and $M$ is $n$-solvequivalent to
$N$ via $W$. Then $L$ is $n$-solvequivalent to $N$ via $V\cup_M
W$.
\end{prop}
\begin{proof}
Let $n\in {\mathbb N}$. Let $X \equiv V\cup_M W$. Then $\partial X = L
\coprod -N$. We need to show that $X$ is an $n$-cylinder. The
proof for the condition on the first homology groups is not hard,
hence left for the readers. Denote the inclusion map from $A$ to
$B$ by $i_{A,B}$ for two topological spaces $A \subset B$. Using
the long exact sequences of pairs, one can prove that
$H_2(V)/(i_{\partial V,V})_*(H_2(\partial V)) \cong
H_2(V)/(i_{L,V})_*(H_2(L)) \cong H_2(V)/(i_{M,V})_*(H_2(M))$. We
have similar isomorphisms for $(W,M\coprod -N)$. Now from the
Mayer-Vietoris sequence for $V$ and $W$ along $M$, one can see
that
$$
\(H_2(V)/(i_{\partial V, V})_*(H_2(\partial V))\) \oplus
\(H_2(W)/(i_{\partial W,W})_*(H_2(\partial W))\) \cong
H_2(X)/(i_{\partial X,X})_*(H_2(\partial X)).
$$
Since $(i_{V,X})_*$ and $(i_{W,X})_*$ map $\ensuremath{\pi_1}(V)^{(n)}$ and
$\ensuremath{\pi_1}(W)^{(n)}$ into $\ensuremath{\pi_1}(X)^{(n)}$, respectively, and intersection
forms are natural, the ``union" of the $n$-Lagrangian with
$n$-duals for $V$ and the $n$-Lagrangian with $n$-duals for $W$
constitutes an $n$-Lagrangian with $n$-duals for $X$.
The proof for the case when $n$ is a half-integer is left to the
reader.
\end{proof}
\noindent The notion of $n$-solvability defined by COT suggests an
equivalence relation on $\mathcal C$ wherein $K_0 \thicksim K_1$ if and
only if $K_0\#(-K_1)$ is $n$-solvable. This relation has not been
given a name. The following proposition reveals the close
connection, for knots, between $n$-solvequivalence and the
equivalence relation arising form COT's $n$-solvability. We
initially expected that these two equivalence relations were in
fact identical. However, we have not proved this and now believe
that they may be slightly different.
\begin{prop}
\label{prop:old and new} Let $n\in \frac12{\mathbb N}_0$. For knots
$K_0$ and $K_1$, if $K_0\#(-K_1)$ is $n$-solvable, then $K_0$ is
$n$-solvequivalent to $K_1$.
\end{prop}
\begin{proof}
Suppose $n \in {\mathbb N}_0$. Let $M_0$, $M_1$, and $M$ denote the zero
surgeries on $S^3$ along $K_0$, $K_1$, and $K_0\#(-K_1)$,
respectively. We construct a (standard) cobordism between
$M_0\coprod (-M_1)$ and $M$. Take a product $(M_0\coprod -M_1)
\times [0,1]$. By attaching a 1-handle along $(M_0\coprod
-M_1)\times \{1\}$ we get a 4-manifold whose upper boundary is
$M_0\# (-M_1)$. Note that $M_0\# (-M_1)$ is also obtained by
taking zero framed surgery on $S^3$ along a split link $K_0\coprod
-K_1$. Next add a zero framed 2-handle along the upper boundary of
this 4-manifold such that the attaching map is an unknotted circle
which links $K_0$ and $-K_1$ once. Then the resulting 4-manifold
has upper boundary $M$ as may be seen by sliding the 2-handle
represented by zero surgery on $K_0$ over that of $K_1$. This
4-manifold is the cobordism $C$ between $M_0\coprod -M_1$ and $M$.
Since $K_0\#(-K_1)$ is $n$-solvable, $M$ bounds an $n$-solution
$V$. Let $W \equiv C\cup_M V$. We claim that $W$ is an
$n$-cylinder with $\partial W = M_0 \coprod -M_1$. This will
complete the proof.
Note that $\partial W = M_0\coprod -M_1$. One sees that $H_2(M_0)$
and $H_2(M_1)$ are generated by capped-off Seifert surfaces for
$K_0$ and $K_1$, respectively. It is easy to show that $H_2(C) =
H_2(M_0) \oplus H_2(M_1)$ $(\cong {\mathbb Z} \oplus {\mathbb Z})$ and $H_1(C)
\cong H_1(M_1) \cong H_1(M_0) \cong H_1(M)$ $(\cong {\mathbb Z})$ using
Mayer-Vietoris sequences. Also one sees that the inclusion from
$M$ to $C$ induces an injection $\operatorname{inc}_* : H_2(M) \longrightarrow H_2(C)$
where $H_2(M) \cong {\mathbb Z}$ and $\operatorname{inc}_*$ sends 1 $(\in {\mathbb Z})$ to
(1,1) $(\in {\mathbb Z} \oplus {\mathbb Z})$. Consider the following
Mayer-Vietoris sequence :
$$
\cdots \to H_2(M) \xrightarrow{f} H_2(C)\oplus H_2(V) \to H_2(W)
\xrightarrow{g} H_1(M) \xrightarrow{h} H_1(C)\oplus H_1(V) \to
\cdots .
$$
Since $h$ is an injection, $g$ is a zero map. Since $H_3(V,M) \to
H_2(M)$ is a dual map of $H^1(V) \to H^1(M)$ which is an
isomorphism, by the long exact sequence of homology groups of the
pair $(V,M)$, the inclusion induced homomorphism $H_2(M) \to
H_2(V)$ is a zero map. Thus the image of $f$ in $H_2(V)$ is zero
and by our previous observation the image of $f$ in $H_2(C)$ is
isomorphic to ${\mathbb Z}$ generated by $(1,1)$ in $H_2(C)$. Therefore
$H_2(W) \cong H_2(V)\oplus {\mathbb Z}$. Furthermore the last ${\mathbb Z}$
summand on the right hand side is exactly $\operatorname{Image} \{\operatorname{inc}_* :
H_2(\partial W) \longrightarrow H_2(W)\}$, which is denoted by $I$. Hence
$H_2(W)/I \cong H_2(V)$. Since $V$ is a subspace of $W$,
$\ensuremath{\pi_1}(V)^{(n)}$ is a subgroup of $\ensuremath{\pi_1}(W)^{(n)}$. This implies that
$k$-surfaces in $V$ are also $k$-surfaces in $W$ for every integer
$k$. Now by naturality of (equivariant) intersections forms, one
can prove that an $n$-Lagrangian with $n$-duals for $V$ maps to an
$n$-Lagrangian with $n$-duals for $W$.
Clearly $W$ is a compact, connected 4-manifold, and it only
remains to show $W$ is spin. But this is obvious since $C$ and $V$
are spin and when we take the union of $C$ and $V$ we can adjust
spin structure of either of $C$ and $V$ to make $W$ spin.
Finally, in the case $K_0\# (-K_1)$ is $(n.5)$-solvable via $V$,
$K_0$ is $(n.5)$-solvequivalent to $K_1$ via $W$ where $W$ is
constructed as above. The argument for a proof for this goes the
same as above, and one just needs to notice that an
$(n+1)$-Lagrangian with $n$-duals for $V$ maps to an
$(n+1)$-Lagrangian with $n$-duals for $W$.
\end{proof}
\begin{rem}
\label{rem:old and new} There are a couple of natural arguments
for attempting to prove the converse of Proposition~\ref{prop:old
and new}, but they do not work completely since capped-off Seifert
surfaces for knots do not lift to higher covers of 4-manifolds in
the arguments.
\end{rem}
\section{Homology groups of an $n$-cylinder and its boundary}
\label{sec:homology} Let $\Gamma$ be a poly-(torsion-free-abelian)
group (abbreviated PTFA). Then ${\mathbb Z}\Gamma$ is an Ore Domain and thus
embeds in its classical right ring of quotients $\mathcal K_\Gamma$, which is
a skew field \cite[Proposition 2.5]{COT1}. The skew field $\mathcal K_\Gamma$
is a ${\mathbb Z}\Gamma$-bimodule and has useful properties. In particular,
$\mathcal K_\Gamma$ is flat as a left ${\mathbb Z}\Gamma$-module (see \cite[Proposition
II.3.5]{Ste}), and every module over $\mathcal K_\Gamma$ is a free module
with a well defined rank over $\mathcal K_\Gamma$. The rank of any
${\mathbb Z}\Gamma$-module $M$ is then defined to be the $\mathcal K_\Gamma$-rank of
$M\otimes_{\mathbb{Z}\Gamma} \mathcal K_\Gamma$. Now we investigate $H_0$, $H_1$,
and $H_2$ of an $n$-cylinder $W$ and its boundary with
coefficients in ${\mathbb Z}\Gamma$ or $\mathcal K_\Gamma$. Throughout this section,
$\Gamma$ denotes a PTFA group and $\mathcal K_\Gamma$ its (skew) quotient field
of fractions. The following two basic propositions are due to
\cite{COT1}.
\begin{prop}\cite[Proposition 2.9]{COT1}
\label{prop:H_0} Suppose $X$ is a $CW$-complex and there is a
homomorphism $\phi : \ensuremath{\pi_1}(X) \longrightarrow \Gamma$. Suppose $\psi : {\mathbb Z}\Gamma \to
\mathcal R$ defines $\mathcal R$ as a ${\mathbb Z}\Gamma$-bimodule and some element of the
augmentation ideal of ${\mathbb Z}[\ensuremath{\pi_1}(X)]$ is invertible in $\mathcal R$. Then
$H_0(X;\mathcal R)=0$. In particular, if $\phi : \ensuremath{\pi_1}(X) \to \Gamma$ is a
nontrivial coefficient system, then $H_0(X;\mathcal K_\Gamma) = 0$.
\end{prop}
\begin{prop}\cite[Proposition 2.11]{COT1}
\label{prop:H_1} Suppose $X$ is a $CW$-complex such that $\ensuremath{\pi_1}(X)$
is finitely generated, and $\phi : \ensuremath{\pi_1}(X) \longrightarrow \Gamma$ is a nontrivial
coefficient system. Then
$$
\operatorname{rk}_{\mathcal K_\Gamma} H_1(X;\mathcal K_\Gamma)\le \beta_1(X)-1.
$$
In particular, if $\beta_1(X) = 1$, then $H_1(X;\mathcal K_\Gamma) = 0$. That
is, $H_1(X;{\mathbb Z}\Gamma)$ is a ${\mathbb Z}\Gamma$-torsion module.
\end{prop}
Since a rational $n$-cylinder $W$ has $\beta_1(W)=1$ we have:
\begin{cor}
\label{cor:homology} Suppose $W$ is a rational $n$-cylinder with
$\partial W = \coprod^\ell_{i=1}M_i$ where each $M_i$ is a
connected component of $\partial W$. If $\phi : \ensuremath{\pi_1}(W) \longrightarrow \Gamma$ is
a nontrivial coefficient system, then $H_0(W;\mathcal K_\Gamma) =
H_1(W;\mathcal K_\Gamma) = 0$ and $H_0(M_i;\mathcal K_\Gamma) = H_1(M_i;\mathcal K_\Gamma) = 0$ for
all $i$. Moreover, $H_2(M_i;\mathcal K_\Gamma)=0$ for all $i$.
\end{cor}
\begin{proof} A short technical argument shows that the restriction
of the coefficient system to each $\pi_1(M_i)$ is non-trivial (see
the proof of \cite[Proposition 2.11]{COT1}). The proof for $H_0$
and $H_1$ follows from the previous two propositions and the
definition of a rational $n$-cylinder. Notice that
$H_2(M_i;\mathcal K_\Gamma) \cong H^1(M_i;\mathcal K_\Gamma) \cong H_1(M_i;\mathcal K_\Gamma) = 0$
by Poincar$\acute{\text{e}}$ duality and the universal coefficient
theorem.
\end{proof}
\noindent The following proposition about $H_2$ plays an essential
role in showing that von Neumann $\rho$-invariants obstruct
$n$-solvequivalence. It extends Proposition 4.~3 of
\cite{COT1}. For the case that $W$ has one
boundary component, refer
to Proposition 4.~3 in \cite{COT1}. Let $I_{\mathbb Q} \equiv \operatorname{Image}
\{\operatorname{inc}_* : H_2(\partial W;{\mathbb Q}) \to H_2(W;{\mathbb Q})\}$.
\begin{prop}
\label{prop:H_2} Suppose $W$ is a compact, connected, oriented
$4$-manifold with $\partial W = M_1 \coprod M$ where $M =
\coprod^\ell_{i=2}M_i$ and $M_i$ are connected, $i=1,2,\ldots ,
\ell$ $(\ell\ge 2)$, and, for all $i$, $H_1(M_i;\mathbb{Q})\cong
H_1(W;\mathbb{Q})\cong\mathbb{Q}$, induced by inclusion. Suppose
$\phi : \ensuremath{\pi_1}(W) \longrightarrow \Gamma$ is a nontrivial (PTFA) coefficient system.
Then
$$
\operatorname{rk}_{\mathcal K_\Gamma}\!H_2(W;\mathcal K_\Gamma) = \operatorname{rk}_{\mathbb Q} (H_2(W;{\mathbb Q})/I_{\mathbb Q}) =
(\operatorname{rk}_{\mathbb Q} H_2(W;{\mathbb Q})) - (\ell -1).
$$
Moreover, suppose there are 2-dimensional surfaces $S_j$ and
continuous maps $f_j : S_j \longrightarrow W$ $(j\in J)$ that are lifted to
$\tilde{f}_j \longrightarrow W_\Gamma$ where $W_\Gamma$ is a regular $\Gamma$-cover
associated to $\phi$. If $\{[f_j]\,|\,j\in J\}$ is linearly
independent in $H_2(W,M;{\mathbb Q})$, then $\{[\tilde{f}_j]\,|\,j\in
J\}$ is ${\mathbb Q}\Gamma$-linearly independent in $H_2(W,M;{\mathbb Q}\Gamma)$.
\end{prop}
\begin{proof}
Since $W$ has nontrivial boundary, we can choose a 3-dimensional
$CW$-complex structure for $(W,\partial W)$. Let $C_*(W)$ be a
cellular chain complex associated to a chosen $CW$-complex
structure with coefficients in ${\mathbb Q}$. Let $C_*(W_\Gamma)$ be the
corresponding free ${\SK_\G}$ chain complex of $W_\Gamma$ that is freely
generated on cells of $W$. Let $b_i
\equiv \operatorname{rk}_{\SK_\G}\! H_i(W;{\SK_\G})$ and $\beta_i \equiv \operatorname{rk}_{\mathbb Q}
H_i(W;{\mathbb Q})$ for $1\le i \le 4$. In this proof all chain complexes
and homology groups are with coefficients in ${\mathbb Q}$ unless
specified otherwise.
By Corollary~\ref{cor:homology}, $b_0 = b_1 = 0$. By Poincar$\acute{\text{e}}$
duality and the universal coefficient theorem, $H_3(W;{\SK_\G}) \cong
H^1(W,\partial W;{\SK_\G}) \cong H_1(W,\partial W;{\SK_\G})$. Using the long
exact sequence for the pair $(W,\partial W)$ with coefficients in
${\SK_\G}$, Proposition \ref{prop:H_0} and Proposition \ref{prop:H_1},
one sees that $H_1(W,\partial W;{\SK_\G})= 0$. Here we also need that
the coefficient system restricted to each $M_i$ is non-trivial as
mentioned previously (see proof of \cite[Proposition 2.11]{COT1}).
Hence $b_3 = 0$.
On the other hand, $H_3(W) \cong H^1(W,\partial W) \cong
H_1(W,\partial W)$. In the following long exact sequence
$$
\cdots \to H_1(\partial W) \xrightarrow{f} H_1(W) \to
H_1(W,\partial W) \to H_0(\partial W) \xrightarrow{g} H_0(W) \to
H_0(W,\partial W) \to 0,
$$
the maps $f$ and $g$ are surjective. Therefore $H_1(W,\partial W)
\cong {\mathbb Q}^{\ell-1}$ and $\beta_3 = \ell -1$. It is clear that $\beta_0
= \beta_1 = 1$. Since the Euler characteristics of $C_*(W)$ and
$C_*(W_\Gamma)$ are equal, we deduce that $b_2 = \beta_2 - (\ell -1)$.
For $\operatorname{rk}_{\mathbb Q} (H_2(W)/I_{\mathbb Q})$, use the following long exact
sequence :
$$
H_2(\partial W) \to H_2(W) \to H_2(W,\partial W) \to H_1(\partial
W) \xrightarrow{f} H_1(W).
$$
Since $f$ is a surjection from ${\mathbb Q}^\ell$ to ${\mathbb Q}$, we have an
exact sequence as follows.
$$
0 \to H_2(W)/I_{\mathbb Q} \to H_2(W,\partial W) \to {\mathbb Q}^{\ell -1} \to
0.
$$
Thus $\operatorname{rk}_{\mathbb Q} (H_2(W)/I_{\mathbb Q}) = \beta_2 - (\ell -1)$. This completes
the first part of the proof.
For the second part, let $X$ be the one point union of $S_j$ using
some base paths. Let $f : X \longrightarrow W$ and $\tilde{f} : X \longrightarrow W_\Gamma$
be maps which restrict to $f_j$ and $\tilde{f}_j$, respectively.
By taking mapping cylinders, we may think of $X$ as a
(2-dimensional) subcomplex of $W$ and $C_*(X)$ as a subcomplex of
$C_*(W)$. If we denote by $X_\Gamma$ the regular $\Gamma$-cover associated
to $\phi\circ f_*$ (which can be thought of as a subcomplex of
$W_\Gamma$), then $C_*(X_\Gamma)$ is a subcomplex of $C_*(W_\Gamma)$. In fact,
by our hypothesis $f_j$ lifts to $\tilde{f}_j$, and this implies
that $\phi\circ f_*$ is trivial on $\ensuremath{\pi_1}(X)$. Hence $X_\Gamma$ is a
trivial cover which consists of $\Gamma$ copies of $X$. Consider the
following commutative diagram where each row is exact.
$$
\begin{diagram}\dgARROWLENGTH=1.5em
\node{H_3(W,M;{\mathbb Q}\Gamma)} \arrow{e}
\arrow{s} \node{H_3(W,M\cup X;{\mathbb Q}\Gamma)}
\arrow{e,t}{\tilde{\partial}} \arrow{s} \node{H_2(X;{\mathbb Q}\Gamma)}
\arrow{e,t}{\tilde{f}_*} \arrow{s}
\node{H_2(W,M;{\mathbb Q}\Gamma)}\arrow{s}\\
\node{H_3(W,M)} \arrow{e}
\node{H_3(W,M\cup X)}
\arrow{e,t}{\partial} \node{H_2(X)}
\arrow{e,t}{f_*}
\node{H_2(W,M)}
\end{diagram}
$$
Since $X_\Gamma$ is a trivial cover, $H_2(X;{\mathbb Q}\Gamma)$ is a free
${\mathbb Q}\Gamma$-module on $\{[S_j]\}$. Thus to complete the proof, we
need to show $\tilde{f}_*$ is injective. Since $H_3(W,M) \cong
H^1(W,M_1) \cong H_1(W,M_1) = 0$ and $C_*(W,M)$ is a 3-dimensional
chain complex, $\partial_\# : C_3(W,M) \longrightarrow C_2(W,M)$ is
injective. Notice that $C_*(W,M)$ can be identified with
$C_*(W_\Gamma,M_\Gamma)\otimes_{{\mathbb Q}\Gamma}{\mathbb Q}$ (similarly for $C_*(X)$ and
$C_*(W,M\cup X)$). Then by \cite[pg.305]{Str} (or see
\cite[Proposition 2.4]{COT1}), $\partial_\# : C_3(W_\Gamma,M_\Gamma) \longrightarrow
C_2(W_\Gamma,M_\Gamma)$ is also injective. Hence we obtain
$H_3(W,M;{\mathbb Q}\Gamma)=0$. Now it suffices to show $H_3(W,M\cup X
;{\mathbb Q}\Gamma)=0$. Since $f_*$ is injective by our hypothesis,
$H_3(W,M\cup X) = 0$. Since $C_*(W,M\cup X)$ is a 3-dimensional
chain complex, this implies that $\partial_\# : C_3(W,M\cup X)
\longrightarrow C_2(W,M\cup X)$ is injective. Once again by
\cite[pg.305]{Str}, $\partial_\# : C_3(W_\Gamma,M_\Gamma\cup X_\Gamma) \longrightarrow
C_2(W_\Gamma,M_\Gamma\cup X_\Gamma)$ is injective. Therefore $H_3(W,M\cup
X;{\mathbb Q}\Gamma) = 0$.
\end{proof}
We now investigate the relationship between the first homology
groups of an $n$-cylinder and its boundary components. The
following lemma generalizes \cite[Lemma 4.5]{COT1}. It is the
linchpin in proving Theorem~\ref{thm:rank}, which establishes the
crucial connection between $n$-solvequivalence and homology.
\begin{lem}
\label{lem:exact} Let $W$ be a rational $n$-cylinder with $M$ as
one of it boundary components. Let $\Gamma$ be an $(n-1)$-solvable
(PTFA) group and $\mathcal R$ be a ring such that ${\mathbb Q}\Gamma \subset \mathcal R
\subset {\SK_\G}$. Suppose $\phi : \ensuremath{\pi_1}(M) \longrightarrow \Gamma$ is a nontrivial
coefficient system that extends to $\psi : \ensuremath{\pi_1}(W) \longrightarrow \Gamma$. Then
$$
TH_2(W,M;\mathcal R) \xrightarrow{\partial} H_1(M;\mathcal R) \xrightarrow{i_*}
H_1(W;\mathcal R)
$$
is exact. (Here for an $\mathcal R$-module $\mathcal M$, $T\mathcal M$ denotes the
$\mathcal R$-torsion submodule of $\mathcal M$.)
\end{lem}
\begin{proof}
We need to show that every element of $\operatorname{Ker} (i_*)$ is in the image of an
element of $TH_2(W,M;\mathcal R)$. Let $m =
\frac12 \operatorname{rk}_{\mathbb Q}(H_2(W;{\mathbb Q})/I_{\mathbb Q})$ where $I_{\mathbb Q} \equiv \operatorname{Image}
\{\operatorname{inc}_* : H_2(\partial W;{\mathbb Q}) \to H_2(W;{\mathbb Q})\}$. By
Proposition~\ref{prop:H_2}, $\operatorname{rk}_{\SK_\G} H_2(W;\mathcal R) = 2m$. Let
$\{\ell_1,\ell_2,\ldots, \ell_m\}$ generate a rational
$n$-Lagrangian for $W$ and $\{d_1,d_2,\ldots, d_m\}$ be its
$n$-duals. Since $\Gamma$ is $(n-1)$-solvable, $\psi$ descends to
$\psi' : \ensuremath{\pi_1}(W)/\ensuremath{\pi_1}(W)^{(n)} \longrightarrow \Gamma$. We denote by $\ell_i'$ and
$d_i'$ the images of $\ell_i$ and $d_i$ in $H_2(W;\mathcal R)$. By
naturality of intersection forms, the intersection form $\lambda$
defined on $H_2(W;\mathcal R)$ vanishes on the module generated by
$\{\ell_1',\ell_2',\ldots, \ell_m'\}$. Let $\mathcal R^m\oplus \mathcal R^m$ be the
free module on $\{\ell_i',d_i'\}$.
The following composition
$$
\mathcal R^m\oplus \mathcal R^m \xrightarrow{j_*} H_2(W;\mathcal R) \xrightarrow{\lambda}
H_2(W;\mathcal R)^* \xrightarrow{j^*} (\mathcal R^m\oplus \mathcal R^m)^*
$$
is represented by a block matrix
$$
\left(\begin{matrix} 0 & I\cr I & X
\end{matrix}\right).
$$
This matrix has an inverse which is
$$
\left(\begin{matrix} -X & I\cr I & 0
\end{matrix}\right).
$$
Thus the composition is an isomorphism. This implies that $j_*$ is
a monomorphism and $j^*$ is a (split) epimorphism. Since $j^*$ is
a split epimorphism between the free $\mathcal R$-modules of the same
${\SK_\G}$-rank, and $\mathcal R$ is an integral domain, $j^*$ is an
isomorphism. Hence $\lambda$ is a surjection. Now consider the
following diagram where the row is the exact sequence of the pair
$(W,M)$.
$$
\begin{diagram}\dgARROWLENGTH=1.2em
\node[4]{H_2(\partial W,M;\mathcal R)} \arrow{s,r}{f_*}\\
\node{H_2(W;\mathcal R)} \arrow[3]{e,t}{\pi_*} \arrow[3]{se,r}{\lambda}
\node[3]{H_2(W,M;\mathcal R)}
\arrow[3]{e,t}{\partial} \arrow{s,r}{g_*} \node[3]{H_1(M;\mathcal R)}
\arrow[3]{e,t}{i_*} \node[3]{H_1(W;\mathcal R)}\\
\node[4]{H_2(W,\partial W;\mathcal R)} \arrow{s,r}{\operatorname{PD}}\\
\node[4]{H^2(W;\mathcal R)} \arrow{s,r}{\kappa}\\
\node[4]{H_2(W;\mathcal R)^*}.
\end{diagram}
$$
We claim that $\operatorname{Ker} (\kappa\circ \operatorname{PD} \circ g_*)$ is $\mathcal R$-torsion.
$\operatorname{Ker}(\kappa)$ is $\mathcal R$-torsion since it is a split surjection between
$\mathcal R$-modules of the same rank over ${\SK_\G}$. $\operatorname{PD}$ is an isomorphism
by Poincar$\acute{\text{e}}$ duality. $\operatorname{Ker}(g_*) = \operatorname{Image} (f_*)$ and
$H_2(\partial W, M;\mathcal R) = H_2(M';\mathcal R)$ where $M'$ is the disjoint
union of the boundary components of $W$ except for $M$. Since
$H_2(M';\mathcal R)$ is $\mathcal R$-torsion by the flatness of ${\SK_\G}$ over $\mathcal R$
and Corollary~\ref{cor:homology} (once again we need to know that
the restricted coefficient system is non-trivial), $\operatorname{Ker} (g_*)$ is
$\mathcal R$-torsion. Combining these, one can deduce that $\operatorname{Ker} (\kappa\circ
\operatorname{PD} \circ g_*)$ is $\mathcal R$-torsion.
Suppose $p\in \operatorname{Ker}(i_*) \subset H_1(M;\mathcal R)$. Then there exists $x
\in H_2(W,M;\mathcal R)$ such that $\partial(x) = p$. Let $y\in
\lambda^{-1}((\kappa\circ\operatorname{PD}\circ g_*)(x))$. Then $x-\pi_*(y) \in \operatorname{Ker}(\kappa
\circ \operatorname{PD} \circ g_*)$. Hence $x-\pi_*(y)$ is $\mathcal R$-torsion and
$\partial(x-\pi_*(y)) = \partial(x) = p$.
\end{proof}
Under the same hypotheses as in Lemma~\ref{lem:exact}, there
exists a non-singular linking form $\mathcal B\ell : H_1(M;\mathcal R) \longrightarrow
H_1(M;\mathcal R)^\# \equiv \overline{\operatorname{Hom}_\mathcal R(H_1(M;\mathcal R), {\SK_\G}/\mathcal R)}$ by
\cite[Theorem 2.13]{COT1}. This definition, and a proof of
nonsingularity will be included in our proof of
Proposition\ref{prop:P and P perb}. For an $\mathcal R$-submodule $P$ of
$H_1(M;\mathcal R)$, we define $P^\bot \equiv \{x\in H_1(M;\mathcal R)\,\,|\,\,
\mathcal B\ell(x)(y) = 0, \forall y \in P\}$, clearly an $\mathcal R$-submodule.
\begin{prop}
\label{prop:P and P perb} Suppose the same hypotheses as in
Lemma~\ref{lem:exact}. Suppose $\mathcal R$ is a PID. Then for $P\equiv
\operatorname{Ker} \{i_* : H_1(M;\mathcal R) \longrightarrow H_1(W;\mathcal R)\}$, $P\subset P^\bot$.
Moreover, if $M = \partial W$, then $ P = P^\bot$.
\end{prop}
\begin{proof}
Consider the following commutative diagram.
$$
\begin{diagram}\dgARROWLENGTH=1.2em
\node{TH_2(W,M;\mathcal R)} \arrow{e,t}{\partial} \arrow{s,r}{g_*}
\node{H_1(M;\mathcal R)} \arrow{e,t}{i_*} \arrow[2]{s,r}{\operatorname{PD}}
\node{H_1(W;\mathcal R)}\\
\node{TH_2(W,\partial W;\mathcal R)} \arrow{s,r}{\operatorname{PD}}\\
\node{TH^2(W;\mathcal R)} \arrow{e,t}{i^*} \arrow{s,r}{B^{-1}}
\node{H^2(M;\mathcal R)} \arrow{s,r}{B^{-1}}\\
\node{H^1(W;{\SK_\G}/\mathcal R)} \arrow{e,t}{j^*} \arrow{s,r}{\kappa}
\node{H^1(M;{\SK_\G}/\mathcal R)} \arrow{s,r}{\kappa}\\
\node{H_1(W;\mathcal R)^\#} \arrow{e,t}{i^\#} \node{H_1(M;\mathcal R)^\#}
\end{diagram}
$$
Let $\b_\text{rel} : TH_2(W,M;\mathcal R) \longrightarrow H_1(W;\mathcal R)^\#$ be the composition of
the maps on the left column and $B\ell : H_1(M;\mathcal R) \longrightarrow
H_1(M;\mathcal R)^\#$ the composition of the maps on the right column.
$\operatorname{PD}$ are induced by Poincar$\acute{\text{e}}$ duality (hence isomorphisms),
and $\kappa$ are the Kronecker evaluation maps. $B^{-1}$ are the
inverses of the Bockstein homomorphisms. For the existence of
$B^{-1}$, we need to show that the Bockstein homomorphisms are
isomorphisms. First, we consider the following exact sequence.
$$
H^1(W;{\SK_\G}) \to H^1(W;{\SK_\G}/\mathcal R) \xrightarrow{B} H^2(W;\mathcal R) \to
H^2(W;{\SK_\G}).
$$
Notice that $H^1(W;{\SK_\G}) \cong \operatorname{Hom}_{\SK_\G}(H_1(W;{\SK_\G}),{\SK_\G}) = 0$ by
Corollary~\ref{cor:homology}. Since $H^2(W;{\SK_\G})$ is $\mathcal R$-torsion
free and $H^1(W;{\SK_\G}/\mathcal R)$ is $\mathcal R$-torsion, one sees that $B$ is
an isomorphism onto $TH^2(W;\mathcal R)$. Secondly, one can prove that
$H^1(M;{\SK_\G}) = H^2(M;{\SK_\G}) = 0$ using Corollary~\ref{cor:homology}.
Hence $B : H^1(M;{\SK_\G}/\mathcal R) \longrightarrow H^2(M;\mathcal R)$ is an isomorphism in
the following long exact sequence.
$$
H^1(M;{\SK_\G}) \to H^1(M;{\SK_\G}/\mathcal R) \xrightarrow{B} H^2(M;\mathcal R) \to
H^2(M;{\SK_\G}).
$$
If $x\in P$, then $x = \partial(y)$ for some $y \in TH_2(W,M;\mathcal R)$
by Lemma~\ref{lem:exact}. Thus $B\ell(x) = (i^\#\circ\b_\text{rel})(y)$. For
every $p\in P$, $B\ell(x)(p) = (i^\#\b_\text{rel})(y)(p) = \b_\text{rel}(y)(i_*(p)) =
\b_\text{rel}(y)(0) = 0$. Therefore $x\in P^\bot$, and $P\subset P^\bot$.
The proof for the case when $M = \partial W$ follows from
\cite[Theorem 4.4]{COT1}.
\end{proof}
\begin{rem}
\label{rem:rank} If $W$ has more than one boundary component, then
in general $P \ne P^\bot$. For example, if $W= M\times [0,1]$, $P
= 0$ and $P^\bot = H_1(M;\mathcal R)$.
\end{rem}
Suppose $\Gamma$ is a PTFA group with $H_1(\Gamma) \cong \Gamma/[\Gamma,\Gamma] \cong
{\mathbb Z}$. Then its commutator subgroup, $[\Gamma,\Gamma]$, is also PTFA and
${\mathbb Z}[\Gamma,\Gamma]$ embeds into its (skew) quotient field of fractions
denoted by ${\mathbb K}$. Then we have a PID ${\mathbb K}[t^{\pm 1}]$ such that
${\mathbb Z}\Gamma \subset {\mathbb K}[t^{\pm 1}] \subset {\SK_\G}$ where $t$ is
identified with the generator of $\Gamma/[\Gamma,\Gamma]$.
The following generalizes \cite[Theorem 6.4]{CT}. The proof is the
same once equipped with Proposition~\ref{prop:P and P perb}.
\begin{thm}
\label{thm:rank} Let $M$ be zero surgery on a knot $K$ in $S^3$.
Let $W$ be an $n$-cylinder with $M$ as one of its boundary
components. Suppose $\Gamma$ is an $(n-1)$-solvable PTFA group with
$H_1(\Gamma) \cong {\mathbb Z}$. Suppose $\phi : \ensuremath{\pi_1}(W) \longrightarrow \Gamma$ induces an
isomorphism upon abelianization. Let $d \equiv \operatorname{rk}_{\mathbb Q}
H_1(M_\infty;{\mathbb Q})$ where $M_\infty$ is the (universal) infinite
cyclic cover of $M$. Then
$$
\operatorname{rk}_{\mathbb K} \operatorname{Image}\{i_* : H_1(M;{\mathbb K}[t^{\pm 1}]) \longrightarrow
H_1(W;{\mathbb K}[t^{\pm 1}])\} \ge (d-2)/2 \phantom{a} \text{if}
\phantom{a} n>1
$$
and this rank is at least $d/2$ if $n=1$.
\end{thm}
\begin{proof}
Let $\mathcal R \equiv {\mathbb K}[t^{\pm 1}]$. By \cite[Theorem 2.13]{COT1},
there exists a non-singular Blanchfield linking form $B\ell :
H_1(M;\mathcal R) \longrightarrow H_1(M;\mathcal R)^\# \equiv
\overline{\operatorname{Hom}_\mathcal R(H_1(M;\mathcal R),{\SK_\G}/\mathcal R)}$. Let $P\equiv \operatorname{Ker} \{i_*
: H_1(M;\mathcal R) \longrightarrow H_1(W;\mathcal R)\}$, $Q \equiv \operatorname{Image}\{i_* :
H_1(M;\mathcal R) \longrightarrow H_1(W;\mathcal R)\}$, and $\mathcal A \equiv H_1(M;\mathcal R)$. By
Proposition~\ref{prop:P and P perb}, $P\subset P^\bot$ with
respect to the Blanchfield linking form. This gives us a
well-defined map $f : P \longrightarrow (\mathcal A/P)^\#$ induced from the
Blanchfield linking form. Since the Blanchfield linking form is
non-singular, $f$ is a monomorphism. Hence $\operatorname{rk}_{\mathbb K} P \le
\operatorname{rk}_{\mathbb K} (\mathcal A/P)^\#$.
We claim that $\operatorname{rk}_{\mathbb K} \mathcal M = \operatorname{rk}_{\mathbb K} (\mathcal M)^\#$ for every
finitely generated $\mathcal R$-module $\mathcal M$. (Here $(\mathcal M)^\# \equiv
\overline{\operatorname{Hom}_\mathcal R(\mathcal M,{\SK_\G}/\mathcal R)}$.) Since $\mathcal R$ is a PID, it is
enough to show this for the case when $\mathcal M$ is a cyclic
$\mathcal R$-module, i.~e.~, $\mathcal M = \mathcal R/p(t)$ where $p(t) \in \mathcal R$. For
this cyclic case, $(\mathcal R/p(t))^\# \cong \mathcal R/\overline{p}(t)$ where
$\overline{p}(t)$ is obtained by taking involution of $p(t)$.
Since $\operatorname{rk}_{\mathbb K} \mathcal R/p(t)$ is the degree of $p(t)$ for every $p(t)
\in \mathcal R$ and $p(t)$ and $\overline{p}(t)$ have the same degree,
$\operatorname{rk}_{\mathbb K} \mathcal R/p(t) = \operatorname{rk}_{\mathbb K} \mathcal R/\overline{p}(t)$. Using this we
can deduce that
$$
\operatorname{rk}_{\mathbb K} P \le \operatorname{rk}_{\mathbb K} (\mathcal A/P)^\# = \operatorname{rk}_{\mathbb K} \mathcal A/P = \operatorname{rk}_{\mathbb K} \mathcal A
- \operatorname{rk}_{\mathbb K} P
$$
Hence $\operatorname{rk}_{\mathbb K} P \le \frac12 \operatorname{rk}_{\mathbb K}\mathcal A$. Since $Q \cong \mathcal A/P$
as $\mathcal R$-modules, we have
$$
\operatorname{rk}_{\mathbb K} Q = \operatorname{rk}_{\mathbb K} \mathcal A - \operatorname{rk}_{\mathbb K} P \ge
\operatorname{rk}_{\mathbb K} \mathcal A - \frac12 \operatorname{rk}_{\mathbb K} \mathcal A = \frac12 \operatorname{rk}_{\mathbb K} \mathcal A.
$$
Thus we only need to show $\operatorname{rk}_{\mathbb K} \mathcal A \ge d-2$ if $n>1$ and
$\operatorname{rk}_{\mathbb K} \mathcal A = d$ if $n=1$.
If $n=1$, $\Gamma \cong {\mathbb Z}$, ${\mathbb K} = {\mathbb Q}$, and $\mathcal R = {\mathbb Q}[t^{\pm
1}]$. In this case $\mathcal A$ is the rational Alexander module, which
is $H_1(M_\infty;{\mathbb Q})$. Hence $\operatorname{rk}_{\mathbb K} \mathcal A = d$. Suppose $n>1$.
Observe that $\mathcal A$ is obtained from $H_1(\ensuremath{S^3\backslash K};\mathcal R)$ by killing the
$\mathcal R$-submodule generated by the longitude, say $\ell$. By
\cite[Corollary 4.8]{C}, $\operatorname{rk}_{\mathbb K} H_1(\ensuremath{S^3\backslash K};\mathcal R) \ge d-1$. Since
$(t-1)_*\ell = 0$ in $H_1(\ensuremath{S^3\backslash K};\mathcal R)$, the submodule generated by
$\ell$ is isomorphic with ${\mathbb K}[t^{\pm 1}]/(t-1) \cong {\mathbb K}$.
Hence $\operatorname{rk}_{\mathbb K} \mathcal A \ge d-2$.
\end{proof}
\section{Obstructions to $n$-solvequivalence}
\label{sec:obstruction} We use the information about twisted
homology groups obtained in Section~\ref{sec:homology} to get
obstructions to $n$-solvequivalence. Our obstructions will be
vanishing of von Neumann $\rho$-invariants. The von Neumann
$\rho$-invariants were firstly used by Cochran-Orr-Teichner to
give obstructions to $n$-solvability (see Theorem $4.2$ and $4.6$
in \cite{COT1}). We begin by giving a very brief explanation about
von Neumann $\rho$-invariants. For more details on von Neumann
$\rho$-invariants, the readers are referred to Section 5 in
\cite{COT1} and Section 2 in \cite{CT}. Let $M$ be a compact,
oriented 3-manifold. If we have a representation $\phi : \ensuremath{\pi_1}(M)
\longrightarrow \Gamma$ for a group $\Gamma$, then the \emph{von Neumann
$\rho$-invariant} or \emph{reduced $L^{(2)}$-signature}
$\rho(M,\phi)$ ($\in {\mathbb R}$) is defined. If $(M,\phi) = \partial
(W,\psi)$ for some compact, oriented 4-manifold $W$ and a
homomorphism $\psi : \ensuremath{\pi_1}(W) \longrightarrow \Gamma$, then we have $\rho(M,\phi) =
\sigma^{(2)}_\Gamma(W,\psi) - \sigma_0(W)$ where $\sigma^{(2)}_\Gamma(W,\psi)$ is the
$L^{(2)}$-signature of the intersection form defined on
$H_2(W;{\mathbb Z}\Gamma)$ twisted by $\psi$ and $\sigma_0(W)$ is the ordinary
signature of $W$. If $\Gamma$ is a PTFA group, then as we have seen
before $\Gamma$ embeds into the (skew) quotient field of fractions
${\SK_\G}$ and $\sigma^{(2)}_\Gamma$ can be thought of as a homomorphism from
$L^0({\SK_\G})$ to ${\mathbb R}$. Some useful properties of von Neumann
$\rho$-invariants due to COT are given below. One can find
detailed proofs or explanations in \cite[Section 5]{COT1}.
\begin{prop}
\label{prop:rho invariants}Let $M$ be as above and $\Gamma$ a PTFA
group. Suppose we have a homomorphism $\phi : \ensuremath{\pi_1}(M) \longrightarrow \Gamma$.
\begin{itemize}
\item [(1)] If $(M,\phi) = \partial (W,\psi)$ for some compact,
spin 4-manifold $W$ and $H_2(W;{\SK_\G})$ has a half-rank summand on
which the (equivariant) intersection form vanishes, then
$\rho(M,\phi) = 0$. In fact, $\sigma^{(2)}_\Gamma(W,\psi) = \sigma_0(W) = 0$.
\item [(2)] If $\phi$ factors through $\phi' : \ensuremath{\pi_1}(M) \longrightarrow \Gamma'$ where
$\Gamma'$ is a subgroup of $\Gamma$, then $\rho(M,\phi') = \rho(M,\phi)$.
\item [(3)] If $\phi$ is trivial, then $\rho(M,\phi) = 0$.
\item [(4)] If $\Gamma={\mathbb Z}$ and $\phi$ extends over $W$ nontrivially
for some compact, spin 4-manifold $W$, then $\rho(M,\phi) =
\int_{\omega \in S^1} \sigma(h(\omega)) \mathrm{d}\omega - \sigma_0(W)$ where $h$
is the matrix representing the intersection form on
$H_2(W;{\mathbb C}[t,t^{-1}])/\text{(torsion)}$.
\end{itemize}
\end{prop}
\noindent Now we are ready to give obstructions to
$n$-solvequivalence. In fact, we give a more general theorem.
\begin{thm}
\label{thm:rho=0} Let $M_i$ ($i=1,2,\ldots ,\ell)$ be closed,
connected, oriented 3-manifolds with $H_1(M_i;{\mathbb Q}) \cong {\mathbb Q}$
for all $i$. Let $\Gamma$ be an $n$-solvable group. Suppose there
exists a rational $(n.5)$-cylinder $W$ with $\partial W =
\coprod^\ell_{i=1} M_i$ (of arbitrary multiplicity) and we have
representations $\phi_i : \ensuremath{\pi_1}(M_i) \longrightarrow \Gamma$. If $\phi_i$ extends to
the same homomorphism $\phi : \ensuremath{\pi_1}(W) \longrightarrow \Gamma$ for all $i$, then
$$
\sum^\ell_{i=1}\rho(M_i,\phi_i) = 0.
$$
\end{thm}
\noindent The following is an easy corollary of
Theorem~\ref{thm:rho=0}, hence the proof is omitted.
\begin{cor}
\label{cor:rho=0} Suppose $K_0$ is $(n.5)$-solvequivalent to $K_1$
via $W$ and $\Gamma$ is an $n$-solvable group. Suppose we have
representations $\phi_i : \ensuremath{\pi_1}(M_i) \longrightarrow \Gamma$ for $i=0,1$ where $M_i$
is zero surgery on $S^3$ along $K_i$, $i=0,1$. If $\phi_i$ extends
to the same homomorphism $\phi : \ensuremath{\pi_1}(W) \longrightarrow \Gamma$ for $i=1,2$, then
$$
\rho(M_0,\phi_0) = \rho(M_1,\phi_1).
$$
\end{cor}
\begin{proof}[Proof of Theorem~\ref{thm:rho=0}]
Since $W$ is spin, $\sigma_0(W) = 0$. Since
$\sum^\ell_{i=1}\rho(M_i,\phi_i) = \sigma^{(2)}_\Gamma(W,\phi) -\sigma_0(W)$,
we need to show $\sigma^{(2)}_\Gamma(W,\phi)=0$. Let $\operatorname{rk}_{\mathbb Q}
(H_2(W;{\mathbb Q})/I_{\mathbb Q}) = 2m$ where $I_{\mathbb Q} \equiv \operatorname{Image}\{\operatorname{inc}_* :
H_2(\partial W;{\mathbb Q}) \longrightarrow H_2(W;{\mathbb Q})\}$. Let $L$ be a rational
$(n+1)$-Lagrangian for $W$ generated by $(n+1)$-surfaces
$\{\ell_1,\ell_2,\ldots ,\ell_m\}$. Since $\Gamma$ is $n$-solvable,
$\phi$ factors through $\ensuremath{\pi_1}(W)/\ensuremath{\pi_1}(W)^{(n+1)}$. Thus if we let ${\SK_\G}$
denote the (skew) quotient field of fractions of ${\mathbb Q}\Gamma$, then we
can take the image of $L$ in $H_2(W;{\SK_\G})$, which is denoted by
$L'$ (generated by $\{\ell'_1,\ell'_2,\ldots ,\ell'_m\}$). By
naturality of intersection form, the intersection form $\lambda' :
H_2(W;{\SK_\G})\times H_2(W;{\SK_\G}) \to {\SK_\G}$ vanishes on $L'$. By
Proposition~\ref{prop:H_2}, $\operatorname{rk}_{\SK_\G}\! H_2(W;{\SK_\G}) = 2m$. Therefore
if we show that $\{\ell'_1,\ell'_2,\ldots ,\ell'_m\}$ is linearly
independent in $H_2(W;{\SK_\G})$, then $\operatorname{rk}_{\SK_\G} L' = m = \frac12\operatorname{rk}_{\SK_\G}
H_2(W;{\SK_\G})$, which implies that $\sigma^{(2)}_\Gamma(W,\phi) = 0$ by
Proposition~\ref{prop:rho invariants}(1).
For convenience we let $M\equiv \coprod^\ell_{i=2} M_i$ and $S
\equiv \{\ell_1,\ell_2,\ldots ,\ell_m\}$. Since $S$ generates a
rational $(n+1)$-Lagrangian and $\#(S) = m = \frac12\operatorname{rk}_{\mathbb Q}
(H_2(W;{\mathbb Q})/I_{\mathbb Q})$, the image of $S$ in $H_2(W;{\mathbb Q})/I_{\mathbb Q}$ is
linearly independent. By investigating the long exact sequences of
homology groups of the pairs $(W,\partial W)$ and $(W,M)$, one
easily sees that $I_{\mathbb Q} = \operatorname{Image}\{\operatorname{inc}_* : H_2(M;{\mathbb Q}) \longrightarrow
H_2(W;{\mathbb Q})\}$. It follows that $H_2(W;{\mathbb Q})/I_{\mathbb Q}$ can be
identified with a ${\mathbb Q}$-subspace of $H_2(W,M;{\mathbb Q})$. Hence the
image of $S$ in $H_2(W,M;{\mathbb Q})$ is linearly independent. By the
second part of Proposition~\ref{prop:H_2}, the image of $S$ in
$H_2(W,M;{\mathbb Q}\Gamma)$ is linearly independent. Hence the image of $S$
in $H_2(W,M;{\SK_\G})$ is also linearly independent. By
Corollary~\ref{cor:homology}, $H_1(M;{\SK_\G}) = H_2(M;{\SK_\G}) = 0$. This
implies that $H_2(W;{\SK_\G}) \cong H_2(W,M;{\SK_\G})$. Therefore
$\{\ell'_1,\ell'_2,\ldots ,\ell'_m\}$ is linearly independent in
$H_2(W;{\SK_\G})$.
\end{proof}
\section{Main Theorem}
\label{sec:main} In this section we prove the main theorem, that
simultaneously generalizes the main theorems of \cite{CT} and
\cite{L}\cite{K}. The strength of our theorem lies in being able
to work within the class of knots that have the same fixed
classical Seifert matrix (and Alexander module) and the same
\emph{higher-order} analogues of these. Therefore, after stating
the theorem, we review the higher-order Alexander modules. Then we
state and prove two theorems that are used at the end of the
section to prove the main theorem. The first of these,
Theorem~\ref{thm:injectivity}, is an important technical result
that significantly generalizes \cite[Theorem 4.3]{CT}. Due to its
difficult technical nature, the proof of our
Theorem~\ref{thm:injectivity} is not completed until the next
section.
\begin{thm}[{\bf Main Theorem}]
\label{thm:main} Let $n\in{\mathbb N}$. Let $K$ be a knot whose Alexander
polynomial has degree greater than 2 (if $n=1$ then degree equal to
$2$ is allowed). Then there is an infinite family of knots $\{K_i
| i\in {\mathbb N}_0\}$ with $K_0 = K$ such that:
\begin{itemize}
\item [(1)] For each $i$, $K_i - K$ is $n$-solvable. In
particular, $K_i$ is $n$-solvequivalent to $K$. Moreover, $K_i$
and $K$ cobound, in $S^3\times\[0,1\]$, a smoothly embedded symmetric
Grope of height $n+2$.
\item [(2)] If $i\ne j$, $K_i$ is not $(n.5)$-solvequivalent to
$K_j$. In particular, $K_i-K_j$ is not $(n.5)$-solvable, and $K_i$
and $K_j$ do not cobound, in $S^3\times\[0,1\]$, any embedded
symmetric Grope of height $(n+2.5)$.
\item [(3)] Each $K_i$ has the same $m$-th integral higher-order
Alexander module as $K$, for $m=0,1,\ldots, n-1$. Indeed, if $G_i$
and $G$ denote the knot groups of $K_i$ and $K$ respectively, then
there is an isomorphism $G_i/(G_i)^{(n+1)}\to G/G^{(n+1)}$ that
preserves the peripheral structures.
\item [(4)] Each $K_i$ has the same \emph{$m^{th}$-order Seifert
presentation} as $K$,
for $m=0,1,\ldots, n-1$.
\item [(5)] For each $i>j$, $K_i - K_j$ is of infinite order in
$\mathcal F_{(n)}/\mathcal F_{(n.5)}$.
\item [(6)] If $i,j>0$, $s(K_i)=s(K_j)$ and $\tau(K_i)=\tau(K_j)$,
where $s$ is Rasmussen's smooth concordance invariant and $\tau$
is the smooth concordance invariant of P.~Ozsv{\'a}th and
Z.~Szab{\'o} \cite{Ra}\cite{OS}.
\end{itemize}
\end{thm}
\noindent Since there certainly exist knots whose Alexander
polynomial has degree $4$, from ($5$) above we recover the result
of Cochran and Teichner.
\begin{cor}
\label{cor:cot filtration} \cite{CT} For every positive integer $n$,
$\mathcal F_{(n)}/\mathcal F_{(n.5)}$ has positive rank.
\end{cor}
\begin{cor} Let $n$ be a positive integer.
\label{cor:gropefiltrations}
\begin{itemize}
\item [1.] $\mathcal G_{n+2}^{smooth}/\mathcal G_{n+2.5}^{smooth}$ has
elements of infinite order represented by knots $K$ with $s(K)=0$
and $\tau(K)=0$.
\item [2.] The kernel of the homomorphism $ST:\mathcal G_{n+2}^{smooth}\to
\mathbb{Z}\times \mathbb{Z}$ given by $ST(K)=(s(K),\tau(K))$ is
infinite.
\end{itemize}
\end{cor}
\begin{proof}[Proof of Corollary~\ref{cor:gropefiltrations}]
For fixed $n$, consider the knot $K_2-K_1$, that is $K_2\#(-K_1)$.
Note that, by part $(6)$ above, $\tau(K_2-K_1)=s(K_2-K_1)=0$. By
part $(1)$ above (applied first to $K_1$ and $K$ and then to $K_2$
and $K$), $K_1$ and $K_2$ cobound, in $S^3\times\[0,1\]$, a
smoothly embedded symmetric Grope of height $n+2$. It follows that
$K_1\#(-K_2)$ bounds a smoothly embedded symmetric Grope of height
$n+2$ in $B^4$. This is seen by choosing an embedded arc from
$K_1$ to $K_2$ in $S^3\times\[0,1\]$, contained in the first-stage
surface of the Grope and avoiding the boundaries of the
higher-stage surfaces. Deleting a tubular neighborhood of this
arc, one gets the desired Grope in $B^4$. But if some multiple of
$K_2\#(-K_1)$ were the boundary of a smoothly (or even
topologically) embedded symmetric Grope of height $n+2.5$ in
$B^4$, then by \cite[Theorem 8.11]{COT1}, it would be of finite
order in $\mathcal F_{(n)}/\mathcal F_{(n.5)}$ contradicting part $(5)$ above.
This completes the proof of the first part of the corollary.
For the second part of the corollary, just consider multiples of
$K_2-K_1$. Alternatively, note that the analysis above applies
equally well to any of the infinite set of knots $\{K_i-K_1$,
$i\geq 1\}$.
\end{proof}
\noindent The higher-order Alexander modules of a knot were
defined by the first author, and we give brief explanations here.
For more details refer to \cite{C}.
\begin{defn}\cite{C}
\label{defn:alexander module} Let $n$ be a nonnegative integer.
Let $K$ be a knot and $G \equiv \ensuremath{\pi_1}(S^3\setminus K)$. The
\emph{$n$-th (integral) higher-order Alexander module}, ${\SA^\bbz_n}(K)$,
of a knot $K$ is the first (integral) homology group of the
covering space of $S^3\setminus K$ corresponding to $G^{(n+1)}$,
considered as a right ${\mathbb Z}[G/G^{(n+1)}]$-module, i.~e.~,
$G^{(n+1)}/G^{(n+2)}$ as a right module over ${\mathbb Z}[G/G^{(n+1)}]$.
\end{defn}
\noindent Using the local coefficient system $\pi:G\to
G/G^{(n+1)}$, one can see that ${\SA^\bbz_n}(K) \cong H_1(S^3\setminus
K;{\mathbb Z}[G/G^{(n+1)}])$ as right ${\mathbb Z}[G/G^{(n+1)}]$-modules. Notice
that $\mathcal A_0^{\mathbb Z}(K)$ is the classical Alexander module of $K$. Let
$\Gamma_n \equiv G/G^{(n+1)}$. Let $\Gamma'_n$ denote the commutator
subgroup of $\Gamma_n$ (that is $G^{(1)}/G^{(n+1)})$. Note that when
$n=0$, $\Gamma'_n=\{e\}$. Let $\Sigma$ be a Seifert surface for $K$, let
$E_K \equiv \ensuremath{S^3\backslash K}$ and let $Y \equiv E_K - (\Sigma \times (-1,1))$. We
denote the two inclusions $\Sigma \to \Sigma\times\{\pm 1\} \to
\partial Y \subset Y$ by $i_+$ and $i_-$. Notice that the coefficient
system $\pi$, when restricted to $\pi_1(\Sigma)$ or $\pi_1(Y)$, has
image in $\Gamma_n'$. Thus $\Sigma$ and $Y$ have naturally induced
$\mathbb{Z}\Gamma_n'$-local coefficient systems and so we have the
inclusion-induced maps $(i_{\pm})_*:H_1(\Sigma;\mathbb{Z}\Gamma'_n)\to
H_1(Y;\mathbb{Z}\Gamma'_n)$. In the classical case $n=0$ these are
just the usual maps $(i_{\pm})_*:H_1(\Sigma;\mathbb{Z})\to
H_1(Y;\mathbb{Z})$ that determine the Seifert matrix and Seifert
form. For $n>0$, no analogue of the Seifert form has been found,
so these maps, though quite complicated, play a crucial role. In
the absence of a true analogue of a higher-order Seifert
\emph{form} they contain the relevant (integral) information
necessary to reconstruct the Alexander modules (but, as in the
classical case, contain \emph{more} information that the modules
alone). Therefore we make the following definition.
\begin{defn}
\label{defn:Seifert presentation} An \emph{$n^{th}$-order Seifert
presentation} of a knot $K$ is the ordered pair
$((i_+)_*,(i_-)_*)$ of maps
$(i_{\pm})_*:H_1(\Sigma;\mathbb{Z}\Gamma'_n)\to
H_1(Y;\mathbb{Z}\Gamma'_n)$ induced by the inclusion maps
$(i_{\pm}):\Sigma\to Y$ for some choice of Seifert surface
$\Sigma$ for $K$.
\end{defn}
Therefore, the case $n=1$ of our main theorem precisely recovers
that of \cite{Li} and the second author \cite{K}. In general,
since $H_1(\Sigma;\mathbb{Z}\Gamma'_n)$ and $H_1(Y;\mathbb{Z}\Gamma'_n)$
may not be free modules (or even finitely-generated), we cannot
speak of a Seifert \emph{matrix}. However after some localization
this situation can be remedied.
Recall from \cite{COT1}\cite{C} that there is a localized version
of ${\SA^\bbz_n}(K)$. We set $S_n \equiv {\mathbb Z}\Gamma_n' - \{0\}$, $n\ge 0$.
Then $S_n$ is a right divisor set of ${\mathbb Z}\Gamma_n$ and we let $\mathcal R_n
\equiv ({\mathbb Z}\Gamma_n)(S_n)^{-1}$. It also follows that ${\mathbb Z}\Gamma_n'$ is
itself an Ore domain and embeds into its classical (skew) quotient
field of fractions, which we denote by ${\mathbb K}_n$. By
\cite[Proposition 3.2]{COT1}, $\mathcal R_n$ is canonically identified
with the (skew) Laurent polynomial ring ${\mathbb K}_n[t^\pm]$, which is
a PID. In the classical case, $S_0=\mathbb{Z}-\{0\}$,
${\mathbb K}_0=\mathbb{Q}$ and $\mathcal R_0$ is $\mathbb{Q}[t^{\pm 1}]$. In
general, Cochran defines the localized higher-order Alexander
module \cite{C}.
\begin{defn}
\label{defn:localized Alexander module} The \emph{$n$-th localized
Alexander module} of a knot $K$ is $\mathcal A_n(K) \equiv
H_1(\ensuremath{S^3\backslash K};\mathcal R_n)$.
\end{defn}
Therefore the case $n=0$ gives the classical (rational) Alexander
module. Moreover the $n^{th}$-order Seifert presentation maps
$(i_{\pm})_*:H_1(\Sigma;\mathbb{Z}\Gamma'_n)\to H_1(Y;\mathbb{Z}\Gamma'_n)$
determine their localized counterparts
$(i_{\pm})_*:H_1(\Sigma;{\mathbb K}_n)\to H_1(Y;{\mathbb K}_n)$. The following
proposition is due to Cochran and Harvey (for example see
\cite[Proposition 6.1]{C}).
\begin{prop}
\label{prop:Seifert form} The following sequence is exact.
$$
H_1(\Sigma;{\mathbb K}_n)\otimes_{{\mathbb K}_n}{\mathbb K}_n[t^{\pm 1}] \xrightarrow{d}
H_1(Y;{\mathbb K}_n)\otimes_{{\mathbb K}_n}{\mathbb K}_n[t^{\pm 1}] \to \mathcal A_n(K) \to 0
$$
where $d(\alpha\otimes 1) = (i_+)_*\alpha\otimes t - (i_-)_*\alpha\otimes 1$.
\end{prop}
\begin{cor}\cite[Corollary 6.2]{C}
\label{cor:Seifert form} If the classical Alexander module of $K$
is not 1, then $\mathcal A_n(K)$, $n>0$, has a square presentation matrix
of size $r = \text{max}\{0,-\chi(\Sigma)\}$ each entry of which is a
Laurent polynomial of degree at most 1. Specifically, we have the
presentation
$$
({\mathbb K}_n[t^{\pm 1}])^r \xrightarrow{\partial} ({\mathbb K}_n[t^{\pm 1}])^r
\to \mathcal A_n(K) \to 0
$$
where $\partial$ arises from the above proposition. If $n=0$, then
the same holds with $r$ replaced by $\beta_1(\Sigma)$.
\end{cor}
\begin{defn}
\label{defn:Seifert form} The above presentation matrix is called
an \emph{order $n$ localized Seifert presentation matrix} for
$K$.
\end{defn}
\begin{prop}\label{prop:failure} Theorem~\ref{thm:main} is false
for $n\geq 2$ if the hypothesis on the degree of the Alexander
polynomial is weakened to allow knots whose Alexander polynomials
have degree $2$.
\end{prop}
\begin{proof}
Let $K$ denote the knot shown in Figure~\ref{fig:ribbon}, well
known as the ``simplest'' ribbon knot.
\begin{figure}[htbp]
\setlength{\unitlength}{1pt}
\begin{picture}(191,156)
\put(0,0){\includegraphics{sribbon.eps}}
\end{picture}
\caption{}\label{fig:ribbon}
\end{figure}
We claim that the conclusions of Theorem~\ref{thm:main} for $n\geq
2$ are false for $K$ because, loosely speaking, any knot with the
same classical Alexander module and first higher-order Alexander
module as $K$ is topologically slice by recent work of Friedl and
Teichner. Details follow.
One easily checks from direct calculation that if $\Delta$ is one
of the two obvious ribbon disks obtained by ``cutting one of the
bands'' in the Figure, then $G=\pi_1(B^4-\Delta)$ is isomorphic to
$\mathbb{Z}[\frac{1}{2}]\rtimes \mathbb{Z}$ and in particular is
$1$-solvable, that is $G^{(2)}=0$. Let $P$ denote $\pi_1(M_K)$
where $M_K$ is the zero surgery on $K$. Then the inclusion map
induces an epimorphism $\phi:P\to G$. Since $G^{(3)}=0$, this
factors through $P/P^{(3)}$.
Suppose the conclusions of the main theorem were true for $K=K_0$
and let $K_1$ denote a knot other than $K$ that satisfies the
conclusions for some $n\geq 2$. By property $(1)$ of
Theorem~\ref{thm:main} $K_1$ is $(0)$-solvable and hence has Arf
invariant zero (this also follows from property $(4)$). Let $P_1$
denote $\pi_1(M_{K_1})$ where $M_{K_1}$ is the zero surgery on
$K_1$. As a consequence of property $(3)$, there is an isomorphism
$f: P_1/(P_1)^{(3)}\to P/P^{(3)}$ (since the conjugacy classes of
the longitudes correspond under the isomorphism given by $(3)$).
Hence there is an epimorphism $\phi_1:P_1\to G$ factoring through
$f$ (essentially $\phi \circ f$). Recall that for any coefficient
system $\psi:Q\to G$, where $Q$ is a group, we have the
isomorphism
$$
H_1(Q;\mathbb{Z}G)\cong \frac{\ker\psi}{[\ker\psi,\ker\psi]}
$$
and so in particular
$$
H_1(P_1;\mathbb{Z}G)\cong
\frac{\ker\phi_1}{[\ker\phi_1,\ker\phi_1]} \text{ and }
H_1(P;\mathbb{Z}G)\cong \frac{\ker\phi}{[\ker\phi,\ker\phi]}.
$$
Since $G^{(2)}=0$, $(P_1)^{(2)}\subset \ker\phi_1$ and so
$(P_1)^{(3)}\subset [\ker\phi_1,\ker\phi_1]$. Similarly for $P$.
It follows that
$$
H_1(P_1;\mathbb{Z}G)\cong H_1(P_1/(P_1)^{(3)};\mathbb{Z}G)\cong
H_1(P/(P)^{(3)};\mathbb{Z}G)\cong H_1(P;\mathbb{Z}G)
$$
Since $K_1$ and $K$ are nontrivial, $M_{K_1}$ and $M_K$ are
aspherical and so
$$ H_1(M_{K_1};\mathbb{Z}G)\cong
H_1(P_1;\mathbb{Z}G) \text{ and } H_1(M_{K};\mathbb{Z}G)\cong
H_1(P;\mathbb{Z}G).
$$
In summary, $H_1(M_{K_1};\mathbb{Z}G)\cong H_1(M_{K};\mathbb{Z}G)$
and thus it follows from \cite[Theorem 8.1]{FT} that $K_1$ is a
(topologically) slice knot, contradicting property $(2)$ of
Theorem~\ref{thm:main}.
\end{proof}
We explain our method to construct desired examples. This
technique is called \emph{genetic modification} which results in a
satellite of a knot. One can find a detailed explanation of
genetic modification in \cite{COT2}. We briefly review this
construction. Let $K$ be a knot and $\{\eta_1,\eta_2,\ldots,
\eta_m\}$ be an oriented trivial link in $S^3$ which misses $K$.
Suppose $\{J_1,J_2,\ldots, J_m\}$ is an $m$-tuple of auxiliary
knots. For each $\eta_i$, remove a tubular neighborhood of
$\eta_i$ in $S^3$ and glue in a tubular neighborhood of $J_i$
along their common boundary, which is a torus, in such a way that
the longitude of $\eta_i$ is identified with the meridian of $J_i$
and the meridian of $\eta_i$ with the longitude of $J_i$. The
resulting knot is denoted by $K(\eta_i,J_i)$ and called the result
of \emph{genetic modification performed on the seed knot $K$ with
the infection $J_i$ along the axis $\eta_i$}. This construction
can also be described in the following way. For each $\eta_i$,
take an embedded disk in $S^3$ bounded by $\eta_i$ such that it
meets with $K$ transversally. Cut off $K$ along the disk, grab the
cut strands of $K$, tie them into the knot $J_i$ with 0-framing as
in Figure~\ref{fig:infection}.
\begin{figure}[htbp]
\setlength{\unitlength}{1pt}
\begin{picture}(262,71)
\put(10,37){$\eta_1$} \put(120,37){$\eta_m$} \put(52,39){$\dots$}
\put(206,36){$\dots$} \put(183,37){$J_1$} \put(236,38){$J_m$}
\put(174,9){$K(\eta_1,\dots,\eta_m,J_1,\dots,J_m)$}
\put(29,7){$K$} \put(82,7){$K$}
\put(20,20){\includegraphics{sinfection1.eps}}
\end{picture}
\caption{$K(\eta_1,\dots,\eta_m,J_1,\dots,J_m)$: Genetic
modification of $K$ by $J_i$ along $\eta_i$}\label{fig:infection}
\end{figure}
Compare the following proposition with \cite[Proposition
3.1]{COT2}.
\begin{prop}
\label{prop:genetic modification} Let $K$ be a knot and $M$ be
zero surgery on $K$. Suppose $n\geq 1$. Suppose $[\eta_i]\in
\ensuremath{\pi_1}(M)\2n$ and $J_i$ has vanishing Arf invariant for $i=1,2,\ldots,
m$. Then $K$ is $n$-solvequivalent to $K'\equiv K(\eta_i,J_i)$.
\end{prop}
\begin{proof}
Since the $J_i$ have vanishing Arf invariant, they are $0$-solvable
(see \cite[Remark 1.3.2]{COT1}). Let $W_i$ be a 0-solution for
$J_i$. By doing surgery along $\ensuremath{\pi_1}(W_i)^{(1)}$ we may assume
$\ensuremath{\pi_1}(W_i) \cong {\mathbb Z}$, generated by the meridian of $J_i$.
Denote zero surgery on $K'$ in $S^3$ by $M'$. We construct an
$n$-cylinder between $M$ and $M'$. Take $M\times[0,1]$. Note that
$\partial W_i = M_i = (S^3\setminus N(J_i)) \cup_{S^1\times S^1}
S^1\times D^2$ where $N(J_i)$ denotes a tubular neighborhood of
$J_i$ in $S^3$ and $\{*\}\times D^2$ in $S^1\times D^2$ denotes
the surgery disk. Take the union of $M\times [0,1]$ and $W_i$'s in
such a way that we identify a product neighborhood of
$\eta_i\times \{1\}$ in $M\times \{1\}$, with the $i^{th}$ copy of
$S^1\times D^2$ in $\partial W_i$ for each $i$. We denote the
4-manifold resulted from this construction by $W$. One easily sees
that $\partial W = M \coprod -M'$. We claim that $W$ is an
$n$-cylinder. Using a Mayer-Vietoris sequence ($M\times [0,1]$
union $W_i$ intersecting along $\eta_i\times D^2$), one easily
sees that $H_1(W) \cong {\mathbb Z}$ and the inclusions from $M$ to $W$
and $M'$ to $W$ induce isomorphisms on the first homology. Also
from the Mayer-Vietoris sequence one observes that $H_2(W) \cong
H_2(M)\oplus (\oplus^m_{i=1} H_2(W_i)) \cong H_2(M')\oplus (\oplus^m_{i=1}
H_2(W_i))$. Note that the generator of $H_2(M)$ under the map
$H_2(M) \longrightarrow H_2(W)$ maps to the image of the generator of
$H_2(M')$ under the map $H_2(M') \longrightarrow H_2(W)$ since they are
represented by capped-off Seifert surfaces. Thus
$H_2(W)/i_*(H_2(\partial W)) \cong \oplus^m_{i=1}H_2(W_i)$ where
$i_*$ is the homomorphism induced from the inclusion $i:
\partial W \longrightarrow W$. Recall that $\ensuremath{\pi_1}(W_i)$ $(\cong {\mathbb Z})$ is
generated by the meridian of $J_i$ which is identified with the
longitude of $\eta_i$. Hence $\ensuremath{\pi_1}(W_i)$ maps into $\ensuremath{\pi_1}(W)^{(n)}$
since $[\eta_i]$ lies in $\ensuremath{\pi_1}(M)^{(n)}$ by hypothesis and
$\ensuremath{\pi_1}(M)^{(n)}$ maps into $\ensuremath{\pi_1}(W)^{(n)}$. This implies that
0-surfaces in $W_i$ are $n$-surfaces in $W$. Now by naturality of
intersection forms, one can see that the union of the
0-Lagrangians (with 0-duals for $W_i$) ($0\le i \le m$)
constitutes an $n$-Lagrangian (with $n$-duals) for $W$. One checks
easily that $W$ is spin since each $W_i$ was spin.
\end{proof}
Suppose $K,\eta_i,J_i$ and $K'$ are as in
Proposition~\ref{prop:genetic modification}. Denote zero surgery
on $J_i$ in $S^3$ by $M_i$. Suppose $W$ is the $n$-cylinder with
$\partial W= M \coprod -M'$ constructed above. Suppose
$\psi:\ensuremath{\pi_1}(W)\longrightarrow \Gamma$ is a map to an arbitrary $n$-solvable PTFA
group $\Gamma$. Let $\phi, \phi'$, and $\phi_i$ denote the induced
maps on $\ensuremath{\pi_1}(M), \ensuremath{\pi_1}(M')$, and $\ensuremath{\pi_1}(M_i)$ respectively. For each
$J_i$, we denote by $\rho_{\mathbb Z}(J_i)$ the $\rho$-invariant
$\rho(M_i,\zeta_i)$ where $\zeta_i : \ensuremath{\pi_1}(M_i) \longrightarrow {\mathbb Z}$ is the
abelianization. The following lemma reveals the additivity of the
$\rho$-invariant in our current setting.
\begin{lem}
\label{lem:additivity}
$$
\rho(M,\phi) - \rho(M',\phi') = \sum^m_{i=1}\epsilon_i\rho_{\mathbb{Z}}(J_i)
$$
where $\epsilon_i=0$ or $1$ according as $\phi(\eta_i)=e$ or not.
\end{lem}
\begin{proof}
It follows from \cite[Proposition 3.2]{COT2} that
$$
\rho(M,\phi) - \rho(M',\phi') = \sum^m_{i=1}\rho(M_i,\phi_i)
$$
Since $\pi_1(W_i)\cong \mathbb{Z}$, $\phi_i$ factors through
$\mathbb{Z}$ generated by $\eta_i$. Thus its image is zero if
$\phi(\eta_i)=e$, and $\mathbb{Z}$ if not ( recall a PTFA group is
torsion-free). In the former case, $\rho(M_i,\phi_i)=0$ by
property ($3$) of Proposition~\ref{prop:rho invariants}. In the
latter case, $\rho(M_i,\phi_i)=\rho_{\mathbb Z}(J_i)$ by property ($2$)
of Proposition~\ref{prop:rho invariants}.
\end{proof}
The following two theorems are the key theorems that show the
existence of a knot that is $n$-solvequivalent to a given knot $K$
but not $(n.5)$-solvequivalent to $K$. The first significantly
generalizes \cite[Theorem 4.3]{CT} in two ways. Firstly, their
theorem applies only when $K$ is a genus $2$ fibered knot.
Secondly their theorem only covers the case when $\partial W=M$.
For a group $G$, let $G\2k_r$ be the \emph{$k$-th rational derived
group of $G$} (see \cite[Section 3]{Ha}).
\begin{thm}
\label{thm:injectivity} Let $n\in{\mathbb N}$. Let $K$ be a knot for
which the degree of the Alexander polynomial is greater than $2$
(if $n=1$, degree equal to $2$ is allowed). Let $M$ be the zero
framed surgery on $K$ in $S^3$. Suppose $\Sigma$ is a Seifert
surface for $K$. Then there exists an oriented trivial link
$\{\eta_1,\eta_2,\dots,\eta_m\}$ in $S^3-\Sigma$ that satisfies
the following:
\begin{itemize}
\item [(1)] $\eta_i\in\ensuremath{\pi_1}(M)\2n$ for all $i$. Moreover, the
$\eta_i$ bound (smoothly embedded) symmetric capped gropes of
height $n$, disjointly embedded in $S^3-K$ (except for the caps,
which will hit $K$).
\item [(2)] For every $n$-cylinder $W$ with $M$ as one of its boundary
components, there exists {\bf some} $i$ such that
$j_*(\eta_i)\notin\ensuremath{\pi_1}(W)\ensuremath{^{(n+1)}}_{r}$ where $j_*:\ensuremath{\pi_1}(M)\longrightarrow\ensuremath{\pi_1}(W)$ is
induced from the inclusion $j:M \longrightarrow W$. The number of such $i$'s
is at least $\frac12 (d-2)$ if $n>1$ or at least $\frac12 d$ if
$n=1$ where $d$ denotes the degree of the Alexander polynomial of
$K$.
\end{itemize}
\end{thm}
\begin{proof}
Let $S \equiv \ensuremath{\pi_1}(M)^{(1)}$. Suppose $W$ is an $n$-cylinder with
$M$ as one of its boundary components. The inclusion $j:M\longrightarrow W$
induces a map $j:S\longrightarrow\ensuremath{\pi_1}(W)\21$. Let $G \equiv
\ensuremath{\pi_1}(W)\21=\ensuremath{\pi_1}(W)\21_{r}$. The last equality holds since $H_1(W)
\cong {\mathbb Z}$. For convenience, let us use the same notation
$\Sigma$ for the capped-off Seifert surface. Let $g \equiv \frac12
\operatorname{rk}_{\mathbb Q} H_1(M\setminus \Sigma;{\mathbb Q})$ (which is equal to the genus
of $\Sigma$). Let $F$ be the free group of rank $2g$. Choose a map
$F\longrightarrow \ensuremath{\pi_1}(M\setminus \Sigma)$ that induces an isomorphism on
$H_1(F;{\mathbb Q})$. Let $i$ be the composition $F\longrightarrow \ensuremath{\pi_1}(M\setminus
\Sigma)\subset S$. For a group $G$, let $G_k$, for $k\in {\mathbb N}_0$,
denote $G/G\2k_r$. By \cite[Corollary 3.6]{Ha} $G_k$ is PTFA. By
\cite[Proposition 2.5]{COT1} ${\mathbb Z} G_k$ embeds into the (skew)
quotient field of fractions, which is denoted by ${\mathbb K}(G_k)$. By
\cite[Proposition II.3.5]{Ste} ${\mathbb K}(G_k)$ is flat over $G_k$.
Since $\ensuremath{\pi_1}(W)/\ensuremath{\pi_1}(W)\21 \cong {\mathbb Z}$, if we let $\Gamma_k \equiv
\ensuremath{\pi_1}(W)/\ensuremath{\pi_1}(W)_r^{(k+1)}$, there is a short exact sequence $1\to G_k
\to \Gamma_k \xrightarrow{\pi} {\mathbb Z} \to 1$ where $\pi$ is the
abelianization. Then we have a PID ${\mathbb K}(G_k)[t^{\pm 1}]$ such
that ${\mathbb Z}\Gamma_k \subset {\mathbb K}(G_k)[t^{\pm 1}] \subset \mathcal K$ where
$\mathcal K$ is the (skew) quotient field of fractions of ${\mathbb Z}\Gamma_k$
($\Gamma_k$ is PTFA by \cite[Corollary 3.6]{Ha}, hence ${\mathbb Z}\Gamma_k$
embeds into the skew quotient field of fractions).
Apply Theorem~\ref{thm:paininbutt} to find a finite collection
$\mathcal P_{n-1}$ of $2g-1$-tuples ($2g$ if $n=1$) of elements of
$F^{(n-1)}$. Since $S^{(n-1)}=\ensuremath{\pi_1}(M)\2n$, the image of the union of
the elements of $\mathcal P_{n-1}$ under $i : F \longrightarrow S$ is a finite set
$\{\alpha_1,\dots,\alpha_m\}$ of elements of $\ensuremath{\pi_1}(M)\2n$ as required by the
part (1) in the statement. By Proposition~\ref{prop:geomalg} the
induced map $j:S\longrightarrow G$ is an algebraic $n$-solution, hence an
algebraic $(n-1)$-solution (see Remark~\ref{rem:algebraic}(3)). By
our choice of $\mathcal P_{n-1}$ from Theorem~\ref{thm:paininbutt}, at
least one tuple $\{w_1,w_2,\ldots ,w_{2g-1}\}\in\mathcal P_{n-1}$
($\{w_1,w_2,\ldots ,w_{2g}\}$ if $n=1$) maps to a generating set
of $H_1(S;{\mathbb K}(G_{n-1}))$. Notice that $H_1(S;{\mathbb K}(G_{n-1})) \cong
H_1(M;{\mathbb K}(G_{n-1})[t^{\pm 1}])$ and $H_1(G;{\mathbb K}(G_{n-1})) \cong
H_1(W;{\mathbb K}(G_{n-1})[t^{\pm 1}])$ as ${\mathbb K}(G_{n-1})$-modules. By
Theorem~\ref{thm:rank} (with $\Gamma = \ensuremath{\pi_1}(W)/\ensuremath{\pi_1}(W)\2n_r, [\Gamma,\Gamma] =
\ensuremath{\pi_1}(W)\21/\ensuremath{\pi_1}(W)\2n_r = G_{n-1}$), we see that at least $\frac12
(d-2)$ of $\{w_1,...,w_{2g-1}\}$ (if $n>1$) or at least $\frac12
d$ of $\{w_1,...,w_{2g}\}$ (if $n=1$) map non-trivially under
$F^{(n-1)}\to S^{(n-1)}\overset{j}{\to}G^{(n-1)}_{r}/G\2n_{r}$
$(\cong H_1(G;{\mathbb Z} G_{n-1})$ modulo $\mathbb{Z}$-torsion). Hence
at least $\frac12 (d-2)$ (if $n>1$) or at least $\frac12 d$ (if
$n=1$) of the $\alpha_i$ have the property that $j_*(\alpha_i)\notin
G\2n_{r}=(\ensuremath{\pi_1}(W))^{\ensuremath{^{(n+1)}}}_{r}$. Since each $\alpha_i$ actually lies in
$\ensuremath{\pi_1}(S^3\setminus \Sigma)^{(n)}$ we can represent the $\alpha_i$ by
simple closed curves in the complement of the chosen Seifert
surface $\Sigma$ for the knot $K$. We can alter these by crossing
changes until the collection of $\alpha_i$ forms a trivial link in
$S^3$. Moreover, by \cite[Lemma 3.8]{CT} and its proof, we may
choose representatives in the same homotopy classes that have the
much stronger property that they bound (smoothly embedded)
symmetric capped gropes of height $n$, disjointly embedded in
$S^3-K$ except for the caps. This is the collection $\{\eta_i\}$
required.
\end{proof}
Let $c_M$ denote the universal bound for all $\rho$-invariants of
a fixed 3-manifold $M$ given by \cite[Theorem 4.10]{ChG} and
Ramachandran \cite[Theorem 3.1.1]{R}. That is, $|\rho(M,\phi)| <
c_M$ for every representation $\phi : \ensuremath{\pi_1}(M) \longrightarrow \Gamma$ where $\Gamma$ is
a group. For a detailed discussion see \cite{CT}. The final
conclusion of the following theorem was proved in \cite{CT} in the
case that $K$ is itself an $n$-solvable knot.
\begin{thm}
\label{thm:infection} Let $n\in {\mathbb N}$. Let $K$ be a knot in $S^3$.
Suppose $\{\eta_1,\eta_2,\dots,\eta_m\}$ is an oriented trivial
link in $S^3$ that misses $K$ and has properties ($1$) and ($2$)
of Theorem~\ref{thm:injectivity}. Then, for every $m$-tuple
$\{J_1,J_2,\dots,J_m\}$ of Arf invariant zero knots for which
$\rho_{\mathbb Z}(J_i)> 2c_M$, the knot $K' \equiv K(\eta_i,J_i)$ (formed
by genetic modification) is $n$-solvequivalent to $K$ but not
$(n.5)$-solvequivalent to $K$. Moreover $K'-K$ is of infinite
order in $\ensuremath{\mathcal F_{(n.0)}/\mathcal F_{(n.5)}}$. (In particular, $K'\#(-K)$ is $(n)$-solvable.)
\end{thm}
\begin{proof}
By Proposition~\ref{prop:genetic modification}, $K'$ is
$n$-solvequivalent to $K$. Let $M$ be zero surgery on $K$ in $S^3$
and $M'$ zero surgery on $K'$ in $S^3$. Let $M_i$ be zero surgery
on $J_i$ in $S^3$, $1\le i\le m$. Let $W$ be an $n$-cylinder with
$\partial W = M \coprod -M'$ constructed as in the proof of
Proposition~\ref{prop:genetic modification}. Suppose $M'$ is
$(n.5)$-solvequivalent to $M$ via $V$. We will show that this
leads us to a contradiction. Let $X\equiv W\cup_{M'} V$. Thus
$\partial X = M\coprod -M$. We assume $\partial_+X = M$ and
$\partial_-X = -M$. Since $V$ is an $(n.5)$-cylinder, it is an
$n$-cylinder. As in Proposition~\ref{prop:transitivity}, one can
show that $X$ is an $n$-cylinder. Let $\Gamma \equiv
\ensuremath{\pi_1}(X)/\ensuremath{\pi_1}(X)\ensuremath{^{(n+1)}}_r$, an $n$-solvable PTFA group. Let $\psi : \ensuremath{\pi_1}(X)
\longrightarrow \Gamma$ be the projection. Let $\phi_+$, $\phi_-$, and $\phi'$
denote the induced maps on $\ensuremath{\pi_1}(\partial_+X)$, $\ensuremath{\pi_1}(\partial_-X)$,
and $\ensuremath{\pi_1}(M')$, respectively. By Lemma~\ref{lem:additivity} we have
$$
\rho(M,\phi_+)-\rho(M',\phi')=\sum^m_{i=1}\epsilon_i\rho_{{\mathbb Z}}(J_i)
$$
where $\epsilon_i=0$ or $1$ according as $\phi_+(\eta_i)=e$ or not. On
the other hand, since $V$ is an $(n.5)$-cylinder,
Theorem~\ref{thm:rho=0} applies to say that
$$\rho(M',\phi')-\rho(M,\phi_-)=0.
$$
Hence
$$
\rho(M,\phi_+)-\rho(M,\phi_-)=\sum^m_{i=1}\epsilon_i\rho_{\mathbb Z}(J_i).
$$
Note that since $X$ is an $n$-cylinder and the collection
$\{\eta_i\}$ was chosen to satisfy property $(2)$ of
Theorem~\ref{thm:injectivity}, there exists at least one $i$ such
that $\psi(\eta_i)\neq e$ from which it follows that
$\phi_+(\eta_i)\neq e$. Thus
$$
\rho(M,\phi_+) - \rho(M,\phi_-) \ge \rho_{{\mathbb Z}}(J_i)> 2c_M.
$$
which is a contradiction. Therefore $K'$ is not
$(n.5)$-solvequivalent to $K$.
We will now show that $K'-K\in \mathcal F_{(n)}$, that is, $K'\#(-K)$ is
$n$-solvable. Note that since $K'$ was obtained from $K$ by
genetic modification along the $\eta_i$, $K'\#(-K)$ can be
obtained from $K\#(-K)$ by performing a genetic modification along
the ``same'' axes $\eta_i$. It is only necessary to observe that
we can perform the connected sum in such a way as to preserve the
fact that the $\eta_i$ lie in the $n^{th}$-derived group of
$\pi_1$. This is clear because $S^3-K'$ can be viewed as a
\emph{subspace} of $S^3-(K\#(-K))$. Notice that $K\#(-K)$ is a
slice knot, hence $n$-solvable. By \cite[Proposition 3.1]{COT2},
performing a genetic modification on an $n$-solvable knot using
Arf invariant zero knots $J_i$, along axes that lie in the
$n^{th}$-derived group results in another $n$-solvable knot. Hence
$K'\#(-K)$ is $n$-solvable.
Suppose $K'-K$ were of order $k > 0$ in $\ensuremath{\mathcal F_{(n.0)}/\mathcal F_{(n.5)}}$. We will show that
this yields a contradiction, implying that $K'-K$ is of infinite
order. Our assumption is equivalent to the fact that
$(\#^k_{i=1}K')\#(\#^k_{j=1}-K)$ is $(n.5)$-solvable. Let $N$
denote the zero surgery on $(\#^k_{i=1}K')\#(\#^k_{j=1}-K)$. Let
$V$ be an $(n.5)$-solution for $N$ and let $W$ be as above. Let
$M'_i$ be the $i$-th copy of $M'$ and $M_j$ be the $j$-th copy of
$M$, $1\le i,j\le k$. Take a standard (spin) cobordism $C$ from
$(\coprod^k_{i=1}M'_i)\coprod (\coprod^k_{j=1}-M_j)$ to $N$ which
is obtained from $((\coprod^k_{i=1}M'_i)\coprod
(\coprod^k_{j=1}-M_j)) \times [0,1]$ by adding $(2k-1)$ 1-handles
and then $(2k-1)$ 2-handles (see \cite[Lemma 4.2]{COT2} for more
detail). Let $X \equiv (\coprod^k_{i=1}W_i)\cup_{\coprod M'_i}C
\cup_{N} V$ where $W_i$ is the $i$-th copy of $W$. Then $\partial
X = (\coprod^k_{i=1}M_i) \coprod (\coprod^k_{j=1}-M_j)$. Let $\Gamma
\equiv \ensuremath{\pi_1}(X)/\ensuremath{\pi_1}(X)\ensuremath{^{(n+1)}}_r$, an $n$-solvable PTFA group. Let $\psi
:\ensuremath{\pi_1}(X) \longrightarrow \Gamma$ be the projection. Let $\phi_j$, $\phi'_i$,
$\phi^+_i$, and $\phi$ denote the restrictions of $\psi$ to
$\ensuremath{\pi_1}(-M_j)$, $\ensuremath{\pi_1}(M'_i)$, $\ensuremath{\pi_1}(M_i)$ (the upper boundary of $W_i$),
and $\ensuremath{\pi_1}(N)$ respectively. One sees that $C$ is an $(n.5)$-cylinder
since $H_2(C)/i_*(H_2(\partial C)) =0$. Thus by
Theorem~\ref{thm:rho=0}
$$
\sum^k_{i=1}\rho(M'_i,\phi'_i) + \sum^k_{j=1}\rho(-M_j,\phi_j) -
\rho(N,\phi)= 0.
$$
But since $V$ is an $(n.5)$-solution for $N$, $\rho(N,\phi) = 0$
by Theorem~\ref{thm:rho=0}. Therefore
$\sum^k_{i=1}\rho(M'_i,\phi'_i) = \sum^k_{j=1}\rho(M_j,\phi_j)$.
By Lemma~\ref{lem:additivity} (applied for each $i$ and then
summed),
$$
\sum^k_{i=1}\rho(M_i,\phi^+_i) - \sum^k_{i=1}\rho(M'_i,\phi'_i) =
\sum^k_{i=1}\sum^m_{l=1}\epsilon_{il}\rho_{\mathbb Z}(J_l)
$$
where $\epsilon_{il} = 0$ if $\phi^+_i(\eta_{il}) = e$ and $\epsilon_{il} = 1$
if $\phi^+_i(\eta_{il}) \ne e$. (By $\eta_{il}$ we mean the $l$-th
axis $\eta_{l}$ for the $i$-th copy of $K$.) Thus
$$
\sum^k_{i=1}\rho(M_i,\phi^+_i) - \sum^k_{j=1}\rho(M_j,\phi_j) =
\sum^k_{i=1}\sum^m_{l=1}\epsilon_{il}\rho_{\mathbb Z}(J_l).
$$
Now we claim that $X$ is an $n$-cylinder. Using a Mayer-Vietoris
sequence and \cite[Lemma 4.2]{COT2}, one can show that
$H_2(X)/i_*(H_2(\partial X)) \cong (\oplus^k_{i=1}H_2(W_i)) \oplus
H_2(V)$ and the union of the $0$-Lagrangians with 0-duals for
$W_i$ ($1\le i\le k$) and the $n$-Lagrangian with $n$-duals for
$V$ constitutes an $n$-Lagrangian with $n$-duals for $X$. Since,
for any $i$, $X$ is an $n$-cylinder with $M_i$ as one of its
boundary components, by Theorem~\ref{thm:injectivity}, for each
$i$ there exists some $l_i$ such that $\psi(\eta_{il_i}) \ne e$.
Hence $\phi^+_i(\eta_{il_i})\ne e$. Then the right-hand side of
the last equation is greater than $2kc_M$, while the left-hand
side is less than $2kc_M$, a contradiction.
\end{proof}
Finally we prove the main theorem.
\begin{proof}[{\bf Proof of Theorem~\ref{thm:main} (Main Theorem)}]
Let $K$ be a knot for which the degree of the Alexander polynomial
is greater than 2 (if $n=1$, degree equal to $2$ is allowed). Let
$M$ denote the zero surgery on $K$. Apply
Theorem~\ref{thm:injectivity} to choose an oriented trivial link
$\{\eta_1,\eta_2,\ldots, \eta_m\}$ in $S^3$ that misses $K$
(except for the caps, which will hit $K$), and such that the
$\eta_i$ bound height $n$ (smoothly embedded) symmetric capped
gropes disjointly embedded in $S^3-K$. Let $c_M$ be the universal
bound for $\rho$-invariants for $M$ as mentioned above. Choose
knots $J^i$ ($i\in {\mathbb N})$ with vanishing Arf invariants
inductively as follows. First, choose $J^1$ such that
$\rho_{\mathbb Z}(J^1) > 2c_M$. Suppose $J^{k-1}$ has been constructed.
Then choose $J^k$ such that $\rho_{\mathbb Z}(J^k) > 2c_M +
2m\rho_{\mathbb Z}(J^{k-1})$. These $J^i$ are easily found by taking the
connected sum of a suitably large even number of copies of the
left-handed trefoil. (Note that for the left-handed trefoil $J$,
$\rho_{\mathbb Z}(J)=4/3$ by Proposition~\ref{prop:rho invariants}(4).)
However, to achieve the final conclusion of Part $(1)$, we must
choose each $J^i$ to be a suitably large connected sum of the knot
shown in \cite[Figure 1.7]{CT}. This knot has the same $\rho_{\mathbb Z}$
as the left-handed trefoil knot \cite[Lemma 4.4]{CT}. For each
$i$, let $J^i_j$ be the $j$-th copy of $J^i$, $1\le j \le m$. Now
define $K_i \equiv K(\eta_1,\dots,\eta_m,J^i_1,\dots,J^i_m)$, the
result of genetic modification performed on $K$ with the
infections $J^i_j$ along the axes $\eta_j$ $(1\le j\le m)$. Set
$K_0 \equiv K$.
\noindent {\bf Part (1)} : It follows from the last sentence of
Theorem~\ref{thm:infection} that $K_i-K \in \mathcal F_{(n)}$, $i>0$ and
hence that $K_i$ is $n$-solvequivalent to $K$ (or apply
Proposition~\ref{prop:genetic modification}). To show that $K_i$
and $K$ cobound, in $S^3\times\[0,1\]$, a (smoothly embedded)
embedded symmetric Grope of height $n+2$, merely apply the proof
of \cite[Theorem 3.7]{CT}.
\noindent {\bf Part (2)} : Suppose $i>j\ge 0$. Let $M_i$ be the
zero surgery on $K_i$ in $S^3$. Suppose $M_i$ is
$(n.5)$-solvequivalent to $M_j$ via $U$ ($\partial U = M_i\coprod
-M_j$). Suppose $V$ be an $n$-cylinder with $\partial V = M
\coprod -M_i$ and $W$ an $n$-cylinder with $\partial W =
M_j\coprod -M$ such that $V$ and $W$ are constructed as in the
proof of Proposition~\ref{prop:genetic modification}. Let $X
\equiv V\cup_{M_i}U \cup_{M_j} W$. Then $X$ is an $n$-cylinder
with $\partial_+ X = M$ and $\partial_-X = -M$ (see
Proposition~\ref{prop:transitivity}). Let $\Gamma \equiv
\ensuremath{\pi_1}(X)/\ensuremath{\pi_1}(X)\ensuremath{^{(n+1)}}_r$, an $n$-solvable PTFA group. Let $\psi : \ensuremath{\pi_1}(X)
\longrightarrow \Gamma$ be the projection. Let $\phi_+$, $\phi_i$, $\phi_j$, and
$\phi_-$ denote the restrictions of $\psi$ to $\ensuremath{\pi_1}(\partial_+ X) (
= \ensuremath{\pi_1}(M))$, $\ensuremath{\pi_1}(M_i)$, $\ensuremath{\pi_1}(M_j)$, and $\ensuremath{\pi_1}(\partial_-X) ( = \ensuremath{\pi_1}(-M))$
respectively. Since $U$ is an $(n.5)$-cylinder, by
Corollary~\ref{cor:rho=0}, $\rho(M_i,\phi_i) = \rho(M_j,\phi_j)$.
On the other hand, by Lemma~\ref{lem:additivity},
$$
\rho(M,\phi_+) - \rho(M_i,\phi_i) =
\sum^m_{k=1}\epsilon_k\rho_{\mathbb Z}(J^i_k)
$$
where $\epsilon_k = 0$ if $\phi_+(\eta_k) = e$ and $\epsilon_k = 1$ if
$\phi_+(\eta_k) \ne e$. Similarly,
$$
\rho(M,\phi_-) - \rho(M_j,\phi_j) =
\sum^m_{k=1}\epsilon'_k\rho_{\mathbb Z}(J^j_k)
$$
where $\epsilon'_k = 0$ if $\phi_-(\eta_k) = e$ and $\epsilon'_k = 1$ if
$\phi_-(\eta_k) \ne e$. Thus we have
$$
\rho(M,\phi_+) - \rho(M,\phi_-) =
\sum^m_{k=1}\epsilon_k\rho_{{\mathbb Z}}(J^i_k) -
\sum^m_{k=1}\epsilon'_k\rho_{{\mathbb Z}}(J^j_k).
$$
Since $J^j_k$ is merely a copy of $J^j$,
$\rho_{\mathbb Z}(J^j_k)=\rho_{\mathbb Z}(J^j)$, and similarly
$\rho_{{\mathbb Z}}(J^i_k)=\rho_{\mathbb Z}(J^i)$. Moreover, by
Theorem~\ref{thm:injectivity} applied to $M=\partial_+ X$, there
is some $k$ such that $\phi_+(\eta_k) = \psi(\eta_k) \ne e$. Thus
the right hand side above is greater than
$\rho_{\mathbb Z}(J^i)-m\rho_{\mathbb Z}(J^j)$, which, by our choice of $J^i$
and $J^j$ is greater than $2c_M$. This is a contradiction since
the left-hand side above has absolute value less than $2c_M$. To
show that $K_i$ and $K_j$ do not cobound, in $S^3\times\[0,1\]$,
an embedded symmetric Grope of height $(n+2.5)$, merely note that
the proof of \cite[Theorem 8.11]{COT1} clearly applies to show
that if they did bound such a Grope then they would be
$(n.5)$-solvequivalent.
\noindent {\bf Part (3)} : This follows from \cite[Theorem
8.1]{C}.
\noindent {\bf Part (4)} : We shall employ the terminology set out
just above Proposition~\ref{prop:Seifert form}. Let $K_*$ denote
one of the $K_i$ and let $G_*$ denote its knot group. Let $G$
denote the knot group of $K$. By part $(3)$, these two knots share
the same higher-order (integral) Alexander modules up to order
$n-1$ and $G/G^{(i+1)}$ is isomorphic to $G_*/(G_*)^{(i+1)}$ for
all $i\leq n$. Therefore in considering the $i^{th}$ higher-order
Alexander modules of these knots, we may consider that they are
modules over the same ring, $\mathbb{Z}\Gamma_i\equiv
\mathbb{Z}[G/G^{(i+1)}]$, as long as $i\leq n$. We shall show that
there exist Seifert surfaces $\Sigma_*$ for $K_*$ and $\Sigma$ for
$K$ with respect to which the $i^{th}$ order Seifert
presentations, $i_{\pm}:H_1(\Sigma;\mathbb{Z}\Gamma'_i)\to
H_1(Y;\mathbb{Z}\Gamma'_i)$ and
$i_{\pm}:H_1(\Sigma_*;\mathbb{Z}\Gamma'_i)\to
H_1(Y_*;\mathbb{Z}\Gamma'_i)$ are identical up to isomorphisms
identifying the domain and range of each, as long as $i\leq n-1$.
(Recall that $\Gamma_i'$ is the commutator subgroup of $\Gamma_i$.) A
Seifert presentation determines the map $d$ in
Proposition~\ref{prop:Seifert form} and hence there are bases with
respect to which the localized $i^\textrm{th}$-order Seifert
matrices for $K_*$ and $K$ are identical. We may assume that
$n\geq1$.
Let $E(K)$ denote the exterior of $K$ in $S^3$. The continuous map
$f:E(K_*)\to E(K)$ that induces all of these isomorphisms is
described as follows (see \cite[Theorem 8.1]{C}). Recall that
$E(K_*)$ is constructed from $E(K)$ by replacing a collection of
solid tori $\eta_j \times D^2$ by a collection of knot exteriors
$E(J_j)$. Since it is well known that there is always a degree one
map, $f_j$, relative boundary from $E(J_j)$ to $E(unknot)\equiv
\eta_j \times D^2$, there is a degree one map relative boundary,
$f$, from $E(K_*)$ to $E(K)$. Recall also that $\eta_j\in
\pi(M)^{(n)}$ by choice. But in fact, the $\eta_j$ actually
produced by Theorem~\ref{thm:paininbutt}, lie in $F^{(n-1)}$ where
$F\to \pi_1(M-\Sigma^*)$ where $\Sigma^*$ is a capped-off Seifert
surface. Hence, the circles $\eta_j$ can be chosen to miss the
given Seifert surface $\Sigma$ and actually represent elements of
$\pi_1(Y)^{(n-1)}\subset G^{(n)}$. Therefore we can use the `same'
Seifert surface for $K_*$ as for $K$. For simplicity for the rest
of this proof we shall assume that there is just one circle,
$\eta$. The proof is no different for a collection. Note that $f$
is the `identity' on a neighborhood of $\Sigma_*$ (mapping to
$\Sigma$) and restricts to a degree one map relative boundary
$f:Y_*\to Y$. We need only show that this map carries
$H_1(Y_*;\mathbb{Z}\Gamma'_i)$ isomorphically to
$H_1(Y;\mathbb{Z}\Gamma'_i)$. Briefly, this is true because $\eta\in
G^{(n)}$ and thus goes to zero in $\Gamma_i$ if $i\leq n-1$. It
follows that the entire group $\pi_1(E(J))$, where $J$ is the
infection knot, maps to zero in $\Gamma'_i$. Thus in a Mayer-Vietoris
analysis, $H_1(E(J);\mathbb{Z}\Gamma'_i)$ really has untwisted
coefficients and thus is not distinguishable from the case that
$J$ is a trivial knot. But if $J$ were trivial then $K_*=K$ and
$Y_*=Y$. For more details, the proof is the same as the proof of
\cite[Theorem 8.2]{C} where it is shown that each of
$H_1(Y_*;\mathbb{Z}\Gamma_i)$ and $H_1(Y;\mathbb{Z}\Gamma_i)$ is
isomorphic to the quotient of $H_1(Y\setminus (\eta \times
D^2);\mathbb{Z}\Gamma_i)$ by the submodule generated by the meridian
of $\eta$.
\noindent {\bf Part (5)} : For $j=0$ this follows directly from
Theorem~\ref{thm:infection}. For general $j$ the proof is
essentially the same as the proof of part $(2)$. We outline the
proof. Here $i$ and $j$ are fixed.
Let $M_i$, $M_j$, and $M$
denote the zero surgeries on $K_i$, $K_j$, and $K$ as usual.
Suppose $K_i-K_j$ were of order $k>0$, that is to say
$(\#^k_{l=1}K_i)\#(\#^k_{l=1}-K_j)$ is $(n.5)$-solvable. Then by
Proposition~\ref{prop:old and new}, $M_{\#K_i}$ is
$(n.5)$-solvequivalent to $M_{\#K_j}$ via some $U$ as in the proof
of part $(2)$. Refer to the schematic Figure~\ref{fig:cobordism} below.
\begin{figure}[h]
\setlength{\unitlength}{1pt}
\begin{picture}(118,203)
\put(58,87){$U$}
\put(58,116){$C$}
\put(58,57){$D$}
\put(14,155){$V^1$}
\put(58,155){$V^l$}
\put(105,155){$V^k$}
\put(13,20){$W^1$}
\put(58,20){$W^l$}
\put(102,20){$W^k$}
\put(108,71){$-M_{\#K_j}$}
\put(110,106){$M_{\#K_i}$}
\put(-6,143){$M^1_i$}
\put(81,143){$M_i^l$}
\put(35,170){$\dots$}
\put(35,11){$\dots$}
\put(79,11){$\dots$}
\put(79,170){$\dots$}
\put(14,177){$M^{1_v}$}
\put(57,177){$M^{l_v}$}
\put(100,177){$M^{k_v}$}
\put(123,143){$M_i^k$}
\put(76,35){$-M_j^l$}
\put(123,35){$-M_j^k$}
\put(-14,35){$-M_j^1$}
\put(10,-2){$-M^{1_w}$}
\put(53,-2){$-M^{l_w}$}
\put(98,-2){$-M^{k_w}$}
\put(10,10){\includegraphics{scobordism.eps}}
\end{picture}
\caption{}\label{fig:cobordism}
\end{figure}
Now let $C$ be the standard cobordism between
$M_{\#K_i}$ and the disjoint union of $k$ copies of $M_i$, which
we denote by $M_i^l$ for $1\leq l\leq k$, and let $D$ be the
standard cobordism between $-M_{\#K_j}$ and the disjoint union of
$k$ copies of $-M_j$, which we denote by $-M_j^l$ for $1\leq l\leq
k$. Let $V^l$ denote the $l^{th}$ copy of the standard n-cylinder
between $M_i$ and $M$, constructed in the proof of
Proposition~\ref{prop:genetic modification}, so that $\partial
V^l=M^{l_v}\coprod -M_i^l$ where we use $M^{l_v}$ to denote the
copy of $M$ that occurs in the boundary of $V^l$. Similarly let
$W^l$ denote the $l^{th}$ copy of the standard $n$-cylinder
between $-M_j$ and $-M$ so that $\partial W^l=M^{l_w}\coprod
-M_j^l$ where we use $M^{l_w}$ to denote the copy of $M$ that
occurs in the boundary of $W^l$. Following the proof of Part
$(2)$, let
$$
X\equiv (\coprod_{l=1}^k V^l)\cup C\cup_{M_{\#K_i}}U
\cup_{M_{\#K_j}}D\cup (\coprod_{l=1}^k W^l).
$$
Recall that we saw in the proof of Theorem~\ref{thm:injectivity}
that $C$ and $D$ are $(n.5)$-cylinders since
$H_2(C)/i_*(H_2(\partial C))=0$ and $H_2(D)/i_*(H_2(\partial
D))=0$. It follows that $X$ is an $n$-cylinder with $\partial_+ X
= \coprod_{l=1}^k M^{l_v}$ and $\partial_-X = \coprod_{l=1}^k
-M^{l_w}$ (see also Proposition~\ref{prop:transitivity}). Let $\Gamma
\equiv \ensuremath{\pi_1}(X)/\ensuremath{\pi_1}(X)\ensuremath{^{(n+1)}}_r$, an $n$-solvable PTFA group. Let $\psi :
\ensuremath{\pi_1}(X) \longrightarrow \Gamma$ be the projection. Let $\phi^{l_v}$ and
$\phi^{l_w}$ denote the restrictions of $\psi$ to $ \ensuremath{\pi_1}(M^{l_v})$,
and $\ensuremath{\pi_1}(M^{l_w})$ respectively. Since $U$ is an $(n.5)$-cylinder
and otherwise $H_2(X)$ comes from the $V^l$ and the $W^l$, we
arrive at an expression similar to that of the proof of Part
$(2)$, except summed from $l=1$ to $k$:
$$
\sum_{l=1}^k(\rho(M^{l_v},\phi^{l_v}) - \rho(M^{l_w},\phi^{l_w}))
=\sum_{l=1}^k( \sum^m_{s=1}\epsilon_{sl}\rho_{{\mathbb Z}}(J^i) -
\sum^m_{s=1}\epsilon'_{sl}\rho_{{\mathbb Z}}(J^j)),
$$
where $\epsilon_{sl} = 0$ if $\phi^{l_v}(\eta_{sl}) = e$ (where
$\eta_{sl}$ means the $s$-th axis $\eta_s$ for $K$ for
constructing the $l$-th copy of $K_i$) and $\epsilon_{sl} = 1$ if
$\phi^{l_v}(\eta_{sl}) \ne e$, and where $\epsilon'_{sl} = 0$ if
$\phi^{l_w}(\eta'_{sl}) = e$ (where $\eta'_{sl}$ means the $s$-th
axis $\eta_s$ for $K$ for constructing the $l$-th copy of $K_j$)
and $\epsilon'_{sl} = 1$ if $\phi^{l_w}(\eta'_{sl}) \ne e$. By property
$(2)$ of Theorem~\ref{thm:injectivity} applied, for \emph{each}
$l$, to $M^{l_v}$, there is some $s$ such that
$\phi^{l_v}(\eta_{sl}) = \psi(\eta_{sl}) \ne e$. Thus, for each
$l$, the $l^{th}$ term of the summation on the right hand side
above is greater than $\rho_{\mathbb Z}(J^i)-m\rho_{\mathbb Z}(J^j)$, which, by
our choice of $J^i$ and $J^j$ is greater than $2c_M$. Thus the
right hand side is greater than $2kc_M$. This is a contradiction
since the left-hand side above has absolute value less than
$2kc_M$, since $M^{l_v}\cong M^{l_w}\cong M$.
\noindent {\bf Part (6)} : Let $g$ be the genus of $K$. Note that
in the construction of $K_i$ we could have chosen \emph{any}
Seifert surface for $K$, in particular one of genus $g$. Recall
that the circles $\eta_i$ are chosen in the complement of this
Seifert surface. Hence, by construction, genus$(K_i)\leq g$. It
follows that $g_s(K)\leq g$, where $g_s$ denotes the smooth slice
genus, and thus that $\mid s(K_i)\mid \leq 2g$ and $\mid
\tau(K_i)\mid \leq g$ by \cite{Ra} and \cite{OS} respectively.
Consider the function $ST$ from $\{K_i\}$ to
$\mathbb{Z}\times\mathbb{Z}$ given by
$ST(K_i)=(s(K_i),\tau(K_i))$. Since the image of this function is
a finite set, there is a subsequence $K_{i_j}$ on which $ST$ is
constant. Redefining our $K_i$ to be this subsequence gives the
claimed result.
\end{proof}
\section{Algebraic $n$-solution}
\label{sec:algebraic} The purpose of this section is to complete
the proof of Theorem~\ref{thm:injectivity}, which relies on
Theorem~\ref{thm:paininbutt}. In this section we define and
investigate an \emph{algebraic $n$-solution}. One might think of
this as an algebraic abstraction of an $n$-cylinder (or an
$n$-solution). In \cite{CT}, Cochran and Teichner defined an
algebraic $n$-solution. Our notion is much more general. In
particular, an algebraic $n$-solution in \cite{CT} is defined
using only the free group of rank 4, but our version is defined
using the free group of (arbitrary) even rank. Moreover, the proof
of \cite{CT} that a (geometric) $n$-solution induces an algebraic
$n$-solution is valid only for fibered knots of genus $2$. For a
group $G$, let $G_k$ denote $G/G\2k_r$ where $G\2k_r$ is the
$k$-th rational derived group of $G$. Then, as noted before, $G_k$
is a $(k-1)$-solvable PTFA group and embeds into its skew quotient
field of fractions which is denoted by ${\mathbb K}(G_k)$.
\begin{defn}
\label{defn:algebraic} Let S be a group such that $H_1(S;{\mathbb Q})\neq
0$. Suppose $F\overset{i}{\longrightarrow} S$ is a fixed homomorphism from
the free group of rank $2g$. A nontrivial homomorphism $r:S \to G$
is called an {\em algebraic $n$-solution} ($n\ge0$) (for
$F\overset{i}{\longrightarrow} S$) if the following hold :
\begin{enumerate}
\item For each $0\le k\le n-1$ the
image of the following composition, after tensoring with the
quotient field ${\mathbb K}(G_k)$ of ${\mathbb Z} G_k$, is nontrivial:
$$
H_1(S;{\mathbb Z} G_k)\overset{r_*}{\longrightarrow} H_1(G;{\mathbb Z} G_k)\cong
G\2k_{r}/[G\2k_{r},G\2k_{r}]\twoheadrightarrow G\2k_{r}/G^{(k+1)}_{r}.
$$
\item For each $0\le k\le n$, the map $ H_1(F;{\mathbb Z}
G_k)\overset{i_*}{\longrightarrow} H_1(S;{\mathbb Z} G_k)$, after tensoring with the
quotient field ${\mathbb K}(G_k)$, is surjective.
\end{enumerate}
\end{defn}
\begin{rem}
\label{rem:algebraic}
\begin{enumerate}
\item If $n\geq 0$, then, by combining conditions $(1)$ and $(2)$
for $k=0$, we conclude that $ H_1(F;\mathbb{Q})\longrightarrow
H_1(G;\mathbb{Q})$ is non-trivial. Thus, for some $i$,
$r_*(x_{i})$ is non-trivial in $G_1=G/G^{(1)}_{r}$ where
$\{x_1,x_2,\ldots , x_m\}$ is a generating set for $F$.
\item In all the applications the image of the map in $(1)$ above
has rank at least one half the rank of $H_1(S;{\mathbb Z} G_k)$.
\item If $r:S\to G$ is an algebraic $n$-solution
then, for any $k<n$ it is an algebraic $k$-solution.
\end{enumerate}
\end{rem}
\noindent Of course we need to establish that the primary
geometric examples do in fact satisfy the algebraic conditions
above.
\begin{prop}
\label{prop:geomalg} Suppose $K$ is a knot for which the degree of
the classical Alexander polynomial is greater than 2 (if $n=1$
then degree equal to $2$ is allowed). Suppose $M$ is the zero
surgery on $K$ in $S^3$, $\Sigma$ is a capped-off Seifert surface
(of genus $g$) for $K$, and $S \equiv \ensuremath{\pi_1}(M)^{(1)}$. Suppose $F\to
\ensuremath{\pi_1}(M-\Sigma)$ is any map inducing an isomorphism on
$H_1(F;{\mathbb Q})$. Let $i$ be the composition $F\to \ensuremath{\pi_1}(M-\Sigma)\to
S$. Suppose W is an $n$-cylinder one of whose boundary components
is $M$. Let $G \equiv \ensuremath{\pi_1}(W)^{(1)}$ Then the map $j:S\longrightarrow G$
(induced by inclusion) is an algebraic $n$-solution for $i:F\to
S$.
\end{prop}
\begin{proof}
First we will establish property (1) of
Definition~\ref{defn:algebraic}. Since property ($1$) is vacuous
if $n=0$, we assume $n\geq 1$. Fix an arbitrary integer $k$,
$0\leq k < n$. We need to consider the map $H_1(S;{\mathbb Z}
G_k)\overset{j_*}{\longrightarrow} H_1(G;{\mathbb Z} G_k)$ induced by the inclusion
map from $M$ into $W$. Let $M_\infty$ be the infinite cyclic cover
of $M$ and $W_\infty$ be the infinite cyclic cover of $W$. First
note that since $\pi_1(M_\infty)=S$ and $\pi_1(W_\infty)=G$, the
map $j_*$ is identical to $H_1(M_\infty;\mathbb{Z}
G_k)\overset{j_*}{\longrightarrow} H_1(W_\infty;\mathbb{Z} G_k)$. Now let
$\Gamma=\ensuremath{\pi_1}(W)/\ensuremath{\pi_1}(W)^{(k+1)}_{r}$ so $\Gamma$ is PTFA. So we have the
inclusion induced map of $\mathbb{Z}\Gamma$-modules
$j_*:H_1(M;\mathbb{Z}\Gamma)\longrightarrow H_1(W;\mathbb{Z}\Gamma)$. But $G_k$ is
equal to the commutator subgroup of $\Gamma$ since $H_1(W)\cong
\mathbb{Z}$ implying that $\Gamma_r ^{(1)}=\Gamma ^{(1)}=G/G\2k_{r}\equiv
G_k$. Therefore this map can be also viewed as a map of
$\mathbb{Z}G_k$-modules. We claim that, as a map of $\mathbb{Z}
G_k$ modules, this is identical to our original map
$j_*:H_1(M_\infty;\mathbb{Z} G_k)\longrightarrow H_1(W_\infty;\mathbb{Z}
G_k)$. This is because $H_1(M;\mathbb{Z}\Gamma)$ is merely the first
homology group of the total space of the $\Gamma$-covering space of
$M$ viewed as a $\mathbb{Z}\Gamma$-module and this total space is the
same as the total space of the $\Gamma^{(1)}$-covering space of
$M_\infty$. Thus, as $Z\Gamma^{(1)}$-modules, $H_1(M;\mathbb{Z}\Gamma)$ is
the same as $H_1(M_\infty;\mathbb{Z}\Gamma^{(1)})$. Hence we are
reduced to studying $j_*:H_1(M;\mathbb{Z}\Gamma)\longrightarrow
H_1(W;\mathbb{Z}\Gamma)$ as a map of $\mathbb{Z}G_k$-modules and we
need to show that it is non-trivial, even after tensoring with
$\mathbb{K}(G_k)$, i.e. after localizing with respect to the set
$R=\mathbb{Z}\Gamma^{(1)}-\{0\}$. Since $\mathbb{Z}\Gamma$ is an Ore
Domain, $R^{-1}\mathbb{Z}\Gamma$ is a flat $\mathbb{Z}\Gamma$-module and
so we just need to show that the map
$j_*:H_1(M;R^{-1}\mathbb{Z}\Gamma)\longrightarrow H_1(W;R^{-1}\mathbb{Z}\Gamma)$ is
nontrivial as a map of $\mathbb{K}(G_k)$ modules. For the
remainder of this proof, let $\mathbb{K}$ be the quotient field of
$\mathbb{Z}\Gamma^{(1)}=\mathbb{Z}G_k$, which is what we have called
$\mathbb{K}(G_k)$. Finally, note that, since $H_1(\Gamma)\cong
\mathbb{Z}$, $\mathbb{Z}\Gamma$ can be identified with the twisted
Laurent polynomial ring $\mathbb{Z}\Gamma^{(1)}[t^{\pm 1}]$ and,
similarly, the localized ring $R^{-1}\mathbb{Z}\Gamma$ can be
identified with $\mathbb{K}[t^{\pm 1}]$. In summary, we are
reduced to studying $j_*:H_1(M;\mathbb{K}[t^{\pm 1}])\longrightarrow
H_1(W;\mathbb{K}[t^{\pm 1}])$ as a map of $\mathbb{K}$-modules,
and we seek to show that it is non-trivial.
We now claim that Theorem~\ref{thm:rank} applies to the map
$j_*:H_1(M;\mathbb{K}[t^{\pm 1}])\longrightarrow H_1(W;\mathbb{K}[t^{\pm
1}])$. Observe that $\Gamma ^{n}=0$ since $\Gamma^{k+1}=\{e\}$ and, since
$k+1 \leq n$, $\Gamma^{n}=\{e\}$. Thus $\Gamma$ is an $(n-1)$-solvable
PTFA group as required. We conclude that the $\mathbb{K}$-rank of
$j_*:H_1(M;\mathbb{K}[t^{\pm 1}])\longrightarrow H_1(W;\mathbb{K}[t^{\pm
1}])$ is at least $(d-2)/2$ if $n>1$ and is precisely $d/2$ if
$n=1$, where $d\equiv rk_{\mathbb{Q}}H_1(M_\infty;\mathbb{Q})$. It
is well known that $d$ is equal to the degree of the classical
Alexander polynomial of $K$ which is, by hypothesis, at least $4$
(or, if $n=1$ at least $2$). Hence in all cases the above rank is
positive and therefore we have shown that our original map ,
$H_1(S;{\mathbb Z} G_k)\overset{j_*}{\longrightarrow} H_1(G;{\mathbb Z} G_k)$, is
non-trivial even after tensoring with $\mathbb{K}(G_k)$.
Since the kernel of $G\2k_{r}/[G\2k_{r},G\2k_{r}]\longrightarrow
G\2k_{r}/G^{(k+1)}_{r}$ is $\mathbb{Z}$-torsion, this map is an
isomorphism after tensoring with $\mathbb{K}(G_k)$ (which contains
$\mathbb{Q})$. Combining this with our previous conclusion, we
see that $j$ satisfies property $(1)$ of the definition of an
algebraic $n$-solution.
Now we establish property $(2)$ of an algebraic $n$-solution. We
consider an arbitrary $k$ with $0\leq k \leq n$. By flatness, what
we need to establish is the surjectivity of
$i_*:H_1(F;\mathbb{K})\longrightarrow H_1(S;\mathbb{K})$ (recall we are
sometimes abbreviating $\mathbb{K}(G_k)$ by $\mathbb{K}$). Let
$Y=M-\Sigma$ and $W$ be a wedge of $2g$ circles. By choice of $f$
there is a continuous map $W\to Y$ inducing $F\to \pi_1(Y)$ that
is $1$-connected on rational homology. It follows from
\cite[Proposition 2.10]{COT1} that this map induces a
$1$-connected map on homology with $\mathbb{K}(G_k)$ coefficients
for \emph{any} PTFA group $G_k$. Hence it is surjective. Since the
map $i_*$ factors through $H_1(Y;\mathbb{K})$, it suffices to
prove that the map $H_1(Y;\mathbb{K})\to
H_1(S;\mathbb{K})=H_1(M_\infty;\mathbb{K})$ is surjective. If $S$
were finitely generated, this would follow as above from
\cite[Proposition 2.10]{COT1}. In other words, for finitely
generated groups $S$ property $(2)$ follows from just knowing
property $(2)$ for the base case $k=0$. Unfortunately this
Proposition fails for general non-finitely generated groups and,
for us, $S$ will \emph{not} be finitely generated unless $K$ is
fibered. Also note that the rank of $H_1(M_\infty;\mathbb{K})$ may
well increase as the integer $k$ increases. Thus, to have a hope
of establishing property $(2)$, the rank of $H_1(Y;\mathbb{K})$
(which, since $Y$ has the homotopy type of a finite $2$-complex,
is bounded above, independently of $k$, by $2g$ \cite[Prop.2.10
and 2.11]{COT1}) must be a universal upper bound for the ranks of
$H_1(M_\infty;\mathbb{K})$ for \emph{any k} and \emph{any}
coefficient system. Fortunately, the technology to establish this
``universal'' upper bound essentially already exists due to work
of Shelly Harvey. We indicate how her work indeed establishes the
result we need. For intuition, first consider the case $k=0$ where
$G_k=0$ and $\mathbb{K}=\mathbb{Q}$. Then the result we claim is
that any finite generating set for the vector space
$H_1(Y;\mathbb{Q})$ generates the Alexander module of $K$ (or $M$)
as a \emph{rational vector space} (a well-known result in
classical knot theory). We proceed with the proof of the general
case. We observed above that, as $\mathbb{Z}G_k$-modules,
$H_1(M;\mathbb{Z}\Gamma)$ is the same as $H_1(M_\infty;\mathbb{Z}G_k)$
Thus, as $\mathbb{K}$-modules, $H_1(M;\mathbb{K}[t^{\pm 1}])$ is
$H_1(M_\infty;\mathbb{K})$ (see also the discussion above
Definition~\ref{defn:localized Alexander module}). In
\cite[Proposition 7.4]{Ha} (see also \cite[section 6]{C}), Harvey
shows that $H_1(M;\mathbb{K}[t^{\pm 1}])$ has a square
presentation matrix (as a $\mathbb{K}[t^{\pm 1}]$-module) whose
entries are polynomials of degree at most $1$, with respect to
generators that are the images under inclusion of an arbitrary
generating set for $H_1(Y;\mathbb{K}[t^{\pm 1}])$. The latter is a
free module isomorphic to $H_1(Y;\mathbb{K})\otimes_{\mathbb{K}}
\mathbb{K}[t^{\pm 1}]$ (see the discussion in \cite[section
7]{Ha}). Choose a generating set $\{e_1,...,e_m\}$ for
$H_1(Y;\mathbb{K})$ as a $\mathbb{K}$-vector space. Then use the
set $\{e_i\otimes 1\}$ as a $\mathbb{K}[t^{\pm 1}]$ generating set
for the module $H_1(Y;\mathbb{K}[t^{\pm 1}])$. Harvey's
presentation result shows that these elements generate
$H_1(M;\mathbb{K}[t^{\pm 1}])$ \emph{as a $\mathbb{K}[t^{\pm
1}]$-module}. We claim that a closer analysis of some other work
of Harvey shows the stronger fact that these actually generate as
a $\mathbb{K}$-module! This will finish the verification that the
map $H_1(Y;\mathbb{K})\to H_1(M_\infty;\mathbb{K})$ is surjective
and thus finish the verification of property $(2)$. In
\cite[Proposition 9.1]{Ha}, Harvey shows that any such matrix as
above (whose entries are polynomials of degree at most $1$)
presents a module of rank $m$ over $\mathbb{K}$. It is only
necessary to examine her proof carefully to see that this stronger
statement is true: The given set of $m$ generators is a
$\mathbb{K}$-generating set for the module. Harvey's proof
involves changing the presentation matrix by certain allowable
matrix operations and finally arriving at a simple matrix which
she explicitly shows satisfies this stronger statement. Therefore
it is only necessary to check that all the matrix operations she
uses do in fact preserve the veracity of this statement. For
example, clearly $\{x_1,...,x_m\}$ is a $\mathbb{K}$-generating
set for the (right) module if and only if
$\{x_1-(x_2)k,x_2...,x_m\}$ is also, where $k$ is any non-zero
element of $\mathbb{K}$. This translates into the fact that adding
a non-zero left $\mathbb{K}$-multiple of a row of Harvey's matrix
to another row is an allowable operation for our purposes. The key
point is to \emph{not} allow a $\mathbb{K}[t^{\pm 1}]$-multiple.
In fact the only matrix operation from Harvey's list that could
lead to problems is : add to any row a left $\mathbb{K}[t^{\pm
1}]$-linear combination of the other rows (because this
corresponds to a change of generators). One only needs to notice
that, in fact, in her proof she never uses the full generality of
this operation. She only adds to any row a left multiple of
another row by a non-zero element of $\mathbb{K}$ (not
$\mathbb{K}[t^{\pm 1}]$). This completes the proof.
\end{proof}
The following theorem greatly generalizes \cite[Theorem 6.3]{CT}
because our definition of an algebraic $n$-solution is much more
general. Our proof follows the proof of \cite[Theorem 6.3]{CT}
closely. In doing so we had to make decisions about what to
include and what to reference. Because the result is already
(logically) extremely demanding on the reader, for the reader's
convenience, we have erred on the side of duplicating material and
have included a complete proof. In addition, our proof is simpler
in certain places.
\begin{thm}
\label{thm:paininbutt} Suppose we are given $F\overset{i}{\longrightarrow} S$
as in Definition~\ref{defn:algebraic}. For each $n\ge0$ there is a
finite collection $\mathcal P_n$ of sets consisting of $2g-1$, if $n>0$,
or $2g$, if $n=0$, elements of $F\2n$ (we refer to such a set as a
tuple even though it is unordered), with the following property:
For any algebraic $n$-solution $r$ for $F\overset{i}{\longrightarrow} S$, at
least one such tuple (which will be called a {\bf special tuple}
for $r$) maps to a generating set under the composition:
$$
F\2n\longrightarrow S\2n/S\ensuremath{^{(n+1)}}\cong H_1(S;{\mathbb Z} S_n)\overset{r_*}{\longrightarrow}
H_1(S;{\mathbb Z} G_n)\longrightarrow H_1(S;{\mathbb Z} G_n)\otimes_{{\mathbb Z} G_n} {\mathbb K}(G_n)
$$
where ${\mathbb K}(G_n)$ is the skew quotient field of fractions of ${\mathbb Z}
G_n$.
\end{thm}
\begin{proof} We remark that to be used for the proof of the main
theorem, it is very important to define the collections $\mathcal P_n$ so
as to depend only on the knot $K$. In particular, they must not
depend on the existence of any particular $n$-cylinder.
Let $\{x_1,...,x_{2g}\}$ be a generating set of the free group
$F$. Set $\mathcal P_0 \equiv \{\{x_1,...,x_{2g}\}\}$, the collection
consisting of a single $2g$-tuple. Set $\mathcal P_1 \equiv
\{\{[x_i,x_1],\dots,
[x_i,x_{i-1}],[x_i,x_{i+1}],\ldots,[x_i,x_{2g}]\}\,\,|\,\,1\le i
\le 2g\}$, the collection consisting of $(2g-1)\cdot (2g)$ number
of $(2g-1)$-tuples. Supposing $\mathcal P_k$ $(k\ge 1)$ has been defined,
define $\mathcal P_{k+1}$ recursively as follows. For each
$\{w_1,...,w_{2g-1}\}\in$ $\mathcal P_k$ include the $2g-1$-tuple
$\{z_1,...,z_{2g-1}\}$ in $\mathcal P_{k+1}$ if $z_i=[w_i,w_i^{x_j}]$
($1\le i \le 2g-1, 1\le j \le 2g$, and $i\ne j$) or if
$z_i=[w_i,w_k]$ for some $1\le i,k\leq 2g-1$ and $i\ne k$. Here
$w_i^{x_j} \equiv x_j^{-1}w_i x_j$. Clearly $z_i\in F^{(k+1)}$.
Now we fix $n$ and show $\mathcal P_n$ satisfies the conditions of the
theorem. Fix an algebraic $n$-solution $r:S\longrightarrow G$. We must show
that there exists a special tuple in $\mathcal P_n$ corresponding to $r$.
This is trivially true for $n=0$ using property $(2)$ of the
definition of an algebraic $n$-solution, so we assume $n\ge1$. Now
we need some preliminary definitions.
Recall that $F$ is the free group on $\{x_1,\dots,x_{2g}\}$. Its
classifying space has a standard cell structure as a wedge of $2g$
circles $W$. Our convention is to consider its universal cover
$\widetilde W$ as a right $F$-space as follows. Choose a preimage of the
$0$-cell as basepoint denoted $*$. For each element $w\in
F\equiv\ensuremath{\pi_1}(W)$, lift $w^{-1}$ to a path $(\tilde w^{-1})$ beginning at
$*$. There is a unique deck translation $\Phi(w)$ of $\widetilde W$ which
sends $*$ to the endpoint of this lift. Then $w$ acts on $\widetilde W$
by $\Phi(w)$. This is the conjugate action of the usual left
action as in \cite{Ma}. Taking the induced cell structure on $\widetilde
W$ and tensoring with an arbitrary left ${\mathbb Z} F$-module $A$ gives
an exact sequence
\begin{equation}
0\longrightarrow H_1(F;A)\overset{d}{\longrightarrow} A^{2g}\longrightarrow A\longrightarrow H_0(F;A)\lra0.
\end{equation}
Specifically consider $A={\mathbb Z} G$ where ${\mathbb Z} F$ acts by left
multiplication via a homomorphism $\phi:F\longrightarrow G$. From the
interpretation of $H_1(F;{\mathbb Z} G)$ as $H_1$ of a $G$-cover of $W$,
one sees that an element $g$ of $\operatorname{Ker}(\phi)$ can be considered as
an element of $H_1(F;{\mathbb Z} G)$. We claim that the composition
$\operatorname{Ker}(\phi)\longrightarrow H_1(F;{\mathbb Z} G)\overset{d}{\longrightarrow}({\mathbb Z} G)^{2g}$ can
be calculated using the ``free differential calculus''
$\partial=(\partial_1,\dots,\partial_{2g})$ where $\partial_i:F\longrightarrow{\mathbb Z} F$. Specifically
we assert that the diagram below commutes
$$
\begin{diagram}\dgARROWLENGTH=1.2em
\node{F} \arrow[2]{e,t}{\partial} \node[2]{({\mathbb Z} F)^{2g}}
\arrow{s,r}{\phi}\\
\node{\operatorname{Ker}(\phi)} \arrow{n} \arrow{e} \node{H_1(F;{\mathbb Z} G)}
\arrow{e,t}{d} \node{({\mathbb Z} G)^{2g}}
\end{diagram}
$$
where $\partial_i(x_j)=\delta_{ij}$, $\partial_i(e)=0$ and
$\partial_i(gh)=\partial_ig+(\partial_ih)g^{-1}$ for each $1\le i\le{2g}$. Note that the
usual formula for the standard left action is
$d_i(gh)=d_i(g)+gd_i(h)$. Our formula is obtained by setting
$\partial_i=\bar d_i$ where $^-$ is the involution on the group ring.
This is justified in more detail in \cite[Section 6]{CT}.
Henceforth we abbreviate maps of the form $(r,...,r):({\mathbb Z}
F_n)^{2g}\longrightarrow({\mathbb Z} G_n)^{2g}$ as $r$. Note that $r\circ\pi_k:F\to
F_k\to G_k$ is the same as $\pi_k\circ r:F\to S\to G\to G_k$.
\begin{lem}
\label{lem:good} Given an algebraic $n$-solution $r:S\longrightarrow G$ for
$F\to S$, for each $k$, $1\le k\le n$ there is an ordering of the
basis elements $\{x_1,...,x_{2g}\}$ of $F$ such that there exists
at least one tuple $\{w_1,...,w_{2g-1}\}$$\in\mathcal P_k$ with the
following {\bf good} property: The set of 2g-1 vectors (obtained
as $i$ varies from $1$ to $2g-1$)
$(r\pi_k\partial_1w_i,...,r\pi_k\partial_{2g-1}w_i)$ (i.e. the vectors
consisting of the first $2g-1$ coordinates of the images of the
$w_i$ under the composition $F\2k\overset{\pi_k\partial}{\longrightarrow}({\mathbb Z}
F_k)^{2g}\overset{r}{\longrightarrow}({\mathbb Z} G_k)^{2g})$ is right linearly
independent over ${\mathbb Z} G_k$.
\end{lem}
\begin{proof}[Proof that Lemma~\ref{lem:good} $\Longrightarrow$
Theorem~\ref{thm:paininbutt}] The set $\mathcal P_n$ has been defined.
Recall that we are assuming that $n\geq 1$. Given an algebraic
$n$-solution the Lemma provides a tuple
$\{w_1,...,w_{2g-1}\}$$\in\mathcal P_n$ which has the {\bf good}
property. We verify that any \textbf{good} tuple
$\{w_1,...,w_{2g-1}\}$ for $r$ is a {\bf special} tuple for $r$.
Consider the diagram below. Recall $G_n=G/G\2n_{r}$.
$$
\begin{diagram}\dgARROWLENGTH=1.2em
\node{F\2n} \arrow{e} \node{H_1(F;{\mathbb Z} F_n)} \arrow{e,t}{d}
\arrow{s,r}{r_*} \node{({\mathbb Z} F_n)^{2g}} \arrow{s,r}{r}\\
\node[2]{H_1(F;{\mathbb Z} G_n)} \arrow{e,t}{d'} \arrow{s,r}{i_*}
\node{({\mathbb Z} G_n)^{2g}}\\
\node[2]{H_1(S;{\mathbb Z} G_n)}
\end{diagram}
$$
The horizontal composition on top is $\pi_n\circ\partial$. The right-top
square commutes by naturality of the sequence (1) above. The rank
of $H_1(F;{\mathbb Z} G_n)$ over ${\mathbb K}(G_n)$ is $2g-1$ by \cite[Lemma
3.9]{C}. Since $d'$ is a monomorphism, and $i_*$ is an epimorphism
after tensoring with ${\mathbb K}(G_n)$, to show $\{w_1,...,w_{2g-1}\}$
is special, it suffices to show that the set
$\{r\pi_n\partial(w_1),...,r\pi_n\partial(w_{2g-1})\}$ is ${\mathbb Z} G_n$-linearly
independent in $({\mathbb Z} G_n)^{2g}$. This follows immediately from
the {\bf good} property of the tuple $\{w_1,...,w_{2g-1}\}$.
This completes the verification that the Lemma implies the
Theorem.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:good}] The integer $n$
is fixed throughout. Let $r:S\to G$ be a fixed algebraic
$n$-solution. Recall that, by Remark~\ref{rem:algebraic}(1), for
any ordering of the basis elements of $F$, there is some $i$ such
that $r_*(x_{i})$ is non-trivial in $G_1=G/G^{(1)}_{r}$. Thus by
reordering we may assume that $r_*(x_{2g})$ is non-trivial in
$H_1(G;{\mathbb Q})=G_1\otimes{\mathbb Q}$. Since $G_1\equiv G/G\21_{r}$ is
torsion-free, this means that $r\ensuremath{\pi_1}(x_{2g})$ is non-trivial in
$G_1$.
We will prove the Lemma by induction on $k$. We begin with $k=1$.
Consider the $2g-1$-tuple
$\{z_1,...,z_{2g-1}\}$$=\{[x_{2g},x_1],...,[x_{2g},x_{2g-1}]\}\in\mathcal P_1$.
We claim that it has the {\bf good} property.
Using $d$ as shorthand for $\partial_j$, one computes that
$d([g,h])=dg+(dh)g^{-1}-(dg)gh^{-1}g^{-1}-(dh)hgh^{-1}g^{-1}$
Using this we compute that, for any $j$, $1\leq j\leq 2g-1$,
$\partial_jz_i$ is $x_{2g}^{-1}-[x_i,x_{2g}]$ if $j=i$ and is otherwise
zero. Thus, only the $i^{th}$ coordinate of
$(r\pi_k\partial_1z_i,...,r\pi_k\partial_{2g-1}z_i)$ is (possibly) non-zero.
Therefore the square matrix whose columns are these vectors is a
diagonal matrix and to establish the {\bf good} property, it
suffices to show that $r\ensuremath{\pi_1}(x_{2g}^{-1}-[x_i,x_{2g}])\neq 0$ since
this certainly implies that $r\pi_k(x_{2g}^{-1}-[x_i,x_{2g}])\neq
0$. This follows since $r\ensuremath{\pi_1}([x_i,x_{2g}])=e$ and
$r\ensuremath{\pi_1}(x_{2g}^{-1})$ is non-trivial by assumption. Therefore the
base of the induction $(k=1)$ is established.
Now suppose the conclusions of the Lemma have been established for
$1,\dots,k$ where $k<n$. We establish them for $k+1$. Note that
$r:S\to G$ is also an algebraic $k$-solution for $i:F\to S$. By
induction the Lemma holds for $k<n$, so there is a tuple
$\{w_1,...,w_{2g-1}\}$$\in\mathcal P_k$ that has the {\bf good} property.
First, we claim that there is at least one of the $w_i$ (which by
relabelling we will assume is $w_1$)
such that $r\pi_{k+1}(w_1)\neq e$ in
$G_{k+1}$. For, by the \emph{proof} of Lemma~\ref{lem:good}
$\Longrightarrow$ Theorem~\ref{thm:paininbutt}, we know that
$\{w_1,...,w_{2g-1}\}$$\in\mathcal P_k$ is \textbf{special} for $r$. Thus
under the composition $ F\2k\longrightarrow H_1(S;{\mathbb Z}
S_k)\overset{r_*}{\longrightarrow}H_1(S;{\mathbb Z} G_k){\longrightarrow}H_1(S;{\mathbb Z} G_k)\otimes
{\mathbb K}(G_k)$ the set $\{w_i\}$ maps to a generating set. Combined
with the fact that $r$ is also an algebraic $n$-solution, we see
that the composition of the above with the map
$$
H_1(S;{\mathbb Z} G_k)\otimes {\mathbb K}(G_k) \overset{r_*}{\longrightarrow}H_1(G;{\mathbb Z}
G_k)\otimes {\mathbb K}(G_k) \cong G\2k_{r}/G^{(k+1)}_{r}\otimes
{\mathbb K}(G_k)
$$
is nontrivial when restricted to $\{w_i\}$. On the other hand the
combined map $F\2k\longrightarrow G\2k_{r}/G^{(k+1)}_{r}$ is clearly given by
$w_i\mapsto r\pi_{k+1}(w_i)$ so it is not possible that all of the
elements $r\pi_{k+1}(w_i)$ lie in $G^{(k+1)}_{r}$. Hence we may
assume that $r\pi_{k+1}(w_1)\neq e$ in $G_{k+1}$.
Consider the tuple $\{z_1,...,z_{2g-1}\}\in \mathcal P_{k+1}$, where
$z_i=[w_i,w_i^{x_{2g}}]$ if $r\pi_{k+1}(w_i)\neq e$ in $G_{k+1}$,
and $z_i=[w_i,w_1]$ if $r\pi_{k+1}(w_i)= e$ in $G_{k+1}$. We will
show that this tuple has the {\bf good} property, finishing the
inductive proof of Lemma~\ref{lem:good}.
For the remainder of this proof we write $x$ for $x_{2g}$,
suppressing the subscript. We need to show that the set of $2g-1$
vectors (obtained as $i$ varies from $1$ to $2g-1$)
$(r\pi_{k+1}\partial_1z_i,...,r\pi_{k+1}\partial_{2g-1}z_i)$ is ${\mathbb Z}
G_{k+1}$-linearly independent. For this purpose we compute, for
each $i$, $\partial_jz_i$. This computation falls into two cases.
\noindent{\bf Case 1}: $r\pi_{k+1}(w_i) \neq e$.
Let $d$ be shorthand for $\partial_j$ for $1\leq j \leq 2g-1$. One has
$dx=0$, $d(g^{-1})=(dg)g$, and
$dg^x=(dg)x$. Using these one computes that
$d([w_i,w_i^x])=(dw_i)p_i$ where $p_i$ is independent of $j$ and
is equal to $1+xw_i^{-1}-(w_i^x)^{-1}[w_i^x,w_i]-x[w_i^x,w_i]$.
Thus for any value of $i$ that falls under Case $1$, the vector
$(\partial_1z_i,...\partial_{2g-1}z_i)$ is a right multiple of the vector
$(\partial_1w_i,...\partial_{2g-1}w_i)$ by the element $p_i$. Hence the vector
in question, $(r\pi_{k+1}\partial_1z_i,...,r\pi_{k+1}\partial_{2g-1}z_i)$ is a
right multiple of the vector
$(r\pi_{k+1}\partial_1w_i,...,r\pi_{k+1}\partial_{2g-1}w_i)$ by the element
$r\pi_{k+1}p_i$ which lies in ${\mathbb Z} G_{k+1}$. The right factor
$r\pi_{k+1}p_i$ is seen to be non-trivial in ${\mathbb Z} G_{k+1}$ as
follows. Note $r\pi_{k+1}p_i$ is a linear combination of $4$ group
elements $e$, $r\pi_{k+1}(xw_i^{-1})$, $r\pi_{k+1}((w_i^x)^{-1})$,
and $r\pi_{k+1}(x)$ in $G_{k+1}$. For $r\pi_{k+1}p_i$ to vanish in
${\mathbb Z} G_{k+1}$ the group elements would have to pair up in a
precise way and in particular in such a way that
$r\pi_{k+1}(x)=r\pi_{k+1}(xw_i^{-1})$ in $G_{k+1}$. This is a
contradiction since $r\pi_{k+1}(w_i)\neq e$ by hypothesis. No
other pairing is possible because the projections of the $4$
elements to $G_1$ are $e$, $r\ensuremath{\pi_1}(x)$, $e$ and $r\ensuremath{\pi_1}(x)$ and we have
already noted that $r\ensuremath{\pi_1}(x)=r\ensuremath{\pi_1}(x_{2g})$ is non-trivial in $G_1$.
Since these right factors are non-trivial and since ${\mathbb Z} G_{k+1}$
has no zero divisors, the right linear independence of the
collection of vectors obtained by ignoring the $p_i$,
$(r\pi_{k+1}\partial_1w_i,...,r\pi_{k+1}\partial_{2g-1}w_i)$,
would be sufficient to imply the right linear independence of the
original set of vectors. We denote these new vectors by
$\mathbf{v_i^{k+1}}$. So far we have only dealt with those values
of $i$ that fall under Case $1$.
\noindent{\bf Case 2}: $r\pi_{k+1}(w_i)=e$.
As above one computes that $d([w_i,w_1])=(dw_i)q_i +
(dw_1)(w_i^{-1}-[w_1,w_i])$
where $q_i$ is independent of $j$ and is equal to
$1-w_1^{-1}[w_1,w_i]$. Note that under the map $r\pi_{k+1}$ the
factor $(w_i^{-1}-[w_1,w_i])$ goes to zero. Thus for any value of
$i$ that falls under Case $2$, the vector in question,
$(r\pi_{k+1}\partial_1z_i,...,r\pi_{k+1}\partial_{2g-1}z_i)$ is a right
multiple of the vector
$(r\pi_{k+1}\partial_1w_i,...,r\pi_{k+1}\partial_{2g-1}w_i)$ by the element
$r\pi_{k+1}q_i$. We denote the latter vector by
$\mathbf{v_i^{k+1}}$ as above. An argument just as in Case $1$
shows that the right factor $r \pi_{k+1}q_i$ is non-trivial, using
the nontriviality of $r\pi_{k+1}(w_1)$.
Now our objective is to show that the set of vectors
$\mathbf{v_i^{k+1}}$ is ${\mathbb Z} G_{k+1}$-linearly independent.
Recall that our hypothesis is that the set of vectors
$(r\pi_{k}\partial_1w_i,...,r\pi_{k}\partial_{2g-1}w_i)$, which we denote by
$\mathbf{v_i^{k}}$, is ${\mathbb Z} G_{k}$-linearly independent. Note
that, for each $i$, $\mathbf{v_i^{k}}$ is the image of
$\mathbf{v_i^{k+1}}$ under the canonical projection $({\mathbb Z}
G_{k+1})^{2g-1}\longrightarrow ({\mathbb Z} G_k)^{2g-1}$. We assert that the linear
independence of $\mathbf{v_i^{k}}$ implies the linear independence
of $\mathbf{v_i^{k+1}}$ since the kernel of $G_{k+1}\longrightarrow G_k$ is a
torsion free abelian group and hence a $D(\mathbf{Z})$-group in
the sense of R. Strebel. Details follow. This will complete the
verification, that $\{z_1,...z_{2g-1}\}$ has the {\bf good}
property.
To establish our assertion above, consider that the vectors
$\mathbf{v_i^{k+1}}$ describe an endomorphism $f$ of free right
${\mathbb Z} G_{k+1}$ modules $f:({\mathbb Z} G_{k+1})^{2g-1}\longrightarrow({\mathbb Z}
G_{k+1})^{2g-1}$ given by multiplying on the left by the matrix
$M$ whose columns are the $\mathbf{v_i^{k+1}}$. Then $f$ induces
such a map $\bar f:({\mathbb Z} G_k)^{2g-1}\longrightarrow({\mathbb Z} G_k)^{2g-1}$ given
by the matrix $\bar M$ obtained by projecting each entry of $M$.
Thus the columns are the $\mathbf{v_i^{k}}$. Let
$H=G\2k_{r}/G^{(k+1)}_{r}$ and note that ${\mathbb Z}
G_{k+1}\otimes_{{\mathbb Z}[H]}{\mathbb Z}\cong{\mathbb Z} G_k$ where $a\ox1\mapsto\bar a$,
as ${\mathbb Z} G_{k+1}-{\mathbb Z}$ bimodules. Moreover (under this
identification) $f$ descends to $f\otimes\operatorname{id}:({\mathbb Z}
G_k)^{2g-1}\longrightarrow({\mathbb Z} G_k)^{2g-1}$ sending $\bar v$ (for $v\in({\mathbb Z}
G_{k+1})^{2g-1})$ to $\bar f (\bar v)$, thus agreeing with $\bar
f$ above. Our hypothesis on $\mathbf{v_i^{k}}$ guarantees that the
columns of $\bar M$ are right linearly independent and hence that
$\bar f$ is injective. Since $H$ is torsion-free-abelian, a
theorem of Strebel \cite[Section 1]{Str} ensures that the
injectivity of $f\otimes\operatorname{id}=\bar f$ implies the injectivity of $f$.
This in turn implies the linear independence of the columns of
$M$, which are the $\mathbf{v_i^{k+1}}$.
\end{proof}
\end{proof}
|
{
"timestamp": "2005-02-03T18:45:50",
"yymm": "0411",
"arxiv_id": "math/0411641",
"language": "en",
"url": "https://arxiv.org/abs/math/0411641"
}
|
\section{Introduction}
The idea that we could live on a brane embedded in a higher-dimensional
space dates back to~Refs.~\cite{shap,akama} where it was proposed that the universe
be modeled as a topological defect (a domain wall or a string) in a higher-dimensional space-time. The key
feature of these models is the localization of matter on the brane, which
makes the theory look four-dimensional at low energies. In the setup of Ref.~\cite{shap}
this was achieved by means of Yukawa interaction between the domain wall and the
fermion field. The latter has a zero mode localized on the brane, as well
as the continuum modes separated from the zero mode by a gap. At low
energies, only the zero mode remains in the spectrum and plays the role of a
four-dimensional particle. It was suggested later~\cite{visser,squires} that the gravitational
attraction may also trap particles on the brane.
This idea has been the subject of renewed interest since it has been
shown~\cite{rs} that the graviton itself can be trapped on the brane by the
gravitational force, leading to effectively four-dimensional gravity.
Considering a brane embedded in a 5D anti-de Sitter (AdS$_5$) space-time,
the authors of~Ref.~\cite{rs} have found that in this ``warped'' geometry
the 5D graviton has a zero-energy bound state, which, being localized on
the brane, can be interpreted as the ordinary 4D graviton. Even though
there is no gap separating the zero mode of graviton from the continuum,
the law of gravity on the brane is essentially Newtonian, the
corrections yielded by light continuum modes modifying this behavior only at
very short distances.
Soon after, a model was proposed~\cite{grs} in which 4D Newton's law is reproduced
only at intermediate scales and gets modified at very large scales.
The geometry considered consisted of a flat 5D space glued onto a
portion of~AdS$_5$ space. It has been shown that, even though there is no
localized graviton, Newton's gravity can still be recovered on the brane at intermediate
scales. At short distances, the gravitational potential receives
corrections similar to those encountered in the model of~Ref.~\cite{rs}. At
very large scales, the extra dimensions ``open up'' and the gravitational potential receives ``five-dimensional'' contributions from the tensor modes, that is contributions scaling as~$r^{-2}$, $r$ being the distance between sources.
This behavior was related~\cite{ceh,dgp-comment_1} to the
presence of a resonant mode -- a ``quasi-bound state'' -- at zero energy,
with lifetime proportional to the long-distance scale of the model (see
also~Ref.~\cite{grs} itself). Subsequent works~\cite{dgp-comment_1,dgp-comment_2,grs_comment,prz} made manifest the importance of the scalar sector (and namely the radion mode) in the description of the effective gravity on the brane in models with metastable gravitons. Indeed, being constructed from the massive modes, the propagator of the effective 4D graviton is likely to have an incorrect tensor structure,
leading to predictions inconsistent with the observations~\cite{dgp-comment_1,prz}. Coupling to the trace of the energy-momentum tensor, radion contributes to the graviton propagator and thus makes it possible to recover Einstein gravity at the intermediate distances~\cite{grs_comment,prz}. Model presented in~\cite{grs} has however a serious drawback: it requires the use of the negative tension branes -- or, in other words, violation of the condition of positivity of energy~\cite{witten}. It was shown~\cite{prz} that this forces radion to become a ghost~\cite{dgp-comment_2,prz} and causes the effective gravity at very large distances to become scalar anti-gravity~\cite{grs_comment}. While this behavior occurs at the experimentally inaccessible scales, presence of negative norm states certainly raises doubts concerning the consistency of the model.
An alternative brane model producing an infrared modification of gravity
was constructed in~\cite{dgp_1} using a thin brane embedded in five-dimensional Minkowski space. The action included a four-dimensional Einstein term on the brane, induced by radiative
corrections due to the matter fields localized there. This term was shown
to dominate the gravitational interactions of the brane matter at
intermediate scales, giving four-dimensional gravity at intermediate scales, while
at large scales it became five-dimensional.
This setup was subsequently
extended to higher dimensions~\cite{dgp_2} and its generalizations to
smooth branes were also considered~\cite{dgp_follows_2,kolanovic}.
Recently, the idea of induced gravity was applied to the warped space-times
\cite{porrati-strong_coupling,dubovsky,padilla}.
As yet, the setups with induced gravity
are not entirely satisfactory: most of
the models suffer from ghosts and/or
strong gravity
problems~\cite{porrati-strong_coupling,rub-strong_coupling,dub-rubakov}.
Nevertheless, the idea of infrared modification of gravity remains very
appealing as it is thought~\cite{witten,dgp_follows_3} that it could be a solution
to the cosmological constant problem.
The feature shared by these models
is that at ultra-large scales gravity becomes higher-dimensional, that is the gravitational potential scales as
$r^{-(1+N)}$, $r$ being the distance between sources and $N$ the number of
extra dimensions (at least as far as the tensor sector is concerned). It is natural to wonder whether this feature is inherent
to the quasilocalized gravity on branes embedded in
higher dimension space-times or if it could exhibit some different, more
general, behaviors. More generally, one may ask what kinds of large-distance
modifications of gravity are in principle possible in this setup and under
what conditions they can be realized.
In order to cast some light on this issue, we study in this paper tensor perturbations of the metric around a fairly generic five-dimensional warped background, in which the graviton is not fully localized on the brane (becoming a metastable state). To some extent, our model can be considered as a toy model of quasilocalized gravity. Of course, as shows the example of~\cite{grs}, studying only the tensor sector of metric perturbations is unlikely to give a full description of effective gravity on branes and we do not aspire to present a realistic solution for quasilocalization of gravity. The aim of this work is rather to signal a peculiar behavior of the potential yielded by the tensor perturbations in such kinds of backgrounds. For the sake of simplicity, throughout the paper we will loosely refer to the contribution of the tensor modes to the gravitational potential as to the potential itself, bearing in mind that in any concrete model one should account for the contribution from the scalar sector as well. Same applies to our description of gravitational waves.
We find that our model gives 4D gravitational potential at intermediate scales -- quite similarly to the situation encountered in the model of~Ref.~\cite{grs}. This ``quasilocalization'' of gravity is a generic feature of the geometry which we consider -- whether or not the
embedding space-time is asymptotically flat. At large distances, we observe a novel behavior: the
potential gets modified and is of type~$1/r^{1+\alpha}$, $0<\alpha\leq 1$ being a function of the
parameters of the model. It becomes~5D ($\alpha=1$) only when the geometry is
asymptotically flat. Thus, the gravitational potential
does not necessarily
exhibit~5D behavior at very large distances.
Although throughout this paper we use the example and the language of gravity, our analysis is in fact quite universal, in that it is not constrained to the gravity sector, but can apply to the matter sector as well. Indeed, our considerations are based on studying the properties of a Shr\"odinger-like equation governing the behavior of the metric perturbations and, by consequence, the effective physics on the brane. Actually, irrespective of the spin of the matter fields present in the bulk, the mass spectrum of the effective four-dimensional theory is determined by a Schr\"odinger-like equation of some type (or a system of thereof). Using this fact, several authors have studied the localization of different matter fields in the AdS$_5$ background of Ref.~\cite{rs} and its variations: It was shown that, similarly to gravitons, the spectrum of massless bulk scalars~\cite{bajc,giddings} consists of a localized zero mode, followed by the continuum of arbitrarily light states. Massless fermions~\cite{bajc} can also be localized on the brane and mechanisms to localize gauge fields were proposed \cite{drt_2,oda,akhmedov}. Localization of massive scalars and fermions was discussed in Ref.~\cite{drt} (see also Ref.~\cite{ringeval}), where it was shown that while there are no truly localized states (massive or massless), the setup allows for metastable massive particles on the brane, having a small, but finite, probability of tunneling into the bulk. In the scalar case, these resonant states were shown to give $1/r$ contribution to the effective static potential on the brane, while the light states of the continuum give a power-law behavior at large distances. Our analysis completes the picture, revealing that, under certain conditions, in the absence of true bound states and massive resonances, it is possible to have an ``almost localized'' massless state, inducing a $1/r$ potential on the brane evolving towards a (fractional) power-law behavior at large distances.
\section{General Setup}\label{setup}
Let us begin with the general setup. We consider 5D ``warped'' space-times
preserving~4D Poincar\'e symmetry:
\begin{equation}
ds^2=e^{-A(z)}\left(\eta_{\alpha\beta} dx^\alpha dx^\beta-dz^2\right) \ ,
\label{metric}
\end{equation}
where the ``warp factor''~$A(z)$ in an even and non-decreasing function of~$z$.
We suppose that the ordinary matter is localized on the brane centered at
$z=0$. In order to study the nature of the gravitational interactions
experienced by this matter, we need to study perturbations around the
background metric. We will consider exclusively the tensor perturbation~$h_{\alpha\beta}(x,z)=g(z)^{3/2}\tilde{h}_{\alpha\beta}(x)u(z)$, $\alpha$ and~$\beta$ being the four-dimensional tensor indices and~$g(z)$ metric determinant.
The behavior of this perturbation
is governed by the Schr\"odinger-like equation
\begin{equation}
\label{schro}
-\frac{d^2u_\mu(z)}{dz^2}+V(z)u_\mu(z)=\mu^2u_\mu(z)\ ,
\end{equation}
where~$-\partial^\rho\partial_\rho \tilde{h}_{\alpha\beta}(x)=\mu^2 \tilde{h}_{\alpha\beta}(x)$ and the potential is given in terms of the warp factor:
\begin{equation}
\label{pot}
V(z)=\frac{9}{16}A^\prime(z)^2-\frac{3}{4}A^{\prime\prime}(z) \ ,
\end{equation}
which is a form familiar from the supersymmetric quantum mechanics with the
``superpotential'' $ W(z)=\frac{3}{4}A^{\prime}(z)\ . $ The factorization
of the Hamiltonian:
\begin{equation}
-\frac{d^2}{dz^2}+V(z)=\left(-\frac{d}{dz}+W(z)\right)\left(\frac{d}{dz}+W(z)\right)
\end{equation}
ensures that the zero-energy mode
\begin{equation}
u_0(z)=\exp\left[-\frac{3}{4}A(z)\right]
\end{equation}
is the ground state of the system.
In this way, studying
the influence of the tensor modes on the effective gravity on the brane
is reduced to studying the properties of the spectrum of the one-dimensional
quantum mechanical system~$(\ref{schro})$. The quantity of particular
interest is the spectral density~$\rho(\mu)=|u_\mu(0)|^2$, as it enters the
retarded Green's function~$G_R(x,z=0;x^\prime,z^\prime=0)$ which determines
the gravitational interactions on the brane and consequently the observable
physical quantities, such as the induced gravitational potential and
radiation of gravitational waves. Let us remind that the
(five-dimensional) retarded Green's
function~$G_R(x,z=0;x^\prime,z^\prime=0)$ of the tensor sector of the linearized Einstein
equations can be constructed from the full set of eigenmodes of
Eqn.~$(\ref{schro})$ via spectral decomposition:
\begin{widetext}
\begin{equation}
\label{green_f}
G_R(x,z;x^\prime,z^\prime)=\int_0^\infty d\mu \ u_{\mu}(z)
u_{\mu}(z^\prime)\int \frac{d^4
p}{(2\pi)^4}\frac{e^{-ip\cdot(x-x^\prime)}}{\mu^2-p^2-i\varepsilon p^0} \ .
\end{equation}
\end{widetext}
\section{Model}
The purpose of this work is not to present any particular realistic setup
for the quasilocalized gravity, but rather to study general properties of
warped space-times in which this phenomenon can arise and determine what are the possible
large distance behaviors of the gravitational potential (induced by the tensor modes).
With this aim in mind, we construct and study a toy model for a generic geometry allowing
(quasi)localization of gravity. Concretely, we consider a class of
potentials $V(z)$ with ``volcano'' shape, possibly with tails -- for which
we choose~$1/z^2$ fall-off, familiar from the Randall-Sundrum
model~\cite{rs}. Knowing that the potential in question necessarily follows from a
``superpotential'', we start our study by defining one of the following
form:
\begin{equation}
\label{w}
W(z)=-W(-z)=\begin{cases}
a\tan(az)\displaystyle{\frac{}{}} & z<z_0\ , \\
\displaystyle{\frac{}{}}b \tanh\left[b\left(z_2-z\right)\right] & z_0\leq z \leq z_c\ ,\\
\displaystyle{\frac{c}{z-z_1}} & z\geq z_c\ .\\
\end{cases}
\end{equation}
The motivation for choosing this particular form of superpotential is twofold: not only does it yield a
simple potential for which an exact solution of the Schr\"odinger equation can be determined, but also allows
a unified description of the (quasi)localization, naturally including regimes already discussed in the literature~\cite{rs,ceh}.
Our choice of superpotential corresponds in general to geometries which are not
asymptotically flat -- except when the limit~\mbox{$c=0$} is considered.
To ensure that the resulting potential does not have singularities stronger
than finite jumps we demand that~$W(z)$ be continuous. This requirement
entails relations between the constants:
\begin{equation}
\label{cont_w_1}
\begin{cases}
\displaystyle{b\tanh\left[b\left(z_2-z_0\right)\right]=a\tan(az_0)} \ , \\
\displaystyle{b\tanh\left[b\left(z_2-z_c\right)\right]=\frac{c}{z_c-z_1}} \ .
\end{cases}
\end{equation}
The potential generated by the superpotential~$(\ref{w})$ is symmetric and
has the desired ``volcano-like'' shape. It is given by:
\begin{equation}
V(z)=-W^\prime(z)+W(z)^2=
\begin{cases}
-a^2 \displaystyle{\frac{}{}} & |z|<z_0\ , \\
\displaystyle{\frac{}{}}b^2 & z_0<|z|<z_c\ , \\
\displaystyle{\frac{c(1+c)}{(|z|-z_1)^2}} & |z|>z_c\ .\\
\end{cases}
\end{equation}
\begin{figure*}
\includegraphics{potential.eps}
\caption{\small Potential $V(z)$.}
\label{fig:pot}
\end{figure*}
\noindent The parameters of the potential must satisfy the condition following from the constraints~$(\ref{cont_w_1})$ by eliminating~$z_2$:
\begin{equation}
\label{cont_w_2}
c\left[1-\frac{a}{b}\tan(az_0)\tanh\left[b\left(z_c-z_0\right)\right]\right]=\left(z_c-z_1\right)\left[a\tan(az_0)-b\tanh\left[b\left(z_c-z_0\right)\right]\right] \ .
\end{equation}
In order for~$(\ref{cont_w_1})$ to remain valid we must also have
$$
b\geq a\tan(az_0)\qquad \qquad \mbox{and}\qquad \qquad b\geq\frac{c}{z_c-z_1} \ .
$$
In spite of its simplicity, the potential~$V(z)$ presents a range of behaviors, depending on the choice of parameters, and thus generates different localization schemes for gravity.
To start, in the limit~$z_c\to \infty $, $V(z)$ becomes a simple rectangular potential well, allowing a localized symmetric zero-mode of the form:
\begin{equation}
u^\infty_{0}(z) ={\cal{C}}^\infty_0
\begin{cases}
\displaystyle{\cos(az)\frac{}{}} & z<z_0 \\
\displaystyle{\cos(az_0)e ^{-b (z-z_0)}\frac{}{}} & z\geq z_0
\end{cases}
\end{equation}
where
\begin{equation}
\label{C_infty}
\left({\cal{C}}^\infty_0\right)^{-2} =z_0 +\frac{1}{a}\cot(az_0)\ ,
\end{equation}
followed by the continuum spectrum starting at~$\mu=b$.
For finite $z_c$,
the asymptotics of the potential makes the continuum descend towards~$\mu=0$. Condition~$(\ref{cont_w_2})$ guarantees the absence of growing contribution to the~$\mu=0$ wave function in the region~$z>z_c$, where it consequently falls off as~$(z-z_1)^{-c}$.
When $c> 1/2$, it is therefore still normalizable and, at the bottom of the continuum, we still have a localized zero-mode. The resulting gravity on the brane will then be of the type described in~Ref.~\cite{rs}.
When $c\leq 1/2$, $u_{0}(z)$ is no longer normalizable and the corresponding state can be qualified as ``quasi-bound''.
In the particular case~$\nu=1/2$, $V(z)$ is identical to the ``volcano box'' potential used in~\cite{ceh} in order to explain the quasilocalization of gravity in the asymptotically flat model~\cite{grs}. We claim that quasilocalization of gravity is present for the whole family of potentials with~$c\leq 1/2$. Let us from now on concentrate on this family.
\section{Wave-function and spectral density}
\noindent The solution for a symmetric continuum wave function has the form:
\begin{widetext}
\begin{equation}
\label{wave_f}
u_{\mu}(z) = u_{\mu}(0)
\begin{cases}
\displaystyle{\cos(kz)\frac{}{}} & z\leq z_0\ , \\
\displaystyle{D_1(\mu) e ^{-\kappa z}+ D_2(\mu)e^{\kappa z}\frac{}{}} & z_0<z<z_c\ , \\
\displaystyle{\sqrt{z-z_1}\left[\frac{}{}B_1(\mu)J_\nu\left(\mu(z-z_1)\right)+B_2(\mu)N_\nu\left(\mu(z-z_1)\right)\right]} & z\geq z_c\ .
\end{cases}
\end{equation}
where
\begin{equation}
k=\sqrt{\mu^2+a^2}\ ,\quad \kappa=\sqrt{b^2-\mu^2}\quad \mbox{and} \quad \nu=c+\frac{1}{2}\ .
\end{equation}
\end{widetext}
The coefficients in~$(\ref{wave_f})$ are determined, as usual,
by the requirement of continuity of the wave function and its
derivative at~$z=z_0$ and~$z=z_c$:
\begin{widetext}
\begin{subequations}
\begin{align}
B_1(\mu)&=\frac{\pi}{2}\sqrt{z_c-z_1}\left\{\left[\kappa f_{-}(\mu)+\frac{2\nu+1}{2(z_c-z_1)}f_{+}(\mu)\right]N_\nu(\mu(z_c-z_1))-\mu f_{+}(\mu)\, N_{\nu+1}(\mu(z_c-z_1))\right\} \ , \\
B_2(\mu)&=-\frac{\pi}{2}\sqrt{z_c-z_1}\left\{\left[\kappa f_{-}(\mu)+\frac{2\nu+1}{2(z_c-z_1)}f_{+}(\mu)\right]J_\nu(\mu(z_c-z_1))-\mu f_{+}(\mu)\, J_{\nu+1}(\mu(z_c-z_1))\right\}\ ,\\
f_{+}(\mu)&= D_1(\mu)e^{-\kappa z_c} + D_2(\mu)e^{\kappa z_c}=\cos(kz_0)\cosh\left[\kappa(z_c-z_0)\right]-\frac{k}{\kappa}\sin(kz_0)\sinh\left[\kappa(z_c-z_0)\right]\ ,\\
f_{-}(\mu)&= D_1(\mu)e^{-\kappa z_c} - D_2(\mu)e^{\kappa z_c}=-\cos(kz_0)\sinh\left[\kappa(z_c-z_0)\right]+\frac{k}{\kappa}\sin(kz_0)\cosh\left[\kappa(z_c-z_0)\right]\ .
\end{align}
\end{subequations}
\end{widetext}
The remaining overall constant $u_\mu(0)$, and consequently the spectral density~$\rho(\mu)=|u_{\mu}(0)|^2$ can be determined from the normalization condition:
\begin{equation}
\int_{-\infty}^\infty dz\ u_{\mu}(z) u^{*}
_{\mu^\prime}(z)=\delta(\mu-\mu^\prime)\ ,
\end{equation}
which gives
\begin{equation}
\label{rho}
\rho(\mu)=\frac{\mu}{2}\frac{1}{|B_1(\mu)|^2+|B_2(\mu)|^2}\ .
\end{equation}
As we have already stressed in section~\ref{setup},~$\rho(\mu)$ is the quantity relevant to the effective gravity on the brane and it is therefore useful to examine its behavior as a function of~$\mu$ in some detail.
In the following, we assume that~$b(z_c-z_0)\gg 1$, that is that the tunneling probability of the light modes through the potential barriers is very small. We suppose also that the constant~$z_1$ is adjusted in such a way that the ratio~$(z_c-z_1)/(z_c-z_0)\ll 1$.
For~$\mu\ll \mu_1\equiv\min\left(a,b,(z_c-z_1)^{-1}\right)$, we can approximate~$B_1(\mu)$ and~$B_2(\mu)$ by their power series in~$\mu$:
\begin{eqnarray}
B_1(\mu) &=& a_0 \mu^{-\nu}+a_1\mu^{-\nu+2}+\dots+b_0 \mu^{\nu}+\dots \\
B_2(\mu) &=& c_0\mu^\nu+\dots
\end{eqnarray}
The condition of continuity of the superpotential~$(\ref{cont_w_2})$ forces~$a_0$ to vanish and guarantees that~$c_0$ and $b_0$ are small (of order $\exp[-b(z_c-z_0)]$).
As a consequence~we have for $\rho(\mu)$:
\begin{equation}
\label{C_approx}
\rho(\mu)\approx\frac{\sin^2(\nu\pi)}{\pi^2}\frac{1}{\mu (z_c-z_1)}\frac{A(\mu)}{\alpha_0^2+\alpha_1^2A(\mu)+\alpha_2^2 A(\mu)^2}\ ,
\end{equation}
where $A(\mu)$ stands for:
\begin{equation}
A(\mu)=\left[\mu(z_c-z_1)\right]^{2-2\nu}e^{2b(z_c-z_0)}
\end{equation}
and
\begin{equation}
\alpha_0^2 = \frac{2^{3-2\nu}}{\Gamma(\nu)^2}\frac{a^2b^2}{a^2+b^2}\left[b+\frac{c}{z_c-z_1}\right]^{-2}\ , \qquad
\alpha_1^2 = -\frac{\sin(2\nu\pi)}{2\pi b}\frac{1+bz_0}{z_c-z_1} \ , \qquad
\alpha_2^2 = \frac{1}{4\cos^2(\nu\pi)}\frac{\alpha_1^4}{\alpha_0^2} \ .
\end{equation}
The equality~$(\ref{C_approx})$ shows that for small $\mu$'s the spectral density~$\rho(\mu)$ is a uniformly decreasing function of~$\mu$, strongly peaked at~$\mu=0$ where it has a singularity of the form $\mu^{(1-2\nu)}$. The bulk of the weight of~$\rho(\mu)$ lies in the region between~$\mu=0$ and $\mu\sim\mu_1\exp\left(-\frac{b}{1-\nu}(z_c-z_0)\right)$ where~$A(\mu)\sim {\cal O}(1)$. We have therefore in our problem two characteristic mass scales, $\mu_1=\min\left(a,b,(z_c-z_1)^{-1}\right)$ and~$\mu_2=\mu_1\exp\left(-\frac{b}{1-\nu}(z_c-z_0)\right)$. The assumption of small tunneling probability ensures that these scales are well separated,~$\mu_2\ll\mu_1$.
\noindent Let us remark that in the limit~$\nu=1/2$ ,
we recover the results of~\cite{ceh} and $\rho(\mu)$ assumes the Breit-Wigner form:
\begin{equation}
\rho(\mu)\approx\frac{\cal{A}}{\mu^2+\Delta\mu^2}\ ,
\end{equation}
with the resonance width given by
\begin{equation}
\Delta\mu\approx \frac{8}{\left(1+b^2/a^2\right)\left(z_0+1/b\right)}e^{-2b(z_c-z_0)}\ .
\end{equation}
To conclude this chapter, let us discuss briefly the connection between the spectral density~$\rho(\mu)$ and
the scattering matrix of our one-dimensional problem.
The relevant eigenvalue of the $S$-matrix is:
\begin{equation}
\label{s-matrix}
S_+(\mu)=\frac{B_1(\mu)-iB_2(\mu)}{B_1(\mu)+i B_2(\mu)}e^{-2i\left(\frac{\pi}{2}\nu+\frac{\pi}{4}+\mu z_1\right)}=\frac{\Phi(\mu)}{{\Phi}^{*}(\mu)}\ ,
\end{equation}
where the Jost function $\Phi(\mu)$ is defined in such a way that tends to one in the non-interaction limit and is given by:
\begin{eqnarray}
\Phi(\mu)&=&\sqrt{\frac{2}{\pi\mu}}\,e^{-i\left(\frac{\pi}{2}\nu+\frac{\pi}{4}+\mu z_1\right)}\left[B_1(\mu)-iB_2(\mu)\right]\ .
\end{eqnarray}
Comparing the expression for the spectral density~$(\ref{rho})$ to~$(\ref{s-matrix})$, we see that $\rho(\mu)$ can be written as
$$
\rho(\mu)=\frac{\pi}{\Phi(\mu)\Phi^{*}(\mu)}\ .
$$
It is known from the formal scattering theory~\cite{taylor} that physical properties of a quantum system are closely related with the analytic properties of its scattering matrix. In particular, poles of the $S$-matrix in the complex $\mu$-plane correspond, depending on their location, to bound states, virtual states or resonances of the system.
In our problem, $S$-matrix has in general a branch point singularity at $\mu$=0, which it inherits from the Bessel functions $J_\nu(\mu(z_c-z_1))$ and~$N_\nu(\mu(z_c-z_1))$ entering the expression $(\ref{s-matrix})$. The results of the next section will demonstrate that this type of singularity can also strongly influence the physics.
\section{Effective gravity}
Let us now turn to the contribution of the tensor modes to the effective gravity on the brane.
The static potential between two sources on the brane, separated by the distance~$r\equiv|\vec{x}-\vec{x}^\prime|$
receives Yukawa-type contributions from all the modes and is given by:
\begin{equation}
V_G(r)=\frac{G_5}{4\pi}\int_0^\infty d\mu\ \frac{e^{-\mu r}}{r} |u_{\mu}(0)|^2 =\frac{G_5}{4\pi}\int_0^\infty d\mu\ \frac{e^{-\mu r}}{r} \rho(\mu)\ .
\end{equation}
It is convenient to divide the integral into two parts:
\begin{equation}
V_G(r)=\frac{G_5}{4\pi}\int_0^\infty d\mu\ \frac{e^{-\mu r}}{r}\rho(\mu) =\frac{G_5}{4\pi}\int_0^{\mu_1} d\mu\ \frac{e^{-\mu r}}{r}\rho(\mu)+\frac{G_5}{4\pi}\int_{\mu_1}^\infty d\mu\ \frac{e^{-\mu r}}{r} \rho(\mu) \ .
\end{equation}
For distances~$r\gg r_{1}={\mu}_1^{-1}$ the second integral is negligible
and in the first one we can replace~$\rho(\mu)$ by its approximate form~$(\ref{C_approx})$, which gives us:
\begin{equation}
\label{V_approx}
V_G(r)=G_5\frac{\sin^2(\nu\pi)}{4\pi^3 (z_c-z_1)}\int_0^{\mu_1} d\mu\ \frac{e^{-\mu r}}{\mu r}\frac{A(\mu)}{\alpha_0^2+\alpha_1^2A(\mu)+\alpha_2^2 A(\mu)^2}\ .
\end{equation}
This integral is always saturated for~$A(\mu)\sim {\cal O}(1)$, that is for
$\mu\sim \mu_2\ll \mu_1$.
We can therefore extend the integration to infinity and evaluate the resulting integral in two regimes:
$r_{1}\ll r\ll r_2=\mu_2^{-1}$ and $r\gg r_2$.
In the first regime, that is for distances $r_{1}\ll r\ll r_2$, the exponential in~$(\ref{V_approx})$ can be set to 1 and we obtain:
\begin{equation}
V_G(r)=\frac{G_5}{4 \pi r}\int_0^\infty \frac{d\mu}{\mu}\ \frac{\sin^2(\nu\pi)}{\pi^2 (z_c-z_1)}\frac{A(\mu)}{\alpha_0^2+\alpha_1^2A(\mu)+\alpha_2^2 A(\mu)^2} =\frac{\left({\cal{C}}^\infty_0\right)^2}{4\pi}\frac{G_5}{r} \ ,
\end{equation}
where ${\cal{C}}^\infty_0$ is defined by~$(\ref{C_infty})$. Therefore, in this (large) interval of distances, light modes of the continuum spectrum reproduce precisely the same~$1/r$ potential we would get in the presence of a massless boson (or, in other words, a normalizable zero mode).
In the second regime, that is for large distances, $r\gg r_2$, only $A\ll 1$ will give significant contributions to the integral. We can therefore neglect $A(\mu)$ and $A(\mu)^2$ terms in the denominator of the integrand in~$(\ref{V_approx})$, thus obtaining:
\begin{align}
V_G(r)&=\frac{\sin^2(\nu\pi)}{4\pi^3\alpha_0^2}(z_c-z_1)^{1-2\nu}e^{2b(z_c-z_0)}\frac{1}{r}\int_0^\infty d \mu \ e^{-\mu r}\mu^{1-2\nu}\nonumber \\
&=\frac{\sin^2(\nu\pi)}{4\pi^3\alpha_0^2}(z_c-z_1)^{1-2\nu}e^{2b(z_c-z_0)}\Gamma(2-2\nu)\frac{G_5}{r^{3-2\nu}}\ .
\end{align}
Hence, we have a modification of the~$1/r$ law for large distances, with the potential of type~$r^{-\beta}$,where~$\beta\in(1,2)$. Thus, the gravity does not in general become five-dimensional, except in an asymptotically flat geometry ($\nu=1/2$), where the potential is:
\begin{equation}
V_G(r)=\frac{a^2+b^2}{16\pi^2 a^2}e^{2b(z_c-z_0)}\frac{G_5}{r^{2}} \ .
\end{equation}
Following~\cite{grs}, we can also study the propagation of gravitational waves generated by a periodic point-like source on the brane, $T(x,z)=T(\vec{x})e^{-i\omega t}\delta(z)$ (we omit the four-dimensional indices). The field induced on the brane is given by the convolution of the source with the Green's function~$(\ref{green_f})$
\begin{equation}
G_R(\vec{x}-\vec{x}^\prime;\omega)=\int_{-\infty}^{\infty}d(t-t^\prime) \ G_R(x,z=0;x^\prime,z^\prime=0)\ e^{-i\omega (t-t^\prime)}\ ,
\end{equation}
which after inserting~$(\ref{green_f})$
and simplifying becomes
\begin{equation}
\label{G_omega}
G_R(\vec{x}-\vec{x}^\prime;\omega)=\frac{1}{4\pi}\int_0^\infty d\mu\ \frac{e^{i\omega_\mu r}}{r}\ \rho(\mu) \ ,
\end{equation}
where again $r\equiv|\vec{x}-\vec{x}^\prime|$ and~$\omega_\mu=\sqrt{\omega^2-\mu^2}$ when~$\mu<\omega$ and~$\omega_\mu=i\sqrt{\mu^2-\omega^2}$ when~$\mu>\omega$. Only modes with~$\mu<\omega$ are actually radiated; the other ones exponentially fall off from the source. Being interested in the propagation of waves, we can integrate in~$(\ref{G_omega})$ only up to~$\mu=\omega$. Considering the range of frequencies $r_2^{-1}\ll\omega\ll r_1^{-1}$
allows us to replace~$\rho(\mu)$ by~$(\ref{C_approx})$ and approximate~$\omega_\mu$ by~$\omega-\frac{\mu^2}{2\omega}$:
\begin{equation}
\label{G_omega_approx}
G_R(\vec{x}-\vec{x}^\prime;\omega)=\frac{\sin^2(\nu\pi)}{4\pi^3}\frac{e^{i\omega r}}{r}\int_0^\infty \ d\mu \ e^{-i(\mu^2/2\omega)r}\frac{A(\mu)}{\alpha_0^2+\alpha_1^2A(\mu)+\alpha_2^2 A(\mu)^2} \ ,
\end{equation}
where we have extended the integration to infinity, using the fact that the integral is saturated for~$\mu\sim \mu_2\ll\omega$.
For distances $r\gg 2\omega\, r_2/r_1$
the phase factor in~$(\ref{G_omega_approx})$ varies very slowly and can be set to 1. We then obtain the usual~$1/r$ dependence of wave amplitude on the distance to the source:
\begin{equation}
G_R(\vec{x}-\vec{x}^\prime;\omega)= \frac{\left({\cal{C}}^\infty_0\right)^2}{4\pi}\frac{e^{i\omega r}}{r}\ .
\end{equation}
For distances $r\gg 2\omega\, r_2/r_1$
we can neglect $A(\mu)$ and $A(\mu)^2$ terms in the denominator of the integrand in~$(\ref{G_omega_approx})$, obtaining
\begin{eqnarray}
G_R(\vec{x}-\vec{x}^\prime;\omega)&=&\frac{\sin^2(\nu\pi)}{4\pi^3\alpha_0^2}\Gamma(1-\nu)(z_c-z_1)^{1-2\nu}(2\omega)^{1-\nu}e^{2b(z_c-z_0)} \frac{e^{i\omega r+ i \pi(\nu -1)/2}}{r^{2-\nu}} \nonumber\\
&=& (z_c-z_1)^{1-2\nu}\frac{(2\omega)^{1-\nu}}{\Gamma(1-\nu)}\frac{e^{2b(z_c-z_0)}}{4\pi\cos^2(az_0)}\frac{e^{i\omega r+ i \pi(\nu -1)/2}}{r^{2-\nu}} \ ,
\end{eqnarray}
that is, the amplitude is proportional to~$r^{-\sigma}$, where~$\sigma\in(1,\frac{3}{2})$. This behavior indicates dissipation of the gravitational waves into the extra dimension.
\section{Conclusions}
In this paper we have studied a toy model for quasilocalization of gravity
on a brane embedded in an asymptotically warped space-time, restricting our attention to the
tensor sector of metric perturbations. While, admittedly, our model is only of limited use as a model for gravity, it allows to uncover quite peculiar behavior of the gravitational potential
(and related quantities) induced by the tensor modes. In the absence of a normalized zero-mode, the effective potential is of the 4D form $1/r$, regardless of
whether or not the embedding space-time is asymptotically flat. At large scales, the presence of the extra dimensions becomes significant and the potential gets modified. The nature of
this modification depends on the asymptotic geometry of the space-time and potential does not necessarily assume the 5D form $1/r^2$.
Although we did not present a concrete model which would possess
asymptotically non-flat solutions with quasilocalized gravity, some
features inherent to such models can be derived. By Einstein equations, the
warped geometry~(\ref{metric}) corresponds to the matter energy-momentum
tensor which obeys
\begin{equation}
\kappa^2\left(\rho + p_z\right) = {\rm e}^{A(z)}\left( \frac{3}{4} A'^2 + \frac{3}{2} A''\right),
\label{energy-cond}
\end{equation}
where $\rho$ and $p_z$ are the energy density and pressure in the
$z$ direction, respectively. Making use of Eq.~(\ref{w}) at large $z$ we
find that our asymptotically non-flat ansatz requires $\rho + p_z<0$ and
thus violates the weak energy condition. In this respect, our model is
quite similar to (and may share the problems of) that of
Ref.~\cite{grs}. It remains yet to be seen whether these problems can be
overcome. This is related to an important question, namely the impact of the scalar sector
of metric perturbations on the effective gravity in a theory where the embedding space-time is not
asymptotically flat and the graviton is quasi-localized. It is legitimate to suspect
that it could be as important as in the model of~Ref.~\cite{grs} and this matter should be
examined on a concrete brane setup.
Finally, it is worth pointing out that Eq.~(\ref{schro}) which is the
starting point of our analysis
is quite universal, in that it describes -- with a suitably adjusted potential --
the localization of matter fields as well. Therefore, our conclusions are directly applicable to the case of
matter fields whenever the corresponding potential can be cast in a
``supersymmetric'' form~$V(z)=W^2(z)-W^\prime(z)$.
The latter condition is equivalent to
the requirement that the four-dimensional particle is massless (the massive
case was treated in Ref.~\cite{drt}). In the case of matter,
the potential in Eq.~(\ref{schro}) is no longer related to the metric of
space-time through the relation~(\ref{pot})
and the problem of violation of the weak energy condition does not need to arise.
\begin{acknowledgments}
The work of P. T. and M. S. is supported in part by Swiss National Science Foundation.
\end{acknowledgments}
|
{
"timestamp": "2004-11-02T11:21:42",
"yymm": "0411",
"arxiv_id": "hep-th/0411031",
"language": "en",
"url": "https://arxiv.org/abs/hep-th/0411031"
}
|
\section{introduction}
In brane world scenarios, matter fields are confined to a four dimensional
dimensional hypersurface, or brane, embedded in a higher dimensional bulk.
Brane cosmology can reproduce the most important features of the the more
usual four dimensional cosmology, but sometimes the extra dimensions make
themselves known and one such effect is considered here.
Many brane world scenarios are built around the Randall-Sundrum models
\cite{randall99b,randall99a}, where there is a five dimensional anti-de Sitter
bulk spacetime and either a double or a single brane. Inflationary solutions
to the single brane Randall-Sundrum model can be obtained by simply
`detuning' the vacuum energy of the brane \cite{Kaloper:1999sm,Nihei:1999mt}.
This can be done with the extra vacuum energy of a brane or a bulk inflaton
field. The former case is called matter field inflation \cite{Lukas:1999yn}.
In 4-dimensional models, hot big bang radiation is produced by the decay of the
inflaton field. In 5 dimensions, the inflaton can decay into ordinary matter
fields or into bulk fields. In most models of inflation, the coupling between
the inflaton and other matter fields is very small, and may be smaller even
than the coupling to gravity or other bulk fields. Some of the inflaton's
vacuum energy can easily end up in the bulk.
In the most extreme case, nearly all of the vacuum energy of the inflaton field
may decay into bulk radiation. This happens according to one scenario, put
forward by Enqvist et al. \cite{Enqvist:2003qc,enqvist04}. In their scenario,
the entropy of the universe arises from fields associated with the flat
directions of the Minimally Supersymmetric Standard Model (MSSM). For this
scenario to work, the energy released onto the brane by these fields has to
dominate over the energy produced by the decay of the inflaton, which can
happen if the inflaton decays into bulk degrees of freedom.
The transfer of energy from a brane to the bulk is always associated with a
geometrical effect on the brane which mimics a radiation source
term in the cosmological evolution equations
\cite{kraus99,kehagias99,cline99,binetruy99,flanagan99,hebecker01,langlois02,langlois03}.
The size of this dark radiation term in relation to the real radiation
is an important feature, because we have strong evidence that the real
radiation provides the dominant proportion of the radiation density driving
the expansion rate at the nucleosynthesis era.
The purpose of this paper is to construct a simple model of the inflaton decay
process with a single brane and a massless scalar bulk field. The bulk
field might occur as a hypothetical superpartner of the graviton or a bulk
dilaton field, for example. We shall construct the reheating terms in the
inflaton's equation of motion and solve for the case where bulk radiation is
the predominant form of reheating. We shall then examine the cosmological
evolution to determine the residual amount of dark radiation.
The action for the model takes the form
\begin{equation}
S_g=-{1\over \kappa_5^2}\int_{\cal M}\left (R-2\Lambda\right)\,dv
+{2\over \kappa_5^2}\int_{\cal\partial M}K\,dv
+\int_{\cal M}{\cal L}_b\,dv
+\int_{\cal \partial M}\left({\cal L}_m-\sigma\right)\,dv
\end{equation}
where we have a curved space ${\cal M}$ with scalar curvature $R$ and a
boundary ${\cal \partial M}$ with extrinsic curvature $K$. The covariant
volume integral is denoted by $dv$. We will take coordinates
$x^\mu$, $\mu=0,1,2,3$, along the brane and $x^5=z$ perpendicular to the
brane. In the final state after inflation, the vacuum energy $\sigma$ and the
5-dimensional cosmological constant $\Lambda$ satisfy the fine tuning
relationship $6\Lambda=\kappa_5^4\sigma^2$
\cite{randall99b,randall99a}. The effective four dimensional Newton's constant
$G$ is given by $48\pi G=\kappa_5^4\sigma$.
The inflaton field $\phi(x)$ is restricted to the brane, with
Lagrangian density
\begin{equation}
\mathcal{L}_m = -\frac{1}{2}(\partial_{\mu}\phi)^{2} -V(\phi).
\label{Lphi}
\end{equation}
In the bulk, there is a massless conformally coupled scalar field $\Phi(x)$
with Lagrangian density
\begin{equation}
\mathcal{L}_b = -\frac{1}{2}(\partial_a\Phi)^{2}-\frac3{32}R\Phi^2.
\end{equation}
Conformally invariant boundary conditions fix the normal derivative of the Bulk
scalar. We consider an interaction which allows $\phi$ to spontaneously
decay into $\Phi$. Since the $\phi$ field is confined on the brane, the two
fields can only interact at $z=0$. A simple model for the Lagrangian density of
the interaction is
\begin{equation}
\mathcal{L}_{I} = -\frac{\lambda}{2}\phi(x)\Phi(x)^{2}\delta(z),
\end{equation}
where $\lambda$ is the coupling constant \cite{enqvist04}.
The effect of the radiation on the inflaton field equation is a damping term.
The general form of this term is identical to the four dimenisonal case which
can be found in ref. \cite{gleiser94}. For a flat
brane,
\begin{equation}
{d^2\phi\over dt^2}+{d V\over d\phi}=J
\end{equation}
where the leading order contribution to the dissipation is
\begin{equation}
J(t)=\lambda^{2}\textrm{Im}\int d^{4}x' G_F(x,x')^{2}
\phi(t')\theta(t-t')\Big{|}_{z=z'=0}, \label{integral}
\end{equation}
and $G_F(x,x')$ is the Feynman propagator of the bulk field $\Phi$ with Robin
boundary conditions on the brane.
For flat extra dimensions,
\begin{equation}
G_F(x,x') = \int \frac{d^{3}k}{(2\pi)^{3}}
\frac{dq}{2\pi}\,4\cos\,qz\cos\,qz' e^{i\vec{k}\cdot(\vec{x}-\vec{x}')}
\frac{i}{2\omega}e^{-i\omega(t-t')}, \label{G>}
\end{equation}
where $\omega=(\vec{k}^{2}+q^{2})^{1/2}$. The intergral (\ref{integral}) is
divergent, but we can apply dimensional regularisation. In five dimensions we
find
\begin{equation}
J(t) = -\frac{\lambda^{2}}{512\pi^{2}}\ln\left(\frac{1}{\mu}
\frac{d}{dt}\right)\left(\frac{d}{dt}\right)^{2}\phi(t), \label{resultS}
\end{equation}
where $\mu$ is the one-loop renormalisation scale. The same renormalisation
scale appears in the one-loop corrections to the potential. In order for these
corrections to remain relatively small, $\mu$ has to be large compared to
other mass scales in potential.
The logarithmic time derivative appearing in eq. (\ref{resultS}) is a
feature of radiation back reaction effects in odd spacetime dimensions
\cite{Moss:2004dp}. For practical applications, we can use an integral kernel
to define the logarithmic derivative (see \cite{Moss:2004dp}),
\begin{equation}
\ln\left({1\over\mu}{d\over dt}\right)\phi=
-\int_{-\infty}^t\ln\left(e^\gamma\mu(t-t')\right){d\phi\over dt'}dt'
\label{logdt}
\end{equation}
where $\gamma$ is Euler's constant.
This result which we have found applies to flat branes and flat extra
dimensions. However, we can make some simple generalisations. In the first
place, for the special case of the conformal scalar field, we can use the same
result for any bulk spacetime which is conformal to flat space. In particular,
the result also applies to anti-de Sitter extra dimensions.
Another generalisation can be made if the brane is spatially homogeneous with
expansion rate $H$ in the range $H^2\ll \dot\phi$. In four dimensions, the
amplitude of density perturbations is of order $H^2/\dot\phi$, and $H^2\ll
\dot\phi$ is consistent with having small amplitude density fluctuations. In
this case the time evolution of the inflaton is the dominant effect producing
the dissipative term in the inflaton equation and we can use the flat space
result (\ref{resultS}).
The complete form for inflaton equation of motion on an expanding brane with
$H^2\ll\dot\phi$ is
\begin{equation}
{d^2\phi\over dt^2}+3H{d\phi\over dt}+{dV\over d\phi}=
-c\ln\left({1\over \mu}{d\over dt}\right){d^2\phi\over dt^2}\label{infeq}
\end{equation}
where $c=\lambda^2/(512\pi^2)$.
The form of the damping term is strongly dependent on the fact that we have
not introduced any explicit mass terms into the bulk fields. When the same
calculation is repeated for a bulk field of mass $M$, there is no damping of
the inflaton when $\dot\phi<M^2$. This does not, however, rule out the
possibility of higher loop or non-perturbative damping effects.
Inflation is associated with a slow roll approximation in which the
leading order terms in the inflaton equation have the smallest number of time
derivatives. The damping term has too many time derivatives to appear at
leading order. The effects of damping will start to appear after inflation, in
the period which is usually associated with reheating. The situation is
similar to reheating, appart from the fact that, according to our assumptions,
the radiation is radiated into the bulk.
To investigate the period following inflation, consider a quadratic form of
potential with minimum at $\phi=0$,
\begin{equation}
V(\phi)=\frac12 m^2\phi^2
\end{equation}
An approximate solution can be constructed by taking
\begin{equation}
\phi(t)=A(t)\sin\theta(t)
\end{equation}
where $A(t)$ is a slowly varying function of $t$ and $\theta\approx mt$. From
the definition (\ref{logdt}),
\begin{equation}
\ln\left({1\over \mu}{d\over dt}\right){d^2\phi\over dt^2}\approx
m^2\ln\left({m\over \mu}\right)A\sin(\theta)+m^2\left(Si(\theta)\cos(\theta)
-Ci(\theta)\sin(\theta)\right)A
\end{equation}
where $Si$ and $Ci$ are sine and cosine integrals. After solving (\ref{infeq})
for an initial value $\phi=\phi_i$ at $t=t_i$, we obtain
\begin{equation}
A(t)\approx\phi_i\left({a_i\over a}\right)^{3/2}e^{-\pi cm(t-t_i)/4}
\end{equation}
If we also set the initial condition to coincide with the end of inflation,
then $\phi_i\sim M_p$, where $M_p^2=(8\pi G)^{-1}$.
The effect of the inflaton on the expansion of the universe can be obtained
from the brane cosmology equations
\cite{kraus99,kehagias99,cline99,binetruy99,flanagan99,shiromizu99}.
We shall only consider the case where the intrinsic brane vacuum energy is
large compared to the density, $\sigma\ll\rho$, when
\begin{equation}
6\dot H+12 H^2=8\pi G(\rho-3p)-2\kappa_5^2T^r_{55}\label{branecos}
\end{equation}
where $T^r_{ab}$ is the stress-energy of the bulk radiation. Since we are
assuming that the radiation in mostly into the extra dimensions,
the density $\rho$ and pressure $p$ are those of the inflaton field,
\begin{eqnarray}
\rho&=&\frac12\dot\phi^2+\frac12m^2\phi^2\\
p&=&\frac12\dot\phi^2-\frac12m^2\phi^2
\end{eqnarray}
The radiation term can be obtained by similar means to the dissipative term
\cite{Moss:2004dp}. The terms up to order $\lambda^2$ are
\begin{equation}
T^r_{55}=
-{3\lambda\over 2048\pi^2}
\ln\left(\frac{1}{\mu}\frac{d}{dt}\right){d^4\phi\over dt^4}+
c'{d\phi\over dt}\ln\left(\frac{1}{\mu}\frac{d}{dt}\right){d^2\phi\over dt^2}
\end{equation}
where $c'=O(\lambda^2)$. The leading term averages out to zero over a period of
the oscillating inflaton field.
The dark radiation $\rho_{dark}$, defined by
\begin{equation}
3H^2=8\pi G(\rho+\rho_{dark}),\label{friedman}
\end{equation}
can be obtained by integrating (\ref{branecos}). We shall express the result in
terms of the radiation flux $T^r_{05}$, using the conservation of energy,
$T^r_{05}=-\frac12J\dot\phi$.
If we assume that
$\rho_{dark}\to 0$ as $t\to-\infty$, then
\begin{equation}
\rho_{\rm dark}=
-{1\over a^4}\int_{-\infty}^\infty\,a(t')^4\,T^r_{05}\,dt'
-{\kappa_5^2\over 8\pi G}
{1\over a^4}\int_{-\infty}^\infty H(t')a(t')^4T^r_{55}dt'
\end{equation}
The first term in identical to the amount of radiation which would be produced
if the inflaton's energy was decaying into brane radiation rather than bulk
radiation. The second term turns out to be less important than the first term
when $\sigma<\rho$.
In the early stages of reheating, the inflaton field energy density dominates
the Friedman equation but decays faster than the dark radiation density, which
will eventually come to dominate. There is some intermediate time $t_{eq}$ at
which the scalar field energy density is equal to the dark energy density.
In the WKB approximation, after averaging over the phase, the radiation flux is
given by
\begin{equation}
T^r_{05}\approx \frac14cA^2m^3\ln\left({m\over\mu}\right)
\end{equation}
The inflaton looses energy provided that $m<\mu$. Most of the the dark energy
is generated whilst the scalar field dominates eq. (\ref{friedman}). In this
regime,
\begin{equation}
3H^2\approx4\pi Gm^2A^2
\end{equation}
We can solve this equation and evaluate the integral to obtain the dark energy
at the time $t_{eq}$,
\begin{equation}
\rho_{dark}\approx-{3\pi\over 40}c^2m^2M_p^2\ln\left({m\over\mu}\right)
\end{equation}
where $M_p^2=(8\pi G)^{-1}$. This is also the maximum value reached by the dark
radiation density.
For comparison, the value of the potential at the end of the inflationary era
would be of order $V_i\sim m^2M_p^2$. Therefore
$\rho_{dark}/V_i\sim10^{-8}\lambda^4$. A typical inflationary model,
which was conventional in every respect appart from the reheating, might have
$V_i\sim 10^{16}$ GeV. There is no particular reason why $\lambda$ should be
very small, and a dark radiation `temperature' as high as $10^{14}$ GeV is a
possibility.
We have so far said nothing about the generation of real radiation. Let us
suppose that this is produced from a field which lies along a flat direction in
the potential. These fields have a non-vanishing potential $V_F$ as a result
of supersymmetry breaking and possible non-renormalisable terms
\cite{Enqvist:2003gh}.
If the dark energy density $\rho_{dark}<V_F$ at $t_{eq}$, then reheating (or
more specifically preheating \cite{Kofman:1997yn}) would give a
radiation energy density $\rho_r\sim V_F$, and the ratio
\begin{equation}
{\rho_{dark}\over \rho_r}\sim 10^{-8}\lambda^4{V_i\over V_F}
\end{equation}
If, on the other hand, $\rho_{dark}>V_F$ and $H\gg V_F''{}^{1/2}$, then the
field evolves very slowly along the flat direction. One of two things can
happen. The expansion rate $H$ can fall below $V_F''{}^{1/2}$
whilst $\rho_{dark}> V_F$. The field will then oscillate about the minimum of
the potential and reheat to give a radiation energy density
$\rho_r<\rho_{dark}$. The second possibility is that $\rho_{dark}$ falls below
$V_F$ whilst $H> V_F''{}^{1/2}$. In this case, we can have reheating with
$\rho_r>\rho_{dark}$ and one has to analyse the combination of
dark reheating with real reheating for the particular model to obtain the
ratios of dark radiation to real radiation.
|
{
"timestamp": "2004-11-01T17:51:00",
"yymm": "0411",
"arxiv_id": "hep-ph/0411021",
"language": "en",
"url": "https://arxiv.org/abs/hep-ph/0411021"
}
|
\chapter*{Prologue}
What is this book about and why is it so long? Parametrized homotopy theory concerns systems of spaces and spectra that are parametrized as fibers over points of a given base space $B$. Parametrized spaces, or ``ex-spaces'', are just spaces over and under $B$, with a projection, often a fibration, and a section. Parametrized spectra are analogous but considerably more sophisticated objects. They provide a world in which one can apply the methods of stable homotopy theory without losing track of fundamental groups and other unstable information. Parametrized homotopy theory is a natural and important part of homotopy theory that is implicitly central to all of bundle and fibration theory. Results that make essential use of it are widely scattered throughout the literature. For classical examples, the theory of transfer maps is
intrinsically about parametrized homotopy theory, and Eilenberg-Moore type spectral sequences are parametrized K\"unneth theorems. Several new and
current directions, such as ``twisted'' cohomology theories
and parametrized fixed point theory cry out for the rigorous foundations that
we shall develop.
On the foundational level, homotopy theory, and especially stable homotopy theory, has undergone a thorough reanalysis in recent years. Systematic use of Quillen's theory of model categories has illuminated the structure of the subject and has done so in a way that makes the general methodology widely applicable to other branches of mathematics. The discovery of categories of spectra with associative and commutative smash products has revolutionized stable homotopy theory. The systematic study and application of equivariant algebraic topology has greatly enriched the subject.
There has not been a thorough and rigorous study of parametrized homotopy theory that takes these developments into account. It is the purpose of this monograph to provide such a study, although we shall leave many interesting loose ends. We shall also give some direct applications, especially to equivariant stable homotopy theory where the new theory is particularly essential. The reason this study is so lengthy is that, rather unexpectedly, many seemingly trivial nonparametrized results fail to generalize, and many of the conceptual and technical obstacles to a rigorous treatment have no nonparametrized counterparts. For this reason, the resulting theory is considerably more subtle than its nonparametrized precursors. We indicate some of these problems here.
The central conceptual subtlety in the theory enters when we try to prove that structure enjoyed by the point-set level categories of parametrized spaces descends to their homotopy categories. Many of our basic functors occur in Quillen adjoint pairs, and such structure
descends directly to homotopy categories. Recall that an adjoint pair of functors $(T,U)$ between model categories is a Quillen adjoint pair, or a Quillen adjunction, if the left adjoint $T$ preserves cofibrations and acyclic cofibrations or, equivalently, the right adjoint $U$ preserves fibrations and acyclic fibrations. It is a Quillen equivalence if, further, the induced adjunction on homotopy categories is an adjoint equivalence. We originally
hoped to find a model structure on para\-me\-trized spaces in which all of the relevant adjunctions are Quillen adjunctions. It eventually became clear that there can be no such model structure, for altogether trivial reasons. Therefore,
it is intrinsically impossible to lay down the basic foundations of parametrized homotopy theory using only the standard methodology of model category theory.
The force of parametrized theory largely comes from base change functors associated to maps $f\colon A\longrightarrow B$. The existing literature on fiberwise homotopy theory says surprisingly little about such functors.
This is especially strange since they are the most important feature that makes parametrized homotopy theory useful for the study of ordinary homotopy theory:
such functors are used to transport information from the parametrized context
to the nonparametrized context. One of the goals of our work is to
fill this gap.
On the point-set level,
there is a pullback functor $f^*$ from ex-spaces (or spectra) over $B$ to ex-spaces (or spectra) over $A$. That functor has a left adjoint $f_!$ and a right adjoint $f_*$. We would like both of these to be Quillen adjunctions, but that is not possible unless the model structures lead to trivial homotopy categories. We mean literally trivial: one object and one morphism. We explain why. It will be clear that the explanation is generic and applies equally well to a variety of sheaf theoretic situations where one encounters analogous base change functors.
\begin{ouch0}\mylabel{noway} Consider the following diagram.
$$\xymatrix{
\emptyset \ar[r]^-{\phi} \ar[d]_{\phi} & B \ar[d]^{i_{0}} \\
B \ar[r]_-{i_{1}} & B\times I}$$
Here $\emptyset$ is the empty set and $\phi$ is the initial (empty) map into $B$. This diagram is a pullback since $B\times\{0\}\cap B\times\{1\} = \emptyset$. The category of ex-spaces over $\emptyset$ is the trivial category with one object, and it admits a unique model structure. Let $*_B$ denote the ex-space $B$ over $B$, with section and projection the identity map. Both $(\phi_!,\phi^*)$ and $(\phi^*,\phi_*)$ are Quillen adjoint pairs for any model structure on the category of ex-spaces over $B$. Indeed, $\phi_!$ and $\phi_*$ preserve weak equivalences, fibrations, and cofibrations since both take $*_{\emptyset}$ to $*_B$. We have $(i_0)^*\com (i_1)_! \iso \phi_!\com\phi^*$ since both composites take any ex-space over $B$ to $*_B$. If $(i_1)_!$ and $(i_0)^*$ were both Quillen left adjoints, it would follow that this isomorphism descends to homotopy categories. If, further, the functors $(i_1)_!$ and $(i_0)^*$ on homotopy categories were equivalences of categories, this would imply that the homotopy category of ex-spaces over $B$ with respect to the given model structure is equivalent to the trivial category.
\end{ouch0}
Information in ordinary homotopy theory is derived from results in parametrized homotopy theory by use of the base change functor $r_!$ associated to the trivial map $r\colon B\longrightarrow *$. For this and other reasons, we choose our basic model structure to be one such that $(f_!,f^*)$ is a Quillen adjoint pair for every map $f\colon A\longrightarrow B$ and is a Quillen equivalence when $f$ is a homotopy equivalence. Then $(f^*,f_*)$ cannot be a Quillen adjoint pair in general. However, it is essential that we still have the adjunction $(f^*,f_*)$ after passage to homotopy categories. For example, taking $f$ to be the diagonal map on $B$, this adjunction is used to obtain the adjunction on homotopy categories that relates the fiberwise smash product functor $\sma_B$ on ex-spaces over $B$ to the function ex-space functor $F_B$. To construct the homotopy category level right adjoints $f_*$, we shall have to revert to more classical methods, using Brown's representability theorem. However, it is not clear how to verify the hypotheses of Brown's theorem in the model theoretic framework.
\myref{noway} also illustrates the familiar fact that a commutative diagram of functors on the point-set level need not induce a commutative diagram of functors on homotopy categories. When commuting left and right adjoints,
this is a problem even when all functors in sight are parts of Quillen
adjunctions. Therefore, proving that compatibility relations that hold on the point-set level descend to the homotopy category level is far from automatic.
In fact, proving such ``compatibility relations'' is often a highly
non-trivial problem, but one which is essential to the applications.
We do not know how to prove the most interesting compatibility relations
working only model theoretically.
Even in the part of the theory in which model theory works, it does not work as expected. There is an obvious naive model structure on ex-spaces over $B$ in which the weak equivalences, fibrations, and cofibrations are the ex-maps whose maps of total spaces are weak equivalences, fibrations, and cofibrations of spaces in the usual Quillen model structure. This ``$q$-model structure'' is the natural starting point for the theory, but it turns out to have severe drawbacks that limit its space level utility and bar it from serving as the starting point for the development of a useful spectrum level stable model structure. In fact, it has two opposite drawbacks. First, it has too many cofibrations. In particular, the model theoretic cofibrations need not be cofibrations in the intrinsic homotopical sense. That is, they fail to satisfy the fiberwise homotopy extension property (HEP) defined in terms of parametrized mapping cylinders. This already fails for the sections of cofibrant objects and for the inclusions of cofibrant objects in their cones. Therefore the classical theory of cofiber sequences fails to mesh with the model category structure.
Second, it also has too many fibrations. The fibrant ex-spaces are Serre fibrations, and Serre fibrations are not preserved by fiberwise colimits. Such colimits are preserved by a more restrictive class of fibrations, namely the well-sectioned Hurewicz fibrations, which we call ex-fibrations. Such preservation properties are crucial to resolving the problems with base change functors that we have indicated.
In model category theory, decreasing the number of cofibrations increases the number of fibrations, so that these two problems cannot admit a solution in common. Rather, we require two different equivalent descriptions of our homotopy categories of ex-spaces. First, we have another model structure, the ``$qf$-model structure'', which has the same weak equivalences as the $q$-model structure but has fewer cofibrations, all of which satisfy the fiberwise HEP. Second, we have a description in terms of the classical theory of ex-fibrations, which does not fit naturally into a model theoretic framework. The former is vital to the development of the stable model structure on parametrized spectra. The latter is vital to the solution of the intrinsic problems with base change functors.
Before getting to the issues just discussed, we shall have to resolve various others that also have no nonparametrized analogues. Even the point set topology requires care since function ex-spaces take us out of the category of compactly generated spaces. Equivariance raises further problems, although most of our new foundational work is already necessary nonequivariantly. Passage to the spectrum level raises more serious problems. The main source of difficulty is that the underlying total space functor is too poorly behaved, especially with respect to smash products and fibrations, to give good control of homotopy groups as one passes from parametrized spaces to parametrized spectra. Moreover, the resolution of base change problems requires a different set of details on the spectrum level than on the space level.
The end result may seem intricate, but it gives a very powerful framework in which to study homotopy theory. We illustrate by showing how fiberwise duality and transfer maps work out and by showing that the basic change of groups isomorphisms of equivariant stable homotopy theory, namely the generalized Wirthm\"uller and Adams isomorphisms, drop out directly from the foundations. Costenoble and Waner \cite{CWNew} have already given other applications in equivariant stable homotopy theory, using our foundations to study Poincar\'e duality in ordinary $RO(G)$-graded cohomology. Further applications are work
in progress.
The theory here gives perhaps the first worked example in which a model theoretic approach to derived homotopy categories is intrinsically insufficient and must be blended with a quite different approach even to establish the essential structural features of the derived category. Such a blending of techniques seems essential in analogous sheaf theoretic contexts that have
not yet received a modern model theoretic treatment.
Even nonequivariantly, the basic results on base change, smash products, and function ex-spaces that we obtain do not appear in the literature. Such results are essential to serious work in parametrized homotopy theory.
Much of our work should have applications beyond the new parametrized theory.
The model theory of topological enriched categories has received much less attention in the literature than the model theory of simplicially enriched
categories. Despite the seemingly equivalent nature of these variants,
the topological situation is actually quite different from the simplicial one,
as our applications make clear. In particular, the interweaving of $h$-type
and $q$-type model structures that pervades our work seems to have no simplicial
counterpart. This interweaving does also appear in algebraic contexts
of model categories enriched over chain complexes, where foundations analogous
to ours can be developed. One of our goals is to give a thorough analysis and axiomatization of how this interweaving works in general in topologically enriched model categories.
\vspace{.7mm}
{\em History.} This project began with unpublished notes, dating from the summer of 2000, of the first author \cite{May}. He put the project aside and returned to it in the fall of 2002, when he was joined by the second author. Some of Parts I and II was originally in a draft of the first author that was submitted and accepted for publication, but was later withdrawn. That draft was correct, but it did not include the ``$qf$-model structure'', which comes from the second author's 2004 PhD thesis \cite{Sig}. The first author's notes \cite{May} claimed to construct the stable model structure on parametrized spectra starting from the $q$-model structure on ex-spaces. Following \cite{May}, the monograph \cite{Hu} of Po Hu also takes that starting point and makes that claim. The second author realized that, with the obvious definitions, the axioms for the stable model structure cannot be proven from that starting point and that any
naive variant would be disconnected with cofiber sequences and other essential
needs of a fully worked out theory. His $qf$-model structure is the crucial new ingredient that is used to solve this problem. Although implemented quite differently, the applications of Chapter 16 were inspired by Hu's work.
\pagebreak
{\em Thanks.} We thank the referee of the partial first version for several helpful suggestions, Gaunce Lewis and Peter Booth for help with the point set topology, Mike Cole for sharing his remarkable insights about model
categories, and Mike Mandell for much technical help. We thank Kathleen
Lewis for working out the counterexample in Theorem 1.1.1 and Victor Ginzburg for giving us the striking Counterexample 11.6.2.
We are especially grateful to Kate Ponto for a meticulously
careful reading that uncovered many obscurities and infelicities.
Needless to say, she is not to blame for those that remain.
\part{Point-set topology, change functors, and proper actions}
\chapter{The point-set topology of parametrized spaces}
\section*{Introduction}
We develop the basic point-set level properties of the category of ex-spaces over a fixed base space $B$ in this chapter. In \S1.1, we discuss convenient categories of topological spaces. The usual category of compactly generated spaces is not adequate for our study of ex-spaces, and we shall see later that the interplay between model structures and the relevant convenient categories is quite subtle. In \S1.2, we give basic facts about based and unbased topologically bicomplete categories. This gives the language that is needed to describe the good formal properties of the various categories in
which we shall work. We discuss convenient categories of ex-spaces in \S1.3,
and we discuss convenient categories of ex-$G$-spaces in \S1.4.
As a matter of recovery of lost
folklore, \S\ref{sec:topass} is an appendix, the substance of
which is due to Kathleen Lewis. It is only at her insistence that she is not
named as its author. It documents the nonassociativity of the smash product
in the ordinary category of based spaces, as opposed to the category
of based $k$-spaces. When writing the historical paper \cite{History},
the first author came across several 1950's references to this
nonassociativity,
including an explicit, but unproven, counterexample in a 1958 paper of
Puppe \cite{Puppe}. However, we know of no reference that gives details, and we
feel that this nonassociativity should be documented in the modern literature.
We are very grateful to Gaunce Lewis for an extended correspondence
and many details about the material of this chapter, but he is not to be blamed for the point of view that we have taken. We are also much indebted to Peter Booth. He is the main pioneer of the theory of fibered mapping spaces (see \cite{B1, B2, B3}) and function ex-spaces, and he sent us several detailed proofs about them.
\section{Convenient categories of topological spaces}\label{sec:pt}
We recall the following by now standard definitions.
\begin{defn}
Let $B$ be a space and $A$ a subset. Let $f\colon K\longrightarrow B$ run
over all continuous maps from compact Hausdorff spaces $K$ into $B$.
\begin{enumerate}[(i)]
\item $A$ is \emph{compactly closed}\index{compactly closed} if each $f^{-1}(A)$ is closed.
\item $B$ is \emph{weak Hausdorff}\index{weak Hausdorff} if each $f(K)$ is closed.
\item $B$ is a \emph{$k$-space}\index{kspace@$k$-space} if each compactly closed subset is closed.
\item $B$ is \emph{compactly generated}\index{compactly generated!space} if it is a weak Hausdorff $k$-space.
\end{enumerate}
Let ${\scr{T}}\!op$\@bsphack\begingroup \@sanitize\@noteindex{Top@${\scr{T}}op$}\index{category!of spaces} be the category of all topological spaces and let $\scr{K}$,\@bsphack\begingroup \@sanitize\@noteindex{K@$\scr{K}$} $w\scr{H}$, and $\scr{U} = \scr{K}\cap w\scr{H}$\@bsphack\begingroup \@sanitize\@noteindex{U@$\scr{U}$}
be its full subcategories of $k$-spaces, weak Hausdorff spaces, and compactly generated
spaces. The \emph{$k$-ification}\index{kification@$k$-ification functor} functor $k\colon {\scr{T}}\!op\longrightarrow \scr{K}$ assigns to a space $X$ the
same set with the finer topology that is obtained by requiring all compactly closed
subsets to be closed. It is right adjoint to the inclusion $\scr{K}\longrightarrow {\scr{T}}op$. The
\emph{weak Hausdorffication}\index{weak Hausdorffication functor} functor $w\colon {\scr{T}}op\longrightarrow w\scr{H}$ assigns to a space $X$ its
maximal weak Hausdorff quotient. It is left adjoint to the inclusion $w\scr{H}\longrightarrow {\scr{T}}\!op$.
\end{defn}
>From now on, we work in $\scr{K}$, implicitly $k$-ifying any space that
is not a $k$-space to begin with. In particular, products and function spaces are understood to be $k$-ified. With this convention, $B$ is weak Hausdorff
if and only if the diagonal map embeds it as a closed subspace of $B\times B$.
Let $A\times_c B$ denote the classical cartesian product in ${\scr{T}}op$ and
recall that $B$ is Hausdorff if and only if the diagonal embeds it as a
closed subspace of $B\times_c B$. The following result is proven in
\cite[App.\S2]{Lewis0}.
\begin{prop}\mylabel{HauswHaus}
Let $A$ and $B$ be $k$-spaces. If one of them is locally
compact or if both of them are first countable, then
$$A\times B = A\times_c B.$$
Therefore, if $B$ is either locally compact or first countable, then
$B$ is Hausdorff if and only if it is weak Hausdorff.
\end{prop}
We would have preferred to work in $\scr{U}$ rather than $\scr{K}$, since there are
many counterexamples which reveal the pitfalls of working without a separation
property. However, as we will explain in \S1.3, several inescapable facts
about ex-spaces force us out of that convenient category.
Like $\scr{U}$, the category $\scr{K}$ is closed cartesian monoidal. This means
that it has function spaces $\text{Map}(X,Y)$\@bsphack\begingroup \@sanitize\@noteindex{MapXY@$\text{Map}(X,Y)$} with homeomorphisms
$$ \text{Map}(X\times Y,Z)\iso \text{Map}(X,\text{Map}(Y,Z)).$$
This was proven by Vogt \cite{Vogt}, who uses the term compactly
generated for our $k$-spaces. See also \cite{Wylie}. An early unpublished preprint by Clark \cite{Clark} also showed
this, and an exposition of ex-spaces based on \cite{Clark} was given by Booth \cite{B2}.
Philosophically, we can justify a preference for $\scr{K}$ over $\scr{U}$ by remarking
that the functor $w$ is so poorly behaved that we prefer to minimize its use.
In $\scr{U}$, colimits must be constructed by first constructing them in $\scr{K}$ and then applying the functor $w$, which changes the underlying point set and loses
homotopical control. However, this justification would be more persuasive were it not that colimits in $\scr{K}$ that are not colimits in $\scr{U}$ can already be quite badly behaved topologically. For example, $w$ itself is a colimit construction in $\scr{K}$. We describe a relevant situation in which colimits behave better in $\scr{U}$ than in $\scr{K}$ in \myref{Umod} below.
More persuasively, $w$ is a formal construction that only retains formal
control because both colimits and the functor $w$ are left adjoints. We will encounter right adjoints constructed in $\scr{K}$ that do not preserve the weak Hausdorff property when restricted to $\scr{U}$, and in such situations we cannot apply $w$ without losing the adjunction. In fact, when restricted to $\scr{U}$,
the relevant left adjoints do not commute with colimits and so cannot be left adjoints there. We shall encounter other reasons for working in $\scr{K}$ later. An obvious advantage of $\scr{K}$ is that $\scr{U}$ sits inside it, so that we can use $\scr{K}$ when it is needed, but can restrict to the better behaved category $\scr{U}$ whenever possible. Actually, the situation is more subtle than a simple dichotomy. In our study of ex-spaces, it is essential to combine use of the
two categories, requiring base spaces to be in $\scr{U}$ but allowing
total spaces to be in $\scr{K}$.
We have concomitant categories $\scr{K}_*$\@bsphack\begingroup \@sanitize\@noteindex{Kpt@$\scr{K}_*$}\index{category!of based spaces} and $\scr{U}_*$\@bsphack\begingroup \@sanitize\@noteindex{Upt@$\scr{U}_*$} of based spaces in $\scr{K}$ and in $\scr{U}$. We generally write $\scr{T}$\@bsphack\begingroup \@sanitize\@noteindex{T@$\scr{T}$} for $\scr{U}_*$ to mesh with
a number of relevant earlier papers. Using duplicative notations, we write $\text{Map}(X,Y)$ for the space $\scr{K}(X,Y)$ of maps $X\longrightarrow Y$
and $F(X,Y)$\@bsphack\begingroup \@sanitize\@noteindex{FXY@$F(X,Y)$} for the based space $\scr{K}_*(X,Y)$ of based maps $X\longrightarrow Y$ between based spaces. Both $\scr{K}_*$ and $\scr{T}$ are closed symmetric monoidal categories under $\sma$ and $F$ \cite{Lewis0, Vogt, Wylie}. This means that the smash
product is associative, commutative, and unital up to coherent natural isomorphism and that $\sma$ and $F$ are related by the usual adjunction homeomorphism
$$ F(X\sma Y, Z) \iso F(X,F(Y,Z)).$$
The need for $k$-ification is illustrated by the nonassociativity of the smash product in ${\scr{T}}\!op_*$; see \S1.5.
We need a few observations about inclusions and colimits.
Recall that a map is an inclusion if it is a homeomorphism onto its image.
Of course, inclusions need not have closed image. As noted by Str{\o}m \cite{Strom1},
the simplest example of a non-closed inclusion in $\scr{K}$ is the inclusion $i\colon \{a\}\subset \{a,b\}$, where $\{a,b\}$ has the indiscrete topology.
Here $i$ is both the inclusion of a retract and a Hurewicz cofibration (satisfies the homotopy extension property, or HEP). As is well-known,
such pathology cannot occur in $\scr{U}$.
\begin{lem}\mylabel{coflemma} Let $i\colon A\longrightarrow X$ be a map in $\scr{K}$.
\begin{enumerate}[(i)]
\item If there is a map $r\colon X\longrightarrow A$ such that $r\com i = \text{id}$,
then $i$ is an inclusion. If, further, $X$ is in $\scr{U}$, then $i$ is a closed inclusion.
\item If $i$ is a Hurewicz cofibration, then $i$ is an inclusion. If, further, $X$ is in $\scr{U}$, then $i$ is a closed inclusion.
\end{enumerate}
\end{lem}
\begin{proof}
Inclusions $i\colon A\longrightarrow X$ are characterized by the property that a function $j\colon Y\longrightarrow A$ is continuous if and only if $i\com j$ is continuous. This implies the first statement in (i).
Alternatively, one can note that a map in $\scr{K}$ is an inclusion if and only if it is an equalizer in $\scr{K}$, and a map in $\scr{U}$ is a closed inclusion if and only if it is an equalizer in $\scr{U}$ \cite[7.6]{Lewis0}. Since $i$ is the equalizer of $i\com r$ and the identity map of $X$, this implies both statements in (i).
For (ii), let $Mi$ be the mapping cylinder of $i$. The canonical map $j\colon Mi\longrightarrow X\times I$ has a left inverse $r$ and is thus an inclusion or closed inclusion in the respective cases. The evident closed inclusions $i_1\colon A\longrightarrow Mi$ and $i_1\colon X\longrightarrow X\times I$ satisfy $j\com i_1 = i_1\com i$, and the conclusions of (ii) follow.
\end{proof}
The following remark, which we learned from Mike Cole \cite{Cole3}
and Gaunce Lewis, compares certain colimits in $\scr{K}$ and $\scr{U}$.
It illuminates the difference between these categories and will
be needed in our later discussion of $h$-type model structures.
\begin{rem}\mylabel{Umod}
Suppose given a sequence of inclusions $g_n\colon X_n\longrightarrow X_{n+1}$ and maps $f_n\colon X_n\longrightarrow Y$ in $\scr{K}$ such that $f_{n+1} g_n = f_{n}$. Let $X=\text{colim}\, X_n$ and let
$f\colon X\longrightarrow Y$ be obtained by passage to colimits. Fix a map $p\colon Z\longrightarrow Y$.
The maps $Z\times_Y X_n\longrightarrow Z\times_Y X$ induce a map
$$\alpha\colon \text{colim}\, (Z\times_Y X_n) \longrightarrow Z\times_Y X.$$
Lewis has provided counterexamples showing that $\alpha$ need not
be a homeomorphism in general. However, if $Y\in \scr{U}$, then a
result of his \cite[App.\,10.3]{Lewis0} shows that $\alpha$ is a
homeomorphism for any $p$ and any maps $g_n$. In fact,
as in \myref{second} below, if $Y\in\scr{U}$, then the pullback
functor $p^*\colon \scr{K}/Y\longrightarrow \scr{K}/Z$ is a left adjoint and
therefore commutes with all colimits. To see what goes wrong
when $Y$ is not in $\scr{U}$, consider the diagram
$$\xymatrix{
\text{colim}\,(Z\times_Y X_n)\ar[r]^-{\alpha} \ar[d]_{\iota} & Z\times_Y X \ar[d] \\
\text{colim}\,(Z\times X_n) \ar[r] & Z\times X.}
$$
Products commute with colimits, so the bottom arrow is a homeomorphism,
and the top arrow $\alpha$ is a continuous bijection. The right
vertical arrow is an inclusion by the construction of pullbacks. If the
left vertical arrow $\iota$ is an inclusion, then the diagram implies that $\alpha$
is a homeomorphism. The problem is that $\iota$ need not be an inclusion.
One point is that the maps $Z\times_Y X_n\longrightarrow Z\times X_n$ are closed inclusions if $Y$ is weak Hausdorff, but not in general otherwise.
Now assume that all spaces in sight are in $\scr{U}$. Since
the $g_n$ are inclusions, the relevant colimits, when computed in $\scr{K}$, are
weak Hausdorff and thus give colimits in $\scr{U}$. Therefore the commutation of
$p^*$ with colimits (which is a result about colimits in $\scr{K}$) applies to
these particular colimits in $\scr{U}$ to show that $\alpha$ is a homeomorphism.
\end{rem}
The following related observation will be needed for applications of
Quillen's small object argument to $q$-type model structures in \S4.5
and elsewhere.
\begin{lem}\mylabel{little}
Let $X_n\longrightarrow X_{n+1}$, $n\geq 0$, be a
sequence of inclusions in $\scr{K}$ with colimit $X$. Suppose that $X/X_0$
is in $\scr{U}$. Then, for a compact Hausdorff space $C$, the natural map
$$\text{colim}\, \scr{K}(C,X_n)\longrightarrow \scr{K}(C,X)$$
is a bijection.
\end{lem}\begin{proof}
The point is that $X_0$ need not be in $\scr{U}$. Let $f\colon C\longrightarrow X$ be a map.
Then the composite of $f$ with the quotient
map $X\longrightarrow X/X_0$ takes image in some $X_n/X_0$, hence $f$ takes image in $X_n$.
The conclusion follows.\end{proof}
\begin{sch}
One might expect the conclusion to hold for colimits of
sequences of closed inclusions $X_{n-1}\longrightarrow X_n$ such that $X_n-X_{n-1}$
is a $T_1$ space. This is stated as \cite[4.2]{IJ}, whose authors
got the statement from May. However, Lewis has shown us a counterexample.
\end{sch}
\section{Topologically bicomplete categories and ex-objects}\label{Sbicat}
We need some standard and some not quite so standard categorical language.
All of our categories $\scr{C}$ will be topologically enriched, with the enrichment given by a topology on the underlying set of morphisms. We therefore agree to write $\scr{C}(X,Y)$\@bsphack\begingroup \@sanitize\@noteindex{CXY@$\scr{C}(X,Y)$} for the space of morphisms $X\longrightarrow Y$ in $\scr{C}$. Enriched category theory would have us distinguish notationally between morphism spaces and morphism sets, but we shall not do that. A topological category $\scr{C}$ is said to be \emph{topologically bicomplete}\index{topologically bicomplete category} if, in addition to being bicomplete in the usual sense of having all limits and colimits, it is bitensored in the sense that it is tensored and cotensored over $\scr{K}$. We shall denote the tensors\index{tensor!with spaces} and cotensors\index{cotensor!with spaces} by $X\times K$\@bsphack\begingroup \@sanitize\@noteindex{XK@$X\times K$} and $\text{Map}(K,X)$\@bsphack\begingroup \@sanitize\@noteindex{MapXY@$\text{Map}(K,X)$} for a space $K$ and an object $X$ of $\scr{C}$. The defining adjunction homeomorphisms are
\begin{equation}\label{tencoten1}
\scr{C}(X\times K,Y)\iso \scr{K}(K,\scr{C}(X,Y))\iso \scr{C}(X,\text{Map}(K,Y)).
\end{equation}
By the Yoneda lemma, these have many standard implications.
For example,
\begin{equation}\label{tenunit}
X\times * \iso X \quad\text{and}\quad \text{Map}(*,Y) \iso Y,
\end{equation}
\begin{equation}\label{tenass}
X\times (K\times L)\iso (X\times K)\times L \ \ \text{and}\ \
\text{Map}(K,\text{Map}(L,X))\iso \text{Map}(K\times L, X).
\end{equation}
We say that a bicomplete topological category $\scr{C}$ is \emph{based}\index{based bicomplete category}\index{category!based bicomplete} if the unique map from the initial object $\emptyset$ to the terminal object $*$ is an isomorphism. In that case, $\scr{C}$ is enriched in the category $\scr{K}_*$ of based $k$-spaces, the basepoint of $\scr{C}(X,Y)$ being the unique map that factors through $*$. We then say that $\scr{C}$ is \emph{based topologically bicomplete}\index{based topologically bicomplete category}\index{category!based topologically bicomplete} if it is tensored and cotensored over $\scr{K}_*$. We denote the tensors\index{tensor!with based spaces} and cotensors\index{cotensor!with based spaces} by $X\sma K$\@bsphack\begingroup \@sanitize\@noteindex{XK@$X\sma K$} and $F(K,X)$\@bsphack\begingroup \@sanitize\@noteindex{FKX@$F(K,X)$} for a based space $K$ and an object $X$ of $\scr{C}$. The defining adjunction homeomorphisms are
\begin{equation}\label{tencoten20}
\scr{C}(X\sma K,Y)\iso \scr{K}_*(K,\scr{C}(X,Y))\iso \scr{C}(X,F(K,Y)).
\end{equation}
The based versions of (\ref{tenunit}) and (\ref{tenass}) are
\begin{equation}\label{tenunit0}
X\sma S^0 \iso X \quad \text{and}\quad F(S^0,Y) \iso Y,
\end{equation}
\begin{equation}\label{tenass0}
X\sma (K\sma L) \iso (X\sma K)\sma L \quad \text{and}\quad
F(K,F(L,X))\iso F(K\sma L,X).
\end{equation}
Although not essential to our work, a formal comparison between
the based and unbased notions of bicompleteness is illuminating.
The following result allows us to interpret topologically bicomplete
to mean based topologically bicomplete whenever $\scr{C}$ is based, a
convention that we will follow throughout.
\begin{prop}\mylabel{ped1}
Let $\scr{C}$ be a based and bicomplete topological category. Then $\scr{C}$ is topologically bicomplete if and only if it is based topologically bicomplete.\index{based topologically bicomplete category}\index{category!based topologically bicomplete}
\end{prop}
\begin{proof}
Suppose given tensors and cotensors for unbased spaces $K$ and write them as $X\ltimes K$ and $\text{Map}(K,X)_{*}$ as a reminder that they take values in a based category. We obtain tensors and cotensors $X\sma K$ and $F(K,X)$ for based spaces $K$ as the pushouts and pullbacks displayed in the respective diagrams
\[\xymatrix{X\ltimes {*} \ar[r] \ar[d] & X\ltimes K \ar[d]\\
{*} \ar[r] & X\sma K}
\qquad\text{and}\qquad
\xymatrix{F(K,X) \ar[r] \ar[d] & \text{Map}(K,X)_{*} \ar[d]\\
{*} \ar[r] & \text{Map}({*},X)_{*}.}\]
Conversely, given tensors and cotensors $X\sma K$ and $F(K,X)$ for based spaces $K$, we obtain tensors and cotensors $X\ltimes K$ and $\text{Map}(K,X)_{*}$ for unbased spaces $K$ by setting
$$ X\ltimes K = X\sma K_+\ \ \ \text{and}\ \ \ \text{Map}(K,X)_{*} = F(K_+,X),$$
where $K_{+}$ is the union of $K$ and a disjoint basepoint.
\end{proof}
As usual, for any category $\scr{C}$ and object $B$ in $\scr{C}$, we let $\scr{C}/B$\@bsphack\begingroup \@sanitize\@noteindex{CBa@$\scr{C}/B$} denote the category of objects over $B$.\index{category!of objects over B@of objects over $B$} An object $X=(X,p)$ of $\scr{C}/B$ consists of a total object $X$ together with a projection map
$p\colon X\longrightarrow B$ to the base object $B$. The morphisms of $\scr{C}/B$ are the maps of total objects that commute with the projections.
\begin{prop}\mylabel{topbicomp}
If $\scr{C}$ is a topologically bicomplete category, then so is $\scr{C}/B$.
\end{prop}
\begin{proof} The product of objects $Y_i$ over $B$, denoted $\times_B Y_i$, is constructed by taking
the pullback of the product of the projections $Y_i\longrightarrow B$ along the
diagonal $B\longrightarrow \times_i B$. Pullbacks and arbitrary colimits of objects
over $B$ are constructed by taking pullbacks and colimits on total objects
and giving them the induced projections. General limits are constructed as
usual from products and pullbacks. If $X$ is an object over $B$ and
$K$ is a space, then the tensor
$X\times_B K$ is just $X\times K$ together with the projection
$X\times K \longrightarrow B\times *\cong B$ induced by the projection
of $X$ and the projection of $K$ to a point. Note that this makes
sense even though the tensor $\times$ need have nothing to do with
cartesian products in general; see \myref{mislead} below.
The cotensor $\text{Map}_B(K,X)$ is the pullback of the diagram
\[\xymatrix{B\ar[r]^-\iota & \text{Map}(K,B) & \text{Map}(K,X)\ar[l]}\]
where $\iota$ is the adjoint of $B\times K \longrightarrow B\times * \cong B$.
\end{proof}
The terminal object in $\scr{C}/B$ is $(B,\text{id})$. Let $\scr{C}_B$\@bsphack\begingroup \@sanitize\@noteindex{CB@$\scr{C}_B$} denote the category of based objects in $\scr{C}/B$,\index{category!of ex-objects} that is, the category of objects under
$(B,\text{id})$ in $\scr{C}/B$. An object $X=(X,p,s)$ in $\scr{C}_B$, which we call an \emph{ex-object over $B$}, consists of on object $(X,p)$ over $B$ together with a section $s\colon B\longrightarrow X$. We can therefore think of the ex-objects as retract diagrams
\[\xymatrix{B\ar[r]^s & X \ar[r]^p & B.}\]
The terminal object in $\scr{C}_B$ is $(B,\text{id},\text{id})$, which we denote
by $*_B$; it is also an initial object. The morphisms in $\scr{C}_B$ are the maps
of total objects $X$ that commute with the projections and sections.
\begin{prop}\mylabel{btopbicomp}
If $\scr{C}$ is a topologically bicomplete category, then the category $\scr{C}_B$ is based topologically bicomplete.
\end{prop}
\begin{proof} The coproduct of objects $Y_i\in \scr{C}_B$, which we shall refer to as the ``wedge over $B$'' of the $Y_i$ and denote by $\wed_B Y_i$, is constructed by taking the pushout of the coproduct $\amalg B\longrightarrow \amalg Y_i$ of the sections along the codiagonal $\amalg_i B\longrightarrow B$. Pushouts and arbitrary limits of objects in $\scr{C}_B$ are constructed by taking pushouts and limits on total objects and giving them the evident induced sections and projections.
The tensor $X\sma_B K$ of $X=(X,p,s)$ and a based space $K$ is the pushout
of the diagram
\[\xymatrix{
B & (X\times {*}) \cup_B (B \times K) \ar[r] \ar[l]
& X\times K,\\}\]
where the right map is induced by the basepoint of $K$
and the section of $X$. The cotensor $F_B(K,X)$ is the pullback
of the diagram
\[\xymatrix{
B \ar[r]^-{s} & X &\text{Map}_B(K,X), \ar[l]_-{\varepsilon}\\}\]
where $\varepsilon$ is evaluation at the basepoint of $K$, that is, the
adjoint of the evident map $X\times K\longrightarrow X$ over $B$.
\end{proof}
\begin{rem}\mylabel{mislead}
Notationally, it may be misleading to write $X\times K$ and $X\sma K$ for unbased and based tensors. It conjures up associations that are appropriate for the examples on hand but that are inappropriate in general. The tensors in a topologically bicomplete category $\scr{C}$ may bear very little relationship to cartesian products or smash products. The standard uniform notation would be $X\otimes K$. However, we have too many relevant examples to want a uniform notation. In particular, we later use the notations $X\times_B K$ and $X\sma_B K$ in the parametrized context, where a notation such as $X\otimes_B K$ would conjure up its own misleading associations.
\end{rem}
\section{Convenient categories of ex-spaces}
We need a convenient topologically bicomplete category of ex-spaces\footnote{Presumably the prefix ``ex''
stands for ``cross'', as in ``cross section''.
The unlovely term ``ex-space'' has been replaced in some
recent literature by ``fiberwise pointed space''. Used repetitively, that is
not much of an improvement. The term ``retractive space'' has also been used.}\index{ex-space}
over a space $B$, where ``convenient'' requires that we have smash
product and function ex-space functors $\sma_B$ and $F_B$ under
which our category is closed symmetric monoidal.
Denoting the unit $B\times S^0$ of $\sma_B$ by $S^0_B$, a
formal argument shows that we will then have isomorphisms
\begin{equation}\label{spsp}
X\sma_B K\iso X\sma_B (S^0_B\sma_B K) \quad \text{and}\quad F_B(K,Y)\iso F_B(S^0_B\sma_B K,Y)
\end{equation}
relating tensors and cotensors to the smash product and function ex-space functors. In particular, $S_B^0\sma_B K$ is just the product ex-space $B\times K$ with section determined by the basepoint of $K$.
The point-set topology leading to such a convenient category is delicate,
and there are quite a few papers devoted to this subject. They do not give exactly what we need, but they come close enough that
we shall content ourselves with a summary. It is based on the papers
\cite{B1, B2, B3, BB1, BB2, Lewis} of Booth, Booth and Brown, and Lewis;
see also James \cite{James, James2}.
We assume once and for all that our base spaces $B$ are in $\scr{U}$.
We allow the total spaces $X$ of spaces over $B$ to be in $\scr{K}$.
We let $\scr{K}/B$\@bsphack\begingroup \@sanitize\@noteindex{K/B@$\scr{K}/B$}\index{category!of spaces
over B@of spaces over $B$} and $\scr{U}/B$\@bsphack\begingroup \@sanitize\@noteindex{U/B@$\scr{U}/B$} denote the categories of spaces over $B$ with total spaces in $\scr{K}$ or $\scr{U}$. Similarly,
we let $\scr{K}_B$\@bsphack\begingroup \@sanitize\@noteindex{KB@$\scr{K}_B$} and $\scr{U}_B$\@bsphack\begingroup \@sanitize\@noteindex{UB@$\scr{U}_B$}\index{category!of ex-spaces} denote the respective categories of ex-spaces over $B$.
\begin{rem}\mylabel{reasonable}
The section of an ex-space in $\scr{U}_B$ is closed, by \myref{coflemma}.
Quite reasonably, references such as \cite{CJ, James} make the blanket
assumption that sections of ex-spaces must be closed. We have not done
so since we have not checked that all constructions in sight preserve
this property.
\end{rem}
Both the separation property on $B$ and the lack of a separation
property on $X$ are dictated by consideration of the function spaces $\text{Map}_B(X,Y)$ over $B$ that we shall define shortly. These are
only known to exist when $B$ is weak Hausdorff. However, even when $B$,
$X$ and $Y$ are weak Hausdorff, $\text{Map}_B(X,Y)$
is generally not weak Hausdorff unless the projection $p\colon X\longrightarrow B$
is an open map. Categorically, this means that the cartesian monoidal
category $\scr{U}/B$ is not closed cartesian monoidal. Wishing to retain the separation property, Lewis \cite{Lewis} proposed the following as convenient
categories of spaces and ex-spaces over a compactly generated space $B$.
\begin{defn}
Let $\scr{O}(B)$\@bsphack\begingroup \@sanitize\@noteindex{OB@$\scr{O}(B)$} and $\scr{O}_*(B)$\@bsphack\begingroup \@sanitize\@noteindex{O*B@$\scr{O}_*(B)$} be the categories of those compactly generated
spaces and ex-spaces over $B$ whose projection maps are open.
\end{defn}
\begin{rem}
Bundle projections over $B$ are open maps. Hurewicz fibrations
over $B$ are open maps if the diagonal $B\longrightarrow B\times B$ is a
Hurewicz cofibration \cite[2.3]{Lewis}; this holds, for example,
if $B$ is a CW complex.
\end{rem}
However, the categories $\scr{O}(B)$ and $\scr{O}_*(B)$ are insufficient for our purposes. Working in these categories, we only have the base change
adjunction $(f^*,f_*)$ of \S2.1 below for open maps
$f\colon A\longrightarrow B$, which is unduly restrictive. For example, we need the adjunction $(\Delta^*,\Delta_*)$, where $\Delta\colon B\longrightarrow B\times B$ is the
diagonal map. Moreover, the generating cofibrations of our $q$-type model
structures do not have open projection maps. This motivates
us to drop the weak Hausdorff condition on total spaces and to focus
on $\scr{K}_B$ as our preferred convenient category of ex-spaces over $B$.
The cofibrant ex-spaces in our $q$-type model structures are weak Hausdorff, hence this separation property is recovered upon cofibrant approximation. Therefore, use of $\scr{K}$ can be viewed as scaffolding in the foundations that can be removed when doing homotopical work.
We topologize the set of ex-maps $X\longrightarrow Y$ as a
subspace of the space $\scr{K}(X,Y)$ of maps of total spaces. It is based,
with basepoint the unique map that factors through $*_B$. Therefore the
category $\scr{K}_B$ is enriched over $\scr{K}_*$. It is based topologically bicomplete by \myref{topbicomp}. Recall that we write $\times_B Y_i$ and $\wed_B Y_i$ for
products and wedges over $B$. We also write $Y/\!_BX$ for quotients, which are
understood to be pushouts of diagrams $*_B \longleftarrow X \longrightarrow Y$. We give
a more concrete description of the tensors and cotensors in $\scr{K}/B$ and $\scr{K}_B$ given by \myref{topbicomp} and \myref{btopbicomp}. For a space $X$ over $B$,
we let $X_b$ denote the fiber $p^{-1}(b)$.
If $X$ is an ex-space, then $X_b$ has the basepoint $s(b)$.
\begin{defn}\mylabel{exctp}
Let $X$ be a space over $B$ and $K$ be a space. Define $X\times_{B}K$\@bsphack\begingroup \@sanitize\@noteindex{XxBK@$X\times_B K$}\index{tensor!for spaces over B@for spaces over $B$} to be the space $X\times K$ with projection the product of the
projections $X\longrightarrow B$ and $K\longrightarrow *$. Define $\text{Map}_B(K,X)$\@bsphack\begingroup \@sanitize\@noteindex{MapBKX@$\text{Map}_B(K,X)$}\index{cotensor!for spaces over B@for spaces over $B$} to be the subspace of $\text{Map}(K,X)$
consisting of those maps $f\colon K\longrightarrow X$ that factor through some
fiber $X_b$; the projection sends such a map $f$ to $b$.
\end{defn}
\begin{defn}\mylabel{exct}
Let $X$ be an ex-space over $B$ and $K$ be a based space. Define $X\sma_B K$\@bsphack\begingroup \@sanitize\@noteindex{XBK@$X\sma_B K$}\index{tensor!for ex-spaces}
to be the quotient of $X\times_B K$ obtained by taking fiberwise smash products, so that $(X\sma_B K)_b = X_b\sma K$; the basepoints of fibers prescribe the section. Define $F_B(K,X)$\@bsphack\begingroup \@sanitize\@noteindex{FBKX@$F_B(K,X)$}\index{cotensor!for ex-spaces} to be the subspace of $\text{Map}_B(K,X)$ consisting of the based maps $K\longrightarrow X_b\subset X$ for some $b\in B$, so that
$F_B(K,X)_b = F(K,X_b)$; the section sends $b$ to the constant map at $s(b)$.
\end{defn}
\begin{rem}
As observed by Lewis \cite[p.\,85]{Lewis}, if $p$ is an open map,
then so are the projections of $X\sma_B K$ and $F_B(K,Y)$. Therefore
$\scr{O}_*(B)$ is tensored and cotensored over $\scr{T}$.
\end{rem}
The category $\scr{K}/B$ is closed cartesian monoidal under the fiberwise cartesian
product $X\times_B Y$ and the function space $\text{Map}_B(X,Y)$ over $B$. The category $\scr{K}_B$ is closed symmetric monoidal under the fiberwise smash product $X\sma_B Y$ and the function ex-space $F_B(X,Y)$. We recall the definitions.
\begin{defn} For spaces $X$ and $Y$ over $B$,
$X\times_B Y$\@bsphack\begingroup \@sanitize\@noteindex{XxBY@$X\times_B Y$}\index{fiberwise product} is the pullback of the projections $p\colon X\longrightarrow B$ and $q\colon Y\longrightarrow B$, with the evident projection $X\times_B Y\longrightarrow B$. When $X$ and $Y$ have sections $s$ and $t$, their pushout $X\vee_B Y$ specifies the coproduct, or wedge, of $X$ and $Y$ in $\scr{K}_B$, and $s$ and $t$ induce a map $X\vee_B Y\longrightarrow X\times_B Y$ over $B$ that sends $x$ and $y$ to $(x,tp(x))$ and $(sq(y),y)$. Then $X\sma_B Y$\@bsphack\begingroup \@sanitize\@noteindex{XBY@$X\sma_B Y$}\index{fiberwise smash product} is the pushout in $\scr{K}/B$ displayed in the diagram
$$\xymatrix{
X\vee_B Y \ar[r] \ar[d] & X\times_B Y \ar[d]\\
{*}_B \ar[r] & X\sma_B Y.}$$
This arranges that $(X\sma_B Y)_b = X_b\sma Y_b$, and the section and projection are evident.
\end{defn}
The following result is \cite[8.3]{BB2}.
\begin{prop}\mylabel{prop:smaB} If $X$ and $Y$ are weak Hausdorff ex-spaces over $B$, then so is $X\sma_BY$. That is, $\scr{U}_B$ is closed under $\sma_B$.
\end{prop}
Function objects are considerably more subtle, and we need a preliminary definition in order to give the cleanest description.
\begin{defn}\mylabel{partialtilde}
For a space $Y\in \scr{K}$, define the \emph{partial map classifier}\index{partial map classifier}
$\tilde Y$\@bsphack\begingroup \@sanitize\@noteindex{tY@$\tilde{Y}$} to be the union of $Y$ and a disjoint point $\omega$, with the topology
whose closed subspaces are $\tilde Y$ and the closed subspaces of $Y$. The point $\omega$ is not a closed subset, and $\tilde{Y}$ is not weak Hausdorff. The name
``partial map classifier'' comes from the observation that, for any space $X$, pairs $(A,f)$ consisting of a closed subset $A$ of $X$ and a continuous map $f\colon A\longrightarrow Y$ are in bijective correspondence with continuous maps $\tilde{f}\colon X\longrightarrow \tilde Y$. Given $(A,f)$, $\tilde f$ restricts
to $f$ on $A$ and sends $X-A$ to $\omega$; given $\tilde{f}$, $(A,f)$ is
$\tilde{f}^{-1}(Y)$ and the restriction of $\tilde{f}$.
\end{defn}
\begin{defn}\mylabel{MapB}
Let $p\colon X\longrightarrow B$ and $q\colon Y\longrightarrow B$ be spaces over $B$.
Define $\text{Map}_B(X,Y)$\@bsphack\begingroup \@sanitize\@noteindex{MapBXY@$\text{Map}_B(X,Y)$}\index{mapping space!of spaces over B@of spaces over $B$} to be the pullback displayed in the diagram
$$\xymatrix{
\text{Map}_B(X,Y) \ar[r] \ar[d] & \text{Map}(X,\tilde{Y}) \ar[d]^{\text{Map}(\text{id},\tilde{q})} \\
B \ar[r]_-{\lambda}& \text{Map}(X, \tilde{B}).\\}$$
Here $\lambda$ is the adjoint of the map $X\times B\longrightarrow \tilde{B}$
that corresponds to the composite of the inclusion $\text{Graph}(p)\subset X\times B$
and the projection $X\times B\longrightarrow B$ to the second coordinate. The graph of $p$ is the
inverse image of the diagonal under $p\times \text{id}\colon X\times B\longrightarrow B\times B$, and
the assumption that $B$ is weak Hausdorff ensures that it is a closed subset of
$X\times B$, as is needed for the definition to make sense. Explicitly, $\lambda(b)$ sends
$X_b$ to $b$ and sends $X-X_b$ to the point $\omega\in \tilde{B}$.
\end{defn}
This definition gives one reason that we require the base spaces of ex-spaces
to be weak Hausdorff. On fibers, $\text{Map}_B(X,Y)_b = \text{Map}(X_b,Y_b)$. The space of
sections of $\text{Map}_B(X,Y)$ is $\scr{K}/B(X,Y)$. We have (categorically equivalent)
adjunctions
\begin{gather}\label{maptimes1}
\text{Map}_B(X\times_B Y, Z)\iso \text{Map}_B(X,\text{Map}_B(Y,Z)),\\[1ex]
\label{maptimes2}
\scr{K}/B\,(X\times_B Y, Z)\iso \,\scr{K}/B\,(X,\text{Map}_B(Y,Z)).
\end{gather}
These results are due to Booth \cite{B1, B2, B3}; see also \cite[\S7]{BB1},
\cite[\S8]{BB2}, \cite[II\S9]{James}, \cite{Lewis}.
Examples in \cite[5.3]{BB1} and \cite[1.7]{Lewis} show that $\text{Map}_B(X,Y)$
need not be weak Hausdorff even when $X$ and $Y$ are. The question of when
$\text{Map}_B(X,Y)$ is Hausdorff or weak Hausdorff was studied in \cite[\S5]{BB1}
and later in \cite{James, James2}, but the definitive criterion was given by Lewis
\cite[1.5]{Lewis}.
\begin{prop}\mylabel{open} Consider a fixed map $p\colon X\longrightarrow B$ and varying maps \linebreak
$q\colon Y\longrightarrow B$, where $X$ and the $Y$ are weak Hausdorff. The map $p$ is
open if and only if the space $\text{Map}_B(X,Y)$ is weak Hausdorff for all $q$.
\end{prop}
\begin{prop}\mylabel{Mapfib}
If $p\colon X\longrightarrow B$ and $q\colon Y\longrightarrow B$ are Hurewicz fibrations,
then the projections $X\times_B Y\longrightarrow B$ and $\text{Map}_B(X,Y)\longrightarrow B$
are Hurewicz fibrations. The second statement is false with Hurewicz
fibrations replaced by Serre fibrations.
\end{prop}\begin{proof}
The statement about $X\times_BY$ is clear. The statements about \linebreak
$\text{Map}_B(X,Y)$
are due to Booth \cite[6.1]{B1} or, in the present formulation \cite[3.4]{B2};
see also \cite[23.17]{James}.\end{proof}
\begin{defn}
For ex-spaces $X$ and $Y$ over $B$, define $F_B(X,Y)$\@bsphack\begingroup \@sanitize\@noteindex{FBXY@$F_B(X,Y)$}\index{mapping space!of ex-spaces} to be the
subspace of $\text{Map}_B(X,Y)$ that consists of the points that restrict to based maps
$X_b\longrightarrow Y_b$ for each $b\in B$; the section sends $b$ to the constant map from
$X_b$ to the basepoint of $Y_b$. Formally, $F_B(X,Y)$ is the pullback
displayed in the diagram
$$\xymatrix{
F_B(X,Y) \ar[r] \ar[d] & \text{Map}_B(X,Y) \ar[d]^{\text{Map}_B(s,\text{id})} \\
B\ar[r]_-{t} & Y\iso \text{Map}_B(B,Y),}\\$$
where $s$ and $t$ are the sections of $X$ and $Y$.
\end{defn}
The space of maps $S^0_B\longrightarrow F_B(X,Y)$ is $\scr{K}_B(X,Y)$, and we have adjunctions
\begin{gather}\label{maptimes1*}
F_B(X\sma_B Y, Z)\iso F_B(X,F_B(Y,Z)),\\[1ex]
\label{maptimes2*}
\scr{K}_B\,(X\sma_B Y, Z)\iso \,\scr{K}_B\,(X,F_B(Y,Z)).
\end{gather}
\myref{open} implies the following analogue of \myref{prop:smaB}.
\begin{prop} If $X$ and $Y$ are weak Hausdorff ex-spaces over $B$ and
$X\longrightarrow B$ is an open map, then $F_B(X,Y)$ is weak Hausdorff.
\end{prop}
We record the following analogue of \myref{Mapfib}. The second part is again
due to Booth, who sent us a detailed write-up. The argument is similar to his proofs
in \cite[6.1(i)]{B1} or \cite[3.4]{B2}, but a little more complicated, and a general
result of the same form is given by Morgan \cite{Morgan}.
\begin{prop}\mylabel{Ffib}
If $X$ and $Y$ are ex-spaces over $B$ whose sections are Hurewicz
cofibrations and whose projections are Hurewicz fibrations, then the
projections of $X\sma_B Y$ and $F_B(X,Y)$ are Hurewicz fibrations.
\end{prop}
\section{Convenient categories of ex-$G$-spaces}
The discussion just given generalizes readily to the equivariant context. Let $G$ be a compactly generated topological group. Subgroups of $G$ are understood to be closed. Let $B$ be a compactly generated $G$-space (with $G$ acting from the left). We consider $G$-spaces over $B$ and ex-$G$-spaces $(X,p,s)$. The total space $X$ is a $G$-space in $\scr{K}$, and the section and projection are $G$-maps. The fiber $X_b$ is a based $G_b$-space with $G_b$-fixed basepoint $s(b)$, where $G_b$ is the isotropy group of $b$.
Recall from \cite[II\S1]{MM} the distinction between the category $\scr{K}_G$\@bsphack\begingroup \@sanitize\@noteindex{KG@$\scr{K}_G$}\index{category!of G spaces@of $G$-spaces} of $G$-spaces and non\-equi\-var\-iant maps and the category $G\scr{K}$\@bsphack\begingroup \@sanitize\@noteindex{GK@$G\scr{K}$} of $G$-spaces and equivariant maps; the former is enriched over $G\scr{K}$, the latter over $\scr{K}$. We have a similar dichotomy on the ex-space level. Here we have a conflict of notation with our notation for categories of ex-spaces, and we agree to let $\scr{K}_{G,B}$ denote the category whose objects are the ex-$G$-spaces over $B$ and whose morphisms are the maps of underlying ex-spaces over $B$, that is, the maps $f\colon X\longrightarrow Y$ such that $f\com s = t$ and $q\com f = p$. Henceforward, we call these maps ``arrows'' to distinguish them from $G$-maps, which we often abbreviate to maps. For $g\in G$, $gf$ is also an arrow of ex-spaces over $B$, so that $\scr{K}_{G,B}(X,Y)$\@bsphack\begingroup \@sanitize\@noteindex{KGBXY@$\scr{K}_{G,B}(X,Y)$} is a $G$-space. Moreover, composition is given by $G$-maps
$$\scr{K}_{G,B}(Y,Z)\times \scr{K}_{G,B}(X,Y)\longrightarrow \scr{K}_{G,B}(X,Z).$$
We obtain the category $G\scr{K}_B$\@bsphack\begingroup \@sanitize\@noteindex{GKB@$G\scr{K}_B$}\index{category!of exGspaces@of ex-$G$-spaces} by restricting to $G$-maps $f$, and we may view it as the $G$-fixed point category of $\scr{K}_{G,B}$. Of course, $G\scr{K}_B(X,Y)$ is a space and not a $G$-space. The pair $(\scr{K}_{G,B}, G\scr{K}_B)$ is an example of a \emph{$G$-category}\index{category!G-@$G$- ---}, a structure that we shall recall formally in \S10.2.
Since $*_B$ is an initial and terminal object in both $\scr{K}_{G,B}$ and $G\scr{K}_B$, their morphism spaces are based. Thus $\scr{K}_{G,B}$ is enriched over the category $G\scr{K}_*$ of based $G$-spaces and $G\scr{K}_B$ is enriched over $\scr{K}_*$. As discussed in \cite[II.1.3]{MM}, if we were to think exclusively in enriched category terms, we would resolutely ignore the fact that the $G$-spaces $\scr{K}_{G,B}(X,Y)$ have elements (arrows), thinking of these $G$-spaces as enriched hom objects. From that point of view, $G\scr{K}_B$ is the ``underlying category'' of our enriched $G$-category. While we prefer to think of $\scr{K}_{G,B}$ as a category, it must be kept in mind that it is not a very well-behaved one. For example, because its arrows are not equivariant, it fails to have limits or colimits.
In contrast, the category $G\scr{K}_B$ is bicomplete. Its limits and colimits are constructed in $\scr{K}_B$ and then given induced $G$-actions. The category $\scr{K}_{G,B}$, although not bicomplete, is tensored and cotensored over $\scr{K}_{G,*}$. The tensors $X\sma_B K$ and cotensors $F_B(K,X)$ are constructed in $\scr{K}_B$ and then given induced $G$-actions. They satisfy the adjunctions
\begin{gather}\label{tensored1}
\scr{K}_{G,B}(X\sma_B K, Y)\iso \scr{K}_{G,*}(K, \scr{K}_{G,B}(X,Y))\iso \scr{K}_{G,B}(X, F_B(K,Y))\\
\intertext{and, by passage to fixed points,}
\label{tensored2}
G\scr{K}_B(X\sma_B K, Y)\iso G\scr{K}_*(K,\scr{K}_{G,B}(X,Y))\iso G\scr{K}_B(X, F_B(K,Y)).
\end{gather}
It follows that $G\scr{K}_{B}$ is tensored and cotensored over $G\scr{K}_*$ and, in particular, is topologically bicomplete.
The category $\scr{K}_{G,B}$ is closed symmetric monoidal via the fiberwise smash products $X\sma_B Y$ and function objects $F_B(X,Y)$. Again, these are defined in $\scr{K}_B$ and then given induced $G$-actions. The unit is the ex-$G$-space $S^0_B=B\times S^0$. The category $G\scr{K}_B$ inherits a structure of closed symmetric monoidal category. We have homeomorphisms of based $G$-spaces
\begin{gather}\label{ad1}
\scr{K}_{G,B}(X\sma_B Y, Z)\iso \scr{K}_{G,B}(X, F_B(Y,Z))
\intertext{and, by passage to $G$-fixed points, homeomorphisms of based spaces}
\label{ad2}
G\scr{K}_{B}(X\sma_B Y, Z)\iso G\scr{K}_{B}(X, F_B(Y,Z)).
\intertext{The first of these implies an associated homeomorphism of ex-$G$-spaces}
\label{ad1'}
F_B(X\sma_B Y, Z)\iso F_B(X, F_B(Y,Z)).
\end{gather}
Nonequivariantly, the functor that sends an ex-space $X$ over $B$ to the fiber $X_b$ has a left adjoint, denoted $(-)^b$. It sends a based space $K$ to the wedge $K^b = B\wed K$, where $B$ is given the basepoint $b$; the section and projection are evident. Nonobviously, the same set $B\wed K$ admits a quite different topology under which it gives a {\em right}\, adjoint to the fiber functor $X\mapsto X_b$. We shall prove the equivariant analogue conceptually in \myref{Johann0}, but we describe the left adjoint to the fiber functor explicitly here.
\begin{con}\mylabel{Fibad}
Let $b\in B$. Then the functor
$G\scr{K}_{B} \longrightarrow G_b\scr{K}_{*}$ that sends $Y$ to $Y_b$ has a
left adjoint. It sends a based $G_b$-space $K$ to the
ex-$G$-space $K^b$ given by the pushout
$$K^b = (G\times_{G_b} K)\cup_G B.$$
Here $G\times_{G_b}K$ is the (left) $G$-space $(G\times K)/\sim$,
where $(gh,k)\sim (g,hk)$ for $g\in G$, $h\in G_b$, and $k\in K$.
The pushout is defined with respect to the map $G\longrightarrow B$
that sends $g$ to $gb$ and the map $G\longrightarrow G\times_{G_b}K$ that
sends $g$ to $(g,k_0)$, where $k_0$ is the ($G_b$-fixed)
basepoint of $K$.
The section is given by the evident inclusion of $B$ and
the projection is obtained by passage to pushouts from the identity
map of $B$ and the $G$-map $\pi_b\colon G\times_{G_b} K\longrightarrow B$
given by $\pi_b(g,k) = gb$. Thus we first extend the group action
on $K$ from $G_b$ to $G$ and then glue the orbit of the basepoint
of $K$ to the orbit of $b$. If $K$ is an unbased $G_b$-space, then
$(K_+)^b = (G\times_{G_b}K)\amalg B$.
\end{con}
\begin{rem}
There is an alternative parametrized view of equivariance that is important in torsor theory but that we shall not study. It focuses on ``topological groups $G_B$ over $B$'' and ``$G_B$-spaces $E$ over $B$'',
where $G_B$ is a space over a nonequivariant space $B$ with a product
$G_B\times _B G_B\longrightarrow G_B$ that restricts on fibers to the products of topological groups $G_b$ and $E$ is a space over $B$ with
an action $G_B\times_B E\longrightarrow E$ that restricts on fibers to actions
$G_b\times E_b\longrightarrow E_b$. That theory intersects ours in the special case
$G_B = G\times B$ for a topological group $G$. Since, at least implicitly, all
of our homotopy theory is done fiberwise, our work adapts without essential difficulty to give a development of parametrized equivariant homotopy theory
in that context.
\end{rem}
\section{Appendix: nonassociativity of smash products in ${\scr{T}}op_*$}
\label{sec:topass}
In a 1958 paper \cite{Puppe}, Puppe asserted the following result,
but he did not give a proof. It was the subject of a series of e-mails
among Mike Cole, Tony Elmendorf, Gaunce Lewis and the first author. Since we know of no published source that gives the details of this
or any other counterexample to the associativity of the smash product
in ${\scr{T}}op_*$, we include the following proof. It is due
to Kathleen Lewis.
Let $\mathbb{Q}$ and $\mathbb{N}$ be the rational numbers and the nonnegative integers,
topologized as subspaces of $\mathbb{R}$ and given the basepoint zero. Consider
smash products as quotient spaces, without applying the $k$-ification
functor. Then we have the following counterexample to associativity.
\begin{thm} $(\mathbb{Q}\sma \mathbb{Q})\sma \mathbb{N}$ is not homeomorphic to $\mathbb{Q}\sma(\mathbb{Q}\sma \mathbb{N})$.
\end{thm}\begin{proof}
Consider the following diagram.
$$\xymatrix{
& \mathbb{Q}\times \mathbb{Q}\times \mathbb{N} \ar[dr]^{p\times\text{id}} \ar[dl]_{\text{id}\times p'} \ar[dd]^{q} & \\
\mathbb{Q}\times (\mathbb{Q}\wedge \mathbb{N}) \ar[d]_{s} &
& (\mathbb{Q}\wedge \mathbb{Q}) \times \mathbb{N} \ar[d]^-{r} \\
\mathbb{Q}\wedge(\mathbb{Q}\wedge \mathbb{N})
& \mathbb{Q}\wedge \mathbb{Q}\wedge \mathbb{N} \ar[l]_-{t} \ar[r]^-{\cong}
& (\mathbb{Q}\wedge \mathbb{Q}) \wedge \mathbb{N}}$$
Here $\mathbb{Q}\sma \mathbb{Q}\sma \mathbb{N}$ denotes the evident quotient space of $\mathbb{Q}\times \mathbb{Q}\times \mathbb{N}$.
The maps $p$, $p'$, $q$, $r$, and $s$ are quotient maps. Since $\mathbb{N}$ is locally compact,
$p\times \text{id}$ is also a quotient map, hence so is $r\com(p\times \text{id})$. The universal
property of quotient spaces then gives the bottom right homeomorphism. Since $\mathbb{Q}$
is not locally compact, $\text{id}\times p'$ need not be a quotient map, and in fact it
is not. The map $t$ is a continuous bijection given by the universal property of
the quotient map $q$, and we claim that $t$ is not a homeomorphism. To show this,
we display an open subset of $\mathbb{Q}\sma\mathbb{Q}\sma N$ whose image under $t$ is not open.
Let $\beta$ be an irrational number, $0<\beta<1$, and let $\gamma = (1-\beta)/2$. Define
$V'(\beta)$ to be the open subset of $\mathbb{R}\times \mathbb{R}$ that is the union of the
following four sets.
\begin{enumerate}[(1)]
\item The open ball of radius $\beta$ about the origin
\item The tubes $[1,\infty)\times (-\gamma,\gamma)$, $(-\infty,-1]\times (-\gamma,\gamma)$,
$(-\gamma,\gamma)\times [1,\infty)$, and $(-\gamma,\gamma)\times (-\infty,-1]$ of width $2\gamma$
about the axes.
\item The open balls of radius $\gamma$ about the four points $(\pm 1,0)$, $(0,\pm 1)$.
\item For each $n\geq 1$, the open ball of radius $\gamma/2^n$ about the four points
$(\pm \gamma_n,0)$, $(0,\pm \gamma_n)$, where $\gamma_n = 1 - \sum_{k=0}^{k=n-1}\gamma/2^k$.
\end{enumerate}
To visualize this set, it is best to draw a picture. It is symmetric with respect
to $90$ degree rotation. Consider the part lying along the positive $x$-axis. A tube
of width $2\gamma$ covers the part of the $x$-axis to the right of $(1,0)$. A ball of
radius $\beta$ centers at the origin. A ball of radius $\gamma$ centers at $(1,0)$. Its
vertical diagonal is the edge of the tube going off to the right. On the left, by
the choice of $\gamma$, this ball reaches halfway from its center $(1,0)$ to the point
$(\beta,0)$ at the right edge of the ball centered at the origin. The point $(1-\gamma,0)$
at the left edge of the ball centered at $(1,0)$ is the center of another ball, which
reaches half the distance from $(1-\gamma,0)$ to $(\beta,0)$. And so on:
the point where the left edge of the $n$th ball crosses the $x$-axis is the
center point of the $(n+1)$st ball, which reaches half the distance from its
center to the edge of the ball centered at the origin.
Define $V(\beta)=V'(\beta)\cap (\mathbb{Q}\times \mathbb{Q})$. Note that the only points of
the coordinate axes of $\mathbb{R}\times \mathbb{R}$ that are not in $V'(\beta)$ are $(\pm \beta,0)$
and $(0,\pm \beta)$. Since $\beta$ is irrational, $V(\beta)$ contains the coordinate axes
of $\mathbb{Q}\times\mathbb{Q}$. Because the radii of the balls in the sequence are decreasing,
for each $\varepsilon > \beta$, there is no $\delta >0$ such that
$((-\varepsilon,\varepsilon)\times (-\delta,\delta))\cap (\mathbb{Q}\times\mathbb{Q})$ is contained in $V(\beta)$.
Now let $\alpha$ be an irrational number, $0<\alpha<1$. Let $\bullet$ be the basepoint of
$\mathbb{Q}\sma \mathbb{N}$ and $*$ be the basepoint of $\mathbb{Q}\sma\mathbb{Q}\sma\mathbb{N}$. Let $U$ be the
union of $\{*\}$ and the image under $q$ of $\cup_{n\geq 1} V(\alpha/n)\times\{n\}$.
This is an open subspace of $\mathbb{Q}\sma\mathbb{Q}\sma \mathbb{N}$ since
$$q^{-1}(U) = \mathbb{Q}\times \mathbb{Q}\times \{0\} \cup (\cup_{n\geq 1} V(\alpha/n)\times \{n\})$$
is an open subset of $\mathbb{Q}\times\mathbb{Q}\times \mathbb{N}$. We claim that $t(U)$ is not open in
$\mathbb{Q}\sma(\mathbb{Q}\sma\mathbb{N})$. Assume that $t(U)$ is open. Then
$$s^{-1}(t(U)) = (\text{id}\times p')(q^{-1}(U))$$
is an open subset of $\mathbb{Q}\times (\mathbb{Q}\sma \mathbb{N})$, hence it contains an open neighborhood
$V$ of $(0,\bullet)$. Now $V$ must contain $((-\varepsilon,\varepsilon)\cap \mathbb{Q}) \times W$ for some $\varepsilon>0$
and some open neighborhood $W$ of $\bullet$ in $\mathbb{Q}\sma \mathbb{N}$. Since $\mathbb{Q}\sma\mathbb{N}$ is
homeomorphic to the wedge over $n\geq 1$ of the spaces $\mathbb{Q}\times \{n\}$,
$W$ must contain the wedge over $n\geq 1$ of subsets $((-\delta_n,\delta_n)\cap \mathbb{Q})\times\{n\}$,
where $\delta_n>0$. By the definition of $U$, this implies that
$$((-\varepsilon,\varepsilon)\times (-\delta_n,\delta_n))\cap(\mathbb{Q}\times \mathbb{Q}) \subset V(\alpha/n).$$
However, for $n$ large enough that $\varepsilon > \alpha/n$, there is no $\delta_n$ for which this holds.
\end{proof}
\chapter{Change functors and compatibility relations}
\section*{Introduction}
In the previous chapter, we developed the internal properties of the category $G\scr{K}_B$ of ex-$G$-spaces over $B$. As $B$ and $G$ vary, these categories are related by various functors, such as base change functors, change of groups functors, orbit and fixed point functors, external smash product and function space functors, and so forth. We define these ``change functors''
and discuss various compatibility relations among them in this chapter.
We particularly emphasize base change functors. We give a general
categorical discussion of such functors in \S2.1, illustrating
the general constructions with topological examples. In \S2.2, we
discuss various compatibility relations that relate
these functors to smash products and function objects.
In \S2.3 and \S2.4 we turn to equivariant phenomena and study restriction of group actions along homomorphisms. As usual, we break this into the study of
restriction along inclusions and pullback along quotient homomorphisms.
In \S2.3, we discuss restrictions of group actions to subgroups, together
with the associated induction and coinduction functors. We also consider their compatibilities with base change functors. In particular, this gives us a convenient way of thinking about passage to fibers and allows us to reinterpret restriction to subgroups in terms of base change and coinduction. That is the starting point of our generalization of the Wirthm\"uller isomorphism in
Part IV.
In \S2.4, we consider pullbacks of group actions from a quotient group
$G/N$ to $G$, together with the associated quotient and fixed point functors.
Again, we also consider compatibilities with base change functors. For an $N$-free base space $E$, we find a relation between the quotient functor $(-)/N$ and the fixed point functor $(-)^N$ that involves base change along the quotient map $E\longrightarrow E/N$. The good properties of the bundle construction in Part IV
can be traced back to this relation, and it is at the heart of the Adams isomorphism in equivariant stable homotopy theory.
In \S2.5, we describe a different categorical framework, one appropriate to ex-spaces with varying base spaces. We show that the relevant category of retracts over varying base spaces is closed symmetric monoidal under external smash product and function ex-space functors. The internal smash product and function ex-space functors are obtained from these by use of base change along diagonal maps. The external smash products are much better behaved homotopically than the internal ones, and homotopical analysis of base change functors will therefore play a central role in the homotopical analysis of smash products.
In much of this chapter, we work in a general categorical framework. In some places where we restrict to spaces, more general categorical formulations are undoubtedly possible. When we talk about group actions, all groups are
assumed to be compactly generated spaces but are otherwise unrestricted.
\section{The base change functors $f_{!}$, $f^*$, and $f_*$}
Let $f\colon A\longrightarrow B$ be a map in a bicomplete subcategory $\scr{B}$ of a
bicomplete category $\scr{C}$. We are thinking of $\scr{U}\subset \scr{K}$
or $G\scr{U}\subset G\scr{K}$. We wish to define functors
$$ f_{!}\colon \scr{C}_A\longrightarrow \scr{C}_B,\qquad f^*\colon \scr{C}_B\longrightarrow \scr{C}_A, \qquad f_*\colon \scr{C}_A\longrightarrow \scr{C}_B,$$
such that $f_{!}$ is left adjoint and $f_{*}$ is right adjoint to $f^*$. The definitions
of $f^*$ and $f_{!}$ are dual and require no further hypotheses. The definition of $f_*$
does not work in full generality, but it only requires the further hypothesis that $\scr{C}/B$
be cartesian closed. Thus
we assume given internal hom objects $\text{Map}_B(Y,Z)$ in $\scr{C}/B$ that satisfy the usual
adjunction, as in (\ref{maptimes2}). One reason to work in this generality is to
emphasize that no further point-set topology is needed to construct these base
change functors in the context of ex-spaces. This point is not clear from the literature, where the functor $f_*$
is often given an apparently different, but naturally isomorphic, description.
We work with generic ex-objects
\[\xymatrix{A\ar[r]^s & X \ar[r]^p & A}
\qquad\text{and}\qquad
\xymatrix{B\ar[r]^t & Y \ar[r]^q & B}\]
in this section.
\begin{defn}\mylabel{retract1}
Define $f_{!}X$\@bsphack\begingroup \@sanitize\@noteindex{fXl@$f_{"!}X$} and its structure maps $q$ and $t$ by means of the map of retracts in the following diagram on the left, where the top square is a pushout and the bottom square is defined by the universal property of pushouts and the requirement that $q\com t= \text{id}$. Define $f^*Y$\@bsphack\begingroup \@sanitize\@noteindex{fYm@$f^*Y$} and its structure maps $p$ and $s$ by means of the map of retracts in the following middle diagram, where the bottom square is a pullback and the top square is defined by the universal property of pullbacks and the requirement that $p\com s = \text{id}$.
\[\xymatrix{
A\ar[d]_{s} \ar[r]^-{f} & B\ar[d]^{t}\\
X\ar[d]_{p} \ar[r] & f_{!}X \ar[d]^{q}\\
A\ar[r]_-{f} & B}
\qquad\qquad
\xymatrix{
A \ar[d]_{s} \ar[r]^-{f} & B\ar[d]^{t} \\
f^*Y \ar[r] \ar[d]_{p} & Y \ar[d]^{q} \\
A \ar[r]_-{f} & B}
\qquad\qquad
\xymatrix{
B\ar[d]_{t} \ar[r]^-{\iota} & \text{Map}_B(A,A)\ar[d]^{\text{Map}(\text{id},s)}\\
f_*X \ar[d]_{q} \ar[r] & \text{Map}_B(A,X)\ar[d]^{\text{Map}(\text{id},p)}\\
B \ar[r]_-{\iota} & \text{Map}_B(A,A)}\]
Thinking of $X$ and $A$ as objects over $B$ via $f\com p$ and $f$ and observing that the adjoint of the identity map of $A$ gives a map $\iota\colon B\longrightarrow \text{Map}_B(A,A)$, define $f_*X$\@bsphack\begingroup \@sanitize\@noteindex{fXr@$f_*X$} and its structure maps $q$ and $t$ by means of the map of retracts in the above diagram on the right, where the bottom square is a pullback and the top square is defined by the universal property of pullbacks and the requirement that $q\com t =\text{id}$.
\end{defn}
\begin{prop}\mylabel{first} $(f_{!},f^*)$ is an adjoint pair of functors:
$$\scr{C}_B(f_{!}X,Y)\iso \scr{C}_A(X,f^*Y).$$
\end{prop}\begin{proof}
Maps in both hom sets are specified by maps $k\colon X\longrightarrow Y$ in $\scr{C}$
such that $q\com k = f\com p$ and $k\com s = t\com f$.\end{proof}
\begin{prop}\mylabel{second} $(f^*,f_*)$ is an adjoint pair of functors:
$$\scr{C}_A(f^*Y,X)\iso \scr{C}_B(Y,f_*X).$$
\end{prop}\begin{proof}
A map $k\colon f^*Y = Y\times_B A \longrightarrow X$ such that $p\com k = p$ and $k\com s = s$
has adjoint $\tilde{k} \colon Y\longrightarrow \text{Map}_B(A,X)$ such that
$\text{Map}(\text{id},p)\com \tilde k= \iota\com q$ and $\tilde{k}\com t = \text{Map}(\text{id},s)\com \iota$.
The conclusion follows directly.\end{proof}
\begin{rem}
Writing these proofs diagrammatically, we see that the adjunction
isomorphisms are given by homeomorphisms in our context of topological categories.
\end{rem}
We specialize to ex-spaces (or ex-$G$-spaces), in the rest of the section.
Observe that the fiber $(f_*X)_b$ is the space of sections $A_b\longrightarrow X_b$ of $p\colon X_b\longrightarrow A_b$.
\begin{rem}
If $f\colon A\longrightarrow B$ is an open map and $X$
is in $\scr{U}$, then $f_*X$ is in $\scr{U}$ and $\scr{U}_A(f^*Y,X)\iso \scr{U}_B(Y,f_*X)$
for $Y\in \scr{U}$, by \cite[1.5]{Lewis}.
\end{rem}
\begin{exmp}\mylabel{incsting}
Let $f\colon A\longrightarrow B$ be an inclusion. Then $f^*Y$
is the restriction of $Y$ to $A$ and $f_!X = B\cup_A X$. The ex-space
$f_*X$ over $B$ is analogous to the prolongation by zero of a sheaf
over $A$. The fiber $(f_*X)_b$ is $X_a$ if $a\in A$ and a point
$\{b\}$ otherwise. To see this from the definition, recall that
$\text{Map}(\emptyset, K)$ is a point for any space $K$ and that
$\text{Map}_B(A,X)_b = \text{Map}(A_b,X_b)$. As a set, $f_*X \iso B\cup_A X$,
but the topology is quite different. It is devised so that the
map $Y\longrightarrow f_*f^*Y$ that restricts to the identity on $Y_a$ for
$a\in A$ but sends $Y_b$ to $\{b\}$ for $b\notin A$ is continuous.
\end{exmp}
\begin{exmp}\mylabel{r!ex}
Let $r\colon B\longrightarrow *$ be the unique map. For
a based space $X$ and an ex-space $E = (E,p,s)$ over $B$, we have
$$r^*X = B\times X, \qquad r_{!}E = E/s(B), \qquad \text{and}\qquad r_*E= \text{Sec}(B,E),$$
where $\text{Sec}(B,E)$ is the space of maps $t\colon B\longrightarrow E$ such that
$p\com t=\text{id}$,
with basepoint the section $s$. These elementary base change functors are the key to
using parametrized homotopy theory to obtain information in ordinary homotopy theory.
Let $\varepsilon\colon r_!r^*\longrightarrow \text{id}$ and $\eta\colon \text{id}\longrightarrow r^*r_!$ be the counit
and unit of the adjunction $(r_!,r^*)$. Then $r_!r^*X\iso B_+\sma X$ and $\varepsilon$
is $r_+\sma\text{id}$. Similarly, $r_!r^*r_! E\iso B_+\sma E/B$, and
$r_!\eta\colon r_!E\longrightarrow r_!r^*r_!E$ is the ``Thom diagonal''\index{Thom diagonal} $E/B\longrightarrow B_+\sma E/B$.
If $p\colon E\longrightarrow B$ is a spherical fibration with section, such as the fiberwise
one-point compactification
of a vector bundle, then $r_{!}E$ is the Thom complex\index{Thom complex} of $p$.
\end{exmp}
\section{Compatibility relations}
The term ``compatibility relation'' has been used in algebraic geometry in
the context of Grothendieck's six functor formalism that relates base change
functors to tensor product and internal hom functors in sheaf theory. We describe how
the analogous, but simpler, formalism appears in our categories of ex-objects.
We recall some language. We are especially interested in the behavior of base change functors with respect to closed symmetric monoidal structures that, in
our topological context, are given by smash products and function objects. Relevant categorical observations are given in \cite{FHM}. We say that a functor $T\colon\scr{B}\longrightarrow \scr{A}$ between closed symmetric monoidal categories
is {\em closed symmetric monoidal} if
\[TS_{\scr{B}}\iso S_{\scr{A}},\quad T(X\sma_{\scr{B}} Y)\iso TX\sma_{\scr{A}} TY,
\quad\text{and}\quad
TF_\scr{B}(X,Y)\iso F_{\scr{A}}(TX,TY),\]
where $S_{\scr{B}}$, $\sma_{\scr{B}}$ and $F_{\scr{B}}$ denote the unit object, product, and internal hom of $\scr{B}$, and similarly for $\scr{A}$. These isomorphisms must satisfy appropriate coherence conditions. In the language of \cite{FHM}, the following result states that any map $f$ of base spaces gives rise to a ``Wirthm\"uller context",\index{Wirthmuller context@Wirthm\"uller context}
which means that the functor $f^*$ is closed symmetric monoidal and has both a left adjoint and a right adjoint.
\begin{prop}\mylabel{Wirth0}
If $f\colon A\longrightarrow B$ is a map of base $G$-spaces, then the functor $f^*\colon G\scr{K}_B\longrightarrow G\scr{K}_A$ is closed symmetric monoidal. Therefore, by definition and implication, $f^*S^0_B\iso S^0_A$ and there are natural isomorphisms
\begin{gather}\label{oneo}
f^*(Y\sma_B Z)\iso f^*Y\sma_A f^*Z,\\[1ex]
\label{twoo}
F_B(Y,f_*X) \iso f_*F_A(f^*Y,X),\\[1ex]
\label{three0}
f^*F_B(Y,Z)\iso F_A(f^*Y,f^*Z),\\[1ex]
\label{four0}
f_{!}(f^*Y\sma_A X)\iso Y\sma_B f_{!}X,\\[1ex]
\label{five0}
F_B(f_{!}X,Y)\iso f_*F_A(X,f^*Y),
\end{gather}
where $X$ is an ex-$G$-space over $A$ and $Y$ and $Z$ are ex-$G$-spaces over $B$.
\end{prop}
\begin{proof}
The isomorphism $f^*S^0_B\iso S^0_A$ is evident since $f^*(B\times K)\iso A\times K$ for based $G$-spaces $K$. The isomorphism (\ref{oneo}) is obtained by passage to quotients from the evident homeomorphism
$$(Y\times_B A)\times _A (Z\times_B A) \iso (Y\times_BZ)\times_B A $$
As explained in \cite[\S\S2, 3]{FHM}, the isomorphism (\ref{oneo}) is equivalent to the
isomorphism (\ref{twoo}), and it determines natural maps from left to right in (\ref{three0}),
(\ref{four0}), and (\ref{five0}) such that all three are isomorphisms if any one is. By a
comparison of definitions, we see that the categorically defined map in (\ref{three0}),
which is denoted $\alpha$ in \cite[3.3]{FHM}, coincides in the present situation with the map,
also denoted $\alpha$, on \cite[p.\,167]{BB2}. As explained on \cite[p.\,178]{BB2}, in the point-set
topological framework that we have adopted, that map $\alpha$ is a homeomorphism.\end{proof}
\begin{rem}
Only the very last statement refers to topology.
The categorically defined map $\alpha$ should quite generally
be an isomorphism in analogous contexts, but we have not pursued
this question in detail. An alternative self-contained proof of the previous proposition is given in \myref{fgext} below by using \myref{Mackey0} to prove (\ref{four0}) instead of (\ref{three0}). In that argument, the only non-formal ingredient is the fact that the functor $D\times_B(-)$ commutes with pushouts.
\end{rem}
We shall later need a purely categorical coherence observation
about the categorically defined map $\alpha$ of (\ref{three0}). In fact, it will play a key role in the proof of the fiberwise duality theorem of \S15.1. It is
convenient to insert it here.
\begin{rem}\mylabel{coherence}
Let $T\colon \scr{B}\longrightarrow \scr{A}$ be a symmetric monoidal functor.
We are thinking of $T$ as, for example, a base change functor $f^*$.
The map
$$\alpha\colon TF_\scr{B}(X,Y)\longrightarrow F_\scr{A}(TX,TY)$$
is defined to be the adjoint of
\[\xymatrix{TF_\scr{B}(X,Y)\sma_\scr{A} TX \iso T(F_\scr{B}(X,Y)\sma_\scr{B} X)\ar[r]^-{T\text{ev}} & TY.}\]
The dual of $X$ is $D_{\scr{B}}X = F_{\scr{B}}(X,S_{\scr{B}})$, where
$S_{\scr{B}}$ is the unit of $\scr{B}$. Taking $Y=S_{\scr{B}}$, the definition of
$\alpha$ implies that the top triangle commutes in the diagram
\[\xymatrix{
TD_\scr{B} X\sma_\scr{A} TX \ar[r]^\iso\ar[d]_{\alpha\sma_\scr{A} \text{id}} &
T(D_\scr{B} X\sma_\scr{B} X) \ar[r]^-{T\text{ev}} & TS_\scr{B}\ar[d]^\iso\\
F_\scr{A}(TX,TS_\scr{B}) \sma_\scr{A} TX \ar[urr]_{\text{ev}}\ar[r]_-\iso & D_\scr{A} f^*X\sma_\scr{A} f^*X \ar[r]_-{\text{ev}} & S_\scr{A}.}\]
The bottom triangle is a naturality diagram. The outer
rectangle is \cite[3.7]{FHM}, but its commutativity in general was not
observed there. However, it was observed in \cite[3.8]{FHM} that its commutativity implies the commutativity of the diagram
\[\xymatrix{TD_\scr{B} X \sma_\scr{A} TY \ar[d]_{\alpha\sma_\scr{A} TY}\ar[r]^\iso
& T(D_\scr{B} X\sma_\scr{B} Y) \ar[r]^-{T\nu}
& TF_\scr{B}(X,Y)\ar[d]^\alpha\\
D_\scr{A} TX \sma_\scr{A} TY \ar[rr]_-\nu && F_\scr{A}(TX,TY),}\]
where $\nu\colon D_{\scr{B}}X\sma_{\scr{B}}Y\longrightarrow F_{\scr{B}}(X,Y)$ is the adjoint of
\[\xymatrix@1{
D_{\scr{B}}X\sma_{\scr{B}} Y\sma_{\scr{B}} X \iso D_{\scr{B}}X\sma_{\scr{B}} X\sma_{\scr{B}} Y \ar[r]^-{\text{ev}\sma\text{id}}
& S_{\scr{B}}\sma_{\scr{B}} Y\iso Y.}\]
\end{rem}
In other contexts, the analogue of (\ref{four0}) is called the ``projection formula'',\index{projection formula} and we shall also use that term. The following base change commutation relations with respect to pullbacks are also familiar from other contexts. We state the result for spaces but, apart from use of the fact that the functor $D\times_B (-)$ commutes with pushouts,
the proof is formal.
\begin{prop}\mylabel{Mackey0}
Suppose given a pullback diagram of base spaces
$$\xymatrix{
C \ar[r]^-{g} \ar[d]_{i} & D \ar[d]^{j} \\
A \ar[r]_{f} & B.}$$
Then there are natural isomorphisms of functors
\begin{equation}\label{bases0}
j^*f_{!} \iso g_{!}i^*, \qquad f^*j_* \iso i_*g^*, \qquad f^*j_{!}\iso i_!g^*, \qquad j^*f_*\iso g_*i^*.
\end{equation}
\end{prop}\begin{proof} The first isomorphism is one of left adjoints, and the second is the corresponding
``conjugate'' isomorphism of right adjoints. Similarly for the third and fourth isomorphisms.
By symmetry, it suffices to prove the first isomorphism.
The functor $j^* = D\times_B(-)$ commutes with pushouts. For a space $X$ over $A$ regarded by
composition with $f$ as a space over $B$, $C\times_AX\iso D\times_BX$. This gives
\[ j^*f_{!} X = D\times _B(B\cup_A X) \iso D\cup_C(C\times_A X) = g_{!}i^*X.\qedhere\]
\end{proof}
\section{Change of group and restriction to fibers}
This section begins the study of equivariant phenomena that have no
non\-equivariant counterparts. In particular, using a conceptual reinterpretation of the adjoints of the fiber functors $(-)_b$,
we relate restriction to subgroups to restriction to fibers. Recall that subgroups of $G$ are understood to be closed and fix an inclusion $\iota\colon H\subset G$ throughout this section. Parametrized theory gives a convenient way of studying restriction along $\iota$ without changing the ambient group from $G$ to $H$.
\begin{prop}\mylabel{homog} The category $G\scr{K}_{G/H}$ of ex-$G$-spaces over
$G/H$ is equivalent to the category $H\scr{K}_*$ of based $H$-spaces.
\end{prop}\begin{proof}
The equivalence sends an ex-$G$-space $(Y,p,s)$ over $G/H$
to the $H$-space $p^{-1}(eH)$ with basepoint the $H$-fixed point $s(eH)$.
Its inverse sends a based $H$-space $X$ to the induced
$G$-space $G\times_H X$, with the evident structure maps.\end{proof}
More formally, recall that there are ``induction''\index{induction} and ``coinduction''\index{coinduction} functors
$\iota_!$ and $\iota_*$ from $H$-spaces to $G$-spaces that are left and right adjoint
to the forgetful functor $\iota^*$ that sends a $G$-space $Y$ to $Y$ regarded
as an $H$-space. Explicitly, for an $H$-space $X$,
\begin{equation}\label{GH1}
\iota_! X = G\times_H X \qquad \text{and}\qquad \iota_*X = \text{Map}_H(G,X).
\end{equation}
The latter is the space of maps of (left) $H$-spaces, with (left)
action of $G$ induced by the right action of $G$ on itself. Similarly,
when $X$ is a based $H$-space, we have the based analogues
\begin{equation}\label{GH2}
\iota_! X = G_+\sma_H X \qquad \text{and}\qquad \iota_*X = F_H(G,X).
\end{equation}
With this notation, some familiar natural isomorphisms take the forms
\begin{gather}\label{GH3}
\iota_!(\iota^*Y\times X)\iso Y\times \iota_!X \qquad \text{and} \qquad
\iota_*\text{Map}(\iota^*Y,X)\iso \text{Map}(Y,\iota_*X)
\intertext{and, in the based case,}
\label{GH4}
\iota_!(\iota^*Y\sma X)\iso Y\sma \iota_!X \qquad\text{and} \qquad
\iota_* F(\iota^*Y,X)\iso F(Y,\iota_*X).
\end{gather}
By the uniqueness of adjoints, or inspection of definitions, we see that these
familiar change of groups functors are change of base functors along $r\colon G/H\longrightarrow *$.
\begin{cor} The change of group and change of base functors associated to
$\iota$ and $r$ agree under the equivalence of categories between $H\scr{K}_*$ and $G\scr{K}_{G/H}$:
$$\iota^*\iso r^*, \qquad \iota_!\iso r_{!}, \qquad \text{and} \qquad \iota_*\iso r_*.$$
\end{cor}
We can generalize this equivalence of categories, using the following definitions. We have a forgetful functor
$\iota^*\colon G\scr{K}_{B} \longrightarrow H\scr{K}_{\iota^*B}$.\@bsphack\begingroup \@sanitize\@noteindex{io@$\iota^*$}
It doesn't have an obvious left or right adjoint, but we have obvious
analogues of induction and coinduction that involve changes of base spaces.
The first will lead to a description of $\iota^*$ as a base change functor and
thus as a functor with a left and right adjoint.
\begin{defn}\mylabel{changes0}
Let $A$ be an $H$-space and $X$ be an $H$-space over $A$.
Define $\iota_!\colon H\scr{K}_A\longrightarrow G\scr{K}_{\iota_! A}$\@bsphack\begingroup \@sanitize\@noteindex{iol@$\iota_{"!}$} by letting $\iota_!X$ be the $G$-space
$G\times_H X$ over $\iota_!A = G\times_H A$. Define $\iota_*\colon H\scr{K}_A\longrightarrow G\scr{K}_{\iota_* A}$\@bsphack\begingroup \@sanitize\@noteindex{ior@$\iota_*$}
by letting $\iota_*X$ be the $G$-space $\text{Map}_H(G,X)$ over $\iota_*A = \text{Map}_H(G,A)$.
\end{defn}
For an $H$-space $A$ and a $G$-space $B$, let
\begin{equation}\mylabel{munuin}
\mu\colon G\times_H \iota^*B = \iota_!\iota^*B\longrightarrow B
\ \, \text{and}\ \, \nu\colon A\longrightarrow \iota^*\iota_!A = \iota^*(G\times_H A)
\end{equation}
be the counit and unit of the $(\iota_!,\iota^*)$ adjunction.
The following result says that ex-$H$-spaces over an $H$-space
$A$ are equivalent to ex-$G$-spaces over the $G$-space $\iota_!A$.
\begin{prop}\mylabel{ishriek}
The functor $\iota_!\colon H\scr{K}_A\longrightarrow G\scr{K}_{\iota_!A}$ is a
closed symmetric monoidal equivalence of categories with inverse
the composite
$$G\scr{K}_{\iota_!A}\stackrel{\iota^*}{\longrightarrow}
H\scr{K}_{\iota^*\iota_!A}\stackrel{{\nu}^*}{\longrightarrow} H\scr{K}_A.$$
\end{prop}
Applied to $A =\iota^*B$, this equivalence leads to the promised description
of $\iota^*\colon G\scr{K}_B\longrightarrow H\scr{K}_{\iota^*B}$ as a base change functor.
\begin{prop}\mylabel{ishriekb}
The functor $\iota^*\colon G\scr{K}_B\longrightarrow H\scr{K}_{\iota^*B}$ is the composite
\[\xymatrix@1
{G\scr{K}_B \ar[r]^-{\mu^*} & G\scr{K}_{\iota_!\iota^*B} \iso H\scr{K}_{\iota^*B}\\}\]
\end{prop}
Change of base and change of groups are related by various further
consistency relations. The following result gives two of them.
\begin{prop}\mylabel{changerel}
Let $f\colon A\longrightarrow \iota^*B$ be a map of $H$-spaces and $\tilde{f}\colon
\iota_! A\longrightarrow B$ be its adjoint map of $G$-spaces. Then the following
diagrams commute up to natural isomorphism.
\[\xymatrix{
G\scr{K}_{\iota_!A} \ar[r]^-{\tilde{f}_!} & G\scr{K}_B \\
H\scr{K}_{A} \ar[r]_-{f_!}\ar[u]^{\iota_!} &
H\scr{K}_{\iota^*B}\ar[u]_{\mu_!\com\iota_!}}
\quad \ \
\xymatrix{
G\scr{K}_B \ar[r]^-{\tilde{f}^*} \ar[d]_{\iota^*}
& G\scr{K}_{\iota_!A} \ar[d]^{{\nu}^*\com\iota^*}\\
H\scr{K}_{\iota^*B} \ar[r]_-{f^*} & H\scr{K}_{A}}\]
\end{prop}
\begin{proof} Since $\tilde{f} = \mu\com \iota_!f$, we have
\[\tilde{f}_!\com \iota_!
\cong (\mu\com \iota_!f)_!\com \iota_!
\cong \mu_!\com (\iota_!f)_!\com \iota_!
\cong \mu_!\com \iota_!\com f_!,\]
where the last isomorphism holds because $G\times_H(-)$ commutes with
pushouts. Since $f = \iota^*\tilde{f}\com \nu$, we have
\[ f^*\com \iota^* \cong (\iota^*\tilde{f}\com \nu)^*\com \iota^*
\cong \nu^*\com (\iota^*\tilde{f})^*\com \iota^*
\cong \nu^*\com \iota^*\com \tilde{f}^*, \]
where the last isomorphism holds because pulling the $G$ action back
to an $H$-action commutes with pullbacks.
\end{proof}
The reader may find it illuminating to work out these isomorphisms in the
context of \myref{homog}. That result leads to the promised conceptual
reinterpretation of \myref{Fibad}.
\begin{exmp}\mylabel{Johann0}
For $b\in B$, we also write $b\colon *\longrightarrow B$ for the map that sends $*$ to $b$, and we write $\tilde{b}\colon G/G_b\longrightarrow B$ for the induced inclusion of orbits. Thus $b$ is a $G_b$-map and $\tilde{b}$ is a $G$-map. Under the equivalence
$G\scr{K}_{G/G_b}\iso G_b\scr{K}_*$ of \myref{homog}, ${\tilde{b}}^*$ may be
interpreted as the fiber functor $G\scr{K}_B\longrightarrow G_b\scr{K}_*$ that sends $X$ to
$X_b$, ${\tilde{b}}_{!}$ may be interpreted as the left adjoint of \myref{Fibad} that sends $K$ to $K^b$, and ${\tilde{b}}_*$ specifies a right adjoint to the fiber functor,
which we denote by ${^{b}}K$. With these notations, the isomorphisms of
\myref{Wirth0} specialize to the following natural isomorphisms,
where $Y$ and $Z$ are in $G\scr{K}_B$ and $K$ is in $G_b\scr{K}_*$.
\begin{gather*}
(Y\sma_B Z)_b\iso Y_b\sma Z_b,\\[1ex]
F_B(Y,\, ^bK) \iso\, {^{b}}F(Y_b,K),\\[1ex]
F_B(Y,Z)_b\iso F(Y_b,Z_b),\\[1ex]
(Y_b\sma K)^b\iso Y\sma_B K^b,\\[1ex]
F_B(K^b,Y)\iso \, {^{b}}F(K,Y_b).
\end{gather*}
\end{exmp}
\begin{exmp}\mylabel{Johann1}
Several earlier results come together in the following
situation. Let $f\colon A\longrightarrow B$ be a $G$-map. For
$b\in B$, let $b\colon \{b\}\longrightarrow B$ and $i_b\colon A_b\longrightarrow A$
denote the evident inclusions of $G_b$-spaces. We have the following compatible
pullback squares, the first of $G_b$-spaces and the second of $G$-spaces.
$$\xymatrix{
A_b \ar[r]^-{f_b} \ar[d]_{i_b} & \{b\} \ar[d]^{b} \\
A \ar[r]_{f} & B}
\qquad
\xymatrix{
G\times_{G_b} A_b \ar[r]^-{G\times_{G_b} f_b} \ar[d]_{\tilde{\imath}_b} & G/G_b \ar[d]^{\tilde{b}} \\
A \ar[r]_-{f} & B}$$
Applying \myref{Mackey0} to the right-hand square and
interpreting the conclusion in terms of fibers by \myref{changes0},
we obtain canonical isomorphisms of $G_b$-spaces
$$(f_!X)_b \iso {f_{b}}_!i_b^*X \qquad\text{and}\qquad(f_*X)_b \iso {f_{b}}_*i_b^*X,$$
where $X$ is an ex-$G$-space over $A$, regarded on the right-hand sides as an ex-$G_b$-space over $A$ by pullback along $\iota\colon G_b\longrightarrow G$.
\end{exmp}
\section{Normal subgroups and quotient groups}
Observe that any homomorphism $\theta\colon G\longrightarrow G'$ factors as the composite of a quotient homomorphism $\varepsilon$, an isomorphism, and an inclusion $\iota$. We
studied change of groups along inclusions in the previous section. Here we consider a quotient homomorphism $\epsilon\colon G\longrightarrow J$ of $G$ by a normal subgroup $N$. We still have a restriction functor
\[\epsilon^*\colon J\scr{K}_A\longrightarrow G\scr{K}_{\epsilon^*A},\]
and we also have the functors
\[(-)/N\colon G\scr{K}_B\longrightarrow J\scr{K}_{B/N}\qquad\text{and}\qquad
(-)^N\colon G\scr{K}_B\longrightarrow J\scr{K}_{B^N}\]
obtained by passing to orbits over $N$ and to $N$-fixed points. When $B$ is a point, these last two functors are left and right adjoint to $\epsilon^*$, but in general change of base must enter in order to obtain such adjunctions. The following observation follows directly by inspection of the definitions.
\begin{prop}\mylabel{factor0}
Let $j\colon B^N\longrightarrow B$ be the inclusion and $p\colon B\longrightarrow B/N$ be the quotient map. Then the following factorization diagrams commute.
\[\xymatrix{
G\scr{K}_B \ar[d]_{p_!}\ar[r]^{(-)/N} & J\scr{K}_{B/N} \\
G\scr{K}_{B/N} \ar[ur]_{(-)/N} }
\qquad\text{and}\qquad
\xymatrix{G\scr{K}_B \ar[d]_{j^*}\ar[r]^{(-)^N} & J\scr{K}_{B^N} \\
G\scr{K}_{B^N} \ar[ur]_{(-)^N}}\]
It follows that $((-)/N,p^*\epsilon^*)$ and $(j_!\epsilon^*,(-)^N)$ are adjoint pairs.
\end{prop}
We have the following analogue of \myref{changerel}.
\begin{prop}\mylabel{fixorbbase}
Let $f\colon A\longrightarrow B$ be a map of $G$-spaces. Then the following diagrams commute up to natural isomorphisms.
{\[\xymatrix{
G\scr{K}_A \ar[r]^-{f_!} \ar[d]_{(-)/N} & G\scr{K}_B \ar[d]^{(-)/N}\\
J\scr{K}_{A/N} \ar[r]_-{(f/N)_!} & J\scr{K}_{B/N}}
\quad
\xymatrix{
G\scr{K}_B \ar[r]^-{f^*} \ar[d]_{(-)^N} & G\scr{K}_A \ar[d]^{(-)^N}\\
J\scr{K}_{B^N} \ar[r]_-{(f^N)^*} & J\scr{K}_{A^N}}
\quad
\xymatrix{
G\scr{K}_A \ar[r]^-{f_!} \ar[d]_{(-)^N} & G\scr{K}_B \ar[d]^{(-)^N}\\
J\scr{K}_{A^N} \ar[r]_-{(f^N)_!} & J\scr{K}_{B^N}}\]}
\end{prop}
\begin{proof}
For ex-$G$-spaces $X$ over $A$ and $Y$ over $B$, these isomorphisms
are given by the homeomorphisms $$(X\cup_A B)/N\iso X/N\cup_{A/N}B/N,$$
$$(Y\times_B A)^N\iso Y^N\times_{B^N}A^N,$$
and
$$(X\cup_A B)^N\iso X^N\cup_{A^N}B^N.$$
As a quibble, the last requires $A\longrightarrow X$ to be a closed inclusion, but this will hold for the sections of compactly generated ex-$G$-spaces over $A$ by \myref{coflemma}(i).
\end{proof}
Specializing to $N$-free $G$-spaces, we obtain a factorization result that is analogous to those in \myref{factor0}, but is less obvious. It is a precursor
of the Adams isomorphism, which we will derive in \S16.4.
\begin{prop}\mylabel{ouch0}
Let $E$ be an $N$-free $G$-space, let $B = E/N$, and let $p\colon E\longrightarrow B$ be the quotient map. Then the diagram
\[\xymatrix{
G\scr{K}_E \ar[r]^{(-)/N} \ar[d]_{p_*} & J\scr{K}_B\\
G\scr{K}_B \ar[ur]_{(-)^N} }\]
commutes up to natural isomorphism. Therefore the left adjoint $(-)/N$ of the functor $p^*\varepsilon^*$ is also its right adjoint.
\end{prop}
\begin{proof}
Let $X$ be an ex-$G$-space over $E$ with projection $q$. Comparing the
pullbacks that are used to define the functors $p_*$ and $\text{Map}_B$ in Definitions \ref{retract1} and \ref{MapB}, we find that $p_*X$ fits into a pullback diagram
\[\xymatrix{
p_*X \ar[r] \ar[d] & \text{Map}(E,\tilde{X})\ar[d]^{\tilde{q}}\\
B\ar[r]_-{\nu} & \text{Map}(E,\tilde{E}).}\]
Here $\nu(b)$, $b = Ne$, corresponds as in \myref{partialtilde} to the inclusion of the closed subset $Ne$ in $E$.
Passing to $N$-fixed points, we see that it suffices to prove that the
following commutative diagram is a pullback.
\[\xymatrix{
X/N \ar[r]^-{\mu} \ar[d]_{q/N} & \text{Map}_N(E,\tilde{X})\ar[d]^{\tilde{q}}\\
E/N = B\ar[r]_-{\nu} & \text{Map}_N(E, \tilde{E})}\]
Here $\mu$ is induced from the adjoint of the map $X\times E\longrightarrow \tilde{X}$ that sends $(x,e)$ to $nx$ if $e = nq(x)$ and sends $(x,e)$ to $\omega$ otherwise.
With this description, $\mu$ is well-defined since $E$ is $N$-free. It suffices
to give a continuous inverse to the induced map
$$\phi\colon X/N \longrightarrow \text{Map}_N(E,\tilde{X})\times_{\text{Map}_N(E,\tilde{E})} E/N.$$
If $(f,Ne)$ is a point in the pullback, then $f$ corresponds to a map
$Ne\longrightarrow X$, and $\phi^{-1}(f,Ne) = Nf(e)$ in $X/N$. For continuity, note that
$\phi^{-1}$ is obtained from the evaluation map
$\text{Map}(E,\tilde{X})\times E\longrightarrow \tilde{X}$ by passage to subquotient spaces.
\end{proof}
\begin{rem}\mylabel{iotaalt}
This leads to a useful alternative description of the functor $\iota_!\colon H\scr{K}_A\longrightarrow G\scr{K}_{\iota_!A}$, where $A$ is an $H$-space and
$\iota_!A = G\times_H A$. We have the projection $\pi\colon G\times A \longrightarrow A$
of $(G\times H)$-spaces, where the
$G\times H$ actions on the source and target are given by
$$(g,h)(g',a) = (gg'h^{-1},ha) \qquad \text{and}\qquad (g,h)a = ha.$$
Consider ex-$H$-spaces $X$ over $A$ as $(G\times H)$-spaces with $G$
acting trivially and let $\epsilon\colon G\times H\longrightarrow H$ be the
projection. We see from the definition that $\iota_!X=({\pi}^*\varepsilon^*X)/H$.
Since $G\times A$ is an $H$-free $(G\times H)$-space, we conclude from the
previous result that $\iota_!X \iso (p_*{\pi}^*\varepsilon^*X)^H$, where
$p\colon G\times A \longrightarrow G\times_H A=\iota_!A$ is the quotient map.
\end{rem}
\section{The closed symmetric monoidal category of retracts}
Let $\scr{B}$ be a topologically bicomplete full subcategory of a topologically bicomplete
category $\scr{C}$. We are thinking of $\scr{U}\subset \scr{K}$ or $G\scr{U}\subset G\scr{K}$.
We have the category of retracts $\scr{C}_{\scr{B}}$.\@bsphack\begingroup \@sanitize\@noteindex{CB@$\scr{C}_{\scr{B}}$}\index{category!of retracts} The objects of $\scr{C}_{\scr{B}}$ are the
retractions $B\stackrel{s}{\longrightarrow}X\stackrel{p}{\longrightarrow}B$
with $B\in\scr{B}$ and $X\in\scr{C}$, abbreviated $(X,p,s)$ or just $X$. The morphisms of
$\scr{C}_{\scr{B}}$ are the evident commutative diagrams. When $\scr{B}=\scr{C}$, this is just a diagram
category for the evident two object domain category.
The importance of the category $\scr{C}_{\scr{B}}$ is apparent from its role in
\myref{retract1}: focus on this category is natural when we consider
base change functors. In our examples, $\scr{B}$ and $\scr{C}$ are enriched and topologically bicomplete over the appropriate category of spaces, $\scr{U}$ for $\scr{B}$ and $\scr{K}$ for $\scr{C}$. For a space $K\in \scr{K}$,
the tensors $-\times K$ and cotensors $\text{Map}(K,-)$ applied to retractions give retractions, and we have the adjunction homeomorphisms
\begin{equation}\label{Bsilly}
\scr{C}_{\scr{B}}(X\times K,Y)\iso \scr{K}(K,\scr{C}_{\scr{B}}(X,Y))\iso \scr{C}_{\scr{B}}(X,\text{Map}(K,Y)).
\end{equation}
The category $G\scr{K}_{G\scr{U}}$ is closed symmetric
monoidal under an external smash product functor, denoted $X\barwedge Y$,\@bsphack\begingroup \@sanitize\@noteindex{XYe@$X\barwedge Y$}\index{smash product!external} and an external \index{mapping space!external}
function ex-space functor, denoted $\bar{F}(Y,Z)$.\@bsphack\begingroup \@sanitize\@noteindex{FYZe@$\bar{F}(Y,Z)$} If $X$, $Y$, and $Z$ are ex-spaces over $A$, $B$, and $A\times B$, respectively, then $X\barwedge Y$ is an
ex-space over $A\times B$ and $\bar{F}(Y,Z)$ is an ex-space over $A$. We have
\begin{equation}\label{exad}
G\scr{K}_{A\times B}(X\barwedge Y, Z)\iso G\scr{K}_A(X, \bar{F}(Y,Z)),
\end{equation}
which gives the required adjunction in $G\scr{K}_{G\scr{U}}$. It specializes to parts of (\ref{tensored2}) when
$A$ or $B$ is a point. The ex-space $X\barwedge Y$ is the evident fiberwise smash product, with
$(X\barwedge Y)_{(a,b)} = X_a\sma Y_b$. The fiber $\bar{F}(Y,Z)_a$ is
$F_B(Y,Z_a)$, where $Z_a$ is the ex-space over $B$ whose fiber $Z_{a,b}$ over $b$
is the inverse image of $(a,b)$ under the projection $Z\longrightarrow A\times B$.
Rather than describe the topology of the ex-space $\bar{F}(Y,Z)$ directly,
we give alternative descriptions of $X\barwedge Y$ and $\bar{F}(Y,Z)$
in terms of internal smash products and internal function ex-spaces.
Let $\pi_A$ and $\pi_B$ be the projections of $A\times B$ on $A$ and $B$ and
observe that $\pi_A^*X \iso X\times B$ and $\pi_B^*Y \iso A\times Y$.
If one likes, the following results can be taken as a definition of the external operations
and a characterization of the internal operations, or vice versa.
\begin{lem}\mylabel{exin} The external smash product and function ex-space functors
are determined by the internal functors via natural isomorphisms
$$ X\barwedge Y \iso \pi_A^*X \sma_{A\times B} \pi_B^*Y
\qquad \text{and} \qquad \bar{F}(Y,Z) \iso {\pi_{A}}_{*}F_{A\times B}(\pi_{B}^{*}Y,Z),$$
where $X$, $Y$, and $Z$ are ex-spaces over $A$, $B$, and $A\times B$, respectively.
\end{lem}
With these isomorphisms taken as definitions, the adjunction (\ref{exad}) follows
from the adjunctions $(\pi^*_A,{\pi_{A}}_{*})$, $(\pi^*_B,{\pi_{B}}_{*})$, and
$(\sma_{A\times B}, F_{A\times B})$.
\begin{lem}\mylabel{internalize} The internal smash product and function ex-space functors
are determined by the external functors via natural isomorphisms
$$ X\sma_B Y \iso \Delta^*(X\barwedge Y) \qquad \text{and} \qquad
F_B(X,Y) \iso \bar{F}(X,\Delta_*Y),$$
where $X$ and $Y$ are ex-spaces over $B$ and $\Delta\colon B\longrightarrow B\times B$ is the diagonal map.
\end{lem}
With these isomorphisms taken as definitions, the adjunction $(\sma_{B}, F_{B})$
follows from the adjunctions $(\Delta^*,\Delta_*)$ and (\ref{exad}). Since $\Delta^*$ is
symmetric monoidal and the composite of either projection
$\pi_i\colon B\times B\longrightarrow B$ with $\Delta$ is the identity map of $B$, we see that,
if we have constructed both internal and external smash products, then they
must be
related by natural isomorphisms as in Lemmas \ref{exin} and \ref{internalize}.
\begin{rem}
The first referee suggests that we point out another consistency check.
The fiber $(\Delta_*Y)_{(b,c)}$ is a point if $b\neq c$ and is
$Y_b$ if $b=c$. Therefore the fiber over $b$ of the restriction
$(\Delta_*Y)_{b}$ of $\Delta_* Y$ to $\{b\}\times B$ is $Y_b\cup (B-\{b\})$,
suitably topologized, and
$$\bar{F}(X,\Delta_*Y)_b = F_B(X,(\Delta_*Y)_{b})_b \iso F(X_b,Y_b) = F_B(X,Y)_b.$$
\end{rem}
\begin{rem}\mylabel{fgext}
The description of the internal smash product in terms
of the external smash product sheds light on the basic compatibility
isomorphisms (\ref{oneo}) and (\ref{four0}). For maps $f\colon A\longrightarrow B$
and $g\colon A'\longrightarrow B'$ and for ex-spaces $X$ over $B$ and $Y$ over
$B'$, it is easily checked that
\begin{equation}\label{fgext1}
f^*Y\barwedge g^*Z\iso (f\times g)^*(Y\barwedge Z).
\end{equation}
Similarly, for ex-spaces $W$ over $A$ and $X$ over $A'$,
\begin{equation}\label{fgext2}
f_!W\barwedge g_!X\iso (f\times g)_!(W\barwedge X).
\end{equation}
Now take $A=A'$, $B=B'$ and $f=g$. For ex-spaces $Y$ and $Z$ over $B$,
$$f^*(Y\sma_B Z)\iso f^*\Delta_B^*(Y\barwedge Z) \iso (\Delta_B\com f)^*(Y\barwedge Z).$$
On the other hand, using (\ref{fgext1}),
$$f^*Y\sma_A f^*Z\iso \Delta_A^*(f\times f)^*(Y\barwedge Z)
\iso ((f\times f)\com\Delta_A)^*(Y\barwedge Z).$$
The right sides are the same since $\Delta_B\com f = (f\times f)\com\Delta_A$.
Similarly,
$$f_!(f^*Y\sma_A X)\iso f_!\Delta_A^*(f\times\text{id})^*(Y\barwedge X)
\iso f_!((f\times\text{id})\com \Delta_A)^*(Y\barwedge X),$$
while
$$Y\sma_B f_!X \iso \Delta_B^*(\text{id}\times f)_!(Y\barwedge X).$$
Since the diagram
$$\xymatrix{
A\ar[d]_f \ar[r]^-{\Delta_A} & A\times A \ar[r]^-{f\times \text{id}}
& B\times A \ar[d]^{\text{id}\times f}\\
B \ar[rr]_{\Delta_B} & & B\times B\\}$$
is a pullback, the right sides are isomorphic by \myref{Mackey0}.
\end{rem}
It is illuminating conceptually to go further and consider group
actions from an external point of view. For groups $H$ and $G$,
an $H$-space $A$, and a $G$-space $B$, we have an evident
external smash product
\begin{equation}\label{GBexex}
\barwedge\colon H{\scr{K}}_A \times G{\scr{K}}_B \to (H\times G)\mathcal{K}_{A\times B}.
\end{equation}
For an ex-$H$-space $X$ over $A$ and an ex-$G$-space $Y$ over $B$,
$X\barwedge Y$ is just the internal smash product over the
$(H\times G)$-space $A\times B$
of $\pi_H^*\pi_A^* X$ and $\pi_G^*\pi_B^* Y$, where the $\pi's$
are the projections from $H\times G$ and $A\times B$ to their
coordinates. It is easily seen that this definition leads to
another $(\barwedge,\bar{F})$ adjunction.
When $H = G$, the diagonal
$\Delta\colon G\longrightarrow G\times G$ is a
closed inclusion since $G$ is compactly generated. We can pull back
along $\Delta$, and then our earlier external smash product $X\barwedge Y$
over the $G$-space $\Delta^*(A\times B)$ is given in terms of (\ref{GBexex})
as the pullback $\Delta^*(X\barwedge Y)$. Note that, by \myref{ishriekb},
$\Delta^*$ here can be viewed as a base change functor.
\chapter{Proper actions, equivariant bundles and fibrations}
\section*{Introduction}
Much of the work in equivariant homotopy theory has focused on compact Lie groups. However, as was already observed by Palais \cite{Palais}, many results can be generalized to arbitrary Lie groups provided that one restricts to proper actions. These are well-behaved actions whose isotropy groups are compact, and all actions by compact Lie groups are proper. The classical definition of a Lie group \cite[p. 129]{Chev} includes all discrete groups (even though they need not be second countable) and, for discrete groups, the proper actions are the properly discontinuous ones.
In the parametrized world, the homotopy theory is captured on fibers. When we restrict to proper actions on base spaces, the fibers have actions by the compact isotropy groups of the base space. So even though our primary interest is still in compact Lie groups of equivariance, proper actions on the base space provide the right natural level of generality. We set the stage for such a theory in this chapter by generalizing various classical results about equivariant bundles and fibrations to a setting focused on proper actions by Lie groups. The reader interested primarily in the nonequivariant theory should skip this chapter since only some very standard material in it is relevant nonequivariantly.
In \S3.1, we recall some basic results about proper actions of locally compact groups. We use this discussion to generalize some results about equivariant bundles in \S3.2. We generalize Waner's equivariant versions of Milnor's results on spaces of the homotopy types of CW complexes in \S3.3. In \S3.4, we recall and generalize classical theorems of Dold and Stasheff about Hurewicz fibrations. We also recall an important but little known result of Steinberger and West that relates Serre and Hurewicz fibrations. We recall the definition of equivariant quasifibrations in \S3.5.
\section{Proper actions of locally compact groups}
We recall relevant definitions and basic results about proper actions in this section. For appropriate generality and technical convenience, we let $G$ be a locally compact topological group whose underlying topological space is compactly generated. Local compactness means that the identity element, hence any point, has a compact neighborhood. We see from \myref{HauswHaus} that $G$ is Hausdorff and, since all compact subsets are closed, it follows that each neighborhood of any point contains a compact neighborhood.
\begin{rem}
We comment on the assumptions we make for $G$. If $G$ is any topological group whose underlying space is in $\scr{K}$, then an action of $G$ on $X$ in $\scr{K}$ may not come from an action in ${\scr{T}}op$. The point is that the product $G\times X$ in $\scr{K}$ is defined by applying the $k$-ification functor to the product $G\times_c X$ in ${\scr{T}}op$, and not every action $G\times X\longrightarrow X$ need be continuous when viewed as a function $G\times_c X\longrightarrow X$. However, when $G$ is locally compact, $G\times_c X$ is already in $\scr{K}$ by \myref{HauswHaus}, and $k$-ification is not needed. There is then no ambiguity about what we mean by a $G$-space, and we need not worry about refining the topology on products with $G$.
Another reason for restricting to locally compact groups is that many useful properties of proper actions only hold in that case. In the literature, such results are usually derived for actions on Hausdorff spaces, but we shall see that weak Hausdorff generally suffices.
\end{rem}
We begin with some standard equivariant terminology.
\begin{defn} Let $X$ be a $G$-space and let $H\subset G$.
\begin{enumerate}[(i)]
\item An \emph{$H$-tube} $U$ in $X$ is an open $G$-invariant
subset of $X$ together with a $G$-map $\pi\colon U\longrightarrow G/H$.
If $x\in U$ and $H = G_x$, then $U$ is a \emph{tube around $x$}.
A tube is \emph{contractible} if $\pi$ is a $G$-homotopy equivalence.
\item An \emph{$H$-slice} $S$ in $X$ is an $H$-invariant subset such
that the canonical $G$-map $G\times_H S\longrightarrow GS\subset X$ is an embedding onto
an open subset. Then $GS$ is an $H$-tube with $S=\pi^{-1}(eH)$. Conversely, if $(U,\pi)$ is an $H$-tube in $X$, then $S=\pi^{-1}(eH)$ is an $H$-slice and $U=GS$.
On isotropy subgroups, we then have $G_y = H_y \subset H$ for all
$y\in S$, but equality need not hold. If $x\in S$ and $H = G_x$, then
$S$ is a \emph{slice through $x$}.
\item We say that $X$ \emph{has enough slices}
if every point $x\in X$ is contained in an $H$-slice for some \emph{compact} subgroup $H$. This implies that every point $x$ has compact isotropy group,
but in general it does not imply that there must be a slice \emph{through}
every point $x$.
\item A \emph{$G$-numerable cover of $X$} is a cover $\{U_j\}$ by tubes
such that there exists a locally finite partition of unity by $G$-maps $\lambda_j\colon X\longrightarrow[0,1]$ with support $U_j$.
\end{enumerate}
\end{defn}
The following is the equivariant generalization of \cite[6.7]{Dold0}.
\begin{prop}\mylabel{doldcover}
Any $G$-CW complex admits a $G$-numerable
cover by contractible tubes.
\end{prop}
\begin{proof}
The proof given by Dold \cite{Dold0} in the nonequivariant case goes through with only a minor change in the initial construction, which we sketch. From there, the technical details are unchanged. Let $X^n$ be the $n$-th skeletal filtration of a $G$-CW complex $X$. Let $\dot{X}^n$ denote the subspace obtained by deleting the centers $G/H\times 0$ of all $n$-cells in $X^n$ and let $r_n\colon \dot{X}^n\longrightarrow X^{n-1}$ denote the obvious retract. Starting
from the interior $e_n=G/H\times (D^n-S^{n-1})$ of an $n$-cell $c_n$, define $V_n^m$ inductively for $m\geq n$ by setting $V_n^n=e_n$ and $V_n^{m+1}=r_{m+1}^{-1}(V_n^m)$. Then the union $V_n^\infty=\bigcup_{m\geq n}V^m_n$ is a contractible tube, where the projection to $G/H$ is induced by
the projection of $e_n$ to $G/H\times 0$.
\end{proof}
We now give the definition of a proper group action in $\scr{K}$. We shall see
that the definition could equivalently be made in $\scr{U}$. For further details, but in ${\scr{T}}op$, see for example \cite{Bour, tomDieck}. Recall that a continuous map is {\em proper} if it is a closed map with compact fibers.
\begin{defn} A $G$-space $X$ in $G\scr{K}$ is \emph{proper} (or \emph{$G$-proper})
if the map
\[\theta\colon G\times X\longrightarrow X\times X\]
specified by $\theta(g,x) = (x,gx)$ is proper.
\end{defn}
We warn the reader that the definition is not quite the standard one. We are working in the category $\scr{K}$, and the product $X\times X$ on the right hand side is the $k$-space obtained by $k$-ifying the standard product topology on $X\times_c X$. In ${\scr{T}}op$ there are various other notions of a proper group action; see \cite{Biller} for a careful discussion. They all agree for actions of locally compact groups on completely regular spaces. If $X$ is proper, then the isotropy groups $G_x$ are compact since they are the fibers
$\theta^{-1}(x,x)$. Moreover, since points are closed subsets of $G$, the diagonal $\Delta_X =\theta(\{e\}\times X)$ must be a closed subset of $X\times X$ and thus $X$ must be weak Hausdorff. This means that proper $G$-spaces must be in $\scr{U}$. Since $G$ is locally compact, we have the following useful characterizations.
\begin{prop}\mylabel{properchar}
For a $G$-space $X$ in $G\scr{K}$ the following are equivalent.
\begin{enumerate}[(i)]
\item The action of $G$ on $X$ is proper.
\item The isotropy groups $G_x$ are compact and for all $(x,y )\in X\times X$
and all neighborhood $U$ of $\theta^{-1}(x,y)$ in $G\times X$, there is a neighborhood $V$ of $(x,y)$ in $X\times X$ such that $\theta^{-1}(V)\subset U$.
\item The isotropy groups $G_x$ are compact and for all $(x,y)\in X\times X$ and all neighborhoods $U$ of $\{ g \mid gx=y \}$ in $G$, there is a neighborhood $V$ of $(x,y)$ such that
\[\{g\in G\mid \text{$ga=b$ for some $(a,b)\in V$}\}\subset U.\]
\item The space $X$ is weak Hausdorff and every point $(x,y)\in X\times X$ has a neighborhood $V$ such that
\[\{g\in G\mid \text{$ga=b$ for some $(a,b)\in V$}\}\]
has compact closure in $G$.
\end{enumerate}
\end{prop}
\begin{proof}
This holds by essentially the same proof as \cite[1.6(b)]{Biller}. One must only keep in mind that we are now working in $\scr{K}$ rather than in ${\scr{T}}op$ and adjust the argument accordingly.
\end{proof}
\begin{cor} If $G$ is discrete, then a $G$-space $X$ is proper if and
only if any point $(x,y)\in X\times X$ has a neighborhood $V$ such that
\[\{g\in G\mid \text{$ga=b$ for some $(a,b)\in V$}\}\]
is finite.
\end{cor}
\begin{cor}
If $G$ is compact, then any $G$-space in $G\scr{U}$ is proper.
\end{cor}
\begin{rem}\mylabel{Hausprop}
There is an alternative description of the set displayed in \myref{properchar}
that may clarify the characterization. Define
\[\phi\colon G\times X\times X\longrightarrow X\times X\]
by $\phi(g,x,y)=(gx,y)$. For $V\subset X\times X$, let $\phi_V$ be the restriction of $\phi$ to $G\times V$ and let $\pi\colon G\times V\longrightarrow G$
be the projection, which is an open map since $G\times V$ has the
product topology. Then the displayed set is $\pi\phi_V^{-1}(\Delta_X)$.
If $X\times X = X\times_c X$, then the condition in \myref{properchar} is equivalent to the more familiar one that any two points $x$ and $y$
in $X$ have neighborhoods $V_x$ and $V_y$ such that
\[\{g\in G\mid gV_x\cap V_y \neq \emptyset\}\]
has compact closure in $G$.
\end{rem}
\begin{prop}\mylabel{propproper}
Proper actions satisfy the following closure properties.
\begin{enumerate}[(i)]
\item The restriction of a proper action to a closed subgroup is proper.
\item An invariant subspace of a proper $G$-space is also proper.
\item Products of proper $G$-spaces are proper.
\item If $X$ is a proper Hausdorff $G$-space in $G\scr{K}$ and $C$ is a compact Hausdorff $G$-space, then the $G$-space $\text{Map}(C,X)$ is proper.
\item An $H$-space $S$ is $H$-proper if and only if $G\times_H S$ is
$G$-proper.
\end{enumerate}
\end{prop}
\begin{proof}
The first three are standard and elementary; see for example \cite[I.5.10]{tomDieck}. The
fifth is \cite[2.3]{Biller}. We prove (iv). We must show that the map
\[\theta\colon G\times \text{Map}(C,X)\longrightarrow \text{Map}(C,X)\times \text{Map}(C,X)\]
is proper, which amounts to showing that it is closed and that the isotropy groups $G_f$ are compact for $f\in\text{Map}(C,X)$. For the latter, let $\{g_i\}$ be a net in $G_f$ and fix $c\in C$. Note that $f(g_ic)=g_if(c)$.
Since $C$ is compact, we can assume by passing to a subnet that $\{g_ic\}$ converges to some $\bar{c}\in C$. Let $V$ be a neighborhood of $(f(c),f(\bar{c}))$ such that
\[B=\{g\in G\mid \text{$ga=b$ for some $(a,b)\in V$}\}\]
has compact closure. Since $C$ is compact, $C\times C\times \text{Map}(C,X)$ has the usual product topology.
Since the map
\[C\times C\times \text{Map}(C,X)\longrightarrow X\times X\]
that sends $(c,d,f)$ to $(f(c),f(d))$ is continuous and the net $\{c,g_ic,f\}$ converges to $(c,\bar{c},f)$, the net $\{(f(c),f(g_ic))\}=\{(f(c),g_if(c))\}$ must converge to $(f(c),f(\bar{c}))$. It follows that a subnet of $\{g_i\}$
lies in $B$ and therefore has a converging sub-subnet.
To show that $\theta$ is closed, let $A$ be a closed subset of $G\times \text{Map}(C,X)$ and let $\{(f_i,g_if_i)\}$ be a net in $\theta(A)$ that converges to $(f, F)$. We must show that $(f, F)$ is in $\theta(A)$. For
$c\in C$, the net $\{g_i^{-1}c\}$ has a subnet that converges to some $\bar{c}$, by the compactness of $C$, so we may as well assume that the original net
converges to $\bar{c}$. Let $V$ be a neighborhood of $(f(\bar{c}),F(c))$
such that
\[B'=\{g\in G\mid \text{$ga=b$ for some $(a,b)\in V$}\}\]
has compact closure. By continuity and the compactness of $C$, there is a compact neighborhood $K_1\times K_2$ of $(\bar{c},c)$ that $(f, F)$ maps into $V$. Since $\{(f_i, g_if_i)\}$ converges to $(f, F)$, there is an $h$ such that $(f_i, g_if_i)(K_1\times K_2)\subset V$ for $i\geq h$. It follows that there is a $k\geq h$ such that $(f_i(g_i^{-1}c), g_if_i(g_i^{-1}c))\in V$ for all $i\geq k$. Then the subnet $\{g_i\}_{i\geq k}$ is contained in $B'$ and therefore has a sub-subnet that converges to some $g\in G$.
We have now seen that our original net $\{(g_i, f_i)\}$ in $A$ has a subnet $\{(g_{i_j}, f_{i_j})\}$ that converges to $(g,f)$, and $(g,f)\in A$ since $A$ is closed. By the continuity of $\theta$, $\{\theta(g_i,f_i)\}$ must converge to $(f, F)=\theta(g,f)\in\theta(A)$. In this last statement, we are using the
uniqueness of limits, which we ensure by requiring $X$ and $C$ to be Hausdorff.
\end{proof}
The following theorem of Palais \cite{Palais}, as generalized by
Biller \cite{Biller}, is fundamental. Those sources work in ${\scr{T}}op$,
but the arguments work just as well in $\scr{U}$.
\begin{thm}[Palais]\mylabel{Palais} Let $X$ be a $G$-space in $G\scr{U}$.
\begin{enumerate}[(i)]
\item If $X$ has enough slices, then it is proper.
\item Conversely, if $X$ is completely regular and proper, then it has enough
slices.
\item If $G$ is a Lie group and $X$ is completely regular and proper, then there is a slice through each point of $X$.
\end{enumerate}
\end{thm}
\begin{proof} Part (i) is given by \cite[2.4]{Biller}. Part (iii) is
given by \cite[2.3.3]{Palais}. Part (ii) is deduced from part (iii)
in \cite[2.5]{Biller}.
\end{proof}
\section{Proper actions and equivariant bundles}
We introduce here the equivariant bundles to which we will apply our basic foundational results in Part IV. As we explain, \myref{Palais} allows us to generalize some basic results about such bundles from actions of compact Lie groups to proper actions of Lie groups.
Let $\Pi$ be a normal subgroup of a Lie group $\Gamma$ such that
$\Gamma/\Pi = G$ and let $q\colon \Gamma\longrightarrow G$ be the quotient homomorphism. By a \emph{principal $(\Pi;\Gamma)$-bundle} we mean the quotient map $p\colon P\longrightarrow P/\Pi$ where $P$ is a $\Pi$-free $\Gamma$-space such that $\Gamma$ acts properly on $P$. It follows that the induced $G$-action on $B=P/\Pi$ is proper. If $F$ is a $\Gamma$-space, then we have the associated $G$-map $E=P\times_\Pi F\longrightarrow P\times_\Pi *\cong P/\Pi$, which we say is a \emph{$\Gamma$-bundle with structure group $\Pi$ and fiber $F$}. For compact Lie groups, bundles of this general form are studied in \cite{LM}, which generalizes the study of the classical case $\Gamma = G\times \Pi$ given in \cite{L}. A summary and further references are given in \cite[Chapter VII]{EHCT}.
We recall an observation about such bundles.
\begin{lem}\mylabel{rho}
For $b\in B$, the action of $\Gamma$ on $F$ induces an action of the isotropy group $G_b$ on the fiber $E_b$ through a homomorphism $\rho_b\colon G_b\longrightarrow \Gamma$ such that $q\com \rho_b$ is the inclusion $G_b\longrightarrow G$ and $E_b\iso \rho_b^*F$.
\end{lem}
\begin{proof}
Choose $z\in P$ such that $\pi(z)=b$. The isotropy group $\Gamma_z$ intersects $\Pi$ in the trivial group, and $q$ maps $\Gamma_z$ isomorphically onto $G_b$. Let $\rho_b$ be the composite of $q^{-1}\colon G_b\longrightarrow \Gamma_z$ and the inclusion $\Gamma_z\longrightarrow \Gamma$. Since the subspace $\{z\}\times F$ of $P\times F$ is $\Gamma_z$--invariant and maps homeomorphically onto $E_b$ on passage to orbits over $\Pi$, the conclusion follows. Note that changing the choice of $z$ changes $\rho_b$ by conjugation by an element of $\Pi$ and changes the identification of $E_b$ with $F$ correspondingly.
\end{proof}
Bundles should be locally trivial. When $P$ is completely regular, local triviality is a consequence of \myref{Palais}(iii), just as in the
case when $\Gamma$ is a compact Lie group \cite[Lemma 3]{LM}, and this
justifies our bundle-theoretic terminology. Note that if $P$ is
completely regular, then so is $B=P/\Pi$.
\begin{lem}
A completely regular principal $(\Pi;\Gamma)$-bundle $P$ is locally trivial.
That is, for each $b\in B$, there is a slice $S_b$ through $b$ and a homeomorphism
\[\xymatrix{
\Gamma\times_{\Lambda} S_b\ar[r]^\cong\ar[d]_{q\times 1} & p^{-1}(GS_b)\ar[d]^{p} \\
G\times_{G_b} S_b \ar[r]^\cong & GS_b}\]
where $\Lambda\subset \Gamma$ only intersects $\Pi$ in the identity element and is mapped isomorphically to $G_b$ by $q$. The $\Lambda$-action on $S_b$ is given by pulling back the $G_b$-action along $q$.
\end{lem}
\section{Spaces of the homotopy types of $G$-CW complexes}
In this section, we recall and generalize the equivariant version of Milnor's results \cite{Milnor} about spaces of the homotopy types of CW complexes. For compact Lie groups, Waner formulated and proved such results
in \cite[\S4]{Waner1}. With a few observations, his proofs generalize to
deal with proper actions by general Lie groups. We first note the
following immediate consequence of \myref{doldcover} and \myref{Palais}.
\begin{thm} For any locally compact group $G$, a $G$-CW complex is proper if
and only if it is constructed from cells of the form $G/K\times D^n$, where $K$ is compact.
\end{thm}
We also note the following recent ``triangulation theorem'' of Illman \cite[Theorem II]{Ill}. It is this result that led us to try to generalize some of our results from compact Lie groups to general Lie groups.
\begin{thm}[Illman]\mylabel{Illman}
If $G$ is a Lie group that acts smoothly and properly on a smooth manifold $M$, then $M$ has a $G$-CW structure.
\end{thm}
Many of our applications of this result are based on the following observation.
\begin{lem}\mylabel{prodproper}
If $H$ and $K$ are closed subgroups of a topological group $G$ and $K$ is
compact, then the diagonal action of $G$ on $G/H\times G/K$ is proper.
\end{lem}
\begin{proof}
The proof given in \cite[I.5.16]{tomDieck} that $G$ acts properly on $G/K$ generalizes directly. Set $X=G/H\times G$. Let $G$ act diagonally from the left
and let $K$ act on the second factor from the right. Note that these actions commute. It suffices to show that $\theta\colon G\times X\longrightarrow X\times X$ is proper. Indeed, consider the commutative square
\[\xymatrix{G\times X \ar[d]\ar[r]^-\theta & X\times X\ar[d]\\
G\times X/K \ar[r]^-{\bar{\theta}} & X/K\times X/K.}\]
The right vertical map is proper and the left vertical map is surjective.
Therefore, by \cite[VI.2.13]{tomDieck}, the bottom horizontal map
is proper if the top horizontal map is proper. Since $X$ is a free $G$-space, $\theta$ is proper if and only if the image $\text{Im}(\theta)$ is a closed subspace of $X\times X$ and the map $\phi\colon \text{Im}(\theta)\longrightarrow G$ specified by $\phi(x,gx)=g$ is continuous. The diagonal subspace of $G/H\times G/H$ is closed, and its preimage under the map
$\zeta\colon X\times X\longrightarrow G/H\times G/H $
specified by
$$\zeta((xH,y),(\bar{x}H,\bar{y}))=(\bar{y}y^{-1}xH,\bar{x}H)$$
is precisely $\text{Im}(\theta)$, which is therefore closed. The function
$\phi$ is the restriction to $\text{Im}(\theta)$ of the continuous map
$\Phi\colon X\times X\longrightarrow G$ specified by
$$\Phi((xH,y),(\bar{x}H,\bar{y}))= \bar{y}y^{-1}$$
and is therefore continuous.
\end{proof}
We shall also make essential use of the following corollary of \myref{Illman}.
\begin{cor}\mylabel{Illcor} If $X$ is a proper $G$-CW complex, then, viewed as an $H$-space for any closed subgroup $H$ of $G$, $X$ has the structure of an $H$-cell complex.
\end{cor}
\begin{proof}
Each cell $G/K\times D^n$ has $K$ compact. Since $G$ acts smoothly and properly on the smooth manifold $G/K$, the closed subgroup $H$ also acts smoothly and properly. We use the resulting $H$-CW structure on all of the cells to obtain an $H$-cell structure. It is homotopy equivalent to an $H$-CW complex obtained by ``sliding down'' cells that are attached to higher dimensional ones, but we
shall not need to use that.
\end{proof}
\begin{thm}[Milnor, Waner]\mylabel{Milnor}
Let $G$ be a Lie group and $(X;X_i)$ be an $n$-ad
of closed sub-$G$-spaces of a proper $G$-space $X$. If $(X;X_i)$ has
the homotopy type of a $G$-CW $n$-ad and $(C;C_i)$ is an $n$-ad of
compact $G$-spaces, then $(X;X_i)^{(C;C_i)}$ has the homotopy type
of a $G$-CW $n$-ad.
\end{thm}
\begin{proof}
We only remark how the proof of Waner for the case of actions by a compact Lie group generalizes to the case of proper actions by a Lie group. Define a $G$-simplicial complex to be a $G$-CW complex such that $X/G$ with the induced cell structure is a simplicial complex. In \cite[\S5]{Waner1}, Waner proves that any $G$-CW complex is $G$-homotopy equivalent to a colimit of finite dimensional $G$-simplicial complexes and cellular inclusions and that a $G$-space dominated by a $G$-CW complex is $G$-homotopy equivalent to a $G$-CW complex. The arguments apply verbatim to any topological group $G$.
The rest of the argument requires two key lemmas. In \cite[4.2]{Waner1}, Waner defines the notion of a $G$-equilocally convex, or $G$-ELC, $G$-space. The first lemma says that every finite dimensional $G$-simplicial complex is $G$-ELC. The essential starting point is that orbits are $G$-ELC, the proof of which uses the Lie group structure just as in \cite[p.358]{Waner1} in the compact case. From there, Waner's proof \cite[\S6]{Waner1} goes through unchanged. The second says that any completely regular, $G$-paracompact, $G$-ELC, proper $G$-space is dominated by a $G$-CW complex. When $G$ is compact Lie, this is proven in \cite[\S7]{Waner1}. However, the hypothesis on $G$ is only used to guarantee the existence of enough slices, hence the proof holds without change for
proper actions of Lie groups, indeed of locally compact groups.
The rest of the proof goes as in \cite[Theorem 3]{Milnor}.
One only needs to make two small additional observations. First, if a $G$-simplicial complex $K$ has the homotopy type of a proper $G$-space $X$,
then it is proper. This holds since if $f\colon K\longrightarrow X$ is a homotopy equivalence, then $G_k\subset G_{f(k)}$ is compact. Second, for an
$n$-ad $(K;K_i)$ of $G$-simplicial complexes and a compact $n$-ad $(C;C_i)$,
$(X;X_i)^{(C;C_i)}$ is proper since it is a subspace of the proper $G$-space $X^C$; see (i) and (iv) of \myref{propproper}. Since it is also completely regular, $G$-paracompact, and $G$-ELC, it is dominated by a $G$-CW complex,
and the result follows from the steps above.
\end{proof}
\section{Some classical theorems about fibrations}
A basic principle of parametrized homotopy theory is that homotopical information is given on fibers. We recall two relevant classical theorems about Hurewicz fibrations and a comparison theorem relating Serre and Hurewicz fibrations. We begin with Dold's theorem \cite[6.3]{Dold0}. The nonequivariant proof in \cite[2.6]{May} is generalized to the equivariant case in Waner \cite[1.11]{Waner2}. Waner assumes throughout \cite{Waner2} that $G$ is a compact Lie group, but that assumption is not used in the cited proof.
\begin{thm}[Dold]\mylabel{Dold}
Let $G$ be any topological group and let
$B$ be a $G$-space that has a $G$-numerable cover by contractible tubes. Let $X\longrightarrow B$ and $Y\longrightarrow B$ be Hurewicz fibrations. Then a map $X\longrightarrow Y$
over $B$ is a fiberwise $G$-homotopy equivalence if and only if each fiber restriction $X_b\longrightarrow Y_b$ is a $G_b$-homotopy equivalence.
\end{thm}
We next recall and generalize a classical result that relates the homotopy types of fibers to the homotopy types of total spaces. Nonequivariantly, it is due to Stasheff \cite{Stash} and, with a much simpler proof, Sch\"on \cite{Schon}. The generalization to the equivariant case, for compact Lie groups, is given by Waner \cite[6.1]{Waner2}. With Theorems \ref{Dold}, \ref{Milnor} and \ref{Illman} in place, Sch{\"o}n's argument generalizes directly to give the following version. Since the result plays an important role in our work and the argument is so pretty, we can't resist repeating it in full.
\begin{thm}[Stasheff, Sch\"on]\mylabel{ss}
Let $G$ be a Lie group and $B$ be a proper $G$-space that has the homotopy type of a $G$-CW complex. Let $p\colon X\longrightarrow B$ be a Hurewicz fibration. Then $X$ has the homotopy type of a $G$-CW complex if and only if each fiber $X_b$ has the homotopy type of a $G_b$-CW complex.
\end{thm}
\begin{proof}
First assume that $X$ has the homotopy type of a $G$-CW complex. For $b\in B$, let $\iota\colon G_b\longrightarrow G$ be the inclusion and consider the $G_b$-map $\iota^*p\colon \iota^*X\longrightarrow \iota^*B$ of $G_b$-spaces. It is still a
Hurewicz fibration, as we see by using the left adjoint $G\times_{G_b}(-)$ of $\iota^*$. By \myref{Illcor}, $\iota^*X$ and $\iota^*B$ have the homotopy types of $G_b$-CW complexes. Factor $\iota^*p$ through the inclusion into its mapping cylinder $i\colon \iota^*X\longrightarrow M\iota^*p$. Since $G_b$ is compact, it follows from \myref{Milnor} that the homotopy fiber $F_bi=(M\iota^*p;\{b\},\iota^*X)^{(I;0,1)}$ has the homotopy type of a
$G_b$-CW complex. Since $F_b i$ is homotopy equivalent to $F_b \iota^*p$, by the gluing lemma, and $F_b \iota^*p$ is homotopy equivalent to the fiber $X_b$, this proves the forward implication.
For the converse, assume that each fiber $X_b$ has the homotopy type of a
$G_b$-CW complex. Let $\gamma\colon \Gamma X\longrightarrow X$ be a $G$-CW approximation of $X$. The mapping path fibration of $\gamma$ gives us a factorization of
$\gamma$ as the composite of a $G$-homotopy equivalence
$\nu\colon \Gamma X\longrightarrow N\gamma$ and a Hurewicz fibration
$q\colon N\gamma\longrightarrow X$. We may view $q$ as a map of fibrations over $B$.
\[\xymatrix{N\gamma \ar[rr]^-{q} \ar[dr]_{p\com q}& & X \ar[dl]^p\\
& B & \\}\]
The fibers of $p\com q$ have the homotopy types of $G_b$-CW complexes by the first part of the proof, since $\Gamma X$ is a $G$-CW complex, and the fibers
of $p$ have the homotopy types of $G_b$-CW complexes by hypothesis. Comparison of the long exact sequences associated to $p\com q$ and $p$ gives that $q$ restricts to a $G_b$-homotopy equivalence on each fiber. Noting that we can pull back a numerable cover by contractible tubes along a homotopy equivalence $B\longrightarrow B'$, where $B'$ is a $G$-CW complex, it follows from \myref{Dold} that $q$ is a homotopy equivalence.
\end{proof}
Although it no longer plays a role in our theory, the following little known result played a central role in our thinking. It shows that the dichotomy between Serre and Hurewicz fibrations diminishes greatly over CW base spaces.
It is due to Steinberger and West \cite{SW}, with a correction by Cauty \cite{Cauty}.
\begin{thm}[Steinberger and West; Cauty]\mylabel{qh}
A Serre fibration whose base and total spaces are CW complexes
is a Hurewicz fibration.
\end{thm}
We believe that this remains true equivariantly for compact Lie groups,
and it certainly remains true for finite groups. Before we understood
the limitations of the $q$-model structure, we planned to use this result
to relate our model theoretic homotopy category of ex-spaces over a
CW complex $B$ to a classical homotopy category defined in terms of
Hurewicz fibrations and thereby overcome the problems illustrated in \myref{noway}. Such a comparison is still central to our theory,
and it is this result that convinced us that such a comparison must hold.
\section{Quasifibrations}
For later reference, we recall the definition of quasifibrations.
Here $G$ can be any topological group.
\begin{defn}\mylabel{quasifib}
A map $p\colon E\longrightarrow Y$ in $\scr{K}$ is a \emph{quasifibration} if the map of pairs $p\colon (E,E_y)\longrightarrow (Y,y)$ is a weak equivalence for all $y$ in $Y$.
A map $p\colon E\longrightarrow Y$ in $\scr{K}/B$ or $\scr{K}_B$ is a \emph{quasifibration}
if it is a quasifibration on total spaces. A $G$-map $p\colon E\longrightarrow Y$
is a quasifibration if each of its fixed point maps $p^H\colon E^H\longrightarrow Y^H$
is a nonequivariant quasifibration.
\end{defn}
The condition that $p\colon (E,E_y)\longrightarrow (Y,y)$ is a weak equivalence means that for all $e\in E_y$ the following two conditions hold.
\begin{enumerate}[(i)]
\item $p_*\colon \pi_n(E,E_y,e)\longrightarrow \pi_n(Y,y)$ is an isomorphism for all $n\geq 1$.
\item For any $x\in E$, $p(x)$ is in the path component of $y$ precisely when the path component of $x$ in $E$ intersects $E_y$. In other words, the sequence
$$\pi_0(E_y,e)\longrightarrow \pi_0(E,e)\longrightarrow \pi_0(Y,y)$$
of pointed sets is exact.
\end{enumerate}
\begin{warn}
In contrast to the usual treatments in the literature, we do not require
$p$ to be surjective and therefore $\pi_0(E,e)\longrightarrow \pi_0(Y,y)$ need not
be surjective. Hurewicz and, more generally, Serre fibrations are examples
of quasifibrations, and they are not always surjective, as the trivial example $\{0\} \longrightarrow \{0,1\}$ illustrates. Model categorically, one point is that
the initial map $\emptyset \longrightarrow Y$ is always a Serre fibration since the
empty lifting problem always has a solution.
\end{warn}
The definition of a quasifibration is arranged so that the long exact sequence of homotopy groups associated to the triple $(E,E_y,e)$ is isomorphic to a long exact sequence
\[\cdots \longrightarrow \pi_{n+1}(Y,y)\longrightarrow \pi_n(E_y,e) \longrightarrow \pi_n(E,e)\longrightarrow \pi_n(Y,y)\longrightarrow\cdots\longrightarrow \pi_0(Y,y).\]
\part{Model categories and parametrized spaces}
\chapter*{Introduction}
In Part III, we shall develop foundations for para\-me\-trized equivariant stable homotopy theory. In making that theory rigorous, it became apparent to us that substantial foundational work was already needed on the level of ex-spaces. That work is of considerable interest for its own sake, and it involves general points about the use of model categories that should be of independent interest. Therefore, rather than rush through the space level theory as just a precursor of the spectrum level theory, we have separated it out in this more leisurely and discursive exposition.
In Chapter 4, which is entirely independent of our parametrized theory, we give general model theoretic background, philosophy, and results. In contrast to the simplicial world, we often have both a classical $h$-type and a derived $q$-type model structure in topologically enriched categories, with respective weak equivalences the homotopy equivalences and the weak homotopy equivalences. We describe what is involved in verifying the model axioms for these two types of model structures.
In Chapter 5, we describe how the parametrized world fits into this general framework. There are several different $h$-type model structures on our categories of parametrized $G$-spaces, with different homotopy equivalences based on different choices of cylinders. These mesh in unexpected ways. Understanding of this particular case leads us to a conceptual axiomatic description of how the classical $h$-type homotopy theory and the $q$-type model structure must be related in order to be able to do homotopy theory satisfactorily in a topologically enriched category.
In Chapter 6, we work nonequivariantly and develop our preferred ``$q$-type'' model category structure, the ``$qf$-model structure'', on the categories
$\scr{K}/B$ and $\scr{K}_B$. This chapter is taken directly from the second author's thesis \cite{Sig}.
In Chapter 7, we give the equivariant generalization of the $qf$-model
structure and begin the study of the resulting homotopy categories by
discussing those adjunctions that are given by Quillen pairs. There
is another new twist here in that we need to use many Quillen
equivalent $qf$-type model structures. In fact, this
is already needed nonequivariantly in the study of base change
along bundles $f\colon A\longrightarrow B$.
In Chapter 8, we discuss ex-fibrations and an ex-fibrant approximation
functor that better serves our purposes than model theoretic fibrant approximation in studying those adjunctions that are not given by
Quillen pairs. In Chapter 9, we describe our parametrized homotopy
categories in terms of classical homotopy categories of ex-fibrations
and use this description to resolve the issues concerning base change
functors and smash products that are discussed in the Prologue.
\chapter{Topologically bicomplete model categories}
\section*{Introduction}
In \S4.1, we describe a general philosophy about the role of different model structures on a given category $\scr{C}$. It is natural and important in many contexts, and it helps to clarify our thinking about topological categories of parametrized objects. In particular, we advertise a remarkable unpublished insight of Mike Cole. It is a pleasure to thank him for keeping us informed of his ideas. We describe how a classical ``$h$-type'' model structure and a suitably related Quillen ``$q$-type'' model structure, can be mixed together to give an ``$m$-type'' model structure such that the $m$-equivalences are the $q$-equivalences and the $m$-fibrations are the $h$-fibrations. This is a completely general phenomenon, not restricted to topological contexts.
In \S\S4.2 and 4.3, we describe classical structure that is present in any topologically bicomplete category $\scr{C}$. Here we follow up a very illuminating paper of Schw\"anzl and Vogt \cite{SVogt}. There are two classes of (Hurewicz) $h$-fibrations and two classes of $h$-cofibrations, ordinary and strong. Taking weak equivalences to be homotopy equivalences, the ordinary $h$-fibrations pair with the strong $h$-cofibrations and the strong $h$-fibrations pair with the ordinary $h$-cofibrations to give two interrelated model like structures. For each choice, all of the axioms for a proper topological model category are satisfied except for the factorization axioms, which hold in a weakened form. To prove that $\scr{C}$ is a model category, it suffices to prove one of the factorization axioms since the other will follow. Again, the theory can easily be adapted to other contexts than our topological one.
We signal an ambiguity of nomenclature. In the model category literature, the term ``simplicial model structure'' is clear and unambiguous, since there is only one model structure on simplicial sets in common use. In the topological context, we understand ``topological model structures'' to refer implicitly to the $h$-model structure on spaces for model structures of $h$-type and to the $q$-model structure on spaces for model structures of $q$-type. The meaning should always be clear from context.
In \S4.4, we give another insight of Cole, which gains power from the work of Schw\"anzl and Vogt. Cole provides a simple hypothesis that implies the missing factorization axioms for an $h$-model structure of either type on a topologically bicomplete category $\scr{C}$. When we restrict to compactly generated spaces, the hypothesis applies to give an $h$-model structure on $\scr{U}$. In $\scr{K}$, this seems to fail, and we give a streamlined version of Str{\o}m's original proof \cite{Strom}, together with his proof that the strong $h$-cofibrations in $\scr{K}$ are just the closed ordinary $h$-cofibrations. This works in exactly the same way for the categories $G\scr{K}$ and $G\scr{U}$, where $G$ is any (compactly generated) topological group.
In \S4.5, we describe how to construct compactly generated $q$-type model structures, giving a slight variant of standard treatments. In particular, $G\scr{K}$ and $G\scr{U}$ have the usual $q$-model structures in which the $q$-equivalences are the weak equivalences and the $q$-fibrations are the Serre fibrations. Again, $G$ can be any topological group. However we only know
that the model structure is $G$-topological when $G$ is a compact Lie group.
\section{Model theoretic philosophy: $h$, $q$, and $m$-model
structures}\label{Sphil}
The point of model categories is to systematize ``homotopy theory''. The homotopy theory present in many categories of interest comes in two flavors. There is a ``classical'' homotopy theory based on homotopy equivalences, and there is a more fundamental ``derived'' homotopy theory based on a weaker notion of equivalence than that of homotopy equivalence. This dichotomy pervades the applications, regardless of field. It is perhaps well understood that both homotopy theories can be expressed in terms of model structures on the underlying category, but this aspect of the classical homotopy theory has usually been ignored in the model theoretical literature, a tradition that goes back to Quillen's original paper \cite{Q}. The ``classical'' model structure on spaces was introduced by Str{\o}m \cite{Strom}, well after Quillen's paper, and the ``classical'' model structure on chain complexes was only introduced explicitly quite recently, by Cole \cite{Cole2} and Schw\"anzl and Vogt \cite{SVogt}.
Perhaps for this historical reason, it may not be widely understood that these two model structures can profitably be used in tandem, with the $h$-model structure used as a tool for proving things about the $q$-model structure. This point of view is implicit in \cite{EKMM, MM, MMSS}, and a variant of this point of view will be essential to our work. In the cited papers, the terms ``$q$-fibration'' and ``$q$-cofibration'' were used for the fibrations and cofibrations in the Quillen model structures, and the term ``$h$-cofibration'' was used for the classical notion of a Hurewicz cofibration specified in terms of the homotopy extension property (HEP). The corresponding notion of an ``$h$-fibration'' defined in terms of the covering homotopy property (CHP) is fortuitously appropriate\footnote{However, the notation conflicts with the notation often used for Dold's notion of a weak or ``halb''-fibration. We shall make no use of that notion, despite its real importance in the theory of fibrations. We do not know whether or not it has a model theoretic role to play.}. Just as the ``$q$'' is meant to suggest Quillen, the ``$h$'' is meant to suggest Hurewicz, as well as homotopy. It is logical to follow this idea further (as was not done in \cite{EKMM, MM, MMSS}) by writing $q$-fibrant, $q$-cofibrant, $h$-fibrant, and $h$-cofibrant for clarity. Following this still further, we should also write ``$h$-equivalence'' for homotopy equivalence and ``$q$-equivalence'' for (Quillen) weak equivalence. The relations among these notions are as follows in all of the relevant categories $\scr{C}$:
\vspace{1mm}
\begin{center}
\begin{tabular}{|r c l|} \hline
$h$-equivalence & $\Longrightarrow$ & $q$-equivalence\\
$h$-cofibration & $\Longleftarrow$ & $q$-cofibration\\
$h$-cofibrant & $\Longleftarrow$ & $q$-cofibrant\\
$h$-fibration & $\Longrightarrow$ & $q$-fibration\\
$h$-fibrant & $\Longrightarrow$ & $q$-fibrant\\\hline
\end{tabular}
\end{center}
\vspace{1mm}
Therefore, {\em the identity functor is the right adjoint of a
Quillen adjoint pair from $\scr{C}$ with its $h$-model structure
to $\scr{C}$ with its $q$-model structure.}\/ It follows
that we have an adjoint pair relating the classical homotopy
category, $h\scr{C}$\@bsphack\begingroup \@sanitize\@noteindex{hC@$h\scr{C}$} say, to the derived homotopy category
$q\scr{C} = \text{Ho}\, \scr{C}$\@bsphack\begingroup \@sanitize\@noteindex{HoC@$\text{Ho}\, \scr{C}$}. This formulation
packages standard information. For example, the Whitehead theorem that a
weak equivalence between cell complexes is a homotopy equivalence,
or its analogue that a quasi-isomorphism between projective complexes
is a homotopy equivalence, is a formal consequence of this adjunction
between homotopy categories.
Recently, Cole \cite{Cole4} discovered a profound new way of thinking
about the dichotomy between the kinds of model structures that we have been
discussing. He proved the following formal model theoretic result.
\begin{thm}[Cole]\mylabel{Colemix}
\index{cofibration!mixed}\index{fibration!mixed}\index{weak equivalence!mixed}
Let $(\scr{W}_h,\scr{F}\!ib_h,\scr{C}\!of_h)$ and $(\scr{W}_q,\scr{F}\!ib_q,\scr{C}\!of_q)$ be two
mo\-del structures on the same category $\scr{C}$. Suppose that $\scr{W}_h\subset
\scr{W}_q$ and $\scr{F}\!ib_h\subset \scr{F}\!ib_q$. Then there is a \emph{mixed model
structure}\index{model structure!mixed} $(\scr{W}_q,\scr{F}\!ib_h,\scr{C}\!of_m)$ on
$\scr{C}$. The mixed cofibrations $\scr{C}\!of_m$ are the maps in $\scr{C}\!of_h$ that
factor as the composite of a map in $\scr{W}_h$ and a map in $\scr{C}\!of_q$. An
object is $m$-cofibrant if and only it is $h$-cofibrant and of the
$h$-homotopy type of a $q$-cofibrant object. If the $h$ and $q$-model
structures are left or right proper, then so is the $m$-model structure.
\end{thm}
By duality, the analogue with the inclusion $\scr{F}\!ib_h\subset \scr{F}\!ib_q$
replaced by an inclusion $\scr{C}\!of_h\subset \scr{C}\!of _q$ also holds. In the
category of spaces with the $h$ and $q$-model structures discussed above,
the theorem gives a mixed model structure whose $m$-cofibrant spaces are
the spaces of the homotopy types of CW-complexes. This $m$-model structure
combines weak equivalences with Hurewicz fibrations, and it might
conceivably turn out to be as important and convenient as the Quillen model
structure. It is startling that this model structure was not discovered
earlier.
The pragmatic point is two-fold. On the one-hand,
there are many basic results that apply to $h$-cofibrations and not just
$q$-cofibrations. Use of $h$-cofibrations limits the need for $q$-cofibrant
approximation and often clarifies proofs by focusing
attention on what is relevant. Many examples appear in
\cite{EKMM, MMSS, MM}, where properties of $h$-cofibrations serve as
scaffolding in the proof that $q$-model structures are in fact model structures. We shall formalize and generalize this idea in the next chapter.
On the other hand, there are many vital results that apply only to
$h$-fibrations (Hure\-wicz fibrations), not to $q$-fibrations (Serre
fibrations). For example, a local Hurewicz fibration is a Hurewicz
fibration, but that is not true for Serre fibrations. The mixed
model structure provides a natural framework in which to make use
of Hurewicz fibrations in conjunction with weak equivalences. While
we shall make no formal use of this model structure, it has provided
a helpful guide to our thinking. The philosophy here applies in
algebraic as well as topological contexts, but we shall focus on
the latter.
\section{Strong Hurewicz cofibrations and fibrations}
Fix a topologically bicomplete category $\scr{C}$ throughout this section
and the next.
With no further hypotheses on $\scr{C}$, we show that it satisfies most of the
axioms for not one but two generally different proper topological $h$-type
model structures. We alert the reader to the fact that we are here using the
term ``$h$-model structure'' in a generic sense. When we restrict
attention to parametrized spaces, we will use the term in a different
specific sense derived from the $h$-model structure on underlying total
spaces. The material of these
sections follows and extends material in Schw\"anzl and Vogt \cite{SVogt}.
We have cylinders $X\times I$\@bsphack\begingroup \@sanitize\@noteindex{XI@$X\times I$} and cocylinders
$\text{Map}(I,X)$.\@bsphack\begingroup \@sanitize\@noteindex{MapIX@$\text{Map}(I,X)$} When $\scr{C}$ is based,
we focus on the based cylinders $X\sma I_+$\@bsphack\begingroup \@sanitize\@noteindex{XIa@$X\sma I_+$}
and cocylinders $F(I_+,X)$.\@bsphack\begingroup \@sanitize\@noteindex{FIX@$F(I_+,X)$} In either case,
these define equivalent notions of homotopy, which we shall sometimes
call \emph{$h$-homotopy}. We will later use these and cognate notations,
but, for the moment, it is convenient to introduce the common notations
$\text{Cyl}(X)$\@bsphack\begingroup \@sanitize\@noteindex{CylX@$\text{Cyl}(X)$}\index{Cylinder object} and
$\text{Cocyl}(X)$\@bsphack\begingroup \@sanitize\@noteindex{CocylX@$\text{Cocyl}(X)$}\index{Cocylinder
object} for these objects. There are obvious classes of maps that one
might hope would specify a model structure.
\begin{defn}\mylabel{hmodel}
Let $f$ be a map in $\scr{C}$.
\begin{enumerate}[(i)]
\item $f$ is an \emph{$h$-equivalence}\index{equivalence!h@$h$-}
if it is a homotopy equivalence in $\scr{C}$.
\item $f$ is a \emph{Hurewicz fibration},\index{Hurewicz fibration}
abbreviated
\emph{$h$-fibration},\index{fibration!h@$h$-} if it
satisfies the CHP\index{CHP}\index{Covering homotopy property} in $\scr{C}$,
that is,
if it has the right lifting property (RLP) with respect
to the maps $i_0: X\longrightarrow \text{Cyl}(X)$ for $X\in\scr{C}$.
\item $f$ is a \emph{Hurewicz cofibration},\index{Hurewicz cofibration}
abbreviated
\emph{$h$-cofibration},\index{cofibration!h@$h$-}
if it satisfies the HEP\index{HEP}\index{Homotopy extension property} in
$\scr{C}$, that is, if it has
the left lifting property (LLP) with respect to the maps
$p_0\colon \text{Cocyl}(X)\longrightarrow X$.
\end{enumerate}
\end{defn}
These sometimes do give a model structure, but then the $h$-cofibrations
must be exactly the maps that satisfy the LLP with respect to the $h$-acyclic $h$-fibrations, and dually. In general, that does not hold. We shall characterize the maps
in $\scr{C}$ that do satisfy the LLP with respect to the $h$-acyclic
$h$-fibrations and, dually, the maps that satify the RLP with
respect to the $h$-acyclic $h$-fibrations. For this, we need the
following relative version of the above notions.
\begin{defn}\mylabel{barhmodel} We define strong Hurewicz fibrations and
cofibrations.
\begin{enumerate}[(i)]
\item A map $p\colon E\longrightarrow Y$ is a \emph{strong Hurewicz
fibration},\index{Hurewicz fibration!strong} abbreviated
\emph{$\bar{h}$-fibration},\index{fibration!h@$\bar{h}$-}
if it satisfies the
\emph{relative CHP}\index{relative CHP}\index{CHP!relative} with
respect to all $h$-cofibrations $i:A\longrightarrow X$, in the sense that a
lift exists in any diagram
\[\xymatrix{A\ar[r]^i\ar[d]_{i_0} & X \ar[r] \ar[d] & E\ar[d]^p\\
\text{Cyl}(A)\ar[r]\ar[urr]|(.55)\hole & \text{Cyl}(X)\ar[r]\ar@{-->}[ur] &
Y.}\]
\item A map $i:A\to X$ is a \emph{strong Hurewicz cofibration},\index{Hurewicz cofibration!strong}
abbreviated \emph{$\bar{h}$-cofibration},\index{cofibration!h@$\bar{h}$-}
if it satisfies the \emph{relative HEP}\index{relative HEP}\index{HEP!relative} with respect to all
$h$-fibrations $p:E\to Y$, in the sense that a lift exists in any diagram
\[\xymatrix{A\ar[r]\ar[d]_i & \text{Cocyl}(E) \ar[r] \ar[d] &
\text{Cocyl}(Y)\ar[d]^{p_0}\\
X\ar[r]\ar@{-->}[ur]\ar[urr]|(.44)\hole & E\ar[r]_p & Y.}\]
\end{enumerate}
\end{defn}
We recall the standard criteria for maps to be $h$-fibrations or $h$-cofibrations. Define the \emph{mapping cylinder}\index{mapping cylinder} $Mf$\@bsphack\begingroup \@sanitize\@noteindex{Mf@$Mf$} and \emph{mapping path fibration}\index{mapping path fibration} $Nf$\@bsphack\begingroup \@sanitize\@noteindex{Nf@$Nf$} by the usual pushout and pullback diagrams
$$\xymatrix{
X \ar[d]_{i_0} \ar[r]^-f & Y \ar[d] \\
\text{Cyl}(X) \ar[r] & Mf}\\
\ \ \ \ \text{and} \ \ \ \
\xymatrix{Nf \ar[d] \ar[r] & \text{Cocyl}(Y) \ar[d]^{p_0} \\
X \ar[r]_{f} & Y.\\}$$
\begin{lem}\mylabel{babyish} Let $f$ be a map in $\scr{C}$.
\begin{enumerate}[(i)]
\item $f$ is an $h$-fibration if and only if
it has the RLP with respect to the map $i_0: Nf\longrightarrow \text{Cyl}(Nf)$.
\item $f$ is an $h$-cofibration if and only if it has
the LLP with respect to the map $p_0: \text{Cocyl}(Mf)\longrightarrow Mf$.
\end{enumerate}
\end{lem}
The $\bar{h}$-fibrations and $\bar{h}$-cofibrations admit similar characterizations. These were taken as definitions in \cite[2.4]{SVogt}.
\begin{lem}\mylabel{h-char} (i) A map $p\colon E\longrightarrow Y$ is an $\bar{h}$-fibration if and only if it has the RLP with respect to the canonical map
$Mi\longrightarrow \text{Cyl}(X)$ for any $h$-cofibration $i:A\to X$;
this holds if and only if the canonical map $\text{Cocyl}(E)\to Np$
has the RLP with respect to all $h$-cofibrations.\\
(ii) A map $i:A\to X$ is an $\bar{h}$-cofibration if and only if
it has the LLP with respect to the canonical map
$\text{Cocyl}(E)\longrightarrow Np$ for any $h$-fibration $p:E\to Y$;
this holds if and only if the canonical map $Mi\to \text{Cyl}(X)$
has the LLP with respect to all $h$-fibrations.
\end{lem}
Observe that the map $i_0:X\longrightarrow\text{Cyl}(X)$ is an $\bar{h}$-cofibration and the map $p_0:\text{Cocyl}(X)\longrightarrow X$ is an $\bar{h}$-fibration. Since the cylinder objects associated to initial objects are initial objects, $\bar{h}$-fibrations are in particular $h$-fibrations. Similarly, $\bar{h}$-cofibrations are $h$-cofibrations. Observe too that every object is both $\bar{h}$-cofibrant and $\bar{h}$-fibrant, hence both $h$-cofibrant and $h$-fibrant.
We shall see in \S4.4 that these distinctions are necessary in $\scr{K}$ but disappear in $\scr{U}$, where the $h$ and $\bar{h}$ notions coincide. Even there, however, the conceptual distinction sheds light on classical arguments.
The results of this section and the next are quite formal. Amusingly, the
main non-formal ingredient is just the use in the following proof of the
standard fact that $\{0,1\}\to I$ has the LLP with respect to all
$h$-acyclic $h$-fibrations.
\begin{lem}\mylabel{h-retract} Let $i\colon A\longrightarrow X$ and $p\colon E\longrightarrow
B$ be maps
in $\scr{C}$.
\begin{enumerate}[(i)]
\item If $i$ is an $h$-acyclic $h$-cofibration, then $i$ is the inclusion
of a strong deformation retraction $r\colon X\longrightarrow A$.
\item If $i$ is the inclusion of a strong deformation retraction $r:X\to A$, then
$i$ is a retract of $Mi\to \text{Cyl}(X)$.
\item If $p$ is an $h$-acyclic $h$-fibration, then $p$ is a strong
deformation retraction.
\item If $p$ is a strong deformation retraction, then $p$ is a retract of
$\text{Cocyl}(E)\longrightarrow Np$.
\end{enumerate}
\end{lem}
\begin{proof} The last two statements are dual to the first two. For (i),
since the $h$-equivalence $i$ is an $h$-cofibration, application of the HEP shows that $i$ has a homotopy inverse $r:X\to A$ such that $ri=\text{id}_A$.
Since $\{0,1\}\longrightarrow I$ has the LLP with respect to $h$-acyclic
$h$-fibrations, an adjunction argument shows that $p_{(0,1)}$ has the RLP
with respect to $h$-cofibrations. Thus a lift
exists in the diagram on the left, which means that $r$ is a strong
deformation retraction with inclusion $i$.
\[\begin{aligned}\xymatrix{
A\ar[d]_i\ar[r]^-{c} & \text{Cocyl}(A) \ar[r]^{\text{Cocyl}(i)} &
\text{Cocyl}(X)\ar[d]^{p_{(0,1)}}\\
X\ar@{-->}[urr]_\beta \ar[rr]_{(i\com r,\text{id}_X)} && X\times X}
\end{aligned}
\quad
\begin{aligned}\xymatrix{
& A \ar[d]_{i_0}\ar[r]^i & X \ar@{-}[d]_{i_0}\ar[dr]^r\\
A\ar[r]^-{i_1}\ar[d]_i &
\text{Cyl}(A)\ar[dr]_{\text{Cyl}(i)}\ar[rr]^(.35){\text{pr}} & \ar[d] &
A\ar[d]^i\\
X \ar[rr]_{i_1} && \text{Cyl}(X) \ar[r]_-\beta & X}
\end{aligned}\]
For (ii), we are given $\beta$ in the diagram on the left displaying $r$ as
a strong deformation retraction with inclusion $i$. Then the diagram on the
right commutes, where the composites displayed in the lower two rows are
identity maps. Using the universal property of $Mi$ to factor the crossing
arrows $i_0$ and $\text{pr}$ through $Mi$, we see that $i$ is a retract
of the canonical map $Mi\to \text{Cyl}(X)$.
\end{proof}
\section{Towards classical model structures in topological
categories}\label{sec:towardh}
We now have two candidates for a classical model structure on $\scr{C}$ based
on the $h$-equivalences. We can either take the $h$-fibrations and the
$\bar{h}$-cofibrations or the $h$-cofibrations and the $\bar{h}$-fibrations.
The following result shows that all of the axioms for a proper topological
model category are satisfied except that, in general, only a weakened form
of the factorization axioms holds.
\begin{thm}\mylabel{h-structure}\index{model structure!generich@generic
$h$-structures} The following versions of the axioms for a proper
topological model category hold.
\begin{enumerate}[(i)]
\item The classes of $h$-cofibrations, $\bar{h}$-cofibrations,
$h$-fibrations and $\bar{h}$-fibrations are closed
under retracts.
\item Let $i$ be an $h$-cofibration and $p$ be an $h$-fibration. The pair
$(i,p)$ has the lifting property if $i$ is strong and $p$ is $h$-acyclic
or if $p$ is strong and $i$ is $h$-acyclic.
\item Any map $f:X\to Y$ factors as
\[\xymatrix{X \ar[r]^-i & Mf \ar[r]^-r & Y}\]
where $i$ is an $\bar{h}$-cofibration and $r$ has a section that is an
$h$-acyclic $\bar{h}$-cofibration and as
\[\xymatrix{X \ar[r]^-s & Nf \ar[r]^-p & Y}\]
where $p$ is an $\bar{h}$-fibration and $s$ has a retraction that is an
$h$-acyclic $\bar{h}$-fibration.
\item Let $i:A\to X$ be an $h$-cofibration and $p:E\to B$ be an
$h$-fibration, where $i$ or $p$ is strong. Then the map
\[\scr{C}^\Box(i,p): \scr{C}(X,E)\to \scr{C}(A,E)\times_{\scr{C}(A,B)}\scr{C}(X,B)\]
induced by $i$ and $p$ is an $h$-fibration of spaces. It is $h$-acyclic if
$i$ or $p$ is acyclic and it is an $\bar{h}$-fibration if both $i$ and $p$
are strong.
\item The $h$-equivalences are preserved under pushouts along
$h$-cofibrations and pullbacks along $h$-fibrations.
\end{enumerate}
\end{thm}
\begin{proof}
Part (i) is clear since all classes are defined in terms of lifting
properties. Part (ii) follows directly from \myref{h-char} and
\myref{h-retract}. The factorizations of part (iii)
are the standard ones. We consider the first. The evident
section $j\colon Y\longrightarrow Mf$ is an $h$-acyclic $\bar{h}$-cofibration since
it is the pushout of one. Consider the lifting problem in the left diagram
below, in which the middle vertical composite is $i$. Here $p$ is an
$h$-acyclic $h$-fibration, and we choose a section $s$ of $p$.
\[\begin{aligned}\xymatrix{ & X\ar[d]_{i_1}\ar[dr]^{\alpha} \\
X\ar[r]^-{i_0}\ar[d]_f & \text{Cyl}(X) \ar@{-->}[r]^-{\lambda'}\ar[d]|(.5)\hole
& E \ar[d]^p\\
Y \ar[r]_j\ar[urr]^(.3){s\beta j} & Mf\ar@{-->}[ur]_{\lambda}\ar[r]_{\beta} & B}
\end{aligned}
\qquad
\begin{aligned}\xymatrix{X\amalg X \ar[rr]^-{s\com \beta\com j\com
f\amalg\alpha}\ar[d]_{i_{(0,1)}} && E\ar[d]^p\\
\text{Cyl}(X) \ar[r]\ar@{-->}[urr]^{\lambda'} & Mf\ar[r]_\beta & B}
\end{aligned}\]
We have a lift $\lambda'$ in the diagram on the right that makes the diagram on
the left commute, and the universal property of $Mf$ then gives us the lift
$\lambda$. Part (iv) is a consequence of the ``pairing theorem'' \cite{SVogt}, which we will state below. Finally we prove the first half of (v). The second half
follows by duality. Assume that $i$ is an $h$-cofibration and
$f$ is an $h$-equivalence in the pushout diagram on the left.
\[\xymatrix{A\ar[r]^f \ar[d]_i & B\ar[d]^j\\
X\ar[r]_g & Y}\qquad\qquad
\xymatrix{B\ar[r]^s\ar[d]_{is} & A\ar[rr]^f\ar[d]\ar[ddr]^(.3)i &&
B\ar[d]\ar[ddr]^j\\
X\ar[r]^{s'}\ar@{=}[drr] & P\ar[rr]|(.25)\hole^{f'}\ar[dr]|(.4)p &&
X \ar[dr]|(.4)q\\
&& X\ar[rr]^g & & Y}\]
We must prove that $g$ is an $h$-equivalence. By (ii), we can factor $f$ as
a composite of an $h$-acyclic $h$-cofibration and a map that has a section
which is an $h$-acyclic $h$-cofibration. Since pushouts preserve $h$-acyclic
$h$-cofibrations, we may assume that $f$ has a section $s\colon B\longrightarrow A$
that is an $h$-acyclic $h$-cofibration. We then obtain the diagram on the
right. Its left back rectangle is a pushout, as is the outer back
rectangle, and therefore the right back rectangle is also a pushout. This
implies that the bottom square is a pushout. The map $s'$ is an $h$-acyclic
$h$-cofibration since $s$ is one, and therefore $p$ is an $h$-equivalence.
The map $f'$ is also $h$-acyclic since it has the $h$-acyclic section $s'$.
Just as we could assume that $f$ has a section that is an $h$-acyclic
$h$-cofibration, we find that we may assume that $p$ has a section $t$ that
is an $h$-acyclic $h$-cofibration and is a map under $A$. Chasing pushout
diagrams, we find that $g$ is a retract of $f'$ and is therefore an
$h$-equivalence.
\end{proof}
The following result is the pairing theorem\index{pairing theorem} of
\cite[2.7 and 3.6]{SVogt}. We shall not repeat the proof, which consists
of careful but formal adjunction arguments. Its general statement is framed
so as to apply to cartesian products in the unbased situation, smash products
in the based situation, and tensors in either situation.
\begin{thm}[Schw\"anzl and Vogt]\mylabel{SVogt} Let $\scr{A}$, $\scr{B}$, and $\scr{C}$
be topologically
bicomplete categories and let
$$ T\colon\scr{A}\times \scr{B}\longrightarrow \scr{C},\ \ U\colon\scr{A}^{\text{op}}\times \scr{C}\longrightarrow
\scr{B},\ \ \text{and} \ \ V\colon\scr{B}^{\text{op}}\times \scr{C}\longrightarrow \scr{A}$$
be continuous functors that satisfy adjunctions
$$ \scr{C}(T(A,B),C) \iso \scr{B}(B, U(A,C))\iso \scr{A}(A,V(B,C)).$$
Let $i\colon A\longrightarrow X$ be an $h$-cofibration in $\scr{A}$,
$j\colon B\longrightarrow Y$ be an $h$-cofibration in $\scr{B}$,
and $p\colon E\longrightarrow Z$ be an $h$-fibration in $\scr{C}$.
\begin{enumerate}[(i)]
\item Assume that $i$ or $j$ is strong. Then the map
$$T(A,Y)\cup_{T(A,B)}T(X,B)\longrightarrow T(X,Y)$$
induced by $i$ and $j$ is an $h$-cofibration in $\scr{C}$. It is
$h$-acyclic if $i$ or $j$ is $h$-acyclic and it is strong if
both $i$ and $j$ are strong.
\item Assume that $j$ or $p$ is strong. Then the map
$$V(Y,E)\longrightarrow V(B,E)\times_{V(B,Z)}V(Y,Z)$$
induced by $j$ and $p$ is an $h$-fibration in $\scr{A}$. It is
$h$-acyclic if $j$ or $p$ is $h$-acyclic and it is strong if
both $j$ and $p$
are strong.
\end{enumerate}
\end{thm}
As Schw\"anzl and Vogt observe, these results imply that the canonical map
$Mi\longrightarrow\text{Cyl}(X)$ is an $h$-acyclic $\bar{h}$-cofibration for any
$h$-cofibration $i:A\longrightarrow X$ and, dually, the canonical map
$\text{Cocyl}(X)\longrightarrow Np$ is an $h$-acyclic $\bar{h}$-fibration for any
$h$-fibration $p:E\longrightarrow Y$. Together with \myref{h-retract} and the
retract and factorization axioms of \myref{h-structure}, this implies that
all of the various classes of maps are characterized by the expected
lifting properties, just as if we had actual model categories.
\begin{prop}\mylabel{charstrong} The following characterizations hold.
\begin{enumerate}[(i)]
\item The $h$-fibrations are the maps that have the RLP
with respect to the $h$-acyclic $\bar{h}$-cofibrations and the
$h$-acyclic $\bar{h}$-cofibrations are the maps that have the LLP
with respect to the $h$-fibrations.
\item The $h$-cofibrations are the maps that have the LLP
with respect to the $h$-acyclic $\bar{h}$-fibrations and the
$h$-acyclic $\bar{h}$-fibrations are the maps that have the RLP
with respect to the $h$-cofibrations.
\item The $\bar{h}$-fibrations are the maps that have the RLP
with respect to the $h$-acyclic $h$-cofibrations and the
$h$-acyclic $h$-cofibrations are the maps that have the LLP
with respect to the $\bar{h}$-fibrations.
\item The $\bar{h}$-cofibrations are the maps that have the LLP
with respect to the $h$-acyclic $h$-fibrations and the
$h$-acyclic $h$-fibrations are the maps that have the RLP
with respect to the $\bar{h}$-cofibrations.
\end{enumerate}
\end{prop}
To show that $\scr{C}$ has an $h$-type model structure, it suffices
to prove the factorization axioms, and it is unnecessary to prove
them both.
\begin{lem}\mylabel{halffact} For either proposed $h$-model structure,
if one of the factorization axioms holds, then so does the other.
\end{lem}
\begin{proof} For definiteness, consider the case of $h$-fibrations and
$\bar{h}$-cofibrations. By \myref{h-structure}(ii), we can factor
any map $f\colon X\longrightarrow Y$ as the composite of an
$\bar{h}$-cofibration $i\colon X\longrightarrow Mf$ and an $h$-equivalence
$r\colon Mf\longrightarrow Y$. Suppose that we can factor $r$ as the composite
of an $h$-acyclic $\bar{h}$-cofibration $j\colon Mf\longrightarrow Z$
and an $h$-fibration $q\colon Z\longrightarrow Y$. Then $q$ must be
$h$-acyclic, hence $f = q\com (j\com i)$ factors $f$ as the
composite of an $\bar{h}$-cofibration and an $h$-acyclic $h$-fibration.
\end{proof}
A homotopy $X\longrightarrow Y$ in $\scr{C}$ can be specified by a path $h\colon I\longrightarrow \scr{C}(X,Y)$. If $i\colon A\longrightarrow X$ and $p\colon Y\longrightarrow B$ are maps in $\scr{C}$, then we say that $h$ is a homotopy \emph{relative to} $i$ or
\emph{corelative to} $p$ if the composite
\[\xymatrix{I\ar[r]^-h & \scr{C}(X,Y) \ar[r]^-{\scr{C}(i,Y)} & \scr{C}(A,Y)}
\qquad\text{or}\qquad
\xymatrix{I\ar[r]^-h & \scr{C}(X,Y) \ar[r]^-{\scr{C}(X,p)} & \scr{C}(X,B)}\]
is constant. When $i$ or $p$ is understood, we also refer to these as homotopies under $A$ or over $B$. The following result is well known and holds in any (based) topologically bicomplete category. Although we preferred to give a direct proof, we could have derived \myref{h-retract} from this result.
\begin{prop}\mylabel{relhtpy}
Let $f\colon X\longrightarrow Y$ be an $h$-equivalence.
\begin{enumerate}[(i)]
\item If $i\colon A\longrightarrow X$ and $j\colon A\longrightarrow Y$ are $h$-cofibrations such that $j=f\circ i$, then $f$ is an $h$-equivalence under $A$.
\item If $p\colon Y\longrightarrow B$ and $q\colon X\longrightarrow B$ are $h$-fibrations such that $q=p\circ f$, then $f$ is an $h$-equivalence over $B$.
\end{enumerate}
\end{prop}
\begin{proof}
For (i), see for example \cite[p.44]{Concise}. The proof there, although written for spaces, goes through without change. Part (ii) follows by a dual proof.
\end{proof}
\begin{rem}
The current section, as well as the previous and the following one, applies verbatim to the $G$-topologically bicomplete $G$-categories of \S10.2, where $G$ is any topological group. Of course, $(\scr{K}_{G,B}, G\scr{K}_B)$ is an example. The only changes occur in \myref{h-structure}(iv), where one must take the arrow $G$-spaces $\scr{C}_G(-,-)$ rather than the non-equivariant spaces $G\scr{C}(-,-)$, and in \myref{SVogt}, where the adjunction hypothesis requires a similar equivariant interpretation. See \S10.3 for a discussion of
$G$-topological model $G$-categories.
\end{rem}
\section{Classical model structures in general and in $\scr{K}$ and
$\scr{U}$}\label{sec:classmod}
Again, fix a topologically bicomplete category $\scr{C}$. Independent of the
work of Schw\"anzl and Vogt \cite{SVogt}, Cole \cite{Cole3} proved a general
result concerning when $\scr{C}$ has an $h$-type model structure. As we now
see is inevitable, the core of his argument concerns the verification
of one of the factorization axioms. That requires a hypothesis.
\begin{hyp}\mylabel{hyp}
Let $j_n: Z_n\longrightarrow Z_{n+1}$ and $q_n: Z_n\longrightarrow Y$ be maps in $\scr{C}$ such
that $q_{n+1}\com j_n = q_{n}$ and the $j_n$ are $h$-acyclic
$h$-cofibrations.
Let $Z=\text{colim} Z_n$ and let $q\colon Z\longrightarrow Y$ be obtained by
passage to colimits. Then the canonical map $\text{colim}\, Nq_n \longrightarrow Nq$ is an isomorphism in $\scr{C}$.
\end{hyp}
\begin{thm}[Cole]\mylabel{Cole}\index{Cole's hypothesis}
If $\scr{C}$ is a topologically bicomplete category which
satisfies \myref{hyp}, then the $h$-equivalences, $h$-fibrations,
and $\bar{h}$-cofibrations specify a proper topological $h$-model
structure\index{hmodel structure@$h$-model structure} on $\scr{C}$.
\end{thm}
\begin{proof}
It suffices to show that a map $f\colon X\longrightarrow Y$ factors
as the composite of an $h$-acyclic $\bar{h}$-cofibration $j\colon X\longrightarrow Z$
and an $h$-fibration $q\colon Z\longrightarrow Y$. Let $Z_0=X$ and $q_0=f$.
Inductively, given $q_n\colon Z_n\longrightarrow Y$, construct the following
diagram, in which $Z_{n+1}$ is the displayed pushout.
$$\xymatrix{
Nq_n\ar[d]_{i_0}\ar[r] & Z_n\ar[ddrr]^{q_n}\ar[d]^{j_n} & &\\
\text{Cyl}(Nq_n) \ar[r]^-{\lambda_n} \ar[rrrd] &
Z_{n+1}\ar@{-->}[drr]^(0.3){q_{n+1}} & &\\
&&& Y \\}$$
The map $\text{Cyl}(Nq_n)\longrightarrow Y$ is the adjoint of the projection
$Nq_n\longrightarrow \text{Cocyl}(Y)$ given by the definition of $Nq_n$, and
$q_{n+1}$ is the induced map. The maps $j_n$ are $h$-acyclic
$\bar{h}$-cofibrations
since they are pushouts of such maps. Let $Z$ be the colimit of the $Z_n$
and $j$ and $q$ be the colimits of the $j_n$ and $q_n$. Certainly
$f = q\com j$ and $j$ is an $h$-acyclic $\bar{h}$-cofibration.
By \myref{hyp}, $Nq$ is the colimit of the $Nq_n$. Since the cylinder
functor preserves colimits, we see by \myref{babyish} that $q$ is an
$h$-fibration since the $\lambda_n$ give a lift $\text{Cyl}(Nq)\longrightarrow Z$
by passage to colimits.
\end{proof}
The dual version of \myref{Cole} admits a dual proof.
\begin{thm}[Cole]\mylabel{Coledual}
If $\scr{C}$ is a topologically bicomplete category which
satisfies the dual of \myref{hyp}, then the $h$-equivalences,
$\bar{h}$-fibrations, and $h$-cofibrations specify a proper topological
$h$-model structure\index{hmodel structure@$h$-model structure}
on $\scr{C}$.
\end{thm}
From now on, we break the symmetry by focusing on $h$-fibrations and
$\bar{h}$-cofibrations. These give model structures in $\scr{K}$ and $\scr{U}$.
Everything in the rest of the section works equally in $G\scr{K}$ and $G\scr{U}$.
The following theorem combines several results of Str{\o}m
\cite{Strom1, Strom2, Strom}.
\begin{thm}[Str{\o}m]\mylabel{hmodelis}\index{model structure!on spaces}
The following statements hold.
\begin{enumerate}[(i)]
\item The $h$-equivalences, $h$-fibrations, and $\bar{h}$-cofibrations
give $\scr{K}$ a proper topological $h$-model structure. Moreover, a map in
$\scr{K}$ is an $\bar{h}$-cofibration if and only if it is a closed
$h$-cofibration.
\item The $h$-equivalences, $h$-fibrations, and $\bar{h}$-cofibrations
give $\scr{U}$ a proper topological $h$-model structure. Moreover, a map in
$\scr{U}$ is an $\bar{h}$-cofibration if and only if it is an $h$-cofibration.
\end{enumerate}
\end{thm}
\begin{proof}
\myref{Cole} applies to prove the first statement in $(ii)$, but it
does not seem to apply to prove the first statement in $(i)$.
The reasons are explained in \myref{Umod}. Taking $Z = Y^I$ and $p=p_0$ there,
the comparison map $\alpha$ specializes to the map
$\text{colim}\, Nf_n \longrightarrow Nf$ of \myref{hyp}.
It may be that $\alpha$ is a homeomorphism in this special case, but
we do not have a proof. It is a homeomorphism when we work in $\scr{U}$.
The characterization of the $\bar{h}$-cofibrations in $\scr{U}$ follows from
\myref{coflemma} and their characterization in $\scr{K}$.
For (i), we give a streamlined version of Str{\o}m's original
arguments that uses the material of the previous section to
prove both statements together. We proceed in four steps.
The first step is Str{\o}m's key observation, the second and
third steps give the second statement, and the fourth step proves
the needed factorization axiom. Consider an inclusion $i\colon A\longrightarrow X$.
Step 1. By Str{\o}m's \cite[Thm.\,3]{Strom1}, if $i$ is the inclusion of a
strong deformation retract and there is a map $\psi\colon X\longrightarrow I$ such
that $\psi^{-1}(0) = A$, then $i$ has the LLP with respect to all
$h$-fibrations. By \myref{charstrong}(i), this means that $i$ is an
$h$-acyclic $\bar{h}$-cofibration.
Step 2. If $i$ is an $h$-cofibration, then the canonical map
$j\colon Mi\longrightarrow X\times I$ is an $h$-acyclic $h$-cofibration and
therefore, by \myref{h-retract}, the inclusion of a strong deformation
retract. If $i$ is closed, then $(X,A)$ is an NDR-pair and there exists $\phi\colon X\longrightarrow I$
such that $\phi^{-1}(0) = A$. Define $\psi\colon X\times I\longrightarrow I$ by
$\psi(x,t) = t\phi(x)$. Then $\psi^{-1}(0) = Mi$. Applying Step 1, we
conclude that $j$ has the LLP with respect to all $h$-fibrations. By
\myref{h-char}, this means that $i$ is an $\bar{h}$-cofibration.
Step 3. We can factor any inclusion $i$ as the composite
$$\xymatrix@1{
A\ar[r]^-{i_0} & E \ar[r]^-{\pi} & X,}$$
where $E$ is the subspace $X\times (0,1]\cup A\times I$ of $X\times I$ and
$\pi$ is the projection. Note that $A = \psi^{-1}(0)$, where
$\psi\colon E\longrightarrow I$ is the projection on the second coordinate.
By direct verification of the CHP \cite[p.\,436]{Strom},
$\pi$ is an $h$-fibration. If $i$ is an $\bar{h}$-cofibration,
then it has the LLP with respect to $\pi$, hence we can lift the identity
map of $X$ to a map $\lambda\colon X\longrightarrow E$ such that $\lambda\com i = i_0$. It
follows that $i(A)$ is closed in $X$ since $i_0(A)$ is closed in $E$.
Step 4. Let $f\colon X\longrightarrow Y$ be a map. Use \myref{h-structure}(ii) to
factor $f$ as $p\com s$, where $s\colon X\longrightarrow Nf$ is the inclusion of a
strong deformation retract and $p$ is an $\bar{h}$-fibration. Use Step 3
to factor $s$ as
$$\xymatrix@1{
X\ar[r]^-{i_0} & Nf\times (0,1]\cup X\times I \ar[r]^-{\pi} & Nf.\\}$$
Here $i_0$ is the inclusion of a strong deformation retract and
$X = \psi^{-1}(0)$, as in Step 3. By Step 1, $i_0$ is an $h$-acyclic
$\bar{h}$-cofibration. By Step 3, $p\com \pi$ is an $h$-fibration.
\end{proof}
There are several further results of Str{\o}m about
$h$-cofibrations that deserve to be highlighted. In order, the following
results are \cite[Theorem 12]{Strom2}, \cite[Lemma 5]{Strom},
and \cite[Corollary 5]{Strom2}.
\begin{prop}\mylabel{StrPull}
If $p\colon E\longrightarrow Y$ is an $h$-fibration and the inclusion
$X\subset Y$ is an $\bar{h}$-cofibration, then the induced map
$p^{-1}(X)\longrightarrow E$ is an $\bar{h}$-cofibration.
\end{prop}
\begin{prop}\mylabel{factorher} If $i\colon A\longrightarrow B$ and $j\colon B\longrightarrow
X$ are maps in $\scr{K}$ such that $j$ and
$j\com i$ are $h$-cofibrations, then $i$ is an $h$-cofibration.
\end{prop}
\begin{prop}\mylabel{closure} If an inclusion $A\subset X$ is an
$h$-cofibration, then so is the induced inclusion $\bar{A}\subset X$.
\end{prop}
In view of the characterization of $\bar{h}$-cofibrations in
\myref{hmodelis}, it is natural to ask if there is an analogous
characterization of $\bar{h}$-fibrations. Only the following
sufficient condition is known. It is stated without proof in \cite[4.1.1]{SVogt}, and it gives another reason for requiring
the base spaces of ex-spaces to be in $\scr{U}$.
\begin{prop}\mylabel{hfibshfib}
An $h$-fibration $p\colon E\longrightarrow Y$ with $Y\in\scr{U}$ is an
$\bar{h}$-fibration.
\end{prop}
\begin{proof}
Let $k\colon A\longrightarrow X$ be an $h$-acyclic $h$-cofibration and let
$j\colon \overline{A}\longrightarrow X$ be the induced inclusion. By Propositions \ref{closure} and \ref{factorher}, $j$ and the inclusion $i\colon A\subset \overline{A}$
are $h$-cofibrations. By \myref{h-retract}(i), $k$ is the inclusion of a deformation retraction $r\colon X\longrightarrow A$ and the deformation restricts to a homotopy from $(i\circ r)\circ j$ to the identity on $\overline{A}$.
It follows that $j$ and hence also
$i$ are $h$-acyclic. Since $j$ is an $h$-acyclic $\bar{h}$-cofibration,
it has the
LLP with respect to $p$, and we see by a little diagram chase that it
suffices to verify that $i$ has the LLP with respect to $p$. Factor $p$
as the composite of $s\colon E\longrightarrow Np$ and $q\colon Np\colon \longrightarrow Y$, as
usual. Since $q$ is an
$\bar{h}$-fibration, $(i,q)$ has the lifting property, and it suffices to
show that $(i,s)$ has the lifting property. Suppose given a lifting problem
$f\colon A\longrightarrow E$ and $g\colon \overline{A}\longrightarrow Np$ such that
$s\com f = g\com i$. Note that $s(e) = (e,c_{p(e)})$ for $e\in E$,
where $c_y$ denotes the constant path at $y$. Since $Y$ is weak Hausdorff,
the constant paths give a closed subset of $Y^I$ and $Np = Y^I\times_Y E$
is a closed subset of $Y^I\times E$. Therefore $s(E)$ is closed in $Np$.
We conclude that
$$g(\overline{A})\subset \overline{g(A)} = \overline {s(f(A)} \subset
\overline{s(E)} = s(E),$$
which means that there is a lift $\overline{A}\longrightarrow E$.
\end{proof}
\section{Compactly generated $q$-type model structures}
We give a variant of the standard procedure for constructing $q$-type model
structures. The exposition prepares the way for a new variant that we will
explain in \S5.4 and which is crucial to our work. Although our discussion is adapted to topological examples, $\scr{C}$ need not be topological until otherwise specified. We first recall the small object argument in settings where compactness allows use of sequential colimits.
\begin{defn}\mylabel{cofhyp} Let $I$ be a set of maps in $\scr{C}$.
\begin{enumerate}[(i)]
\item A \emph{relative $I$-cell complex}\index{relative cell
complex}\index{cell complex!relative} is a map $Z_0\longrightarrow Z$, where
$Z$ is the colimit of a sequence of maps $Z_n\longrightarrow Z_{n+1}$ such that
$Z_{n+1}$ is the pushout $Y\cup_X Z_n$ of a coproduct $X\longrightarrow Y$ of
maps in $I$ along a map $X\longrightarrow Z_n$.
\item $I$ is \emph{compact}\index{compact!set of maps} if for every domain
object $X$ of a map in $I$ and every relative $I$-complex $Z_0\longrightarrow Z$,
the map $\text{colim}\,\scr{C}(X,Z_n)\longrightarrow \scr{C}(X,Z)$ is a bijection.
\item An \emph{$I$-cofibration}\index{cofibration!I@$I$-} is a map that
satisfies the LLP with respect to any map that satisfies the RLP with
respect to $I$.
\end{enumerate}
\end{defn}
\begin{lem}[Small object argument]\mylabel{small}\index{small object
argument}
Let $I$ be a compact set of maps in $\scr{C}$, where $\scr{C}$ is co\-com\-plete.
Then any map $f:X\longrightarrow Y$ in $\scr{C}$ factors functorially as a composite
\[\xymatrix{X \ar[r]^i & W \ar[r]^p & Y}\]
such that $p$ satisfies the RLP with respect to $I$ and $i$ is a relative
$I$-cell complex and therefore an $I$-cofibration.
\end{lem}
\begin{defn}\mylabel{compgendef}
A model structure on $\scr{C}$ is \emph{compactly generated}\index{compactly
generated!model structure}\index{model structure!compactly generated} if
there are compact sets $I$ and $J$ of maps in $\scr{C}$ such that the following
characterizations hold.
\begin{enumerate}[(i)]
\item The fibrations are the maps that satisfy the RLP with respect to $J$,
or equivalently, with respect to retracts of relative $J$-cell complexes.
\item The acyclic fibrations are the maps that satisfy the RLP with respect
to $I$, or equivalently, with respect to retracts of relative $I$-cell
complexes.
\item The cofibrations are the retracts of relative $I$-cell complexes.
\item The acyclic cofibrations are the retracts of relative $J$-cell
complexes.
\end{enumerate}
The maps in $I$ are called the \emph{generating
cofibrations}\index{generating cofibration}\index{cofibration!generating}
and the maps in $J$ are
called the \emph{generating acyclic cofibrations}.
\end{defn}
We find it convenient to separate out properties of classes of maps in a
model category, starting with the weak equivalences.
\begin{defn}\mylabel{weakeqsub} A subcategory of $\scr{C}$ is a
\emph{subcategory of weak equivalences}\index{weak equivalence!subcategory
of --s} if it satisfies the following closure properties.
\begin{enumerate}[(i)]
\item All isomorphisms in $\scr{C}$ are weak equivalences.
\item A retract of a weak equivalence is a weak equivalence.
\item If two out of three maps $f$, $g$, $g\com f$ are weak
equivalences, so is the third.
\end{enumerate}
\end{defn}
\begin{thm}\mylabel{compgen}
Let $\scr{C}$ be a bicomplete category with a subcategory of weak equivalences.
Let $I$ and $J$ be compact sets of maps in $\scr{C}$. Then $\scr{C}$ is a compactly
generated model category with generating cofibrations $I$ and generating
acyclic cofibrations $J$ if the following two conditions hold:
\begin{enumerate}[(i)]
\item (Acyclicity condition) Every relative $J$-cell complex is a weak equivalence.
\item (Compatibility condition) A map has the RLP with respect to $I$ if
and only if it is a weak equivalence and has the RLP
with respect to $J$.
\end{enumerate}
\end{thm}
\begin{proof} This is the formal part of Quillen's original proof of the
$q$-model structure on topological spaces and is a variant of
\cite[2.1.19]{Hovey} or \cite[11.3.1]{Hirschhorn}. The fibrations are
defined to be the maps that satisfy the RLP with respect to $J$. The cofibrations are defined to be the $I$-cofibrations and turn out to be the retracts of relative
$I$-cell complexes. The retract axioms clearly hold and, by (ii), the
cofibrations are the maps that satisfy the LLP with respect to the acyclic
fibrations, which gives one of the lifting axioms. The maps in $J$ satisfy
the LLP with respect to the fibrations and are therefore cofibrations,
which verifies something that is taken as a hypothesis in the versions in
the cited sources. Applying the small object argument to $I$, we factor a
map $f$ as a composite of an $I$-cofibration followed by a map that
satisfies the RLP with respect to $I$; by (ii), the latter is an acyclic
fibration. Applying the small object argument to $J$, we factor $f$ as a
composite of a relative $J$-cell complex that is a $J$-cofibration followed
by a fibration. By (i), the first map is acyclic, and it is a cofibration
because it satisfies the LLP with respect to all fibrations, in particular
the acyclic ones. Finally, for the second lifting axiom, if we are given a
lifting problem with an acyclic cofibration $f$ and a fibration $p$, then a
standard retract argument shows that $f$ is a retract of an acyclic
cofibration that satisfies the LLP with respect to all fibrations.
\end{proof}
Using the following companion to \myref{weakeqsub}, we codify the usual
pattern for verifying the acyclicity condition.
\begin{defn}\mylabel{cofsub} A subcategory of a cocomplete category
$\scr{C}$ is a \emph{subcategory of cofibrations}\index{cofibration!subcategory
of --s}
if it satisfies the following closure properties.
\begin{enumerate}[(i)]
\item All isomorphisms in $\scr{C}$ are cofibrations.
\item All coproducts of cofibrations are cofibrations.
\item If $i\colon X \longrightarrow Y$ is a cofibration and $f\colon X\longrightarrow Z$ is
any map, then the pushout $j\colon Y\longrightarrow Y\cup_X Z$ of $f$ along $i$ is a cofibration.
\item If $X$ is the colimit of a sequence of cofibrations
$i_n\colon X_n\longrightarrow X_{n+1}$, then the induced map $i\colon X_0\longrightarrow X$ is
a cofibration.
\item A retract of a cofibration is a cofibration.
\end{enumerate}
\end{defn}
In more general contexts, (iv) should be given a transfinite
generalization, but we shall not have need of that. Note that if a
subcategory of cofibrations is defined in terms of a left
lifting property, then all of the conditions hold automatically.
\begin{lem}\mylabel{cofcat}
Let $\scr{C}$ be a cocomplete category together with a subcategory of
cofibrations, denoted \emph{$g$-cofibrations}, and a subcategory
of weak equivalences, satisfying the following properties.
\begin{enumerate}[(i)]
\item A coproduct of weak equivalences is a weak equivalence.
\item If $i\colon X \longrightarrow Y$ is an acyclic $g$-cofibration and $f\colon
X\longrightarrow Z$ is any map, then the pushout $j\colon Y\longrightarrow Y\cup_X Z$ of $f$
along $i$ is a weak equivalence.
\item If $X$ is the colimit of a sequence of acyclic $g$-cofibrations
$i_n\colon X_n\longrightarrow X_{n+1}$, then the induced
map $i\colon X_0\longrightarrow X$ is a weak equivalence.
\end{enumerate}
If every map in a set $J$ is an acyclic $g$-cofibration, then every
relative $J$-cell complex is a weak equivalence.
\end{lem}
We emphasize that the $g$-cofibrations are not the model category
cofibrations and may or may not be the intrinsic $h$-cofibrations or
$\bar{h}$-cofibrations. They serve as a convenient scaffolding for
proving the model axioms.
\begin{rem} The properties listed in \myref{cofcat} include some of the
axioms for a ``cofibration category'' given by Baues \cite[pp 6, 182]{Baues}.
However, our purpose is to describe features of categories that
are more richly structured than model categories, often with several
relevant subcategories of cofibrations, rather than to describe
deductions from axiom systems for less richly structured categories,
which is his focus. The $g$-cofibrations in \myref{cofcat} need not be
the cofibrations of any cofibration category or model category.
\end{rem}
The $q$-model structures on $\scr{K}$ and $\scr{U}$ are obtained by \myref{compgen},
taking the $q$-equivalences to be the weak equivalences, that is, the maps
that induce isomorphisms on all homotopy groups, and the $q$-fibrations to
be the Serre fibrations. We also have the equivariant generalization, which
applies to any topological group $G$. We introduce the following notations, which will be used throughout.
\begin{defn}\mylabel{UrIJ}
Nonequivariantly, let $I$ and $J$ denote the set of inclusions $i\colon S^{n-1} \longrightarrow D^n$ (where $S^{-1}$ is empty) and the set of maps $i_0\colon D^n\longrightarrow D^n\times I$. Equivariantly, let $I$ and $J$ denote the set of all maps of the form $G/H\times i$, where $H$ is a (closed) subgroup of $G$ and $i$ runs through the maps in the nonequivariant sets $I$ and $J$. In the based categories $\scr{K}_*$ and
$G\scr{K}_*$ we continue to write $I$ and $J$ for the sets obtained
by adjoining disjoint base points to the specified maps.
\end{defn}
A map $f\colon X\longrightarrow Y$ of $G$-spaces is said to be a weak equivalence or Serre fibration if all fixed point maps $f^H\colon X^H\longrightarrow Y^H$ are weak equivalences or Serre fibrations. Just as nonequivariantly, we also call these $q$-equivalences and $q$-fibrations. Observe that $q$-equivalences are defined in terms of the equivariant homotopy groups $\pi_n^H(X,x) = \pi_n(X^H,x)$ for $H\subset G$ and $x\in X^H$ and that $q$-fibrations are defined in terms of the RLP with respect to the cells in $J$.
If $X_0 \longrightarrow X$ is a relative $I$ or $J$-cell complex, then $X/X_0$ is in $G\scr{U}$ and \myref{little} gives all that is needed to verify the compactness hypothesis in \myref{cofhyp}(ii). Taking the $g$-cofibrations to be the $h$-cofibrations, \myref{cofcat} applies to verify the acyclicity condition of \myref{compgen}. With considerable simplification, our verification of the compatibility condition for the $qf$-model structure in Chapter 6 specializes to verify it here. Nonequivariantly, the $q$-model structure is discussed in \cite[\S8]{DS} and, with somewhat different details, in \cite[2.4]{Hovey}
(where the details on transfinite sequences are unnecessary).
Equivariantly, a detailed proof of the following result is given in \cite[III\S1]{MM}. The argument there is given for based $G$-spaces, in $G\scr{T}$, but it works equally well for unbased $G$-spaces, in $G\scr{K}$.
\begin{thm}\mylabel{Gold}
For any $G$, $G\scr{K}$ is a compactly generated proper mo\-del category whose
$q$-equivalences, $q$-fibrations, and $q$-co\-fib\-rations are the weak equivalences, the Serre fibrations, and the retracts of relative $G$-cell complexes. The sets $I$ and $J$ are the generating $q$-cofibrations and the generating acyclic $q$-cofibrations, and all $q$-cofibrations are $\bar{h}$-cofibrations. If $G$ is a compact Lie group, then the model structure is
$G$-topological.
\end{thm}
The notion of a $G$-topological model category is defined
in the same way as the notion of a simplicial or topological
model category and is discussed formally in \S10.3 below.
The point of the last statement is that if $H$ and $K$ are
subgroups of a compact Lie group $G$, then $G/H\times G/K$
has the structure of a $G$-CW complex. By \myref{Illman},
this remains true when $G$ is a Lie group and $H$ and $K$
are compact subgroups. We shall see how to use this fact
model theoretically in Chapter 7.
\chapter{Well-grounded topological model categories}
\section*{Introduction}
It is essential to our theory
to understand the interrelationships among the various model structures that appear naturally in the parametrized context, both in topology and in general.
This understanding leads us more generally to an axiomatization of the properties that are required of a good $q$-type model structure in order
that it relate well to the classical homotopy theory on a topological category. The obvious $q$-model structure on ex-spaces over $B$ does not satisfy the axioms, and in the next chapter we will introduce a new model structure, the $qf$-model structure, that does satisfy the axioms.
As we recall in \S5.1, any model structure on a category $\scr{C}$ induces
a model structure on the category of objects over, under, or over and under
a given object $B$. When $\scr{C}$ is topologically bicomplete, so are these over and under categories. They then have their own intrinsic $h$-type model structures, which differ from the one inherited from $\scr{C}$. This leads to
quite a few different model structures on the category $\scr{C}_B$ of objects over and under $B$, each with its own advantages and disadvantages. Letting $B$ vary, we also obtain a model structure on the category of retracts. We shall only
be using most of these structures informally, but the
plethora of model structures is eye opening.
In \S5.2, we focus on spaces and compare the various classical notions of fibrations and cofibrations that are present in our over and under categories. Although elementary, this material is subtle, and it is nowhere presented accurately in the literature. In particular, we discuss $h$-type, $f$-type and $fp$-type model structures, where $f$ and $fp$ stand for ``fiberwise'' and ``fiberwise pointed''. For simplicity, we discuss this material nonequivariantly, but it applies verbatim equivariantly.
The comparisons among the $q$, $h$, $f$, and $fp$ classes of maps and model structures guide our development of parametrized homotopy theory. We think of the $f$-notions as playing a transitional role, connecting the $fp$ and
$h$-notions. In the rest of the chapter, we work in a general topologically bicomplete category $\scr{C}$, and we sort out this structure and its
relationship to a desired $q$-type model structure axiomatically.
Here we shift our point of view. We focus on three basic types of cofibrations that are in play in the general context, namely the Hurewicz cofibrations determined by the cylinders in $\scr{C}$, the ground cofibrations that come in practice from a given forgetful functor to underlying spaces, and the $q$-type model cofibrations. The first two are intrinsic, but we think of the $q$-type cofibrations as subject to negotiation. In $\scr{K}_B$, the Hurewicz cofibrations are the $fp$-cofibrations and the ground cofibrations are the $h$-cofibrations, which is in notational conflict with the point of view taken in the previous chapter.
In \S\S5.3 and 5.4, we ignore model theoretic considerations entirely. We describe how the two intrinsic types of cofibrations relate to each other
and to colimits and tensors, and we explain how this structure relates to weak equivalences.
We define the notion of a ``well-grounded model structure'' in \S5.5. We believe that this notion captures exactly the right blend of classical and model categorical homotopical structure in topological situations. It describes what is needed for a $q$-type model structure in a topologically bicomplete category to be compatible with its intrinsic $h$-type model structure and its ground structure. Crucially, the $q$-type cofibrations should be ``bicofibrations'', meaning that they are {\em both} Hurewicz cofibrations and ground cofibrations. To illustrate the usefulness of the axiomatization, and for later reference, we derive the long exact sequences associated to cofiber sequences and the $\text{lim}^1$ exact sequences associated to colimits in \S5.6.
A clear understanding of the desiderata for a good $q$-type model structure reveals that the obvious over and under $q$-model structure is essentially worthless for serious work in parametrized homotopy theory. This will lead us to introduce the new $qf$-model structure, with better behaved $q$-type cofibrations, in the next chapter. The formalization given in \S\S5.3--5.6 might seem overly pedantic were it only to serve as motivation for the definition of the $qf$-model structure. However, we will encounter exactly the same structure in Part III when we construct the level and stable model structures on parametrized spectra. We hope that the formalization will help guide the reader through the rougher terrain there.
We note parenthetically that there is still another interesting model structure on the category of ex-spaces over $B$, one based on local considerations. It is due to Michelle Intermont and Mark Johnson \cite{IJ}. We shall not discuss their model structure here, but we are indebted to them for illuminating discussions. It is conceivable that their model structure could be used in an alternative development of the stable theory, but that has not been worked out. Their structure suffers the defects that it is not known to be left proper and that, with their definition of weak equivalences, homotopy equivalences of base spaces need not induce equivalences of homotopy categories.
We focus mainly on the nonequivariant context in this chapter, but $G$
can be any topological group in all places where equivariance is considered.
\section{Over and under model structures}
Recall from \S1.2 that, for any category $\scr{C}$ and object $B$ in $\scr{C}$, we let $\scr{C}/B$\@bsphack\begingroup \@sanitize\@noteindex{CBa@$\scr{C}/B$} and $\scr{C}_B$ denote the categories of objects over $B$ and of ex-objects over $B$. We also have the category $B\backslash \scr{C}$ of objects under $B$. If $\scr{C}$ is bicomplete, then so are $\scr{C}/B$, $B\backslash \scr{C}$ and $\scr{C}_B$. We begin with some general observations about over and under model categories before returning to topological categories.
We have forgetful functors $U\colon \scr{C}/B\longrightarrow \scr{C}$\@bsphack\begingroup \@sanitize\@noteindex{U@$U$} and $V\colon \scr{C}_B\longrightarrow \scr{C}/B$.\@bsphack\begingroup \@sanitize\@noteindex{V@$V$} The first is left adjoint to the functor that sends an object $Y$ to the object $B\times Y$ over $B$:
\begin{equation}\label{silly}
\scr{C}(UX,Y) \iso \,\scr{C}/B\,(X, B\times Y).
\end{equation}
The second is right adjoint to the functor that sends an object $X$ over $B$
to the object $X\amalg B$ over and under $B$:
\begin{equation}\label{CexC}
\scr{C}_B(X\amalg B,Y) \iso \,\scr{C}/B\,(X,VY).
\end{equation}
As a composite of a left and a right adjoint, the total object functor
$UV\colon \scr{C}_B\longrightarrow \scr{C}$ does not enjoy good formal properties. This
obvious fact plays a significant role in our work. For example, it limits
the value of
the model structures on $\scr{C}_B$ that are given by the following result.
\begin{prop}\mylabel{under}\index{model structure!over and under}
Let $\scr{C}$ be a model category. Then $\scr{C}/B$, $B\backslash\scr{C}$, and $\scr{C}_B$
are model categories in which the weak equivalences, cofibrations, and fibrations are the maps over $B$, under $B$, or over and under $B$ which
are weak equivalences, fibrations, or cofibrations
in $\scr{C}$. If $\scr{C}$ is left or right proper, then so are $\scr{C}/B$,
$B\backslash\scr{C}$, and $\scr{C}_B$.
\end{prop}\begin{proof}
As observed in \cite[p.\,5]{Hovey} and \cite[3.10]{DS}, the
statement about $\scr{C}/B$ is a direct verification from the definition of a
model category. By the self-dual nature of the axioms, the statement about
$B\backslash\scr{C}$ is equivalent. The statement about $\scr{C}_B$ follows since
it is the category of objects under $(B,\text{id})$ in
$\scr{C}/B$. The last statement holds since pushouts and pullbacks in these
over and under categories are constructed in $\scr{C}$. \end{proof}
When considering $q$-type model structures, we start with a compactly
generated model category $\scr{C}$. Using the adjunctions (\ref{silly})
and (\ref{CexC}), we then obtain the following addendum to \myref{under}.
\begin{prop}\mylabel{cg}
If $\scr{C}$ is a compactly generated model category, then $\scr{C}/B$ and $\scr{C}_B$ are compactly generated. The generating (acyclic) cofibrations in $\scr{C}/B$ are the maps $i$ such that $Ui$ is a generating (acyclic) cofibration in $\scr{C}$. The generating (acyclic) cofibrations in $\scr{C}_B$ are the maps $i\amalg B$ where $i$ is a generating (acyclic) cofibration in $\scr{C}/B$.
\end{prop}
We now return to the case when $\scr{C}$ is topologically bicomplete. Then it has the resulting ``classical'', or $h$-type, structure that was discussed in \S\ref{sec:towardh} and \S\ref{sec:classmod}. If our philosophy in \S\ref{Sphil} applies to $\scr{C}$, then it also has $q$ and $m$-structures and the categories $\scr{C}/B$ and $\scr{C}_B$ both inherit over and under model structures that are related as we discussed there. However, since $\scr{C}$ is topologically bicomplete, so is $\scr{C}/B$ by \myref{topbicomp}, and $\scr{C}_B$ is based topologically bicomplete by \myref{btopbicomp}. These categories therefore have classical
$h$-type structures when they are regarded in their own right as topologically bicomplete categories. To fix notation and avoid confusion we give an overview of all of these structures.
We start with the $h$-classes of maps in $\scr{C}$ that are given in \myref{hmodel} and \myref{h-char}. As in our discussion of spaces, we work assymmetrically, ignoring the $\bar{h}$-fibrations and focusing on the candidates for $h$-type model structures given by the $h$-fibrations and $\bar{h}$-cofibrations. We agree to use the letter $h$ for the inherited classes of maps in $\scr{C}/B$ and $\scr{C}_B$, although that contradicts our previous use of $h$ for the classical classes of maps in an arbitrary topologically bicomplete category, such as $\scr{C}/B$ or $\scr{C}_B$. We shall resolve that ambiguity shortly by introducing new names for the classes of ``classical'' maps in those categories.
\begin{defn}
A map $g$
in $\scr{C}/B$ is an $h$-equivalence,\index{equivalence!h@$h$-}
$h$-fibration,\index{fibration!h@$h$-}
$h$-co\-fi\-bra\-tion,\index{cofibration!h@$h$-} or
$\bar{h}$-cofibration\index{cofibration!hs@$\bar{h}$-} if $Ug$ is
such a map in $\scr{C}$. A map $g$ in $\scr{C}_B$
is an $h$-equivalence, $h$-fibration, $h$-cofibration, or
$\bar{h}$-cofibration if $Vg$ is such a map in $\scr{C}/B$ or, equivalently,
$UVg$ is such a map in $\scr{C}$.
\end{defn}
The $\bar{h}$-cofibrations are $h$-cofibrations, but not conversely in
general. Since the object $*_B=(B,\text{id},\text{id})$ is initial and
terminal in $\scr{C}_B$, an object of $\scr{C}_B$ is $h$-cofibrant (or $\bar{h}$-cofibrant) if its section is an $h$-cofibration (or $\bar{h}$-cofibration) in $\scr{C}$. It is $h$-fibrant if its projection is an $h$-fibration in $\scr{C}$.
In $\scr{C}/B$, we have the notion of a homotopy over $B$, defined in terms of
$X\times_B I$ or, equivalently, $\text{Map}_B(I,X)$. The adjective
``fiberwise'' is generally used in the literature to describe these
homotopies. See, for example, the books \cite{CJ, James} on fiberwise
homotopy theory. To distinguish from the $h$-model structure, we agree to
write $f$ rather than $h$ for the fiberwise specializations of \myref{hmodel} and \myref{h-char}. To avoid any possible confusion, we formalize this,
making use of \myref{charstrong}.
\begin{defn}\mylabel{fmodel}
Let $g$ be a map in $\scr{C}/B$.
\begin{enumerate}[(i)]
\item $g$ is an \emph{$f$-equivalence}\index{equivalence!f@$f$-} if it is a
fiberwise homotopy equivalence.
\item $g$ is an \emph{$f$-fibration}\index{fibration!f@$f$-} if it satisfies
the fiberwise CHP, that is, if it has the RLP with respect to the maps
$i_0\colon X\longrightarrow X\times_B I$ for $X\in\scr{C}/B$.
\item $g$ is an \emph{$f$-cofibration}\index{cofibration!f@$f$-} if it
satisfies the fiberwise HEP, that is, if it has the LLP with
respect to the maps $p_0\colon \text{Map}_B(I,X) \longrightarrow X$.
\item $g$ is an \emph{$\bar{f}$-cofibration}\index{equivalence!fs@$\bar{f}$-}
if it has the LLP with respect to the $f$-acyclic $f$-fibrations.
\end{enumerate}
A map $g$ in $\scr{C}_B$ is an $f$-equivalence, $f$-fibration, $f$-cofibration,
or $\bar{f}$-cofibration if $Vg$ is one in $\scr{C}/B$.
\end{defn}
Again, $\bar{f}$-cofibrations are $f$-cofibrations, but not conversely in general. \myref{Cole} often applies to show that the $f$-fibrations and $\bar{f}$-cofibrations define an $f$-model structure on $\scr{C}/B$ and therefore, by \myref{under}, on $\scr{C}_B$. As is always the case for an intrinsic classical model structure, every object of $\scr{C}/B$ is both $f$-cofibrant and $\bar{f}$-cofibrant as well as $f$-fibrant. While this is obvious from the definitions, it may seem counterintuitive. It does not follow that every object of $\scr{C}_B$ is $f$-cofibrant since the two categories have different initial objects.
In $\scr{C}_B$, we also have the notion of a homotopy over and under $B$,
defined in terms of $X\sma_B I_+$ or, equivalently, $F_B(I_+,X)$.
The adjective ``fiberwise pointed'' is used in \cite{CJ, James} to
describe these homotopies. Again, for notational clarity, we agree to
write $fp$ rather than $h$ for the fiberwise pointed specializations of
\myref{hmodel} and \myref{h-char}, and we formalize this to avoid any
possible confusion.
\begin{defn}\mylabel{fpmodel}
Let $g$ be a map in $\scr{C}_B$.
\begin{enumerate}[(i)]
\item $g$ is an \emph{$fp$-equivalence} if it is a fiberwise pointed
homotopy equivalence.
\item $g$ is an \emph{$fp$-fibration} if it satisfies the fiberwise
point\-ed CHP, that is,
if it has the RLP with respect to the maps $i_0\colon X\longrightarrow X\sma_B I_+$.
\item $g$ is a \emph{$fp$-cofibration} if it satisfies the fiberwise
point\-ed HEP, that is, if it has the LLP with
respect to the maps $p_0\colon F_B(I_+,X) \longrightarrow X$.
\item $g$ is an \emph{$\overline{fp}$-cofibration} if it has the LLP
with respect to the $fp$-acyclic $fp$-fibrations.
\end{enumerate}
\end{defn}
Again, $\overline{fp}$-cofibrations are $fp$-cofibrations, but not conversely
in general, and \myref{Cole} often applies to show that the $fp$-fibrations
and $\overline{fp}$-cofibrations define an $fp$-model structure on $\scr{C}_B$.
We summarize some general formal implications relating our classes of maps.
\begin{prop}\mylabel{compare} Let $\scr{C}$, $\scr{C}/B$ and $\scr{C}_B$ be
topologically bicomplete
categories with $h$, $f$, and $fp$-classes of maps defined as above.
Then the following implications hold for maps in $\scr{C}_B$.
\vspace{1mm}
\begin{center}
\begin{tabular}{|r c c c l|} \hline
$fp$-equivalence & $\Longrightarrow$ & $f$-equivalence & $\Longrightarrow$
& $h$-equivalence\\
$fp$-cofibration & $\Longleftarrow$ & $f$-cofibration &$\Longrightarrow$
& $h$-cofibration\\
$\Uparrow$ \ \ \ \ \ \ \ \ & & $\Uparrow$ & & $\ \ \ \ \ \ \ \ \Uparrow$\\
$\overline{fp}$-cofibration & $\Longleftarrow$ & $\bar{f}$-cofibration &
$\Longrightarrow$ & $\bar{h}$-cofibration\\
$fp$-fibration & $\Longrightarrow$ & $f$-fibration & $\Longleftarrow$ &
$h$-fibration\\\hline
\end{tabular}
\end{center}
\vspace{1mm}
\noindent
Moreover, every object of $\scr{C}_B$ is both $fp$-fibrant and $fp$-cofibrant.
\end{prop}\begin{proof}
Trivial inspections of lifting diagrams show that an $h$-fibration is an
$f$-fibration, an $f$-cofibration is an $fp$-cofibration, and an
$\bar{f}$-co\-fi\-bra\-tion is an $\overline{fp}$-cofibration. Use of the
adjunctions (\ref{silly}) and (\ref{CexC}) shows that
an $f$-cofibration is an $h$-cofibration, an $\bar{f}$-cofibration is an
$\bar{h}$-cofibration, and an $fp$-fibration is an $f$-fibration. The last
statement holds since fiberwise pointed homotopies with domain or target
$B$ are constant at the section or projection of the target or source.
\end{proof}
\begin{rem} Assume that these classes of maps define model structures.
Then the implications in \myref{compare} lead via \myref{Colemix} and its
dual version to two new mixed model structures on $\scr{C}_B$, one with weak
equivalences the $f$-equivalences and fibrations the $fp$-fibrations and
one with weak equivalences the $h$-equivalences and cofibrations the
$\bar{f}$-cofibrations.
\end{rem}
The category $\scr{C}_{\scr{B}}$ of retracts introduced in \S2.5 suggests an alternative model theoretic point of view. We give the basic definitions, but we shall not pursue this idea in any detail. Again, \myref{Cole} often applies to verify the model category axioms. Note that the intrinsic homotopies are given by homotopies of total objects over and under homotopies of base objects.
\begin{defn}\mylabel{rmodel}
Assume that $\scr{C}_{\scr{B}}$ is topologically bicomplete and let $g$ be a map in
$\scr{C}_{\scr{B}}$.
\begin{enumerate}[(i)]
\item $g$ is an \emph{$r$-equivalence}\index{equivalence!r@$r$-} if it is a
homotopy equivalence of retractions.
\item $g$ is an \emph{$r$-fibration}\index{fibration!r@$r$-} if it satisfies
the retraction CHP, that is, if it has the RLP with respect to the maps
$i_0\colon X\longrightarrow X\times I$ for $X\in\scr{C}_{\scr{B}}$.
\item $g$ is an \emph{$r$-cofibration}\index{cofibration!r@$r$-} if it
satisfies the retraction HEP,
that is, if it has the LLP with respect to the maps
$p_0\colon \text{Map}(I,X) \longrightarrow X$.
\item $g$ is an \emph{$\bar{r}$-cofibration}\index{cofibration!r@$\bar{r}$-}
if it has the LLP
with respect to the $r$-acyclic $r$-fibrations.
\end{enumerate}
\end{defn}
\begin{rem}
The initial and terminal object of $\scr{C}_{\scr{B}}$ are the identity retractions of the initial and terminal objects of $\scr{B}$ and every object is both $r$-cofibrant and $r$-fibrant. It might be of interest to characterize the retractions for which the map $*_B\longrightarrow (X,p,s)$ induced by $s$ is an $r$-cofibration or for which the map $(X,p,s)\longrightarrow *_B$ induced by $p$ is an $r$-fibration. By specialization of the lifting properties, an ex-map over $B$ that is an $r$-cofibration or $r$-fibration is an $fp$-cofibration or $fp$-fibration in $\scr{C}_B$, but we have not pursued this question further.
\end{rem}
\section{The specialization to over and under categories of spaces}
Now we take $\scr{C}$ to be $\scr{K}$ or $\scr{U}$. We discuss the relationships
among our various classes of fibrations and cofibrations in this
special case, and we consider when the $f$ and $fp$ classes of maps
give model structures. Everything in this section applies equally
well equivariantly.
We first say a bit about based spaces, which are ex-spaces over $B=\{*\}$.
Here the fact that $*$ is a terminal object greatly simplifies matters.
All of the $f$-notions coincide with the corresponding $h$-notions, and
our trichotomy reduces to the familiar dichotomy between free (or $h$)
notions and based (or $fp$) notions. Recall that a based space is \emph{well-based},\index{space!well-based --} or \emph{nondegenerately based}, if the inclusion of the
basepoint is an $h$-cofibration. Every based space is $fp$-cofibrant,
and an $fp$-cofibration between well-based spaces is an $h$-cofibration \cite[Prop.\,9]{Strom}. Every based space is $fp$-fibrant, and an
$h$-fibration of based spaces satisfies the based CHP
with respect to well-based source spaces. Of course, the over and
under $h$-model structure differs from the intrinsic $fp$-model structure.
None of the reverse implications in \myref{compare} holds in general.
We gave details of that result since it is easy to get confused and
think that more is true than we stated.
\begin{sch}
On \cite[p.\,66]{CJ}, it is stated that a fiberwise
pointed cofibration which is a closed inclusion is a fiberwise
cofibration. That is false even when $B$ is a point, since
it would imply that every point of a $T_1$-space is a nondegenerate
basepoint. On \cite[p.\,69]{CJ}, it is stated that a fiberwise pointed
map (= ex-map) is a fiberwise pointed fibration if and only if it is
a fiberwise fibration. That is also false when $B$ is a point, since
the unbased CHP does not imply the based CHP.
\end{sch}
However, as for based spaces, the reverse implications in parts of
\myref{compare} often do hold under appropriate additional hypotheses.
\begin{prop}\mylabel{reverse}
The following implications hold for an arbitrary topologically bicomplete category $\scr{C}$.
\begin{enumerate}[(i)]
\item A map in $\scr{C}/B$ between $h$-fibrant objects over $B$ is an
$h$-equivalence if and only if it is an $f$-equivalence.
\item An ex-map between $f$-cofibrant ex-objects over $B$ is an
$f$-equivalence if and only if it is an $fp$-equivalence.
\end{enumerate}
\end{prop}
\begin{proof}
The first part follows from \myref{relhtpy}(ii) since an $f$-equivalence in $\scr{C}/B$ is the same as an $h$-equivalence over $B$ in $\scr{C}$. The second part follows similarly from \myref{relhtpy}(i) since an $fp$-equivalence in $\scr{C}_B$ is the same as an $f$-equivalence under $B$ in $\scr{C}/B$.
\end{proof}
The following results hold for spaces. We are doubtful that they hold in general.
\begin{prop}\mylabel{reverse2}
The following implications hold in both $G\scr{K}$ and $G\scr{U}$.
\begin{enumerate}[(i)]
\item An ex-map between $\bar{f}$-cofibrant ex-spaces is an $f$-cofibration
if and only if it is an $fp$-cofibration.
\item An ex-map whose source is $\bar{f}$-cofibrant is an $f$-fibration if and
only if it is an $fp$-fibration.
\end{enumerate}
\end{prop}\begin{proof}
Part (ii) is \cite[16.3]{CJ}.
Part (i) is stated on \cite[p.\,441]{Strom} and the proof given there for based spaces generalizes using the following lemma.
\end{proof}
It is easy to detect $f$-cofibrations by means of the following result,
whose proof is the same as that of the standard characterization of
Hurewicz cofibrations (e.g.\ \cite[p.\,43]{Concise}, see also
\cite[Thm.\,2]{Strom1}, \cite[Lem.\,4]{Strom2} and \cite[4.3]{CJ}).
\begin{lem}\mylabel{fNDR}\index{NDR}
An inclusion $i\colon X\longrightarrow Y$ in $\scr{K}/B$ is an $f$-cofibration if and
only if $(Y,X)$ is a fiberwise NDR-pair in the sense that there is a map
$u\colon Y\longrightarrow I$ such that $X \subset u^{-1}(0)$ and a homotopy $h\colon
Y\times_B I \longrightarrow Y$ over $B$ such that $h_0 = \text{id}$, $h_t = \text{id}$
on $X$ for $0\leq t\leq 1$, and $h_1(y)\in X$ if $u(y)<1$.
A closed inclusion $i: X\longrightarrow Y$ in $\scr{K}/B$ is an $\bar{f}$-cofibration if
and only if the map $u$ above can be chosen so that $X=u^{-1}(0)$.
\end{lem}
We introduce the following names here, but we defer a full discussion
to \S8.1.
\begin{defn}\mylabel{names} An ex-space is said to be \emph{well-sectioned}\index{well-sectioned!ex-space}\index{ex-space!well-sectioned}
if it is $\bar{f}$-cofibrant. An ex-space is said to be {\em ex-fibrant}
or, synonomously, to be an \emph{ex-fibration}\index{ex-fibration} if it is both $\bar{f}$-cofibrant and $h$-fibrant. Thus an ex-fibration is a well-sectioned ex-space whose projection is an $h$-fibration.
\end{defn}
The term ex-fibrant is more logical than ex-fibration, since we are defining
a type of object rather than a type of morphism of $\scr{K}_B$, but the term
ex-fibration goes better with Serre and Hurewicz fibration and is standard
in the literature. We have the following implication of Propositions \ref{compare} and \ref{reverse}. It helps explain the usefulness of ex-fibrations.
\begin{cor}\mylabel{fpequiv} Let $g$ be an ex-map between ex-fibrations
over $B$.
\begin{enumerate}[(i)]
\item $g$ is an $h$-equivalence if and only if $g$ is an $f$-equivalence,
and this hold if and only if $g$ is an $fp$-equivalence.
\item $g$ is an $f$-cofibration if and only if $g$ is an $fp$-cofibration,
and then $g$ is an $h$-cofibration.
\item $g$ is an $f$-fibration if and only if $g$ is an $fp$-fibration,
and this holds if $g$ is an $h$-fibration.
\end{enumerate}
\end{cor}
\begin{rem}\mylabel{guess} The model theoretic significance of ex-fibrations
over $B$ is unclear. They are fibrant and cofibrant objects in the mixed
model structure on ex-spaces over $B$ whose weak equivalences are the
$h$-equivalences
and whose cofibrations are the $\bar{f}$-cofibrations. However, the
converse fails since there are well-sectioned $f$-fibrant ex-spaces that
are $f$-equivalent to $h$-fibrant ex-spaces, hence are mixed fibrant, but
are not themselves $h$-fibrant.
\end{rem}
The previous remark anticipated the following result on over and under
model structures in the categories of spaces and ex-spaces over $B$.
Note that \myref{coflemma} applies to $\scr{K}/B$ and $\scr{K}_B$ as well
as to $\scr{K}$ to show that both $f$-cofibrations and $fp$-cofibrations are
inclusions which are closed when the total spaces are in $\scr{U}$.
\begin{thm}\mylabel{ffpmodel}\index{model structure!fstructure@$f$-structure
on $\scr{K}/B$, $\scr{U}/B$ and $\scr{U}_B$} The following statements hold.
\begin{enumerate}[(i)]
\item The $f$-equivalences, $f$-fibrations, and $\bar{f}$-cofibrations
give $\scr{K}/B$ a proper topological model structure. Moreover, a map in
$\scr{K}/B$ is an $\bar{f}$-cofibration if and only if it is a closed
$f$-cofibration.
\item The $f$-equivalences, $f$-fibrations, and $\bar{f}$-cofibrations
give $\scr{U}/B$ a proper topological model structure. Moreover, a map in
$\scr{U}/B$ is an $\bar{f}$-cofibration if and only if it is an $f$-cofibration.
\item The $fp$-equivalences, $fp$-fibrations, and
$\overline{fp}$-cofibrations
give $\scr{U}_B$ an $fp$-model structure.
\item The $r$-classes of maps give the category $\scr{U}_{\scr{U}}$ of retracts
a proper topological $r$-model structure.
\end{enumerate}
\end{thm}
\begin{proof}
Apart from the factorization axioms, the model structures follow from the
discussion in \ref{sec:towardh}. In particular, the lifting axioms, the
properness, and the topological property of all of these model structures
are given by \myref{h-structure}. In (ii), (iii), and (iv), the
factorization axioms follow from \myref{Cole} since the argument in
\myref{Umod} verifies \myref{hyp}. The rest of (i) can be proven by direct mimicry of the proof of \myref{hmodelis}, using \myref{fNDR}, and the
characterization of the $\bar{f}$-cofibrations in (ii) follows.
\end{proof}
\begin{rem}\mylabel{fpmodel?}
We do not know whether or not $\scr{K}_B$ is an $fp$-model category
or whether the $\overline{fp}$-cofibrations in $\scr{K}_B$ are
characterized as the closed $fp$-cofibrations. We also
do not know whether or not $\scr{K}_{\scr{U}}$ is an $r$-model category.
The problem here is related to the fact that, while the sections
of ex-spaces are always inclusions, they need not be closed inclusions
unless the total spaces are in $\scr{U}$. Steps 1 and 3 of the proof
of \myref{hmodelis} fail in $\scr{K}_B$, and we also do not see how
to carry over Str{\o}m's original proofs in \cite{Strom2, Strom}.
Theorem \ref{h-structure} still applies, giving much
of the information carried by a model structure.
Observe too that if $i\colon A \longrightarrow X$ is a map of well-sectioned
ex-spaces over $B$, then $i$ is an $fp$-cofibration if and only if it
is an $f$-cofibration, by \myref{reverse}(iii). For ex-spaces that
are not well-sectioned, we have little understanding of $fp$-cofibrations,
even when $B$ is a point. We have little understanding of
$\overline{fp}$-cofibrations that are not $\overline{f}$-cofibrations
in any case.
\end{rem}
There is a certain tension between the $fp$ and $h$-notions, with the $f$-notions serving as a bridge between the two. Fiberwise pointed homotopy is the intrinsically right notion of homotopy in $\scr{K}_B$, hence the $fp$-structure is the philosopically right classical $h$-type model structure on $\scr{K}_B$, or at least on $\scr{U}_B$. It is the one that is naturally related to fiber and cofiber sequences, the theory of which works formally in any based topologically bicomplete category in exactly the same way as for based spaces, as we will
recall in \S5.6. A detailed exposition in the case of ex-spaces is given in \cite{CJ, James, James2}.
However, with $h$ replaced by $fp$, we do not have the implications that
we emphasized in the general philosophy of \S\ref{Sphil}. In particular,
with the over and under $q$-model structure,
$q$-cofibrations need not be $fp$-cofibrations and $fp$-fibrations need
not be $q$-fibrations, let alone $h$-fibrations. The $q$-model structure
is still related to the $h$-model structure as in \S4.1, but this does not serve to relate the $q$-model structure to parametrized fiber and cofiber sequences in the way that we are familiar with in the nonparametrized context.
This already suggests that the $q$-model structure might not be appropriate
in parametrized homotopy theory. In the following four sections, we explore conceptually what is required of a $q$-type model structure to connect it up with the intrinsic homotopy theory in a topologically bicomplete category.
\section{Well-grounded topologically bicomplete categories}\label{sec:backstr}
Let $\scr{C}$ be a topologically bicomplete category in either the based or the unbased sense; we use the notations of the based context. In our work here, and in other topological contexts, $\scr{C}$ is \emph{topologically concrete} in the sense that there is a faithful and continuous forgetful functor from $\scr{C}$ to spaces. In practice, appropriate ``ground cofibrations'' can then be specified in terms of underlying spaces. These cofibrations should be thought
of as helpful background structure in our category $\scr{C}$.
To avoid ambiguity, we use the term ``Hurewicz cofibration'', abbreviated
notationally to \emph{$cyl$-cofibration},\index{cofibration!cyl- --@$cyl$- --} for the maps that satisfy the HEP with respect to the cylinders in $\scr{C}$. We
also have the notion of a strong Hurewicz cofibration, which we abbreviate
notationally to \emph{$\overline{cyl}$-cofibration}.\index{cofibration!cyl- --@$\overline{cyl}$- --}
For example, the $cyl$-cofibrations in $\scr{K}$, $\scr{K}/B$, and $\scr{K}_B$ are the $h$-cofibrations, the $f$-cofibrations, and the $fp$-cofibrations, respectively, and similarly for $\overline{cyl}$-cofibrations. As we have seen, it often happens that $cyl$-cofibrations between suitably nice objects of $\scr{C}$, which we shall call ``well-grounded'', are also ground cofibrations. We introduce language to describe this situation. The following definitions codify the behavior of the well-grounded objects with respect to the $cyl$-cofibrations, colimits, and tensors in $\scr{C}$. It is convenient to build in the appropriate equivariant generalizations of our notions, although we defer a formal discussion of $G$-topologically bicomplete $G$-categories to \S10.2; see \myref{defn:enrichBG}. The examples in \S1.4 give the idea.
\begin{defn}\mylabel{spaceback} An unbased space is {\em well-grounded}
if it is compactly generated. A based space is {\em well-grounded}
if it is compactly generated and well-based. The same definitions
apply to $G$-spaces for a topological group $G$.
\end{defn}
Let $\scr{C}$ be a topologically bicomplete category.
\begin{defn}\mylabel{back}
A full subcategory of $\scr{C}$ is said to be a subcategory of \emph{well-grounded objects}\index{well-grounded!object}\index{category!sub-- of
well-grounded objects} if the following properties hold.
\begin{enumerate}[(i)]
\item The initial object of $\scr{C}$ is well-grounded.
\item All coproducts of well-grounded objects are well-grounded.
\item If $i\colon X \longrightarrow Y$ is a $cyl$-cofibration and $f\colon X\longrightarrow Z$ is any map, where $X$, $Y$, and $Z$ are well-grounded, then the pushout
$Y\cup_X Z$ is well-grounded.
\item The colimit of a sequence of $cyl$-cofibrations between well-grounded
objects is well-grounded.
\item A retract of a well-grounded object is well-grounded.
\item If $X$ is a well-grounded object and $K$ is a well-grounded space, then
$X \sma K$ ($X\times K$ in the unbased context) is well-grounded.
\end{enumerate}
When $\scr{C}$ is $G$-topologically bicomplete, we replace spaces by
$G$-spaces in (vi).
\end{defn}
\begin{defn}\mylabel{moreback}
A \emph{ground structure}\index{ground structure} on $\scr{C}$ is a (full) subcategory of well-grounded objects together with a subcategory of cofibrations, called the \emph{ground cofibrations}\index{ground cofibrations}\index{cofibration!ground --s} and denoted \emph{$g$-cofibrations}\index{gcofibration!$g$-cofibration}\index{cofibration!g@$g$-}, such that every $cyl$-cofibration between well-grounded objects is a $g$-cofibration. A map that is both a $g$-cofibration and a $cyl$-cofibration
is called a \emph{bicofibration}.\index{bicofibration!bicofibration}
\end{defn}
Thus a $cyl$-cofibration between well-grounded objects is a bicofibration.
The need for focusing on bicofibrations and the force of the definition
come from the following fact.
\begin{warn} In practice, (iii) often fails if $i$ is a $g$-cofibration
between well-grounded objects that is {\em not}\, a $cyl$-cofibration,
as we shall illustrate in \S6.1. In particular, in $G\scr{K}_B$ with the
canonical ground structure described below, it can already fail for an
inclusion $i$ of $I$-cell complexes, where $I$ is the standard set of
generators for the $q$-cofibrations.
\end{warn}
In the next chapter, we will construct a $q$-type model structure for
$G\scr{K}_B$ with a set of generating cofibrations to which the following
implication of Definitions \ref{cofsub} and \ref{back} applies.
\begin{lem}\mylabel{cofgcof}
Let $I$ be a set of $cyl$-cofibrations between well-grounded objects and let $f\colon X\longrightarrow Y$ be a retract of a relative $I$-cell complex $W\longrightarrow Z$. Then $f$ is a bicofibration. If $W$ is well-grounded, then so are $X$, $Y$,
and $Z$.
\end{lem}
Our categories of equivariant parametrized spaces have canonical ground structures. Recall that the classes of $f$ and $\bar{f}$-cofibrations in
$G\scr{U}/B$ and $G\scr{U}_B$ coincide.
\begin{defn}\mylabel{exbackdef}
A space over $B$ is \emph{well-grounded}\index{space!well-grounded} if
its total space is compactly generated. An ex-space over $B$ is \emph{well-grounded}\index{ex-space!well-grounded} if
it is well-sectioned and its total space is compactly generated. In both
$G\scr{K}/B$ and $G\scr{K}_B$, define the $g$-cofibrations to be the $h$-cofibrations.
\end{defn}
Note that the only distinction between well-sectioned and well-grounded
ex-spaces is the condition on total spaces. The distinction is relevant
when we consider relative $I$-cell complexes $X_0\longrightarrow X$
in $G\scr{K}_B$. If $X_0$ is well-sectioned, then so is $X$, whereas $X/X_0$
is an $I$-cell complex and is therefore well-grounded for any $X_0$.
\begin{prop}\mylabel{exback} These definitions specify
ground structures on $G\scr{K}/B$ and on $G\scr{K}_B$.
\end{prop}
\begin{proof}
For $G\scr{K}/B$, the Hurewicz cofibrations are the $f$-cofibrations, and these
are $h$-cofibrations. It is standard that $G\scr{U}/B$ has the closure properties
specified in \myref{back}. For $G\scr{K}_B$, the Hurewicz cofibrations are the
$fp$-cofibrations. Between well-sectioned ex-spaces, these are $f$-cofibrations and therefore $h$-cofibrations by \myref{reverse2}(i).
Parts (i)--(v) of \myref{back} are clear since well-sectioned means
$\bar{f}$-cofibrant, which is a lifting property. Finally we consider
part (vi). Recall that $X\sma_B K$ can be constructed as the pushout of
\[\xymatrix{{*}_B & X\amalg (B\times K) \ar[l]\ar[r] & X\times K}\]
in the category of spaces over $B$. By the equivariant version of
the NDR-pair characterization of $f$-cofibrations in \myref{fNDR},
these spaces are $f$-cofibrant and the map on the right is an
$f$-cofibration. This implies that $X\sma_B K$ is $f$-cofibrant.
\end{proof}
\section{Well-grounded categories of weak equivalences}
The following definition describes how the weak equivalences and the ground structure are related in practice.
\begin{defn}\mylabel{hproper}
Let $\scr{C}$ be a topologically bicomplete category with a given ground structure.
A subcategory of weak equivalences in $\scr{C}$ is \emph{well-grounded}\index{well-grounded!weak equivalence}\index{weak-equivalence!well-grounded}
if the following properties hold (where acyclicity refers to the weak equivalences).
\begin{enumerate}[(i)]
\item A homotopy equivalence is a weak equivalence.
\item A coproduct of weak equivalences between
well-grounded objects is a weak equivalence.
\item (Gluing lemma)\mylabel{glue}\index{Gluing lemma}
Assume that the maps $i$ and $i'$ are bicofibrations and the vertical
arrows are weak equivalences in the following diagram.
\[\xymatrix{Y \ar[d] & X \ar[d] \ar[l]_-{i} \ar[r]^f & Z \ar[d] \\
Y' & X' \ar[l]^-{i'} \ar[r]_{f'} & Z'}\]
Then the induced map of pushouts is a weak equivalence. In particular,
pushouts of weak equivalences along bicofibrations are weak equivalences.
\item (Colimit lemma) Let $X$ and $Y$ be the colimits of sequences of bicofibrations $i_n\colon X_n\longrightarrow X_{n+1}$ and $j_n\colon Y_n\longrightarrow Y_{n+1}$ such that both $X/X_0$ and $Y/Y_0$ are well-grounded. If $f\colon X\longrightarrow Y$ is the colimit of a sequence of compatible weak equivalences $f_n\colon X_n\longrightarrow Y_n$, then $f$ is a weak equivalence. In particular, if each $i_n$ is a weak equivalence, then the induced map $i\colon X_0\longrightarrow X$ is a weak equivalence.
\item For a map $i\colon X\longrightarrow Y$ of well-grounded objects in $\scr{C}$ and a
map $j\colon K\longrightarrow L$ of well-grounded spaces, $i\Box j$ is a weak
equivalence if $i$ is a weak equivalence or $j$ is a $q$-equivalence.
\end{enumerate}
\end{defn}
Here, in the based context, $i\Box j$ is the evident induced map
$$ (X\sma L) \cup_{X\sma K} (Y\sma K) \longrightarrow Y\sma L.$$
The gluing lemma implies that acyclic bicofibrations are preserved
under push\-outs, as of course holds for pushouts of acyclic cofibrations in
model categories. The special case mentioned in (iii) corresponds to the
left proper axiom in model categories. As there, it can be used to prove
the general case of the gluing lemma provided that we have suitable
factorizations.
\begin{lem}\mylabel{gluederiv} Assume the following hypotheses.
\begin{enumerate}[(i)]
\item Weak equivalences are preserved under pushouts along bicofibrations.
\item Every map factors as the composite of a bicofibration and a weak equivalence.
\end{enumerate}
Then the gluing lemma holds.
\end{lem}
\begin{proof} We use the notations of \myref{hproper}(iii) and proceed in
three cases.
If $f$ and $f'$ are both weak equivalences, then, by (i), so are the horizontal arrows in the commutative diagram
$$ \xymatrix{
Y \ar[d] \ar[r] & Y\cup_X Z \ar[d] \\
Y' \ar[r] & Y'\cup_{X'} Z'.\\}$$
Since $Y\longrightarrow Y'$ is a weak equivalence, the right arrow is a weak equivalence by the two out of three property of weak equivalences.
If $f$ and $f'$ are both bicofibrations, consider the commutative diagram
\[\xymatrix@=.6cm{
&& X \ar[rr]^-{i} \ar[dll]_{f} \ar@{-}[d] & & Y \ar[dd] \ar[ddr] \ar[dl] \\
Z \ar[rrr] \ar[dd] && \ar[d] & Y\cup_XZ \ar[dd] \ar[ddr] \\
&& X' \ar@{-}[r] \ar[dll]_{f'} & \ar[r]|(.5)\hole & Y\cup_X X' \ar[dl]|(.35)\hole \ar[r] & Y' \ar[dl] \\
Z' \ar[rrr] && & Y\cup_X Z' \ar[r] & Y'\cup_{X'} Z'.}\]
The back, front, top, and two bottom squares are pushouts, and the
middle composite $X'\longrightarrow Y'$ is $i'$. Since $f$ and $f'$ are
bicofibrations, so are the remaining three arrows from the back to the front. Similarly, $i$ and its pushouts are bicofibrations.
Since $X\longrightarrow X'$, $Y\longrightarrow Y'$, and $Z\longrightarrow Z'$ are weak equivalences,
(i) and the two out of three property imply that
$Y\longrightarrow Y\cup_X X'$, $Y\cup_X X'\longrightarrow Y'$,
$Y\cup_X Z\longrightarrow Y\cup_X Z'$, and $Y\cup_X Z'\longrightarrow Y'\cup_{X'}Z'$ are weak equivalences. Composing the last two, $Y\cup_X Z\longrightarrow Y'\cup_{X'}Z'$ is a
weak equivalence.
To prove the general case, construct the following commutative diagram.
\[\xymatrix@=.6cm{
Y \ar[dd] & & X \ar[dd] \ar[ll]_-{i} \ar[rr]^(.35){f} \ar[dr] && Z \ar[dd] \\
& & & W \ar[ur]_{\bar{f}} \ar[dd] & \\
Y' & & \ar[ll]_-{i'} X' \ar@{-}[r]^(.7){f'} \ar[dr] & \ar[r] & Z' \\
& & & X'\cup_X W \ar[ur]_{\bar{f}'} }\]
Here we first factor $f$ as the composite of a bicofibration and a weak equivalence $\bar{f}$ and then define a map $\bar{f}'$ by the universal property of pushouts. By hypothesis (i), $W\longrightarrow X'\cup_X W$ is a weak equivalence, and by the two out of three property, so is $\bar{f}'$. By the second case,
$$Y\cup_X W \longrightarrow Y'\cup_{X'} (X'\cup_X W) \iso Y'\cup_X W $$
is a weak equivalence and by the first case, so is
\[Y\cup_X Z\iso (Y\cup_X W)\cup_W Z \longrightarrow (Y'\cup_X W)\cup_{(X'\cup_X W)} Z'
\iso Y'\cup_{X'} Z'. \qedhere\]
\end{proof}
\begin{rem} Clearly the previous result applies to any categories of
weak equivalences and cofibrations that satisfy (i) and (ii). The
essential point is that, in practice, we often need bicofibrations
in order to verify (i).
\end{rem}
Similarly, but more simply, the following observation reduces the verification of \myref{hproper}(v) to special cases. Here we assume that $\scr{C}$ is based.
\begin{lem}\mylabel{boxacy} Let $i\colon X\longrightarrow Y$ be a map in $\scr{C}$ and
$j\colon K\longrightarrow L$ be a map of based spaces. Display $i\Box j$ in the diagram
$$\xymatrix@=.6cm{
X\sma K \ar[rr]^{\text{id}\sma j} \ar[dd]_{i\sma\text{id}} & & X\sma L \ar[dd]^{i\sma\text{id}} \ar[dl]_{k}\\
& (X\sma L)\cup_{X\sma K}(Y\sma K)\ar[dr]_-{i\Box j} & \\
Y\sma K \ar[rr]_{\text{id}\sma j} \ar[ur]
& & Y\sma L.}$$
If the maps $i\sma\text{id}$ and the pushout $k$ of $i\sma\text{id}$ along $\text{id}\sma j$ are weak equivalences, then so is $i\Box j$, and
similarly with the roles of $i$ and $j$ reversed.
\end{lem}
Together with \myref{cofgcof}, the notion of a well-grounded category of weak equivalences encodes a variant of \myref{cofcat} that often applies when the latter does not.
\begin{lem}\mylabel{veriiii}
If $J$ is a set of acyclic $cyl$-cofibrations between well-grounded objects, then all relative $J$-cell complexes are weak equivalences.
\end{lem}
\begin{proof} This follows from (ii), (iii), and (iv) of \myref{hproper}, together with the observation that if $X_0\longrightarrow X$ is a relative $J$-cell complex, then $X/X_0$ is a $J$-cell complex and is therefore well-grounded,
so that (iv) applies.
\end{proof}
There is an analogous reduction of the problem of determining when a
functor preserves weak equivalences.
\begin{lem}\mylabel{reducts}
Let $F\colon \scr{C}\longrightarrow \scr{D}$ be a functor between topologically bicomplete categories that come equipped with subcategories of well-grounded weak equivalences with respect to given ground structures. Let $J$ be a set of acyclic $cyl$-cofibrations between well-grounded objects in $\scr{C}$. Assume
that $F$ has a continuous right adjoint and that $F$ takes maps in $J$ to
weak equivalences between well-grounded objects. Then $F$ takes a retract
of a relative $J$-cell complex to an acyclic map in $\scr{D}$.
\end{lem}
\begin{proof}
The functor $F$ preserves $cyl$-cofibrations since it has a continuous right adjoint and hence $FJ$ consists of acyclic $cyl$-cofibrations between well-grounded objects. The conclusion follows from \myref{veriiii} and the fact
that left adjoints commute with colimits and therefore the construction of cell complexes.
\end{proof}
Similarly, cell complexes are relevant to the verification of \myref{hproper}(v). Recall that the $cyl$-cofibrations in $\scr{K}_*$
are the $fp$-cofibrations, that is, the based cofibrations.
\begin{lem}\mylabel{topmod}
Let $I$ be a set of $cyl$-cofibrations between well-grounded objects of $\scr{C}$ and let $J$ be a set of $fp$-cofibrations between well-based spaces. If $i$ is a retract of a relative $I$-cell complex, $j$ is a retract of a relative $J$-cell complex, and either $I$ or $J$ consists of weak equivalences, then $i\Box j$ is a weak equivalence.
\end{lem}
\begin{proof}
Assume that $I$ consists of weak equivalences; the proof of the other case is symmetric. Since the functor $-\sma K$ commutes with coproducts, pushouts, sequential colimits, and retracts, we can construct $j\sma K$ by first applying $-\sma K$ to the generators, then construct the cell complex, and finally pass to retracts. Since $-\sma K$ preserves $cyl$-cofibrations and well-grounded objects by \myref{back}(vi), it takes maps in $I$ to $cyl$-cofibrations between well-grounded objects. By \myref{veriiii}, the resulting cell complex is acyclic and therefore so also is any retract of it. Thus $j\sma K$ is an acyclic bicofibration. Since such maps are preserved under pushouts, Lemma \ref{boxacy} applies to give the conclusion.
\end{proof}
The following classical example is implicit in the literature.
\begin{prop}
The $q$-equivalences in $G\scr{K}$ are well-grounded with respect to the
ground structure whose well-grounded objects are the compactly
generated spaces and whose $g$-cofibrations are the $h$-cofibrations.
\end{prop}
\begin{proof} Parts (i), (ii), and, here in the unbased case, (v)
of \myref{hproper} are clear, and (iv) follows easily from
\myref{little}. The essential point is the gluing lemma of (iii). By
passage to fixed point spaces, it suffices to prove this nonequivariantly.
Using the gluing lemma for the proper $h$-model structure on $\scr{K}$, we
see that $f$ and $f'$ can be replaced by their mapping cylinders. Then
the induced map of pushouts is the map of double mapping cylinders
induced by the original diagram. This map is equivalent to a map of excisive
triads, and in that case the result is \cite[1.3]{weak}, whose proof is
corrected in \cite{Wit}.
\end{proof}
\begin{prop}\mylabel{exwellgr}
The $q$-equivalences in $G\scr{K}/B$ and $G\scr{K}_B$ are well-ground\-ed with respect to the ground structures of \myref{exback}. In these cases, one need only assume that the relevant maps in the gluing and colimit lemmas are ground cofibrations (= $h$-cofibrations), not both ground and Hurewicz cofibrations.
\end{prop}
\begin{proof}
We verify this for $G\scr{K}_B$. Part (i) of \myref{hproper} holds since any $fp$-equivalence is a $q$-equivalence and part (iii) follows directly from the gluing lemma in $G\scr{K}$. For part (ii), the total space of $\vee_B X_i$ is the pushout in $G\scr{K}$ of
\[\xymatrix{{*}_B & \amalg {*}_B \ar[l]\ar[r] & \amalg X_i.}\]
Since the $X_i$ are well-grounded, the map on the right is an $h$-cofibration,
hence (ii) also follows from the gluing lemma in $G\scr{K}$. In part (iv), the relevant quotient in $G\scr{K}_B$ is given by the pushout, $X/\!_BX_0$, of the diagram $*_B \longleftarrow X_0 \longrightarrow X$. Since $X/\!_BX_0$ is well-grounded, the quotient total space is in $\scr{U}$ and one can apply \myref{little} just as on the space level. Finally consider (v). As in the proof of \myref{exback}(vi), $X\sma_B K$ can be constructed as the pushout of the following diagram of $f$-cofibrant spaces over $B$.
\[\xymatrix{{*}_B & X\amalg (B\times K) \ar[l]\ar[r] & X\times K}\]
The map on the right is an $f$-cofibration. By the gluing lemma in $G\scr{K}$,
it suffices to observe that $X\times K$ preserves $q$-equivalences in both variables since homotopy groups commute with products.
\end{proof}
\section{Well-grounded compactly generated model structures}\label{sec:wellgr}
Let $\scr{C}$ be a topologically bicomplete category or, equivariantly, a
$G$-topologically bicomplete $G$-category. In the notion of a ``well-grounded
model structure'', we formulate the properties that a compactly generated model structure on $\scr{C}$ should have in order to mesh well with the intrinsic $h$-structure on $\scr{C}$ described in \S\ref{sec:towardh}. When $\scr{C}$ has such a model structure, and when the classical $h$-structure actually is a model structure, the identity functor on $\scr{C}$ is a Quillen left adjoint from the well-grounded model structure to the $h$-model structure. Thus this notion gives a precise axiomatization for the implementaton of the philosophy that
we advertised in \S4.1. We begin with a variant of \myref{compgen}.
\begin{thm}\mylabel{Newcompgen}
Let $\scr{C}$ be a topologically bicomplete category with a ground structure, a subcategory of well-grounded weak equivalences, and compact sets $I$ and $J$ of maps
that satisfy the following conditions.
\begin{enumerate}[(i)]
\item (Acyclicity condition) Every map in $J$ is a weak equivalence.
\item (Compatibility condition) A map has the RLP with respect to $I$ if
and only if it is a weak equivalence and has the RLP
with respect to $J$.
\item Every map in $I$ and $J$ is a $\overline{cyl}$-co\-fib\-ration
between well-grounded objects.
\end{enumerate}
Then $\scr{C}$ is a compactly generated model category with generating sets
$I$ and $J$ of cofibrations and acyclic cofibrations. Every cofibration
is a bicofibration and every cofibrant object is well-grounded. A
pushout of a weak equivalence along a bicofibration is a weak
equivalence and, in particular, the model structure is left proper.
The model structure is topological or, equivariantly, $G$-topological
if the following condition holds.
\begin{enumerate}
\item[(iv)] $i\Box j$ is an $I$-cell complex if $i\colon X\longrightarrow Y$ is a map
in $I$ and $j\colon K\longrightarrow L$ is a map of spaces (or $G$-spaces) in $I$. \end{enumerate}
\end{thm}
\begin{proof} By \myref{veriiii}, \myref{compgen} applies to verify the model
axioms. Condition (iii) implies the statements about cofibrations and
cofibrant objects by \myref{cofgcof}, and the gluing lemma implies the statement about pushouts of weak equivalences. In the last statement, the set $I$ of
generating cofibrations in the relevant category of (based or unbased)
spaces is as specified in \myref{UrIJ}. By passage to coproducts, pushouts, sequential colimits, and retracts, (iv) implies that $i\Box j$ is a
cofibration if $i\colon X\longrightarrow Y$ is a cofibration in $\scr{C}$ and
$j\colon K\longrightarrow L$ is a $q$-cofibration of spaces (or $G$-spaces).
Together with \myref{topmod}, this implies that the model structure is topological.
\end{proof}
We emphasize the difference between the acyclicity conditions stated in
\myref{compgen} and in \myref{Newcompgen}. In the applications of the
former, it is the verification of the acyclicity of $J$-cell complexes
that is problemmatic, but in the latter our axiomatization has built
in that verification. Similarly, our axiomatization has built in the verification of the acyclicity condition required for the model
structure to be topological.
\begin{defn}\mylabel{wellmodel}
A compactly generated model structure on $\scr{C}$ is said to be
\emph{well-ground\-ed}\index{well-grounded!model category}\index{model category!well-grounded} if it is right proper and satisfies all of the hypotheses of the preceding theorem. It is therefore proper and topological
or, equivariantly, $G$-topological.
\end{defn}
\section{Properties of well-grounded model categories}
Assume that $\scr{C}$ is a well-grounded model category. To derive properties
of its homotopy category $\text{Ho}\scr{C}$, we must sort out the relationship between homotopies defined in terms of cylinders and homotopies in the model theoretic sense, which we call ``model homotopies''. Of course, the cylinder objects $\text{Cyl}(X)$ in $\scr{C}$ have maps $i_0$, $i_1\colon X\longrightarrow \text{Cyl}(X)$ and $p\colon \text{Cyl}(X) \longrightarrow X$, and $i_0$ (or $i_1$) and $p$ are inverse homotopy equivalences since tensors with spaces preserve homotopies in the space variable. \myref{hproper}(i) ensures that $p$ is therefore a weak equivalence. This means that $\text{Cyl}(X)$ is a model theoretic cylinder object in $\scr{C}$, provided that we adopt the non-standard definition of \cite{DS}. With the language there, it need not be a {\em good} cylinder object since $i_0\amalg i_1\colon X\amalg X\longrightarrow \text{Cyl}\,(X)$ need not be a cofibration. As pointed out in \cite[p. 90]{DS}, this already fails for spaces, where the inclusion $X\amalg X\longrightarrow X\times I$ is not a $q$-cofibration unless $X$ is $q$-cofibrant. With the standard definition given
in \cite{Hirschhorn, Hovey, Q}, cylinder objects are required to have this cofibration property. Under that interpretation, the cylinder objects $\text{Cyl}(X)$ would not qualify as model theoretic cylinder objects
in general. (We note parenthetically that ``good cylinders''
are defined in \cite{SVogt} in such a way as to include all
standard cylinders in the category of spaces). We record the following observations.
\begin{lem}\mylabel{comphty}
Consider maps $f,g\colon X\longrightarrow Y$ in $\scr{C}$.
\begin{enumerate}[(i)]
\item If $f$ is homotopic to $g$, then $f$ is left model homotopic to $g$.
\item If $X$ is cofibrant, then $\text{Cyl}(X)$ is a good cylinder object.
\item If $X$ is cofibrant and $Y$ is fibrant, then $f$ is homotopic to $g$ if and only if $f$ is left and right model homotopic to $g$.
\end{enumerate}
\end{lem}
\begin{proof}
Part (i) is \cite[4.6]{DS}, part (ii) follows from \myref{back}(iii), and part (iii) follows from \cite[4.23]{DS}.
\end{proof}
Let $[X,Y]$\@bsphack\begingroup \@sanitize\@noteindex{XY@$[X,Y]$} denote the set of morphisms $X\longrightarrow Y$ in $\text{Ho} \scr{C}$ and let $\pi(X,Y)$\@bsphack\begingroup \@sanitize\@noteindex{piXY@$\pi(X,Y)$} denote the set of homotopy classes of maps $X\longrightarrow Y$. We shall only use the latter notation when homotopy and model homotopy coincide.
\begin{lem}[Cofiber sequence lemma]\mylabel{modelcofiber}\index{cofiber sequence}
Assume that $\scr{C}$ is based. Consider the cofiber sequence
\[X\longrightarrow Y \longrightarrow Cf \longrightarrow \Sigma X \longrightarrow \Sigma Y \longrightarrow \Sigma Cf \longrightarrow \Sigma^2X \longrightarrow \cdots\]
of a well-grounded map $f\colon X\longrightarrow Y$. For any object $Z$, the
induced sequence
\[\cdots \longrightarrow [\Sigma^{n+1} X, Z] \longrightarrow [\Sigma^n Cf, Z] \longrightarrow [\Sigma^n Y, Z] \longrightarrow [\Sigma^n X, Z] \longrightarrow \cdots \longrightarrow [X, Z]\]
of pointed sets (groups left of $[\Sigma X,Z]$, Abelian groups left of $[\Sigma^2 X, Z]$) is exact.
\end{lem}
\begin{proof} As usual, giving $I$ the basepoint $1$, we define
$$CX = X\sma I, \qquad \Sigma X = X\sma S^1, \qquad \text{and} \qquad Cf = Y\cup_f CX.$$
If $X$ is cofibrant, then $X$ is well-grounded and $X\longrightarrow CX$ is a cofibration and therefore a bicofibration. If $X$ and $Y$ are cofibrant, then so is $Cf$, as one sees by solving the relevant lifting problem by first using that $Y$ is cofibrant, then using that $X\longrightarrow CX$ is a cofibration, and finally using that $Cf$ is a pushout. Thus, taking $Z$ to be fibrant, the conclusion follows in this case from the sequence of homotopy classes of maps
\[\cdots \longrightarrow \pi(\Sigma X, Z)\longrightarrow \pi(Cf, Z) \longrightarrow \pi(Y, Z) \longrightarrow \pi(X, Z),\]
which is proven to be exact in the same way as on the space level. If $X$ and $Y$ are not cofibrant, let $Qf\colon QX\longrightarrow QY$ be a cofibrant approximation to $f$. The gluing lemma applies to give that the canonical map $CQf\longrightarrow Cf$ is a weak equivalence. Therefore the conclusion follows in general from the special case of cofibrant objects.
\end{proof}
\begin{warn}
While the proof just given is very simple, it hides substantial subtleties.
It is crucial that cofibrant objects $X$ be well-grounded, so that the $cyl$-cofibration $X\longrightarrow CX$ is a bicofibration and the gluing lemma applies.
\end{warn}
Of course, the group structures are defined just as classically.
The pinch maps
\[S^1\cong I/\{0,1\} \longrightarrow I/\{0,\tfrac12,1\}\cong S^1\vee S^1
\qquad\text{and}\qquad
I \longrightarrow I/\{\tfrac12,1\}\cong I\vee S^1\]
induce pinch maps
\[\Sigma X \longrightarrow \Sigma X \vee \Sigma X \qquad\text{and}\qquad Cf\longrightarrow Cf\vee \Sigma X\]
that give $\Sigma X$ the structure of a cogroup object in $\text{Ho}\scr{C}$ and $Cf$ a coaction by $\Sigma X$; $\Sigma^2 X$ is an abelian cogroup object for the same reason that higher homotopy groups are abelian. Therefore $[\Sigma X,Z]$ is a group, $[Cf,Z]$ is a $[\Sigma X,Z]$-set, and $[\Sigma X,Z]\longrightarrow [Cf,Z]$ is a $[\Sigma X,Z]$-map.
\begin{lem}[Wedge lemma]\mylabel{modelwedges}\index{wedge lemma}
For any $X_i$ and $Y$ in $\scr{C}$, $[\amalg X_i, Y]\iso \Pi [X_i,Y]$.
\end{lem}
\begin{proof} This is standard, using that a coproduct of cofibrant
approximations is a cofibrant approximation.
\end{proof}
\begin{lem}[$\text{Lim}^1$ lemma]\mylabel{modellim1}\index{lim1lemma @$\text{lim}^1$ lemma}
Assume that $\scr{C}$ is based.
Let $X$ be the colimit of a sequence of well-ground\-ed $cyl$-cofibrations $i_n\colon X_n \longrightarrow X_{n+1}$. Then, for any object $Y$, there is a ${\text{lim}}^1$ exact sequence of pointed sets
$$* \longrightarrow {\text{lim}}^1\,[\Sigma X_n,Y] \longrightarrow [X,Y] \longrightarrow \text{lim}\,[X_n,Y]\longrightarrow *.$$
\end{lem}
\begin{proof}
The telescope $\text{Tel}\,X_n$ is defined to be $\text{colim}\,T_n$,
where the $T_n$ and a ladder of weak equivalences
$j_n\colon X_n\longrightarrow T_n$ and $r_n\colon T_n\longrightarrow X_n$
are constructed inductively by setting $T_0=X_0$ and letting
$j_{n+1}$ and $r_{n+1}$ be the maps of pushouts induced by
the following diagram.
\[\xymatrix{
X_n\ar[d]_{i_1} \ar@{=}[r] & X_n \ar[d]^{\nu_2} \ar[r]^-{i_n} & X_{n+1}\ar[d]^{\nu_2}\\
\text{Cyl}\,X_n\ar[d]_{p} & X_n\amalg X_n \ar[l]_{i_{(0,1)}}\ar[r]^{j_n\amalg i_n}\ar@{=}[d] & T_n \amalg X_{n+1}\ar[d]^{r_n\amalg\text{id}}\\
X_n & X_n\amalg X_n \ar[l]^-{\nabla}\ar[r]_-{\text{id}\amalg i_n} & X_n \amalg X_{n+1}}\]
Since $j_{n+1}$ is a pushout of the bicofibration
$i_1\colon X_n\longrightarrow \text{Cyl}(X_n)$, the gluing lemma and colimit lemma
specified in \myref{hproper}(iii) and (iv) apply to show that the induced
maps $\text{Tel}\, X_n \longrightarrow \text{colim}\, X_n = X$ are weak equivalences.
As in the cofiber sequence lemma, we can use cofibrant approximation to
reduce to a question about $\pi(-,-)$. Then the telescope admits an alternative description from which the ${\text{lim}}^1$ exact sequence is immediate. It would take us too far afield to go into full details of what should be a standard argument, but we give a sketch since we cannot find our preferred
argument in the literature.
Recall that the classical homotopy pushout, or double mapping cylinder, of
\[\xymatrix{Y & X \ar[l]_-f\ar[r]^-{f'} & Y'}\]
is the ordinary pushout $M(f,f')$ of
\[\xymatrix{\text{Cyl}\, X & X\amalg X \ar[l]_-{i_{0,1}} \ar[r]^{f\amalg f'} & Y\amalg Y'.}\]
It fits into a cofiber sequence
\[Y\amalg Y'\longrightarrow M(f,g)\longrightarrow \Sigma X.\]
There results a surjection from $\pi(M(f,g),Z)$ to the evident pullback,
the kernel of which is the set of orbits of the right action
of $\pi(\Sigma Y,Z)\times \pi(\Sigma Y', Z)$ on $\pi(\Sigma X, Z)$ given by
$x(y,y') = (\Sigma f)^*(y)^{-1}x(\Sigma f')^*(y')$.
The classical homotopy coequalizer $C(f,g)$ of parallel maps $f,g\colon X\longrightarrow Y$ is the homotopy pushout of the coproduct $f\amalg g\colon X\amalg X\longrightarrow Y\amalg Y$ and the codiagonal $\nabla\colon X\amalg X \longrightarrow X$. Using a little algebra, we see that $\pi(C(f,g),Z)$ maps surjectively to the equalizer of $f^*$ and $g^*$ with kernel isomorphic to the set of orbits of $\pi(\Sigma X,Z)$ under the right action of $\pi(\Sigma Y,Z)$
specified by $xy = (\Sigma f)^*(y)^{-1}x(\Sigma g)^*(y)$.
In this language, $\text{Tel}\,X_n$ is the classical homotopy coequalizer of the identity and the coproduct of the $i_n$, both being self maps of the coproduct of the $X_n$. By algebraic inspection, the $\lim^{1}$ exact
sequence follows directly. A quicker, less conceptual, argument is
possible, as in \cite[p.\,146]{Concise} for example.
\end{proof}
\begin{rem}
Let $\scr{C}$ be an arbitrary pointed model category with (for simplicity) a functorial cylinder construction $\text{Cyl}$. If $X$ is cofibrant,
let $\Sigma X$ denote the quotient $\text{Cyl}(X)/(X\vee X)$. Quillen \cite{Q} constructed a natural cogroup structure on $\Sigma X$ in $\text{Ho}\scr{C}$. For
a cofibration $X\longrightarrow Y$ between cofibrant objects, he also constructed a natural coaction of $\Sigma X$ on the quotient $Y/X$. One can then define cofiber sequences in $\text{Ho}\scr{C}$ just as in the homotopy category of a topological model category, and one can define fiber sequences dually.
The cofiber sequences and fiber sequences each give $\text{Ho}\scr{C}$ a
suitably weakened form of the notion of a triangulation, called a
``pretriangulation'' \cite{Hovey, Q}, and they are suitably compatible.
If $\text{Ho}\scr{C}$ is closed symmetric monoidal one can take this a step further and formulate what it means for the pretriangulation to be compatible with that structure, as was done in \cite{Tri} for triangulated categories. However, proving the compatibility axioms from this general
point of view would at best be exceedingly laborious, if it could be done
at all.
These purely model theoretic constructions of the suspension and looping functors $\Sigma$ and $\Omega$ are more closely related to the familiar topological constructions than might appear. The homotopy category of any model category is enriched and bitensored over the homotopy category of spaces (obtained from the $q$-model structure) \cite{DK, Hovey}, and the suspension and loop functors are given by the (derived) tensor and cotensor with the unit circle. That is, $\Sigma X \simeq X\sma S^1$ and $\Omega X \simeq F(S^1,X)$.
This general point of view is not one that we wish to emphasize. For topological model categories, the structure described in this section is far easier to define and work with directly, as in classical homotopy theory, and we have
axiomatized what is required of a model structure in order to allow the use of such standard and elementary classical methods. In our topological context, the homotopy category $\text{Ho}\scr{C}$ is automatically enriched over $\text{Ho}\scr{K}_*$ and $(\Sigma, \Omega)$ is a Quillen adjoint pair that descends to an adjoint pair on homotopy categories that agrees with the purely model theoretic adjoint pair just described.
The crucial point for our stable work is that a large part of this structure exists \emph{before} one constructs the desired model structure. It can therefore be used as a tool for carrying out that construction. This is
in fact how stable model categories were constructed in \cite{EKMM, MM, MMSS},
but there the compatibility between $q$-type and $h$-type structures was too
evident to require much comment. The key step in our construction of the stable model structure on parametrized spectra in Chapter 12 is to show that cofiber sequences induce long exact sequences on stable homotopy groups. That will allow us to verify that the stable equivalences are suitably well-grounded,
and from there the model axioms follow as in the earlier work just cited.
\end{rem}
\chapter{The $qf$-model structure on $\scr{K}_B$}
\section*{Introduction}
In this chapter, we introduce and develop our preferred $q$-type model structure, namely the $qf$-model structure. It is a Quillen equivalent
variant of the $q$-model structure that has fewer, and better structured, cofibrations. For clarity of exposition, we work nonequivariantly
in this chapter, which is taken from \cite{Sig}.
We begin by comparing the homotopy theory of spaces and the homotopy theory of ex-spaces over $B$, starting with a comparison of the $q$-model structures that we have on both. In the category $\scr{K}$ of spaces, we have the familiar situation described in \S4.1. The homotopy category $\text{Ho}\scr{K}$ that we care about is defined in terms of $q$-equivalences, the intrinsic notion of homotopy is given by the classical cylinders, and, since all spaces are $q$-fibrant, the category $\text{Ho}\scr{K}$ is equivalent to the classical homotopy category $h\scr{K}_c$ of $q$-cofibrant spaces (or CW complexes). Since the $q$-cofibrations are $h$-cofibrations, the $q$-model structure and the $h$-model structure on $\scr{K}$ mesh smoothly. Indeed, the classical and model theoretic homotopy theory have been used in tandem for so long that this meshing of structures goes without notice. In particular, although cofiber and fiber sequences are defined in terms of the $h$-model structure while the homotopy category is defined in terms of the $q$-model structure, the compatibility seems automatic.
Now consider the category $\scr{K}_B$. The homotopy category $\text{Ho}\scr{K}_B$ that we care about is defined in terms of $q$-equivalences of total spaces, but we need some justification for making that statement. A map of $q$-fibrant ex-spaces is a $q$-equivalence of total spaces if and only if all of its maps on fibers are $q$-equivalences. This reformulation captures the idea that the homotopical information in parametrized homotopy theory should be encoded on the fibers, and it is such fiberwise $q$-equivalences that we really care about. It is only for $q$-fibrant ex-spaces, or ex-spaces whose projections are at least quasifibrations, that the homotopy groups of total spaces give the ``right answer''. There are three notions of homotopy in sight, $h$, $f$, and $fp$. The last of these is the intrinsic one defined in terms of the relevant cylinders in $\scr{K}_B$, and $\text{Ho}\scr{K}_B$ is equivalent to the classical homotopy category $h{\scr{K}_{B}}_{cf}$ of $q$-cofibrant and $q$-fibrant objects, defined with respect to $fp$-homotopy. It is still true that $q$-cofibrations are $h$-cofibrations. However, it is {\em not} true that $q$-cofibrations are $fp$-cofibrations, and it is the latter that are intrinsic to cofiber sequences. The classical and model theoretic homotopy theory no longer mesh.
Succinctly, the problem is that the $q$-model structure is not an example
of a well-grounded compactly generated model category. The task that lies
before us is to find a model structure which does satisfy
the axioms that we set out in \S5.5 and therefore can be used in tandem with
the $fp$-structure to do parametrized homotopy theory. Before embarking on this, we point out the limitations of the $q$-model structure more explicitly
in \S\ref{sec:danger}. There are two kinds of problems, those that we are focusing on in our development of the model category theory, and the more intrinsic ones that account for \myref{noway} and which cannot be overcome
model theoretically.
Ideally, to define the $qf$-model structure, we would like to take the
$qf$-co\-fi\-bra\-tions to be those $q$-cofibrations that are also $f$-cofibrations. However, with that choice, we would not know how to prove
the model category axioms. We get closer if we try to take as generating
sets of cofibrations and acyclic cofibrations those generators in the
$q$-model structure that are $f$-cofibrations, but with that choice we
still would not be able to prove the compatibility condition \myref{Newcompgen}(ii). However, using this generating set of cofibrations
and a subtler choice of a generating set of acyclic cofibrations, we obtain
a precise enough homotopical relationship to the $q$-equivalences that we
can prove the cited compatibility. The construction of the $qf$-model
structure is given in \S\ref{sec:qfstr}, but all proofs are deferred
to the following three sections.
\section{Some of the dangers in the parametrized world}\label{sec:danger}
We introduce notation for the generating (acyclic) cofibrations for the
$q$-model structures on $\scr{K}/B$ and $\scr{K}_B$. These maps are identified in \myref{cg}, starting from the sets $I$ and $J$ in $\scr{K}$ specified in \myref{UrIJ}. We then make some comments about these maps that
help explain the structure of our theory.
\begin{defn}\mylabel{IJB} For maps $i\colon C\longrightarrow D$ and $d\colon D\longrightarrow B$ of (unbased) spaces, we have the restriction $d\com i\colon C\longrightarrow B$ and may view $i$ as a map over $B$. We agree to write $i(d)$ for either the map $i$ viewed as a map over $B$ or the map $i\amalg\text{id}\colon C\amalg B\longrightarrow D\amalg B$ of ex-spaces over $B$ that is obtained by taking the coproduct with $B$ to adjoin a section. In either $\scr{K}/B$ or $\scr{K}_B$, define $I_B$ to be the set of all such maps $i(d)$ with $i\in I$, and define $J_B$ to be the set of all such maps $j(d)$ with $j\in J$. Observe that in $\scr{K}_B$, each map in $J_B$ is the inclusion of a deformation retract of spaces under, but not over, $B$.
\end{defn}
\begin{warn}
We cannot restrict the maps $d$ to be open here. That is one of the
reasons we chose $\scr{K}_B$ over $\scr{O}_*(B)$ in \S1.3.
\end{warn}
\begin{warn}\mylabel{quien}
The maps in $I_B$ and $J_B$ are clearly not $f$-cofibrations, only $h$-cofibrations. Looking at the NDR-pair characterization of $f$-cofibrations
given in \myref{fNDR}, we see that, with our arbitrary projections $d$,
there is in general no way to carry out the required deformation over $B$.
Since the maps in $I_B$ and $J_B$ are maps between well-sectioned spaces,
they cannot be $fp$-cofibrations in general, by \myref{reverse2}(i).
\end{warn}
\begin{rem}\mylabel{mildhelp} Observe that the maps $i$ in $I_B$ or $J_B$
are closed inclusions in $\scr{U}$, so that those maps in $I_B$ or $J_B$ which
are $f$-cofibrations are necessarily $\bar{f}$-cofibrations and
therefore both $\bar{fp}$-cofibrations and $\bar{h}$-cofibrations,
by \myref{compare} and \myref{ffpmodel}.
\end{rem}
\myref{quien} shows that the $q$-model structure is \emph{not} well-grounded since its generating (acyclic) cofibrations are not ${fp}$-cofibrations. This may sound like a minor technicality, but that is far from the case. We record
an elementary example.
\begin{ouch0} Let $B = I$ and define an ex-map $i\colon X\longrightarrow Y$ over $I$ by letting $X = \{0\}\amalg I$, $Y = I \amalg I$, and $i$ be the inclusion. The second copies of $I$ give the sections, and the projections are given by the identity map on each copy of $I$. This is a typical generating acyclic $q$-cofibration, and it is not an $fp$-cofibration. Let $Z$ be the pushout of $i$ and $p\colon X\longrightarrow I$, where the latter is viewed as a map of ex-spaces over $I$. Then $Z$ is the one-point union $I\vee I$ obtained by identifying the points $0$. The section $I\longrightarrow Z$ is not an $f$-cofibration, so that $Z$ is not well-sectioned. The same is true if we replace $Y$ by $Y'=\{1/(n+1)\mid n\in\mathbb{N}\}\amalg I$ and obtain $Z'$. The map $Z'\longrightarrow C_I Z'$ of $Z'$ into its cone over $I$ is not an $h$-cofibration (and therefore not a $q$-cofibration).
\end{ouch0}
Thus we cannot apply the classical gluing lemma to develop cofiber sequences, as we did in \S5.6. This and related problems prevent use of the $q$-model structure in a rigorous development of parametrized stable homotopy theory. For example, consider $q$-fibrant approximation. If we have a map $f\colon X\longrightarrow Y$ with $q$-fibrant approximation $Rf\colon RX\longrightarrow RY$, there is no reason to believe that $C_BRf$ is $q$-equivalent to $RC_Bf$.
We are about to overcome model-theoretically the problems pointed out in
the warnings above. Turning to the intrinsic problems that must hold in any
$q$-type model structure, we explain why the base change functor $f^*$ and
the internal smash product cannot be Quillen left adjoints.
\begin{warn}
If $f\colon A\longrightarrow B$ is a map and $d\colon D\longrightarrow B$ is a disk over $B$,
we have no homotopical control over the pullback $A\times_B D \longrightarrow A$ in
general.
\end{warn}
\begin{warn}\mylabel{ouchtoo}
In sharp contrast to the nonparametrized case, the generating sets do not behave well with respect to internal smash products, although they do behave well with respect to external smash products. We have
$$(D\amalg A)\barwedge (E\amalg B) \iso (D\times E)\amalg (A\times B).$$
If the projections of $D$ and $E$ are $d$ and $e$, then the projection of $D\times E$ is $d\times e$. However, if $A=B$, then
$$ (D\amalg B)\sma_B (E\amalg B) \iso (d\times e)^{-1}(\Delta B)\amalg (A\times B).$$
We have no homotopical control over the space $(d\times e)^{-1}(\Delta B)$ in general.
\end{warn}
This has the unfortunate consequence that, when we go on to parametrized spectra in Part III, we will not be able to develop a homotopically well-behaved theory of point-set level parametrized ring spectra. However, we will be able to develop a satisfactory point-set level theory of parametrized module spectra
over nonparametrized ring spectra.
\section{The $qf$ model structure on the category $\scr{K}/B$}\label{sec:qfstr}
Rather than start with a model structure on $\scr{K}$ to obtain model structures on $\scr{K}/B$ and $\scr{K}_B$, we can start with a model structure on $\scr{K}/B$ and then apply \myref{under} to obtain a model structure on $\scr{K}_B$. This gives us the opportunity to restrict the classes of generating (acyclic) cofibrations present in the $q$-model structure on $\scr{K}/B$ to ones that are $f$-cofibrations, retaining enough of them that we do not lose homotopical information. This has the effect that the generating (acyclic) cofibrations are $f$-cofibrations between well-grounded spaces over $B$, as is required of a well-grounded model structure. Such maps have closed images, hence are $\bar{f}$-cofibrations, and therefore all of the cofibrations in the resulting model structure on $\scr{K}/B$ are $\bar{f}$-cofibrations.
We call the resulting model structure the ``$qf$-model structure'', where $f$ refers to the fiberwise cofibrations that are used and $q$ refers to the weak equivalences. The latter are the same as in the $q$-model structure, namely the weak equivalences on total spaces, or $q$-equivalences. This model structure restores us to a situation in which the philosophy advertised in \S\ref{Sphil} applies, with the $q$ and $h$-model structures on spaces replaced by the $qf$ and $f$-model structures on spaces over $B$. Since $f$-cofibrations in $\scr{K}_B$ are $fp$-cofibrations, by \myref{compare}, the philosophy also applies to the $qf$ and $fp$-model structures on $\scr{K}_B$, or at least on $\scr{U}_B$ (see \myref{ffpmodel} and \myref{fpmodel?}).
We need some notations and recollections in order to describe the generating (acyclic) $qf$-cofibrations and the $qf$-fibrations.
\begin{notn}\mylabel{euclid}
For each $n\geq 1$, embed $\mathbb{R}^{n-1}$ in $\mathbb{R}^n = \mathbb{R}^{n-1}\times \mathbb{R}$ by sending $x$ to $(x,0)$. Let $e_n=(0,1)\in \mathbb{R}^n$. For
$n\geq 0$, define the following subspaces of $\mathbb{R}^n$.
\begin{alignat*}{2}
\mathbb{R}^n_+ &= \{(x,t)\in\mathbb{R}^n \mid t\geq 0\} &
\mathbb{R}^n_- &= \{(x,t)\in\mathbb{R}^n \mid t\leq 0\}\\
D^n &= \{(x,t)\in\mathbb{R}^n \mid |x|^2+t^2\leq 1\} &\qquad
S^{n-1} &= \{(x,t)\in\mathbb{R}^n \mid |x|^2+t^2=1\}\\
S^{n-1}_+ &= S^{n-1}\cap \mathbb{R}^n_+ &
S^{n-1}_- &= S^{n-1}\cap \mathbb{R}^n_-
\end{alignat*}
Here $\mathbb{R}^{0} = \{0\}$ and $S^{-1}=\emptyset$. We think of $S^n\subset \mathbb{R}^{n+1}$ as having equator $S^{n-1}$, upper hemisphere $S^n_+$ with north pole $e_{n+1}$ and lower hemisphere $S^n_-$.
\end{notn}
We recall a characterization of Serre fibrations.
\begin{prop}
The following conditions on a map $p\colon E\longrightarrow Y$ in $\scr{K}$ are equivalent; $p$
is called a Serre fibration, or $q$-fibration, if they are satisfied.
\begin{enumerate}[(i)]
\item The map $p$ satisfies the covering homotopy property with respect to disks $D^n$; that is, there is a lift in the diagram
\[\xymatrix{D^n \ar[r]^\alpha\ar[d] & E \ar[d]^p\\
D^n\times I \ar[r]_-h\ar@{-->}[ur] & Y.}\]
\item If $h$ is a homotopy relative to the boundary $S^{n-1}$ in the diagram above, then there is a lift that is a homotopy relative to the boundary.
\item The map $p$ has the RLP with respect to the inclusion $S^n_+\longrightarrow D^{n+1}$ of the upper hemisphere into the boundary $S^n$ of $D^{n+1}$; that is,
there is a lift in the diagram
\[\xymatrix{S^n_+ \ar[r]^\alpha\ar[d] & E \ar[d]^p\\
D^{n+1} \ar[r]_-{\bar h}\ar@{-->}[ur] & Y.}\]
\end{enumerate}
\end{prop}
\begin{proof}
Serre fibrations $p\colon E\longrightarrow Y$ are usually characterized by the first condition. Since the pairs $(D^n\times I, D^n)$ and
$(D^n\times I, D^n \cup (S^{n-1}\times I))$ are homeomorphic, one easily obtains that the first condition implies the second. Similarly a homeomorphism of the pairs $(D^{n+1},S^n_+)$ and $(D^n\times I, D^n)$ gives that the first and third conditions are equivalent. A homotopy $h\colon D^n\times I\longrightarrow Y$ relative to the boundary $S^{n-1}$ factors through the quotient map
$D^n\times I\longrightarrow D^{n+1}$ that sends $(x,t)$ to $(x, (2t-1)\sqrt{1-|x|^2})$.
Conversely, any map $\bar{h}\colon D^{n+1} \longrightarrow Y$ gives rise to a homotopy $h\colon D^n\times I\longrightarrow Y$ relative to the boundary $S^{n-1}$. It follows that the second condition implies the third.
\end{proof}
Property (ii) states that Serre fibrations are the maps that satisfy the ``disk lifting property''\index{disk lifting property} and that is the way we shall think about the $qf$-fibrations. In view of property (iii), we sometimes abuse language
by calling a map $h\colon D^{n+1}\longrightarrow Y$ a disk homotopy. The restriction to the upper hemisphere $S^n_+$ gives the ``initial disk'' and the restriction to the lower hemisphere $S^n_-$ gives the ``terminal disk''.
\begin{defn}
A disk $d\colon D^n\longrightarrow B$ in $\scr{K}/B$ is said to be an \emph{$f$-disk} if
$i(d)\colon S^{n-1}\longrightarrow D^n$ is an $f$-cofibration. An $f$-disk
$d\colon D^{n+1}\longrightarrow B$ is said to be a \emph{relative $f$-disk} if
the lower hemisphere $S^n_-$ is also an $f$-disk, so that the restriction $i(d)\colon S^{n-1}\longrightarrow S^n_-$ is an $f$-cofibration; the upper hemisphere $i(d)\colon S^{n-1}\longrightarrow S^n_+$ need not be an $f$-cofibration.
\end{defn}
\begin{defn}\mylabel{IJBf}
Define $I^f_B$\@bsphack\begingroup \@sanitize\@noteindex{IBf@$I^f_B$} to be the set of inclusions $i(d)\colon S^{n-1} \longrightarrow D^n$ in $\scr{K}/B$, where $d\colon D^n\longrightarrow B$ is an $f$-disk. Define $J^f_B$\@bsphack\begingroup \@sanitize\@noteindex{JBf@$J^f_B$} to be the set of inclusions $i(d)\colon S^n_+\longrightarrow D^{n+1}$ of the upper hemisphere into a relative $f$-disk $d\colon D^{n+1}\longrightarrow B$; note that these initial disks are not assumed to be $f$-disks. A map in $\scr{K}/B$ is said to be
\begin{enumerate}[(i)]
\item a \emph{$qf$-fibration}\index{fibration!qf-@$qf$- --} if it has the RLP with respect to $J^f_B$ and
\item a \emph{$qf$-cofibration}\index{cofibration!qf-@$qf$- --} if it has the LLP with respect to all $q$-acyclic $qf$-fibrations, that is, with respect to those $qf$-fibrations that are $q$-equivalences.
\end{enumerate}
Note that $J^f_B$ consists of relative $I^f_B$-cell complexes and that a map is a $qf$-fibration if and only if it has the ``relative $f$-disk lifting property.'' \end{defn}
With these definitions in place, we have the following theorem. Recall
the definition of a well-grounded model category from \myref{wellmodel}
and recall from Propositions \ref{exback} and \ref{exwellgr} that we have ground structures on $\scr{K}/B$ and $\scr{K}_B$ with respect to which the $q$-equivalences
are well-grounded. Also recall the definition of a quasifibration from \myref{quasifib}.
\begin{thm}\mylabel{Thesis}\index{model structure!qf@$qf$- --}
The category $\scr{K}/B$ of spaces over $B$ is a well-grounded model category with respect to the $q$-equivalences, $qf$-fibrations and $qf$-cofibrations. The sets $I^f_B$ and $J^f_B$ are the generating $qf$-cofibrations and the generating acyclic $qf$-cofibrations. All $qf$-cofibrations are also $\bar{f}$-cofibrations and all $qf$-fibrations are quasifibrations.
\end{thm}
Using \myref{under} and \myref{cg}, we obtain the $qf$-model structure on $\scr{K}_B$. We define a $qf$-fibration in $\scr{K}_B$ to be a map which is a $qf$-fibration when regarded as a map in $\scr{K}/B$, and similarly for
$qf$-cofibrations.
\begin{thm}
The category $\scr{K}_B$ of ex-spaces over $B$ is a well-grounded model category with respect to the $q$-equivalences, $qf$-fibrations, and $qf$-cofibrations. The sets $I^f_B$ and $J^f_B$ of generating $qf$-cofibrations and generating acyclic $qf$-cofibrations are obtained by adjoining disjoint sections to the corresponding sets of maps in $\scr{K}/B$. All $qf$-cofibrations are $\bar{f}$-cofibrations and all $qf$-fibrations are quasifibrations.
\end{thm}
Since the $qf$-model structures are well-grounded, they are in particular proper and topological. Furthermore, the $qf$-cofibrant spaces over $B$ are well-grounded and the $qf$-fibrant spaces over $B$ are quasifibrant. Since $qf$-cofibrations are $q$-cofibrations, we have an obvious comparison.
\begin{thm}
The identity functor is a left Quillen equivalence from $\scr{K}/B$ with the $qf$-model structure to $\scr{K}/B$ with the $q$-model structure, and similarly for
the identity functor on $\scr{K}_B$.
\end{thm}
We state and prove two technical lemmas in \S\ref{sec:qflemmas}, prove that $\scr{K}/B$ is a compactly generated model category in \S\ref{sec:qfproof}, and prove that the $qf$-fibrations are quasifibrations and the model structure is right proper in \S\ref{sec:qqfuasi}. The $\Box$-product condition of \myref{Newcompgen}(iv) follows as usual by inspection of what happens on generating (acyclic) cofibrations and, as in the case $A=*$ of \myref{ouchtoo},
the projections cause no problems here.
\section{Statements and proofs of the thickening lemmas}\label{sec:qflemmas}
We need two technical ``thickening lemmas''. They encapsulate the
idea that no
information about homotopy groups is lost if we restrict from the
general disks and cells used in the $q$-model structure to the $f$-disks
and $f$-cells that we use in the $qf$-model structure.
\begin{lem}\mylabel{thicken1}
Let $(S^m,q)$ be a sphere over $B$. Then there is an $h$-equivalence $\mu\colon (S^m,\bar{q})\longrightarrow (S^m,q)$ in $\scr{K}/B$ such that $(S^m,\bar{q})$ is an $I^f_B$-cell complex with two cells in each dimension.
\end{lem}
\begin{lem}\mylabel{thicken2}
Let $(D^n,q)$ be a disk over $B$. Then there is an $h$-equivalence $\nu\colon (D^n, \bar{q})\longrightarrow (D^n, q)$ relative to the upper hemisphere
$S^{n-1}_+$ such that $(D^n,\bar{q})$ is a relative $f$-disk.
\end{lem}
The rest of the section is devoted to the proofs of these lemmas.
The reader may prefer to skip ahead to \S\ref{sec:qfproof} to see
how they are used to prove \myref{Thesis}.
\begin{proof}[Proof of \myref{thicken1}]
To define the map $\mu\colon (S^m,\bar{q})\longrightarrow (S^m,q)$, we begin by defining some auxiliary maps for each natural number $n\leq m$. They will in fact be continuous families of maps, defined for each $s\in [\tfrac12, 1]$. The parameter $s$ will show that $\mu$ is an $h$-equivalence.
First we define the map
\[\phi^n_+\colon D^n\cap \mathbb{R}^n_+\longrightarrow A_s\cup s\cdot S^{n-1}_+\]
from the upper half of the disk $D^n$ to the union of the equatorial annulus
\[A_s=\overline{D^{n-1}-s\cdot D^{n-1}}=\{(x,0)\in \mathbb{R}^{n}\colon s\leq |x|\leq 1\}\]
and the upper hemisphere
\[s\cdot S^{n-1}_+=\{(x,t)\in \mathbb{R}^n\colon \text{$t\geq 0$ and $|(x,t)|=s$}\}\]
to be the projection from the south pole $-e_n$. Similarly, we define
\[ \phi^n_-\colon D^n\cap \mathbb{R}^n_- \longrightarrow A_s\cup s\cdot S^{n-1}_-\]
to be the projection from the north pole $e_n$. The map $\phi^n_+$ is drawn schematically in the following picture. Each point in the upper half of the larger disk lies on a unique ray from $-e_n$. The map $\phi^n_+$ sends it to the intersection of that ray with $A_s\cup s\cdot S^{n-1}$; two such points of
intersection are marked with dots in the picture.
\[\begin{xy}
0;/r3pc/:(1,0),{\ellipse<>{-}}
,(.5,0);(1,0),{\ellipse<>{-}}
,(0,0);(.5,0) **@{-}
,(1.5,0);(2,0) **@{-}
,(1,-1);(.6,1) **@{-}
,(.72,.4) *{\bullet}
,(1,-1);(-.05,.4) **@{-}
,(.25,0) *{\bullet}
,(1.25,0)*{\sb{s\cdot D^n}}
,(1,-1.1)*{\sb{-e_n}}
,(1.75,-1)*{\sb{D^n}}
\end{xy}\]
Next we use the maps $\phi^n_\pm$ to define a continuous family of maps $f^n_s\colon D^n\longrightarrow D^n$ for $s\in[\tfrac12,1]$ by induction on $n$. We let $f^0_s\colon D^0\longrightarrow D^0$ be the unique map and we define $f^1_s\colon D^1\longrightarrow D^1$
by
\[f^1_s(t) = \begin{cases}
t/s & \text{if $|t|\leq s$},\\
1 & \text{if $t\geq s$},\\
-1 & \text{if $t\leq -s$};
\end{cases}\]
it maps $[-s,s]$ homeomorphically to $[-1,1]$. We define
$f^n_s\colon D^n\longrightarrow D^n$ by
\[f^n_s(x,t)=\begin{cases}
s^{-1}\cdot (x,t) & \text{if $|(x,t)|\leq s$},\\
s^{-1}\cdot \phi^n_+(x,t) & \text{if $|(x,t)|\geq s$, $t\geq 0$ and $|\phi^n_+(x,t)|= s$},\\
f^{n-1}_s(\phi^n_+(x,t)) & \text{if $|(x,t)|\geq s$, $t\geq 0$ and $|\phi^n_+(x,t)|\geq s$},\\
s^{-1}\cdot \phi^n_-(x,t) & \text{if $|(x,t)|\geq s$, $t\leq 0$ and $|\phi^n_-(x,t)|= s$},\\
f^{n-1}_s(\phi^n_-(x,t)) & \text{if $|(x,t)|\geq s$, $t\leq 0$ and $|\phi^n_-(x,t)|\geq s$}.
\end{cases}\]
The map $f^n_s$ is drawn in the following picture. The smaller ball $s\cdot D^n$ is mapped homeomorphically to $D^n$ by radial expansion from the origin. Next comes the region in the upper half of the larger ball that is inside the cone and outside the smaller ball. Each segment of a ray from the south pole $-e_n$ that lies in that region is mapped to a point which is determined by where we mapped the intersection of that ray-segment with the smaller ball (which was radially from the origin to the boundary of $D^n$). Third is the region in the upper half of the larger ball that is outside the cone. Each segment of a ray from the south pole $-e_n$ that lies in that region is first projected to the annulus in the equatorial plane of the two balls; we then apply the previously defined map $f^{n-1}_s$ to map the projected points to the equator of $D^n$. The lower half of the ball is mapped similarly.
\[\begin{xy}
0;/r4pc/:(1,0),{\ellipse<>{-}},{\ellipse(,.2) d,^u{-}},{\ellipse(,.2) u,^d{.}}
,(.5,0);(1,0),{\ellipse<>{-}},{\ellipse(,.2) d,^u{-}},{\ellipse(,.2) u,^d{.}}
,(0,1);(1,-1) **@{-};(2,1) **@{-}
,(.2,.6);(1,.6),{\ellipse(,.2) d,^u{-}},{\ellipse(,.2) u,^d{.}}
,(0,1);(1,1),{\ellipse(,.2){-}}
,(1,0)*{\sb{sD^n}}
,(1,-1.1) *{\sb{-e_n}}
,(1.75,-1)*{\sb{D^n}}
\end{xy}\]
It is clear that $f^n_s$ gives a homotopy from $f^n_{1/2}$ to the identity and, given any disk $(D^n,q)$ in $\scr{K}/B$, the map $f^n_s$ induces an $h$-equivalence from the $f$-disk $(D^n, q\circ f^n_{1/2})$ to the disk $(D^n,q)$.
Finally we define the required cell structure on the domain of the desired map $\mu\colon (S^m,\bar{q})\longrightarrow (S^m,q)$. For each $n\leq m$, the boundary sphere $(S^n, q\circ f^{n+1}_{1/2}|S^n)$ is constructed from two copies of the $f$-disk $(D^n, q\circ f^n_{1/2})$ by gluing them along their boundary. The inclusions $(D^n, q\circ f^n_{1/2})\longrightarrow (S^n,q\circ f^{n+1}_{1/2}|S^n)$ of the two cells are given by projecting $D^n$ to the upper hemisphere from the south pole $-e_{n+1}$ and, similarly, by projecting $D^n$ to the lower hemisphere from the north pole $e_{n+1}$. The map
\[\mu=f^{m+1}_{1/2}|S^m\colon (S^m,q\circ f^{m+1}_{1/2}|S^m) \longrightarrow (S^m, q).\]
is then the required $f$-cell sphere approximation.
\end{proof}
\begin{proof}[Proof of \myref{thicken2}]
Define $\nu_s\colon D^n\longrightarrow D^n$ for $s\in[\tfrac12,1]$ by
\[\nu_s(x,t)=\begin{cases}
s^{-1}\cdot (x,t) & \text{if $|(x,t)|\leq s$},\\
|(x,t)|^{-1}\cdot (x,t) & \text{if $|(x,t)|\geq s$, $t\geq 0$ and $|x|\geq s$},\\
s^{-1}\cdot \phi^{n+1}_-(x,t) & \text{if $|(x,t)|\geq s$, $t\leq 0$ and $|\phi^{n+1}_-(x,t)|= s$},\\
|\phi^{n+1}_-(x,t)|^{-1}\cdot \phi^{n+1}_-(x,t) & \text{if $|(x,t)|\geq s$, $t\leq 0$ and $|\phi^{n+1}_-(x,t)|\geq s$},
\end{cases}\]
where $\phi^n_-$ is the projection as in the previous proof. Then $\nu_s$ maps $s\cdot D^n$ homeomorphically to $D^n$, it is radially constant on the region in the upper half space between the disks $D^n$ and $s\cdot D^n$ with respect to projection from the origin, and it is radially constant on the region in the lower half space between the two disks with respect to projection from the north pole.
\end{proof}
\section{The compatibility condition for the $qf$-model structure}\label{sec:qfproof}
This section is devoted to the proof that $\scr{K}/B$ is a compactly generated model category. Since our generating sets $I^f_B$ and $J^f_B$ certainly satisfy conditions (i) and (iii) of \myref{Newcompgen}, it only remains to verify the compatibility condition (ii). That is, we must show that a map has the RLP with respect to $I^f_B$ if and only if it is a $q$-equivalence and has the RLP with respect to $J^f_B$. Let $p\colon E\longrightarrow Y$ have the RLP with respect to $I^f_B$. Since all maps in $J^f_B$ are relative $I^f_B$-cell complexes, $p$ has the RLP with respect to $J^f_B$. To show that $\pi_n(p)$ is injective, let $\alpha\colon S^n\longrightarrow E$ represent an element in $\pi_n(E)$ such that $p\circ \alpha\colon S^n\longrightarrow Y$ is null-homotopic. Then there is a nullhomotopy $\beta\colon CS^n\longrightarrow Y$ that gives rise to a lifting problem
\[\xymatrix{S^n\ar[rr]^\alpha\ar[d]_i && E \ar[d]^{p}\\
D^{n+1}\ar[r]_-\nu & D^{n+1}\cong CS^n \ar[r]_-\beta & Y}\]
where $\nu\colon D^{n+1}\longrightarrow D^{n+1}$ is defined by
$$\nu(x) = \begin{cases}
2x & \text{if $|x|\leq \tfrac12$},\\
|x|^{-1}\cdot x & \text{if $|x|\geq \tfrac12$.}
\end{cases}$$
Then $i$ is an $f$-disk and we are entitled to a lift, which can be viewed as a nullhomotopy of $\alpha$ after we identify $D^{n+1}$ with $CS^n$.
To show that $\pi_n(p)$ is surjective, choose a representative $\alpha\colon S^n\longrightarrow Y$ of an element in $\pi_n(Y)$. The projection of $Y$ induces a projection $q\colon S^n\longrightarrow B$ and by \myref{thicken1} there is an $h$-equivalence $\mu\colon (S^n,\bar{q})\longrightarrow (S^n,q)$ such that $(S^n,\bar{q})$ is an $I^f_B$-complex with two cells in each dimension. We may therefore assume that the source of $\alpha$ is an $I^f_B$-cell complex. Inductively, we can then solve the lifting problems for the diagrams
\[\xymatrix{S^{k-1}\ar[rr]\ar[d]\ar[dr] && E\ar[d]^{p}\\
S^k_\pm \ar[r]_-{i_\pm} & S^k \ar[r]_-{\alpha|S^k} & Y,}\]
where $S^{k-1}\longrightarrow S^k$ is the inclusion of the equator and $i_\pm\colon S^k_\pm\longrightarrow S^k$ are the inclusions of the upper and lower hemispheres. We obtain a lift $S^n\longrightarrow E$.
Conversely, assume that $p\colon E\longrightarrow Y$ is an acyclic $qf$-fibration. We must show that $p$ has the RLP with respect to any cell $i$ in $I^f_B$. We are therefore faced with a lifting problem
\[\xymatrix{S^n\ar[r]^\alpha\ar[d]_i & E \ar[d]^{p}\\
D^{n+1}\ar[r]_\beta & Y.}\]
Identifying $D^{n+1}$ with $CS^n$ we see that $\beta$ gives a nullhomotopy of $p\circ \alpha$. Since $\pi_n(p)$ is injective there is a nullhomotopy $\gamma\colon CS^n\longrightarrow E$ such that $\alpha=\gamma\circ i$, but it may not cover $\beta$. Gluing $\beta$ and $p\circ\gamma$ along $p\circ \alpha$ gives $\delta\colon S^{n+1}\longrightarrow Y$ such that $\delta|S^{n+1}_+ = \beta$ and $\delta|S^{n+1}_-=p\circ\gamma$. Surjectivity of $\pi_{n+1}(p)$ gives a map $\Delta\colon S^{n+1}\longrightarrow E$ and a homotopy $h\colon S^{n+1}\wedge I_+\longrightarrow Y$ from $p\circ \Delta$ to $\delta$. We now construct a diagram
\[\xymatrix{&S^{n+1}_+\ar[d]\ar[r]\ar[dl]_{j} & S^{n+1}_+\cup H \ar[r]^-{(-)/S^n}\ar[d]& S^{n+1}\times 0 \cup S^{n+1}_-\times 1\ar[r]^-{\Delta\cup \gamma}\ar[d] & E \ar[d]^{p}\\
D^{n+2}\ar[r]_\nu & D^{n+2} \ar[r]_\xi & D^{n+2}\ar[r]_-\phi & S^{n+1}\wedge I_+\ar[r]_-h & Y}\]
where the downward maps, except $p$, are inclusions. Part of the bottom row of the diagram is drawn schematically below. Let $H$ be the region on $S^{n+1}_-$ between the equator $S^n$ and the circle through $e_1$ and $-e_{n+2}$ with center on the line $\mathbb{R}\cdot(e_1-e_{n+2})$. Let $\xi$ be a homeomorphism whose restriction to $S^{n+1}_+$ maps it homeomorphically to $S^{n+1}_+\cup H$. Define $\phi\colon D^{n+2}\longrightarrow D^{n+2}/S^n\cong S^{n+1}\wedge I_+$ as the composite of the quotient map that identifies the equator $S^n$ of $D^{n+2}$ to a point and a homeomorphism that maps the upper hemisphere $S^{n+1}_+$ to $S^{n+1}\times 0$, maps $H$ to $S^{n+1}_-\times 1$, and is such that
$(h\circ \phi\circ \xi)|S^{n+1}_-=\beta$. The map $\nu$ is defined as above.
\[
\begin{xy}
0;/r5pc/:(.5,0),{\ellipse<>{-}},{\ellipse(,.2) d,^u{-}},{\ellipse(,.2) u,^d{.}}
,(.5,-.6)*!U{D^{n+2}}
,(.5,.5);(.75,.25),{\ellipse(,.2) d,_r{}},{\ellipse(,.2) u,^l{.}}
,(0,.25)*!R{\sb{p\circ\gamma}}
,(0,-.35)*!UR{\sb\beta}
,(1,.35)*!L{\sb{p\circ\Delta}}
,(.5,0)*{\sb{p\circ\alpha}}
\end{xy}
\quad\xrightarrow{\xi}\quad
\begin{xy}
0;/r5pc/:(.5,0),{\ellipse<>{-}},{\ellipse(,.2) d,^u{-}},{\ellipse(,.2) u,^d{.}}
,(.5,-.6)*!U{D^{n+2}}
,(.5,-.5);(.75,-.25),{\ellipse(,.2) l,_d{.}},{\ellipse(,.2) r,^u{-}}
,(0,-.25)*!R{\sb{p\circ\gamma}}
,(1,-.35)*!UL{\sb\beta}
,(0,.25)*!R{\sb{p\circ\Delta}}
,(.75,-.25)*{\sb{p\circ\alpha}}
,(.35,-.25)*{\sb{H}}
\end{xy}
\quad\xrightarrow{\phi}\quad
\begin{xy}
0;/r5pc/:(-.5,0),{\ellipse<>{-}},{\ellipse(,.2) d,^u{-}},{\ellipse(,.2) u,^d{.}}
;(-.25,0),{\ellipse<>{-}},{\ellipse(,.2) d,^u{-}},{\ellipse(,.2) u,^d{.}}
,(-.5,-.6)*!U{S^{n+1}\wedge I_+}
,(-.25,-.25)*!UR{\sb{p\circ\gamma}}
,(-.25,.25)*!DR{\sb\beta}
,(-1,.35)*!D{\sb{p\circ\Delta}}
,(-.25,0)*{\sb{p\circ\alpha}}
\end{xy}
\]
Since the restriction $S^n\longrightarrow S^{n+1}_-\cong D^{n+1}$ of $j$ agrees with the $f$-cofibration $i$ in our original lifting problem, we see that $j$ is a $J^f_B$-cell. Since $p$ is a $qf$-fibration we get a lift in the outer trapezoid. Denote its restriction to $S^{n+1}_-\cong D^{n+1}$ by $k\colon D^{n+1}\longrightarrow E$. Then $k$ solves our original lifting problem.
\section{The quasifibration and right properness properties}\label{sec:qqfuasi}
We have now established the $qf$-model structures on both $\scr{K}/B$ and $\scr{K}_B$.
We will derive the right properness of $\scr{K}/B$, and therefore of $\scr{K}_B$, from the fact that every $qf$-fibration is a quasifibration.
\begin{prop}\mylabel{qfles}
If $p\colon E\longrightarrow Y$ is a $qf$-fibration in $\scr{K}/B$, then $p$ is a quasifibration. Therefore, for any choice of $e\in E$, there results a long exact sequence of homotopy groups
\[\cdots \longrightarrow \pi_{n+1}(Y,y)\longrightarrow \pi_n(E_y,e) \longrightarrow \pi_n(E,e)\longrightarrow \pi_n(Y,y)\longrightarrow\cdots\longrightarrow \pi_0(Y,y),\]
where $y=p(e)$ and $E_y=p^{-1}(y)$.
\end{prop}
\begin{proof}
We must prove that $p$ induces an isomorphism
\[\pi_n(p)\colon \pi_n(E,E_y,e)\longrightarrow \pi_n(Y,y)\]
for all $n\geq 1$ and verify exactness at $\pi_0(E,e)$. We begin with the latter. Let $e'\in E$ and suppose that $p(e')$ is in the component of $y'$.
Let $\gamma\colon I\longrightarrow Y$ be a path in $Y$ from $p(e')$ to $y'$ such that $\gamma$ is constant at $p(e')$ for time $t\leq \tfrac12$. Let $q$ be the projection of $Y$. Then $(I,q\circ \gamma)$ is a relative $f$-disk, and we obtain a lift $\bar\gamma\colon I\longrightarrow E$ such that $\gamma=p\circ \bar\gamma$. But then $e'$ is in the same component as the endpoint of $\bar\gamma$, which lies in $E_y$.
Now assume that $n\geq 1$. Recall that an element of $\pi_n(X,A,*)$ can be represented by a map of triples $(D^n, S^{n-1}, S^{n-1}_+)\longrightarrow (X,A,*)$. We begin by showing surjectivity. Let $\alpha\colon (D^n,S^{n-1})\longrightarrow(Y,y)$ represent an element of $\pi_n(Y,y)$. We can view $D^n$ as a disk over $B$, and \myref{thicken2} gives an approximation $\nu\colon D^n\longrightarrow D^n$ by a relative $f$-disk. Then we can solve the lifting problem
\[\xymatrix{S^{n-1}_+ \ar[d]\ar[r]^-{c_e} & E\ar[d]^p\\
D^n\ar@{-->}[ur]_{\bar\alpha}\ar[r]_-{\alpha\circ \nu} & Y,}\]
where the top map is the constant map at $e\in E$. A lift is a map of triples $\bar\alpha \colon (D^n, S^{n-1},S^{n-1}_+)\longrightarrow(E,E_y,e)$ such that $p_*([\bar\alpha])=[\alpha]$.
For injectivity, let $\alpha\colon (D^n,S^{n-1},S^{n-1}_+)\longrightarrow (E,E_y,e)$ represent an element of $\pi_n(E,E_y,e)$ such that $p_*([\alpha])=0$. Then there is a homotopy $h\colon D^n\times I\longrightarrow Y$ rel $S^{n-1}$ such that $h|D^n\times 0=p\circ \alpha$ and $h$ maps the rest of the boundary of $D^n\times I$ to $y$. Let $A=D^n\times \{0,1\} \cup S^{n-1}_+\times I\subset \partial (D^n\times I)$ and define $\beta\colon A\longrightarrow E$ by setting $\beta(x)= \alpha(x)$ if $x\in D^n\times 0$ and $\beta(x)=e$ otherwise. We then have a homeomorphism of pairs $\phi\colon (D^n\times I, A)\longrightarrow(D^{n+1},S^n_+)$ and an approximation $\nu\colon D^{n+1}\longrightarrow D^{n+1}$ by an $f$-disk by \myref{thicken2}. We can now solve the lifting problem
\[\xymatrix{S^n_+\ar[d]\ar[r]^-{\beta\circ(\phi|_A)^{-1}} & E \ar[d]\\
D^{n+1}\ar@{-->}[ur]_{\bar\alpha}\ar[r]_-{h\circ\phi^{-1}\circ\nu} & Y,}\]
and this shows that $[\alpha]=0$ in $\pi_n(E,E_y,e)$.
\end{proof}
\begin{cor}\mylabel{fdrp}
The $qf$-model structure on $\scr{K}/B$ is right proper.
\end{cor}
\begin{proof}
Since $qf$-fibrations are preserved under pullbacks, this is a
five lemma comparison of long exact sequences as in \myref{qfles}.
\end{proof}
\chapter{Equivariant $qf$-type model structures}
\section*{Introduction}
We return to the equivariant context in this chapter, letting $G$ be a Lie group throughout. Actually, our definitions of the $q$ and $qf$-model structures work for arbitrary topological groups $G$, but we must restrict to Lie groups to obtain structures that are $G$-topological and behave well with respect to change of groups and smash products. A discussion of details special to the
non-compact Lie case is given in \S7.1, but after that the generalization
from compact to non-compact Lie groups requires no extra work. However,
we alert the reader that passage to {\em stable} equivariant homotopy theory raises new problems in the case of non-compact Lie groups that will not be
dealt with in this book; see \S11.6.
The equivariant $q$-model structure on $G\scr{K}_B$ is just the evident over and under $q$-model structure. However, the equivariant generalization of the
$qf$-model structure is subtle. In fact, the subtlety is already relevant nonequivariantly when we study base change along the projection of a bundle. The problem is that there are so few generating $qf$-cofibrations that many functors that take generating $q$-cofibrations to $q$-cofibrations do not take generating $qf$-cofibrations to $qf$-cofibrations. We show how to get around this in \S7.2. For each such functor that we encounter, we find an enlargement of the obvious sets of (acyclic) generating $qf$-cofibrations on the target of the functor so that it is still a model category, but now the functor does send generating (acyclic) $qf$-cofibrations to (acyclic) $gf$-cofibrations.
The point is that there are many different useful choices of Quillen equivalent $qf$-type model structures, and they can be used in tandem. For all of our choices, the weak equivalences are the $\scr{G}$-equivalences and all cofibrations are both $q$-cofibrations and $f$-cofibrations. Given a finite number of adjoint pairs with composable left adjoints such that each is a Quillen adjunction with its own choice of $qf$-type model structure, we can successively expand generating sets in target categories of the left
adjoints to arrange that the composite be one of Quillen left adjoints
with respect to well chosen $qf$-type model structures.
In \S7.2, we describe the $qf(\scr{C})$-model structure associated to a
``generating set'' $\scr{C}$ of $\scr{G}$-complexes. Each such model structure
is $G$-topological. In \S\ref{sec:qfadj}, we show that external smash products
are Quillen adjunctions when $\scr{C}$ is a ``closed'' generating set, as can always
be arranged, and we show that all base change adjunctions $(f_!,f^*)$ are Quillen adjunctions. We show further that there are generating sets for which $(f^*,f_*)$ is a Quillen adjunction when $f$ is a bundle with cellular fibers. In \S\ref{sec:JHadj}, we show similarly that various change of group functors are given by Quillen adjunctions when the generating sets are well chosen.
In \S7.5, we show that $\text{Ho}G\scr{K}_B$ has the properties required for application of the Brown representability theorem. Those adjunctions between our basic functors that are not given by Quillen adjoint pairs in any choice of $qf$-model structure are studied in Chapter 9.
\section{Families and non-compact Lie groups}
Two sources of problems in the equivariant homotopy theory of general
topology groups $G$ are that we only know that orbit types $G/K$ are
$H$-CW complexes for $H\subset G$ when $G$ is a Lie group and $K$ is
a compact subgroup and we only know that a product of orbits
$G/H\times G/K$ is a $G$-CW complex when $G$ is a Lie group and $K$
(or $H$) is a compact subgroup. This motivates us to restrict to Lie
groups, for which these conclusions are ensured by \myref{Illman} and \myref{prodproper}.
The compactness requirements force us to restrict orbit types when we prove
properties of our model structures, and the family $\scr{G}$ of all compact subgroups of our Lie group $G$ plays an important role. We recall the relevant definitions, which apply to any topological group $G$ and are familiar and important in a variety of contexts. They provide a context that allows us to work with non-compact Lie groups with no more technical work than is required for compact Lie groups.
A {\em family} $\scr{F}$ in $G$ is a set of subgroups that is closed under passage to subgroups and conjugates. An {\em $\scr{F}$-space} is a $G$-space all of whose isotropy groups are in $\scr{F}$. An {\em $\scr{F}$-equivalence} is a $G$-map $f$ such that $f^H$ is a weak equivalence for all $H\in \scr{F}$. If $X$ is an $\scr{F}$-space,
then the only non-empty fixed point sets $X^H$ are those for groups $H\in\scr{F}$. In particular, an $\scr{F}$-equivalence between $\scr{F}$-spaces is the same as a $q$-equivalence. For based $G$-spaces, the definition of an $\scr{F}$-space must be altered to require that all isotropy groups except that of the $G$-fixed
base point must be in $\scr{F}$. The notion of an $\scr{F}$-equivalence remains unchanged.
A map in $G\scr{K}/B$ or $G\scr{K}_B$ is an $\scr{F}$-equivalence if its map of total $G$-spaces is an $\scr{F}$-equivalence. If $B$ is an $\scr{F}$-space, then so is any $G$-space $X$ over $B$ and any fiber $X_b$. The only orbits that can
then appear in our parametrized theory are of the form $G/H$ with $H\in \scr{F}$
and the only non-empty fixed point sets $X^H$ are those for groups $H\in\scr{F}$.
In particular, $H$ must be subconjugate to some $G_b$. An $\scr{F}$-equivalence
of $G$-spaces over an $\scr{F}$-space $B$ is the same as a $q$-equivalence.
It is well-known that equivariant $q$-type model structures generalize naturally to families. One takes the weak equivalences to be the $\scr{F}$-equivalences, and one restricts the orbits $G/H$ that appear as factors in the generating (acyclic) cofibrations to be those such that $H\in \scr{F}$. The resulting cell complexes are called $\scr{F}$-cell complexes. Restricting tensors from $G$-spaces
to $\scr{F}$-spaces, we obtain a restriction of the notion of a $G$-topological model category to an $\scr{F}$-topological model category that applies here; see \myref{Gtopfamily}.
Proper $G$-spaces are particularly well-behaved $\scr{G}$-spaces, where $\scr{G}$ is the family of compact subgroups of our Lie group $G$, and $\scr{G}$-cell complexes are proper $G$-spaces. Restricting base $G$-spaces to be proper, or more generally to be $\scr{G}$-spaces, has the effect of restricting all relevant orbit types
$G/H$ to ones where $H$ is compact. However, this is too restrictive for some purposes. For example, we are interested in developing nonparametrized equivariant homotopy theory for non--compact Lie groups $G$. Here $B=*$ is a
$G$-space which, in the unbased sense, is not a $\scr{G}$-space.
We therefore do not make the blanket assumption that $B$ is a $\scr{G}$-space.
We give the $q$-model structure in complete generality, in \myref{qoverB}, but after that we restrict to $\scr{G}$-model structures throughout. That is, our weak
equivalences will be the $\scr{G}$-equivalences. This ensures that, after cofibrant approximation, our total $G$-spaces are $\scr{G}$-spaces. This convention
enables us to arrange that all of our model categories are $G$-topological. Everything in this chapter applies more generally to the study of parametrized $\scr{F}$-homotopy theory for any family $\scr{F}$; see \myref{Fremark}.
The reader may prefer to think in terms of either the case when $B=*$
or the case when $B$ is proper. Indeed, in order to resolve the problems intrinsic to the parametrized context that are described in the Prologue,
which we do in Chapter 9, it seems essential that we restrict to proper
actions on base spaces. The reason is that Stasheff's \myref{ss} relating
the equivariant homotopy types of fibers and total spaces plays a
fundamental role in the solution. Alternatively, the reader may prefer
to focus just on compact Lie groups, reading $q$-equivalence
instead of $\scr{G}$-equivalence and $G$-space instead of $\scr{G}$-space.
\section{The equivariant $q$ and $qf$-model structures}\label{sec:qfeq}
Recall from \myref{UrIJ} that the sets $I$ and $J$ of generating cofibrations and generating acyclic cofibrations of $G$-spaces are defined as the sets of all maps of the form $G/H\times i$, where $i$ is in the corresponding set $I$ or $J$ of maps of spaces.
\begin{defn}\mylabel{IJBG}
Starting from the sets $I$ and $J$ of maps of $G$-spaces, define sets $I_B$ and $J_B$ of maps of ex-$G$-spaces over $B$ in exactly the same way that their nonequivariant counterparts were defined in terms of the sets $I$ and $J$ of maps of spaces in \myref{IJB}. Note that if $B$ is a $\scr{G}$-space, then only orbits $G/H$ with $H$ compact appear in the sets $I_B$ and $J_B$.
\end{defn}
Taking $Y = B$ in the usual composite adjunction
\begin{equation}\label{adadad}
G\scr{K}(G/H\times T , Y) \iso H\scr{K}(T,Y) \iso \scr{K}(T,Y^H)
\end{equation}
for non-equivariant spaces $T$ and $G$-spaces $Y$, we can translate back and forth between equivariant homotopy groups and cells for $G$-spaces over $B$ on the one hand and nonequivariant homotopy groups and cells for spaces over $B^H$ on the other. Maps in each of the equivariant sets specified in \myref{IJBG} correspond by adjunction to maps in the nonequivariant set with the same name. Systematically using this translation, it is easy to use \myref{compgen} to generalize the $q$-model structures on $\scr{K}/B$ and $\scr{K}_B$ to corresponding model structures on $G\scr{K}/B$ and $G\scr{K}_B$. We obtain the following theorem.
\begin{thm}[$q$-model structure]\mylabel{qoverB}
The categories $G\scr{K}/B$ and $G\scr{K}_B$ are compactly generated proper $\scr{G}$-top\-o\-lo\-gi\-cal model categories whose $q$-equivalences, $q$-fibrations, and $q$-co\-fibrations are the maps whose underlying maps of total $G$-spaces are $q$-equivalences, $q$-fibrations, and $q$-cofibrations. The sets $I_B$ and $J_B$ are the generating $q$-cofibrations and generating acyclic $q$-cofibrations, and all $q$-cofibrations are $\bar{h}$-cofibrations. If $B$ is a $\scr{G}$-space, then the model structure is $G$-topological.
\end{thm}
To show that the $q$-model structures are $\scr{G}$-topological, and $G$-topological if $B$ is a $\scr{G}$-space, we must inspect the maps $i\Box j$ in $G\scr{K}/B$, where $i$ is a generating $q$-cofibration in $G\scr{K}/B$ and $j$ is a generating cofibration in $G\scr{K}$. They have the form
\[i\Box j\colon G/H\times G/K \times \partial(D^m\times D^n)
\longrightarrow G/H\times G/K \times D^m\times D^n\]
given by the product of $G/H\times G/K$ with the inclusion of the boundary of $D^m\times D^n$. By \myref{prodproper}, $G/H\times G/K$ is a proper $G$-space
if $H$ or $K$ is compact. Since we are assuming that $G$ is a Lie group, we can then triangulate $G/H\times G/K$ as a $\scr{G}$-CW complex and use the triangulation to write $i\Box j$ as a relative $I_B$-cell complex. The case when either $i$ or $j$ is acyclic works in the same way. As explained in \myref{ouchtoo}, there is no problem with projection maps in this external context. Moreover, if $i$ is an $f$-cofibration, then so is $i\Box j$, as we see from the fiberwise NDR characterization.
One might be tempted to generalize the $qf$-model structure to the equivariant context in exactly the same way as we just did for the $q$-model structure. This certainly works to give a model structure. However, there is no reason to think that it is either $G$ or $\scr{G}$-topological. The problem is that we need
$i\Box j$ above to be a $qf$-cofibration when $i$ is a generating $qf$-cofibration, and triangulations into $f$-cells are hard to come by. Therefore the $G$-CW structure on $G/H\times G/K$ will rarely produce a relative $I^f_B$-cell complex. This means that we must be careful when selecting the generating (acyclic) $qf$-cofibrations if we want the resulting model structure to be $G$-topological. We will build the solution into our definition of $qf$-type model structures, but we need a few preliminaries.
We shall make repeated use of the adjunction
\begin{equation}\label{keyadj}
G\scr{K}(C \times T , Y)\cong \scr{K}(T,\text{Map}_G(C, Y))
\end{equation}
for non-equivariant spaces $T$ and $G$-spaces $C$ and $Y$. This is a generalization of (\ref{adadad}).
Taking $Y=B$, we note in particular that it gives a correspondence
between maps $f\colon T\longrightarrow T'$ over $\text{Map}_G(C,B)$ and
$G$-maps $\text{id}\times f\colon C\times T\longrightarrow C\times T'$ over $B$.
\begin{lem}\mylabel{mapgood}
If $C$ is a $\scr{G}$-cell complex, then the functor $\text{Map}_G(C,-)\colon G\scr{K} \longrightarrow \scr{K}$ preserves all $q$-equivalences.
\end{lem}
\begin{proof} The functor $\text{Map}(C,-)$ is a Quillen right adjoint since the $q$-model structure on $G\scr{K}$ is $\scr{G}$-topological. The $G$-fixed point functor is also a Quillen right adjoint, for example by \myref{fixedptrQa0} below. The composite $\text{Map}_G(C,-)$ therefore preserves $q$-equivalences between $q$-fibrant $G$-spaces. However, every $G$-space is $q$-fibrant.
\end{proof}
Observe that \myref{prodproper} gives that the collection of $\scr{G}$-cell complexes is closed under products with arbitrary orbits $G/H$ of $G$.
\begin{defn}\mylabel{IJBG2}
Let $\scr{O}_G$ denote the set of all orbits $G/H$ of $G$. Any set $\scr{C}$ of $\scr{G}$-cell complexes in $G\scr{K}$ that contains all orbits $G/K$ with $K\in \scr{G}$ and is closed under products with arbitrary orbits in $\scr{O}_G$ is called a \emph{generating set}. It is a {\em closed} generating set if it is
closed under finite products. The {\em closure} of a generating set
$\scr{C}$ is the generating set consisting of the finite products of the
$\scr{G}$-cell complexes in $\scr{C}$.
We define sets of generating $qf(\scr{C})$-cofibrations and acyclic $qf(\scr{C})$-cofibrations in $G\scr{K}/B$ associated to any generating set $\scr{C}$ as follows.
\begin{enumerate}[(i)]
\item Let $I^f_B(\scr{C})$ consist of the maps
\[(\text{id}\times i)(d)\colon C\times S^{n-1} \longrightarrow C\times D^n\]
such that $C\in \scr{C}$, $d\colon C\times D^n\longrightarrow B$ is a $G$-map, $i$ is the boundary inclusion, and the corresponding map $i$ over $\text{Map}_G(C,B)$ is a generating $qf$-cofibration in $\scr{K}/\text{Map}_G(C,B)$; that is, $i$ must be an $f$-cofibration.
\item Similarly let $J^f_B(\scr{C})$ consist of the maps
$$(\text{id}\times i)(d)\colon C\times S^n_+\longrightarrow C\times D^{n+1}$$
such that $C\in\scr{C}$, $d\colon C\times D^{n+1}\longrightarrow B$ is a $G$-map, $i$ is the inclusion, and the corresponding map $i$ over $\text{Map}_G(C,B)$ is a generating acyclic $qf$-cofibration in $\scr{K}/\text{Map}_G(C,B)$.
\end{enumerate}
Adjoining disjoint sections, we obtain the corresponding sets $I^f_B(\scr{C})$ and $J^f_B(\scr{C})$ in $G\scr{K}_B$.
\end{defn}
Fix a generating set $\scr{C}$. We define a $qf$-type model structure based on $\scr{C}$, called the $qf(\scr{C})$-model structure. Its weak equivalences are the $\scr{G}$-equivalences, which are the same as the $q$-equivalences when $B$ is a $\scr{G}$-space. We define the $qf(\scr{C})$-fibrations.
\begin{defn}\mylabel{qffibdef}
A map $f$ in $G\scr{K}/B$ is a \emph{$qf(\scr{C})$-fibration} if $\text{Map}_G(C,f)$ is a $qf$-fibration in $\scr{K}/\text{Map}_G(C,B)$ for all $C\in\scr{C}$. A map in $G\scr{K}_B$ is a \emph{$qf(\scr{C})$-fibration} if the underlying map in $G\scr{K}/B$ is one. In
either category, a map $f$ is a {\em $\scr{G}$-quasifibration} if $f^H$ is a quasifibration for $H\in \scr{G}$.
\end{defn}
\begin{thm}[$qf$-model structure]\mylabel{Gqfstr}
For any generating set $\scr{C}$, the categories $G\scr{K}/B$ and $G\scr{K}_B$ are well-grounded (hence $G$-topological) model categories. The weak equivalences and fibrations are the $\scr{G}$-equi\-valences and the $qf(\scr{C})$-fibrations. The sets $I^f_B(\scr{C})$ and $J^f_B(\scr{C})$ are the generating $qf(\scr{C})$-cofib\-rations and the generating acyclic $qf(\scr{C})$-cofibrations. All $qf(\scr{C})$-cofibrations are both $q$-cofibrations and $\bar{f}$-cofib\-rations, and all $qf(\scr{C})$-fibrations are $\scr{G}$-quasifibrations.
\end{thm}
\begin{proof} Recall from \myref{exwellgr} that the $q$-equivalences in $G\scr{K}/B$ and $G\scr{K}_B$ are well-grounded with respect to the ground structure given in \myref{exbackdef} and \myref{exback}. It follows that the $\scr{G}$-equivalences
are also well-grounded. It suffices to verify conditions (i)--(iv) of \myref{Newcompgen}. The acyclicity condition (i) is obvious.
Consider the compatibility condition (ii). By the adjunction (\ref{keyadj}), a map $f$ has the RLP with respect to $I^f_B(\scr{C})$ if and only if $\text{Map}_G(C,f)$ has the RLP with respect to $I^f_{\text{Map}_G(C,B)}$ for all $C\in\scr{C}$. By the compatibility condition for the nonequivariant $qf$-model structure, that holds if and only if $\text{Map}_G(C,f)$ is a $q$-equivalence and has the LLP with respect to $J^f_{\text{Map}_G(C,B)}$ for all $C\in\scr{C}$. By \myref{mapgood}, $\text{Map}_G(C,f)$ is a $q$-equivalence if $f$ is one. Conversely, if $\text{Map}_G(C,f)$ is a $q$-equivalence for all $C\in\scr{C}$, then the case $C=G/K$ shows that $f^K$ is a $q$-equivalence for every compact $K$ and thus $f$ is a $\scr{G}$-equivalence. By the adjunction again, we see that $f$ has the RLP with respect to $I^f_B(\scr{C})$ if and only if $f$ is a $\scr{G}$-equivalence which has the RLP with respect to $J^f_B(\scr{C})$.
The fiberwise NDR characterization of $\bar{f}$-cofibrations given in \myref{fNDR} shows that $I^f_B(\scr{C})$ and $J^f_B(\scr{C})$ consist of $\bar{f}$-cofibrations, as stipulated in (iii). More precisely, if $(u,h)$, $u\colon D^n\longrightarrow I$ and $h\colon D^n\times I \longrightarrow D^n$, represents $(D^n,S^{n-1})$ as a fiberwise NDR-pair over $\text{Map}_G(C,B)$, then the map $v=u\circ\pi\colon C\times D^n\longrightarrow D^n\longrightarrow I$ and the homotopy given by the maps $\text{id}\times h_t$ over $B$ corresponding to the $h_t$ represent $(C\times D^n,C\times S^{n-1})$ as a fiberwise NDR pair over $B$.
Since $\text{Map}_G(G/K,f)\cong f^K$ is a nonequivariant $qf$-fibration for any $qf(\scr{C})$-fibration $f$, every $qf(\scr{C})$-fibration is a $\scr{G}$-quasifibration by \myref{qfles}. That the model structure is right proper follows as in \myref{fdrp}.
Finally, we must verify the $\Box$-product condition (iv). The relevant maps $i\Box j$,
\[i\colon C\times S^{m-1}\longrightarrow C\times D^m\quad \text{and}\quad
j\colon G/H\times S^{n-1}\longrightarrow G/H\times D^n,\]
are of the form
\[C\times G/H \times k\colon C\times G/H \times \partial(D^m\times D^n)\longrightarrow
C\times G/H \times D^m\times D^n,\]
where $k$ is the boundary inclusion. Now $C\times G/H\in \scr{C}$ by the closure property of the generating set, so we don't need to triangulate. The projection of the target factors through the projection of the target $C\times D^m$ of $i$. To see that the corresponding map $k$ over $\text{Map}_G(C\times G/H,B)$ is an $\bar{f}$-cofibration, let $(u,h)$ represent $(D^m,S^{m-1})$ as a fiberwise NDR-pair over $\text{Map}_G(C,B)$ and let $(v,j)$ represent $(D^n,S^{n-1})$ as an NDR-pair; we can think of the latter as a fiberwise NDR-pair over $* = \text{Map}_G(G/H,*)$. Then the usual product pair representation (for example, \cite[p. 43]{Concise}) exhibits $k$ as a fiberwise NDR over $\text{Map}_G(C,B)\times \text{Map}_G(G/H,*)$ and thus, by the factorization of the projection of $i\Box j$, also over $\text{Map}_G(C\times G/H,B\times *)$.
\end{proof}
\begin{thm} If $\scr{C}\subset \scr{C}'$ is an inclusion of generating sets, then the identity functor is a left Quillen equivalence from $G\scr{K}/B$ with the $qf(\scr{C})$-model structure to $G\scr{K}/B$ with the $qf(\scr{C}')$-model structure. The identity functor is also a left Quillen equivalence from $G\scr{K}/B$ with the $qf(\scr{C})$-model structure to $G\scr{K}/B$ with the $q$-model structure. Both statements also hold for the identity functor on $G\scr{K}_B$.
\end{thm}
\begin{proof}
The first statement is obvious. For the second, if $\text{id}_C\times i$ is a generating $qf(\scr{C})$-cofibration, then $C$ is a $\scr{G}$-cell complex and we can use the triangulation to write $\text{id}_C\times i$ as a relative $I_B$-cell complex.
\end{proof}
\begin{thm}
For any $\scr{C}$, the identity functor is a left Quillen adjoint from $G\scr{K}/B$ with the $qf(\scr{C})$-model structure to $G\scr{K}/B$ with the $f$-model structure. Similarly, the identity functor is a left Quillen adjoint from $G\scr{K}_B$ with the $qf(\scr{C})$-model structure to $G\scr{K}_B$ with the $fp$-model structure.
\end{thm}
The last result, which implements the philosophy of \S4.1, is false for the
$q$-model structures.
\begin{rem}\mylabel{Theqf}
The smallest generating set $\scr{C}$ is the set of all (non-empty) finite products of orbits $G/H$ of $G$ such that at least one of the factors has $H$ compact.
Clearly it is a closed generating set.
Henceforward, by {\em the} $qf$-model structure, we mean the $qf(\scr{C})$-model structure associated to this choice of $\scr{C}$. In the nonequivariant
case, this is the $qf$-model structure of the previous chapter.
\end{rem}
\begin{rem}
In the nonparametrized setting, the $\scr{G}$-model structure associated to the
$q$-model structure and the $qf(\scr{C})$-model structures on $G\scr{K} = G\scr{K}/*$
coincide, and similarly for $G\scr{K}_*$. This holds since the $f$-cofibrations and $h$-cofibrations over a point coincide and since the $C\in\scr{C}$ for any choice
of $\scr{C}$ are $\scr{G}$-cell complexes. Of course, the $qf(\scr{C})$-model structures have more generating (acyclic) cofibrations.
\end{rem}
\begin{rem}
It might be useful to combine the various $qf(\scr{C})$-model structures by taking the union of the $qf(\scr{C})$-cofibrations over some suitable collection of generating sets $\scr{C}$ and so obtain a ``closure'' of the $qf$-model structure whose cofibrations are as close as possible to being the intersection of the $q$-cofibrations with the $\bar{f}$-cofibrations. We do not know whether or not
that can be done.
\end{rem}
\begin{rem}\mylabel{Fremark}
As noted in the introduction, we can generalize the $q$ and $qf(\scr{C})$-model structures to the context of families $\scr{F}$. We generalize the $q$-model structure to the $\scr{F}$-model structure by taking the $\scr{F}$-equivalences and $\scr{F}$-fibrations and by restricting the sets $I_B$ and $J_B$ to be constructed from orbits $G/H$ with $H\in \scr{F}$. The resulting model structure will then be $(\scr{F}\cap \scr{G})$-topological and $\scr{F}$-topological if the base space $B$ is a $\scr{G}$-space.
To generalize the $qf(\scr{C})$ model structure, we take the weak equivalences to be the $\scr{F}\cap\scr{G}$-equivalences and we require the generating set $\scr{C}$ to consist of $\scr{F}\cap\scr{G}$-cell complexes, to contain the orbits $G/K$ for $K\in \scr{F}\cap \scr{G}$, and to be closed under products with orbits $G/K$ where $K\in \scr{F}$. With that modification, everything else above goes through unchanged.
\end{rem}
\section{External smash product and base change adjunctions}\label{sec:qfadj}
The following results relate the $q$ and $qf(\scr{C})$-model structures to smash products and base change functors and show that various of our adjunctions are given by Quillen adjoint pairs and therefore induce adjunctions on passage to homotopy categories. For uniformity, we must understand the $q$-model structure to mean the associated $\scr{G}$-model structure, although many of the results do apply to the full $q$-model structure. Those results that refer to $q$-equivalences by name work equally well for $\scr{G}$-equivalences. Most of the results in this section and the next apply both to the $\scr{G}$-model structure
and to the $qf(\scr{C})$-model structure for any generating set $\scr{C}$. We agree to omit the $q$ or $qf(\scr{C})$ from the notations in those cases. In other cases, we will have to restrict to well chosen generating sets $\scr{C}$.
With these conventions, our first result is clear from the fact that our
model structures are $G$-topological.
\begin{prop}\mylabel{smaB}
For a based $G$-CW complex $K$, the functor $(-)\sma_B K$ preserves cofibrations and acyclic cofibrations, hence the functor $F_B(K,-)$ preserves fibrations and acyclic fibrations. Thus $((-)\sma_B K, F_B(K,-))$ is a Quillen adjoint pair of endofunctors of $G\scr{K}_B$.
\end{prop}
For the rest of our results, recall from \myref{reducts} that a left adjoint
that takes generating acyclic cofibrations to acyclic cofibrations preserves acyclic cofibrations. The following two results apply to the $qf(\scr{C})$-model structure for any closed generating set $\scr{C}$.
\begin{prop}\mylabel{Boxcof20}
If $i\colon X\longrightarrow Y$ and $j\colon W\longrightarrow Z$ are cofibrations over base $G$-spaces $A$ and $B$, then
$$i\Box j\colon (Y\barwedge W)\cup_{X\barwedge W}(X\barwedge Z)\longrightarrow Y\barwedge Z$$
is a cofibration over $A\times B$ which is acyclic if either $i$ or $j$ is acyclic.
\end{prop}
\begin{proof}
It suffices to inspect $i\Box j$ for generating (acyclic) cofibrations as was done for the case $A=*$ in the proof of \myref{Gqfstr}. For generating
cofibrations, the argument there generalizes without change to this setting.
The assumption that $\scr{C}$ is closed avoids the need for triangulations here.
For the acyclicity, it suffices to work in the $q$-model structure, for which
the conclusion is both more general and easier to prove. There it is easily
checked using triangulations of products of $\scr{G}$-cell complexes that if $i$ is
a generating cofibration and $j$ is a generating acyclic cofibration, then
$i\Box j$ is an acyclic cofibration.
\end{proof}
Of course, by \myref{ouchtoo}, the analogue for internal smash products fails. Taking $W =*_B$ and changing notations, we obtain the following special case.
\begin{cor}\mylabel{smaAB}
Let $Y$ be a cofibrant ex-space over $B$. Then the functor
$(-)\barwedge Y$
from ex-spaces over $A$ to ex-spaces over $A\times B$ preserves cofibrations
and acyclic cofibrations, hence the functor $\bar{F}(Y,-)$
from ex-spaces over $A\times B$ to ex-spaces over $A$ preserves
fibrations and acyclic fibrations. Thus
$((-)\barwedge Y,\, \bar{F}(Y,-))$ is a
Quillen adjoint pair of functors between $G\scr{K}_A$ and $G\scr{K}_{A\times B}$.
\end{cor}
The next two results apply to the $qf(\scr{C})$-model structures for any $\scr{C}$, provided that we use the same generating set $\scr{C}$ for both $G\scr{K}_A$ and $G\scr{K}_B$.
\begin{prop}\mylabel{Qad10}
Let $f\colon A\longrightarrow B$ be a $G$-map. Then the functor $f_!$ preserves cofibrations and acyclic cofibrations, hence $(f_{!},f^*)$ is a Quillen adjoint pair. The functor $f_!$ also preserves $q$-equivalences between well-sectioned ex-spaces. If $f$ is a $q$-fibration, then the functor $f^*$ preserves all
$q$-equivalences.
\end{prop}
\begin{proof}
If $(D,p)$ is a space over $A$, then $f_!((D,p)\amalg A)=(D,f\circ p)\amalg B$. Therefore $f_!$ takes generating (acyclic) $q$-cofibrations over $A$ to such maps over $B$. If $(u, h)$ represents $(D^n, S^{n-1})$ as a fiberwise NDR-pair over $\text{Map}_G(C,A)$, then, after composing the projection maps with
$\text{Map}_G(C,A)\longrightarrow \text{Map}_G(C,B)$,
it also represents $(D^n, S^{n-1})$ as a fiberwise NDR-pair over $\text{Map}_G(C,B)$. It follows that $f_!$ also preserves the generating (acyclic) $qf$-cofibrations. Recall that the well-sectioned ex-spaces are those that are $\bar{f}$-cofibrant and that $f$-cofibrations are $h$-cofibrations. Since $f_!X$ is defined by a pushout in $G\scr{K}$, the gluing lemma in $G\scr{K}$ implies that $f_!$ preserves $q$-equivalences between well-sectioned ex-spaces.
If $f$ is a $q$-fibration and $k\colon Y\longrightarrow Z$ is a $q$-equivalence of ex-spaces over $B$, consider the diagram
\[\xymatrix@=.4cm{
&& f^*Z \ar[rrr]\ar[dddl]|(.4)\hole &&& Z\ar[dddl]\\
f^*Y \ar[urr]^-{f^*k}\ar[ddr]\ar[rrr] &&& Y \ar[urr]^-k\ar[ddr]\\
\\
& A \ar[rrr]_f &&& B.}\]
The relation $(A\times_B Z)\times _Z Y\iso A\times_BY$ shows that the top square is a pullback, and the pullback $f^*Z\longrightarrow Z$ of $f$ is a $q$-fibration. Since the $q$-model structure on the category of $G$-spaces is right proper, it follows that $f^*k$ is a $q$-equivalence.
\end{proof}
\begin{prop}\mylabel{ffequiv0}
If $f\colon A\longrightarrow B$ is a $q$-equivalence, then $(f_{!},f^*)$ is a Quillen equivalence.
\end{prop}
\begin{proof} The conclusion holds if and only if the induced adjunction
on homotopy categories is an adjoint equivalence \cite[1.3.3]{Hovey}, so
it suffices to verify the usual defining condition for a Quillen
adjunction in either model structure. The condition for the other
model structure follows formally. We choose the $q$-model structure.
Let $X$ be a $q$-cofibrant ex-space over $A$ and $Y$ be a $q$-fibrant
ex-space over $B$, so that $A\longrightarrow X$ is a $q$-cofibration and $Y\longrightarrow B$
is a $q$-fibration of $G$-spaces. Since the model structure on the category
of $G$-spaces is left and right proper, inspection of the defining diagrams
in \myref{retract1} shows that the canonical maps $X\longrightarrow f_{!}X$ and $f^*Y\longrightarrow Y$ of total spaces are $q$-equivalences. For an ex-map
$k\colon f_!X\longrightarrow Y$ with adjoint $\tilde k\colon X\longrightarrow f^*Y$, the commutative diagram
$$\xymatrix{
X\ar[r] \ar[d]_{\tilde{k}} & f_!X \ar[d]^{k} \\
f^*Y \ar[r] & Y}$$
of total spaces then implies that $k$ is a $q$-equivalence if and only if $\tilde{k}$ is a $q$-equivalence.
\end{proof}
In view of \myref{noway}, we can at best expect only a partial and restricted analogue of \myref{Qad10} for $(f^*,f_*)$. We first give a result for the
$q$-model structure and then show how to obtain the analogue for the
$qf(\scr{C})$-model structures using well chosen generating sets $\scr{C}$.
\begin{prop}\mylabel{Qad20}
Let $f\colon A\longrightarrow B$ be a $G$-bundle such that $B$ is a $\scr{G}$-space and
each fiber $A_b$ is a $G_b$-cell complex. Then $(f^*,f_*)$ is a Quillen adjoint pair with respect to the $q$-model structures. Moreover, if the total space of an ex-$G$-space $Y$ over $B$ is a $\scr{G}$-cell complex, then so is the total space of $f^*Y$.
\end{prop}
\begin{proof}
Since $f$ is a $q$-fibration, $f^*$ preserves $q$-equivalences. It therefore suffices to show that $f^*$ takes generating cofibrations in $I_B$ to relative $I_A$-cell complexes. Observe first that if $\phi\colon G/H \longrightarrow B$ is a $G$-map with $\phi(eH)= b$, then $H\subset G_b$ and the pullback $G$-bundle $\phi^*f\colon f^*(G/H,\phi)\longrightarrow G/H$ of $f$ along $\phi$ is $G$-homeomorphic to $G\times_H A_b\longrightarrow G/H$. We can triangulate orbits in a $G_b$-cell decomposition of $A_b$ as $H$-CW complexes, by \myref{Illman}, and so give $A_b$ the structure of an $H$-cell complex. Then $G\times_H A_b$ has an induced structure of a $\scr{G}$-cell complex and thus so does $f^*(G/H,\phi)$.
For a space $d\colon E\longrightarrow B$ over $B$ with associated ex-space $E\amalg B$ over $B$, we have $f^*(E\amalg B) = f^*E\amalg A$. Let $E = G/H\times D^n$ and let $i\colon G/H\longrightarrow G/H\times D^n$ be the inclusion $i(gH)=(gH,0)$. The composite $d\circ i$ is a map $\phi$ as above. Since the identity map on
$G/H\times D^n$ is homotopic to the composite $i\circ \pi\colon G/H\times D^n\longrightarrow G/H\times D^n$, where $\pi$ is the projection, the pullback $G$-bundle $d^*f\colon f^*(E,d)\longrightarrow E$ is equivalent to the pullback bundle $(\phi\circ\pi)^*f\colon f^*(E,\phi\circ\pi)\longrightarrow E$. But the latter is the product of $\phi^*f\colon f^*(G/H,\phi)\longrightarrow G/H$ and the identity map of $D^n$ as we see from the following composite of pullbacks
\[\xymatrix{f^*(G/H\times D^n,\phi\com\pi) \ar[r]\ar[d]_{(\phi\circ\pi)^*f}
& f^*(G/H,\phi) \ar[r]\ar[d]_{\phi^*f} & f^*(G/H\times D^n,d) \ar[r]\ar[d]_{d^*f} & A\ar[d]^f\\
G/H\times D^n\ar[r]_-{\pi} & G/H \ar[r]_-i & G/H\times D^n\ar[r]_-d & B.}\]
The $\scr{G}$-cell structure on $f^*(G/H,\phi)$ gives a canonical decomposition of the inclusion $f^*(G/H,\phi) \times S^{n-1}\longrightarrow f^*(G/H,\phi) \times D^n$ as a relative $\scr{G}$-cell complex. The last statement follows by applying this analysis inductively to the cells of $Y$.
\end{proof}
The previous result fails for the $qf$-model structure. In fact, it already fails nonequivariantly for the unique map $f\colon A\longrightarrow *$, where $A$ is a
CW-complex. The proof breaks down when we try to use a
cell decomposition of $A$ (the fiber over $*$) to decompose cells
$A\times S^{n-1}\longrightarrow A\times D^n$ over $A$ as relative $I^f_A$-cell
complexes. Similarly, the equivariant proof above breaks down when we try to use the $G$-cell structure of $f^*(G/H,\phi)$ to obtain a relative $I^f_A$-cell complex. Note, however, that there is no problem when the fibers are
homogeneous spaces $G/H$; the nonequivariant analogue is just the trivial
case when $f$ is a homeomorphism, but principal bundles and projections
$G/H\times B\longrightarrow B$ give interesting equivariant examples.
For the general equivariant case, we choose a closed generating set $\scr{C}(f)$ that depends on the $G$-bundle $f$ and a given closed generating set $\scr{C}$. Using the $qf(\scr{C})$-model structures on $G\scr{K}_A$ and $G\scr{K}_B$, we then
recover the Quillen adjunction.
\begin{con}
Let $f\colon A\longrightarrow B$ be a $G$-bundle such that $B$ is a $\scr{G}$-space and
each fiber $A_b$ is a $G_b$-cell complex and let $\scr{C}$ be a closed generating set. We construct the set $\scr{C}(f)$ inductively. Let $\scr{C}(f)_0=\scr{C}$ and suppose that we have constructed a set $\scr{C}(f)_n$ of $\scr{G}$-cell complexes in $G\scr{K}$ that is closed under both finite products and products with arbitrary orbits $G/H$ of $G$. Let
\[\scr{A}_n=\{f^*(C,\phi)\mid \text{$C\in \scr{C}(f)_n$ and $\phi\in G\scr{K}(C,B)$}\}.\]
Then let $\scr{C}(f)_{n+1}$ consist of all finite products of spaces in $\scr{C}(f)_n\cup \scr{A}_n$. Note that $\scr{C}(f)_{n+1}$ contains $\scr{C}(f)_n$ and that the $f^*(C,\phi)$ are $\scr{G}$-cell complexes by the last statement of \myref{Qad20}. Finally, let $\scr{C}(f)=\bigcup \scr{C}(f)_n$. Clearly $\scr{C}(f)\supset \scr{C}$ is a closed
generating set that contains $f^*(C,\phi)$ for all $C\in \scr{C}(f)$ and all
$G$-maps $\phi\colon C\longrightarrow B$.
\end{con}
\begin{prop}\mylabel{Qad202}
Let $f\colon A\longrightarrow B$ be a $G$-bundle such that $B$ is a $\scr{G}$-space
and all fibers $A_b$ are $G_b$-cell complexes. Then $(f^*,f_*)$ is a
Quillen adjoint pair with
respect to the $qf(\scr{C}(f))$-model structures on $G\scr{K}_A$ and $G\scr{K}_B$.
\end{prop}
\begin{proof}
Reexamining the proof of \myref{Qad20}, but starting with a map $d\colon E=C\times D^n\longrightarrow B$ where $C\in \scr{C}(f)$, we see that
$$f^*E\iso f^*(C,\phi)\times D^n$$
where $\phi=d\circ i$. Since $f^*(C,\phi)$ is a $\scr{G}$-cell complex in $\scr{C}(f)$, it remains only to show that $f^*(C,\phi)\times S^{n-1}\longrightarrow f^*(C,\phi)\times D^n$ is an $f$-cofibration. Let $(u,h)$ represent $(D^n,S^{n-1})$ as a fiberwise NDR-pair over $\text{Map}_G(C,B)$. Applying $f^*$ to the corresponding maps $h_t\colon C\times D^n\longrightarrow C\times D^n$ over $B$, we obtain maps $f^*h_t\colon f^*E\longrightarrow f^*E$ over $A$. Under the displayed isomorphism, these maps give a homotopy $f^*h\colon D^n\times I\longrightarrow D^n$ that, together with $u$, represents $(D^n,S^{n-1})$ as a fiberwise NDR-pair over $\text{Map}_G(f^*(C,\phi),A)$.
\end{proof}
\section{Change of group adjunctions}\mylabel{sec:JHadj}
We consider change of groups in the $q$ and the $qf$-model structures, starting with the former. The context of the following results is given in \S\S2.3 and 2.4.
\begin{prop}\mylabel{grpres}
Let $\theta\colon G\longrightarrow G'$ be a homomorphism of Lie groups. The restriction of action functor
\[\theta^*\colon G'\scr{K}_B \longrightarrow G\scr{K}_{\theta^*B}\]
preserves $q$-equivalences and $q$-fibrations. If $B$ is a $\scr{G}'$-space, then it also preserves $q$-cofibrations.
\end{prop}
\begin{proof}
Since $(\theta^*A)^H=A^{\theta(H)}$ for any subgroup $H$ of $G$ and a map
$f:X\longrightarrow Y$ of $G$-spaces is a $q$-equivalence or $q$-fibration if and only if each $f^H$ is a $q$-equivalence or $q$-fibration, it is clear that $\theta^*$ preserves $q$-equivalences and $q$-fibrations. To study $q$-cofibrations, recall that $\theta$ factors as the composite of a quotient homomorphism, an isomorphism, and an inclusion. If $\theta$ is an inclusion and $H'$ is a compact subgroup of $G'$, then we can triangulate $G'/H'$ as a $G$-CW complex by \myref{Illman}. If $\theta$ is a quotient homomorphism with kernel $N$ and $H'$ is a subgroup of $G'$, then $H' = H/N$ for a subgroup $H$ of $G$ and $\theta^*(G'/H') = G/H$ so that no triangulations are required. Thus in both of these cases, $\theta^*$ takes generating $q$-cofibrations to $q$-cofibrations. Since $\theta^*$ is also a left adjoint in both cases, it preserves $q$-cofibrations in general.
\end{proof}
\begin{rem}
We did not require $\theta^*B$ to be a $\scr{G}$-space in \myref{grpres}.
However, if the kernel of $\theta$ is compact and $B$ is a $\scr{G}'$-space,
then $\theta^*B$ is a $\scr{G}$-space. Indeed, $\theta$ is then
a proper map and $G_b=\theta^{-1}(G'_b)$ is compact since $G'_b$ is compact.
The restriction to compact kernels is the price we must
pay in order to stay in the context of compact isotropy groups.
We might instead consider $G'$-spaces $B$ such that the isotropy groups
of both $B$ as a $G'$-space and $\theta^*B$ as a $G$-space are compact,
but the assumption on $\theta^*B$ would be unnatural. Note however that
one of the main reasons for restricting to compact isotropy groups is to
obtain $G$-CW structures. If $X$ is a $G'$-CW complex where $G' = G/N$
is a quotient group of $G$, then $\theta^*X$ is a $G$-CW complex with the same cells since the relevant orbits $G'/H'$ can be identified with $G/H$,
where $H' = H/N$.
\end{rem}
For the $qf$-model structures, and to study adjunctions, it is convenient to consider quotient homomorphisms and inclusions separately. For the
former, we consider the adjunctions of \myref{factor0}.
\begin{prop}\mylabel{fixedptrQa0}
Let $\epsilon\colon G\longrightarrow J$ be a quotient homomorphism of $G$ by a
normal subgroup $N$. For a $G$-space $B$,
consider the functors
\[(-)/N\colon G\scr{K}_B \longrightarrow J\scr{K}_{B/N} \ \ \text{and} \ \
(-)^N\colon G\scr{K}_B \longrightarrow J\scr{K}_{B^N}.\]
Let $j\colon B^N\longrightarrow B$ be the inclusion and $p\colon B\longrightarrow B/N$ be the quotient map. Then $((-)/N,p^*\epsilon^*)$ and $(j_!\epsilon^*,(-)^N)$ are Quillen adjoint pairs with respect to the $q$-model structures on
both $G\scr{K}_B$ and $J\scr{K}_{B/N}$. Let $\scr{C}_G$ and $\scr{C}_J$ be generating
sets of $G$-cell complexes and $J$-cell complexes. Consider $G\scr{K}_B$
with the $qf(\scr{C}_G)$-model structure and $J\scr{K}_{B/N}$ and $J\scr{K}_{B^N}$
with the $qf(\scr{C}_J)$-model structure. Then
\begin{enumerate}[(i)]
\item $((-)/N,p^*\epsilon^*)$ is a Quillen adjunction if $C/N\in\scr{C}_J$ for $C\in \scr{C}_G$.
\vspace{1mm}
\item $(j_!\epsilon^*,(-)^N)$ is a Quillen adjunction if $\varepsilon^*C\in \scr{C}_G$ for $C\in \scr{C}_J$.
\end{enumerate}
\end{prop}
\begin{proof} Since $(j_!,j^*)$ and $(p_!,p^*)$ are Quillen adjoint pairs
in both the $q$ and the $qf$ contexts, it suffices to consider the case
when $N$ acts trivially on $B$, so that $j$ and $p$ are identity maps.
Then $\varepsilon^*$ is right
adjoint to $(-)/N$ and left adjoint to $(-)^N$. The properties of $\varepsilon^*$
in the previous result give the conclusion for the $q$-model structures.
The functors $\varepsilon^*$ and $(-)^N$ preserve $q$-equivalences. Since
$$ \text{Map}_G(C,\varepsilon^*f')\cong\text{Map}_G(C/N,f') \ \
\text{and} \ \ \text{Map}_J(C',f^N)\cong\text{Map}_G(\epsilon^*C',f)$$
for a $J$-map $f'$ and a $G$-map $f$, the conditions on generating sets in
(i) and (ii) ensure that $\varepsilon^*$ and $(-)^N$ preserve the relevant $qf$-fibrations.
\end{proof}
\begin{rem} In (i), we can take $\scr{C}_J$ to consist of all finite products of quotients $C/N$ with $C\in \scr{C}_G$ and orbits $J/H$ to arrange that $\scr{C}_J$
be closed and contain these $C/N$. In (ii), we can take $\scr{C}_G$ to
consist of all products of pullbacks $\varepsilon^*C$ for $C\in\scr{C}_J$ with finite
products of orbits $G/H$. This set will be closed if $\scr{C}_J$ is closed
since $\varepsilon^*$ preserves products.
\end{rem}
Using \myref{fixedptrQa0} in conjunction with the additional change of
group relations of Propositions \ref{fixorbbase} and \ref{ouch0}, we
obtain the following compendium of equivalences in homotopy categories.
\begin{prop}\mylabel{orbfixdescend}
Let $A$ and $B$ be $G$-spaces. Let $j\colon B^N\longrightarrow B$ be the inclusion and $p\colon B\longrightarrow B/N$ be the quotient map, and let $f\colon A\longrightarrow B$ be
a $G$-map. Then, for ex-$G$-spaces $X$ over $A$ and $Y$ over $B$,
\begin{alignat*}{3}
&(p_!Y)/N \simeq Y/N, &\qquad\qquad
&(f_!X)/N \simeq (f/N)_!(X/N),\\
&(j^*Y)^N \simeq Y^N, &\qquad\qquad
&(f^*Y)^N \simeq (f^N)^*(Y^N), \\
&(p_*Y)^N \simeq Y/N, &\qquad\qquad
&(f_!X)^N \simeq (f^N)_!(X^N),
\end{alignat*}
where, for the last equivalence on the left, $B$ must be an $N$-free $G$-space.
\end{prop}
\begin{proof}
The equivalences displayed in the first line come from isomorphisms between Quillen left adjoints and are therefore clear. Similarly the equivalences in the second line come from isomorphisms between Quillen right adjoints. The first equivalence in the third line (in which we have changed notations from
\myref{ouch0}) comes from an isomorphism between a Quillen right adjoint on the left hand side, by \myref{Qad20}, and a Quillen left adjoint on the right hand side and therefore also descends directly to an equivalence on homotopy categories. For the last equivalence, note that $(-)^N$ preserves all $q$-equivalences and also preserves well-grounded ex-spaces and that $(f^N)_!$ preserves $q$-equivalences between well-grounded ex-spaces. Letting $Q$ and $R$ denote cofibrant and fibrant replacement functors, as usual, it follows that the maps
\[(R(f_!X))^N \longleftarrow (f_!X)^N \cong (f^N)_!(X^N) \longleftarrow (f^N)_!(Q(X^N)) \]
are $q$-equivalences on ex-spaces $X$ that are $qf$-fibrant and $qf$-cofibrant.
As noted in the proof of \myref{fixorbbase}, the point set level isomorphism $(f_!X)^N\iso (f^N)_!(X^N)$ is only valid for an ex-space $X$ whose section is a closed inclusion. However, if $X$ is $qf$-cofibrant, then it is compactly generated and this holds by \myref{coflemma}(i). Thus the equivalence holds
in general in the homotopy category.
\end{proof}
The context for the next result is given in \myref{changes0} and \myref{ishriek}.
\begin{prop}\mylabel{Lishriek}
Let $\iota\colon H\longrightarrow G$ be the inclusion of a subgroup and let $A$ be
an $H$-space. The adjoint equivalence $(\iota_!,\nu^*\iota^*)$ relating $H\scr{K}_A$ and $G\scr{K}_{\iota_!A}$ is a Quillen equivalence in the $q$-model structures and
also in the $qf(\scr{C}_H)$ and $qf(\scr{C}_G)$-model structures for any generating
sets $\scr{C}_H$ and $\scr{C}_G$ of $H$-cell complexes and $G$-cell complexes such
that $\iota_!C = G\times_H C \in \scr{C}_G$ for $C\in\scr{C}_H$.
If $A$ is proper and completely regular, then the functor $\iota_!$ is
also a Quillen right adjoint with respect to the $q$ and $qf$-model
structures.
\end{prop}
\begin{proof}
Recall that $\nu\colon A\longrightarrow \iota^*\iota_!A= G\times_H A$ is the natural inclusion of $H$-spaces and that $(\nu_!,\nu^*)$ is a Quillen adjunction in both
the $q$ and $qf$ contexts. The functor $\iota^*$ preserves $q$-equivalences and
$q$-fibrations. It takes $qf(\scr{C}_G)$-fibrations to $qf(\scr{C}_H)$-fibrations
when $\iota_!C \in \scr{C}_G$ for $C\in\scr{C}_H$ since
$$\text{Map}_H(C,\iota^*f)\cong\text{Map}_G(\iota_!C,f).$$
To show that $(\iota_!,\nu^*\iota^*)$ is a Quillen equivalence, we may
as well check the defining condition in the $q$-model structure.
Let $X$ be a $q$-cofibrant ex-$H$-space over $A$ and $Y$ be a $q$-fibrant ex-$G$-space over $\iota_!A$. Consider a $G$-map $f\colon \iota_!X\longrightarrow Y$. We must show that $f$ is a $q$-equivalence if and only if its adjoint $H$-map
$\tilde{f}\colon X\longrightarrow \nu^*\iota^*Y$ is a $q$-equivalence. Since $\iota_!$ preserves acyclic $q$-cofibrations, we can extend $f$ to $f'\colon \iota_!RX\longrightarrow Y$, where $RX$ is a $q$-fibrant approximation. Since $f'$ is a $q$-equivalence if and only if $f$ is one, and similarly for their adjoints, we may assume without loss of generality that $X$ is $q$-fibrant. Recall from \myref{ishriek} that $\iota_!$ and $\nu^*\iota^*$ are inverse equivalences of categories and observe that $\nu^*\iota^*Y$ can be viewed as the restriction, $Y|_A$, of $Y$ along the inclusion of $H$-spaces $\nu\colon A\longrightarrow G\times_H A$. From that point of view, $\tilde{f}\colon X \longrightarrow \nu^*\iota^*Y$ is just the map $X\longrightarrow Y|A$ of ex-$H$-spaces over $A$ obtained by restriction of $\iota^*f$ along $\nu$.
Now $f$ is a $q$-equivalence if and only if $f$ restricts to a $q$-equivalence $f_{[g,a]}$ on each fiber, meaning that this restriction is a weak equivalence after passage to fixed points under all subgroups of the isotropy group of $[g,a]$. For $a\in A$, the isotropy subgroup $H_a\subset H$ of $a$ coincides with the isotropy subgroup $G_{[e,a]}\subset G$ of $[e,a]\in G\times_H A$. For $g\in G$, the isotropy subgroup of $[g,a]$ is $gH_a g^{-1}$. Since the action by $g\in G$ induces a homeomorphism between the fibers over $[e,a]$ and over $[g,a]$, we see that $f$ is a $q$-equivalence if and only if each of the restrictions $f_{[e,a]}$ is a $q$-equivalence. But that holds if and only if $\tilde{f}$ is a $q$-equivalence.
For the last statement, recall the description of $\iota_!$ in
\myref{iotaalt} as the composite $(p_*\pi^*\varepsilon^*(-))^H$, where
$\varepsilon\colon G\times H\longrightarrow H$ and $\pi\colon G\times A\longrightarrow A$ are the projections and $p\colon G\times A\longrightarrow G\times_H A$ is the quotient map.
Since $G\times A$ is completely regular, $p$ is a bundle with fiber
$G/H_a$ over $[g,a]$, and $H_a$ is compact since $A$ is
proper. Therefore, by Propositions \ref{Qad20} and \ref{Qad202},
$p_*$ is a Quillen right adjoint with respect to the $q$ and $qf$-model structures. In view of \myref{fixedptrQa0}, this displays $\iota_!$ as a composite of Quillen right adjoints.
\end{proof}
\begin{rem} We can take $\scr{C}_G$ to consist of all finite products of
the $\iota_!C$ with $C\in \scr{C}_H$ and orbits $G/K$ to arrange that
$\scr{C}_G$ be closed and contain these $\iota_!C$.
\end{rem}
We shall prove that $(\iota_!,\nu^*\iota^*)$ descends to a closed symmetric
monoidal equivalence of homotopy categories in \myref{imonoidaldescends} below.
The first statement of \myref{Lishriek} implies that the description of $\iota^*$ in terms of base change that is given in \myref{ishriekb} descends to homotopy categories.
\begin{cor}\mylabel{LishriekCor}
The functor $\iota^*\colon \text{Ho}G\scr{K}_B\longrightarrow \text{Ho}H\scr{K}_{\iota^*B}$ is the composite
\[\xymatrix@1
{\text{Ho} G\scr{K}_B \ar[r]^-{\mu^*} & \text{Ho}G\scr{K}_{\iota_!\iota^*B} \htp \text{Ho} H\scr{K}_{\iota^*B}\\}\]
\end{cor}
\section{Fiber adjunctions and Brown representability}
For a point $b$ in $B$, we combine the special case
$\tilde{b}\colon G/G_b\longrightarrow B$ of \myref{Qad10} with the special
case $\iota\colon G_b\longrightarrow G$ and $A = *$, hence $\nu\colon *\longrightarrow G/G_b$,
of \myref{Lishriek} to obtain the following result concerning passage
to fibers. Recall from
\myref{Johann0} that the fiber functor $(-)_b\colon G\scr{K}_B\longrightarrow G_b\scr{K}_*$ is given by $\nu^*\iota^*\tilde{b}^*=b^*\iota^*$. By conjugation, its left adjoint $(-)^b$ therefore agrees with $\tilde{b}_!\iota_!$.
\begin{prop}\mylabel{FibadQ0}
For $b\in B$, the pair of functors $((-)^b,(-)_b)$ relating $G_b\scr{K}_*$ and $G\scr{K}_B$ is a Quillen adjoint pair.
\end{prop}
We use this result to verify the formal hypotheses of Brown's
representability theorem \cite{Brown} for the category $\text{Ho}G\scr{K}_B$.
Of course, this verification is independent of the choice of model structure.
The category $G\scr{K}_B$ has coproducts and homotopy pushouts, hence
homotopy colimits of directed sequences. The usual constructions of homotopy
pushouts as double mapping cylinders and of directed homotopy colimits as
telescopes makes clear that if the total spaces of their inputs are compactly
generated, as they are after $q$-cofibrant approximation, then so are the
total spaces of their outputs. We need a few preliminaries.
\begin{defn}\mylabel{detectset}
For $n\geq 0$, $b\in B$, and $H\subset G_b$, let $S^{n,b}_H$\@bsphack\begingroup \@sanitize\@noteindex{SHnb@$S^{n,b}_H$} be the ex-$G$-space
$((G_b/H\times S^n)_+)^b$ over $B$. Explicitly, by \myref{Fibad}, $S^{n,b}_H = (G/H\times S^n)\amalg B$, with the obvious section and with the projection that maps $G/H\times S^n$ to the point $b$ and maps $B$ by the identity map. Equivalently, taking $d$ to be the constant map at $b$, $S^{n,b}_H$ is the quotient ex-$G$-space associated to the generating cofibration $i(d)$, $i\colon G/H\times S^{n-1}\longrightarrow G/H\times D^n$. Therefore, $S^{n,b}_H$ is cofibrant in both the $q$ and the $qf$-model structures. Let $\scr{D}_B$\@bsphack\begingroup \@sanitize\@noteindex{DB@$\scr{D}_B$} be the ``detecting set'' of all such ex-$G$-spaces $S^{n,b}_H$.
\end{defn}
Let $[X,Y]_{G,B}$ denote the set of maps $X\longrightarrow Y$ in $\text{Ho}G\scr{K}_B$.
\begin{lem} Each $X$ in $\scr{D}_B$ is compact, in the sense
that
$$\text{colim}\, [X, Y_n]_{G,B}\iso [X, \text{hocolim}\,Y_n]_{G,B}$$
for any sequence of maps $Y_n\longrightarrow Y_{n+1}$ in $G\scr{K}_B$.
\end{lem}
\begin{proof} If $X = S^{n,b}_H$, then
$[X,Y]_{G,B} \iso [G_b\times S^n)_+,Y_b]_{G_b}$.
In $G_b\scr{K}_*$, every object is fibrant and the target is the set of
homotopy classes of $G_b$-maps $(G_b\times S^n)_+ \longrightarrow Y_b$, which
is the set of unbased nonequivariant homotopy classes of maps $S^n\longrightarrow Y_b$.
Using cofibrant replacement, we can arrange that the $(Y_n)_b$ have total spaces
in $\scr{U}$. Then the conclusion follows from \myref{little}.
\end{proof}
The following result says that the set $\scr{D}_B$ detects $q$-equivalences.
\begin{prop} A map $\xi\colon Y\longrightarrow Z$ in $G\scr{K}_B$ is a $q$-equivalence
if and only if the induced map $\xi_*\colon [X,Y]_{G,B}\longrightarrow [X,Z]_{G,B}$ is a bijection for all $X\in \scr{D}_B$.
\end{prop}
\begin{proof} We may assume that $Y$ and $Z$ are fibrant. By the evident long exact sequences of homotopy groups and the five lemma, $\xi$
is a $q$-equivalence if and only if each $Y_b\longrightarrow Z_b$ is a $q$-equivalence.
This is detected by the based $G_b$-spaces $(G_b/H\times S^n)_+$ and the
conclusion follows by adjunction.
\end{proof}
\begin{thm}[Brown]\mylabel{brown0} A contravariant set-valued functor on the
category $\text{Ho}G\scr{K}_B$ is
representable if and only if it satisfies the wedge and Mayer-Vietoris
axioms.
\end{thm}
\chapter{Ex-fibrations and ex-quasifibrations}
To complete the foundations of parametrized homotopy theory,
we are faced with two problems that were discussed in the
Prologue. In our preferred $qf$-model structure, the base change
adjunction $(f_!, f^*)$ is a Quillen pair for any map $f$ and is a
Quillen equivalence if $f$ is an equivalence. As shown by \myref{noway},
this implies that the base change adjunction $(f^*,f_*)$ cannot be a
Quillen adjoint pair. Some such defect must hold for any model structure.
Therefore, we cannot turn to model theory to construct the functor $f_*$
on the level of homotopy categories. The same counterexample illustrates that passage to derived functors is not functorial in general, so that a relation between composites of functors that holds on the point-set level need not
imply a corresponding relation on homotopy categories.
In any attempt to solve those two problems, one runs into a third one that concerns a basic foundational problem in ex-space theory. Model theoretical considerations lead to the use of Serre fibrations as projections, or to the even weaker class of $qf$-fibrations. However, only Hurewicz fibrations are considered in most of the literature. There is good reason for that.
Fiberwise smash products, suspensions, cofibers, function spaces, and other fundamental constructions in ex-space theory do not preserve Serre fibrations.
The solutions to all three problems are obtained by the use of ex-fibrations.
Recall that these are the well-sectioned $h$-fibrant ex-spaces. We study
their properties in \S8.1. They seem to give the definitively right kind of ``fibrant ex-space'' from the point of view of classical homotopy theory, and they behave much better under the cited constructions than do Serre fibrations,
as we show in \S8.2. Many variants of this notion appear in the literature. Precisely this variant, with this name, appears in Monica Clapp's paper \cite{Clapp}, and we are indepted to her work for an understanding of the centrality of the notion. Perversely, as we noted in \myref{guess}, it is unclear how it fits into the model categorical framework.
We construct an elementary ex-fibrant approximation functor in \S8.3. It plays a key role in bridging the gap between the model theoretic and classical worlds. In a different context, the classification of sectioned fibrations, the first author introduced this construction in \cite[\S5]{May}. We record
some its properties in \S8.4.
We define quasifibrant ex-spaces and ex-quasifibrations and show that they inherit some of the good properties of ex-fibrations in \S8.5. They will
play a key role in the stable theory.
Everything in this chapter works just as well equivariantly as nonequivariantly
for any topological group $G$ of equivariance.
\section{Ex-fibrations}
Under various names, the following notions were in common use in the 1970's.
We shall see shortly that these definitions agree with those given in \myref{names}.
\begin{defn}\mylabel{well}
Let $(X,p,s)$ be an ex-space over $B$.
\begin{enumerate}[(i)]
\item $(X,p,s)$ is \emph{well-sectioned}\index{ex-space!well-sectioned}\index{well-sectioned ex-space} if $s$ is a closed inclusion and there is a retraction
$$\rho\colon X\times I\longrightarrow X\cup_B(B\times I) = Ms$$
over $B$.
\item $(X,p,s)$ is \emph{well-fibered}\index{ex-space!well-fibered}\index{well-fibered ex-space} if there is a coretraction, or \emph{path-lifting function},\index{path-lifting function}
$$\iota\colon Np = X\times_B B^I \longrightarrow X^I$$
under $B^I$, where $B^I$ maps to $Np$ via $\alpha \longrightarrow (s\alpha(0),\alpha)$.
\item $(X,p,s)$ is an \emph{ex-fibration}\index{ex-fibration} if it is both well-sectioned
and well-fibered.
\end{enumerate}
\end{defn}
The requirement in (i) that the retraction $\rho$ be a map over $B$ ensures that it restricts on fibers to a retraction that exhibits the nondegeneracy of the basepoint $s(b)$ in $X_b$ for each $b\in B$. In view of \myref{ffpmodel}(i),
we have the following characterization of well-sectioned ex-spaces, in
agreement with \myref{names}.
\begin{lem}\mylabel{charcof}
An ex-space $X$ is well-sectioned if and only if $X$ is $\bar{f}$-cofibrant.
\end{lem}
We use the term ``well-sectioned'' since it goes well with ``well-based''.
The category of well-sectioned ex-spaces is the appropriate para\-me\-trized generalization of the category of well-based spaces, and restricting to well-sectioned ex-spaces is analogous to restricting to well-based spaces.
Note that the section of $X$ provides a canonical way of lifting a path in $B$
that starts at $b$ to a path in $X$ that starts at $s(b)$. The requirement in \myref{well}(ii) that the path-lifting function $\iota$ be a map under $B^I$ says that $\iota(s\alpha(0),\alpha)(t) = s(\alpha(t))$ for all $\alpha\in B^I$ and $t\in I$. That is, $\iota$ is required to restrict to the canonical lifts provided by the section, so that paths in $X$ that start in $s(B)$ remain in $s(B)$. In
contrast with \myref{charcof}, the well-fibered condition does not by itself fit naturally into the model theoretic context of Chapter 5. However, we have the following characterization of ex-fibrations, which again is in agreement with the original definition we gave in \myref{names}.
\begin{lem}\mylabel{charexfib}
If $X$ is well-fibered, then $X$ is $h$-fibrant.
If $X$ is well-sectioned, then $X$ is an ex-fibration if and only
if $X$ is $h$-fibrant.
\end{lem}
\begin{proof}
The first statement is clear since the coretraction $\iota$ is a path-lifting function. This gives the forward implication of the second statement, and the
converse is a special case of the following result of Eggar \cite[3.2]{Eggar}.
\end{proof}
\begin{lem}\mylabel{Eggar}
Let $i\colon X\longrightarrow Y$ be an $\bar{f}$-cofibration of
ex-spaces over $B$, where $Y$ is $h$-fibrant. Then any map
$\iota\colon X\times_B B^I\longrightarrow Y^I$ such that the composite
$$\xymatrix@1{
X\times_B B^I \ar[r]^-{\iota} & Y^I \ar[r] & Y\times_B B^I\\}$$
is the inclusion can be extended to a coretraction $Y\times_B B^I\longrightarrow Y^I$.
\end{lem}
\begin{proof}
The inclusion $X\times_B B^I \longrightarrow Y\times_B B^I$ is an
$\bar{h}$-cofibration by \myref{StrPull}. Therefore
there is a lift ${\nu}$ in the diagram
$$\xymatrix{
(Y\times_B B^I)\times\{0\} \cup (X\times_B B^I)\times I \ar[r]^-{f} \ar[d]
& Y \ar[d]\\
(Y\times_B B^I)\times I \ar[r]_-{g} \ar@{-->}[ur]^{\nu} & B,\\}$$
where $f(y,\omega,0) = y$, $f(x,\omega,t) = \iota(x,\omega)(t)$, and $g(y,\omega,t)= \omega(t)$.
The adjoint $Y\times_B B^I\longrightarrow Y^I$ of $\nu$ is the required extension to a
coretraction.
\end{proof}
\begin{rem}
We comment on the terminology.
(1) We are following \cite{CJ, James} and others in saying that an
$\bar{f}$-cofibrant ex-space is well-sectioned; the term ``fiberwise well-pointed'' is also used. For a based space, the terms ``nondegenerately based'' and ``well-based'' or ``well-pointed'' are used interchangeably to mean that the inclusion of the basepoint is an $h$-cofibration. In contrast, for an ex-space, the term ``fiberwise nondegenerately pointed'' is used in \cite{CJ, James} to indicate a somewhat weaker condition than well-sectioned.
(2) The term ``well-fibered'' is new but goes naturally with well-sec\-tion\-ed. The concept itself is old. We believe that it is due to Eggar \cite[3.3]{Eggar}, who calls a coretraction under $B^I$ a {\em special lifting function}.
(3) Becker and Gottlieb \cite{BG1} may have been the first to use the term ``ex-fibration'', but for a slightly different notion with sensible CW restrictions. As noted in the introduction, precisely our notion is used by Clapp \cite{Clapp}. Earlier, in \cite[\S5]{May} and \cite{May1}, the first author called ex-fibrations ``$\scr{T}$-fibrations'', and he studied their classification and their fiberwise localizations and completions. The equivariant generalization appears in Waner \cite{Waner}. A more recent treatment of the classification of ex-fibrations has been given by Booth \cite{Boothbrag}.
\end{rem}
\section{Preservation properties of ex-fibrations}
We have a series of results that show that ex-fibrations behave well with respect to standard constructions. In some of them, one must use the equivariant version of \myref{fNDR} to verify that the given construction preserves well-sectioned objects. In all of them, if we only assume that the input ex-spaces are well-sectioned, then we can conclude that the output ex-spaces are well-sectioned. It is the fact that the given constructions preserve well-fibered objects that is crucial. Few if any of these results hold with Serre rather
than Hurewicz fibrations as projections.
\begin{prop}\mylabel{pres1}
Ex-fibrations satisfy the following properties.
\begin{enumerate}[(i)]
\item A wedge over $B$ of ex-fibrations is an ex-fibration.
\item If $X$, $Y$ and $Z$ are ex-fibrations and $i$ is an $\bar{f}$-cofibration in the following pushout diagram of ex-spaces over $B$, then $Y\cup_X Z$ is an ex-fibration.
$$\xymatrix{
X \ar[r]^{i} \ar[d] & Y \ar[d] \\
Z \ar[r] & Y\cup_X Z}$$
\item The colimit of a sequence of $\bar{f}$-co\-fi\-bra\-tions $X_i\longrightarrow X_{i+1}$ between ex-\-fi\-bra\-tions is an ex-fibration.
\end{enumerate}
If the input ex-spaces are only assumed to be well-sectioned, then the output ex-spaces are well-sectioned.
\end{prop}
\begin{proof} The last statement is clear. Using it, we see that the
colimits in (i), (ii), and (iii) are well-sectioned, hence it suffices
to prove that they are $h$-fibrant. This is done by constructing path
lifting functions for the colimits from path lifting functions for their
inputs. In (i), we start with path lifting functions under $B^I$ for
the wedge summands and see that they glue together to define a path
lifting function under $B^I$ for the wedge. Part (ii) is due to Clapp \cite[1.3]{Clapp}, and we omit full details. She starts with a path lifting function for $X$ and uses \myref{Eggar} to extend it to a path lifting
function for $Y$. She also starts with a path lifting function for $Z$.
She then uses a representation $(h,u)$ of $(Y,X)$ as a fiberwise NDR pair
to build a path lifting function for the pushout from the given path
lifting function for $Z$ and a suitably deformed version of the path
lifting function for $Y$. In (iii), \myref{Eggar} shows that we can
extend a path lifting function for $X_i$ to a path lifting function
for $X_{i+1}$. Inductively, this allows the construction of compatible
path lifting functions for the $X_i$ that glue together to give a path
lifting function for their colimit.
\end{proof}
Although of little use to us, since the $f$-homotopy category is not the right one for our purposes, many of our adjunctions give Quillen adjoint pairs with respect to the $f$-model structure. For example, the following result, which should be compared with \myref{Qad10}, implies that $(f_!,f^*)$ is a Quillen adjoint pair in the $f$-model structures and that it is a Quillen equivalence
if $f$ is an $h$-equivalence.
\begin{prop}\mylabel{fexpres}
Let $f\colon A\longrightarrow B$ be a map, let $X$ be a well-sectioned ex-space over $A$, and let $Y$ be a well-sectioned ex-space over $B$. Then $f_!X$ and $f^*Y$ are well-sectioned. If $Y$ is an ex-fibration, then so is $f^*Y$, and the functor $f^*$ preserves $f$-equivalences. If $f$ is an $h$-equivalence, then $(f_!,f^*)$ induces an equivalence of $f$-homotopy categories.
\end{prop}
\begin{proof}
It is easy to check that representations of $(X,A)$ and $(Y,B)$ as fiberwise NDR-pairs induce representations of $(f_!X,B)$ and $(f^*Y,A)$ as fiberwise NDR-pairs. As a pullback, the functor $f^*$ preserves both $f$-fibrant and $h$-fibrant ex-spaces, and $f^*$ preserve $f$-equivalences since it preserves $f$-homotopies. For the last statement, if $f$ is a homotopy equivalence with homotopy inverse $g$, then standard arguments with the CHP imply that $f^*$ induces an equivalence of $f$-homotopy categories with inverse $g^*$; see, for example, \cite[2.5]{May}. It follows that $g^*$ is equivalent to $f_!$ and that $(f_!,f^*)$ is a Quillen equivalence.
\end{proof}
The following result appears in \cite{Eggar} and \cite[3.6]{May}. It also leads to a Quillen adjoint pair with respect to the $f$-model structure; compare \myref{smaAB}.
\begin{prop}\mylabel{Hursma}
Let $X$ and $Y$ be well-sectioned ex-spaces over $A$ and $B$. Then
$X\barwedge Y$ is a well-sectioned ex-space over $A\times B$. If $X$
and $Y$ are ex-fibrations, then $X\barwedge Y$ is an ex-fibration.
\end{prop}
\begin{proof}
Representations of $(X,A)$ and $(Y,B)$ as fiberwise NDR-pairs determine a representation of $(X\barwedge Y, A\times B)$ as a fiberwise NDR-pair, by standard formulas \cite[p.\,43]{Concise}. Similarly, path lifting functions for $X$ and $Y$ can be used to write down a path lifting function for
$X\barwedge Y$.
\end{proof}
\begin{cor}
If $X$ and $Y$ are ex-fibrations over $B$, then so is $X\sma_B Y$.
\end{cor}
\begin{cor}\mylabel{HursmaK}
If $X$ is an ex-fibration over $B$ and $K$ is a well-based space, then $X\sma_B K$ is an ex-fibration over $B$.
\end{cor}
\begin{prop}\mylabel{savior}
Let $X$ and $Y$ be well-sectioned and let $f\colon X\longrightarrow Y$ be an ex-map that is an $h$-equivalence. Then $f\sma_B\text{id}\colon X\sma_B Z\longrightarrow Y\sma_B Z$ is an $h$-equivalence for any ex-fibration $Z$. In particular, $f\sma_B\text{id}\colon X\sma_B K\longrightarrow Y\sma_B K$ is an $h$-equivalence for any well-based space $K$.
\end{prop}
\begin{proof}
As observed by Clapp \cite[2.7]{Clapp}, this follows from the gluing lemma by comparing the defining pushouts.
\end{proof}
As in ordinary topology, function objects work less well, but we do have the following analogue of \myref{HursmaK}.
\begin{prop}
If $X$ is an ex-fibration over $B$ and $K$ is a compact well-based space, then $F_B(K,X)$ is an ex-fibration over $B$.
\end{prop}
\begin{proof}
Let $(h,u)$ represent $(X,B)$ as a fiberwise NDR-pair. Then $(j,v)$ represents $(F_B(K,X),B)$ as a fiberwise NDR-pair, where
\[v(f) = \text{sup}_{k\in K}u(f(k)) \qquad\text{and}\qquad
j_t(f)(k) = h_t(f(k))\]
for $f\in F_B(K,X)$. Note for this that $F_B(K,B)=B$ and that, by \myref{Ffib}, $F_B(K,X)$ is $h$-fibrant.
\end{proof}
\section{The ex-fibrant approximation functor}
We describe an elementary ex-fibrant replacement functor ${P}$. It is just
the composite of a whiskering functor $W$ with a version of the mapping path fibration functor $N$. The functor $P$ replaces ex-spaces by naturally $h$-equivalent ex-fibrations. From the point of view of model theory, ${P}$ can be thought of as
a kind of $q$-fibrant replacement functor that gives Hurewicz fibrations rather than just Serre fibrations as projections. The nonequivariant version of $P$ appears in \cite[5.3, 5.6]{May}, and the equivariant version appears in \cite[\S3]{Waner}. With motivation from the theory of transports in fibrations, those sources work with Moore paths of varying length. Surprisingly, that choice turns out to be essential for the construction to work.
We therefore begin by recalling that the space of \emph{Moore paths}\index{Moore paths} in $B$ is given by\@bsphack\begingroup \@sanitize\@noteindex{LB@$\Lambda B$}
\[\Lambda B = \{(\lambda, l)\in B^{[0,\infty]}\times [0,\infty) \mid \text{$\lambda(r)=\lambda(l)$ for $r\geq l$}\}\]
with the subspace topology. We write $\lambda$ for $(\lambda,l)$ and
$l_{\lambda}$ for $l$, which is the length of $\lambda$.
Let $e\colon \Lambda B \longrightarrow B$ be the endpoint projection $e(\lambda)=\lambda(l_{\lambda})$. The composite of Moore paths $\mu$ and $\lambda$ such that $\lambda(l_\lambda)=\mu(0)$ is defined by $l_{\mu\lambda}=l_\mu+l_\lambda$ and
\[(\mu\lambda)(r)=\begin{cases}
\lambda(r) &\text{if $r\leq l_\lambda$},\\
\mu(r-l_\lambda) &\text{if $r\geq l_\lambda$}.
\end{cases}\]
Embed $B$ and $B^I$ in $\Lambda B$ as the paths of length $0$ and $1$.
For a Moore path $\lambda$ in $B$ and real numbers $u$ and $v$ such that
$0\leq u\leq v$, let $\lambda|_u^v$ denote the Moore path $r\mapsto \lambda(u+r)$ of length $v-u$.
\begin{defn}
Consider an ex-space $X = (X,p,s)$ over $B$.
\begin{enumerate}[(i)]
\item Define the \emph{whiskering functor}\index{whiskering functor}\index{functor!whiskering --} $W$\@bsphack\begingroup \@sanitize\@noteindex{WX@$WX$} by letting
\[WX=(X\cup_B (B\times I),q,t),\]
where the pushout is defined with respect to $i_0\colon B\longrightarrow B\times I$.
The projection $q$ is given by the projection $p$ of $X$ and the projection
$B\times I\longrightarrow B$, and the section $t$ is given by $t(b)=(b,1)$.
\item Define the \emph{Moore mapping path fibration functor}\index{Moore mapping path fibration}\index{functor!Moore mapping path fibration --} $L$\@bsphack\begingroup \@sanitize\@noteindex{LX@$LX$} by letting
\[LX=(X\times_B \Lambda B, q,t),\]
where the pullback is defined with respect to the map $\Lambda B\longrightarrow B$
given by evaluation at $0$. The projection $q$ is given by $q(x,\lambda)=e(\lambda)$ and the section $t$ is given by
$t(b)=(s(b), b)$, where $b$ is viewed as a path of length $0$.
\end{enumerate}
\end{defn}
Thus $WX$ is obtained by growing a whisker on each point in the section of $X$, and the endpoints of the whiskers are used to give $WX$ a section. Similarly, $LX$ is obtained by attaching to $x\in X$ all Moore paths in $B$ starting at $p(x)$. The endpoints of the paths give the projection. In the language of \S\ref{sec:towardh}, $WX$ is the standard mapping cylinder construction of the section of $X$, thought of as a map in $G\scr{K}/B$. The section $t$ of $WX$ is just the $f$-cofibration in the standard factorization $\rho \circ t$ of $s$ through its mapping cylinder. In particular, $WX$ is well-sectioned. Similarly, $LX$ is a modification of the mapping path fibration $Np$ in $G\scr{K}$. The projection $p$ of $X$ factors through the projection $q$ of $LX$, which is an $h$-fibration; a path lifting function $\xi\colon LX\times_B B^I \longrightarrow (LX)^I$ is given by
$\xi((x,\lambda),\gamma)(t)=(x,\gamma|_0^t\lambda)$. Thus $LX$ is $h$-fibrant, but it need not be well-fibered.
We can display all of this conveniently in the following diagram. The third
square on the top is a pushout and the second square on the bottom is a pullback. That defines the maps $\phi$ and $\pi$, and the maps $\rho$ and
$\iota$ are induced by the universal properties from the identity map of $X$.
\[\xymatrix@=.8cm{
&&&B\ar[d]_{i_1}\ar@{=}[dr]\\
B \ar@{=}[r]\ar[d] & B \ar@{=}[r]\ar[d] & B \ar[d]^s \ar[r]_-{i_0} & B\times I \ar[d]\ar[r]_-{\text{pr}} & B\ar[d] \\
X \ar[d]\ar@{-->}[r]^{\iota} & LX \ar[r]^-\pi\ar[d] & X \ar[d]^p \ar[r]^-\phi & WX \ar[d]\ar@{-->}[r]^-{\rho} & X \ar[d] \\
B\ar@{=}[dr]\ar[r] & \Lambda B \ar[d]^{e}\ar[r]^-{p_0} & B\ar@{=}[r] & B \ar@{=}[r] & B\\
& B}\]
Thus $\rho$ projects whiskers on fibers to the original basepoints and $\iota$
is the inclusion $x\mapsto (x,p(x))$, where $p(x)$ is the path of length zero. Note that $\phi$ is not a map under $B$ and $\pi$ is not a map over $B$. They give an inverse $f$-equivalence to $\rho$ and an inverse $h$-equivalence to $\iota$.
\begin{prop}\mylabel{fpequiv2}
The map $\rho\colon WX\longrightarrow X$ is a natural $f$-equivalence of ex-spaces and $WX$ is well-sectioned. The map $\iota\colon X\longrightarrow LX$ is a natural $h$-equivalence of ex-spaces and $LX$ is $h$-fibrant. Therefore $W$ takes $f$-equivalences to $fp$-equivalences and $L$ takes $h$-equivalences to $f$-equivalences.
\end{prop}
The last statement follows from \myref{reverse}. We think of $\rho$ and
$\iota$ as giving a \emph{well-sectioned approximation}\index{well-sectioned approximation}\index{approximation!well-sectioned --} and an
\emph{$h$-fibrant approximation}\index{fibrant approximation@$h$-fibrant approximation}\index{approximation!fibrant@$h$-fibrant --} in the category of ex-spaces. We will combine them to obtain the promised ex-fibrant approximation, but we first insert a technical lemma.
\begin{lem}\mylabel{closedP}
If $X$ is an ex-space with a closed section, then $WLX$ is an ex-fibration.
If $X$ is well-fibered, then $WX$ is an ex-fibration.
\end{lem}
\begin{proof}
A path lifting function $\xi\colon NWLX = WLX\times_B B^I \longrightarrow (WLX)^I$ for $WLX$ is obtained by letting
\[\xi(z,\gamma)(t) =
\begin{cases}
(x,\gamma|_0^t\lambda)\in LX & \text{if $z=(x,\lambda)\in LX$},\\
(\gamma(t),u-t)\in B\times I & \text{if $z=(b,u)$ and $t\leq u$},\\
(s(\gamma(u)), \gamma|_u^t)\in LX & \text{if $z=(b,u)$ and $t\geq u$.}
\end{cases}\]
It is easy to verify that, as a map of sets, $\xi$ gives a well-defined section of the canonical retraction $\pi\colon (WLX)^I\longrightarrow WLX\times_B B^I$. Continuity is a bit more delicate, but if the section of $X$ is closed, then one verifies that
\[\Phi=\{(z,\gamma) \mid \text{$z$ is the equivalence class of $(s(b),b)\sim (b,0)$}\}\]
is a closed subset of WLX and hence $N\Phi$ is a closed subset of $NWLX$.
To see the implication, note that $(-)\times B^I$ preserves closed inclusions and $Z\times_B B^I\subset Z\times B^I$ is a closed inclusion because $B$ is in $\scr{U}$ (see \myref{Umod}).
Continuity follows since we are then piecing together continuous
functions on closed subsets.
If $X$ is well-fibered and $\xi\colon X\times_B B^I\longrightarrow X^I$ is a path-lifting function under $B^I$, we can define a path lifting function $\bar\xi\colon WX\times_B B^I \longrightarrow (WX)^I$ for $WX$ by
\[\bar\xi(x,\gamma)=\begin{cases}
\xi(x,\gamma) & \text{if $x\in X$},\\
(\gamma,u) & \text{if $x=(b,u)$}.
\end{cases}\]
To check that $\bar\xi$ is continuous, we use the fact that the functor
$N(-)=B^I\times_B (-)$ commutes with pushouts to write $NWX$ as a pushout.
We then see that $\bar{\xi}$ is the map obtained by passage to pushouts
from a pair of continuous maps.
\end{proof}
Recall that the sections of ex-spaces in $G\scr{U}_B$ are closed, by \myref{coflemma}. Since we shall only need to apply the constructions
of this section to ex-spaces in $G\scr{U}_B$, the closed section
hypothesis need not concern us.
\begin{defn}\mylabel{exfibapp}
Define the \emph{ex-fibrant approximation functor}\index{approximation!ex-fibrant --} $P$\@bsphack\begingroup \@sanitize\@noteindex{PX@$PX$} by the natural
zig-zag of $h$-equivalences $\phi = (\rho,W\iota)$ displayed in the diagram
$$\xymatrix{
X & WX \ar[l]_-{\rho} \ar[r]^-{W\iota} & WLX = PX.}
$$
By \myref{fpequiv2}, $P$ takes $h$-equivalences between arbitrary ex-spaces to $fp$-equivalences. If $X$ has a closed section, then $PX$ is an ex-fibration. If $X$ is an ex-fibration, then it has a closed section, and the above display is a natural zig-zag of $fp$-equivalences between ex-fibrations.
\end{defn}
\section{Preservation properties of ex-fibrant approximation}
One advantage of ex-fibrant approximation over $q$ or $qf$-fibrant approximation is that there are explicit commutation natural transformations relating it to many constructions of interest. The following result is an elementary illustrative example.
\begin{lem}
Let $\scr{D}$ be a small category, $X\colon \scr{D}\longrightarrow G\scr{K}_B$ be a functor, and
$$ \omega\colon \text{colim}\, WX_d \longrightarrow W\text{colim}\, X_d \ \ \ \text{and}\ \ \
\nu\colon \text{colim}\, LX_d\longrightarrow L\text{colim}\, X_d $$
be the evident natural maps. Then $\omega$ is a map over $\text{colim}\, X_d$ and $\nu$ is a map under $\text{colim}\, X_d$, so that the following diagrams commute. All maps in these diagrams are $h$-equivalences.
$$\xymatrix{
\text{colim}\, WX_d \ar[rr]^{\omega} \ar[dr]_{\text{colim}\,\rho}\, & & W\text{colim}\, X_d \ar[dl]^{\rho} \\
& \text{colim}\, X_d & }$$
$$\xymatrix{
& \text{colim}\, X_d \ar[dl]_{\text{colim}\,\iota} \ar[dr]^{\iota} & \\
\text{colim}\, LX_d \ar[rr]_{\nu} & & L\text{colim}\, X_d}$$
Let $\mu = W\nu\com \omega\colon \text{colim}\, PX_d\longrightarrow P\text{colim}\, X_d$. Then the following diagram of $h$-equivalences commutes.
$$\xymatrix{
\text{colim}\, X_d\ar@{=}[d] & \text{colim}\, WX_d \ar[l]_-{\text{colim}\, \rho}
\ar[r]^-{\text{colim}\, W\iota} \ar[d]^{\omega} & \text{colim}\, PX_d \ar[d]^{\mu}\\
\text{colim}\, X_d & W\text{colim}\, X_d \ar[l]^-{\rho} \ar[r]_-{W\iota} & P\text{colim}\, X_d}$$
The analogous statements for limits also hold.
\end{lem}
\begin{proof}
This is clear from the construction of limits and colimits in
\myref{btopbicomp}. The relevant $h$-equivalences of total
spaces are natural and piece together to pass to limits and colimits.
\end{proof}
\begin{warn}\mylabel{DoubleTrouble}
We would like an analogue of the previous result for tensors. In particular, we would like a natural map $(LX)\sma K\longrightarrow L(X\sma K)$ under $X\sma K$ for ex-spaces $X$ over $B$ and based spaces $K$. Inspection of definitions makes clear that there is no such map. The obvious map that one might write down, as in the erroneous \cite[5.6]{May}, is not well-defined. In Part III, this complicates the extension of $P$ to a functor on spectra over $B$.
\end{warn}
\begin{lem}\mylabel{munu}
Let $f\colon A\longrightarrow B$ be a map.
\begin{enumerate}[(i)]
\item Let $X$ be an ex-space over $A$. Then there are natural maps
$$\omega\colon f_!WX \longrightarrow Wf_!X \ \ \ \text{and} \ \ \ \nu\colon f_!LX\longrightarrow Lf_!X$$
of ex-spaces over $B$ such that $\omega$ is a map over $f_!X$ and $\nu$ is a map under $f_!X$. Let $\mu = W\nu\com \omega \colon f_!PX\longrightarrow Pf_!X$. Then the following diagram commutes.
$$\xymatrix{
f_!X\ar@{=}[d] & f_!WX \ar[l]_-{f_!\rho}
\ar[r]^-{f_!W\iota} \ar[d]^{\omega} & f_!PX \ar[d]^{\mu}\\
f_!X & Wf_!X \ar[l]^-{\rho} \ar[r]_-{W\iota} & Pf_!X }$$
\item Let $Y$ be an ex-space over $B$. Then there are natural maps
$$\omega\colon Wf^*Y \longrightarrow f^*WY \ \ \text{and} \ \ \nu\colon Lf^*Y\longrightarrow f^*LY$$ of ex-spaces over $A$, the first an isomorphism, such that $\omega$ is a map over $f^*Y$ and $\nu$ is a map under $f^*Y$. Let $\mu = \omega\com W\nu \colon Pf^*Y\longrightarrow f^*PY$. Then the following diagram commutes.
$$\xymatrix{
f^*Y\ar@{=}[d] & Wf^*Y \ar[l]_-{\rho}
\ar[r]^-{W\iota} \ar[d]^{\omega} & Pf^*Y \ar[d]^{\mu}\\
f^*Y & f^*WY \ar[l]^-{f^*\rho} \ar[r]_-{f^*W\iota} & f^*PY}$$
If $Y$ is an ex-fibration, then $\mu$ is an $fp$-equivalence.
\item Let $X$ be an ex-space over $A$. Then there are natural maps
$$\omega\colon Wf_*X \longrightarrow f_*WX \ \ \ \text{and} \ \ \ \nu\colon Lf_*X\longrightarrow f_*LX$$
of ex-spaces over $B$ such that $\omega$ is a map over $f_*X$ and $\nu$ is a map under $f_*X$. Let $\mu = \omega\com W\nu \colon Pf_*X\longrightarrow f_*PX$. Then the following diagram commutes.
$$\xymatrix{
f_*X\ar@{=}[d] & Wf_*X \ar[l]_-{\rho}
\ar[r]^-{W\iota} \ar[d]^{\omega} & Pf_*X \ar[d]^{\mu}\\
f_*X & f_*WX \ar[l]^-{f_*\rho} \ar[r]_-{f_*W\iota} & f_*PX}$$
\end{enumerate}
\end{lem}
\begin{proof} Again, the proof is by inspection of definitions.
Since $f_!$ does not preserve ex-fibrations, we do not have an
analogue for $f_!$ of the last statement about $f^*$ in (ii).
\end{proof}
\begin{warn}\mylabel{PffP}
We offer another example of the technical dangers lurking in this subject. The maps $\mu$ in the previous proposition are {\em not} $h$-equi\-va\-lences in general, the problem in (ii), say, being that $f^*$ does not preserve $h$-equivalences in general. If $\mu\colon Pf^*Y\longrightarrow f^*PY$ were always an $h$-equivalence, then one could prove by the methods in \S9.3 below that the relations (\ref{bases0}) descend to homotopy categories for all pullbacks of the form displayed in \myref{Mackey0}. In view of \myref{noway}, that conclusion is false. This is another pitfall we fell into, and it invalidated much work in an earlier draft.
\end{warn}
\section{Quasifibrant ex-spaces and ex-quasifibrations}
By analogy with the fact that an ex-fibration is a well-sectioned
$h$-fibrant ex-space, we adopt the following terminology.
\begin{defn}
An ex-space $X$ is \emph{quasifibrant}\index{ex-space!quasifibrant} if its projection $p$ is a quasifibration. An \emph{ex-quasifibration}\index{ex-quasifibration} is a well-sectioned
quasifibrant ex-space.
\end{defn}
If $X$ is quasifibrant, there is a long exact sequence of homotopy groups
\[\cdots\longrightarrow \pi^H_{q+1}(B,b)\longrightarrow \pi^H_q(X_b,x)\longrightarrow \pi^H_q(X,x) \longrightarrow \pi^H_q(B,b) \longrightarrow \cdots \longrightarrow \pi^H_0(B,b)\]
for any $b\in B$, $x\in X_b$ and $H\subset G_b$. Using this and the long exact sequences of
the pairs $(X,X_b)$, five lemma comparisons give the following observations.
\begin{lem}\mylabel{quasichar} Let $f\colon X\longrightarrow Y$ be a
$q$-equivalence of ex-spaces over $B$. Then each map of fibers
$f\colon X_b\longrightarrow Y_b$ is a $q$-equivalence if and only if each map
of pairs $f\colon (X,X_b)\longrightarrow (Y,Y_b)$ is a $q$-equivalence. If $X$
and $Y$ are quasifibrant, then these maps of pairs are $q$-equivalences. Conversely, if these maps of pairs are $q$-equivalences and either $X$ or
$Y$ is quasifibrant, then so is the other.
\end{lem}
Working in $G\scr{U}_B$, we obtain the following result. The same pattern of
proof gives many other results of the same nature that we leave to the reader.
\begin{prop}\mylabel{quasicof}
The following statements hold.
\begin{enumerate}[(i)]
\item A wedge over $B$ of ex-quasifibrations is an ex-quasifibration.
\item If $f:X\longrightarrow Y$ is a map such that $X$ is an ex-quasifibration
and $Y$ is quasifibrant, then the cofiber $C_B f$ is quasifibrant.
\item If $X$ is an ex-quasifibration and $K$ is a well-based space,
then $X\sma_B K$ is an ex-quasifibration.
\end{enumerate}
\end{prop}
\begin{proof}
This follows from \myref{quasichar}, the natural zig-zag
\[\xymatrix{X & WX \ar[l]\ar[r] & PX }\]
of $h$-equivalences, the corresponding preservation properties for
ex-fibrations, and the properties of $q$-equivalences given by the
statement that they are well-grounded; see \myref{hproper} and
\myref{exwellgr}. It is also relevant that in each case passage to
fibers gives the nonparametrized analogue of the construction under consideration. Since this result plays a vital role in our work, we
give more complete details of (ii) and (iii); (i) works the same way.
The cofiber $C_Bf$ is the pushout of the diagram
\[ \xymatrix{C_B X & X \ar[l]\ar[r]^-f & Y.}\]
If $X$ is well-sectioned, then the left arrow is an $h$-cofibration
and $WX$ and $PX$ are well-sectioned. Replacing $f$
by $Wf$ and $Pf$ we obtain three such cofiber
diagrams. Together with our original zig-zag this gives a
$3\times 3$-diagram. Applying the gluing lemma, \myref{hproper}(iii),
we obtain a zig-zag of $q$-equivalences
\[\xymatrix{C_B f & C_B Wf \ar[l]\ar[r] & C_B Pf.}\]
Similarly, on fibers we obtain zig-zags of $q$-equivalences
\[\xymatrix{Cf_b & C(Wf)_b \ar[l]\ar[r] & CW(Lf)_b.}\]
There results a zig-zag of $q$-equivalences of pairs
\[\xymatrix{(C_B f, Cf_b) & (C_B Wf, CWf_b)\ar[l]\ar[r] &
(C_B Pf, CW(Lf)_b).}\]
Since $C_BPf$ is ex-fibrant and in particular quasifibrant, $C_Bf$ is quasifibrant.
Similarly, by \myref{hproper}(v), we have natural zig-zags of $q$-equivalences
\[\xymatrix{X\sma_B K & WX\sma_B K \ar[l]\ar[r] & PX\sma_B K}\]
and
\[\xymatrix{X_b\sma K & WX_b\sma K \ar[l]\ar[r] & W(LX)_b\sma K.}\]
We therefore have a zig-zag of $q$-equivalences of pairs
\[\xymatrix@=.6cm{(X\sma_B K, X_b\sma K) & (WX\sma_B K, WX_b\sma K)\ar[l]\ar[r]
& (PX \sma_B K, W(LX)_b\sma K).}\]
Since $PX\sma_B K$ is ex-fibrant and in particular quasifibrant, $X\sma_B K$ is quasifibrant.
\end{proof}
\chapter{The equivalence between $\text{Ho}\, G\scr{K}_B$ and $hG\scr{W}_B$}
\section*{Introduction}
We developed the point-set level properties of the category $G\scr{K}_B$
of ex-$G$-spaces over $B$ in Chapters 1 and 2, and we developed those homotopical properties that are accessible to model theoretic techniques
in Chapter 4 -- 7. In this chapter, we use ex-fibrations to prove that
certain structure on the point-set level that seems inaccessible from the
point of view of model category theory nevertheless descends to homotopy categories. In particular, we prove that $\text{Ho}\, G\scr{K}_B$ is closed symmetric monoidal and that the right derived functor $f^*$ of the Quillen adjunction $(f_!,f^*)$ in the $qf$-model structure is closed symmetric monoidal and
has a right adjoint $f_*$.
In \S9.1 we use the ex-fibrant approximation functor to prove that our model theoretic homotopy category of ex-$G$-spaces over $B$ is equivalent to the classical homotopy category of ex-$G$-fibrations over $B$. In \S9.2, we discuss how to pass to derived functors on either side of that equivalence in certain general cases. Replacing the model-theoretic method of constructing derived functors by a more classical method given in terms of ex-fibrant approximation, we construct the functors $f_*$ and $F_B$ on homotopy categories in \S9.3. By a combination of methods, we prove that $\text{Ho}\, G\scr{K}_B$ is a symmetric monoidal category and that the base change functor $f^*$ descends to a closed symmetric monoidal functor on homotopy categories in \S9.4. We also obtain such descent
to homotopy categories results for change of group adjunctions and for passage to fibers in that section. These results are central to the theory, and there
seem to be no shortcuts to their proofs.
Everything is understood to be equivariant in this chapter, and we abbreviate ex-$G$-fibration and ex-$G$-space to ex-fibration and ex-space throughout.
We shall retreat just a bit from all--embracing generality. We assume that $G$ is a Lie group and that all given base $G$-spaces $B$ are proper and
are of the homotopy types of $G$-CW complexes. The reader may prefer to
assume that $G$ is compact, but there is no gain in simplicity. In view of
the properties of the base change adjunction $(f_!,f^*)$ given in
\myref{Qad10}, there would be no real loss of generality if we restricted further to base spaces that are actual $G$-CW complexes, but that would be inconveniently restrictive.
\section{The equivalence of $\text{Ho}\, G\scr{K}_B$ and $hG\scr{W}_B$}
Recall that $X\sma_B I_+$ is a cylinder object in the sense of the $qf$-model structure. When we restrict to $qf$-fibrant and $qf$-cofibrant objects, homotopies in the $qf$-model sense are the same as $fp$-homotopies, by \myref{comphty}. The morphism set $[X,Y]_{G,B}$ in $\text{Ho}\, G\scr{K}_B$ is naturally isomorphic to $[RQX,RQY]_{G,B}$, and this is the set of $fp$-homotopy classes of maps $RQX\longrightarrow RQY$. Here $R$ and $Q$ denote the functorial $qf$-fibrant and
$qf$-cofibrant approximation functors obtained from the small object argument. The total space of $RQX$ has the homotopy type of a $G$-CW complex since $B$
does. This leads us to introduce the following categories.
\begin{defn}
Define $G\scr{V}_B$ to be the full subcategory of $G\scr{K}_B$ whose objects are
well-grounded and $qf$-fibrant with total spaces of the homotopy types
of $G$-CW complexes. Define $G\scr{W}_B$ to be the full subcategory of $G\scr{V}_B$ whose objects are the ex-fibrations over $B$. Let $hG\scr{W}_B$ denote the category obtained from $G\scr{W}_B$ by passage to $fp$-homotopy classes of maps.
\end{defn}
Note that the definition of $G\scr{W}_B$ makes no reference to model category theory. Recall that well-grounded means well-sectioned and compactly generated. When $B=*$, $G\scr{W}_*$ is just the category of well-based compactly generated $G$-spaces of the homotopy types of $G$-CW complexes, and it is standard that its classical homotopy category is equivalent to the homotopy category of based $G$-spaces with respect to the $q$-model structure. We shall prove a parametrized generalization.
We think of $G\scr{V}_B$ as a convenient half way house between $G\scr{K}_B$ and $G\scr{W}_B$. It turns out to be close enough to the category of $qf$-cofibrant and $qf$-fibrant objects in $G\scr{K}_B$ to serve as such for our purposes, while already having some of the properties of $G\scr{W}_B$. The following crucial theorem fails for the $q$-model structure. It is essential for this result that we allow the objects of $\scr{V}_B$ to be well-sectioned rather than requiring them
to be $qf$-cofibrant. This will force an assymmetry when we deal with left and right derived functors in \myref{cderiv} below.
\begin{thm}\mylabel{CWfix}
The $qf$-cofibrant and $qf$-fibrant approximation functor $RQ$ and the ex-fibrant approximation functor $P$, together with the forgetful functors
$I$ and $J$, induce the following equivalences of homotopy categories.
\[\xymatrix{\text{Ho}\, G\scr{K}_B \ar@<.5ex>[r]^-{RQ}
& \text{Ho}\, G\scr{V}_B \ar@<.5ex>[r]^-P\ar@<.5ex>[l]^-I
& hG\scr{W}_B \ar@<.5ex>[l]^-J }\]
\end{thm}
\begin{proof}
For $X$ in $G\scr{K}_B$, we have a natural zig-zag of $q$-equivalences in $G\scr{K}_B$
\[\xymatrix{X & QX\ar[l]\ar[r] & RQX.}\]
Therefore $X$ and $IRQX$ are naturally $q$-equivalent in $G\scr{K}_B$. If $X$ is in $G\scr{V}_B$, then it is $qf$-fibrant and therefore so is $QX$. Then the above zig-zag is in $G\scr{V}_B$ and thus $X$ and $RQIX$ are naturally $q$-equivalent in $G\scr{V}_B$.
Since $q$-equivalences in $G\scr{V}_B$ are $h$-equivalences, and $P$ takes $h$-equivalences to $fp$-equivalences, it is clear that $P$ induces a functor on homotopy categories. Conversely, since $fp$-equivalences are in particular
$q$-equivalences, the forgetful functor $J$ induces a functor in the other direction. For $X$ in $G\scr{V}_B$ we have the natural zig-zag of $h$-equivalences
\[\xymatrix{X & WX\ar[r]^{W\iota}\ar[l]_\rho & PX}\]
of \myref{exfibapp}. However $WX$ may not be in $G\scr{V}_B$ since it may not be
$qf$-fibrant. Applying $qf$-fibrant approximation, we get a natural zig-zag of $q$-equivalences in $G\scr{V}_B$ connecting $X$ and $PX$. It follows that $X$ and $JPX$ are naturally $q$-equivalent in $G\scr{V}_B$. Starting with $X$ in $G\scr{W}_B$, the above display is a zig-zag of $fp$-equivalences in $G\scr{W}_B$, by \myref{fpequiv2}. It follows that $X$ and $JPX$ are naturally
$fp$-equivalent in $G\scr{W}_B$.
\end{proof}
\section{Derived functors on homotopy categories}
Model category theory tells us how Quillen functors $V\colon G\scr{K}_A\longrightarrow G\scr{K}_B$ induce derived functors on the homotopy categories on the left hand side of the equivalence displayed in \myref{CWfix}. We now seek an equivalent way of passing to derived functors on the right hand side. We begin with an informal discussion. We focus on functors of one variable, but functors of several
variables work the same way.
Following the custom in algebraic topology, we have been abusing notation by using the same notation for point-set level functors and for derived homotopy category level functors. We will continue to do so. However, the more accurate notations of algebraic geometry, $LV$ and $RV$ for left and right derived functors, might clarify the discussion. As we have already seen in \myref{noway}, passage to derived functors is not functorial in general, so that a relation between composites of functors that holds on the point-set level need not imply a corresponding relation on passage to homotopy categories.
Recall that, model theoretically, if $V$ is a Quillen right adjoint, then the right derived functor of $V$ is obtained by first applying fibrant approximation $R$ and then applying $V$ on homotopy categories, which makes sense since $V$ preserves weak equivalences between fibrant objects. The left derived functor of a Quillen left adjoint $V$ is defined dually, via $VQ$. Problems arise when one tries to compose left and right derived functors, which is what we must do to prove some of our compatibility relations.
The equivalence of categories proven in \myref{CWfix} gives us a way of putting the relevant left and right adjoints on the same footing, giving a ``straight'' passage to derived functors that is neither ``left'' nor ``right''. We need
mild good behavior for this to work.
\begin{defn}
A functor $V\colon G\scr{K}_A\longrightarrow G\scr{K}_B$ is \emph{good}\index{functor!good}\index{good functor} if it is continuous,
takes well-grounded ex-spaces to well-grounded ex-spaces, and takes ex-spaces
whose total spaces are of the homotopy types of $G$-CW complexes to ex-spaces with that property. Since $V$ is continuous, it preserves $fp$-homotopies.
\end{defn}
\begin{prop}\mylabel{cderiv}
Let $V\colon G\scr{K}_A\longrightarrow G\scr{K}_B$ be a good functor that is a left or
right Quillen adjoint. If $V$ is a Quillen left adjoint, assume further
that it preserves $q$-equivalences between well-grounded ex-spaces. Then,
under the equivalence of categories in \myref{CWfix}, the derived functor
$\text{Ho}\, G\scr{K}_A\longrightarrow\text{Ho}\, G\scr{K}_B$ induced by $VQ$ or $VR$
is equivalent to the functor
$PVJ\colon hG\scr{W}_A\longrightarrow hG\scr{W}_B$ obtained
by passage to homotopy classes of maps. \end{prop}
\begin{proof}
If $V$ is a Quillen right adjoint, then it preserves $q$-equivalences between $qf$-fibrant objects. If $V$ is a Quillen left adjoint, then we are assuming that it preserves $q$-equivalences between well-grounded objects. Since $G\scr{V}_A$ consists of well-sectioned $qf$-fibrant objects, it follows in both cases that $V\colon G\scr{V}_A \longrightarrow G\scr{V}_B$ passes straight to homotopy categories to give $V\colon \text{Ho}G\scr{V}_A \longrightarrow \text{Ho}G\scr{V}_B$. Since $V$ preserves $G$-CW homotopy types on total spaces, $V$ takes $q$-equivalences to $h$-equivalences. Therefore $PV$ takes $q$-equivalences
to $fp$-equivalences and induces a functor $\text{Ho}\, G\scr{V}_A\longrightarrow hG\scr{W}_B$. To show that $PVJ$ and either $VQ$ or $VR$ agree under the equivalence of categories,
it suffices to verify that the following diagram commutes.
\[\xymatrix{
\text{Ho}\, G\scr{K}_A\ar[d]_{RQ} \ar[rr]^-{VQ \ \ \text{or} \ \ VR}
& & \text{Ho}\, G\scr{K}_B \ar[d]^{PRQ}\\
\text{Ho}\, G\scr{V}_A \ar[rr]_{PV} & & hG\scr{W}_B}\]
We have a natural acyclic $qf$-fibration $QX\longrightarrow X$ and a natural
acyclic $qf$-cofibration $X\longrightarrow RX$. If $V$ is a Quillen left adjoint,
then we have a zig-zag of natural $q$-equivalences
\[\xymatrix{RQVQ \ar[r] & RVQ & VQ \ar[l]\ar[r] & VRQ}\]
because $V$ preserves acyclic $qf$-cofibrations.
If $V$ is a Quillen right adjoint, then we have a zig-zag of natural $q$-equivalences
\[\xymatrix{RQVR & RQVRQ \ar[r]\ar[l] & RVRQ & VRQ\ar[l]}\]
because $V$ preserves $q$-equivalences between $qf$-fibrant objects. In both cases, all objects have total spaces of the homotopy types of $G$-CW complexes, so in fact we have zig-zags of $h$-equivalences. Therefore, applying $P$ gives
us zig-zags of $fp$-equivalences in $G\scr{W}_B$, by \myref{fpequiv2}.
\end{proof}
\begin{rem}
When $V$ preserves ex-fibrations, $PV$ is naturally $fp$-equivalent to $V$ on ex-fibrations, by \myref{fpequiv2}. The derived functor of $V$ can then be obtained directly by applying $V$ and passing to equivalence classes of maps under $fp$-homotopy.
\end{rem}
\section{The functors $f_*$ and $F_B$ on homotopy categories}
We use the equivalence between $\text{Ho}\, G\scr{K}_B$ and $hG\scr{W}_B$ to prove that, for any map $f\colon A\longrightarrow B$ between spaces of the homotopy types of $G$-CW complexes, the $(f^*,f_*)$ adjunction descends to homotopy categories.
We begin by verifying that $f^*$ satisfies the hypotheses of \ref{cderiv}.
\begin{prop}\mylabel{fstarderiv}
Let $f\colon A\longrightarrow B$ be a map of base spaces. Then the base change functor $f^*$ restricts to a functor $f^*\colon G\scr{W}_B\longrightarrow G\scr{W}_A$.
\end{prop}
\begin{proof}
Consider $Y$ in $G\scr{W}_B$. Since the total space of $Y$ is of the homotopy type of a $G$-CW complex, the fibers $Y_b$ are of the homotopy types of $G_b$-CW complexes by \myref{ss}. The fiber $(f^*Y)_a$ is a copy of $Y_{f(a)}$, and $G_a$ acts through the evident inclusion $G_a\subset G_{f(a)}$. Therefore $(f^*Y)_a$ is of the homotopy type of a $G_a$-CW complex. The total space of $f^*Y$ is therefore of the homotopy type of a $G$-CW complex, again by \myref{ss}. Moreover, $f^*Y$ is an ex-fibration by \myref{fexpres}. Thus $f^*$ restricts to a functor $f^*\colon G\scr{W}_B\longrightarrow G\scr{W}_A$.
\end{proof}
\begin{thm}\mylabel{descendf0}
For any map $f\colon A\longrightarrow B$ of base spaces, the right derived functor $f^*\colon \text{Ho}\, G\scr{K}_B\longrightarrow \text{Ho}\, G\scr{K}_A$ has a right adjoint $f_*$, so that $$ [f^*Y,X]_{G,A} \iso [Y,f_*X]_{G,B}$$ for $X$ in $G\scr{K}_A$ and $Y$ in $G\scr{K}_B$.
\end{thm}
\begin{proof}
In view of the equivalence of categories in \myref{CWfix} and the fact that $f^*$ descends directly to a functor $f^*\colon hG\scr{W}_B\longrightarrow hG\scr{W}_B$
on homotopy categories, by Propositions \ref{cderiv} and \ref{fstarderiv}, it suffices to construct a right adjoint $f_*\colon hG\scr{W}_A \longrightarrow hG\scr{W}_B$. We do that using the Brown representability theorem. By \myref{brown0}, $\text{Ho}\, G\scr{K}_B$ satisfies the formal hypotheses for Brown representability, and therefore so does $hG\scr{W}_B$. In fact $G\scr{W}_B$ has all of the relevant wedges and homotopy colimits since these constructions preserve ex-fibrations by \myref{pres1} and \myref{HursmaK} and since they clearly preserve $G$-CW homotopy types on the total space level and stay within $G\scr{U}_B$. The objects in the detecting set $\scr{D}_B$ of \myref{detectset} are not in $G\scr{W}_B$, but we can apply the ex-fibrant approximation functor $P$ to them to obtain a detecting set of objects in $hG\scr{W}_B$. Therefore a contravariant set-valued functor on $hG\scr{W}_B$ is representable if and only if it satisfies the wedge and Mayer-Vietoris axoms.
For a fixed ex-fibrant space $X$ over $A$, consider the functor $\pi(f^*Y,X)_{G,A}$ on $Y$ in $G\scr{W}_B$, where $\pi$ denotes $fp$-homotopy classes of maps. Since the functor $\pi(W,X)_{G,A}$ on $W$ in $G\scr{W}_A$ is represented and is computed using homotopy classes of maps, it clearly satisfies the wedge and Mayer-Vietoris axioms. It therefore suffices to show that the functor $f^*$ preserves wedges and homotopy pushouts, since that will imply
that the functor $\pi(f^*Y,X)_{G,A}$ of $Y$ also satisfies the wedge and Mayer-Vietoris axioms. We can then conclude that there is an object $f_*X\in G\scr{W}_B$ that represents this functor. It follows formally that $f_*$ is a functor of $X$ and that the required adjunction holds.
Because $f^*\colon G\scr{K}_B\longrightarrow G\scr{K}_A$ is a left adjoint, it preserves colimits, and this implies that $f^*\colon G\scr{W}_B\longrightarrow G\scr{W}_A$ preserves the relevant homotopy colimits. Moreover, $f^*$ preserves $fp$-homotopies and so induces a functor on homotopy categories that still preserves these homotopy colimits.
\end{proof}
We agree to write $\simeq$ for natural equivalences on homotopy categories.
\begin{rem}
For composable maps $f$ and $g$, $g_*\com f_* \simeq (g\com f)_*$ on homotopy categories since $f^*\com g^*\simeq (g\com f)^*$ on homotopy categories. The latter equivalence is clear since $f^*$ and $g^*$ are derived from Quillen right adjoints. More sophisticated commutation laws are proven in the next section.
\end{rem}
Applying \myref{descendf0} to diagonal maps and composing with the homotopy category level adjunction between the external smash product and function ex-space functors, we obtain the following basic result; compare \myref{internalize}.
\begin{thm}\mylabel{smashing0}
Define $\sma_B$ and $F_B$ on $\text{Ho}\, G\scr{K}_B$ as the composite (derived) functors
$$X\sma_B Y = \Delta^*(X\barwedge Y) \qquad\text{and}\qquad F_B(X,Y) = \bar{F}(X,\Delta_*Y).$$
Then
$$ [X\sma_B Y, Z]_{G,B}\iso [X,F_B(Y,Z)]_{G,B}$$
for $X$, $Y$, and $Z$ in $\text{Ho}\, G\scr{K}_B$.
\end{thm}
\begin{proof}
The displayed adjunction is the composite of adjunctions for the (derived) external smash and function ex-space functors and for the (derived) adjoint pair $(\Delta^*,\Delta_*)$.
\end{proof}
\begin{rem}
The referee points out that the ex-space analogue of \cite[7.2]{BB2} shows that we can work directly with the point-set topology to show that the $(\sma_B,F_B)$ adjunction on the original category $G\scr{K}_B$ is continuous and so descends to (classical) $fp$-homotopy categories to give the adjunction
$$ hG\scr{K}_B(X\sma_B Y,Z)\iso hG\scr{K}_B(X,F_B(Y,Z)).$$
Presumably similar point-set topological arguments work to show that, for a map $f\colon A\longrightarrow B$, we have an adjunction
$$hG\scr{K}_A(f^*X,Y)\iso hG\scr{K}_B(X,f_*Y).$$
These adjunctions do {\em not}\, imply our Theorems \ref{descendf0} and \ref{smashing0}. By definition, our category $hG\scr{W}_B$ is a full subcategory of $hG\scr{K}_B$, but it is not an {\em equivalent}\, full subcategory. The objects of $G\scr{W}_B$ are very restricted, and general function ex-spaces $F_B(Y,Z)$ are not $fp$-homotopy equivalent to such objects. The force of our theorems is that, after restricting to our subcategories $hG\scr{W}_B$, we still have right adjoints {\em in these categories}. It is this fact that we need to obtain right adjoints in our preferred homotopy categories $\text{Ho}\, G\scr{K}_B$.
\end{rem}
\section{Compatibility relations for smash products and base change}
We first prove that $\text{Ho}\, G\scr{K}_B$ satisfies the associativity, commutativity and unity conditions required of a symmetric monoidal category. We then show that all of the isomorphisms of functors in \myref{Wirth0} and some of the isomorphisms of functors in \myref{Mackey0} still hold after passage to homotopy categories. Finally, we relate change of groups and passage to fibers to the
symmetric monoidal structure on homotopy categories. In some of our arguments, it is natural to work in $\text{Ho}\, G\scr{K}_{B}$. In others, it is natural to work in the equivalent category $hG\scr{W}_{B}$.
\begin{prop}\mylabel{fgext'}
For maps $f\colon A\longrightarrow B$ and $g\colon A'\longrightarrow B'$ of base spaces and for ex-spaces $X$ over $B$ and $Y$ over $B'$,
\begin{equation}\label{fgext1'}
(f^*Y\barwedge g^*Z)\simeq (f\times g)^*(Y\barwedge Z)
\end{equation}
in $\text{Ho}\, G\scr{K}_A$. For ex-spaces $W$ over $A$ and $X$ over $A'$,
\begin{equation}\label{fgext2'}
(f_!W\barwedge g_!X)\simeq (f\times g)_!(W\barwedge X)
\end{equation}
in $\text{Ho}\, G\scr{K}_B$.
\end{prop}
\begin{proof}
For (\ref{fgext1'}), we work with ex-fibrations, starting in $hG\scr{W}_{B\times B'}$. By Propositions \ref{fexpres} and \ref{Hursma}, the functors we are dealing with preserve ex-fibrations and therefore descend straight to homotopy categories. The conclusion is thus immediate from its point-set level analogue. For (\ref{fgext2'}), we work with model theoretic homotopy categories, starting in $\text{Ho}\, G\scr{K}_{A\times A'}$. Since $(f\times g)_!\simeq (f\times\text{id})_!\com (\text{id}\times g)_!$, we can proceed in two steps and so assume that $g=\text{id}$. By \myref{smaAB} and \myref{Qad10}, we are then composing Quillen left adjoints. Starting with $qf$-cofibrant objects, we do not need to apply $qf$-cofibrant approximation, and the conclusion follows directly from its point-set level analogue.
\end{proof}
We use this to complete the proof that $\text{Ho}\, G\scr{K}_B$ is symmetric monoidal.
\begin{thm}\mylabel{clsymmon}
The category $\text{Ho}\, G\scr{K}_B$ is closed symmetric monoidal under the functors $\sma_B$ and $F_B$.
\end{thm}
\begin{proof}
In view of \myref{smashing0}, we need only prove the associativity, commutativity, and unity of $\sma_B$ up to coherent natural isomorphism. The external smash product has evident associativity, commutativity, and unity isomorphisms, and these descend directly to homotopy categories since the external smash product of $qf$-cofibrant ex-spaces over $A$ and $B$ is $qf$-cofibrant over $A\times B$. To see that these isomorphisms are inherited after internalization along $\Delta^*$, we use (\ref{fgext1'}). For the associativity of $\sma_B$, we have
\begin{multline*}
\Delta^*(\Delta^*(X\barwedge Y)\barwedge Z)
\simeq \Delta^*(\Delta\times\text{id})^*((X\barwedge Y)\barwedge Z)
\simeq ((\Delta\times\text{id})\Delta)^*((X\barwedge Y)\barwedge Z)\\
\quad \quad \simeq ((\text{id}\times \Delta)\Delta)^*(X\barwedge(Y\barwedge Z))
\simeq \Delta^*(\text{id}\times \Delta)^*(X\barwedge(Y\barwedge Z))
\simeq \Delta^*(X\barwedge \Delta^*(Y\barwedge Z)).
\end{multline*}
\noindent
The commutativity of $\sma_B$ is similar but simpler. For the unit, we observe that $S^0_B\simeq {r^*S^0}$,
$r\colon B\longrightarrow *$. Therefore, since $(\text{id}\times r)\Delta = \text{id}$,
\[X\sma_B S^0_B \simeq \Delta^*(X\barwedge r^*S^0)
\simeq \Delta^*(\text{id}\times r)^*(X\barwedge S^0)
\simeq ((\text{id}\times r)\Delta)^* (X) = X.\qedhere\]
\end{proof}
We turn next to the derived versions of the base change compatibilities of Propositions \ref{Wirth0} and \ref{Mackey0}. Observe that the functor $f_!$ is
good since the section of a well-sectioned ex-space is an $h$-cofibration and
since $G$-CW homotopy types are preserved under pushouts, one leg of which is
an $h$-cofibration. Moreover, $f_!$ preserves $q$-equivalences between well-sectioned ex-spaces by \myref{Qad10}. Therefore \myref{cderiv} applies to $f_!$.
\begin{thm}\mylabel{fclsymmon}
For a $G$-map $f\colon A\longrightarrow B$, $f^*\colon \text{Ho}\, G\scr{K}_B\longrightarrow \text{Ho}\, G\scr{K}_A$ is a closed symmetric monoidal functor.
\end{thm}
\begin{proof}
Since $f^*S^0_B\iso S^0_A$ in $G\scr{K}_A$ and $S^0_B$ is $qf$-fibrant, $f^*S^0_B\simeq S^0_A$ in $\text{Ho}\, G\scr{K}_A$. We must prove that the isomorphisms (\ref{oneo}) through (\ref{five0}) descend to equivalences on homotopy categories. Categorical arguments in \cite[\S\S2, 3]{FHM} show that it suffices to show that the two isomorphisms (\ref{oneo}) and (\ref{four0}) descend to equivalences on homotopy categories. These two isomorphisms do not involve the right adjoints $f_*$ or $\Delta_*$ and are therefore more tractable than the other three. First consider
(\ref{oneo}):
\[f^*(Y\sma_B Z)\iso f^*Y\sma_A f^*Z.\]
If $Y$ and $Z$ are in $G\scr{W}_B$, then the two sides of this isomorphism are both in $G\scr{W}_A$, by \myref{fexpres} and \myref{Hursma}. Therefore the point-set level isomorphism descends directly to the desired homotopy category level equivalence. Next, consider (\ref{four0}):
\[f_{!}(f^*Y\sma_A X)\iso Y\sma_B f_{!}X.\]
Assume that $X$ is in $G\scr{W}_A$ and $Y$ is in $G\scr{W}_B$. The functor $f_!$ does not preserve ex-fibrations so, to pass to derived categories, we must replace
it by $Pf_!$ on both sides. By \myref{savior}, the functor $Y\sma_B(-)$ preserves $h$-equivalences between well-sectioned ex-spaces. Since $P$ sends $h$-equivalences to $fp$-equivalences, we therefore have $fp$-equivalences, natural up to $fp$-homotopy,
\[\xymatrix{
Pf_{!}(f^*Y\sma_A X)\iso P(Y\sma_B f_{!}X)\ \ar[r]^-{P(\text{id}\sma_B\phi)} &
\ P(Y\sma_BPf_!X) & Y\sma_BPf_!X, \ar[l]_-{\phi}}\]
where $\phi = (\rho,W\iota)$ is the zigzag of $h$-equivalences of \myref{exfibapp}. This implies the desired equivalence in the homotopy category.
\end{proof}
The reader is invited to try to prove directly that the projection formula holds in the homotopy category. Even the simple case of $f\colon *\longrightarrow B$, the inclusion of a point, should demonstrate the usefulness of \myref{cderiv}.
\begin{thm}\mylabel{pullbackfix}
Suppose given a pullback diagram of $G$-spaces
$$\xymatrix{
C \ar[r]^-{g} \ar[d]_{i} & D \ar[d]^{j} \\
A \ar[r]_{f} & B}$$
in which $f$ (or $j$) is a $q$-fibration. Then there are natural equivalences of functors on homotopy categories
\begin{equation}\label{basesmore0}
j^*f_{!} \simeq g_{!}i^*, \quad f^*j_* \simeq i_*g^*, \quad f^*j_{!}\simeq i_!g^*, \quad j^*f_*\simeq g_*i^*.
\end{equation}
\end{thm}
\begin{proof}
As in \myref{Mackey0} the second and fourth equivalences are conjugate to the first and third. However, since the situation is no longer symmetric, we must prove both the first and third equivalences, assuming $f$ is a $q$-fibration.
First consider the desired equivalence $f^*j_{!}\simeq i_!g^*$. We work with ex-fibrations, starting with $X\in hG\scr{W}_D$. We must replace $j_!$ and $i_!$ by $Pj_!$ and $Pi_!$ before passing to homotopy categories. By \myref{Qad10}, $f^*$ preserves $q$-equivalences since $f$ is a $q$-fibration. Moreover, our $q$-equivalences are $h$-equivalences since we are dealing with total spaces of the homotopy types of $G$-CW complexes. By the diagram in \myref{munu}(ii), we see that $\mu\colon Pf^*\longrightarrow f^*P$ is a natural $h$-equivalence here. This would be false for arbitrary maps $f$, as observed in \myref{PffP}. Since $\mu$ is an $h$-equivalence between ex-fibrations, it is an $fp$-equivalence. Therefore
$$f^*Pj_{!}X\htp Pf^*j_!X\iso Pi_!g^*X.$$
Now consider the desired equivalence $j^*f_{!}X \simeq g_{!}i^*X$ in $\text{Ho}\, G\scr{K}_D$. Our assumption that $f$ is a $q$-fibration gives us no direct help with this. However, we may factor $j$ as the composite of a homotopy equivalence and an $h$-fibration. Expanding our pullback diagram as a composite of pullbacks, we see that it suffices to prove our commutation relation when $j$ is an $h$-fibration and when $j$ is a homotopy equivalence. The first case is immediate by symmetry from the first part. Thus assume that $j$ is a homotopy equivalence. Then $i$ is also a homotopy equivalence. By \myref{Qad10}, $(i_!,i^*)$ and $(j_!,j^*)$ are adjoint equivalences of homotopy categories. Therefore
\[ j^*f_! \simeq j^*f_!i_!i^* \simeq j^*j_!g_!i^*\simeq g_!i^*. \qedhere\]
\end{proof}
Finally, we turn to a promised compatibility relationship between products
and change of groups. We observed in \myref{Lishriek} that the point-set level closed symmetric monoidal equivalence of \myref{ishriek} is given by a Quillen equivalence. The following addendem shows that the resulting equivalence on homotopy categories is again closed symmetric monoidal.
\begin{prop}\mylabel{imonoidaldescends}
Let $\iota\colon H\longrightarrow G$ be the inclusion of a subgroup and $A$ be an $H$-space. The Quillen equivalence $(\iota_!, \nu^*\iota^*)$ descends to a closed symmetric monoidal equivalence between $\text{Ho}H\scr{K}_A$ and
$\text{Ho}G\scr{K}_{\iota_!A}$.
\end{prop}
\begin{proof}
Let $\Delta\colon A\longrightarrow A\times A$ be the diagonal map. The isomorphisms
\[\iota^*\Delta^*(X\barwedge Y)\cong \Delta^*\iota^* (X\barwedge Y)
\cong \Delta^* (\iota^*X\barwedge \iota^*Y)\]
descend to equivalences on homotopy categories, the first since it is between Quillen right adjoints, the second since $\iota^*$ preserves all $q$-equivalences. It follows that $\nu^*\iota^*$ is a symmetric monoidal
functor on homotopy categories. Since it is also an equivalence, it follows formally that it is closed symmetric monoidal.
\end{proof}
Combined with \myref{fclsymmon} applied to the inclusion
$\tilde{b}\colon G/G_b \longrightarrow B$, this last observation
gives us the following conclusion.
\begin{thm}\mylabel{fiberfun}
The derived fiber functor $(-)_b\colon \text{Ho}\, G\scr{K}_B\longrightarrow \text{Ho}\, G_b\scr{K}_b$ is closed symmetric monoidal, and it has a left adjoint $(-)^b$ and a right adjoint ${^b}(-)$.
\end{thm}
We emphasize that this innocent looking result packages highly non-trivial and important information. It gives in particular that, for ex-$G$-spaces $X$
and $Y$, the (derived) fiber $F_B(X,Y)_b$ of the (derived) function space $F_B(X,Y)$ is equivalent in $\text{Ho}\, G_b\scr{K}_b$ to the (derived) function space $F(X_b,Y_b)$ of the (derived) fibers $X_b$ and $Y_b$. On the point set level, that is what motivated the definition of the internal function ex-space.
That it still holds on the level of homotopy categories is a reassuring consistency result.
\part{Parametrized equivariant stable homotopy theory}
\chapter*{Introduction}
We develop rigorous foundations for parametrized equivariant
stable homotopy theory. The idea is to start with a fixed base
$G$-space $B$ and to build a good category,
here denoted $G\scr{S}_{B}$, of $G$-spectra over $B$. We assume once and
for all that our base spaces $B$ must be compactly generated and must
have the homotopy types of $G$-CW complexes. By ``good'' we mean
that $G\scr{S}_{B}$ is a closed symmetric monoidal topological model category
whose associated homotopy category has properties analogous to those
of the ordinary equivariant stable homotopy category.
Informally, the homotopy theory of $G\scr{S}_{B}$ is specified by the
homotopy theory seen on the fibers of $G$-spectra over $B$.
One compelling reason for taking the parametrized stable homotopy
category seriously, even nonequivariantly, is to build a natural home
in which one can do stable homotopy theory while still keeping track of fundamental groups and groupoids. Stable homotopy theory has tended to
ignore such intrinsically unstable data. This has the effect of losing
contact with more geometric branches of mathematics in which the
fundamental group cannot be ignored.
For example, one basic motivation for the
equivariant theory is that it gives a context in which to better
understand equivariant orientations, Thom isomorphisms, and
Poincar\'e duality. There is no problem for $G$-simply connected
manifolds $M$ \cite[III\S6]{LMS}, but restriction to such $M$ is
clearly inadequate for applications to transformation group theory.
Despite a great deal of work on the subject by Costenoble and Waner,
and some by May, \cite{CMW, CW1, CW2, CW3, MayR}, this circle of
ideas is not yet fully understood. Costenoble and Waner \cite{CWNew}
use our work to study this problem for ordinary equivariant theories,
and for general theories this is work in progress by the second
author.
There are many problems that
make the development far less than an obvious generalization of the
nonparametrized theory. Problems on the space level were dealt
with in Parts I and II, and we deal with the analogous spectrum level
problems here. We give some categorical preliminaries on enriched
equivariant categories in Chapter 10. We define and develop the basic properties of our preferred category of parametrized $G$-spectra in
Chapter 11, study its model structures in Chapter 12, and study
adjunctions and compatibility relations in Chapter 13. All of the
problems that we faced on the space level are still there, but their
solutions are considerably more difficult. In Chapter 14, we go on to
study further such compatibilities that more fundamentally involve
equivariance.
The theory of highly structured spectra is highly cumulative.
We build on the theory of equivariant orthogonal
spectra of Mandell and May \cite{MM}. In turn, that
theory builds on the theory of
nonequivariant orthogonal spectra. A self-contained treatment of
nonequivariant diagram spectra, including orthogonal spectra,
is given by Mandell, May, Schwede, Shipley in \cite{MMSS}.
The treatments of \cite{MM} and \cite{MMSS}, like this one, are topological
as opposed to simplicial. That seems to be essential when dealing with
infinite groups of equivariance. It also allows use of orthogonal
spectra rather than symmetric spectra. These are much more natural
equivariantly and, even nonequivariantly, they have the major
convenience that their weak equivalences are exactly the maps that
induce isomorphisms of homotopy groups.
The theory of equivariant parametrized spectra can be thought
of as the pushout over the theory of spectra of the theories of
equivariant spectra and of nonequivariant parametrized spectra.
However, there is no nonequivariant precursor of the present treatment
of parametrized spectra in the literature. There are preliminary forms
of such a theory \cite{BG1, BG2, Clapp, CP, CJ}, but these either do not
go beyond suspension spectra or are based on obsolescent technology.
None of them go nearly far enough into the theory for the purposes we
have in mind, although the early first approximation of
Monica Clapp \cite{Clapp}, written up in more detail with Dieter Puppe
\cite{CP}, deserves considerable credit. Clapp gave the strongest previous
version of our fiberwise duality theorem, and her emphasis on ex-fibrations,
together with some key technical results about them, have been very helpful.
The reader primarily interested in classical homotopy theory should ignore all
details of equivariance in reading Chapters 11--13. In fact,
given \cite{MM}, the equivariance adds few serious difficulties to the passage from spectra to parametrized spectra, although it does add many interesting new features.
There are at least two possible alternative cumulative approaches. Rather
than building on the theory of orthogonal $G$-spectra of \cite{MM, MMSS},
one can build on the theory of $G$-spectra of \cite{LMS}, the theory of
$S$-modules of \cite{EKMM}, and the pushout of these, the theory of
$S_G$-modules of \cite{MM}. Po Hu \cite{Hu} began work on the first
stage of a treatment along these lines, using parametrized $G$-spectra,
but she did not address the foundational issues concerning smash products, function spectra, base change functors, and compatibility relations considered
here. Moreover, following the first author's misleading unpublished notes \cite{May0}, she took the $q$-model structure on ex-$G$-spaces
as her starting point, and the stable model structure cannot be made rigorous from there. It appears to us that resolving all of these issues in that framework is likely to be more difficult than in the framework that we have adopted. In particular, homotopical control of the parametrized spectrification functor and of cofiber sequences seems problematic.
Alternatively, for finite groups $G$, one can build on the theory of symmetric
spectra of Hovey, Smith, and Shipley \cite{HSS} and its equivariant
generalization due to Mandell \cite{Mandell}. Such an approach would avoid
the point-set topological technicalities of the present approach and would
presumably lead to rather different looking problems with fibrations and
cofibrations. The problems with the stable homotopy category level adjunctions
that involve base change functors, smash products, and function spectra are
intrinsic and would remain. Our solutions to these problems do not seem to
carry over to the simplicial context in an obvious way, and an alternative simplicial treatment could prove to be quite illuminating.
In view of the understanding of unstable equivariant homotopy
theory for proper actions of non-compact Lie groups that was
obtained in Part II, it might seem that there should be no real
difficulty in obtaining a good stable theory along the same lines
as the theory for compact Lie groups. However, in contrast with the
rest of this book, equivariant stable homotopy theory for non-compact Lie
groups is in preliminary and incomplete form, with still unresolved
technical problems. We leave its study to future work, explaining
in \S11.6 where some of the problems lie. Except in that section,
$G$ is asssumed to be a compact Lie group from Chapter 11 onwards.
A few other notes on terminology may be helpful. We shall not use the term ``ex-spectrum over $B$'' since, stably, there is no meaningful unsectioned theory. Instead, we shall use the term ``spectrum over $B$''. This is especially convenient when considering base change. We write out ``orthogonal $G$-spectrum over $B$'' until \S11.4. However, since we consider no other kinds
of $G$-spectra and work equivariantly throughout, we later abbreviate this
to ``spectrum over $B$'' when there is no danger of confusion. That is,
we work equivariantly throughout, but we only draw attention to this fact
when it plays a significant mathematical role.
\chapter{Enriched categories and $G$-categories}
\section*{Introduction}
To give context for the structure enjoyed by the categories of
parametrized orthogonal $G$-spectra that we shall define, we first
describe the kind of equivariant para\-metrized enrichments that we shall encounter. In fact, our categories have several layers of enrichment, and
it is helpful to have a consistent language, somewhat non-standard from a categorical point of view, to keep track of them.
In \S\S10.1 and 10.2, we give some preliminaries on enriched categories,
working non\-equi\-variant\-ly in \S10.1 and adding considerations of
equivariance in \S10.2. We discuss the role of the several enrichments
in sight in our $G$-topological model $G$-categories in \S10.3. In
this chapter, $G$ can be any topological group.
\section{Parametrized enriched categories}
As discussed in \S1.2, all of our categories $\scr{C}$ are topological,
meaning that they are enriched over the category $\scr{K}_*$ of based spaces
(= $k$-spaces). In contrast with general enriched category theory and our further enrichments, the topological enrichment is given just by a topology on the underlying set of morphisms, and we denote the space of morphisms $X\longrightarrow Y$ by $\scr{C}(X,Y)$. We say that a topological category $\scr{C}$ is {\em topologically bicomplete}\, if it is bicomplete and bitensored over
$\scr{K}_*$. In fact, we shall have enrichments and bitensorings over the
category $\scr{K}_B$ of ex-spaces over $B$ that imply the topological
enrichment and bitensoring by restriction to ex-spaces $B\times T$ for $T\in\scr{K}_*$.
Recall from \S1.3 that $\scr{K}_B$ is topologically bicomplete, with tensors
and cotensors denoted by $K\sma_B T$ and $F_B(T,K)$ for $T\in\scr{K}_*$ and $K\in \scr{K}_B$. (Since we shall use letters like $X$, $Y$, and $Z$ for spectra, we
have changed the letters that we use generically for spaces and ex-spaces
from those that we used earlier). It is also closed symmetric monoidal under
its fiberwise smash
product and function space functors, which are also denoted by $\sma_B$ and $F_B$; its unit object is $S^0_B = B\times S^0$. It is therefore enriched and bitensored over itself. The two enrichments are related by natural based homeomorphisms
\begin{equation}\label{eqn:enrich}
\scr{K}_B(K,L) \iso \scr{K}_B(S^0_B, F_B(K,L)).
\end{equation}
This is the case $T = S^0$ of the more general based homeomorphism
\begin{equation}\label{eqn:autoenrich}
\scr{K}_*(T,\scr{K}_B(K,L)) \iso \scr{K}_B(S^0_B\sma_B T, F_B(K,L))
\end{equation}
for $T\in \scr{K}_*$ and $K$, $L\in \scr{K}_B$. The Yoneda lemma, (\ref{eqn:enrich}),
and the bitensoring adjunctions imply that the two bitensorings are related by
the equivalent natural isomorphisms of ex-spaces
\begin{equation}\label{eqn:compatible}
K\wedge_BT \cong K\wedge_B (S^0_B\wedge_B T)
\quad\text{and}\quad F_B(T,K)\cong F_B(S^0_B\wedge_B T, K).
\end{equation}
These in turn imply the equivalent generalizations
\begin{equation}\label{eqn:trans}
K\sma_B (L\sma_B T)\iso (K\sma_B L)\sma_B T
\quad\text{and}\quad F_B(T, F_B(K,L))\iso F_B(K\sma_B T,L).
\end{equation}
Formally, rather than defining the enrichments and bitensorings over $\scr{K}_*$ independently,
we can take (\ref{eqn:autoenrich}) and (\ref{eqn:compatible}) as definitions of these
structures in terms of the enrichment and bitensoring over $\scr{K}_B$. Then (\ref{eqn:trans})
and the bitensoring adjunction homeomorphisms
\begin{equation}
\scr{K}_B(K\sma_B T, L)\iso \scr{K}_*(T,\scr{K}_B(K,L))\iso \scr{K}_B(K,F_B(T,L))
\end{equation}
follow directly.
\begin{rem}\mylabel{confusion}
We shall be making much use of the functor
$S^0_B\sma_B(-)$, and we henceforward abbreviate
notation by setting
$$ T_B = B\times T = S^0_B\sma_B T$$
for a based space $T$, and similarly for maps. Observe that
$K\sma_B T$ and $K\sma_B T_B$ are two names for the same
ex-space over $B$. When working on a formal conceptual level,
it is often best to think in terms of tensors over $\scr{K}_*$ and
use the first name. However, on a pragmatic level, to avoid
confusion, it is best to view based spaces as embedded
in ex-spaces via $S^0_B\sma_B(-)$ and to use the second
notation, working only with tensors over $\scr{K}_B$.
\end{rem}
We generalize and formalize several aspects of the discussion above.
\begin{defn}
A topological category $\scr{C}$ is \emph{topological over $B$}\index{category!topological over B@topological over $B$} if it is enriched and bitensored over $\scr{K}_B$. It is \emph{topologically bicomplete over $B$}\index{category!topologically bicomplete over B@topologically bicomplete over $B$} if it is
also bicomplete. We write $P_B(X,Y)$\@bsphack\begingroup \@sanitize\@noteindex{PBXY@$P_B(X,Y)$} for
the hom ex-space over $B$, and we write $X\sma_B K$\@bsphack\begingroup \@sanitize\@noteindex{XBK@$X\sma_B K$} and $F_B(K,X)$\@bsphack\begingroup \@sanitize\@noteindex{FBKX@$F_B(K,X)$} for the
tensor and cotensor in $\scr{C}$, where $X$, $Y\in\scr{C}$ and $K\in\scr{K}_B$.
Explicitly, we
require bitensoring adjunction homeomorphisms of based spaces
\begin{equation}\label{eqn:adj}
\scr{C}(X\wedge_B K, Y)\cong \scr{K}_B(K, P_B(X,Y)) \cong \scr{C}(X, F_B(K,Y)).
\end{equation}
By Yoneda lemma arguments, these imply unit and transitivity isomorphisms in $\scr{C}$
\begin{equation}\label{eqn:transag}
X\iso X\sma_B S^0_B\quad\text{and}\quad X\sma_B (K\sma_B L) \iso (X\sma_B K)\sma_B L.
\end{equation}
and also bitensoring adjunction isomorphisms of ex-spaces
\begin{equation}\label{eqn:intadj}
P_B(X\wedge_B K, Y)\cong F_B(K, P_B(X,Y)) \cong P_B(X,F_B(K,Y)).
\end{equation}
Conversely, there is a natural homeomorphism
\begin{equation}\label{eqn:under}
\scr{C}(X,Y)\cong \scr{K}_{B}(S^0_B, P_B(X,Y)),
\end{equation}
and the isomorphisms (\ref{eqn:adj}) follow from (\ref{eqn:intadj}) by
applying $\scr{K}_B(S^0_B,-)$.
\end{defn}
If we do not require $\scr{C}$ to be topological to begin with, we can take
(\ref{eqn:under}) as the definition of the space $\scr{C}(X,Y)$ and so recover
the topological enrichment. With the
notation of \myref{confusion}, we obtain tensors and cotensors
with based spaces $T$ by setting
\begin{equation}\label{eqn:biten3}
X\sma_B T = X\sma_B T_{B} \quad\text{and}\quad F_B(T,X) = F_B(T_{B}, X).
\end{equation}
The adjunction homeomorphisms
\begin{equation}
\scr{C}(X\wedge_B T, Y)\cong \scr{K}_*(T, \scr{C}(X,Y)) \cong \scr{C}(X, F_B(T,Y))
\end{equation}
are obtained by replacing $K$ by $T_{B}$ in (\ref{eqn:adj}) and using
(\ref{eqn:autoenrich}) and (\ref{eqn:under}).
In the cases of interest, $\scr{C}$ is closed symmetric monoidal, and then the
hom ex-spaces $P_B(X,Y)$ can be understood in terms of the internal hom in $\scr{C}$ by the following definition and result.
\begin{defn}\mylabel{topsym}
Let $\scr{C}$ be a topological category over $B$ with a closed symmetric
monoidal structure given by a product $\sma_B$ and function object
functor $F_B$, with unit object $S_B$. We say that $\scr{C}$ is a \emph{topological
closed symmetric monoidal category over $B$}\index{category!topological closed symmetric monoidal over B@topological closed symmetric monoidal over $B$} if the tensors and products
are related by a natural isomorphism
\[X\wedge_B K \cong X\wedge_B(S_B\wedge_B K)\]
in $\scr{C}$ for $K\in \scr{K}_B$ and $X\in \scr{C}$.
\end{defn}
\begin{prop}\mylabel{prop:PvsF}
Let $\scr{C}$ be a topological closed symmetric monoidal category over $B$.
Then, for $K\in\scr{K}_B$ and $X$, $Y$, $Z\in \scr{C}$, there are natural isomorphisms
\begin{gather*}
F_B(K, Y) \cong F_B(S_B\wedge_B K, Y),\\
P_B(X, Y)\iso P_B(S_B,F_B(X,Y)),\\
P_B(X\wedge_B Y, Z)\cong P_B(X,F_B(Y,Z))
\end{gather*}
in $\scr{C}$ and a natural homeomorphism of based spaces
\[\scr{K}_B(K,P_B(X,Y)) \cong \scr{C}(S_B\wedge_B K, F_B(X,Y)).\]
\end{prop}
\section{Equivariant parametrized enriched categories}
Turning to the equivariant generalization, we give details of the context
of topological $G$-categories, continuous $G$-functors, and natural $G$-maps
that we first alluded to in \S1.4. The discussion elaborates on that given
in \cite[II\S1]{MM}. Generically, we use notations of the form $\scr{C}_G$ and $G\scr{C}$ to denote a category $\scr{C}_G$ enriched over the category $G\scr{K}_*$ of
based $G$-spaces and its associated ``$G$-fixed category'' $G\scr{C}$ with the same objects and the $G$-maps between them; $G\scr{C}$ is enriched over $\scr{K}_*$. We shall write $(\scr{C}_G,G\scr{C})$ for such a pair, and we shall refer to the pair as a ``$G$-category''.
In the terminology of enriched category theory, $G\scr{C}$ is the
underlying topological category of $\scr{C}_G$. The hom objects of
$\scr{C}_G$ are $G$-spaces $\scr{C}_G(X,Y)$; $G$-functors and natural $G$-maps just
mean functors and natural transformations enriched over $G\scr{K}_*$.
Consistently with enriched category theory, the space
$G\scr{C}(X,Y) = \scr{C}_G(X,Y)^G$ can be identified with the space of $G$-maps
$S^0\longrightarrow \scr{C}_G(X,Y)$. We call the points of $\scr{C}_G(X,Y)$ ``arrows'' to distinguish them from the points of $G\scr{C}(X,Y)$, which we call ``$G$-maps'',
or often just ``maps'', with the equivariance understood.
We cannot expect $\scr{C}_G$ to have limits and colimits, but $G\scr{C}$ is usually
bicomplete. In many of our examples, both $\scr{C}_G$ and $G\scr{C}$ are closed symmetric
monoidal under functors $\sma_B$ and $F_B$. For example, we have the closed
symmetric monoidal $G$-category $(\scr{K}_{G,B},G\scr{K}_B)$ of ex-$G$-spaces over a
$G$-space $B$ described in \S1.4.
\begin{defn}\mylabel{defn:enrichBG}
A $G$-category $(\scr{C}_G,G\scr{C})$\@bsphack\begingroup \@sanitize\@noteindex{CG@$\scr{C}_B$}\@bsphack\begingroup \@sanitize\@noteindex{GC@$G\scr{C}$} is \emph{$G$-topological over $B$}\index{category!G-topological over B@$G$-topological over $B$} if $\scr{C}_G$
is enriched over $G\scr{K}_B$ and bitensored over $\scr{K}_{G,B}$. It follows that
$G\scr{C}$ is enriched over $\scr{K}_B$ and bitensored over $G\scr{K}_B$. We say
that $(\scr{C}_G,G\scr{C})$ is \emph{$G$-topologically bicomplete over $B$}\index{category!G-topologically bicomplete over B@$G$-topologically bicomplete over $B$} if, in addition, $G\scr{C}$ is bicomplete. We write $P_B(X,Y)$ for the hom ex-$G$-space
over $B$, and we write $X\sma_B K$ and $F_B(K,X)$ for the
tensor and cotensor in $\scr{C}_G$, where $X$, $Y\in\scr{C}_G$ and $K\in\scr{K}_{G,B}$.
Explicitly, we require bitensoring adjunction homeomorphisms of based $G$-spaces
\begin{equation}
\scr{C}_G(X\wedge_B K, Y)\cong \scr{K}_{G,B}(K, P_B(X,Y)) \cong \scr{C}_G(X, F_B(K,Y)).
\mylabel{eqn:adjG}
\end{equation}
There result coherent unit and transitivity isomorphisms in $G\scr{C}$
\begin{equation}\label{eqn:transagG}
X\iso X\sma_B S^0_B\quad \text{and}\quad X\sma_B (K\sma_B L) \iso (X\sma_B K)\sma_B L
\end{equation}
and also bitensoring adjunction isomorphisms of ex-$G$-spaces
\begin{equation}\label{eqn:intadjG}
P_B(X\wedge_B K, Y)\cong F_B(K, P_B(X,Y)) \cong P_B(X,F_B(K,Y)).
\end{equation}
Conversely, there is a natural homeomorphism of based $G$-spaces
\begin{equation}\label{eqn:underG}
\scr{C}_G(X,Y)\cong \scr{K}_{G,B}(S^0_B, P_B(X,Y)),
\end{equation}
and the isomorphisms (\ref{eqn:adjG}) follow from (\ref{eqn:intadjG}) by
applying $\scr{K}_{G,B}(S^0_B,-)$. Passage to $G$-fixed points from
(\ref{eqn:adjG})
gives the bitensoring adjunction homeomorphisms of based spaces
\begin{equation}\label{eqn:adjG2}
G\scr{C}(X\wedge_B K, Y)\cong G\scr{K}_B(K, P_B(X,Y)) \cong G\scr{C}(X, F_B(K,Y)).
\end{equation}
\end{defn}
We warn the reader that we shall not always adhere strictly to the notational
pattern of \myref{defn:enrichBG} for our several layers of enrichment.
In particular, in the domain categories for our equivariant diagram spaces
and diagram spectra, only $\scr{C}_G$ is of interest, not $G\scr{C}$, and our notations will reflect that. On the other hand, when studying model categories, it is always the bicomplete category $G\scr{C}$ that is of fundamental interest.
If $(\scr{C}_G,G\scr{C})$ is $G$-topological over $B$, then it is automatically
$G$-topo\-lo\-gic\-al (over $*$). This enrichment is recovered by taking (\ref{eqn:under}),
read equivariantly, as the definition of the based $G$-space $\scr{C}_G(X,Y)$.
Just as in the nonequivariant case, for based $G$-spaces $T$ and objects
$X$ of $\scr{C}_G$, the tensors and cotensors in $\scr{C}_G$ and $G\scr{C}$ are given
on objects by
\begin{equation}\label{SOB}
X\wedge_B T = X\wedge_B T_{B} \quad\text{and}\quad F_B(T,X) = F_B(T_{B}, X),
\end{equation}
using the notation of \myref{confusion} equivariantly. The required $G$-homeomorphisms
\begin{equation}\label{SOB2}
\scr{C}_G(X\wedge_B T, Y)\cong \scr{K}_{G,*}(T, \scr{C}_G(X,Y)) \cong \scr{C}_G(X, F_B(T,Y))
\end{equation}
follow directly.
We have equivariant analogues of \myref{topsym} and \myref{prop:PvsF}.
\begin{defn}\mylabel{topsymG}
Let $(\scr{C}_G,G\scr{C})$ be a $G$-topological $G$-category over $B$ with a closed
symmetric monoidal structure given by a product $G$-functor $\sma_B$ and a
function object $G$-functor $F_B$, with unit object $S_B$. We say that $(\scr{C}_G,G\scr{C})$
is a \emph{$G$-topological closed symmetric monoidal $G$-category over $B$}\index{category!G-toplogical closed symmetric@$G$-topological closed symmetric monoidal over $B$} if the tensors and
products are related by a natural isomorphism
\[X\wedge_B K \cong X\wedge_B(S_B\wedge_B K)\]
in $G\scr{C}$ for $K\in G\scr{K}_B$ and $X\in G\scr{C}$.
\end{defn}
\begin{prop}\mylabel{prop:PvsFG}
Let $(\scr{C}_G,G\scr{C})$ be a $G$-topological closed symmetric mon\-oid\-al $G$-category over $B$.
Then, for $K\in\scr{K}_B$ and $X$, $Y$, $Z\in \scr{C}$, there are natural isomorphisms
\begin{gather*}
F_B(K, Y) \cong F_B(S_B\wedge_B K, Y),\\
P_B(X, Y)\iso P_B(S_B,F_B(X,Y)),\\
P_B(X\wedge_B Y, Z)\cong P_B(X,F_B(Y,Z))
\end{gather*}
in $G\scr{C}$ and there is a natural homeomorphism of based $G$-spaces
\[\scr{K}_{G,B}(K,P_B(X,Y))\cong \scr{C}_G(S_B\wedge_B K, F_B(X,Y)).\]
\end{prop}
\section{$G$-topological model $G$-categories}
We explain what it means for a $G$-topological $G$-category $(\scr{C}_G, G\scr{C})$
over $B$ to have a $G$-topological model structure. This structure implies
in particular that the homotopy category $\text{Ho}G\scr{C}$ is bitensored over
the homotopy category $\text{Ho}G\scr{K}$. We need some notation.
Throughout this section, we consider maps
$$i\colon W\longrightarrow X, \ j\colon V\longrightarrow Z,\ \text{and} \ p\colon E\longrightarrow Y$$
in $G\scr{C}$ and a map $k\colon K\longrightarrow L$ in either $G\scr{K}_B$ or $G\scr{K}_*$; in
the latter case we apply the functor $(-)_B = B\times (-)$ to $k$ and so regard it as a map in $G\scr{K}_B$, as suggested in \myref{confusion}. We shall define the notion of a $G$-topological model category in terms of the induced map
\begin{equation}\label{gbox3}
\scr{C}_G^\Box(i,p)\colon \scr{C}_G(X,E) \longrightarrow \scr{C}_G(W,E)\times_{\scr{C}_G(W,Y)}\scr{C}_G(X,Y)
\end{equation}
of based $G$-spaces. Passing to $G$-fixed points, this gives rise to a map
\begin{equation}\label{gbox3bis}
G\scr{C}^\Box(i,p)\colon G\scr{C}(X,E) \longrightarrow G\scr{C}(W,E)\times_{G\scr{C}(W,Y)}G\scr{C}(X,Y)
\end{equation}
of based spaces, and we have the following motivating observation.
\begin{lem}\mylabel{lemma:lppair}
The pair $(i,p)$ has the lifting property if and only if the function $G\scr{C}^\Box(i,p)$ is surjective.
\end{lem}
\begin{defn}\mylabel{Gtopmodel}
Let $(\scr{C}_G,G\scr{C})$ be a $G$-topological $G$-category over $B$ such that
$G\scr{C}$ is a model category. We say that the model structure is
\emph{$G$-topological}\index{G-topological@$G$-topological}\index{model category!G-top@$G$-topological} if $\scr{C}_G^\Box(i,p)$ is a fibration in
$G\scr{K}_*$ when $i$ is a cofibration and $p$ is a fibration and is acyclic
when, further, either $i$ or $p$ is acyclic.
\end{defn}
\begin{rem}\mylabel{Gtopfamily}
The definition must refer consistently to either $h$-type or $q$-type model
structures. The resulting notions are quite different. We usually have in
mind a $q$-type model structure. In that case, the weak equivalences and
fibrations are often characterized by conditions on the $H$-fixed point maps
$f^H$ of a map $f$. If $\scr{F}$ is a family of subgroups of $G$, such as the
family $\scr{G}$ of compact subgroups, we can restrict attention to those $H\in
\scr{F}$. The resulting $\scr{F}$-equivalences and $\scr{F}$-fibrations usually specify
another model structure on $G\scr{C}$. In particular, we have the $\scr{F}$-model
structure on $G\scr{K}_*$. For the $qf$-type model structures of \S7.2, we must
start with a generating set $\scr{C}$ that contains the orbits $G/H$ with $H\in
\scr{F}\cap\scr{G}$ and consists of $\scr{F}\cap\scr{G}$-cell complexes. We say that an
$\scr{F}$-model structure on $G\scr{C}$ is {\em $\scr{F}$-topological} if the condition
of the previous definition holds when we use the $\scr{F}$-notions of fibration,
cofibration and weak equivalence throughout. The observations of this
section generalize to $\scr{F}$-topological model categories for any family
$\scr{F}$.
\end{rem}
In addition to the map of $G$-spaces displayed in (\ref{gbox3}), we have a map
\begin{equation}\label{PBoxmap}
P^\Box_B(i,p)\colon P_B(X,E)\longrightarrow P_B(W,E)\times_{P_B(W,Y)} P_B(X,Y)
\end{equation}
of ex-$G$-spaces over $B$.
\begin{warn}
We can define what it means for $(\scr{C}_G,G\scr{C})$ to be $G$-topological
{\em over $B$}, using the map $P^{\Box}_B(i,p)$ of ex-spaces rather than the
map $\scr{C}_G^\Box(i,p)$ of spaces. However, we know of no examples where this
condition is satisfied. For example, $(\scr{K}_{G,B},G\scr{K}_B)$ is $G$-topological, by Theorems \ref{qoverB} and \ref{Gqfstr}, but, as \myref{ouchtoo} makes clear by
adjunction, we cannot expect it to be $G$-topological over $B$.
\end{warn}
Just as in the classical theory of simplicial or topological model categories, there are various equivalent reformulations of what it means for $G\scr{C}$ to be $G$-topological. To explain them, observe that the tensors and cotensors with
ex-$G$-spaces over $B$ give rise to induced maps
\begin{equation}\label{gbox1}
\ \ \ \ i\Box_B k\colon (X\wedge_B K) \cup_{W\wedge_B K} (W\wedge_B L)
\longrightarrow X\wedge_B L
\end{equation}
and
\begin{equation}\label{gbox2}
F^\Box_B(k,p)\colon F_B(L,E)\longrightarrow F_B(K,E)\times_{F_B(K,Y)} F_B(L,Y)
\end{equation}
of ex-$G$-spaces over $B$. If $(\scr{C}_G,G\scr{C})$ is closed symmetric monoidal,
then we also have the induced maps
\begin{equation}
i\Box_B j\colon (X\wedge_B V) \cup_{W\wedge_B V} (W\wedge_B Z)
\longrightarrow X\wedge_B Z
\end{equation}
and
\begin{equation}
F^\Box_B(j,p)\colon F_B(Z,E)\longrightarrow F_B(V,E)\times_{F_B(V,Y)} F_B(Z,Y)
\end{equation}
in $G\scr{C}$. We have various adjunction isomorphisms relating these various $\Box$-product maps and $\Box$-function object maps.
\begin{prop}\mylabel{prop:topveri}
If $k$ is a map of ex-$G$-spaces over $B$, then there are adjunction isomorphisms
\begin{equation}
P^\Box_B(i\Box_B k, p)\cong F^\Box_B(k,P^\Box_B(i,p))
\cong P^\Box_B(i,F^\Box_B(k,p))
\end{equation}
of maps of ex-$G$-spaces over $B$ and
\begin{equation}
\scr{C}^\Box_G(i\Box_B k, p)\cong \scr{K}^\Box_{G,B}(k,P^\Box_B(i,p))
\cong \scr{C}^\Box_G(i,F^\Box_B(k,p))
\end{equation}
of maps of based $G$-spaces. If $k$ is a map of based $G$-spaces, then
the last pair of isomorphisms can be rewritten as
\begin{equation}\label{gtopequiv}
\scr{C}^\Box_G(i\Box_B k, p)\cong \scr{K}^\Box_{G,*}(k,\scr{C}^\Box_G(i,p))
\cong \scr{C}^\Box_G(i,F^\Box_B(k,p)).
\end{equation}
When $(\scr{C}_G,G\scr{C})$ is closed symmetric monoidal there are
adjunction isomorphisms
\begin{equation}
P^\Box_B(i\Box_B k, p)\cong P^\Box_B(i,F^\Box_B(k,p))
\end{equation}
of maps of ex-$G$-spaces over $B$ and
\begin{equation}
\scr{C}^\Box_G(i\Box_B k, p)\cong \scr{C}^\Box_G(i,F^\Box_B(k,p))
\end{equation}
of maps of based $G$-spaces.
\end{prop}
Together with \myref{lemma:lppair}, this implies the promised alternative equivalent conditions that describe when a model category is $G$-topological.
\begin{prop}\mylabel{Gtopchar}
Let $(\scr{C}_G,G\scr{C})$ be a $G$-topological $G$-category over $B$ such that $G\scr{C}$ has a model structure. Then the following conditions are equivalent.
\begin{enumerate}[(i)]
\item The map $i\Box_B k$ of (\ref{gbox1}) is a cofibration in $G\scr{C}$ if $i$ is a cofibration in $G\scr{C}$ and $k$ is a cofibration in $G\scr{K}_*$. It is acyclic if either $i$ or $k$ is acyclic.
\item The map $F^\Box_B(k,p)$ of (\ref{gbox2}) is a fibration in $G\scr{C}$ if $p$ is a fibration in $G\scr{C}$ and $k$ is a cofibration in $G\scr{K}_*$. It is acyclic if either $p$ or $k$ is acyclic.
\item The map $\scr{C}^\Box_G(i,p)$ of (\ref{gbox3}) is a fibration in $G\scr{K}_*$ if $i$ is a cofibration in $G\scr{C}$ and $p$ is a fibration in $G\scr{C}$. It is acyclic if either $i$ or $p$ is acyclic.
\end{enumerate}
\end{prop}
\begin{proof}
The third condition is our definition of the model structure being
$G$-topological. We prove that the first condition is equivalent to the
third. A similar argument shows that the second condition is also equivalent
to the third. The map $\scr{C}^\Box_G(i,p)$ is a fibration if and only if $(k,\scr{C}^\Box_G(i,p))$ has the lifting property with respect to all acyclic cofibrations $k$ in $G\scr{K}_*$. By \myref{lemma:lppair} and the first adjunction isomorphism in (\ref{gtopequiv}), that holds if and only if $(i\Box_B k, p)$
has the lifting property, that is, if and only if $i\Box_B k$ is an acyclic cofibration. If either $i$ or $p$ is acyclic, then we take $k$ to be a
cofibration in $G\scr{K}_*$ and argue similarly.
\end{proof}
\chapter{The category of orthogonal $G$-spectra over $B$}
\section*{Introduction}
Intuitively, an orthogonal spectrum $X$ over $B$ consists of ex-spaces
$X(V)$ over $B$ and ex-maps $\sigma\colon X(V)\sma_B S^W\longrightarrow X(V\oplus W)$ for
suitable inner product spaces $V$ and $W$. The orthogonal group $O(V)$
must act on $X(V)$, and $\sigma$ must be $(O(V)\times O(W))$-equivariant. The
orthogonal group actions enable the definition of a good external smash
product. Moreover, they will later allow us to define stable weak equivalences
in terms of homotopy groups, as would not be possible if we only had
actions by symmetric groups.
Similarly, use of general inner product
spaces allows us to build in actions by a compact Lie group $G$ without
difficulty. For non-compact Lie groups, we should ignore inner products
and use linear isomorphisms, replacing the compact
orthogonal group $O(V)$ by the general linear group $GL(V)$.
However, as we explain in \S11.6, there are more serious problems
in generalizing to non-compact Lie groups; except in that section,
we require $G$ to be a compact Lie group.
Working equivariantly, we first describe $X$ as a suitable diagram of
ex-$G$-spaces in \S11.1. The domain category for our diagrams is denoted
$\scr{I}_G$ and is independent of $B$. We then build in the structure maps $\sigma$
in \S11.2, where we define the category of orthogonal $G$-spectra over $B$.
In \S11.3, we show that it too can be described as a category of diagrams
of ex-$G$-spaces. The domain category here is denoted $\scr{J}_{G,B}$. It does depend on $B$, as indicated by the notation. The formal properties of the category of ex-$G$-spaces
over $B$ carry over to the category of orthogonal $G$-spectra over $B$,
but there are some new twists. For example, our category of $G$-spectra
over $B$ is enriched not just over based $G$-spaces, but more generally
over ex-$G$-spaces over $B$. We discussed the relevant formalities in
the previous chapter. This enhanced enrichment is essential to the
definition of function $G$-spectra over $B$.
We show in \S11.4 that the base change functors and their properties also
carry over to these categories of parametrized $G$-spectra, and we discuss change of group functors and restriction to fibers in \S11.5.
\section{The category of $\scr{I}_G$-spaces over $B$}
We recall the $G$-category $(\scr{I}_G,G\scr{I})$ from \cite[II.2.1]{MM}. The objects and arrows of $\scr{I}_G$ are finite dimensional $G$-inner product spaces and linear isometric isomorphisms. The maps of $G\scr{I}$ are $G$-linear isometries. More precisely, as dictated by the general theory
of \cite{MM, MMSS}, we take $\scr{I}_G(V,W)$ as based with basepoint disjoint from the space of linear isometric isomorphisms $V\longrightarrow W$. As in \cite[II.1.1]{MM}, the objects $V$ run over the collection $\scr{V}$ of all representations that embed up to isomorphism in a given ``$G$-universe'' $U$, where a $G$-universe is a sum of countably many copies of representations in a set of representations that includes the trivial representation. We think in terms of a ``complete $G$-universe'', one that contains
all representations of $G$, but the choice is irrelevant until otherwise
stated.
As in \cite[II.2.2]{MM}, we can restrict from $\scr{V}$ to any cofinal
subcollection $\scr{W}$ that is closed under direct sums.
Based $G$-spaces are ex-$G$-spaces over $*$, and $\scr{I}_G$-spaces are defined in
\cite[II.2.3]{MM} as $G$-functors $\scr{I}_G\longrightarrow \scr{T}_G$, where $\scr{T}_G$ is the
$G$-category of compactly generated based $G$-spaces. One can just as well drop the weak Hausdorff condition, which plays no necessary mathematical role in
\cite{MM, MMSS}, and allow general $k$-spaces. With the notations of Part II,
we can thus change the target $G$-category to
$\scr{K}_{G,*}$. Then we generalize the definition to the parametrized context simply
by changing the target $G$-category to the category $\scr{K}_{G,B}$ of ex-$G$-spaces over a $G$-space $B$. Thus we define an $\scr{I}_G$-space $X$ over (and under) $B$
to be a $G$-functor $X\colon \scr{I}_G\longrightarrow \scr{K}_{G,B}$. Using nonequivariant arrows and equivariant maps, we obtain the $G$-category $(\scr{I}_G\scr{K}_B,G\scr{I}\scr{K}_B)$ of $\scr{I}_G$-spaces.
To unravel definitions, for each representation $V\in\scr{V}$ we are given an ex-$G$-space $X(V)$
over $B$, for each arrow (linear isometric isomorphism) $f\colon V\longrightarrow W$ we are given an arrow
(non-equivariant map)
\[X(f)\colon X(V)\longrightarrow X(W)\]
of ex-$G$-spaces over $B$, and the continuous function
\[X\colon \scr{I}_G(V,W)\longrightarrow \scr{K}_{G,B}(X(V),X(W))\]
is a based $G$-map. An arrow $\alpha\colon X\longrightarrow Y$ is just a natural transformation,
and a $G$-map is a $G$-natural transformation, for which each $\alpha_V\colon X(V)\longrightarrow Y(V)$
is a $G$-map. For both arrows and $G$-maps, the naturality diagrams
$$\xymatrix{
X(V)\ar[r]^-{\alpha_V} \ar[d]_{X(f)} & Y(V) \ar[d]^{Y(f)}\\
X(W)\ar[r]_-{\alpha_W} & Y(W)}$$
must commute for all arrows $f\colon V\longrightarrow W$. The group $G$ acts on the space
$\scr{I}_G\scr{K}_B(X,Y)$ of arrows by levelwise conjugation. The $G$-fixed category
is denoted by $G\scr{I}\scr{K}_B$. It has objects the $\scr{I}_G$-spaces $X$ and maps the $G$-maps.
To study the parametrized enrichment of the $G$-category of orthogonal
$G$-spectra over $B$, it is convenient to extend the domain category $\scr{I}_G$,
which is enriched over $\scr{K}_{G,*}$, to a new domain category $\scr{I}_{G,B}$ that
is enriched over $\scr{K}_{G,B}$. Departing from the notational pattern of
\myref{defn:enrichBG} and using \myref{confusion}, we define the hom ex-$G$-spaces over $B$ of $\scr{I}_{G,B}$ by
\begin{equation}\label{IGB}
\scr{I}_{G,B}(V,W) = \scr{I}_G(V,W)_B \equiv B\times \scr{I}_G(V,W).
\end{equation}
If $X\colon \scr{I}_G\longrightarrow \scr{K}_{G,B}$ is an $\scr{I}_G$-space,
then the given based $G$-maps
$$X\colon \scr{I}_G(V,W) \longrightarrow \scr{K}_{G,B}(X(V),X(W))$$
correspond by adjunction (see (\ref{SOB}) and (\ref{SOB2})) to ex-$G$-maps
$$X(V)\sma_B \scr{I}_{G,B}(V,W)\longrightarrow X(W).$$
In turn, these correspond by the internal hom adjunction to ex-$G$-maps
$$X\colon \scr{I}_{G,B}(V,W)\longrightarrow F_B(X(V),X(W)).$$
These give an equivalent version of the original $G$-functor $X$, but
now in terms of categories enriched over the category $G\scr{K}_B$.
\begin{lem}\mylabel{omnilem}
The $G$-category $(\scr{I}_G\scr{K}_B,G\scr{I}\scr{K}_B)$ of $\scr{I}_G$-spaces is equivalent
to the $G$-category of $\scr{I}_{G,B}$-spaces, where an $\scr{I}_{G,B}$-space is a
$G$-functor $X\colon \scr{I}_{G,B}\longrightarrow \scr{K}_{G,B}$ enriched over $G\scr{K}_B$.
\end{lem}
\begin{prop}\mylabel{omnispace}
The $G$-category $(\scr{I}_G\scr{K}_B,G\scr{I}\scr{K}_B)$ is $G$-top\-o\-log\-i\-cal
over $B$ and thus also $G$-topological. Therefore the category
$G\scr{I}\scr{K}_B$ is topologically bicomplete over $B$.
\end{prop}
\begin{proof}
We define tensor and cotensor $\scr{I}_G$-spaces over $B$
$$ X\sma_B K \qquad\text{and}\qquad F_B(K,X)$$
levelwise, where $K$ is an ex-$G$-space and $X$ is an $\scr{I}_G$-space.
For $\scr{I}_G$-spaces $X$ and $Y$, we must define a parametrized morphism
ex-$G$-space $P_B(X,Y)$ over $B$. Parallelling a standard formal description
of the $G$-space $\scr{I}_G\scr{K}_B(X,Y)$, we define $P_B(X,Y)$ to be the end
\begin{equation}\label{P}
P_B(X,Y) = \int_{\scr{I}_{G,B}} F_B(X(V),Y(V)).
\end{equation}
Explicitly, it is the equalizer displayed in the following diagram of ex-$G$-spaces.
$$
\xymatrix{P_B(X,Y) \ar[d]\\
\prod_{V}F_B(X(V),Y(V))
\ar@<1ex>[d]^-{\tilde{\nu}}
\ar@<-1ex>[d]_-{\tilde{\mu}}\\
\prod_{V,W} F_B(\scr{I}_{G,B}(V,W),F_B(X(V),Y(W))).}
$$
The products run over the objects and pairs of objects of a skeleton $sk\scr{I}_G$
of $\scr{I}_G$. The $(V,W)$th coordinate of $\tilde{\mu}$ is given by the composite of the
projection to $F_B(X(W),Y(W))$ and the $G$-map
$$F_{B}(X(W),Y(W))
\longrightarrow F_B(\scr{I}_{G,B}(V,W),F_{B}(X(V),Y(W)))$$
adjoint to the composite ex-$G$-map
\[\xymatrix{
F_{B}(X(W),Y(W))\sma_B \scr{I}_{G,B}(V,W) \ar[d]^-{\text{id}\sma_B X}\\
F_{B}(X(W),Y(W))\sma_B F_{B}(X(V),X(W)) \ar[d]^-{\com}\\
F_B(X(V),Y(W)).}\]
The $(V,W)$th coordinate of $\tilde{\nu}$ is the composite of the
projection to $F_B(X(V),Y(V))$ and the $G$-map
$$\tilde{\nu}_{V,W}\colon F_B(X(V),Y(V))
\longrightarrow F_B(\scr{I}_{G,B}(V,W), F_B(X(V),Y(W))$$
adjoint to the composite ex-$G$-map
$$\xymatrix{
\scr{I}_{G,B}(V,W)\sma_B F_B(X(V),Y(V)) \ar[d]^-{Y\sma_B \text{id}}\\
F_{B}(Y(V),Y(W))\sma_B F_B(X(V),Y(V)) \ar[d]^-{\com}\\
F_B(X(V),Y(W)).\\}$$
Passage to ends from the isomorphisms of ex-$G$-spaces
\[F_B(X(V) \wedge_B K, Y(V))
\cong F_B(K, F_B(X(V),Y(V)))
\cong F_B(X(V), F_B(K,Y(V)))\]
gives natural isomorphisms of ex-$G$-spaces
\begin{equation}\label{Pad0}
P_B(X \wedge_B K, Y)
\cong F_B(K, P_B(X,Y))
\cong P_B(X, F_B(K,Y)).
\end{equation}
With these constructions, we see that $(\scr{I}_G\scr{K}_B,G\scr{I}\scr{K}_B)$
is $G$-top\-o\-log\-i\-cal over $B$; compare \myref{defn:enrichBG} and the
discussion following it. The last statement follows since $G\scr{I}\scr{K}_B$
is complete and cocomplete, with limits and colimits constructed levelwise
from the limits and colimits in $G\scr{K}_{B}$.
\end{proof}
We have several kinds of smash products and function objects in this context.
For $\scr{I}_G$-spaces $X$ and $Y$ over $B$, define the \emph{``external'' smash
product}\index{external smash product} $X \barwedge_B Y$\@bsphack\begingroup \@sanitize\@noteindex{XbwY@$X\barwedge_B Y$} by
$$X \barwedge_B Y = \sma_B\com (X\times Y)\colon \scr{I}_G\times \scr{I}_G \longrightarrow \scr{K}_{G,B}.$$
Thus $(X\barwedge_B Y)(V,W) = X(V)\sma_B Y(W)$. Here we have used the word ``external''
to refer to the use of pairs of representations, as is usual in the theory of diagram
spectra. It is standard category theory \cite{Day, MMSS} to use left Kan extension to
internalize this external smash product over $B$. This gives the internal smash product
$X\sma_B Y$ of
$\scr{I}_G$-spaces over $B$, which is again an $\scr{I}_G$-space over $B$. For an $\scr{I}_G$-space $Y$
over $B$ and an $(\scr{I}_G\times\scr{I}_G)$-space $Z$ over $B$, define the {\em external function
$\scr{I}_G$-space over $B$}, denoted $\bar F_B(Y,Z)$, by
$$\bar F_B(Y,Z)(V) = P_B(Y,Z\langle V\rangle),$$
where $Z\langle V\rangle (W) = Z(V,W)$. It is mainly to allow this definition that we
need the morphism ex-$G$-spaces $P_B(-,-)$. It is also formal to obtain an internal function
$\scr{I}_G$-space functor $F_B$ on $\scr{I}_G$-spaces over $B$ by use of right Kan extension
\cite{Day, MMSS}. Using these internal smash product and function $\scr{I}_G$-space functors,
we obtain the following result. Recall \myref{topsymG} and \myref{prop:PvsFG}.
\begin{thm} $(\scr{I}_G\scr{K}_B,G\scr{I}\scr{K}_B)$ is a $G$-top\-o\-lo\-gi\-cal
closed symmetric mon\-oid\-al $G$-category over $B$.
\end{thm}
\begin{rem}\mylabel{extsmash1}
In the theory of ex-spaces, we also have the ``external smash product'' of ex-spaces over different base spaces defined in \S2.5. Using the two different notions of ``external'' together, we obtain the definition of the ``external external smash product'' of an $\scr{I}_G$-space $X$ over $A$ and an $\scr{I}_G$-space $Y$ over $B$; it is an $(\scr{I}_G\times \scr{I}_G)$-space over $A\times B$. We write $X\barwedge Y$ for the left Kan extension internalization of this smash product. Thus $X\barwedge Y$ is an $\scr{I}_G$-space over $A\times B$. Similarly, using the external function ex-space construction $\bar{F}$ of \S2.5, for an $\scr{I}_G$-space $Y$ over $B$ and an $\scr{I}_G$-space $Z$ over $A\times B$, we obtain the ``internalized external function $\scr{I}_G$-space'' $\bar{F}(Y,Z)$ over $A$. Notationally, use of $\barwedge$ and $\bar{F}$ without an ensuing subscript always denotes these internalized external operations with respect to varying base spaces. We shall return to these functors in \myref{extsmash2}.
Similarly, but more simply, we have the ``external tensor'' $K\barwedge Y$ of an ex-$G$-space $K$ over $A$ and an $\scr{I}_G$-space $Y$ over $B$, which again is an $\scr{I}_G$-space over $A\times B$. When $A = *$, this is just the tensor of based $G$-spaces with $\scr{I}_G$-spaces over $B$. The case $B=*$ shows how to construct an
$\scr{I}_G$-space over $A$ from an ex-$G$-space over $A$ and an $\scr{I}_G$-space.
Since these external tensors can be view as special cases of external smash products, via variants of \myref{topsymG} and (\ref{smasma}) below, we shall not treat them formally and shall not repeat the definitions on the $G$-spectrum level. However, we shall find several uses for them.
\end{rem}
\section{The category of orthogonal $G$-spectra over $B$}
For a representation $V$ of $G$ and an $\scr{I}_G$-space $X$, we define
\begin{equation}\label{SIOM}
\Sigma^V_B X = X\sma_B S^V_B \qquad\text{and}\qquad \Omega^V_B X = F_B(S^V_B,X),
\end{equation}
where $S^V$ is the one-point compactification of $V$.
\begin{defn}\mylabel{spheres}
Define the $G$-sphere $S_B$, written $S_{G,B}$ when necessary for clarity,
to be the $\scr{I}_G$-space over $B$ that sends $V$ to $S_B^V$.
\end{defn}
Clearly $S_B^V\sma_B S_B^W\iso S_B^{V\oplus W}$, and the functor $S_{B}$ is strong symmetric monoidal, where the monoidal structure on $\scr{I}_G$ is given by direct sums. It follows that $S_{B}$ is a commutative monoid in the symmetric monoidal category $G\scr{I}\scr{K}_B$, and we can define $S_{B}$-modules $X$ in terms of (right) actions $X\sma_B S_{B}\longrightarrow X$. These $S_{B}$-modules are our orthogonal $G$-spectra over $B$, but it is more convenient to give the definition using the equivalent reformulation in terms of the external
smash product.
\begin{defn}
An \emph{$\scr{I}_G$-spectrum, or orthogonal $G$-spectrum, over $B$}\index{spectrum!over B@over $B$} is an $\scr{I}_G$-space $X$ over $B$ together with a structure $G$-map
$$\sigma\colon X\barwedge_B S_{B}\longrightarrow X\com \oplus$$
such that the
evident unit and associativity diagrams commute. Thus we have compatible
equivariant structure maps
$$\sigma\colon \Sigma_B^W X(V) = X(V)\sma_B S_B^W\longrightarrow X(V\oplus W).$$
Let $\scr{S}_{G,B}$\@bsphack\begingroup \@sanitize\@noteindex{SGB@$\scr{S}_{G,B}$} denote the topological $G$-category of $\scr{I}_G$-spectra over $B$
and arrows $f\colon X\longrightarrow Y$ that commute with the structure maps, with $G$ acting
by conjugation on arrows. Let $G\scr{S}_{B}$\@bsphack\begingroup \@sanitize\@noteindex{GSB@$G\scr{S}_B$} denote the topological category of
$\scr{I}_G$-spectra over $B$ and $G$-maps (equivariant arrows) between them.
\end{defn}
\begin{defn}
Define the suspension orthogonal $G$-spectrum functor\@bsphack\begingroup \@sanitize\@noteindex{SiB@$\Sigma^\infty_B$}
and the $0$th ex-$G$-space functor\@bsphack\begingroup \@sanitize\@noteindex{OmB@$\Omega^\infty_B$}
$$\Sigma^{\infty}_B\colon \scr{K}_{G,B}\longrightarrow \scr{S}_{G,B} \ \ \text{and}\ \
\Omega^{\infty}_B\colon \scr{S}_{G,B}\longrightarrow \scr{K}_{G,B}$$
by
$(\Sigma^{\infty}_B K)(V) = \Sigma_B^VK$, with the
evident isomorphisms as structure maps, and $\Omega^{\infty}_B X = X(0)$. Then
$\Sigma^{\infty}_B$ and $\Omega^{\infty}_B$ give left and right adjoints between
$\scr{K}_{G,B}$ and $\scr{S}_{G,B}$ and, on passage to $G$-fixed points, between $G\scr{K}_B$ and $G\scr{S}_B$.
\end{defn}
The category $G\scr{S}_{B}$ is our candidate for a good category of para\-me\-trized $G$-spectra
over $B$. It inherits all of the properties of the category $G\scr{I}\scr{K}_B$ of $\scr{I}_G$-spaces
that were discussed in the previous section and, in the case $B=*$, it is exactly the
category $G\scr{S}$ of orthogonal $G$-spectra that is studied in \cite{MM}. We summarize its formal
properties in the following omnibus theorem. In the language of \S10.2, much of it can
be summarized by the assertion that the $G$-category $(\scr{S}_{G,B},G\scr{S}_B)$ is a
$G$-topological closed symmetric monoidal $G$-category over $B$, but we prefer to be
more explicit than that.
\begin{thm}\mylabel{omnigood}
The $G$-category $\scr{S}_{G,B}$ is enriched over $G\scr{K}_{B}$ and is tensored and cotensored over $\scr{K}_{G,B}$. The category $G\scr{S}_{B}$ is enriched over $\scr{K}_{B}$ and is tensored and cotensored over $G\scr{K}_B$. The $G$-category $\scr{S}_{G,B}$ and the category $G\scr{S}_{B}$ admit smash product and function spectrum functors $\sma_B$ and $F_B$ under which they are closed symmetric monoidal with unit object $S_{B}$. Let $X$ and $Y$ be orthogonal $G$-spectra over $B$ and $K$ be an ex-$G$-space over $B$. The morphism ex-$G$-spaces $P_B(X,Y)$ can be specified by
$$P_B(X,Y) = \Omega^{\infty}_B F_B(X,Y),$$
and there are natural isomorphisms
$$\Sigma^{\infty}_B K\iso S_B\sma_B K \qquad\text{and}\qquad \Omega^{\infty}_BX\iso P_B(S_B,X).$$
The tensors and cotensors are related to smash products and function $G$-spectra by natural isomorphisms
\begin{equation}\label{smasma}
X\sma_B K \iso X\sma_B \Sigma^{\infty}_B K
\qquad\text{and}\qquad
F_B(K,X)\iso F_B(\Sigma^{\infty}_B K,X)
\end{equation}
of orthogonal $G$-spectra. There are natural isomorphisms
\begin{equation}\label{Pad!}
P_B(\Sigma^{\infty}_BK,X) \iso F_B(K,\Omega^{\infty}_BX)
\end{equation}
and
\begin{equation}\label{Pad0S}
P_B(X \wedge_B K, Y)
\cong F_B(K, P_B(X,Y))
\cong P_B(X, F_B(K,Y))
\end{equation}
of ex-$G$-spaces,
\begin{equation}\label{Pad1S}
\scr{S}_{G,B}(X \wedge_B K, Y)
\cong \scr{K}_{G,B}(K,P_B(X,Y))
\cong \scr{S}_{G,B}(X, F_B(K,Y))
\end{equation}
of based $G$-spaces, and
\begin{equation}\label{Pad2S}
G\scr{S}_B(X \wedge_B K, Y)
\cong G\scr{K}_B(K, P_B(X,Y))
\cong G\scr{S}_B(X, F_B(K,Y))
\end{equation}
of based spaces. Moreover, $G\scr{S}_{B}$ is $G$-topologically bicomplete over $B$.
\end{thm}\begin{proof} For the enrichment, the $G$-space $\scr{S}_{G,B}(X,Y)$ is the evident sub $G$-space
of $\scr{I}_G\scr{K}_B(X,Y)$, and the space $G\scr{S}_B(X,Y)$ is the evident sub space of $G\scr{I}\scr{K}_B(X,Y)$.
The tensors and cotensors in $\scr{S}_{G,B}$ are constructed in $\scr{I}_G\scr{K}_B$ and given induced
structure maps. The limits and colimits in $G\scr{S}_B$ are constructed in the same way.
As in \cite[II\S3]{MM}, we think of orthogonal $G$-spectra over $B$ as
$S_{B}$-modules, and we construct the smash product and function spectra functors
by passage to coequalizers and equalizers from the smash product and function $\scr{I}_G$-space
functors, exactly as in the definition of tensor products and hom functors in algebra. We
have defined $P_B(X,Y)$ in the statement, but we shall give a more intrinsic alternative
description later. The first isomorphism of (\ref{smasma}) is given by unit and associativity
relations
$$X\sma_B K \iso (X\sma_B S_{B})\sma_B K \iso X \sma_B \Sigma^{\infty}_B K.$$
The second follows from the Yoneda lemma since
\begin{align*}
G\scr{S}_B(X,F_B(K,Y)) & \iso G\scr{S}_B(X\sma_B K,Y)\\
& \cong G\scr{S}_B(X\sma_B\Sigma^{\infty}_B K,Y) \\
& \iso G\scr{S}_B(X,F_B(\Sigma^{\infty}_B K,Y)).
\end{align*}
Now (\ref{Pad!}) and (\ref{Pad0S}) follow from already established adjunctions. For part of the latter, we apply $\Omega^{\infty}_B$ to the composite isomorphism
\begin{align*}
F_B(X\sma_B K,Y) & \iso F_B(X\sma_B \Sigma^{\infty}_B K,Y) \\
& \cong F_B(X,F_B(\Sigma^{\infty}_B K,Y))\\
& \cong F_B(X,F_B(K,Y)).
\end{align*}
Comparisons of definitions, seen more easily from (\ref{altPB}) below, give
\begin{equation}
\scr{S}_{G,B}(X,Y) = \scr{K}_{G,B}(S_B^0,P_B(X,Y))
\end{equation}
and
\begin{equation}
G\scr{S}_B(X,Y)\iso G\scr{K}_B(S^0_B,P_B(X,Y)).
\end{equation}
Therefore the isomorphisms (\ref{Pad1S}) and (\ref{Pad2S}) follow from (\ref{Pad0S}).
\end{proof}
As noted in \S10.1, we obtain the following corollary by replacing
$K$ with $T_B$ for a based $G$-space $T$ in the
tensors and cotensors of the theorem. Of course, these tensors
and cotensors with $G$-spaces could just as well be defined directly.
It will be important in our discussion of model category structures
to keep separately in mind the tensors and cotensors over ex-$G$-spaces
over $B$ and over based $G$-spaces.
\begin{cor}\mylabel{omnigoodcor} The $G$-category $\scr{S}_{G,B}$ is enriched over $G\scr{K}_{*}$ and is
tensored and cotensored over $\scr{K}_{G,*}$. The category $G\scr{S}_{B}$ is enriched over
$\scr{K}_{G,*}$ and is tensored and cotensored over $G\scr{K}_*$. Thus, for orthogonal $G$-spectra $X$
and $Y$ and based $G$-spaces $T$,
\begin{equation}\label{tencoten2}
\scr{S}_{G,B}(X\sma_B T,Y)\iso \scr{K}_{G,*}(T,\scr{S}_{G,B}(X,Y))\iso \scr{S}_{G,B}(X,F_B(T,Y))
\end{equation}
and
\begin{equation}
\label{tencoten2G}
G\scr{S}_{B}(X\sma_B T,Y)\iso G\scr{K}_{*}(T,\scr{S}_{G,B}(X,Y))\iso G\scr{S}_B(X,F_B(T,Y)).
\end{equation}
\end{cor}
We have the parallel definition of $G$-prespectra over $B$.
\begin{defn}
A \emph{$G$-prespectrum $X$ over $B$}\index{prespectrum over B@prespectrum over $B$} consists of ex-$G$-spaces
$X(V)$ over $B$ for $V\in \scr{V}$ together with structure $G$-maps
$\sigma\colon \Sigma_B^WX(V)\longrightarrow X(V\oplus W)$ such that $\sigma$ is the identity
if $W=0$ and the following diagrams commute.
$$\xymatrix{
\Sigma^Z_B\Sigma^W_B X(V)\ar[d]_{\Sigma^Z_B\sigma} \ar[r]^-{\iso}
& \Sigma^{W\oplus Z}_B X(V) \ar[d]^{\sigma}\\
\Sigma^Z_BX(V\oplus W) \ar[r]_-{\sigma} & X(V\oplus W\oplus Z)\\}$$
Let $\scr{P}_{G,B}$\@bsphack\begingroup \@sanitize\@noteindex{PGB@$\scr{P}_{G,B}$} denote the
$G$-category of $G$-prespectra and nonequivariant arrows, and let $G\scr{P}_B$\@bsphack\begingroup \@sanitize\@noteindex{GPB@$G\scr{P}_B$} denote
its $G$-fixed category of $G$-prespectra and $G$-maps. There result
forgetful functors
$$\mathbb{U}\colon \scr{S}_{G,B}\longrightarrow \scr{P}_G \qquad\text{and}\qquad \mathbb{U}\colon G\scr{S}_{B}\longrightarrow G\scr{P}_{B}.$$
\end{defn}
The categories $\scr{P}_{G,B}$ and $G\scr{P}_{B}$ enjoy the same properties that were specified
for $\scr{S}_{G,B}$ and $G\scr{S}_{B}$ in \myref{omnigood} and \myref{omnigoodcor},
except for the statements about smash product and function spectra. Here,
since we do not have the internal hom functor $F_B$, we must give
an alternative direct description of $P_B(X,Y)$, as in (\ref{altPB}) below.
\section{Orthogonal $G$-spectra as diagram ex-$G$-spaces}
Arguing as in \cite[\S2]{MMSS} and \cite[II\S4]{MM}, we
construct a new domain category $\scr{J}_{G,B}$\@bsphack\begingroup \@sanitize\@noteindex{JGB@$\scr{J}_{G,B}$} which
has the same object set $\scr{V}$ as $\scr{I}_G$ and, like $\scr{I}_{G,B}$, is enriched
over $G\scr{K}_{B}$. It builds in spheres in such a way that the category of $\scr{I}_G$-{\em spectra}\, over $B$ is equivalent to the category
of $\scr{J}_{G,B}$-{\em spaces}\, over $B$. Here, just as for $\scr{I}_{G,B}$
in \myref{omnilem}, we understand a $\scr{J}_{G,B}$-space to be an enriched
$G$-functor $X\colon \scr{J}_{G,B}\longrightarrow \scr{K}_{G,B}$. Thus it is specified
by ex-$G$-spaces $X(V)$ and ex-$G$-maps
$$X\colon \scr{J}_{G,B}(V,W)\longrightarrow F_B(X(V), X(W)).$$
To construct $\scr{J}_{G,B}$, recall from \cite[II\S4]{MM} that we have a topological $G$-category
$\scr{J}_G$ with object set $\scr{V}$ such that the category of $\scr{I}_G$-spectra is equivalent to the category of $\scr{J}_G$-spaces. We define
\begin{equation}\label{JG1}
\scr{J}_{G,B}(V,W) = \scr{J}_G(V,W)_B,
\end{equation}
just as we defined $\scr{I}_{G,B}$ in (\ref{IGB}), and the desired equivalence
of categories follows. Rather than repeat either of the different constructions of $\scr{J}_G$ given in \cite{MMSS} and \cite{MM}, we shall shortly give a direct
description of $\scr{J}_{G,B}$. The intuition is that an extension of an $\scr{I}_{G,B}$-space to a $\scr{J}_{G,B}$-space builds in an action by $S_{B}$.
The alternative description of $G\scr{S}_{B}$ as the category of enriched $G$-functors
$\scr{J}_{G,B}\longrightarrow\scr{K}_{G,B}$ and enriched $G$-natural transformations
leads to a more
conceptual proof of \myref{omnigood}: it is a specialization of
general results
about diagram categories of enriched functors. In analogy with (\ref{P}) we could have
defined $P_B(X,Y)$ to be the end
\begin{equation}\label{altPB}
P_B(X,Y) = \int_{\scr{J}_{G,B}} F_B(X(V),Y(V))
\end{equation}
and derived the isomorphism (\ref{Pad0S}) just as we derived (\ref{Pad0})
in the previous section. By the Yoneda lemma, the two definitions of $P_B(X,Y)$
agree. With this description of $P_B$, some of the adjunctions in
\myref{omnigood} become more transparent.
This leads to an alternative description of $\scr{J}_{G,B}$ in terms of
$\scr{I}_{G,B}$, following \cite[2.1]{MMSS}. We have the represented functors
$V^*\colon \scr{I}_{G}\longrightarrow \scr{K}_{G,B}$ specified by $V^*(W) = \scr{I}_{G,B}(V,W)$.
If $X$ is an $\scr{I}_G$-space, such as $V^*$, then the smash product
$X\sma_B S_{B}$ in the category of $\scr{I}_G$-spaces is a ``free'' orthogonal
$G$-spectrum over $B$. Let
\begin{equation}\label{JG2}
\scr{J}_{G,B}(V,W) = P_{B}(W^*\sma_B S_{B}, V^*\sma_B S_{B}),
\end{equation}
with the evident composition. Then we can mimic the arguments of \cite[\S\S2, 23]{MMSS}
to check that the category of $\scr{J}_{G,B}$-spaces is equivalent to the category of
$\scr{I}_{G}$-spectra over $B$. An enriched Yoneda lemma argument \cite[2.4]{Ke} shows that this
description of $\scr{J}_{G,B}$ coincides up to isomorphism with our original one.
Although we will not have occasion to quote it formally, we record the following consequence
of the identification of $\scr{I}_G$-spectra over $B$ with $\scr{J}_{G,B}$-spaces.
\begin{lem}\mylabel{save} For any enriched $G$-functor $T\colon\scr{K}_{G,B}\longrightarrow \scr{K}_{G,B}$
and orthogonal $G$-spectrum $X$ over $B$, the composite functor $T\com X$ is an orthogonal
$G$-spectrum over $B$. Similarly, an enriched natural transformation
$\xi\colon T\longrightarrow T'$ induces a natural $G$-map
$\xi\colon T\com X\longrightarrow T'\com X$.
\end{lem}\begin{proof} The enriched functor $T$ is given by maps
$$T\colon F_B(K,L)\longrightarrow F_B(T(K),T(L)).$$
Composing levelwise with $X$ gives maps
$$\scr{J}_{G,B}(V,W)\longrightarrow F_B(T(X(V)),T(X(W)))$$
that specify $T\com X$. It is a direct categorical
implication of the fact that $T$ is an enriched functor that there are natural maps of
ex-$G$-spaces
$$T(K)\sma_B L\longrightarrow T(K\sma_B L) \qquad\text{and}\qquad TF_B(K,L)\longrightarrow F_B(K,T(L))$$
for ex-$G$-spaces $K$ and $L$. This explains more concretely why the
structure maps of $X$ induce structure maps for $T\com X$. Similarly, since $\xi$ is enriched,
it is given by maps from the unit ex-$G$-space $S^0_B$ to $F_B(T(K),T'(K))$ such that the
appropriate diagrams commute. We specialize to $K=X(V)$ to obtain
$\xi\colon T\com X\longrightarrow T'\com X$.
\end{proof}
The following functors relating ex-$G$-spaces to orthogonal $G$-spectra over $B$ play a central role in our theory. In particular, they give ``negative
dimensional'' spheres $\Sigma^{\infty}_VS^0_B = S^{-V}_B$.
\begin{defn}\mylabel{FVs}
Let $V^* = V^*_B$ denote the represented $\scr{J}_{G,B}$-space
specified by $V^*(W) = \scr{J}_{G,B}(V,W)$. Define the \emph{shift desuspension
functor}\index{functor!shift desuspension --}\index{shift desuspension functor}\@bsphack\begingroup \@sanitize\@noteindex{FV@$F_V$}
$$F_V \colon \scr{K}_{G,B}\longrightarrow \scr{S}_{G,B}$$
by letting
$F_V K = V^*\sma_B K$ for an ex-$G$-space $K$. Let
$\text{Ev}_V\colon \scr{S}_{G,B}\longrightarrow \scr{K}_{G,B}$ be the functor given by evaluation at $V$.\index{functor!evaluation --}\index{evaluation functor}\@bsphack\begingroup \@sanitize\@noteindex{EvV@$\text{Ev}_V$}
The alternative notations
$$\Sigma^{\infty}_V K = F_VK \qquad\text{and}\qquad \Omega^{\infty}_V K = \text{Ev}_V$$
are often used. In particular, $F_0 = \Sigma^{\infty}_0= \Sigma^{\infty}_B$ and
$\text{Ev}_0 = \Omega^{\infty}_0 = \Omega^{\infty}_B$.
\end{defn}
\begin{lem}\mylabel{FVEV} The functors $F_V$ and $\text{Ev}_V$ are left and
right adjoint, and there is a natural isomorphism
$$ F_V K\sma_B F_W L \iso F_{V\oplus W}(K\sma_B L).$$
\end{lem}\begin{proof} The first statement is clear, and the verification of the
second statement is formal, as in \cite[\S1]{MMSS}.
\end{proof}
\section{The base change functors $f^*$, $f_!$, and $f_*$}
From now on, we drop the adjective ``orthogonal'' (or prefix $\scr{I}_G$), and we
generally take the equivariance for granted, referring to orthogonal $G$-spectra
over $B$ just as spectra over $B$. We return $G$ to the notations when
considering change of groups, or for emphasis, but otherwise $G$-actions
are tacitly assumed throughout.
We first show that the results on base change functors proven for
ex-spaces in \S2.2 extend to parametrized spectra. We then show
that the results in \S2.5 relating external and internal smash
product and function ex-spaces also extend to parametrized spectra. Let
$A$ and $B$ be base $G$-spaces.
\begin{thm}\mylabel{Wirth} Let $f\colon A\longrightarrow B$ be a $G$-map. Let $X$ be in
$\scr{S}_{G,A}$ and let $Y$ and $Z$ be in $\scr{S}_{G,B}$. There are $G$-functors
\[f_!\colon \scr{S}_{G,A} \longrightarrow \scr{S}_{G,B},\qquad
f^*\colon \scr{S}_{G,B} \longrightarrow \scr{S}_{G,A}, \qquad
f_*\colon \scr{S}_{G,A} \longrightarrow \scr{S}_{G,B}\]
and $G$-adjunctions
\[\scr{S}_{G,B}(f_!X,Y)\iso \scr{S}_{G,A}(X,f^*Y) \qquad\text{and}\qquad
\scr{S}_{G,A}(f^*Y,X)\iso \scr{S}_{G,B}(Y,f_*X).\]
On passage to $G$-fixed points levelwise, there result functors
\[f_!\colon G\scr{S}_{A} \longrightarrow G\scr{S}_{B},\qquad
f^*\colon G\scr{S}_{B} \longrightarrow G\scr{S}_{A}, \qquad
f_*\colon G\scr{S}_{A} \longrightarrow G\scr{S}_{B}\]
and adjunctions
\[G\scr{S}_{B}(f_!X,Y)\iso G\scr{S}_{A}(X,f^*Y)\qquad\text{and}\qquad
G\scr{S}_{A}(f^*Y,X)\iso G\scr{S}_{B}(Y,f_*X).\]
The functor $f^*$ is closed symmetric monoidal. Therefore,
by definition and implication, $f^*S_{B}\iso S_{A}$ and there
are natural isomorphisms
\begin{gather}\label{one}
f^*(Y\sma_B Z)\iso f^*Y\sma_A f^*Z,\\[1ex]
\label{two}
F_B(Y,f_*X) \iso f_*F_A(f^*Y,X),\\[1ex]
\label{three}
f^*F_B(Y,Z)\iso F_A(f^*Y,f^*Z),\\[1ex]
\label{four}
f_{!}(f^*Y\sma_A X)\iso Y\sma_B f_{!}X,\\[1ex]
\label{five}
F_B(f_{!}X,Y)\iso f_*F_A(X,f^*Y).
\end{gather}
\end{thm}\begin{proof} We define the functors $f^*$, $f_!$, and $f_*$
levelwise. This certainly gives well-defined functors on $\scr{I}_G$-spaces
that satisfy the appropriate adjunctions there. We shall show shortly
that these functors preserve $\scr{I}_G$-spectra. For a based $G$-space $T$, $f^*(T_B) \iso T_A$, and this implies $f^*S_{B}\iso S_{A}$. If we replace $\scr{I}_G$-spectra by
$\scr{I}_G$-spaces and replace the internal smash product and function object
functors ($\sma$ and $F$) by their
external precursors ($\barwedge$ and $\bar F$), then everything is
immediate by levelwise application of the corresponding results for
ex-spaces. Still working with $\scr{I}_G$-spaces, we first show how to
internalize the isomorphisms (\ref{one}) and (\ref{four}) by use of the
universal property of left Kan extension. Indeed, noting that
$(f_*X)\circ \oplus \iso f_*(X\circ \oplus)$, and similarly for $f^*$ and $f_!$,
we have
\begin{align*}
\scr{I}_G\scr{K}_{A}(f^*(Y\wedge_B Z), X)
&\cong \scr{I}_G\scr{K}_{B}(Y\wedge_B Z, f_* X) \\
&\cong (\scr{I}_G\times\scr{I}_G)\scr{K}_{B}(Y\barwedge_B Z, f_*X\circ \oplus)\\
&\cong (\scr{I}_G\times\scr{I}_G)\scr{K}_{A}(f^*(Y\barwedge_B Z), X\circ \oplus )\\
&\cong (\scr{I}_G\times\scr{I}_G)\scr{K}_{A}(f^*Y\barwedge_A f^*Z, X\circ \oplus)\\
&\cong \scr{I}_G\scr{K}_{A}(f^*Y \wedge_A f^*Z, X)
\end{align*}
and
\begin{align*}
\scr{I}_G\scr{K}_{B}(f_!X\wedge_B Y,Z)
&\cong (\scr{I}_G\times\scr{I}_G)\scr{K}_{B}(f_!X\barwedge_B Y,Z\circ \oplus)\\
&\cong (\scr{I}_G\times\scr{I}_G)\scr{K}_{B}(f_!(X\barwedge_A f^*Y),Z\circ \oplus)\\
&\cong (\scr{I}_G\times\scr{I}_G)\scr{K}_{A}(X\barwedge_A f^*Y,f^*Z\circ \oplus) \\
&\cong \scr{I}_G\scr{K}_{A}(X\wedge_A f^*Y,f^*Z)\\
&\cong \scr{I}_G\scr{K}_{A}(f_!(X\wedge_A f^*Y),Z).
\end{align*}
As explained in \cite[\S\S2--3]{FHM}, the remaining isomorphisms on the
$\scr{I}_G$-space level follow formally.
We must show that our functors
on $\scr{I}_G$-spaces preserve $\scr{I}_G$-spectra. The given structure map
$\sigma\colon Y\barwedge_B S_{B}\longrightarrow Y\com \oplus$ gives rise
via the external version of (\ref{one}) to the required structure map
$$f^*Y\barwedge_A S_{A}\iso f^*(Y\barwedge_B S_{B})\longrightarrow f^*Y\com \oplus.$$
Similarly, the given structure map
$\sigma\colon X\barwedge S_{A}\longrightarrow X\com \oplus$
gives rise to the required structure map
$$ f_!X\barwedge_B S_{B} \iso f_!(X\barwedge_A S_{A})\longrightarrow f_!X\com \oplus.$$
As in \cite[(3.6)]{FHM}, there is a canonical natural map, not usually an isomorphism,
$$\pi\colon f_*X\barwedge_B Y\longrightarrow f_*(X\barwedge_A f^*Y).$$
Taking $Y=S_{B}$, we see that $\sigma$ also induces the required structure map
$$ f_*X\barwedge_B S_{B} \longrightarrow f_*(X\barwedge_A S_{A})\longrightarrow f_*X\com \oplus.$$
Now the spectrum level adjunctions follow directly from their $\scr{I}_G$-space analogues.
The spectrum level isomorphisms (\ref{one}) and (\ref{four}) follow from their
$\scr{I}_G$-space analogues by comparisons of coequalizer diagrams, and the remaining isomorphisms again follow formally.
\end{proof}
\begin{rem}\mylabel{FVvsf*}
Since the base change functors are defined levelwise, they
commute with the evaluation functors $\text{Ev}_V$. These commutation
relations for the right adjoints $f_*$ and $f^*$ imply conjugate
commutation isomorphisms
$$f^*F_V\iso F_Vf^* \qquad\text{and}\qquad f_!F_V \iso F_V f_!$$
of left adjoints. In particular,
$$f^*\Sigma^{\infty}_B\iso \Sigma^{\infty}_A f^* \qquad\text{and}\qquad
f_! \Sigma^{\infty}_A \iso \Sigma^{\infty}_B f_!.$$
Via (\ref{smasma}), these isomorphisms
and the isomorphisms of the theorem imply isomorphisms relating base change functors
to tensors and cotensors. For example (\ref{four}) implies isomorphisms
$$ f_!(f^*Y\sma_A K) \iso Y\sma_B f_!K \qquad\text{and}\qquad
f_!(f^*L\sma_A X) \iso L\sma_B f_!X.$$
Here $K$ and $L$ are ex-spaces over $A$ and $B$ and $X$ and $Y$ are spectra over $A$ and $B$.
\end{rem}
The following result is immediate from its precursor \myref{Mackey0} for ex-spaces.
\begin{prop}\mylabel{Mackey}
Suppose given a pullback diagram of $G$-spaces
$$\xymatrix{
C \ar[r]^-{g} \ar[d]_{i} & D \ar[d]^{j} \\
A \ar[r]_{f} & B.\\}$$
Then there are natural isomorphisms of functors
\begin{equation}\label{bases}
j^*f_{!} \iso g_{!}i^*, \qquad f^*j_* \iso i_*g^*,
\qquad f^*j_{!}\iso i_!g^*, \qquad j^*f_*\iso g_*i^*.
\end{equation}
\end{prop}
Returning to \myref{extsmash1}, we have the following important results
on external smash product and function spectra and their internalization by
means of base change along diagonal maps.
\begin{prop}\mylabel{extsmash2} Let $X$ be a spectrum over $A$, $Y$ be
a spectrum over $B$, and $Z$ be a spectrum over $A\times B$. There
is an external smash product functor that assigns a spectrum $X\barwedge Y$
over $A\times B$ to $X$ and $Y$ and an external function spectrum functor
that assigns a spectrum $\bar{F}(Y,Z)$ over $A$ to $Y$ and $Z$, and
there is a natural isomorphism
\[G\scr{S}_{A\times B}(X\barwedge Y, Z)\iso G\scr{S}_A(X, \bar{F}(Y,Z)).\]
The internal smash products are determined from the external ones via
$$ X\sma_B Y \iso \Delta^*(X\barwedge Y) \qquad\text{and}\qquad
F_B(X,Y) \iso \bar{F}(X,\Delta_*Y),$$
where $X$ and $Y$ are spectra over $B$ and $\Delta\colon B\longrightarrow B\times B$ is the
diagonal map.
\end{prop} \begin{proof}
It is not hard to start from \myref{extsmash1} and construct
these functors directly. We instead follow \myref{exin} and observe that
the spectrum level external functors can and, up to isomorphism, must be
defined in terms of the internal functors as
$$ X\barwedge Y \iso \pi_A^*X \sma_{A\times B} \pi_B^*Y
\qquad\text{and}\qquad \bar{F}(Y,Z) \iso \pi_{A\, *}F_{A\times B}(\pi_{B}^{*}Y,Z),$$
where $\pi_A\colon A\times B\longrightarrow A$ and $\pi_B\colon A\times B\longrightarrow B$ are the
projections. The displayed adjunction is immediate from the adjunctions
$(\pi^*_A,\pi_{A\, *})$, $(\pi^*_B,\pi_{B\, *})$, and
$(\sma_{A\times B}, F_{A\times B})$. The second statement follows formally,
as in \myref{internalize}.
\end{proof}
\begin{prop}\mylabel{SISISI} For ex-spaces $K$ over $A$ and $L$
over $B$, there is a natural isomorphism
$$\Sigma^{\infty}_{A\times B} (K\barwedge L)
\iso \Sigma^{\infty}_{A}K\barwedge \Sigma^{\infty}_{B}L. $$
\end{prop}\begin{proof}
This is most easily seen using adjunction and the Yoneda lemma.
Using external function objects, we see that
$\bar{F}(\Sigma^{\infty}_BL,Z)\iso \bar{F}(L,Z)$ for $Z\in G\scr{S}_{A\times B}$.
This has zeroth ex-space $\bar{F}(L,Z(0))$ over $A$.
\end{proof}
\section{Change of groups and restriction to fibers}
We give the analogues for parametrized spectra of the results concerning
change of groups and restriction to fibers that were given for parametrized
ex-spaces in \S2.3. We shall say more about change of groups in Chapter 14.
Fix an inclusion $\iota\colon H\longrightarrow G$ of
a (closed) subgroup $H$ of $G$ and let $A$ be an $H$-space and $B$ be a
$G$-space. We index $H$-spectra over $A$ on the collection $\iota^*{\scr{V}}$
of $H$-representations $\iota^*V$ with $V\in \scr{V}$. As we discuss in \S\S14.2
and 14.3, when $\scr{V}$ is the collection of all representations of $G$, we can change indexing to the collection of all representations of $H$ since our assumption that $G$ is compact ensures that every representation of $H$
is a direct
summand of a representation $\iota^*V$. We have an evident forgetful functor
\begin{equation}
\iota^*\colon G\scr{S}_{B} \longrightarrow H\scr{S}_{\iota^*B}.
\end{equation}
On the space level, we write $\iota_!$ ambiguously for both the based and
unbased induction functors $G_+\sma_H(-)$ and $G\times_H(-)$, and similarly
for coinduction $\iota_*$. Context should make clear which is intended.
Applying the unbased versions to retracts, we defined induction and
coinduction functors $\iota_!$ and $\iota_*$ on ex-spaces in \myref{changes0}.
These functors extend to the spectrum level. Recall that $S_{G,B}$
denotes the $G$-sphere spectrum over $B$.
\begin{prop}\mylabel{eyeeye} Levelwise application of $\iota_!$ and
$\iota_*$ gives functors
\[\iota_!\colon H\scr{S}_A\longrightarrow G\scr{S}_{\iota_! A} \qquad\text{and}\qquad
\iota_*\colon H\scr{S}_A\longrightarrow G\scr{S}_{\iota_* A}.\]
\end{prop}\begin{proof}
We must show that the structure $H$-maps
$\sigma\colon X\barwedge S_{H,A}\longrightarrow X\com \oplus$
of an $H$-spectrum $X$ over $A$ induce structure $G$-maps
for the $\scr{I}_G$-spaces $\iota_!X$ and $\iota_*X$.
It is clear that $\iota_!(X\com \oplus)\iso \iota_!X\com \oplus$
and $\iota_*(X\com \oplus)\iso \iota_*X\com \oplus$.
Using (\ref{GH3}), we see that $S_{G,\iota_!A} \iso \iota_!S_{H,A}$.
Since the functor $\iota_!$ on the ex-space level is symmetric monoidal by
\myref{ishriek}, its levelwise $\scr{I}_G$-space analogue commutes
up to isomorphism with the external smash product $\barwedge$. Thus
$\sigma$ induces a structure $G$-map
$$\iota_!X \barwedge_{\iota_!A} S_{G,\iota_!A}
\iso \iota_!(X\barwedge_A S_{H,A})\longrightarrow \iota_!(X\com \oplus)\iso \iota_!X\com \oplus.$$
For $\iota_*$, let $\mu:\iota^*\iota_*\longrightarrow \text{Id}$ be the counit
of the space
level adjunction $(\iota_*,\iota^*)$ (see (\ref{GH1})). For
an $H$-space $A$, $\mu$ is the $H$-map $\text{Map}_H(G,A)\longrightarrow A$
given by evaluation at the identity element of $G$. Applied to an
ex-space $K$ over $A$, thought of as a retract,
$\mu$ gives a map $\iota^*\iota_*K\longrightarrow K$ of total spaces over and
under the map $\mu:\iota^*\iota_*A\longrightarrow A$ of base spaces in the category
of retracts of \S2.5. We can apply this to $X$ levelwise.
We also have the projection
$\text{pr}:\mu^*S_{H,A}\longrightarrow S_{H,A}$ over $\mu$. Together, these maps give
$$
\xymatrix{
\iota^*(\iota_*X\barwedge_{\iota_*A}S_{G,\iota_*A})
\cong \iota^*\iota_*X\barwedge_{\iota^*\iota_*A} \mu^*S_{H,A}
\ar[r]^-{\mu\barwedge \text{pr}} & X\barwedge_A S_{H,A}.\\}
$$
For the isomorphism, we have used the facts that $\iota^*$ is strong monoidal and
that $\iota^*S_{G,\iota_*A}\cong S_{H,\iota^*\iota_*A}\cong \mu^*S_{H,A}$.
The adjoint of the composite of this map with the structure map
$\sigma:X\barwedge_A S_{H,A}\longrightarrow X\circ \oplus$ gives the required
structure map
$\iota_*X\barwedge_{\iota_*A} S_{G,\iota_*A}\longrightarrow \iota_*X\circ\oplus$.
\end{proof}
As on the ex-space level, the categories $H\scr{S}_A$ and $G\scr{S}_{G\times_H A}$
can be used interchangeably. The following result is immediate from \myref{ishriek}.
\begin{prop}\mylabel{changes}
Let $\nu\colon A\longrightarrow \iota^*\iota_!A$ be the natural inclusion of $H$-spaces.
Then $\iota_!\colon H\scr{S}_A\longrightarrow G\scr{S}_{\iota_{!}A}$ is a closed symmetric monoidal equivalence of categories with inverse the composite
${\nu}^*\com \iota^*\colon G\scr{S}_{\iota_!A}\longrightarrow H\scr{S}_{\iota^*\iota_!A}\longrightarrow H\scr{S}_A$.
\end{prop}
In particular, if $A = *$ then $\nu$ maps $*$ to the identity coset
$eH\in G/H$ and we see that $H\scr{S}$ and $G\scr{S}_{G/H}$ can be used
interchangeably. Arguing as in \myref{homog}, we could more easily
prove this directly.
\begin{cor}\mylabel{changestoo}
The category $H\scr{S}$ is equivalent as a closed symmetric mon\-oid\-al
category to $G\scr{S}_{G/H}$. Under this equivalence,
$$\iota^*\iso r^*, \qquad \iota_!\iso r_{!}, \qquad\text{and}\qquad \iota_*\iso r_*,$$
where $r\colon G/H \longrightarrow *$.
\end{cor}
Looking at the fiber $X_b(V) = X(V)_b$ over $b$ of a $G$-spectrum $X$ over $B$, we see a $G_b$-spectrum $X_b$ of the sort that has been studied in \cite{MM},
where $G_b$ is the isotropy group of $b$. Our homotopical analysis of
parametrized $G$-spectra will be based on the idea of applying the results
of \cite{MM} fiberwise. By the previous result, we can think of this fiber
as a $G$-spectrum over $G/G_b$. The following spectrum level analogues of
\myref{Johann0} and \myref{Johann1} analyze the relationships among passage to
fibers, base change, and change of groups.
\begin{exmp}\mylabel{Johann}
For $b\in B$, we write $b\colon *\longrightarrow B$ for the $G_b$-map that sends $*$ to $b$ and $\tilde{b}\colon G/G_b \longrightarrow B$ for the induced inclusion of orbits. Under the equivalence $G\scr{S}_{G/G_b}\iso G_b\scr{S}$, $\tilde{b}^*$ may be interpreted as the fiber functor $G\scr{S}_B\longrightarrow G_b\scr{S}$ that sends $Y$ to $Y_b$. Its left and right adjoints $\tilde{b}_{!}$ and $\tilde{b}_*$ may be interpreted as the functors that send a $G_b$-spectrum $X$ to the $G$-spectra $X^b$ and $^bX$ over $B$ obtained by levelwise application of the corresponding ex-space level adjoints of \myref{Fibad} and \myref{Johann0}. With these notations, the isomorphisms of \myref{Wirth} specialize to the following natural isomorphisms, where $Y$ and $Z$ are in $G\scr{S}_B$ and $X$ is in $G_b\scr{S}$.
\begin{gather*}
(Y\sma_B Z)_b\iso Y_b\sma Z_b,\\[1ex]
F_B(Y,\, ^bX) \iso\, {^{b}}F(Y_b,X),\\[1ex]
F_B(Y,Z)_b\iso F(Y_b,Z_b),\\[1ex]
(Y_b\sma X)^b\iso Y\sma_B X^b,\\[1ex]
F_B(X^b,Y)\iso \, {^{b}}F(X,Y_b).
\end{gather*}
\end{exmp}
\begin{exmp}\mylabel{Johann2}
Let $f\colon A\longrightarrow B$ be a $G$-map and let $i_b\colon A_b\longrightarrow B$
be the inclusion of the fiber over $b$, which is a $G_b$-map. As in
\myref{Johann1}, we have the compatible pullback squares
$$\xymatrix{
A_b \ar[r]^-{f_b} \ar[d]_{i_b} & \{b\} \ar[d]^{b} \\
A \ar[r]_{f} & B\\}
\qquad\qquad
\xymatrix{
G\times_{G_b} A_b \ar[r]^-{G\times_{G_b} f_b} \ar[d]_{\tilde{\imath}_b}
& G/G_b \ar[d]^{\tilde{b}} \\
A \ar[r]_-{f} & B.\\}$$
Applying \myref{Mackey} to the right-hand square and
interpreting the conclusion in terms of fibers, we obtain canonical
isomorphisms of $G_b$-spectra
$$(f_!X)_b \iso {f_{b}}_!i_b^*X \qquad\text{and}\qquad (f_*X)_b \iso {f_{b}}_*i_b^*X,$$
where $X$ is a $G$-spectrum over $A$, regarded on the right-hand sides as a $G_b$-spectrum
over $A$ by pullback along $\iota\colon G_b\longrightarrow G$.
\end{exmp}
\section{Some problems concerning non-compact Lie groups}
In equivariant stable homotopy theory, the key idea
is that the one-point compactification of a representation $V$ of
dimension $n$ is a $G$-sphere and that smashing with that sphere
should be a self-equivalence of the equivariant stable homotopy category.
That is, the idea is to invert $G$-spheres in just the way that we
invert spheres when constructing the nonequivariant stable homotopy
category. For compact Lie groups of equivariance, the philosophy
and its implementation and applications are well understood. When
we invert representation spheres, we invert other homotopy spheres
as well, and the relevant Picard group is analyzed in \cite{FLM}.
For non-compact Lie groups, the present work seems to be the first
attempt to consider foundations for equivariant stable
homotopy theory. The philosophy is less clear, and its technical
implementation is problematic. The need for such a theory
is evident, however. The focus on finite dimensional representations
is intrinsic to the philosophy but fails to come to grips with basic
features of the representation theory of non-compact Lie groups. A theory
based on finite dimensional representations
should still have its uses, but there are real difficulties to obtaining
even that much. In particular,
a focus on spheres associated to linear representations, rather than
on less highly structured homotopy spheres, may be misplaced.
A non-compact semi-simple Lie group will generally have no non-trivial
finite dimensional unitary or orthogonal representations, hence our
theory of ``orthogonal'' $G$-spectra is clearly too restrictive. This is
easily remedied.
The use of linear isometries in the definition of orthogonal
spectra is a choice dictated more by the
history than by the mathematics. In the alternative
approach to equivariant stable homotopy theory based on
Lewis-May spectra and EKMM \cite{EKMM, LMS, MM}, use of orthogonal
complements is certainly convenient and perhaps essential. However,
the diagram orthogonal spectra of \cite{MM, MMSS} could just as well
have been developed in terms of diagram ``general linear spectra''.
In the few places where complements are used, they can by avoided.
For consistency with the previous literature, we have chosen to give
our exposition in the compact case using the word ``orthogonal'' and
the language from the cited references, but for general
Lie groups of equivariance, we should eliminate all
considerations of isometries.
More precisely, for the complete case, we redefine $\scr{I}$ by taking $\scr{V}$ to be the collection of all finite dimensional representations $V$ of $G$. More generally, we can index on any subcollection that contains the trivial representation and is closed under finite direct sums. Since we are only interested in a skeleton of $\scr{I}$, we may as well restrict to orthogonal representations in $\scr{V}$ when $G$ is compact.
We replace linear isometries by linear isomorphims when defining the $G$-spaces $\scr{I}(V,W)$. Thus we replace orthogonal groups by general linear groups. Otherwise, the formal definitional framework developed in this chapter (or,
in the nonparametrized case, \cite[II]{MM}) goes through verbatim for general
topological groups $G$.
However, we emphasize the formality. When considering change of groups,
for example, the significance changes drastically. As noted at the start
of the previous section, for an inclusion $\iota\colon H\longrightarrow G$ of
a (closed) subgroup $H$ of $G$, we index $H$-spectra on the collection $\iota^*{\scr{V}}$ of $H$-representations $\iota^*V$ with $V\in \scr{V}$. We also
pointed out the relevance of the compact case of the following result.
\begin{prop} If $G$ is either a compact Lie group or a matrix group
and $W$ is a representation of a subgroup $H$, then there is a representation $V$
of $G$ and an embedding of $W$ as a subrepresentation of $\iota^*V$.
\end{prop}
This is clear in the compact case and is given by \cite[3.1]{Palais}
for matrix groups. However, the following striking counterexample,
which we learned from Victor Ginzburg, shows just how badly this
basic result fails in general.
\begin{ouch0}[Ginzburg] Let $\mathbb{H}$ be the Heisenberg group of
$3\times 3$ matrices
\[ \left( \begin{array}{ccc}
1 & a & c \\
0 & 1 & b \\
0 & 0 & 1 \\
\end{array} \right) \]
where $a$, $b$, and $c$ are real numbers. Embed $\mathbb{R}$ in $\mathbb{H}$
as the subgroup of matrices with $a=b=0$. Embed $\mathbb{Z}$ in $\mathbb{R}$
as usual. Then $\mathbb{R}$ is a central subgroup of $\mathbb{H}$. Define
$G = \mathbb{H}/\mathbb{Z}$. Then $T = \mathbb{R}/\mathbb{Z}$ is a circle subgroup of $G$.
Moreover, $T$ is the center of $G$ and coincides with the commutator
subgroup $[G,G]$. Let $V$ be any finite dimensional (complex linear) representation
of $G$. Since $T$ is compact, the action of $T$ on $V$ is semisimple,
and since $T$ is central, any weight space of $T$ is a $G$-submodule.
Therefore $V$ is a direct sum of $G$-submodules $V_i$ such that $T$
acts on each $V_i$ by scalar matrices. Since $T = [G,G]$,
this scalar action of $T$ on $V_i$ is trivial: the determinant
of $g$ is $1$ for any $g\in [G,G]$. Therefore no nontrivial
$1$-dimensional character of $T$ can embed in $V$. Reinterpreting
in terms of real representations, as we may, we conclude that, for
$\iota\colon T\longrightarrow G$, $\iota^*\scr{V}$ is the trivial $T$-universe.
\end{ouch0}
For a compact Lie group $G$ and inclusion $\iota\colon H\subset G$,
$\iota^*X$ is a dualizable $H$-spectrum if $X$ is a dualizable $G$-spectrum,
and an $H$-spectrum indexed on the trivial $H$-universe is dualizable if
and only if it is a retract of a finite $H$-CW spectrum built up from
trivial orbits. We conclude that duality theory (in the nonparametrized
context) cannot work as one would wish in the context of the previous example.
Looking ahead, much of the theory of the following three chapters also
works formally in the context of non-compact Lie groups. However, there is at
least one serious technical difficulty. Our theory is
based on the use of one-point compactifications $S^V$. If $V$ is a linear
representation of a non-compact Lie group $G$, there is no reason to
think that $G$ acts smoothly and properly on $S^V$, even if the isotropy
groups of $V$ are compact. In fact, if Illman's \myref{Illman} were
to apply, then $S^V$ would be a $G$-cell complex, hence it would be built
up from non-compact orbits $G/H$ given by compact subgroups $H$. However, as closed
subsets of $S^V$, the closed cells would have to be compact. That is, the
putative $G$-CW structure would contradict the
compactness of $S^V$. Said another way, we see no reason to believe
that the $S^V$ are $q$-cofibrant $G$-spaces. Therefore, the functors
$(-)\sma S^V$ need not be Quillen left adjoints and the functors $\Omega^V$
and $\Omega^V_B$ need not preserve fibrant objects in the relevant model
structures. Compare, for example, \myref{QuillK} and the derivation
of the long exact sequences (\ref{les2}) and (\ref{les3}) below. What
seems to be needed, for a start, is something like a model structure on
$G$-spaces such that $X$ is cofibrant if $\iota^*X$ is $H$-cofibrant for
all inclusions $\iota\colon H\to G$ of compact subgroups.
\chapter{Model structures for parametrized $G$-spectra}
\section*{Introduction}
We define and study two model structures on the category $G\scr{S}_B$ of (orthogonal) $G$-spectra over $B$. We emphasize that, except for the theory of smash products, everything in this chapter applies equally well to the category $G\scr{P}_B$ of $G$-prespectra over $B$. That fact will become
important in the next chapter.
We start in \S12.1 by defining a ``level model structure'' on $G\scr{S}_B$, based on the $qf$-model structure on $G\scr{K}_B$. In \S12.2, we record analogues for this model structure of the results on external smash product and base change functors that were given for $G\scr{K}_B$ in \S7.2. The level model structure serves as a stepping stone to the stable model structure, which we define in \S12.3. It has the same cofibrations as the level model structure, and we therefore call these ``$s$-cofibrations''. An essential point in our approach is a fiberwise definition of the homotopy groups of a parametrized $G$-spectrum that throws much of our work onto the theory of nonparametrized orthogonal $G$-spectra developed by Mandell and the first author in \cite{MM}. We define homotopy groups using the level $qf$-fibrant replacement functor provided by the level model structure, and we define stable equivalences to be the $\pi_*$-isomorphisms. It is essential to think in terms of fibers and not total spaces since the total spaces of a parametrized spectrum do not assemble into a spectrum. We show in \S12.4 that the $\pi_*$-isomorphisms give a well-grounded subcategory of weak equivalences, and we complete the proofs of the model axioms in \S12.5. We return to the context of \S12.2 in \S12.6, where we prove
that various Quillen adjoint pairs in the level model structures are also
Quillen adjoint pairs in the stable model structures.
The basic conclusion is that $G\scr{S}_B$ is a well-grounded model category under the stable structure. Although not very noticeable on the surface, essential use is made of the $qf$-model structure on $G\scr{K}_B$ throughout this chapter. It is possible to obtain a level model structure on $G\scr{S}_B$ from the $q$-model structure on $G\scr{K}_B$, as we explain in \myref{qnogood}. However this model
structure is not well-grounded and therefore does not provide the necessary tools to work out the technical details of \S12.4. The results there are crucial to prove that the relative cell complexes over $B$ defined in terms of the appropriate generating acyclic $s$-cofibrations are acyclic.\footnote{In \cite[3.4]{Hu}, such acyclicity of relative cell complexes is assumed without proof.} It was our fruitless attempt to obtain a stable model structure starting from the level $q$-model structure that led us to the construction of the $qf$-model structure on $G\scr{K}_B$ and to the notion of a well-grounded model category.
When there are no issues of equivariance, we generally abbreviate
$G$-spectrum over $B$, ex-$G$-space, and $G$-space to spectrum
over $B$, ex-space, and space; $G$ is a compact Lie group throughout.
\section{The level model structure on $G\scr{S}_B$}
After changing the base space from $*$ to $B$, the level model structure works in much the same way as in the nonparametrized case of \cite{MM}.
\begin{defn}\mylabel{never}
Let $f\colon X\longrightarrow Y$ be a map of spectra over $B$. With one exception, for any type of ex-space and any type of map of ex-spaces, we say that $X$ or $f$ is a \emph{level type of spectrum over $B$}\index{spectrum!level type of --} or a \emph{level type of map of spectra over $B$}\index{fibration!level type of --} if each $X(V)$ or $f(V)\colon X(V) \longrightarrow Y(V)$ is that type of ex-space or that type of map. Thus, for example, we have level $h$, level $f$ and level $fp$-fibrations, cofibrations and equivalences from \S5.1 together with the corresponding fibrant and cofibrant objects. We have level $q$-equivalences and level $q$ and $qf$-fibrations from \S7.1 and we have level ex-fibrations and level ex-quasifibrations from \S8.1 and \S8.5. The exceptions concern cofibrations and cofibrant objects. We shall \emph{never} be interested in ``level $q$-cofibrations'' or ``level $qf$-cofibrations'', nor in ``level $q$-cofibrant'' or ``level $qf$-cofibrant'' objects, since these do not correspond
to cofibrations and cofibrant objects in the model structures that we consider. Instead we have the following definitions.
\begin{enumerate}[(i)]
\item $f$ is an \emph{$s$-cofibration}\index{cofibration!s-@$s$- --} if it satisfies the LLP with respect to the level acyclic $qf$-fibrations.
\item $f$ is a \emph{level acyclic $s$-cofibration} if it is both a level $q$-equivalence and an $s$-cofibration.
\end{enumerate}
To reiterate, in the phrase ``level acyclic $qf$-fibration'', the adjective ``level'' applies to ``acyclic $qf$-fibration'', but in the phrase ``level
acyclic $s$-cofibration'' it applies \emph{only} to ``acyclic''; the cofibrations are not defined levelwise.
\end{defn}
\begin{defn}\mylabel{sillybilly}
A spectrum $X$ over $B$ is \emph{well-sectioned} if it is level well-sectioned,\index{spectrum!well-sectioned}\index{well-sectioned!spectrum}
so that each ex-space $X(V)$ is $\bar{f}$-cofibrant. It is \emph{well-grounded}\index{well-grounded!spectrum}\index{spectrum!well-grounded}
if it is level well-grounded, so that each $X(V)$ is well-sectioned
and compactly generated.
\end{defn}
The discussion of \S\ref{sec:towardh} applies to the category $G\scr{S}_B$ of $G$-spectra over $B$ with homotopies defined in terms of the cylinders $X\sma_B I_+$. In particular, we have the notion of a Hurewicz cofibration in $G\scr{S}_B$, abbreviated $cyl$-cofibration,\index{cofibration!cyl-@$cyl$- --} defined in terms of these cylinders, and we also have the notion of strong Hurewicz cofibration, abbreviated $\overline{cyl}$-cofibration.
\begin{lem}\mylabel{levelcofs}
A $cyl$-cofibration of spectra over $B$ is a level $fp$-cofibration and a $cyl$-fibration of spectra over $B$ is a level $fp$-fibration. A $cyl$-cofibration between well-sectioned spectra over $B$ is a level $f$-cofibration and therefore both a level $h$-cofibration and a level $fp$-cofibration.
\end{lem}
\begin{proof}
By the mapping cylinder retraction criterion of Hurewicz cofibrations, a $cyl$-cofibration of spectra over $B$ is a level $fp$-cofibration. The statement about fibrations follows similarly from the path lifting function characterization of Hurewicz fibrations. An $fp$-cofibration between well-sectioned ex-spaces is an $f$-cofibration by \myref{reverse2}, and all $f$-cofibrations are $h$-cofibrations.
\end{proof}
Recall the notions of a ground structure and of a well-grounded
subcategory of weak equivalences from Definitions \ref{back},
\ref{moreback}, and \ref{hproper}.
\begin{prop}\mylabel{levelwellgr}
The well-grounded spectra over $B$ give $G\scr{S}_B$ a ground structure whose
ground cofibrations, or $g$-cofibrations, are the level $h$-cofibrations. The level $q$-equivalences specify a well-grounded subcategory of weak equivalences
with respect to this ground structure. In the gluing and colimit lemmas, one need only assume that the relevant maps are level $h$-cofibrations, not necessarily also $cyl$-cofibrations.
\end{prop}
\begin{proof}
That we have a ground structure follows levelwise from the ground structure on ex-spaces in \myref{exback}. That the level $q$-equivalences
are well-grounded follows levelwise from \myref{exwellgr}.
\end{proof}
We construct the level model structure on $G\scr{S}_{B}$ from the $qf$-model structure on $G\scr{K}_B$ specified in \myref{Theqf}, but all results apply
verbatim starting from the $qf(\scr{C})$-model structure for any closed
generating set $\scr{C}$ (as defined in \myref{IJBG2}). We shall need the extra generality for the reasons discussed in Chapter 7. Recall that $I^f_B$ and $J^f_B$ denote the sets of generating $qf$-cofibrations and generating acyclic $qf$-cofibrations in $G\scr{K}_B$. We use the shift desuspension functors $F_V$ of \myref{FVs} to obtain corresponding sets on the spectrum level. We need the following observations.
\begin{lem}\mylabel{goodFV}
The functor $F_V$ enjoys the following properties.
\begin{enumerate}[(i)]
\item If $K$ is a well-grounded ex-space over $B$, then $F_VK$ is well-grounded. If $K$ is an ex-fibration, then $F_VK$ is a level ex-fibration.
\item If $i\colon K\longrightarrow L$ is an $h$-equivalence between well-grounded ex-spaces over $B$, then $F_Vi$ is a level $h$-equivalence.
\item If $i\colon K\longrightarrow L$ is an $fp$-cofibration, then $F_Vi$ is a $cyl$-cofibration and therefore a level $fp$-cofibration. If, further,
$K$ and $L$ are well-sectioned, then $F_Vi$ is a level $f$-cofibration
and therefore a level $h$-cofibration.
\item If $i\colon K\longrightarrow L$ is an $\overline{fp}$-cofibration, then
$F_Vi$ is a $\overline{cyl}$-cofibration.
\item If $i\colon K\longrightarrow L$ is an $\overline{f}$-cofibration
between well-grounded ex-spaces over $B$,
then $F_Vi$ is a $\overline{cyl}$-cofibration which is a level
$\overline{f}$-cofibration and therefore both a level
$\overline{fp}$-cofibration and a level $\overline{h}$-cofibration.
\end{enumerate}
\end{lem}
\begin{proof}
By \myref{FVs}, $(F_V K)(W) = \scr{J}_G(V,W)_B\wedge_B K$, and the $G$-space $\scr{J}_G(V,W)$ is well-based. Now (i) holds by \myref{HursmaK} and (ii) holds
by \myref{savior}. Since $F_V$ is left adjoint to the evaluation functor $\text{Ev}_V$ and since $cyl$-fibrations are level $fp$-fibrations, (iv)
and the first statement of (iii) follow from the definitions by adjunction.
The second statement of (iii) follows from \myref{reverse2}. The first
half of (v) follows from (iv) since $\overline{f}$-cofibrations are $\overline{fp}$-cofibrations, and the second half follows from (iii) since $F_Vi$ is a level $f$-cofibration between well-grounded spectra and therefore a level $\overline{f}$-cofibration by \myref{ffpmodel}(ii).
\end{proof}
\begin{defn}\mylabel{FBJB}
Define $FI^f_B$\@bsphack\begingroup \@sanitize\@noteindex{FIBf@$FI^f_B$} to be the set of maps $F_V i$ with $V$ in a skeleton $sk \scr{I}_G$ of $\scr{I}_G$ and $i$ in $I^f_B$. Define $FJ^f_B$\@bsphack\begingroup \@sanitize\@noteindex{FJBf@$FJ^f_B$} to be the set of maps $F_Vj$ with
$V$ in $sk \scr{I}_G$ and $j$ in $J^f_B$.
\end{defn}
Recall the notion of a well-grounded model structure from \myref{wellmodel}.
Among other properties, such model structures are compactly generated, proper,
and $G$-topological.
\begin{thm}\mylabel{levelqf}\index{model structure!level -- on parametrized spectra}\index{level model structure}
The category $G\scr{S}_B$ is a well-grounded model category with respect to the level $q$-equivalences, the level $qf$-fibrations and the $s$-cofibrations. The sets $FI^f_B$ and $FJ^f_B$ give the generating $s$-co\-fi\-bra\-tions and the generating level acyclic $s$-co\-fi\-bra\-tions. All $s$-cofibrations are level $\overline{f}$-cofibrations, hence level $\overline{fp}$ and level $\overline{h}$-cofibrations, and all $s$-cofibrant spectra over $B$ are well-grounded.
\end{thm}
\begin{proof}
By \myref{goodFV}, the maps in $FI^f_B$ and $FJ^f_B$ are
$\overline{cyl}$-co\-fi\-bra\-tions between well-grounded
objects and $\overline{f}$-co\-fi\-bra\-tions. Moreover, the
maps in $FJ^f_B$ are level acyclic. Therefore, to prove the
model axioms, we need only verify the compatibility condition (ii) in \myref{Newcompgen}. Adjunction arguments show that a map is a level $qf$-fibration if and only if it has the RLP with respect to $FJ^f_B$ and that it is a level acyclic $q$-fibration if and only if it has the RLP with respect to $FI^f_B$. This implies that the classes of $s$-cofibrations and of $FI^f_B$-cofibrations (in the sense of \myref{cofhyp}(iii)) coincide. Therefore, if a map has the RLP with respect to $FI^f_B$, then it is a level acyclic $qf$-fibration. The required compatibility condition now follows from its analogue for $G\scr{K}_B$. Condition (iv) in \myref{Newcompgen} holds by its ex-space level analogue and the fact that $(F_V K)\sma_B T\iso F_V(K\sma_B T)$ for an ex-space $K$ over $B$ and a based space $T$. Right properness follows directly from the space level analogue.
\end{proof}
\begin{rem}\mylabel{qnogood}
Just as in \myref{FBJB}, we can also define sets $FI_B$ and $FJ_B$ based on
the generating sets $I_B$ and $J_B$ for the $q$-model structure on $G\scr{K}_B$.
We can then use \myref{compgen} to prove the analogue of \myref{levelqf}
stating that $G\scr{S}_B$ is a cofibrantly generated model category under the
level $q$-model structure. Since the compatibility condition holds by the same proof as for the level $qf$-model structure, we need only verify the acyclicity condition to show this.
For a generating acyclic $q$-cofibration $j\in J_{B}$, we have
$F_Vj=V^*\sma_B j$, where $V^*(W)=\scr{J}_{G,B}(V,W)$. This map is a level
$h$-equivalence by \myref{goodFV}(ii). Although $j$ is an $h$-cofibration,
it is not immediate that $F_Vj$ is a level $h$-cofibration. (This holds for $j\in J_B^f$ by \myref{goodFV}(iii), since $j$ is then an $fp$-cofibration). Indeed, for general ex-spaces $K$ and $h$-cofibrations $f$, $K\sma_B f$ need
not be an $h$-cofibration. However, since $\scr{J}_{G,B}(V,W) = \scr{J}_G(V,W)_B$, we see directly that $F_Vj$ is indeed a level $h$-cofibration. By inspection of
the definition of wedges over $B$ in terms of pushouts, the gluing lemma in
$\scr{K}$ then applies to show that wedges over $B$ of maps in $FJ_B$ are level
acyclic $h$-cofibrations. Since pushouts and colimits in $\scr{S}_B$ are
constructed levelwise on total spaces, it follows that relative $FJ_B$
complexes are acyclic $h$-cofibrations since the $q$-model structure on
$\scr{K}$ is well-grounded.
\end{rem}
\begin{rem}\mylabel{plus1} As in the nonparametrized case \cite{MM}, ``positive'' model structures would be needed to obtain a comparison
with the as yet undeveloped alternative approach to parametrized stable
homotopy theory based on \cite{EKMM, LMS}. Such model structures can be
defined as in \cite[p.\, 44]{MM}, starting from the subsets $(FI^f_B)^+$
and $(FJ^f_B)^+$ that are obtained by restricting to those $V$ such that $V^G\neq 0$. One then defines the \emph{positive level} versions of all
of the types of maps specified in \myref{never} by restricting to those
levels $V$ such that $V^G\neq 0$. The positive level analogue of
\myref{levelqf} holds, where the positive $s$-cofibrations are the
$s$-cofibrations that are isomorphisms at all levels $V$ such that
$V^G=0$; compare \cite[III.2.10]{MM}. However, we shall make no use of the positive model structure in this paper, and we will make little further reference to it.
\end{rem}
The same proof as in \cite[I.2.10, II.4.10, III.2.12]{MM} gives the following result.
\begin{thm}\mylabel{QPU}
The forgetful functor $\mathbb{U}$ from spectra over $B$ to prespectra over $B$ has a left adjoint $\mathbb{P}$ such that $(\mathbb{P},\mathbb{U})$ is a Quillen equivalence.
\end{thm}
\section{Some Quillen adjoint pairs relating level model structures}
This section gives the analogues for the level model structure of some of the ex-space level results in \S\S7.2-7.4. These results are also analogues of
results in \cite[III.\S2]{MM}, which in turn have non-equivariant precursors
in \cite[\S6]{MMSS}. They admit essentially the same proofs as in Chapter 7
or in the cited references. The level $qf$-model structure is understood throughout. More precisely, where a $qf(\scr{C})$-model structure was used in
Chapter 7, we must use the corresponding level $qf(\scr{C})$-model structure here.
Since we want our model structures to be $G$-topological, we only use generating
sets $\scr{C}$ that are closed under finite products.
Our first observation is immediate from the fact that equivalences and
fibrations are defined levelwise, the next follows directly from its
ex-space analogue Proposition \ref{smaB}, and the third and fourth
are proven in the same way as their ex-space analogues \ref{Boxcof20} and
\myref{smaAB}. All apply to the level $qf(\scr{C})$-model structures for any
choice of $\scr{C}$.
\begin{prop}
The pair of adjoint functors $(F_V,\text{Ev}_V)$ between $G\scr{K}_B$ and $G\scr{S}_B$ is a Quillen adjoint pair.
\end{prop}
\begin{prop}\mylabel{QuillK}
For a based $G$-CW complex $T$, $((-)\sma_B T, F_B(T,-))$ is a Quillen adjoint pair of endofunctors of $G\scr{S}_B$.
\end{prop}
\begin{prop}\mylabel{Boxcof2}
If $i\colon X\longrightarrow Y$ and $j\colon W\longrightarrow Z$ are $s$-cofibrations
of spectra over base spaces $A$ and $B$, then
\[i\Box j\colon (Y\barwedge W)\cup_{X\barwedge W}(X\barwedge Z)\longrightarrow Y\barwedge Z\]
is an $s$-cofibration over $A\times B$ which is level
acyclic if either $i$ or $j$ is acyclic.
\end{prop}
As in \S7.2, we cannot expect this result to hold for internal smash
products over $B$. The case $A=*$, which relates spectra to spectra
over $B$, is particularly important. As we explain in \S14.1, it
leads to a fully satisfactory theory of \emph{parametrized} module
spectra over \emph{nonparametrized} ring spectra.
\begin{cor}\mylabel{ext}
If $Y$ is $s$-cofibrant over $B$, then the functor $(-)\barwedge Y$ from $G\scr{S}_A$ to $G\scr{S}_{A\times B}$ is a Quillen left adjoint with Quillen right adjoint $\bar{F}(Y,-)$.
\end{cor}
Again the next result is a direct consequence of its
ex-space analogue \myref{Qad10} and applies with any
choice of $\scr{C}$.
\begin{prop}\mylabel{Qad1}
Let $f\colon A\longrightarrow B$ be a $G$-map. Then $(f_{!},f^*)$ is a Quillen
adjoint pair. The functor $f_!$ preserves level $q$-equivalences
between well-sectioned $G$-spectra over $B$. If $f$ is a $qf$-fibration,
then $f^*$ preserves all level $q$-equivalences.
\end{prop}
\begin{prop}\mylabel{ffequiv}
If $f\colon A\longrightarrow B$ is a $q$-equivalence, then $(f_{!},f^*)$ is a
Quillen equivalence.
\end{prop}
\begin{proof} We mimic the proof of \myref{ffequiv0}, but with $X$
and $Y$ taken to be an $s$-cofibrant $G$-spectrum over $A$ and
a level $qf$-fibrant $G$-spectrum over $B$. It is clear that
$f^*Y\longrightarrow Y$ is a level $q$-equivalence since $A\longrightarrow B$ is
a $q$-equivalence. Since $X$ is $s$-cofibrant, $*_A\longrightarrow X$ is a
level $h$-cofibration. Note that it is essential for this statement
that we start from the $qf$ and not the $q$-model structure on ex-spaces.
Since pushouts along level $h$-cofibrations preserve level $q$-equivalences, $X\longrightarrow f_!X$ is a level $q$-equivalence. The conclusion follows as in
\myref{ffequiv0}.
\end{proof}
\begin{prop}\mylabel{Qad2}
Let $f\colon A\longrightarrow B$ be a $G$-bundle whose fibers $A_b$ are $G_b$-CW complexes. Then $f^*$ preserves level $q$-equivalences and
$s$-cofibrations. Therefore $(f^*,f_*)$ is a Quillen adjoint pair.
\end{prop}
\begin{proof} Here we must use a generating set $\scr{C}(f)$ as specified
in \myref{Qad202}. The proof that $f^*$ preserves $s$-cofibrations
reduces to showing that the maps $f^*F_Vi \cong F_Vf^*i$ are
$s$-cofibrations for generating $s$-cofibrations $i$. Since $F_V$ is a
Quillen left adjoint it takes $qf$-cofibrations to $s$-cofibrations,
so we are reduced to the ex-space level, where $f^*i$ is shown to be a $qf$-cofibration in \myref{Qad202}.
\end{proof}
Now consider the change of groups functors of \S11.5. The following result
shows that the equivalence of \myref{changes} descends to homotopy categories.
It is proven by levelwise application of its ex-space analogue
\myref{Lishriek}, together with change of universe considerations
that are deferred until \S14.2 and \S14.3.
\begin{prop}\mylabel{Lchanges}
Let $\iota\colon H\longrightarrow G$ be the inclusion of a subgroup. The pair of functors $(\iota_!,\nu^*\iota^*)$ relating $H\scr{S}_A$ and $G\scr{S}_{\iota_!A}$ give a Quillen equivalence. If $A$ is completely regular, then $\iota_!$ is also a Quillen right adjoint.
\end{prop}
For a point $b$ in $B$, we combine the special case $\tilde{b}\colon G/G_b\longrightarrow B$ of \myref{Qad1} with \myref{Lchanges}, where $\iota\colon G_b\longrightarrow G$ and $\nu\colon *\longrightarrow G/G_b$, to obtain the following analogue of
\myref{FibadQ0}. Recall from \myref{Johann} that the fiber functor
$(-)_b\colon G\scr{S}_B\longrightarrow G_b\scr{S}$ is given by $\nu^*\iota^*\tilde{b}^*=b^*\iota^*$. Its left adjoint $(-)^b$
therefore agrees with $\tilde{b}_!\iota_!$.
\begin{prop}\mylabel{FibadQ}
For $b\in B$, the pair of functors $((-)^b,(-)_b)$ relating $G_b\scr{S}_*$ and $G\scr{S}_B$ is a Quillen adjoint pair.
\end{prop}
\section{The stable model structure on $G\scr{S}_B$}
The essential point in the construction of the stable model structure is to define the appropriate (stable) homotopy groups. The weak equivalences will then be the maps of parametrized spectra that induce isomorphisms on all homotopy groups. We refer to them as the $\pi_*$-isomorphisms or $s$-equivalences, using
these terms interchangeably. There are several motivating observations for our definitions. We return the group $G$ to the notations for the moment.
First, a $G$-spectrum $X$ over $B$ is level $qf$-fibrant if and only if each
projection $X(V)\longrightarrow *_B(V)=B$ is a $qf$-fibration of ex-$G$-spaces. It is equivalent that each fixed point map $X(V)^H\longrightarrow B^H$ be a non-equivariant $qf$-fibration, and, by \myref{qfles}, we have resulting long exact sequences of homotopy groups
\begin{equation}\label{les1}
\cdots \longrightarrow \pi_{q+1}^H(B)
\longrightarrow \pi^H_q(X_b(V))
\longrightarrow \pi^H_q(X(V))
\longrightarrow \pi_q^H(B)
\longrightarrow \cdots
\end{equation}
for each $b\in B^H$. Here, for a $G$-space $T$, $\pi_q^H(T)$ denotes $\pi_q(T^H)$.
Second, as we have already discussed in \S11.4, the fibers $X_b$ of a $G$-spectrum $X$ are $G_b$-spectra, and our guiding principle is to use these nonparametrized spectra to encode the homotopical information about our parametrized spectra. \myref{FibadQ} allows us to encode levelwise
information in the level homotopy groups of fibers, and it is plausible that
we can similarly encode the full structure of our parametrized $G$-spectrum
$X$ in the spectrum level homotopy groups of the fiber $G_b$-spectra $X_b$.
However, we can only expect to do so when $X$ is level $qf$-fibrant and we have
the long exact sequences (\ref{les1}).
Recall that the homotopy groups $\pi_q^H(Y)$ of a nonparametrized
$G$-spectrum $Y$ are defined in \cite[III.3.2]{MM} as the colimits of the groups $\pi_q^H(\Omega^VY(V))$, where the maps of the colimit system are induced in the
evident way by the adjoint structure maps $\tilde\sigma\colon Y(V)\longrightarrow \Omega^{W-V}Y(W)$ of $Y$. The functor $\Omega^V$ on based $G$-spaces preserves
$q$-fibrations and the functor $\Omega^V_B=F_B(S^V,-)$ on $G$-spectra over $B$ preserves level $qf$-fibrations. Formally, these hold since $S^V$ is a $q$-cofibrant $G$-space and the relevant model structures are $G$-topological.
This leads to two families
of long exact sequences relating the homotopy groups $\pi_q^H(\Omega^VX_b(W)$ of
fibers to the homotopy groups of the base space $B$ and of the total spaces
$X(W)$. First, if $X$ is a level $q$-fibrant $G$-spectrum over $B$, then, using basepoints determined by a point $b\in B^H$ for any $H\subset G_b$, the $q$-fibrations $\Omega^VX(W) \longrightarrow \Omega^VB$ of based $G$-spaces with fibers $\Omega^VX_b(W)$ induce long exact sequences
\begin{equation}\label{les2}
\cdots \longrightarrow \pi_{q+1}^H(\Omega^V B)\longrightarrow \pi^H_q(\Omega^VX_b(W))
\longrightarrow \pi^H_q(\Omega^VX(W)) \longrightarrow \pi_q^H(\Omega^V B)\longrightarrow \cdots.
\end{equation}
Second, if $X$ is level $qf$-fibrant, then the $qf$-fibrations $(\Omega^V_BX)(W)\longrightarrow *_B$ of ex-$G$-spectra over $B$ with
fibers $\Omega^VX_b(W)$ induce long exact sequences
\begin{equation}\label{les3}
\cdots \longrightarrow \pi_{q+1}^H(B)
\longrightarrow \pi^H_q(\Omega^VX_b(W))
\longrightarrow \pi^H_q((\Omega^V_BX)(W))
\longrightarrow \pi_q^H(B)
\longrightarrow \cdots.
\end{equation}
The first allows us to relate the homotopy groups of the
$X_b$ to the homotopy groups of the ordinary loops $\Omega^VX(W)$
on total spaces. The second
allows us to relate the homotopy groups of the $X_b$ to
the homotopy groups of the parametrized loop ex-spaces $(\Omega_B^VX)(W)$.
It is the second that is most relevant to our work.
\begin{defn}\mylabel{htygps}
The \emph{homotopy groups}\index{homotopy groups!of parametrized spectra} of a level $qf$-fibrant $G$-spec\-trum over $B$, or of a level $qf$-fibrant $G$-prespectrum $X$, are all of the homotopy groups $\pi_q^H(X_b)$ of all of the fibers $X_b$, where $H\subset G_b$. The homotopy groups of a general $G$-spectrum, or $G$-prespectrum, $X$ over $B$ are the homotopy groups $\pi_q^H((RX)_b)$ of a level $qf$-fibrant approximation $RX$ to $X$. We still denote these homotopy groups by $\pi_q^H(X_b)$.
In either category, a map $f\colon X\longrightarrow Y$ is said to be a \emph{$\pi_*$-isomorphism}\index{equivalence!s-@$s$- --}\index{equivalence!p-iso@$\pi_*$-isomorphism}\index{p-iso@$\pi_*$-isomorphism} or, synonymously, an \emph{$s$-equi\-va\-lence}, if, after level $qf$-fibrant approximation, it induces an isomorphism on all homotopy groups.
\end{defn}
There are also homotopy groups specified in terms of maps out of sphere spectra over $B$, but we choose to ignore them in setting up our model theoretic foundations. Our choice captures the intuitive idea that spectra over $B$ should be {\em parametrized spectra}\/: the fiber spectra should carry all of the homotopy theoretical information. With this choice, a good deal of the work needed to set up the stable model structure reduces to work that has already been done in \cite{MM}. The following observation is a starting point that illustrates the pattern of proof. Now that we have seen how the equivariance
appears in the definition of homotopy groups, we revert to our custom of
generally deleting $G$ from the notations.
\begin{lem}\mylabel{levelpi}
A level $q$-equi\-va\-lence of spectra over $B$ is a $\pi_*$-iso\-mor\-phism. \end{lem}
\begin{proof}
A level $qf$-fibrant approximation to the given level $q$-equi\-va\-lence is a level acyclic $qf$-fibration, and it induces a level $q$-equi\-va\-lence on fibers over points of $B$ by \myref{FibadQ}. This allows us to apply {\cite[III.3.3]{MM}}, which gives the same conclusion for nonparametrized spectra, one fiber at a time.
\end{proof}
To exploit our definition of homotopy groups, we need the following accompanying definition and proposition.
\begin{defn}
An \emph{$\Omega$-prespectrum over $B$}\index{prespectrum!Om@$\Omega$- --} is a level $qf$-fibrant prespectrum $X$ over $B$ such that each of its adjoint structure maps $\tilde{\sigma}\colon X(V)\longrightarrow \Omega_B^{W-V}X(W)$ is a $q$-equi\-va\-lence of ex-spaces over $B$, that is, a $q$-equi\-va\-lence of total spaces. An (orthogonal) \emph{$\Omega$-spectrum over $B$}\index{spectrum!Om@$\Omega$- -- over $B$} is a level $qf$-fibrant spectrum over $B$ such that each of its adjoint structure maps is a $q$-equi\-va\-lence; equivalently, its underlying prespectrum must be an $\Omega$-prespectrum over $B$.
\end{defn}
Since we are omitting the adjective ``orthogonal'' from ``orthogonal spectrum over $B$'', we must use the term ``$\Omega$-prespectrum over $B$'' on the prespectrum level to avoid confusion; the more standard term ``$\Omega$-spectrum'' was used in \cite{MM}.
\begin{prop}\mylabel{OmOm}
A level fibrant $G$-spectrum $X$ over $B$ is an $\Omega$-$G$-spectrum over $B$ if and only if each fiber $X_b$ is an $\Omega$-$G_b$-spectrum. The $G$-prespectrum analogue also holds.
\end{prop}
\begin{proof}
By the five lemma, this is immediate from a comparison of the long exact sequences in (\ref{les1}) and (\ref{les3}).
\end{proof}
This result leads to the following partial converse to \myref{levelpi}.
\begin{thm}\mylabel{bombsaway}
A $\pi_*$-isomorphism between $\Omega$-spectra over $B$ is a level $q$-equi\-va\-lence.
\end{thm}
\begin{proof}
The analogue for nonparametrized $\Omega$-spectra is \cite[III.3.4]{MM}. In view of \myref{OmOm}, we can apply that result on fibers and then use that $\Omega$-spectra over $B$ are required to be level $qf$-fibrant to deduce the claimed level
$q$-equi\-va\-lence on total spaces from (\ref{les1}).
\end{proof}
Technically, the real force of our definition of homotopy groups is that this result describing the $\pi_*$-isomorphisms between $\Omega$-spectra over $B$ is an immediate consequence of the work in \cite{MM}. Given this relationship between $\Omega$-spectra and homotopy groups, many of the arguments of \cite{MM} apply fiberwise to allow the development of the \emph{stable model structure}. However, as discussed in the next section, careful use of level fibrant approximation is required. We shall use the terms ``stable model structure'' and ``$s$-model structure'' interchangeably. The $s$-cofibrations are the same as those of the level $qf$-model structure and the $s$-fibrant spectra over $B$ turn out to be the $\Omega$-spectra over $B$.
\begin{defn}\mylabel{Def5}
A map of spectra or prespectra over $B$ is
\begin{enumerate}[(i)]
\item an \emph{acyclic $s$-cofibration} if it is a $\pi_*$-isomorphism and an $s$-cofibration,
\item an \emph{$s$-fibration}\index{fibration!s-@$s$- --} if it satisfies the RLP with respect to the acyclic $s$-co\-fi\-bra\-tions,
\item an \emph{acyclic $s$-fibration} if it is a $\pi_*$-isomorphism and an $s$-fibration.
\end{enumerate}
\end{defn}
We shall prove the following basic theorem in the next two sections.
\begin{thm}\mylabel{modelS}
The categories $G\scr{S}_{B}$ and $G\scr{P}_B$ are well-grounded model categories with respect to the
$\pi_*$-iso\-mor\-phisms (= $s$-equi\-va\-lences),
$s$-fibrations and $s$-cofibrations. The $s$-fibrant
objects are the $\Omega$-spectra over $B$.
\end{thm}
\begin{rem}\mylabel{plus2} Recall \myref{plus1}. We can define positive
$\Omega$-prespectra and positive analogues of our $s$-classes of maps,
starting with the positive level $qf$-model structure.
As in \cite[III\S5]{MM}, the positive analogue of the previous theorem also holds, with the same proof. The identity functor is the left adjoint of a Quillen equivalence from $G\scr{S}_{B}$ or $G\scr{P}_B$ with its positive stable
model structure to $G\scr{S}_{B}$ or $G\scr{P}_B$ with its stable model structure.
\end{rem}
The proof of the following result is virtually the same as the proof of its
nonparametrized precursor \cite[III.4.16 and III.5.7]{MM} and will not be
repeated.
\begin{thm}\mylabel{modelPU}
The adjoint pair $(\mathbb{P},\mathbb{U})$ relating the categories $G\scr{P}_{B}$ and $G\scr{S}_B$ of prespectra and spectra over $B$ is a Quillen equivalence with respect to either the stable model structures or the positive stable model structures.
\end{thm}
As in \cite[III.\S6]{MM}, \myref{modelS} leads to the following definition and theorem, whose proof is the same as the proof of \cite[III.6.1]{MM}.
\begin{defn}
Let $[X,Y]^{\ell}$ denote the morphism sets in the homotopy category associated to the level $qf$-model structure on $G\scr{P}_B$ or $G\scr{S}_B$. A map $f\colon X\longrightarrow Y$ is a \emph{stable equivalence}\index{equivalence!stable --}\index{stable equivalence} if $f^*\colon [Y,E]^{\ell}\longrightarrow [X,E]^{\ell}$ is an isomorphism for all $\Omega$-spectra $E$ over $B$. Define the positive analogues similarly. Let $[X,Y]$ denote the morphism sets in the stable homotopy category $\text{Ho}\, G\scr{S}_B$ of spectra over $B$.
\end{defn}
\begin{thm}\mylabel{modelT}
The following are equivalent for a map $f\colon X\longrightarrow Y$ of spectra or prespectra over $B$.
\begin{enumerate}[(i)]
\item $f$ is a stable equivalence.
\item $f$ is a positive stable equivalence.
\item $f$ is a $\pi_*$-isomorphism.
\end{enumerate}
Moreover $[X,E] = [X,E]^{\ell}$ if $E$ is an $\Omega$-spectrum.
\end{thm}
\myref{compaR} below should make it clear why the last statement is true.
\section{The $\pi_*$-isomorphisms}
In the main, the proof of \myref{modelS} is obtained by applying the results in \cite{MM} fiberwise. Since total spaces are no longer assumed to be weak Hausdorff, we have to be a little careful: we are quoting results proven for $\scr{T}$ and using them for $\scr{K}_*$. However, we can just as well interpret \cite{MM} in terms of $\scr{K}_*$. The total spaces $X(V)$ of an $s$-cofibrant
spectrum over $B$ are weak Hausdorff, hence $s$-cofibrant approximation
places us in a situation where total spaces are in $\scr{U}$ and therefore fibers are in $\scr{T}$.
There is a more substantial technical problem to overcome in adapting the proofs of \cite{MM, MMSS} to the present setting. In the situations encountered in those references, all objects were level $q$-fibrant, and that simplified matters considerably. Here, level $qf$-fibrant approximation entered into our definition of homotopy groups, and for that reason the results of this section are considerably more subtle than their counterparts in the cited sources.
We begin by noting that any level ex-quasifibrant approximation, not necessarily
a $qf$-fibrant approximation, can be used to calculate the homotopy groups of parametrized spectra.
\begin{lem}\mylabel{zigzag}
A zig-zag of level $q$-equivalences connecting a spectrum $X$ over $B$ to a level ex-quasifibrant spectrum $Y$ over $B$ induces an isomorphism between the homotopy groups of $X$ and of $Y$, and the latter can be computed directly in terms of the fibers of $Y$.
\end{lem}
\begin{proof}
This follows from \myref{levelpi} by applying a level $qf$-fibrant approximation functor to the zig-zag.
\end{proof}
\begin{thm}\mylabel{exact}
Let $f\colon X\longrightarrow Y$ be a map between $G$-spectra over $B$. For any $H\subset G$ and $b\in B^H$, there is a natural long exact sequence
\[ \cdots \longrightarrow \pi^H_{q+1}(Y_b) \longrightarrow \pi^H_q((F_B f)_b)\longrightarrow \pi^H_q(X_b)\longrightarrow \pi^H_q(Y_b)\longrightarrow \cdots\]
and, if $X$ is well-sectioned, there is also a natural long exact sequence
\[ \cdots \longrightarrow \pi^H_q(X_b)\longrightarrow \pi^H_q(Y_b)\longrightarrow \pi^H_{q}((C_B f)_b)\longrightarrow \pi^H_{q-1}(X_b)\longrightarrow \cdots.\]
\end{thm}
\begin{proof}
For the first long exact sequence, let $R$ be a level $qf$-fibrant approximation functor and consider $Rf$. We claim that the induced map $F_Bf\longrightarrow F_BRf$ is a level $q$-equivalence and that $F_BRf$ is level $qf$-fibrant. This means that $F_BRf$ is a level $qf$-fibrant approximation to $F_Bf$, so that the homotopy
groups of the fibers $(F_BRf)_b \iso F((Rf)_b)$ are the homotopy groups of $F_Bf$. When restricted to fibers over $b$, the parametrized fiber sequence
$RX\longrightarrow RY\longrightarrow F_BRf$ of spectra over $B$ gives the
nonparametrized fiber sequence $(RX)_b \longrightarrow (RY)_b \longrightarrow F((Rf)_b)$, and the long exact sequence follows from \cite[III.3.5]{MM}. To prove the claim, observe that since $F_B(I,Y)\longrightarrow Y$ is a Hurewicz fibration, it has a path-lifting function which levelwise shows that $F_B(I,Y)\longrightarrow Y$ is a level $fp$-fibration and therefore a level $qf$-fibration (since all $qf$-cofibrations are $fp$-cofibrations in $G\scr{K}_B$). The dual gluing lemma (see \myref{hproper}(iii)) then gives that the induced map $F_Bf\longrightarrow F_BRf$ is a level $q$-equivalence. Since $F_B(I,-)$ preserves level $qf$-fibrant objects and since pullbacks of level $qf$-fibrant objects along a level $qf$-fibration are level $qf$-fibrant, $F_BRf$ is level $qf$-fibrant.
Since the maps $X\longrightarrow C_B X$ and $RX\longrightarrow C_B RX$ are $cyl$-cofibrations between well-sectioned spectra and therefore level $h$-cofibrations by \myref{levelcofs}, the gluing lemma gives that $C_B f\longrightarrow C_B Rf$ is a level $q$-equivalence. Since $RX$ and $RY$ are level well-sectioned and level $qf$-fibrant, they are level ex-quasifibrations. It follows from \myref{quasicof} that $C_B Rf$ is a level ex-quasifibration. We cannot conclude that $C_BRf$ is
level $qf$-fibrant, but by \myref{zigzag} we can nevertheless use $C_B Rf$ to calculate the homotopy groups of $C_B f$. On fibers over $b$, the cofiber sequence of $Rf$ is just the cofiber sequence of $(Rf)_b$, and the long exact sequence follows from \cite[III.3.5]{MM}.
\end{proof}
Recall \myref{levelwellgr}, which specifies the ground structure
in $G\scr{S}_B$ and shows that the level $q$-equivalences give a well-grounded subcategory of weak equivalences; the $g$-cofibrations are just the level
$h$-cofibrations. The following result shows that the same
is true for the $\pi_*$-isomorphisms. However, in contrast to \myref{levelwellgr}, it is crucial to assume that the relevant maps
in the gluing and colimit lemmas are both $cyl$-cofibrations and
$g$-cofibrations, as prescribed in \myref{hproper}.
\begin{thm}\mylabel{piwellgr}
The $\pi_*$-isomorphisms in $G\scr{S}_B$ give a well-grounded
subcategory of weak equivalences. In detail, the following
statements hold.
\noindent\begin{enumerate}[(i)]
\item A homotopy equivalence is a $\pi_*$-isomorphism.
\item The homotopy groups of a wedge of well-grounded spectra over $B$ are the direct sums of the homotopy groups of the wedge summands.
\item The $\pi_*$-isomorphisms are preserved under pushouts along maps that are both $cyl$ and $g$-cofibrations.
\item Let $X$ be the colimit of a sequence $i_n\colon X_n\longrightarrow X_{n+1}$ of maps that are both $cyl$ and $g$-cofibrations and assume that $X/\!_B X_0$ is well-grounded. Then the homotopy groups of $X$ are the colimits of the homotopy groups of the $X_n$.
\item For a map $i\colon X\longrightarrow Y$ of well-grounded spectra over $B$ and a map $j\colon K\longrightarrow L$ of well-based spaces, $i\Box j$ is a $\pi_*$-isomorphism if either $i$ is a $\pi_*$-isomorphism or $j$ is a $q$-equivalence.
\end{enumerate}
\end{thm}
\begin{proof}
The conclusion that the $\pi_*$-isomorphisms give a well-grounded
subcategory of weak equivalences, as prescribed in \myref{hproper},
follows directly from the listed properties, using \myref{gluederiv} to derive the gluing lemma. Since level $q$-equivalences are $\pi_*$-isomorphisms,
$s$-cofibrant approximation in the level $qf$-model structure gives the
factorization hypothesis \myref{gluederiv}(ii).
A homotopy equivalence of spectra is a level $fp$-equivalence and hence a level $q$-equivalence, so (i) follows from \myref{levelpi}. For finite wedges, (ii) is immediate from the evident split cofiber sequences and \myref{exact}. For arbitrary wedges of well-grounded spectra over $B$, $\vee_B X_i \longrightarrow \vee_B RX_i$ is a level $q$-equivalence since the level $q$-equivalences are well-grounded and $\vee_B RX_i$ is level quasifibrant by \myref{quasicof}.
By \myref{zigzag} we can use $\vee_B RX_i$ to calculate the homotopy groups of $\vee_B X_i$. Over a point $b$ in $B$, $\vee_B RX_i$ is just $\vee (RX_i)_b$ and the result follows from the nonparametrized analogue \cite[III.3.5]{MM}.
Now consider (iii). Let $i\colon X\longrightarrow Y$ be both a $cyl$-cofibration and
a $g$-cofibration and let $f\colon X\longrightarrow Z$ be a $\pi_*$-isomorphism. Since
$i$ and its $s$-cofibrant approximation $Qi$ are both $cyl$ and
$g$-cofibrations and since the level $q$-equivalences give a well-grounded
subcategory of weak equivalences, the gluing lemma shows that we may
approximate our given pushout diagram by one in which all objects are well-sectioned. Let $j\colon Z\longrightarrow Y\cup_X Z$ be the pushout of $i$ along $f$. Since $i$ and $j$ are $cyl$-cofibrations and $j$ is the pushout of $i$, their
cofibers are homotopy equivalent. Comparing the long exact sequences of
homotopy groups associated to the cofiber sequences of $i$ and $j$ gives
that the pushout $Y\longrightarrow Y\cup_X Z$ of $f$ along $i$ is a $\pi_*$-isomorphism.
For (iv), we may use $s$-cofibrant approximation in the level model structure
to replace our given tower by one in which all objects are well-sectioned. We note as in the proof of \myref{modellim1} that the natural map $\text{Tel} X_n \longrightarrow \text{colim} X_n$ is a level $q$-equivalence and therefore a
$\pi_*$-isomorphism. Relating the telescope to a classical homotopy
coequalizer as in the cited proof, we reduce the calculation of the
homotopy groups of the telescope to an algebraic inspection based on (ii).
Alternatively, one can commute double colimits to reduce the verification
to its space level analogue.
For (v), it suffices to show that the tensor $X\sma_B T$ preserves
$\pi_*$-isomorphisms in either variable, by \myref{boxacy}. That follows
from \myref{pismash} below.
\end{proof}
\begin{prop}\mylabel{pismash}
Let $f\colon X\longrightarrow Y$ be a map between well-grounded spectra over $B$.
\begin{enumerate}[(i)]
\item If $f$ is a level $q$-equivalence and $g\colon T\longrightarrow T'$ is a $q$-equivalence of well-based spaces, then
$$ \text{id}\sma_B g\colon X\sma_B T\longrightarrow X\sma_B T'$$
is a level $q$-equivalence and therefore a $\pi_*$-isomorphism.
\item If $f$ is a $\pi_*$-iso\-mor\-phism, then
$$ f\sma_B\text{id}\colon X\sma_B T\longrightarrow Y\sma_B T$$
is a $\pi_*$-isomorphism for any well-based space $T$ and
$$F_B(\text{id},f)\colon F_B(T,X)\longrightarrow F_B(T,Y)$$
is a $\pi_*$-isomorphism for any finite based CW complex $T$.
\item For a representation $V$ in $\scr{V}$, the map $f$ is a $\pi_*$-iso\-mor\-phism if and only if $\Sigma^V_Bf$ is a $\pi_*$-isomorphism.
\end{enumerate}
\end{prop}
\begin{proof}
Part (i) holds since the level $q$-equivalences are well-grounded.
Therefore, for the first part of (ii), we may assume by $q$-cofibrant approximation in the space variable that $T$ is a based CW complex.
Using \myref{quasicof}, it also implies that $-\sma_B T$ preserves approximations of well-grounded spectra over $B$ by level ex-quasifibrations.
Now the first part of (ii) follows fiberwise from its nonparametrized analogue \cite[III.3.11]{MM} and (iii) follows fiberwise from its nonparametrized analogue \cite[III.3.6]{MM}. Since $F_B(-,X)$ takes cofiber sequences of based spaces to fiber sequences of spectra over $B$, the second part of (iii) follows from the first exact sequence in \myref{exact}, as in the proof of \cite[III.3.9]{MM}.
\end{proof}
This leads to the following result, which shows that we are in a stable situation.
\begin{prop}\mylabel{SIOMV}
For all well-grounded spectra $X$ over $B$ and all representations $V$ in $\scr{I}_G$, the unit $\eta\colon X\longrightarrow \Omega_B^V\Sigma_B^VX$ and counit $\varepsilon\colon \Sigma_B^V\Omega_B^VX\longrightarrow X$ of the $(\Sigma_B^V,\Omega^V_B)$ adjunction are $\pi_*$-isomorphisms. Therefore, if $f\colon X\longrightarrow Y$ is a map between well-grounded spectra over $B$, then the natural maps $\eta\colon F_B f\longrightarrow \Omega_B C_B f$ and $\epsilon\colon \Sigma_BF_Bf\longrightarrow C_B f$ are $\pi_*$-isomorphism.
\end{prop}
\begin{proof}
For $\eta$, after approximation of $X$ by an ex-quasifibration, the
conclusion follows fiberwise from its nonparametrized analogue \cite[III.3.6]{MM}. Using the two out of three property and the
triangle equality for the adjunction, it follows that $\Omega^V_B\varepsilon$
is a $\pi_*$-isomorphism, hence so is $\varepsilon$. For the last statement,
the maps $\eta$ and $\varepsilon$ are the parametrized analogues of the maps
defined for ordinary loops and suspensions in \cite[p. 61]{Concise},
and they fit into diagrams relating fiber and cofiber sequences
like those displayed there. Now the last statement follows from the
five lemma and the exact sequences in \myref{exact}.
\end{proof}
\section{Proofs of the model axioms}
We need some $G$-spectrum level recollections from \cite{MM} and their
analogues for $G$-spectra over $B$ to describe the generating acyclic
$s$-cofibrations. Let $(\scr{S}_G,G\scr{S})$ denote the $G$-category of
$G$-spectra. To keep track of enrichments, we return $G$ to the
notations for the moment.
We have a shift desuspension functor
$F_V$ from based $G$-spaces to $G$-spectra given by $F_VT = V^*\sma T$,
where $V^*(W) = \scr{J}_G(V,W)$ \cite[III.4.6]{MM}. It is left adjoint to
evaluation at $V$. For $G$-spectra $X$, the adjoint structure $G$-map
$$\tilde{\sigma}\colon X(V)\longrightarrow \Omega^WX(V\oplus W)$$
may be viewed by
adjunction as a $G$-map
$$\tilde{\sigma}\colon \scr{S}_{G}(F_VS^0,X) \longrightarrow \scr{S}_{G}(F_{V\oplus W}S^W,X).$$
Passing to $G$-fixed points and taking $X= F_VS^0$, the image of the
identity map gives a map of $G$-spectra
$$ \lambda^{V,W}\colon F_{V\oplus W}S^W\longrightarrow F_VS^0.$$
(The notation $\lambda_{V,W}$ was used in \cite{MM}, but we need room for a
subscript). A Yoneda lemma argument then shows that the map of $G$-spaces
$$ \scr{S}_G(\lambda^{V,W},\text{id})\colon \scr{S}_G(F_VS^0,X)\longrightarrow \scr{S}_G(F_{V\oplus W}S^W,X)$$
can be identified with $\tilde{\sigma}\colon X(V)\longrightarrow \Omega^WX(V\oplus W)$.
We need the analogue for $G$-spectra over $B$. Recall from \myref{FVs} that, for an ex-$G$-space $K$ over $B$,
$(F_V K)(W) = V^*(W)\wedge_B K$, where
$$V^*(W) = \scr{J}_{G,B}(V,W) = \scr{J}_G(V,W)_B = (F_VS^0)(W)\sma_B S^0_B.$$
It follows that we can identify $F_VK$ with the evident external tensor
$F_VS^0 \sma_B K$ of the $G$-spectrum $F_VS^0$ and the ex-$G$-space $K$ over $B$; compare \myref{extsmash1}. We have used the notation $\sma_B$ for this generalized tensor, but viewing it as a special case of the external smash product of spectra over $*$ and over $B$ would suggest the alternative notation $\barwedge$.
\begin{defn}\mylabel{lambdas}
For ex-$G$-spaces $K$ over $B$, we define a natural map
$$ \lambda^{V,W}_B\colon F_{V\oplus W}\Sigma^W_B K\longrightarrow F_VK.$$
Namely, identifying the source and target with external tensor products, define
$$\lambda_B^{V,W} = \lambda^{V,W}\sma_B \text{id}\colon (F_{V\oplus W}S^W)\sma_B K \longrightarrow (F_VS^0)\sma_B K.$$
\end{defn}
We can describe the adjoint structure maps of $G$-spectra over $B$ in
terms of these maps $\lambda^{V,W}_B$.
\begin{lem}\mylabel{eqn:lambdaadj}
Under the adjunctions
\[P_B(F_VS^0_B, X)\cong F_B(S^0_B, X(V))\cong X(V)\]
and
\[P_B(F_{V\oplus W}S^W_B, X)\cong F_B(S^0_B, \Omega^W_B X(V\oplus W))
\cong \Omega^W_BX(V\oplus W),\]
the map
$$P_B(\lambda^{V,W}_B, \text{id})\colon P_B(F_VS^0_B, X)\longrightarrow
P_B(F_{V\oplus W}S^W_B, X)$$
corresponds to
$$\tilde\sigma\colon X(V)\longrightarrow \Omega_B^W X(V\oplus W).$$
\end{lem}\begin{proof}
When $X=F_VS^0_B$, the conclusion holds by comparison with the case
of $G$-spectra. The general case follows from the Yoneda lemma of
enriched category theory. See, for example, \cite[6.3.5]{Borceux}.
\end{proof}
We could have started off by defining $\lambda^{V,W}_B$ in a conceptual manner
analogous to our definition of $\lambda^{V,W}$, but we want the explicit
description of $\lambda^{V,W}_B$ in terms of $\lambda^{V,W}$ in order to deduce homotopical properties in the parametrized context from homotopical properties in the nonparametrized context. For that and other purposes, we need the following observation. We return to our convention of deleting $G$ from
the notations, on the understanding that everything is equivariant.
\begin{lem}\mylabel{gentensor}
If $\phi\colon X\longrightarrow Y$ is an $s$-equivalence of level well-based
non\-par\-a\-me\-trized spectra and $K$ is a well-grounded ex-space with total space of the homotopy type of a $G$-CW complex, then
$\phi\sma_B \text{id}\colon X\sma_B K\longrightarrow Y\sma_B K$ is an $s$-equivalence.
\end{lem}
\begin{proof} We use the ex-fibrant approximation functor $P$ of
\myref{exfibapp}. We have a natural zig-zag of $h$-equivalences between
$K$ and $PK$.
By \myref{savior}, it induces a zig-zag of level $h$-equivalences between $X\sma_B K$ and $X\sma_B PK$ and, by \myref{HursmaK}, $X\sma_B PK$ is a level ex-fibration. Therefore, by \myref{zigzag}, it suffices to consider the case when $K$ is an ex-fibration. Since $(X\sma_B K)_b = X\sma K_b$ and $K_b$ is of the homotopy type of a $G_b$-CW complex, by \myref{ss}, each $(\phi\sma_B \text{id})_b$ is an $s$-equivalence by \cite[III.3.11]{MM}.
\end{proof}
The following result is crucial.
\begin{prop}\mylabel{key}
Let $K$ be a well-grounded ex-space with total space of the homotopy
type of a CW complex. Then
\[\lambda^{V,W}_B\colon F_{V\oplus W} \Sigma^W_B K\longrightarrow F_VK\]
and
\[\lambda^{V,W}\barwedge \text{id}\colon
F_{V\oplus W}S^W \barwedge F_ZK \longrightarrow F_VS^0\barwedge F_ZK\]
are $\pi_*$-isomorphisms of spectra over $B$.
\end{prop}
\begin{proof}
Since $\lambda^{V,W}_B = \lambda^{V,W}\sma_B \text{id}$, \myref{gentensor} and the corresponding nonparametrized statement \cite[III.4.5]{MM} imply the first statement. For the second statement, observe that for spectra $X$ we have the associativity relation
$$X\barwedge F_ZK \cong X\barwedge (F_Z S^0\sma_B K)\cong
(X\sma F_Z S^0)\sma_B K.$$
Taking $X = F_V T$ for a based space $T$ and using \myref{FVEV}, we see that
\[F_VT \barwedge F_ZK \cong F_{V\oplus Z}(T \sma_B K).\]
Using equivalences of this form and checking definitions, we conclude that the map $\lambda^{V,W}\barwedge \text{id}$ of the statement can be identified with the map
$$\lambda^{V\oplus Z,W}\sma_B\text{id}\colon
(F_{V\oplus Z\oplus W}S^W)\sma_B K \longrightarrow
(F_{V\oplus Z}S^0)\sma_B K.$$
Thus the second $\pi_*$-isomorphism is a special case of the first.
\end{proof}
From here, the proof of \myref{modelS} closely parallels arguments in \cite[III.\S4]{MM}, but simplified a little by \myref{Newcompgen}. The generating set of $s$-cofibrations is again $FI^f_B$. The generating set $FK^f_B$ of acyclic $s$-cofibrations is given by a variant of the definition in the nonparametrized case \cite[III.4.6]{MM}.
\begin{defn}\mylabel{Def6}
Recall the factorization of $\lambda^{V,W}$ through the mapping cylinder (in the category of spectra) as
\[\xymatrix{\lambda^{V,W}\colon F_{V\oplus W} S^W \ar[r]^-{k^{V,W}} & M\lambda^{V,W}\ar[r]^-{r^{V,W}} & F_VS^0.}\]
Here $k^{V,W}$ is an $s$-cofibration and $r^{V,W}$ is a deformation retraction. For $i\colon C\longrightarrow D$ in $I^f_B$, the map
\[i\Box k^{V,W}\colon C\sma_B M\lambda^{V,W} \cup_{C\sma_B F_{V\oplus W} S^W} D\sma_B F_{V\oplus W} S^W \longrightarrow D\sma_B M\lambda^{V,W}\]
is an $s$-cofibration in $G\scr{S}_B$ by \myref{Boxcof2}, and it is therefore also a $cyl$-cofibration by \myref{levelqf}. It is a $\pi_*$-isomorphism by \myref{key} and inspection of definitions. The $s$-cofibrations in $FJ^f_B$ are level acyclic and are therefore also $\pi_*$-isomorphisms. Restricting to $V$ and $W$ in $\text{sk}\scr{I}_G$, define the generating set $FK^f_B$ of acyclic $s$-cofibrations to be the union of $FJ^f_B$ and the set of all maps of the form $i\Box k^{V,W}$ with $i\in I^f_B$.
\end{defn}
A fortiori, the following result identifies the $s$-fibrations, but it must be proven a priori as a first step towards the verification of the model axioms.
\begin{prop}\mylabel{RLPL}
A map $f\colon X\longrightarrow Y$ satisfies the RLP with respect to $FK^f_B$ if and only if $f$ is a level $qf$-fibration and the diagrams
\begin{equation}\label{OMpb}
\xymatrix{
X(V) \ar[r]^-{\tilde{\sigma}} \ar[d]_{f(V)} & \Omega^W_B X(V\oplus W)
\ar[d]^{\Omega^W_B f(V\oplus W)} \\
Y(V) \ar[r]_-{\tilde{\sigma}} & \Omega^W_B Y(V\oplus W) \\}
\end{equation}
are homotopy pullbacks for all $V$ and $W$.
\end{prop}
\begin{proof}
As in \cite[III.4.7]{MM}, the homotopy pullback property must be interpreted as requiring a $q$-equi\-va\-lence from $X(V)$ into the pullback in the displayed diagram. Recall that $FJ^f_B$ is contained in $FK^f_B$ and that a map has the RLP with respect to $FJ^f_B$ if and only if it is a level $qf$-fibration. This gives part of both implications. It remains to show that a level $qf$-fibration $f$ has the RLP with respect to $i\Box k^{V,W}$ for all $i\in I^f_B$ if and only if the displayed diagram is a homotopy pullback. This is a formal but not altogether trivial exericise from the fact that the level $qf$-model structure is $G$-topological in the sense characterized in \myref{Gtopchar}. Notice that the map
$i\Box k^{V,W}$ is isomorphic to the map $i\Box k^{V,W}_B$, where $k^{V,W}_B = k^{V,W}\sma_B S^0_B$. With notation as in (\ref{PBoxmap}), $f$ has the RLP with respect to $i\Box k^{V,W}_B$ for all $i\in I^f_B$ if and only if the pair
$(i, P_B^\Box(k^{V,W}_B,f))$ has the lifting property for all $i\in I^f_B$, which holds
if and only if the map $P_B^\Box(k^{V,W}_B,f)$ of ex-spaces over $B$ is an acyclic $qf$-fibration. This map is a $qf$-fibration since, for $j\in J^f_B$, the map $j\Box k^{V,W} \iso j\Box k^{V,W}_B$ is a level acyclic $s$-cofibration of spectra over $B$ by \myref{Boxcof2}. Since $f$ is a level $qf$-fibration, $(j\Box k^{V,W}_B, f)$ has the lifting property, hence, by adjunction, so does $(j, P_B^\Box(k^{V,W}_{B},f))$. Finally, $P_B^\Box(k^{V,W}_{B},f)$ is homotopy equivalent to $P_B^\Box(\lambda^{V,W}_B,f)$
so one is a $q$-equi\-va\-lence if and only if the other is. Under the isomorphisms in \myref{eqn:lambdaadj}, the map $P_B^\Box(\lambda^{V,W}_{B},f)$ coincides with the map from $X(V)$ into the pullback in the displayed diagram and is thus a $q$-equi\-va\-lence if and only if that diagram is a homotopy pullback.
\end{proof}
Let $*_B$ be the terminal spectrum over $B$, so that each $*_B(V)$ is the terminal ex-space $*_B$. Observe that $*_B$ is an $\Omega$-spectrum with trivial homotopy groups.
\begin{cor}
The terminal map $F\longrightarrow *_B$ satisfies the RLP with respect to $FK_B$ if and only if $F$ is an $\Omega$-spectrum over $B$.
\end{cor}
\begin{cor}\mylabel{stabislevel}
If $f\colon X\longrightarrow Y$ is a $\pi_*$-isomorphism that satisfies the RLP with respect to $FK_B$, then $f$ is a level acyclic $qf$-fibration.
\end{cor}
\begin{proof}
Since $f$ is a level $qf$-fibration by \myref{RLPL}, the dual of the gluing lemma applied to the diagram
\[\xymatrix{
{*}_B\ar[r]\ar[d] & Y\ar@{=}[d] & X\ar[l]_f\ar@{=}[d] \\
F_B(I, Y) \ar[r] & Y & X\ar[l]^f}\]
gives that the induced map $F\longrightarrow F_Bf$ of pullbacks is a level $q$-equivalence. Since $f$ has the RLP with respect to $FK_B$, so does its pullback $F\longrightarrow *_B$. By the previous corollary, $F$ is thus an $\Omega$-spectrum over $B$. In particular, it is level $qf$-fibrant. We conclude that $F$ is a level $qf$-fibrant approximation for $F_Bf$. Since $f$ is a $\pi_*$-isomorphism, \myref{exact} gives that $F$ is acyclic. By \myref{bombsaway}, this implies that $F\longrightarrow *_B$ is a level $q$-equi\-va\-lence. Thus the fibers $F(V)_b$ all have trivial homotopy groups. We conclude (with a bit of extra argument as in \cite[9.8]{MMSS} to handle $\pi_0$) that each map of fibers $f(V)_b$ induces an isomorphism on homotopy groups. Therefore, since each $f(V)$ is a $qf$-fibration, each $f(V)$ induces an isomorphism on homotopy groups.
\end{proof}
The proof of the model axioms for the stable model structure is now immediate.
\begin{proof}[Proof of \myref{modelS}]
The $\pi_*$-isomorphisms give a well-grounded subcategory of weak
equivalences, by \myref{piwellgr}. Conditions (i), (iii), and (iv) in \myref{Newcompgen} are clear from our specification of the generating
acyclic $s$-cofibrations and the result for the level $qf$-model
structure. For condition (ii), a $\pi_*$-isomorphism that satisfies
the RLP with respect to $FK_B$ has the RLP with respect to $FI_B$ by \myref{stabislevel}. Conversely, a map that has the RLP with respect to $FI_B$ is a level acyclic $qf$-fibration and therefore has the RLP with respect to $FK_B$ by \myref{RLPL}. It is a $\pi_*$-isomorphism since it is level acyclic.
Since all $s$-fibrations are level $qf$-fibrations, right properness follows from the slightly stronger observation in the following result.
\end{proof}
\begin{prop}\mylabel{s-rightproper}
The $\pi_*$-isomorphisms in $G\scr{S}_B$ are preserved under pullbacks along level $qf$-fibrations.
\end{prop}
\begin{proof}
Let $g$ be the pullback of a level $qf$-fibration $f$ along a $\pi_*$-isomorphism. Then $g$ is a level $qf$-fibration and the fibers of $g(V)$
are isomorphic to the fibers of $f(V)$. Therefore the homotopy fibers
$F_Bg$ are level $q$-equivalent to the homotopy fibers $F_Bf$.
The result follows by comparison of the first long exact sequence in \myref{exact} for $f$ and $g$.
\end{proof}
\section{Some Quillen adjoint pairs relating stable model structures}
We prove here that all of the adjoint pairs that were shown to be Quillen adjoints with respect to the level model structure in \S12.2 are still Quillen adjoints with respect to the stable model structure. In view of the role played by level $qf$-fibrant approximation in our definition of homotopy groups, it is helpful to first understand the relationship between $s$-fibrant approximation and level $qf$-fibrant approximation. Now that the model structures have been established, we henceforward use the term $s$-equivalence rather than the synonymous term $\pi_*$-isomorphism.
\begin{lem}\mylabel{compaR}
Let $\nu\colon X\longrightarrow RX$ and $\nu_{\ell}\colon X\longrightarrow R_{\ell}X$ be an $s$-fibrant approximation of $X$ and a level $qf$-fibrant approximation of $X$. Then there is an $s$-equivalence $\xi\colon R_{\ell} X\longrightarrow RX$ under $X$.
\end{lem}
\begin{proof}
Since $\nu_{\ell}$ is a level acyclic $s$-cofibration, it is an acyclic $s$-cofibration by \myref{levelpi}. Since $RX$ is $s$-fibrant, the RLP gives a map $\xi$ under $X$, and it is an $s$-equivalence since $\nu$ and $\nu_{\ell}$ are $s$-equivalences.
\end{proof}
We have the following relationship between the homotopy categories of ex-spaces over $B$ and of spectra over $B$.
\begin{prop}\mylabel{stablepair}
The pair $(\Sigma^{\infty}_B,\Omega^{\infty}_B)$ is a Quillen adjunction relating $G\scr{S}_B$ and $G\scr{K}_B$. More generally, $(\Sigma^{\infty}_V,\Omega^{\infty}_V) = (F_V,Ev_V)$ is a Quillen adjunction for any representation $V\in \scr{V}$.
\end{prop}
\begin{proof}
The maps $\Sigma^{\infty}_V i$, where $i\in I^f_B$ is a generating cofibration for the $qf$-model structure on $G\scr{K}_B$, are among the generating cofibrations of the $s$-model structure on $G\scr{S}_B$, and it follows that $\Sigma^{\infty}_V$ preserves cofibrations. Since $\Sigma^{\infty}_V$ takes acyclic $qf$-cofibrations to level acyclic $qf$-cofibrations, and these are acyclic by \myref{levelpi}, $\Sigma^{\infty}_V$ also preserves acyclic cofibrations.
\end{proof}
Now consider an adjoint pair $(F,V)$ between categories of parametrized spectra that is a Quillen adjunction with respect to the level model structures. Since the cofibrations are the same in the level model structure and in the stable model structure, the left adjoint $F$ certainly preserves cofibrations. Thus,
to show that $(F,V)$ is also a Quillen adjunction with respect to the stable model structures, we need only show that $F$ carries acyclic $s$-cofibrations
to $s$-equivalences. When $F$ preserves all $s$-equivalences, this is obvious;
otherwise, by \myref{reducts}, it suffices to verify this for the generating acyclic $s$-cofibrations. The cited result applies in general to subcategories of well-grounded weak equivalences, and in our context it applies to both the
level $q$-equivalences and the $s$-equivalences. Recall that a Quillen left adjoint in any model structure preserves weak equivalences between cofibrant objects, by Ken Brown's lemma \cite[1.1.12]{Hovey}. The following parenthetical observation applies to give a stronger conclusion for the Quillen left adjoints that we shall encounter. It will play a crucial role in exploiting
the equivalence of homotopy categories that we will establish in
the next chapter. Note that the $s$-cofibrant spectra are the cofibrant
objects in both the level and the stable model structures, and they are
well-grounded.
\begin{prop}\mylabel{neat}
Let $F$ be a Quillen left adjoint between categories of parametrized
spectra with their stable model structures and suppose that $F$ preserves
level $q$-equivalences between well-grounded spectra. Then $F$ preserves
$s$-equivalences between well-grounded spectra.
\end{prop}
\begin{proof}
If $g\colon X\longrightarrow Y$ is an $s$-equivalence, where $X$ and $Y$ are well-grounded, factor $g$ in the level model structure as
\[\xymatrix{X\ar[r]^{g'} & W \ar[r]^{g''} & Y,}\]
where $g'$ is an $s$-cofibration and $g''$ is a level acyclic $qf$-fibration. Then $W$ is well-grounded and $Fg''$ is a level $q$-equivalence by assumption. Since $F$ is a Quillen left adjoint in the $s$-model structures, $Fg'$ is an $s$-equivalence. Since level $q$-equivalences are $s$-equivalences it follows that $Fg=Fg''\circ Fg'$ is an $s$-equivalence.
\end{proof}
The following sequence of results consists of analogues for the stable
model structures of results proven for the level model structures in \S12.2.
Recall that we actually have well-grounded stable model structures $s(\scr{C})$
for any closed generating set $\scr{C}$.
As in \S12.2, wherever a $qf(\scr{C})$-model structure was used in Chapter 7
for some particularly well chosen $\scr{C}$, we must use the corresponding
$s(\scr{C})$-model structure here.
\begin{prop}\mylabel{spacesmashpair}
Let $T$ be a based $G$-CW complex. Then $(-\sma_B T, F_B(T,-))$ is a Quillen adjunction on $G\scr{S}_B$. When $T = S^V$, it is a Quillen equivalence.
\end{prop}
\begin{proof}
This is immediate from the fact that the stable model structure is
$G$-topological, together with Propositions \ref{pismash} and \ref{SIOMV}.
\end{proof}
\begin{prop}\mylabel{Boxcof2too}
If $i\colon X\longrightarrow Y$ and $j\colon W\longrightarrow Z$ are $s$-cofibrations
of spectra over base spaces $A$ and $B$, then
\[i\Box j\colon (Y\barwedge W)\cup_{X\barwedge W}(X\barwedge Z)\longrightarrow Y\barwedge Z\]
is an $s$-cofibration over $A\times B$ which is $s$-acyclic if either $i$ or $j$ is $s$-acyclic.
\end{prop}
\begin{proof}
The statement about $s$-cofibrations is part of the analogue, \myref{Boxcof2},
for the level model structure.
As usual, it suffices to show that $i\Box j$ is an $s$-equivalence if
$i\in FI^f_B$ and $j\in FK^f_B$, where $FK^f_B$ is the set of generating
acyclic $s$-cofibrations specified in \myref{Def6}. Arguing
as in \myref{boxacy} and using properness, this will hold if smashing the
source and the target of $i$ with $j$ give $s$-equivalences. The reduction so far would work just as well for internal smash products. The required last step
reduces via inspection of \myref{Def6} to an application of \myref{key}, with base space taken to be $A\times B$. The reason that this last step works for external smash products but fails for internal smash products is made clear in \myref{ouchtoo}.
\end{proof}
\begin{cor}\mylabel{exttoo}
If $Y$ is an $s$-cofibrant spectrum over $B$, then the functor $(-)\barwedge Y$ from $G\scr{S}_A$ to $G\scr{S}_{A\times B}$ is a Quillen left adjoint with Quillen right adjoint $\bar{F}(Y,-)$.
\end{cor}
\begin{prop}\mylabel{Qad1too}
Let $f\colon A\longrightarrow B$ be a $G$-map. Then $(f_{!},f^*)$ is a Quillen adjoint pair. If $f$ is a $q$-equivalence, then $(f_{!},f^*)$ is a Quillen equivalence.
\end{prop}
\begin{proof}
We must show that $f_!$ takes acyclic $s$-co\-fi\-bra\-tions to $s$-equi\-va\-lences. Since $f_!$ preserves well-grounded objects and level $q$-equivalences between well-grounded objects by \myref{Qad1},
it suffices by \myref{reducts} to prove that $f_!k$ is an $s$-equivalence for each map $k$ in $FK^f_A$. This follows from the corresponding Quillen adjunction with respect to the level model structure if $k\in FJ^f_A$, so assume that $k$ is of the form $i\Box k^{V,W}\iso i\Box k^{V,W}_A$. We claim that $f_!k$ is a map in $FK^f_B$ and is therefore an $s$-equivalence. Observe that $k^{V,W}_A\iso f^*k^{V,W}_B$. Using (\ref{four}) and the fact that $f_!$ preserves pushouts, we see from the definition of the $\Box$-product that $f_!(i\Box f^*k^{V,W}_B)\iso (f_!i)\Box k^{V,W}_B$. Since $i$ is obtained from a map over $A$ by adjoining a disjoint section, $f_!i$ is obtained from a map over $B$ by adjoining a disjoint section and is thus in $I^f_B$.
Now assume that $f$ is a $q$-equivalence. By \cite[1.3.16]{Hovey}, $(f_!,f^*)$ is a Quillen equivalence if and only if $f^*$ reflects $s$-equivalences between $s$-fibrant objects and the composite $X\longrightarrow f^*f_!X\longrightarrow f^*Rf_!X$ given by the unit of the adjunction and $s$-fibrant approximation is an $s$-equivalence for all $s$-cofibrant $X$. Since the $s$-fibrant objects are the $\Omega$-spectra over $B$ and the $s$-equivalences between $\Omega$-spectra over $B$ are the level $q$-equivalences, the reflection property follows directly from the corresponding Quillen equivalence with respect to the level model structure. That result also gives that the composite $X\longrightarrow f^*f_!X\longrightarrow f^*R_{\ell}f_!X$ is a level $q$-equivalence and hence an $s$-equivalence. Applying \myref{compaR} with $X$ replaced by $f_!X$ and observing that $f^*$ preserves $s$-equivalences between level $qf$-fibrant $G$-spectra over $B$ since $(f^*Y)_a\iso Y_{f(a)}$, a little diagram chase shows that the composite $X\longrightarrow f^*f_!X\longrightarrow f^*Rf_!X$ is an $s$-equivalence.
\end{proof}
Observe that \myref{neat} applies to $f_!$.
\begin{prop}\mylabel{Qad2too}
Let $f\colon A\longrightarrow B$ be a $G$-bundle whose fibers $A_b$ are $G_b$-CW complexes. Then $(f^*,f_*)$ is a Quillen adjoint pair.
\end{prop}
\begin{proof}
We must show that $f^*$ preserves acyclic $s$-cofibrations. Again it suffices by \myref{reducts} to prove that $f^*k$ is an $s$-equivalence between well-grounded spectra for each map $k\in FK^f_B$. That $f^*k$ is a map between well-grounded spectra follows from the fact that if $K\amalg B$ is a space over $B$ with a disjoint section, then $f^* F_V (K\amalg B) = F_V f^*K \amalg A$ is well-grounded. To see that $f^*k$ is an $s$-equivalence, it is enough, as in the proof of \myref{Qad1too}, to consider $k = i\Box k^{V,W}_B$ with $i\in I^f_B$. We have that $f^*k^{V,W}_B = k^{V,W}_A$ and, since $f^*$ preserves pushouts, smash products, and factorizations through mapping cylinders, we see as in the cited proof that $f^*k\iso f^*i\Box k^{V,W}_A$, which is an acyclic $s$-cofibration.
\end{proof}
\begin{prop}\mylabel{Lchanges2}
Let $\iota\colon H\longrightarrow G$ be the inclusion of a subgroup. The pair of functors $(\iota_!,\nu^*\iota^*)$ relating $H\scr{S}_A$ and $G\scr{S}_{\iota_!A}$ gives a Quillen equivalence. If $A$ is completely regular, then $\iota_!$ is also a Quillen right adjoint.
\end{prop}
\begin{proof}
By \myref{grprestrrQa} below, $(\iota_!,\nu^*\iota^*)$ is a Quillen adjoint pair. The proof that it is a Quillen equivalence is the same as the proof of
the ex-space level analogue in \myref{Lishriek}. The last statement is less
obvious. As in the proof of the corresponding statement in
\myref{Lishriek}, it follows from the spectrum level analogue of
\myref{iotaalt}, which in turn requires the spectrum level analogue
of \myref{ouch0}, and the analogue in the stable model structure of \myref{fixedptrQa0}. The required analogues are proven in \S14.4 below.
\end{proof}
We shall see that $(\iota_!,\nu^*\iota^*)$ descends to a closed symmetric
monoidal equivalence of homotopy categories in \myref{Symmoni} below.
\begin{cor}\mylabel{LishriekCor2}
The functor $\iota^*\colon \text{Ho}G\scr{S}_B\longrightarrow \text{Ho}H\scr{S}_{\iota^*B}$ is the composite
\[\xymatrix@1
{\text{Ho}G\scr{S}_B \ar[r]^-{\mu^*} & \text{Ho} G\scr{K}_{\iota_!\iota^*B} \htp \text{Ho} H\scr{K}_{\iota^*B}\\}\]
\end{cor}
Using \myref{Johann} as in \myref{FibadQ}, the
following result is now a special case of
Propositions \ref{Lchanges2} and \ref{Qad1too}.
\begin{prop}\mylabel{FibadQtoo}
For $b\in B$, the pair of functors $((-)^b,(-)_b)$ relating $G_b\scr{S}$ and $G\scr{S}_B$ is a Quillen adjoint pair.
\end{prop}
\chapter{Adjunctions and compatibility relations}
\section*{Introduction}
The utility of the stable homotopy category $\text{Ho}\, G\scr{S}_B$ depends on the fact that the usual functors and adjunctions descend to it and still satisfy appropriate commutation relations. We consider such matters in this chapter.
Many of our basic adjunctions are Quillen adjunctions in the stable model structure. We recorded those in \S12.6. The crucial adjunction missing from \S12.6 is $(f^*,f_*)$ for a general map $f$ of base spaces. This cannot be a Quillen adjoint pair by the argument in \myref{noway}. We used Brown representability to construct the right adjoint $f_*$ between homotopy categories of ex-spaces in \myref{descendf0}. Analogously, in \S13.1 we use Brown representability to construct $f_*$ between homotopy categories of
parametrized spectra, and we use base change along diagonal maps to internalize smash products and function spectra. There is an interesting twist here. It is not easy to verify the Mayer-Vietoris axiom directly. Rather, we use the triangulated category variant of the Brown representability theorem, whose hypotheses turn out to be easier to check.
In \S13.7, we complete the proof that our stable homotopy categories are symmetric monoidal and prove some basic compatibility relations among smash products and base change functors. These results involve commutation of Quillen left and right adjoints, and we would not know how to prove them using only model theoretic fibrant and cofibrant replacement functors.
Rather, their proofs depend on an equivalence between our model theoretic
stable homotopy category of parametrized $G$-prespectra and a classical
homotopy category of what we call ``excellent'' parametrized $G$-prespectra.
We used an analogous, but more elementary, equivalence of categories in Chapter 9. It is essential to use parametrized $G$-prespectra rather than parametrized $G$-spectra to make the comparison since the relevant constructions do not all preserve functoriality on linear isometries; that is, they do not preserve $\scr{I}_G$-spaces. Results proven using the comparison are then translated to parametrized $G$-spectra along the Quillen equivalence between
parametrized $G$-prespectra and parametrized $G$-spectra.
These equivalences of categories allow us to use a prespectrum level analogue $T$ of the ex-fibrant approximation functor $P$ to study derived functors.
We define excellent parametrized $G$-prespectra in \S13.2. We lift the ex-fibrant approximation functor $P$ from ex-$G$-spaces to parametrized $G$-spectra
in \S13.3. There are several further twists here. First, the functor $P$ on
ex-$G$-spaces does not behave well with respect to tensors, so extending it to
a functor on parametrized $G$-prespectra is subtle. Second, with the extension, the zig-zag of $h$-equivalences connecting $P$ to the identity functor is no longer given by honest maps of parametrized $G$-prespectra, only weak maps. Third, the functor $P$ does not take parametrized $G$-prespectra to excellent ones. To remedy this, we introduce two auxiliary functors $K$ and $E$ in \S13.4. The composite $T=KEP$ does land in excellent parametrized $G$-prespectra, and $K$ converts weak maps to honest maps. In \S\S13.5 and 13.6 we use $T$ to
prove the promised equivalence of homotopy categories and show how to study derived funtors in this context.
There are few issues of equivariance in this chapter, and we generally continue to omit the (compact Lie) group $G$ from the notations. We adopt the convention of calling isomorphisms in homotopy categories \emph{equivalences} and we denote them by $\simeq$ rather than $\cong$.
\section{Brown representability and the functors $f_*$ and $F_B$}
We need some preliminaries about the two versions of Brown representability
that are applicable in stable situations. Recall \myref{Johann}.
\begin{defn}\mylabel{detecting}
For $n\in \mathbb{Z}$ and $H\subset G$, we have an $s$-cofibrant sphere $G$-spectrum $S^n_H$ such that $\pi_n^H(X) = [S^n_H,X]_G$ for all $G$-spectra $X$. Explicitly,
$$S^n_H = \begin{cases}
\Sigma^{\infty}(G/H_+ \sma S^n) & \text{if $n\geq 0$,}\\
F_{-n}(G/H_+\sma S^0) & \text{if $n<0$},
\end{cases}$$
as in \cite[II.4.7]{MM}, where $F_{-n}$ is the shift desuspension by $\mathbb{R}^n$.
We may allow the ambient group to vary. Replacing $G$ by $G_b$ for $b\in B$ and letting $H\subset G_b$, define $S^{n,b}_H$ to be the $G$-spectrum $(S^n_H)^b$ over $B$. Note that $S^{n,b}_H$ is $s$-cofibrant, by \myref{FibadQ}. By adjunction, for $G$-spectra $X$ over $B$, $\pi_n^H(X_b)$ is isomorphic to $[S^{n,b}_H,X]_{G,B}$. Let $\scr{D}_B$ be the set of all such $G$-spectra $S^{n,b}_H$ over $B$.
\end{defn}
From here, the following three results work in exactly the same way as their ex-space analogues in \S7.4. Observe that the category $\text{Ho}\, G\scr{K}_B$ has coproducts and homotopy pushouts, hence homotopy colimits of directed sequences.
\begin{lem}\mylabel{compact}
Each $X\in \scr{D}_B$ is compact, in the sense that
$$\text{colim}\, [X, Y_n]_{G,B}\iso [X, \text{hocolim}\, Y_n]_{G,B}$$
for any sequence of maps $Y_n\longrightarrow Y_{n+1}$ in $G\scr{S}_B$.
\end{lem}
\begin{prop}\mylabel{detect}
A map $\xi\colon Y\longrightarrow Z$ in $G\scr{S}_B$ is an $s$-equivalence if and only if the induced map $\xi_*\colon [X,Y]_{G,B}\longrightarrow [X,Z]_{G,B}$ is a bijection for all $X\in \scr{D}_B$.
\end{prop}
\begin{proof}
This is a tautology since as $X$ ranges through the $S^{n,b}_H$,
$[X,Y]_{G,B}$ ranges through the homotopy groups $\pi_n^H(Y_b)$
that define the $s$-equivalences.
\end{proof}
\begin{thm}[Brown]\mylabel{brown}
A contravariant set-valued functor on the cat\-egory $\text{Ho}\, G\scr{S}_B$ is representable if and only if it satisfies the wedge and Mayer-Vietoris axioms.
\end{thm}
Since we have the Quillen adjoint pair $(f_!,f^*)$, we have the right derived
functor $f^*\colon \text{Ho}\, G\scr{S}_B\longrightarrow \text{Ho}\, G\scr{S}_A$. As in the proof of the analogous result on the level of ex-spaces, \myref{descendf0}, we can obtain the desired right adjoint $f_*$ to $f^*$ by use of Brown's theorem provided that we can show that $f^*$ preserves the relevant homotopy colimits. However, since $f^*\colon G\scr{S}_B\longrightarrow G\scr{S}_A$ does not preserve $s$-cofibrant objects, this is not obvious. We will later give results that would allow us to carry out the proof in a manner analogous to the proof of \myref{descendf0}, but it is instructive to switch gears and give a more direct proof. It is based on the use of triangulated categories and would not have applied on the ex-space level.
\begin{lem}\mylabel{yestrian}
The category $\text{Ho}\, G\scr{S}_B$ is triangulated.
\end{lem}
\begin{proof}
The treatment of triangulated categories in \cite{Tri} gives a general pattern of proof for showing that homotopy categories associated to appropriate model categories are triangulated. It applies here. The distinguished triangles are those equivalent in $\text{Ho}\, G\scr{S}_B$ to cofiber sequences that start with a well-grounded spectrum or, equivalently by \myref{SIOMV}, those equivalent to the negatives of fiber sequences. Note that, by the proof of \myref{exact}, every cofiber sequence is equivalent in
$\text{Ho}\, G\scr{S}_B$ to a cofiber sequence of level ex-quasifibrations.
\end{proof}
In triangulated categories, there is an alternative version of Brown's representability theorem due to Neeman \cite{Nee2}. It requires a ``detecting set of compact objects''. In triangulated categories with coproducts (or sums), an object $X$ is said to be compact if
$\bigoplus [X, Y_i] \iso [X,\bigoplus Y_i]$ for any set of objects $Y_i$. In our topological situations, this reduces to the compactness of spheres, exactly as the proof of \myref{compact}. A {\em detecting} set of objects is one that detects equivalences, in the sense suggested by \myref{detect}. We have the following result.
\begin{lem}
$\scr{D}_B$ is a detecting set of compact objects in $\text{Ho}\, G\scr{S}_B$.
\end{lem}
Recall that an additive functor between triangulated categories is said to be {\em exact} if it commutes with $\Sigma$ up to a natural equivalence and preserves distinguished triangles. The following theorems are proven in
\cite[3.1, 4.1]{Nee2}; they are discussed with an eye to applications
such as ours in \cite[\S8]{FHM}.
\begin{thm}\mylabel{BR}
Let $\scr{A}$ be a compactly detected triangulated category. A functor $H\colon \scr{A}^{op}\longrightarrow \scr{A} b$ that takes distinguished triangles to long exact sequences and converts coproducts to products is representable.
\end{thm}
\begin{thm}\mylabel{TAFT}
Let $\scr{A}$ be a compactly detected triangulated category and $\scr{B}$ be any triangulated category. An exact functor $F\colon \scr{A} \longrightarrow \scr{B}$ that
preserves coproducts has a right adjoint $G$.
\end{thm}
\begin{thm}\mylabel{descendf}
For any $G$-map $f\colon A\longrightarrow B$, there is a right adjoint
$f_*$ to the functor $f^*\colon \text{Ho}\, G\scr{S}_B\longrightarrow \text{Ho}\, G\scr{S}_A$,
so that
$$[f^*Y,X]_{G,A} \iso [Y,f_*X]_{G,B}$$
for $X$ in $G\scr{S}_A$ and $Y$ in $G\scr{S}_B$.
\end{thm}
\begin{proof} The left adjoint $f_!$ commutes with $\Sigma$ and preserves
cofiber sequences, and this remains true after passage to derived homotopy
categories. Therefore the derived functor $f_!$ is exact. Since $f^*$ is Quillen right adjoint to $f_!$, the derived functor $f^*$ is right adjoint to $f_!$ and is therefore also exact; see, for example, \cite[3.9]{Nee1}.
If $X$ is in $\scr{D}_A$, then $f_!X$ is compact in $\text{Ho}\, G\scr{S}_B$, as we
see from commutation relations between relevant Quillen left adjoints
given in \myref{FVvsf*}. It follows formally that $f^*$ preserves
coproducts, by \cite[5.1]{Nee2} or \cite[7.4]{FHM}.
\end{proof}
\begin{rem}
For composable maps $f$ and $g$, there is a natural equivalence $g_*\com f_* \simeq (g\com f)_*$ on homotopy categories since $f^*\com g^*\simeq (g\com f)^*$.
\end{rem}
Exactly as for ex-spaces in \myref{smashing0}, we apply change of base along the diagonal map $\Delta\colon B\longrightarrow B\times B$ to obtain internal smash product and function spectra functors in $\text{Ho}\, G\scr{S}_B$.
\begin{thm}\mylabel{smashing}
Define $\sma_B$ and $F_B$ on $\text{Ho}\, G\scr{S}_B$ to be the composite (derived) functors
$$X\sma_B Y = \Delta^*(X\barwedge Y) \quad\text{and}\quad F_B(X,Y) = \bar{F}(X,\Delta_*Y).$$
Then
$$ [X\sma_B Y, Z]_{G,B}\iso [X,F_B(Y,Z)]_{G,B}$$
for $X$, $Y$ and $Z$ in $\text{Ho}\, G\scr{S}_B$.
\end{thm}
\begin{proof}
The displayed adjunction is the composite of the adjunction for the external smash product and function spectra functors given by \myref{exttoo} and the adjunction $(\Delta^*,\Delta_*)$.
\end{proof}
\section{The category $G\scr{E}_B$ of excellent prespectra over $B$}
We must still prove that $\text{Ho}\, G\scr{S}_B$ is a closed symmetric monoidal category under the derived internal smash product, that the derived functor $f^*$ is closed symmetric monoidal, and that various compatibility relations that hold on the point-set level descend to homotopy categories. In particular, since our right adjoints $f_*$, $\Delta_*$, and therefore $F_B$ come from Brown's representability theorem, it is not at all obvious how to prove that they are well-behaved homotopically. In Chapter 9, we solved the corresponding ex-space level problems by proving that $\text{Ho}\, G\scr{K}_B$ is equivalent to the more classical and elementary homotopy category $hG\scr{W}_B$. Here $G\scr{W}_B$ is the category of ex-fibrations over $B$ whose total spaces are compactly generated and of the homotopy types of $G$-CW complexes, and $hG\scr{W}_B$ is obtained from $G\scr{W}_B$ simply by passage to homotopy classes of maps. This equivalence allowed us to exploit the ex-fibrant approximation functor $P$ of \S8.3 to resolve the
cited problems.
We shall resolve our spectrum level problems similarly, and the
following definitions give the appropriate analogues of $G\scr{W}_B$ and
$hG\scr{W}_B$. However, to keep closer to the ex-space level, it is essential
to work with parametrized prespectra rather than parametrized spectra. It is safe to do so in view of the Quillen equivalence $(\mathbb{P},\mathbb{U})$ of \myref{modelPU} relating $G\scr{P}_B$ and $G\scr{S}_B$.
\begin{defn}\mylabel{Sigma}
Let $X$ be a $G$-prespectrum over $B$.
\begin{enumerate}[(i)]
\item $X$ is \emph{well-structured} if each level $X(V)$ is in $G\scr{W}_B$.
\item $X$ is \emph{$\Sigma$-cofibrant} if it is well-grounded and each structure map \[\sigma\colon \Sigma^W_BX(V)\longrightarrow X(V\oplus W)\] is an $fp$-cofibration.
\end{enumerate}
\end{defn}
We can now give the definition of excellent $G$-prespectra over $B$ and of the associated classical homotopy category. Working with classical nonequivariant
and nonparametrized coordinatized prespectra $\{E_n\}$, it has been known
since the 1960's that the following definition gives the simplest
quick and dirty rigorous construction of the stable homotopy category.
\begin{defn}\mylabel{excel}
The category $G\scr{E}_B$ of \emph{excellent} $G$-prespectra over $B$ is the full subcategory of $G\scr{P}_B$ whose objects are the well-structured $\Sigma$-cofibrant $\Omega$-$G$-prespectra over $B$. Let $hG\scr{E}_B$ denote the classical homotopy category obtained from $G\scr{E}_B$ by passage to homotopy classes of maps.
\end{defn}
We comment on the conditions we require of excellent prespectra over $B$. We require that they be well-structured so that we can exploit levelwise our equivalence of homotopy categories on the ex-space level. We require that
they be $\Sigma$-cofibrant since that provides ``homotopical glue'' that is necessary for the transition from the known equivalence on the ex-space level
to the desired equivalence on the prespectrum level. We shall make this idea
precise shortly, in \myref{CPhelp}. We require that they be $\Omega$-prespectra over $B$ since it is clearly sensible to restrict attention to $s$-fibrant objects in $G\scr{S}_B$ if we hope to compare homotopy categories.
Recall that $X$ is an $\Omega$-prespectrum if it is a level $qf$-fibrant prespectrum over $B$ whose adjoint structure maps
$$\tilde\sigma\colon X(V) \longrightarrow \Omega_B^{W-V}X(W)$$
are $q$-equivalences. Since excellent prespectra over $B$ are required to be
level ex-fibrations, they are automatically level $qf$-fibrant. The condition
on the adjoint structure maps is stronger than it appears on the surface.
\begin{lem}\mylabel{fpOM}
For excellent $G$-prespectra $X$ over $B$, the adjoint structure maps
$$\tilde{\sigma}\colon X(V)\longrightarrow \Omega^W_BX(V\oplus W)$$
are $fp$-equivalences.
\end{lem}\begin{proof} The $\tilde{\sigma}$ are $q$-equivalences between
$G$-CW homotopy types and are therefore $h$-equivalences.
Since they are maps between ex-fibrations, they are
$fp$-equivalences by \myref{reverse}.
\end{proof}
This implies, for example, that homotopy-preserving functors $G\scr{E}_B\longrightarrow G\scr{P}_B$ that may not preserve level $q$-equivalences nevertheless do
preserve the equivalence property required of the adjoint structure maps.
\begin{rem} Our definition of excellent parametrized prespectra is close to
that used by Clapp and Puppe \cite{Clapp, CP}, who in turn were influenced by definitions in \cite{MQR}. Curiously, while Clapp \cite{Clapp} focuses on ex-fibrations, Clapp and Puppe \cite{CP} never mention fibration conditions. These papers are nonequivariant, but the second is written in terms of what the authors call ``coordinate-free spectra'' over $B$. These are the same as our nonequivariant prespectra over $B$, except that their adjoint structure maps $\tilde{\sigma}$ are required to be closed inclusions, which holds automatically for $\Sigma$-cofibrant prespectra. Clapp and Puppe \cite{CP} use the term ``cofibrant'' for our notion of $\Sigma$-cofibrant.
\end{rem}
A crucial result of Clapp and Puppe makes the idea
of homotopical glue precise. It is stated nonequivariantly in
\cite[6.1]{CP}, but it works just as well equivariantly.
Translated to our language, it reads as follows.
\begin{prop}[Clapp-Puppe]\mylabel{CPhelp}
If $f\colon X\longrightarrow Y$ is a level $fp$-equivalence between $\Sigma$-cofibrant prespectra over $B$, then $f$ is a homotopy equivalence of prespectra
over $B$. Therefore, if $f\colon X\longrightarrow Y$ is a level $h$-equivalence
between well-structured $\Sigma$-cofibrant prespectra over $B$, then $f$ is a homotopy equivalence of prespectra over $B$.
\end{prop}
\begin{proof}[Sketch proof]
The proof is analogous to the proof that a ladder of homotopy equivalences
connecting sequences of cofibrations induces a homotopy equivalence on passage
to colimits. The point is that, for $\Sigma$-cofibrant parametrized prespectra $Y$, we can carry out inductive arguments just as if $Y$ were just such a colimit.
Using standard cofibration arguments, carried over to the parametrized case, we can extend an $fp$-homotopy inverse of $\Sigma_B^{W_i}X(V_i)\longrightarrow \Sigma_B^{W_i}Y(V_i)$ to an $fp$-homotopy inverse of $X(V_{i+1})\longrightarrow Y(V_{i+1})$ and proceed inductively. The last statement follows by \myref{fpequiv}(i),
which shows that a level $h$-equivalence between well-structured prespectra
over $B$ is a level $fp$-equivalence.
\end{proof}
\section{The level ex-fibrant approximation functor $P$ on prespectra}
We seek an approximation functor to play the role on the parametrized prespectrum level that the functor $P$ played on the ex-space level functor.
We shall introduce three approximation functors, $P$, $E$ and $K$, that successively build in the properties of being well-structured, being an $\Omega$-prespectrum, and being $\Sigma$-cofibrant, each preserving the properties already obtained. We define $P$ in this section and $E$ and
$K$ in the next.
Lifting the ex-space level functor $P$ of \S8.3 to the prespectrum level requires care. Recall that $P$ is the composite of the whiskering functor
$W$ and the Moore mapping path space functor $L$, together with the
natural zig-zag of $h$-equivalences
\begin{equation}\label{repeat}
\xymatrix{K & WK \ar[l]_{\rho}\ar[r]^-{W\iota} & WLK=PK}
\end{equation}
of \myref{exfibapp} for ex-spaces $K$ over $B$. The functors $W$ and $L$ do not commute with tensors with based spaces, hence cannot be enriched over $G\scr{K}_B$, by \myref{save}. There is therefore no canonical way of inducing structure maps after applying $P$ levelwise to a prespectrum, as one might at first hope. We shall resolve this by constructing by hand certain non-canonical but natural maps
\begin{equation}\label{alphamap}
\alpha_V\colon WK\sma_B S^V\longrightarrow W(K\sma_B S^V)\end{equation}
and
\begin{equation}\label{betamap}
\beta_V\colon LK\sma_B S^V\longrightarrow L(K\sma_B S^V)
\end{equation}
such that
$\alpha_0 = \text{id}$, $\beta_0=\text{id}$ and the following associativity
diagram commutes, where $(F,f_V)$ stands for either $(W,\alpha_V)$ or $(L,\beta_V)$.
\begin{equation}\label{eqn:assoc}
\xymatrix{FK \sma_B S^V\sma_B S^{V'}\ar[r]^{f_V\sma \text{id}}\ar[d]_{\iso} & F(K\sma_B S^V)\sma_B S^{V'} \ar[r]^{f_{V'}} & F(K\sma_B S^V \sma_B S^{V'})\ar[d]^{\iso}\\
FK \sma_B S^{V\oplus V'} \ar[rr]^{f_{V\oplus V'}} && F(K\sma_B S^{V\oplus V'})}
\end{equation}
The definitions of these maps and the proofs that these diagrams commute depend
on chosen decompositions of $V$ and $V'$ as direct sums of indecomposable
representations, and we cannot choose compatible decompositions for all
representations $V$ and $V'$ at once. For this reason, and for other reasons
that will become apparent later, we must switch gears and work with sequentially indexed prespectra.
Thus, to be precise about the constructions in this section and the next, we restrict our original collection $\scr{V}$ of indexing representations to a countable cofinal sequence $\scr{W}$ of expanding representations in our given universe $U$. More precisely, $\scr{W}$ consists of representations $V_i$ for $i\geq 0$ such that $V_0=0$ and $V_i\subset V_{i+1}$. We set $W_i=V_{i+1}-V_{i}$. Such a sequence can be chosen in any universe. We could just as well start with representations $W_i$ and
define $V_i$ inductively by $V_{i+1} = V_i\oplus W_i$. There is no need to
use orthogonal complements. We shall write in terms of complements, but on
the understanding that that is just a notational convenience.
\begin{rem}\mylabel{indexingreps}
There is a small quibble here since we originally defined our categories of parametrized prespectra only on collections of representations that are closed under finite direct sums, which $\scr{W}$ clearly is not. However, if we let ${\scr{W}'}$ consist of all finite sums of the $W_i$, then we recover such a collection. As in \S11.3 (or \cite[\S2]{MMSS}), we can interpret $G\scr{P}_B^{\scr{W}'}$ as a diagram category indexed on a certain small category,
say $\scr{D}^{\scr{W}'}_G$, with object set $\scr{W}'$, and we can interpret $G\scr{P}_B^{\scr{W}}$
as a diagram category indexed on the full subcategory $\scr{D}^{\scr{W}}_G$ of $\scr{D}^{\scr{W}'}_G$ whose object set is $\scr{W}$. This gives a restriction functor
$\mathbb{U}\colon G\scr{P}_B^{\scr{W}'}\longrightarrow G\scr{P}_B^{\scr{W}}$ that is right adjoint to a prolongation functor $\mathbb{P}$ \cite[\S3]{MMSS}, and $(\mathbb{P},\mathbb{U})$ induces an adjoint equivalence of homotopy categories. We shall study such ``change of universe'' adjunctions in \S14.2. They allow us to lift all results we prove about the categories of parametrized prespectra indexed on cofinal sequences to our
usual ones indexed on collections of representations closed under direct sums.
\end{rem}
\begin{defn}\mylabel{wierddef} Let $X$ be a prespectrum over $B$ indexed on the countable cofinal sequence $\scr{W} = \{V_i\}$, where $V_{0} = 0$ and
$V_{i+1} = V_i\oplus W_i$. Let $X$ have structure maps
$\sigma_i\colon \Sigma_B^{W_i}X(V_i)\longrightarrow X(V_{i+1})$. Then the maps
$$W\sigma_i \circ \alpha\colon WX(V_i)\sma_B S^{W_i}\longrightarrow WX(V_{i+1})$$
and
$$L\sigma_i \circ \beta\colon LX(V_i)\sma_B S^{W_i}\longrightarrow LX(V_{i+1})$$
specify structure maps for prespectra $WX$ and $LX$ over $B$. Therefore
$PX = WLX$ is a prespectrum over $B$.
\end{defn}
Unfortunately, as will be clear from the following construction, the maps in the zig-zag (\ref{repeat}) do not lift to the prespectrum level. They only induce \emph{weak} maps of prespectra, that is, levelwise maps that only commute with the structure maps up to (canonical)
$fp$-homotopy. Fortunately, the last approximation functor $K$, which arranges $\Sigma$-cofibrancy and will be discussed in the next section, turns weak maps
into honest ones.
\begin{con}\mylabel{wierdcon}
We define $\alpha_V$ and $\beta_V$. Fix a decomposition of $V$ into irreducible representations
and let $\scr{P}_V$ be the set of the projections from $V$ to the irreducible subrepresentations in this fixed decomposition. Define three equivariant
maps from $V$ to the real numbers by setting
\[\|v\|_V=\max_{\pi\in \scr{P}_V}|\pi v|, \quad
\mu_V(v)=\prod_{\pi\in \scr{P}_V}(1-|\pi v|), \quad
\nu_V(v)=\prod_{\pi\in \scr{P}_V}\max(1,|\pi v|).\]
Applying the same definitions to another representation $V'$ and to
$V\oplus V'$ with its induced decomposition as a sum of irreducible representations, we see that the following equations hold.
\begin{gather*}
\|v\oplus v'\|_{V\oplus V'}=\max\{\|v\|_V, \|v'\|_{V'}\},\\
\mu_{V\oplus V'}(v\oplus v')=\mu_V(v)\mu_{V'}(v'),\\
\nu_{V\oplus V'}(v\oplus v')=\nu_V(v)\nu_{V'}(v').
\end{gather*}
Define a natural map
\[h_V\colon WK \wedge_B S^V\sma_B [1,\infty)_+ \longrightarrow W(K \wedge_B S^V),\]
by setting
\begin{gather*}
h_V(x\sma v\sma t) =
\begin{cases}
\ x\sma \mu(t^{-1}v)^{-1}\cdot v & \ \ \text{if $\|v\| \leq t$,}\\
(p(x),1-\nu(t^{-1}v)^{-1}) & \ \ \text{if $\|v\| \geq t$,}
\end{cases} \\
h_V((b,s)\sma v\sma t) =
\begin{cases}
(b,s) & \text{if $\|v\| \leq t$,}\\
(b,1-(1-s)\nu(t^{-1}v)^{-1}) & \text{if $\|v\| \geq t$.}
\end{cases}
\end{gather*}
At time $t=1$ this specifies $\alpha_V$ and it is easy to verify that the associativity diagram (\ref{eqn:assoc}) commutes. Further,
the map $\rho\circ h_V$ extends to $t=\infty$ to give an $fp$-homotopy from $\rho\circ\alpha_V$ to $\rho\sma_B \text{id}$. It follows that $\rho$ induces levelwise a weak map of prespectra $WX\longrightarrow X$.
Similarly define
\[k_V\colon LK \sma_B S^V \sma_B [1,\infty)_+ \longrightarrow L(K \wedge_B S^V),\]
by setting
\[k_V((x,\lambda)\sma v\sma t) =
\begin{cases}
(x\sma v,\lambda) & \text{if $\|v\| \leq t$,}\\
(x\sma v,\nu(t^{-1}v)\lambda & \text{if $\|v\| \geq t$.}
\end{cases}\]
Here, if $1\leq a <\infty$, and $\lambda\in \Lambda B$, then $a\lambda$ denotes the Moore path of length $l_\lambda/a$ given by $a\lambda (t)=\lambda(at)$. At time $t=1$ this specifies $\beta_V$, and it is again easy to check the
required associativity. The map $k_V\circ (\iota\sma\text{id})$ extends to an $fp$-homotopy from $\beta_V\circ (\iota\sma_B \text{id})$ to $\iota$, hence
$\iota$ induces levelwise a weak map of prespectra $X\longrightarrow LX$, to which we
can apply $W$ to obtain a weak map $WX\longrightarrow WLX = PX$.
\end{con}
In view of \myref{exfibapp}, naturality arguments from
\myref{wierddef} and \myref{wierdcon} prove the following theorem.
\begin{thm}\mylabel{PPP} There are functors $L$, $W$, and
$P = WL$ on $G\scr{P}_B$ that are given levelwise by the
functors $L$, $W$, and $P$ on $G\scr{K}_B$. There are natural weak maps
$\rho\colon WX \longrightarrow X$ and $\iota\colon X\longrightarrow LX$ that are given
levelwise by the ex-space maps $\rho$ and $\iota$. Therefore,
there is a natural zig-zag of weak maps $\phi = (\rho,W\iota)$ as displayed
in the diagram
$$\xymatrix{
X & WX \ar[l]_-{\rho} \ar[r]^-{W\iota} & WLX = PX.}
$$
These maps are level $h$-equivalences, and $P$ converts level $h$-equivalences
to level $fp$-equivalences. If each $X(V)$ is compactly generated and of the
homotopy type of a $G$-CW complex, then $PX$ is well-structured. If $X$ is well-structured, then the weak maps in the above display are level $fp$-equivalences between well-structured $G$-prespectra over $B$. If, further,
the adjoint structure maps of $X$ are $h$-equivalences or $q$-equivalences,
then so are the adjoint structure maps of $LX$, $WX$, and $PX$.
\end{thm}
\begin{proof}
The only point that may need elaboration is the last clause. For a
weak map $f\colon X\longrightarrow Y$, we have a homotopy commutative diagram
$$\xymatrix{
X(V) \ar[r]^-{\tilde{\sigma}} \ar[d]_f & \Omega^W_B X(V\oplus W) \ar[d]^{\Omega_B^Wf}\\
Y(V) \ar[r]_-{\tilde{\sigma}} & \Omega^W_B Y(V\oplus W). \\}
$$
The functor $\Omega^W_B$ preserves $fp$-equivalences. Therefore, if
$f$ is an $fp$-equivalence, then the $\tilde{\sigma}$ for $X$ are
$h$-equivalences or $q$-equivalences if and only if the $\tilde{\sigma}$
for $Y$ are so. We apply this to $f=\rho$ and $f=W\iota$.
\end{proof}
\section{The auxiliary approximation functors $K$ and $E$}\label{sec:K}
We begin with the parametrized $\Omega$-prespectrum approximation functor $E$.
This is a folklore construction when $B$ is a point. In the parametrized
context, the proof of the following result makes essential use of Stasheff's
theorem, \myref{ss}, and therefore depends on our standing assumption that
$G$ acts properly on $B$.
\begin{prop}\mylabel{EEE}
There is a functor $E\colon G\scr{P}_B\longrightarrow G\scr{P}_B$ and a natural map $\alpha\colon X\longrightarrow EX$ with the following properties.
\begin{enumerate}[(i)]
\item The functor $E$ preserves level $fp$-equivalences and well-grounded prespectra. \item If $X$ is well-structured, then $EX$ is a well-structured $\Omega$-prespectrum and the map $\alpha\colon X\longrightarrow EX$ is an $s$-equivalence.
\end{enumerate}
\end{prop}
\begin{proof}
Define $EX$ by letting $EX(V_i)$ be the telescope over $j\geq i$ of the ex-spaces $\Omega^{V_j-V_i}_BX(V_j)$ with respect to the adjoint structure maps
\[ \Omega^{V_j-V_i}_B\tilde{\sigma}\colon \Omega^{V_j-V_i}_BX(V_j)
\longrightarrow \Omega^{V_j-V_i}_B\Omega^{W_j}_BX(V_{j+1})
\iso \Omega^{V_{j+1}-V_i}_BX(V_{j+1}).\]
Since the functor $\Omega^{W_i}_B$ commutes with telescopes, $\Omega_B^{W_i}EX(V_{i+1})$ is isomorphic to the telescope over $j\geq i+1$ of the ex-spaces $\Omega^{V_j-V_{i+1}}_BX(V_j)$. The adjoint structure map $EX(V_i)\longrightarrow \Omega^{W_i}_B EX(V_{i+1})$ is induced by the maps $\Omega_B^{V_j-V_i}\tilde\sigma_j$ for $j\geq i$. The map $\alpha\colon X\longrightarrow EX$ is given by the inclusion of the bases of the telescopes. If $f\colon X\longrightarrow Y$ is a level $fp$-equivalence, then $Ef\colon EX\longrightarrow EY$ is a level $fp$-equivalence since a standard inductive argument (applicable in any topologically bicomplete category) shows that the telescope of a ladder of $fp$-equivalences is an $fp$-equivalence.
If $X$ is well-grounded or level ex-fibrant, then so is $EX$ since the construction clearly stays in the category of compactly generated spaces and since it preserves the conditions of being well-sectioned or level ex-fibrant by results in \S8.2. To show that $E$ preserves well-structured prespectra, it remains to show that if $X$ has total spaces of the homotopy types of $G$-CW complexes, then so does $EX$. By Stasheff's theorem (\myref{ss}), the fibers $X(V)_b = X_b(V)$ have the homotopy types of $G_b$-CW complexes. We have the analogous construction $E$ in the category of $G_b$-prespectra and, by
Milnor's theorem (\myref{Milnor}) and standard facts about telescopes, the
$(E(X_b))(V)$ have the homotopy types of $G_b$-CW complexes. It is clear from the definition of $E$ that $(E(X_b))(V) = ((EX)(V))_b$. That is, the $G_b$-prespectrum $E(X_b)$ is the fiber $(EX)_b$ of the $G$-prespectrum $EX$ over
$B$. By Stasheff's theorem again, it follows that the $(EX)(V)$ have the homotopy types of $G$-CW complexes.
To check that the adjoint structure maps are $q$-equivalences when $X$ is well-structured, it suffices to check that they induce $q$-equivalences on the fibers over $b$ for all $b\in B$. That holds by inspection of the homotopy groups of the colimits that define $(EX)_b \iso E(X_b)$. Similarly, we see that $\alpha$ is a $\pi_*$-equivalence when $X$ is well-structured by fiberwise comparison of the colimits of homotopy groups of fibers that define the homotopy groups of $X$
and $EX$.
\end{proof}
To approximate parametrized prespectra by level $fp$-equivalent $\Sigma$-cofibrant prespectra, we use the elementary cylinder construction $K$ that
was first defined in \cite{May69} and has been used in various papers since.
We recall the construction and its main properties from \cite[6.8]{LMS}, which carries over verbatim to the parametrized context. A more sophisticated but less convenient treatment is given in \cite{EKMM}.
\begin{prop}\mylabel{KKK}
There is a functor $K\colon G\scr{P}_B\longrightarrow G\scr{P}_B$ and a natural level $fp$-equivalence $\pi\colon KX\longrightarrow X$. Therefore $K$ preserves level $fp$-equivalences. If $X$ is well-grounded, then $KX$ is $\Sigma$-cofibrant. If $X$ is well-structured, then $KX$ is well-structured. If $X$ is a well-structured $\Omega$-prespectrum, then so is $KX$ and thus $KX$ is excellent. There is a natural weak map $\iota\colon X\longrightarrow KX$ that is a right inverse of $\pi$, and $K$ takes weak maps $f$ to honest maps $Kf$ such that $\iota\circ f = Kf \circ \iota$.
\end{prop}
\begin{proof}
Define $KX$, a level inclusion $\iota\colon X\longrightarrow KX$, and a level $fp$-deformation retraction $\pi\colon KX\longrightarrow X$ right inverse to $\iota$ as follows. Let $KX(0) = X(0)$ and $\iota(0) = \pi(0) = \text{id}$. Inductively, suppose given $KX(V_i)$, an inclusion $\iota(V_i)\colon X(V_i)\longrightarrow KX(V_i)$ and an inverse $fp$-deformation retraction $\pi(V_i)\colon KX(V_i)\longrightarrow X(V_i)$. Let $KX(V_{i+1})$ be the double mapping cylinder in $G\scr{K}_B$ of the pair of maps
\[\xymatrix@=.6cm{\Sigma^{W_i}_B KX(V_i) && \Sigma^{W_i}_B X(V_i)\ar[ll]_-{\Sigma^{W_i}_B \iota(V_i)} \ar[rr]^-\sigma && X(V_{i+1})}\]
in $G\scr{K}_B$. Let $\sigma\colon \Sigma^{W_i}_BKX(V_i)\longrightarrow KX(V_{i+1})$ be the inclusion of the left base of the double mapping cylinder, which is an
$fp$-cofibration and let $\iota(V_{i+1})\colon X(V_{i+1})\longrightarrow KX(V_{i+1})$ be the inclusion of the right base. Let $\pi(V_{i+1})\colon KX(V_{i+1})\longrightarrow X(V_{i+1})$ be the map obtained by first using the $fp$-equivalence $\Sigma^{W_i}_B\pi(V_i)$ on the left base to map to the mapping cylinder of $\sigma$ and then using the evident deformation retraction to the right base. There is an equivalent description as a finite telescope. Certainly $\pi$ is a map of prespectra over $B$ and a level $fp$-deformation retraction with level inverse the weak map $\iota$. The functoriality of the construction is clear.
If $X$ is well-grounded, then $KX$ is clearly also well-grounded and thus $KX$ is $\Sigma$-cofibrant. If $X$ is well-structured, then so is $KX$ by Propositions \ref{pres1} and \ref{Hursma}. If, further, the adjoint structure maps of $X$ are $q$-equivalences, then they are $fp$-equivalences since $X$ is well-structured. Since $K$ preserves $fp$-homotopies, it follows that $KX$ is also an $\Omega$-prespectrum. Alternatively, since $\Omega_B^V$ is a Quillen right adjoint in the $qf$-model structure, it preserves $q$-equivalences between $qf$-fibrant ex-spaces. In particular, the maps $\Omega^W_B\pi(V_i)$ are $q$-equivalences.
If $f\colon X\longrightarrow Y$ is a weak map with $fp$-homotopies
$$h_i\colon \Sigma_B^{W_i}X(V_i)\sma_B I_+ \longrightarrow Y(V_{i+1})$$
from $\sigma_Y\circ \Sigma^{W_i}f(V_i)$ to $f(V_{i+1})\circ \sigma_X$, define $Kf$ inductively by setting $Kf(0)=f(0)$ and letting $Kf(V_{i+1})$ be $\Sigma^{W_j}_BKf(V_i)$ on the left end of the mapping cylinder, $f(V_{i+1})$ on the right end and as follows on the cylinder itself:
\[Kf(V_{i+1})[x,t]=\begin{cases}
[\Sigma^{W_i}_B f(V_i)(x),2t] & \text{if $0\leq t\leq \tfrac12$},\\
h_i(x,2t-1) & \text{if $\tfrac12 \leq t\leq 1$.}
\end{cases}\]
Then $Kf$ is a map of prespectra over $B$ and $\iota\circ f = Kf\circ \iota$.
\end{proof}
The composite approximation functor $T = KEP$ has various good preservation properties. The ex-space level properties of $P$ recorded in \S8.4 are
inherited on the prespectrum level, and we have the following sample result
for $E$ and $K$.
\begin{lem}
For a $G$-map $f\colon A\longrightarrow B$, a prespectrum $Y$ over $B$ and a prespectrum $X$ over $A$, there are natural isomorphisms
\[f^*EY \cong Ef^*Y,\quad f^*KY\cong Kf^*Y \quad\text{and}\quad Kf_!X\cong f_!KX.\]
\end{lem}
\begin{proof}
The relevant telescopes commute with $f^*$ since it is a symmetric monoidal left
adjoint and with $f_!$ since it is a left adjoint and the projection formula (\ref{four0}) holds.
\end{proof}
\section{The equivalence between $\text{Ho}\, G\scr{P}_B$ and $h G\scr{E}_B$}
We can now extend the results of \S9.1 to parametrized prespectra. As in the previous section, our parametrized prespectra are indexed on a countable
cofinal sequence of expanding representations in our given universe. We
begin by collating the results of the previous two sections.
\begin{thm}\mylabel{Tzigzag}
Let $X$ be a well-grounded $G$-prespectrum over $B$ whose total spaces are of the homotopy types of $G$-CW complexes and define $TX = KEP X$.
\begin{enumerate}[(i)]
\item $TX$ is an excellent $G$-prespectrum.
\item $T$ takes level $q$-equivalences between $G$-prespectra over $B$
that satisfy the hypotheses on $X$ to homotopy equivalences of $G$-prespectra.
\item There is a zig-zag of $s$-equivalences between $X$ and $TX$.
\item If $X$ is an excellent $G$-prespectrum over $B$, then the zig-zag
consists of level $fp$-equivalences, and it gives rise to a zig-zag of
homotopy equivalences of $G$-prespectra over $B$ connecting $X$ and $TX$.
\end{enumerate}
\end{thm}
\begin{proof}
We have that $PX$ is well-structured by \myref{PPP}, $EPX$ is a well-structured $\Omega$-prespectrum by \myref{EEE}, and $TX$ is excellent by \myref{KKK}. In (ii), a level $q$-equivalence is a level $h$-equivalence. By the results just quoted, $P$ takes level $h$-equivalences to level $fp$-equivalences, which are preserved by $E$, and $K$ takes level $fp$-equivalences to homotopy equivalences. Since $K$ converts weak maps to genuine maps, we have the following diagram of maps of $G$-presepectra over $B$.
\begin{equation}\label{GROWL}
\xymatrix{KX\ar[d]_\pi & KWX\ar[l]_{K\rho}\ar[d]^\pi & WKX\ar[d]_{W\pi}\ar[r]^{WK\iota} & WKLX\ar[d]^{W\pi} & KEPX\ar[d]^\pi\\
X & WX\ar@{=}[r] & WX & WLX\ar[r]_{\alpha} & EPX}
\end{equation}
The vertical maps $\pi$, hence also the vertical maps $W\pi$, are level $fp$-equivalences. The map $\rho$ is a level $f$-equivalence. The map $\iota$ is a level $h$-equivalence, hence so is $WK\iota$. The map $\alpha$ is an $s$-equivalence because $PX$ is well-structured. Since level $q$-equivalences are also
$s$-equivalences, the diagram displays a zig-zag of $s$-equivalences
between $X$ and $TX$.
For the last statement, observe that all prespectra in the diagram are well-structured $\Omega$-prespectra over $B$. Moreover, $\alpha$ is a level $q$-equivalence by \myref{bombsaway}. It is therefore a level $h$-equivalence since our total spaces have the homotopy types of $G$-CW complexes. Since all prespectra in our diagram are well-structured, our level $h$-equivalences are level $fp$-equivalences, by \myref{fpequiv2}. Applying $K$ where needed, we can expand the diagram to a zig-zag of level $fp$-equivalences between $\Sigma$-cofibrant prespectra. By \myref{CPhelp}, this gives a zig-zag of homotopy equivalences connecting $X$ and $TX$.
\end{proof}
We introduce a category that is intermediate between $G\scr{P}_B$ and $G\scr{E}_B$.
\begin{defn}
Define $G\scr{Q}_B$ to be the full subcategory of $G\scr{P}_B$ consisting of the well-grounded $\Omega$-prespectra over $B$ whose total spaces are of the homotopy types of $G$-CW complexes. Define $\text{Ho} G\scr{Q}_B$ to be the homotopy category obtained by inverting the $s$-equivalences in $G\scr{Q}_B$; by the proof of the next theorem, there are no set-theoretic problems in defining $\text{Ho} G\scr{Q}_B$. Define $T = KEP\colon G\scr{Q}_B \longrightarrow G\scr{E}_B$.
\end{defn}
Since the $\Omega$-prespectra over $B$ are the $s$-fibrant prespectra over $B$ and since $s$-cofibrant spectra are well-grounded and have total spaces of the homotopy types of $G$-CW complexes, all $G$-prespectra over $B$ that are
$s$-cofibrant and $s$-fibrant are in $G\scr{Q}_B$. We prove that $\text{Ho}G\scr{P}_B$ is equivalent to $hG\scr{E}_B$ by proving that these categories are both equivalent to $\text{Ho} G\scr{Q}_B$.
\begin{thm}\mylabel{exceq}
The canonical $s$-cofibrant and $s$-fibrant approximation functor $RQ$ and the composite approximation functor $T=KEP$, together with the forgetful functors, induce the following equivalences of homotopy categories.
\[\xymatrix{\text{Ho}\, G\scr{P}_B \ar@<.5ex>[r]^-{RQ}
& \text{Ho}\, G\scr{Q}_B \ar@<.5ex>[r]^-{T}\ar@<.5ex>[l]^-I
& hG\scr{E}_B \ar@<.5ex>[l]^-J }\]
\end{thm}
\begin{proof}
For $X$ in $G\scr{P}_B$, we have a natural zig-zag of $s$-equivalences in $G\scr{P}_B$
\[\xymatrix{X & QX\ar[l]\ar[r] & RQX.}\]
Therefore $X$ and $IRQX$ are naturally $s$-equivalent in $G\scr{P}_B$. If $X$ is in $G\scr{Q}_B$, then it is $s$-fibrant and therefore so is $QX$. Then the above zig-zag is in $G\scr{Q}_B$ so $X$ and $RQIX$ are naturally $s$-equivalent in $G\scr{Q}_B$.
By \myref{bombsaway}, $s$-equivalences in $G\scr{Q}_B$ are level $q$-equivalences, and $T$ takes level $q$-equivalences to homotopy equivalences by \myref{Tzigzag}. Conversely, since homotopy equivalences are $s$-equivalences, the forgetful functor $J$ induces a functor in the other direction.
For $X$ in $G\scr{Q}_B$ we have the natural zig-zag of $s$-equivalences displayed in (\ref{GROWL}). Applying $s$-fibrant approximation, we get a natural zig-zag of $s$-equivalences in $G\scr{Q}_B$ so $X$ and $JTX$ are naturally $s$-equivalent in $G\scr{Q}_B$. Starting with $X$ in $G\scr{E}_B$, the last statement of \myref{Tzigzag} shows that $X$ and $TJX$ are naturally homotopy equivalent in $G\scr{E}_B$.
\end{proof}
\section{Derived functors on homotopy categories}
With $P$ replaced by $T$, the discussion of derived functors in \S9.2 carries over from the level of ex-spaces to the level of parametrized prespectra indexed on cofinal sequences. In \S13.7 and \S14.2 we will discuss how to pass from there to conclusions on the level of parametrized spectra indexed on our usual collections of representations closed under direct sums. We must show that if $V$ is a Quillen left or right adjoint, then its model theoretic left or right derived functor agrees under our equivalences of categories with the functor obtained simply by passing to homotopy classes of maps from the composite $TV$. As on the ex-space level, we need some mild good behavior for this to work.
\begin{defn}\mylabel{goodfun}
A functor $V\colon G\scr{P}_A\longrightarrow G\scr{P}_B$ is \emph{good} if it is continuous, preserves well-grounded parametrized prespectra, and takes prespectra over $A$ whose levelwise total spaces are of the homotopy types of $G$-CW complexes to prespectra over $B$ with that property. Since $V$ is continuous, it preserves homotopies. There are evident variants for functors $V$ with source or target
$G\scr{K}_*$: $V$ must be continuous, preserve well-grounded objects, and preserve
$G$-CW homotopy type conditions on objects.
\end{defn}
Note that a good functor $V$ need not take $\Omega$-$G$-prespectra to
$\Omega$-$G$-prespectra and recall that a Quillen right adjoint must
preserve fibrant objects and thus, in our context, must preserve
$\Omega$-$G$-prespectra.
\begin{prop}\mylabel{scderiv}
Let $V\colon G\scr{P}_A\longrightarrow G\scr{P}_B$ be a good functor that is a part of a Quillen adjoint pair. If $V$ is a Quillen left adjoint, assume further that it preserves level $q$-equivalences between well-grounded objects. Then the derived functor $\text{Ho}\, G\scr{P}_A\longrightarrow\text{Ho}\, G\scr{P}_B$, induced by $VQ$ or $VR$, is equivalent to the functor $TVJ\colon hG\scr{E}_A\longrightarrow hG\scr{E}_B$ under the equivalence of categories
in \myref{exceq}
\end{prop}
\begin{proof}
If $V$ is a Quillen right adjoint, then it preserves $s$-equivalences between $s$-fibrant objects. If $V$ is a Quillen left adjoint, then it preserves $s$-equivalences between well-grounded objects by \myref{neat}. Therefore, since $G\scr{Q}_A$ consists of well-sectioned $s$-fibrant objects, the functor $V\colon G\scr{Q}_A \longrightarrow G\scr{P}_B$ passes straight to homotopy categories to give $V\colon \text{Ho}G\scr{Q}_A \longrightarrow \text{Ho}G\scr{P}_B$ in both cases.
If $V$ is a Quillen right adjoint, then it takes an $s$-equivalence $f$ in $G\scr{Q}_A$ to an $s$-equivalence since the objects of $G\scr{Q}_A$ are $s$-fibrant. Then $Vf$ is a level $q$-equivalence by \myref{bombsaway} and, since $V$ is good, it is a level $h$-equivalence. On the other hand, if $V$ is a Quillen left adjoint, then \myref{bombsaway} gives that $f$ is a level $q$-equivalence and, by assumption, $Vf$ is then a level $q$-equivalence. Since $V$ is good, $Vf$ is actually a level $h$-equivalence. In both cases it follows that $V$ takes $s$-equivalences to level $h$-equivalences and therefore $TV$ passes to a functor $\text{Ho}\, G\scr{Q}_A\longrightarrow hG\scr{E}_B$.
To show that $TVJ$ and either $VQ$ or $VR$ agree under the equivalence of categories, it suffices to verify that the following diagram commutes.
\[\xymatrix{
\text{Ho}\, G\scr{P}_A\ar[d]_{RQ} \ar[rr]^-{VQ \ \ \text{or} \ \ VR}
& & \text{Ho}\, G\scr{P}_B \ar[d]^{TRQ}\\
\text{Ho}\, G\scr{Q}_A \ar[rr]_{TV} & & h G\scr{E}_B}\]
We have functorial $s$-cofibrant and $s$-fibrant approximation functors $Q$ and $R$, with natural acyclic $s$-fibrations $QX\longrightarrow X$ and acyclic $s$-cofibrations $X\longrightarrow RX$. Clearly $Q$ and $R$ preserve $s$-equivalences. If $V$ is a Quillen left adjoint, then we have a zig-zag of natural $s$-equivalences
\[\xymatrix{RQVQ \ar[r] & RVQ & VQ \ar[l]\ar[r] & VRQ}\]
because $V$ preserves acyclic $s$-cofibrations.
If $V$ is a Quillen right adjoint, then we have a zig-zag of natural $s$-equivalences
\[\xymatrix{RQVR & RQVRQ \ar[r]\ar[l] & RVRQ & VRQ\ar[l]}\]
because $V$ preserves $s$-equivalences between $s$-fibrant objects. In both cases, all objects have total spaces of the homotopy types of $G$-CW complexes,
hence we have zig-zags of level $h$-equivalences. Applying $T$, we obtain a zig-zag of homotopy equivalences in $G\scr{E}_B$ by \myref{Tzigzag}.
\end{proof}
\begin{rem}
If $V$ preserves excellent parametrized prespectra, then $TV$ is naturally homotopy equivalent to $V$ on excellent parametrized prespectra. The derived functor of $V$ can then be obtained directly by applying $V$ and passing to homotopy classes of maps.
\end{rem}
\section{Compatibility relations for smash products and base change}
This section is parallel to \S9.3. The main change is just that we must replace the functor $P$ used there with the functor $T =KEP$ that we have here. This gives us results for the categories $G\scr{P}_B^\scr{W}$ of parametrized prespectra indexed on a collection $\scr{W}$ consisting of a cofinal sequence in some universe $U$. In order to obtain statements about $G\scr{S}_B^\scr{V}$, where $\scr{V}=\scr{V}(U)$, we have two pairs of Quillen equivalences, both of which can be viewed as consisting of a prolongation functor left adjoint to a forgetful functor that creates the weak equivalences; see \cite[1.2]{MM}.
\[\xymatrix@1{G\scr{P}_B^{\scr{W}} \ar@<.5ex>[r]^-{j_*}
& G\scr{P}_B^\scr{V} \ar@<.5ex>[r]^-{\mathbb{P}}\ar@<.5ex>[l]^-{j^*}
& G\scr{S}_B^\scr{V} \ar@<.5ex>[l]^{\mathbb{U}}}\]
We postpone until \S14.2 consideration of the pair $(j_*,j^*)$ and the extension from $G\scr{P}_B^\scr{W}$ to $G\scr{P}_B^\scr{V}$ and deal with the extension from $G\scr{P}_B^\scr{V}$ to $G\scr{S}_B^\scr{V}$ in this section.
One general remark is in order, though. The forgetful functors $j^*$ and $\mathbb{U}$ create weak equivalences and therefore pass directly to homotopy categories. If they commute on the point set level with a functor $V$ which is a part of a Quillen adjoint pair, then they will also commute with its derived functor on the level of homotopy categories. It follows formally that the derived prolongation functors $\mathbb{P}$ and $j_*$ then also commute with the derived functor $V$ and its adjoints. This holds in particular for the base change functor $V=f^*$. Extending commutation results for such functors from $G\scr{P}_B^\scr{W}$ to $G\scr{S}_B^\scr{V}$ is therefore easy. However, some of the functors $V$ that we need to consider only exist on some of the categories in the above display, and such
functors require special care. These include the change of universe functors that we discuss in \S14.2, which don't exist on the level of $G\scr{P}_B^\scr{W}$, and the smash product $\sma_B$, which we have only specified on the spectrum level and which we now discuss on the prespectrum level.
\begin{rem}\mylabel{subtlety}
Because the domain category for the diagram category of (equivariant and parametrized) prespectra is only monoidal, not symmetric mon\-oid\-al,
we cannot use left Kan extension to internalize ``external'' smash products of prespectra; see \cite[4.1]{MMSS}. Here ``external'' is understood in the sense of indexing on pairs of representations. Therefore, on the equivariant parametrized prespectrum level, when we write $X\barwedge Y$ for prespectra
$X$ over $A$ and $Y$ over $B$, we should understand the external external smash product, in the sense of \myref{extsmash1}. When passing from prespectrum level arguments to spectrum level conclusions using $(\mathbb{P}, \mathbb{U})$, we are implicitly using composites of the general form $\mathbb{P} V\mathbb{U}$, and similarly for functors of several variables involving smash products. We can carry out the several variable arguments externally on the prespectrum level, only internalizing
with left Kan extension after passage to spectra, where we have good
homotopical control by \myref{exttoo}.
Alternatively, we can make use of classical ``handicrafted smash products'' of prespectra, which are defined by use of arbitrary choices of sequences of representations. As discussed on the nonequivariant nonparametrized level in \cite[\S11]{MMSS}, handicrafted smash products of prespectra agree under the adjoint equivalence $(\mathbb{P},\mathbb{U})$ with the internalized smash products. Provided that we use external parametrized handicrafted smash products over varying base spaces, only internalizing along diagonal maps at the end, the discussion there adapts readily to give the same conclusion for homotopy categories of equivariant parametrized prespectra and spectra. The advantage of handicrafted smash products is that their definition involves only direct use of ex-space level constructions that enjoy good preservation properties with respect to ex-fibrations. This often allows direct transposition of ex-space level arguments in $h G\scr{W}_B$ to parametrized prespectrum level arguments
in $hG\scr{E}_B$.
\end{rem}
We state the following results in terms of parametrized spectra, and we indicate which parts of the proofs require the use of $hG\scr{E}_B$ and which parts work directly in the stable homotopy category $\text{Ho}G\scr{S}_B$.
\begin{prop}\mylabel{helpme}
Let $f\colon A\longrightarrow B$ and $g\colon A'\longrightarrow B'$ be $G$-maps. If $W$ and $X$ are spectra over $A$ and $A'$, then
$$f_!W\barwedge g_!X \simeq (f\times g)_!(W\barwedge X)$$
in $\text{Ho}\, G\scr{S}_{B\times B'}$. If $Y$ and $Z$ are spectra over $B$ and $B'$, then
$$f^*Y\barwedge g^*Z \simeq (f\times g)^*(Y\barwedge Z)$$
in $\text{Ho}\, G\scr{S}_{A\times A'}$.
\end{prop}
\begin{proof}
Working directly in $\text{Ho}\, G\scr{S}_{B\times B'}$, the first equivalence reduces to its point-set level analogue by consideration of Quillen left adjoints, as in the corresponding proof of \myref{fgext'}. We work in $h G\scr{E}_{A\times A'}$ to prove the second equivalence. Here $f^*$ and $\barwedge$ (understood in the external or handicrafted sense) are both good, and both preserve excellent prespectra. Indeed, they preserve well-structured prespectra by levelwise application of Propositions \ref{fexpres} and \ref{Hursma}, they preserve $\Sigma$-cofibrant prespectra since $f^*$ and $\barwedge$ on ex-spaces preserve $fp$-cofibrations because they are left adjoints that commute with $fp$-homotopies, and they preserve $\Omega$-prespectra by \myref{fpOM} since they preserve $fp$-homotopies. Therefore, using excellent prespectra, we can pass straight to homotopy categories, without use of $T$, as in the corresponding proof of \myref{fgext'}.
\end{proof}
\begin{thm}\mylabel{symmonhtp}
The category $\text{Ho}\, G\scr{S}_B$ is closed symmetric monoidal under the functors $\sma_B$ and $F_B$.
\end{thm}
\begin{proof}
Working in $\text{Ho}\, G\scr{S}_B$, the associativity, commutativity, and unity of $\sma_B$ follow by pullback along diagonal maps from their easily proven external analogues and the second equivalence in the previous result, exactly as in \myref{clsymmon}.
\end{proof}
We have a commutation relation between change of base and suspension spectrum functors that is analogous to the relation between change of base and smash products recorded in \myref{helpme}.
\begin{prop}\mylabel{SIfSI}
For a $G$-map $f\colon A\longrightarrow B$, there are natural equivalences
\[\Sigma^{\infty}_Bf_!\simeq f_!\Sigma^{\infty}_A
\quad\text{and}\quad \Sigma^{\infty}_Af^*\simeq f^*\Sigma^{\infty}_B\]
of (derived) functors. The same conclusion holds more generally for the shift desuspension functors $F_V = \Sigma^{\infty}_V$.
\end{prop}
\begin{proof}
Working in $\text{Ho}\, G\scr{S}_B$, the first equivalence is clear since it is a comparison of Quillen left adjoints that commute on the point-set level. For the second equivalence, we start in $h G\scr{W}_B$ and end in $h G\scr{E}_A$. For $K\in G\scr{W}_B$, the point set level suspension prespectrum $\Sigma^{\infty}_BK$ is both $\Sigma$-cofibrant and well-structured, by \myref{HursmaK}, but of course it is not an $\Omega$-prespectrum over $B$. Since $\Sigma^\infty_B$ is good and takes well-grounded $q$-equivalences to well-grounded level $q$-equivalences, $T\Sigma^{\infty}_B$ is equivalent to the model theoretic left derived functor of the Quillen left adjoint $\Sigma^{\infty}_B$.
Here we may omit $P$ from the composite functor $T$ and, since $f^*$ commutes with both $K$ and $E$, the conclusion follows on passage to homotopy categories.
\end{proof}
Applying this to $\Delta\colon B\longrightarrow B\times B$ and using \myref{SISISI}, we obtain the following consequence.
\begin{prop}\mylabel{SISISI2}
For ex-spaces $K$ and $L$ over $B$,
\[\Sigma^{\infty}_{B} (K\sma_B L) \simeq \Sigma^{\infty}_{B}K \sma_B \Sigma^{\infty}_{B}L\]
in $\text{Ho}\, G\scr{S}_B$.
\end{prop}
For $f\colon A\longrightarrow B$, evident properties of the functor $f_!$ on ex-spaces
imply that the functor $f_!\colon G\scr{P}_A\longrightarrow G\scr{P}_B$ is good, and $f_!$ satisfies the other hypotheses of \myref{scderiv} by \myref{Qad1}. We use
this to prove the following basic result.
\begin{thm}\mylabel{Wirthmore}
For a $G$-map $f\colon A\longrightarrow B$ between base spaces, the derived functor $f^*\colon \text{Ho}\, G\scr{S}_B\longrightarrow \text{Ho}\, G\scr{S}_A$ is closed symmetric monoidal.
\end{thm}
\begin{proof}
Since $S_B$ is not $s$-fibrant, the isomorphism $f^*S_B\iso S_A$ in $G\scr{S}_B$ does not immediately imply the required equivalence $f^*S_B\simeq S_A$ in $\text{Ho}\, G\scr{S}_A$, where $f^*S_B$ means $f^*RS_B$. However, \myref{SIfSI} specializes to give this equivalence. For the rest, we must show that the isomorphisms (\ref{one}) through (\ref{five}) descend to equivalences on homotopy categories. By category theory in \cite{FHM}, it suffices to consider (\ref{one}) and (\ref{four}), and the proofs are similar to those in \myref{fclsymmon}. Since $\barwedge$ and $\Delta^*$ both preserve excellent prespectra, so do the internalized smash products $\sma_A$ and $\sma_B$. For excellent prespectra $Y$ and $Z$ over $B$, it follows that both sides of
\[f^*(Y\sma_B Z) \iso f^*Y \sma_A f^*Z\]
are excellent prespectra over $A$, hence the point-set level isomorphism descends directly to the desired equivalence on the homotopy category level. Next consider
\[f_!(f^*Y\sma_A X)\iso Y\sma_B f_!X,\]
where $X$ is an excellent prespectrum over $A$. Here we must replace $f_!$ by $Tf_!$ on both sides. By \myref{Tzigzag} we have a natural zig-zag $\phi$ of level $h$-equivalences connecting $T$ to the identity functor which, when applied to excellent parametrized prespecra gives rise to a zig-zag $\psi$ of actual homotopy equivalences. We obtain the following zig-zag.
\[\xymatrix{Tf_!(f^*Y\sma_A X)\cong T(Y\sma_B f_!X) \ar[rr]^-{T(\text{id}\sma_B \phi)} && T(Y\sma_B Tf_!X) \ar[r]^-\psi \ar[ll] & Y\sma_B Tf_!X. \ar[l]}\]
Using handicrafted products with their termwise construction in terms of smash products of ex-spaces, it follows from \myref{savior} that $\text{id}\sma_B -$ preserves level $h$-equivalences between well-sectioned spectra. Thus $\text{id}\sma_B \phi$ is a zig-zag of level $h$-equivalences and $T(\text{id}\sma_B\phi)$ is a zig-zag of actual homotopy equivalences.
\end{proof}
\begin{thm}\mylabel{Mackeymore}
Suppose given a pullback diagram of $G$-spaces
\[\xymatrix{
C \ar[r]^-{g} \ar[d]_{i} & D \ar[d]^{j} \\
A \ar[r]_{f} & B}\]
in which $f$ (or $j$) is a $q$-fibration. Then there are natural equivalences of (derived) functors on stable homotopy categories
\begin{equation}\label{basesmore}
j^*f_{!} \simeq g_{!}i^*, \quad f^*j_* \simeq i_*g^*, \quad f^*j_{!}\simeq i_!g^*, \quad j^*f_*\simeq g_*i^*.
\end{equation}
\end{thm}
\begin{proof}
Working in $hG\scr{E}_B$, the proof is similar to that of \myref{pullbackfix} but with $P$ replaced by $T$. Again it suffices to consider the first equivalence, and, as explained there, since $f$ is a $q$-fibration there is a level $fp$-equivalence $\mu\colon Pf^*\longrightarrow f^*P$. Since $f^*$ commutes with both $K$ and $E$, we obtain a level $fp$-equivalence $\mu\colon Tf^*\longrightarrow f^*T$ between $\Sigma$-cofibrant prespectra over $A$ so it is in fact a homotopy equivalence by \myref{CPhelp}. Then $f^*Tj_!X\simeq Tf^*j_!X\cong Ti_!g^*X$.
\end{proof}
The following observation holds by the same proof as the analogous ex-space level result \myref{imonoidaldescends}.
\begin{prop}\mylabel{Symmoni}
Let $\iota\colon H\longrightarrow G$ be the inclusion of a subgroup and $A$ be
an $H$-space. The closed symmetric monoidal Quillen equivalence $(\iota_!, \nu^*\iota^*)$ descends to a closed symmetric monoidal equivalence between
$\text{Ho}H\scr{S}_A$ and $\text{Ho}G\scr{S}_{\iota_!A}$.
\end{prop}
Combined with \myref{Wirthmore}, applied to the inclusion
$\tilde{b}\colon G/G_b \longrightarrow B$, and \myref{SIfSI}, this last
observation gives us the following stable analogue of \myref{fiberfun}.
\begin{thm}\mylabel{reassuring}
The derived fiber functor $(-)_b\colon \text{Ho}\, G\scr{K}_B\longrightarrow \text{Ho}\, G_b\scr{K}_*$ is closed symmetric monoidal and it has both a left adjoint $(-)^b$ and a right adjoint ${^b}(-)$. Moreover, the derived fiber functor commutes with the derived
suspension spectrum functor, $(\Sigma^{\infty}_B K)_b\simeq \Sigma^{\infty}(K_b)$
as $G_b$-spectra.
\end{thm}
For emphasis, we repeat a remark that we made after the analogous ex-space level result. This innocent looking result packages highly non-trivial and important information. In particular, it gives that
$F_B(X,Y)_b\htp F(X_b,Y_b)$ in $\text{Ho}\, G_b\scr{S}$
for $X,Y\in \text{Ho}\, G\scr{S}_B$, where the fiber and function
object functors are understood in the derived sense. This reassuring
consistency result is central to our applications in the last two chapters,
where parametrized duality is studied fiberwise.
\chapter{Module categories, change of universe, and change of groups}
\section*{Introduction}
We first give a discussion of module categories of parametrized
spectra over nonparametrized ring spectra. Although we shall not
go into these applications here, one basic motivation for our work
is to set up the homotopical foundations for studying the generalized
homology and cohomology theories of parametrized spectra that are
represented by such nonparametrized ring spectra. The good behavior
of the external
smash product $G\scr{S}\times G\scr{S}_B\longrightarrow G\scr{S}_B$ makes it easy to do this.
While the mathematics here is evident, it deserves emphasis since
the ideas are likely to be central to future applications.
In the rest of the chapter, we focus on problems that are special to the equivariant context. We give the parametrized generalization of some of the
work in \cite{MM} concerning change of universe, change of groups, and fixed point and orbit spectra. As usual, an essential point is to determine which
of the standard adjunctions are given by Quillen adjoint pairs and to prove
that other adjunctions and compatibilities that are evident on the point set level also descend to homotopy categories. We discuss change of universe
in \S14.2. Here the use of prespectra indexed on cofinal sequences in the
previous chapter introduces some minor difficulties that were not studied in
the nonparametrized theory of \cite[V\S1]{MM} and are already relevant
nonequivariantly. We study subgroups and fixed point spectra in \S14.3. We study quotient groups and orbit spectra in \S14.4. Aside from some analogues for parametrized spectra
of earlier results for parametrized spaces, these sections are precisely
parallel to \cite[V\S\S2 and 3]{MM}. We have not written down the parametrized analogue of \cite[V\S4]{MM}, which gives the theory of geometric fixed point
spectra, since it would be tedious to repeat the constructions given there.
It will be apparent to the interested reader that, mutatis mutandis, the definitions and results in \cite[V\S4]{MM} generalize to the parametrized
context.
\section{Parametrized module $G$-spectra}
We can define a parametrized (strict) ring $G$-spectrum $R$ over $B$ to be a monoid in the symmetric monoidal category $G\scr{S}_B$, and we can then define parametrized $R$-modules and $R$-algebras in the usual way, as has become standard in stable homotopy theory \cite{EKMM, HSS, MM, MMSS}. However, even though the
smash product $\sma_B$ in $G\scr{S}_B$ gives a point-set level symmetric monoidal structure, we cannot expect to obtain Quillen model structures on the categories
of such $R$-modules or $R$-algebras, as was done for orthogonal $G$-spectra in \cite[III\S\S7,8]{MM}. To do that, we would need better homotopical behavior than we can prove here. We have only set up adequate foundations for the classical style theory of up to homotopy parametrized module spectra over up to homotopy parametrized ring spectra. From that point of view, our
homotopical foundations are entirely satisfactory. The source of the problem is \myref{ouchtoo}, which implies that $X\sma_B (-) $ in $G\scr{S}_B$ cannot be a Quillen functor.
However, in applications, it is natural to start with a nonparametrized orthogonal ring $G$-spectrum $R$. We are then interested in understanding
the $R$-homology and $R$-cohomology theories of $G$-spectra over $B$ and their relationships with the $R$-homology and $R$-cohomology of the fibers. For this study, just as in the nonparametrized work of \cite{EKMM, HSS, MM, MMSS}, one
is interested in the theory of $R$-modules. The \emph{external} smash product $\barwedge\colon G\scr{S}\times G\scr{S}_B \longrightarrow G\scr{S}_B$ has enough of the good properties of the nonparametrized smash product $G\scr{S}\times G\scr{S} \longrightarrow G\scr{S}$
to give us homotopical control over parametrized module spectra over nonparametrized ring spectra. We devote this section to
developing the relevant theory, which is parallel to \cite[III\S7]{MM}.
Let $R$ be a ring spectrum in $G\scr{S}$ which is well-grounded when viewed
as a spectrum, meaning that each $R(V)$ is well-based and
compactly generated.
\begin{defn} A {\em (left) $R$-module over $B$} is a $G$-spectrum $M$
over $B$ together with a left action $R\barwedge M\longrightarrow M$ satisfying
the usual associativity and unit conditions. The category $GR\scr{M}_B$ of
left $R$-modules over $B$ consists of the $G$-spectra $M$ over $B$ and
the maps of $G$-spectra over $B$ that preserve the action by $R$.
\end{defn}
Since $(R\barwedge X)_b = R\sma X_b$, a parametrized $R$-module over $B$ is
precisely that: each $X_b$ is an $R$-module $G_b$-spectrum. More
formally, we have the $G$-category $(R\scr{M}_{G,B},GR\scr{M}_B)$, as
discussed in \S\S1.4 and 12.2, and the following result is clear.
\begin{prop}
The $G$-category $(R\scr{M}_{G,B},GR\scr{M}_B)$ is $G$-topologically bicomplete in the sense of \myref{defn:enrichBG}. All of the required limits, colimits,
tensors, and cotensors are constructed in the underlying $G$-category $(\scr{S}_{G,B},G\scr{S}_B)$ and then given induced $R$-module structures in
the evident way. A $\text{cyl}$-cofibration of $R$-modules
is a $\text{cyl}$-cofibration of underlying $G$-spectra
over $B$.
\end{prop}
The last statement holds by the retract of mapping cylinders characterization of $\text{cyl}$-cofibrations. This immediately implies that $GR\scr{M}_B$ inherits a ground structure from $G\scr{S}_B$, in the sense of \myref{back}. Recall that the well-grounded $G$-spectra over $B$ are those that are level well-grounded (well-sectioned and compactly generated) and that the $g$-cofibrations of $G$-spectra over $B$ are the level $h$-cofibrations; see \myref{sillybilly} and \myref{levelwellgr}.
\begin{defn}\mylabel{Rground}
An $R$-module over $B$ is well-grounded if its underlying
$G$-spectrum over $B$ is well-grounded. A map of $R$-modules over $B$
is a $g$-cofibration, level $q$-equivalence, or $s$-equivalence if its
underlying map of $G$-spectra over $B$ is such a map.
\end{defn}
Also recall the notion of a subcategory of well-grounded weak equivalences from \myref{hproper}. Since colimits and tensors for $R$-modules are defined in terms of the underlying $G$-spectra over $B$, the following theorem is immediate from its counterpart for $G$-spectra over $B$, which is given by \myref{levelwellgr} and \myref{piwellgr}.
\begin{thm}
\myref{Rground} specifies a ground structure on $GR\scr{M}_B$
such that the level $q$-equivalences and the $s$-equivalences
both give subcategories of well-grounded weak equivalences.
\end{thm}
Finally, recall the definition of a well-grounded model structure from
\myref{wellmodel}. Such model structures are compactly generated, and we
must define the generators of $GR\scr{M}_B$. The free $R$-module
functor $\mathbb{F}_R=R\barwedge - \colon G\scr{S}_B \longrightarrow GR\scr{M}_B$ is left
adjoint to the forgetful functor $\mathbb{U}\colon GR\scr{M}_B\longrightarrow G\scr{S}_B$.
Adjunction arguments from the definitions show that $\mathbb{F}_R$ preserves
$\text{cyl}$-cofibrations and $\overline{\text{cyl}}$-cofibrations.
\begin{defn}
Define $\mathbb{F}_R FI^f_B$, $\mathbb{F}_R FJ^f_B$ and $\mathbb{F}_R FK^f_B$ by applying the
free $R$-module functor to the maps in the sets specified in \myref{FBJB}
and \myref{Def6}.
A map of $R$-modules over $B$ is
\begin{enumerate}[(i)]
\item a level $qf$-fibration or an $s$-fibration if it is one in $G\scr{S}_B$,
\item an $s$-cofibration if it satisfies the LLP with respect
to the level acyclic $qf$-fibrations,
\end{enumerate}
\end{defn}
\begin{thm}
The category $GR\scr{M}_B$ is a well-grounded model category with respect to the level $q$-equivalences, the level $qf$-fibrations, and the $s$-cofibrations. The sets $\mathbb{F}_R FI^f_B$ and $\mathbb{F}_R FJ^f_B$ give the generating $s$-cofibrations and generating level acyclic $s$-cofibrations. All $s$-cofibrations of $R$-modules over $B$ are $s$-cofibrations of $G$-spectra over $B$.
\end{thm}
We omit the proof since it is virtually the same as the proof of the
following theorem, which gives the starting point for serious work on the homology and cohomology theory of parametrized $G$-spectra.
\begin{thm}
The category $GR\scr{M}_B$ is a well-grounded model category with respect to the $s$-equivalences, the $s$-fibrations, and the $s$-cofibrations; $\mathbb{F}_RFK^f_B$ gives the generating acyclic $s$-cofibrations.
\end{thm}
\begin{proof}
The compatibility condition is automatic by adjunction from the para\-metrized spectrum level, and we have already observed that the free $R$-module functor $\mathbb{F}_R$ preserves $\overline{cyl}$-cofibrations. It also preserves the relevant $\Box$-products, and $\mathbb{F}_R F_VK = (R\sma F_VS^0)\barwedge K$ is well-grounded
if $K$ is a well-grounded ex-space. Only the acyclicity condition remains.
If $R$ is $s$-cofibrant as a ring spectrum, then $R$ is also $s$-cofibrant
as a spectrum, by \cite[III.7.6(iv) and (v)]{MM}. In that case, the functor
$R\barwedge (-)=\mathbb{U}\mathbb{F}_R$ is a Quillen left adjoint by \myref{exttoo} and therefore preserves level acyclic $s$-cofibrations. It follows that the
maps in $\mathbb{F}_RK^f_B$ are $s$-equivalences. The case of a general
well-grounded $R$ reduces to the cofibrant case by use of the next result;
compare \myref{Rinvar} below.
\end{proof}
\begin{prop}\mylabel{barwcof}
The following statements hold.
\begin{enumerate}[(i)]
\item For an $s$-cofibrant spectrum $X$ over $B$, the functor $-\barwedge X\colon G\scr{S} \longrightarrow G\scr{S}_B$ preserves $s$-equivalences between well-grounded spectra in $G\scr{S}$.
\item If $Y$ is well-grounded in $G\scr{S}$, $j\colon A\longrightarrow X$ is an acyclic $s$-cofibration in $G\scr{S}_B$, and $A$ is well-grounded, then $Y\barwedge j\colon Y\barwedge A\longrightarrow Y\barwedge X$ is an $s$-equivalence.
\end{enumerate}
\end{prop}
\begin{proof} Let $\phi\colon Y\longrightarrow Z$ be an $s$-equivalence between
well-grounded spectra. By parts (ii)--(iv) of \myref{hproper}, it
suffices to show that $\phi\barwedge F_V K$ is an $s$-equivalence if
$K$ is the source or target of a map in $I_B^f$. This map is isomorphic
to the map $(\phi\sma F_VS^0)\sma_B K$, where $F_VS^0$ is the shift desuspension
in $G\scr{S}$, not $G\scr{S}_B$. Here $\phi\sma F_VS^0$ is an $s$-equivalence by
the nonparametrized analogue \cite[III.7.3]{MM}, and the
conclusion follows from \myref{gentensor}. (The comment on the notations $\barwedge$ and $\sma_B$ above \myref{lambdas} is relevant: the former
is an external smash product and the latter is a tensor).
For (ii), we apply an argument from \cite[12.5]{MMSS}. We let $Z = X/\!_BA$,
which is $s$-cofibrant, and we let $QY\longrightarrow Y$ be an $s$-cofibrant approximation. Since $j$ is an $s$-cofibration, it is a
$\text{cyl}$-cofibration and $Cj$ is homotopy equivalent to $Z$.
Since $A$ is well-grounded, we can apply the long exact sequence
of homotopy groups of \myref{exact} to conclude that $Z$ is $s$-acyclic.
The map $Z\longrightarrow *_B$ is then an $s$-equivalence between $s$-cofibrant spectra over $B$. Since $QY\barwedge -$ is a Quillen left adjoint, by \myref{Boxcof2}, $QY\barwedge Z\longrightarrow QY\barwedge *_B \cong *_B$ is an $s$-equivalence. Since $QY\barwedge Z \longrightarrow Y\barwedge Z$ is an $s$-equivalence by part (i), $Y\barwedge Z$ is $s$-acyclic. Since the functor $Y\barwedge -$ preserves cofiber
sequences, another application of \myref{exact} shows that $Y\barwedge j$ is an $s$-equivalence.
\end{proof}
\begin{prop}\mylabel{Rinvar}
If $\phi\colon Q\longrightarrow R$ is an $s$-equivalence of well-grounded ring spectra, then the functors
$$\phi_*=R\sma_Q(-)\colon GQ\scr{M}_B\longrightarrow GR\scr{M}_B \ \ \text{and} \ \
\phi^*\colon GR\scr{M}_B\longrightarrow GQ\scr{M}_B $$
given by extension of scalars and restriction of action define a Quillen equivalence $(\phi_*,\phi^*)$ between the categories of
$Q$-modules and of $R$-modules over $B$.
\end{prop}
\begin{proof}
Since $s$-fibrations and $s$-equivalences are created in the underlying
category of spectra over $B$, it is clear that they are preserved by
$\phi^*$, so that we have a Quillen pair. If $M$ is an $s$-cofibrant $Q$-module, then, by the previous result, the unit map
$\phi\sma\text{id}\colon M\cong Q\sma_Q M\longrightarrow \phi^*(R\sma_Q M)$
of the adjunction is an $s$-equivalence of spectra over $B$. Therefore, if $N$ is an $s$-fibrant $R$-module, then a map $M\longrightarrow \phi^*N$ of $Q$-modules is an $s$-equivalence if and only if its adjoint map $R\sma_Q M\longrightarrow N$ of $R$-modules is an $s$-equivalence.
\end{proof}
Implicitly, we have been dealing all along with the case when $R$ is the
sphere spectrum $S$, and we can
mimic all of the model theoretic work that we have done in that case. The
results of \S12.6 and \S13.1 carry over directly. For $f\colon A\longrightarrow B$, base change preserves $R$-modules, $(f_!,f^*)$ gives a Quillen adjoint pair relating the categories of $R$-modules over $A$ and over $B$, and we obtain a Quillen equivalence if $f$ is a $q$-equivalence. If $f$ is a bundle with CW fibres, we obtain a Quillen pair $(f^*,f_*)$, and we can apply the triangulated category version of Brown representability to construct a right adjoint $f_*$ in general.
However, we do not know how to generalize the rest of Chapter 13 to the module context since we have not worked out a theory of excellent $R$-modules with an accompanying excellent $R$-module approximation functor. In view of the retreat to prespectra with their primitive handicrafted smash products in that theory,
it seems unlikely to us that any such construction can be expected.
We also have the notion of a right $R$-module over a nonparametrized ring spectrum $R$. If $M$ and $N$ are right and left $R$-modules over $A$ and
$B$ and $L$ is a left $R$-module over $A\times B$, then we define
spectra $M\barwedge_R N$ over $A\times B$ and $\bar{F}_R(N,L)$ over $A$
by the usual coequalizer
\[\xymatrix{M\barwedge R\barwedge N \ar@<.5ex>[r]\ar@<-.5ex>[r] & M\barwedge N \ar[r]& M\barwedge_R N}\]
and equalizer
\[\xymatrix{\bar{F}_R(N,L)\ar[r] & \bar{F}(N, L) \ar@<.5ex>[r]\ar@<-.5ex>[r] & \bar{F}(R\barwedge N, L).}\]
If $R$ is commmutative, then $M\sma_R N$ and $F_R(N,L)$ are naturally $R$-modules.
We have good homotopical control over these external constructions, as in
Propositions \ref{Boxcof2} and \ref{Boxcof2too}. For example, if we take $A=*$, then we have good homotopical
control over the smash product spectrum $M\sma_R N$ over $B$ and the non-parametrized function spectrum $F_R(N,L)$, where $M$ is a non-parametrized
right $R$-module and $N$ and $L$ are left $R$-modules over $B$. However, if we take $A=B$ and internalize $M\barwedge_R N$ along the diagonal $\Delta\colon B\longrightarrow B\times B$ by setting $M\sma_R N=\Delta^* M\barwedge_R N$ and $F_R(M,N)=\bar{F}_R(M,\Delta_*N)$, we lose homotopical control.
Similarly, when $R$ is commutative, $R\scr{M}_B$ has the structure of a closed symmetric monoidal category, and that allows us to define (commutative) $R$-algebras over $B$ to be (commutative) monoids in $R\scr{M}_B$. However, because of the lack of homotopical control, in the absence of the theory of Chapter 13, we cannot give the categories of $R$-algebras and of commutative $R$-algebras over $B$ model structures.
\begin{rem}
Although we have not pursued the idea, it seems highly likely that there are interesting examples of rings and modules that allow varying base spaces and
are defined in terms of the external smash product.
For example, one might consider $G$-spectra $R_n$ over $B^n$ with products $R_m\barwedge R_n\longrightarrow R_{m+n}$, or one might consider ``globally defined'' parametrized ring spectra $R$ consisting of spectra $R_B$ over $B$ for all $B$ together with appropriate products $R_A\barwedge R_B\longrightarrow R_{A\times B}$. The
$R_B$ would in particular be module spectra over the nonparametrized ring
spectrum $R_*$. As in the nonparametrized theory, one must use the positive stable model structures to study such ring objects model theoretically when
$R_*$ is commutative. The essential point is that the external smash product
is sufficiently well-behaved homotopically that there is no obstacle to
passage from point-set level constructions to homotopy category level conclusions.
\end{rem}
\section{Change of universe}\mylabel{universesec}
Recall that $G$-spectra over $B$ are defined in terms of a chosen collection $\scr{V}$ of representations of $G$. As usual in equivariant stable homotopy theory, we must introduce functors that allow us to change the collection $\scr{V}$. Such functors are usually referred to as ``change of universe'' functors, since $\scr{V}$ is often given as the collection $\scr{V}(U)$ of all representations that embed up to isomorphism in a given $G$-universe $U$. It is however often convenient to restrict $\scr{V}$ to be a cofinal subcollection of $\scr{V}(U)$ that is closed under direct sums, and when we dealt with excellent prespectra it became essential to restrict $\scr{V}$ further to a countable cofinal sequence of expanding representations in $U$. In both cases it is usual to insist that the trivial representation $\mathbb{R}$ is included in $\scr{V}$. In order to deal with the change functors in all of the above cases at once, we adopt a slightly different approach from the one that was used in \cite[V.\S1]{MM}. We then explain how it specializes to the more explicit approach given there.
Let $G\scr{S}_B^{\scr{V}}$ denote the category of $G$-spectra over $B$ indexed on $\scr{V}$. If $\scr{V}$ is not closed under direct sums, then we are thinking of $G\scr{S}_B^\scr{V}$ as the restriction of the diagram category corresponding to $G\scr{S}_B^{\scr{V}'}$,
where $\scr{V}'$ is the closure of $\scr{V}$ under sums, as discussed in \myref{indexingreps}.
Let $i\colon \scr{V}\subset \scr{V}'$ be the inclusion of one collection of representations in another. Thinking of parametrized spectra as
diagram ex-spaces, we see that the evident forgetful functor
\[i^*\colon G\scr{S}^{\scr{V}'} \longrightarrow G\scr{S}^\scr{V}\]
has a left adjoint $i_*$ given by the prolongation, or expansion of universe, functor
\[(i_*X)(V') = \scr{J}^{\scr{V}'}_G(-,V') \otimes_{\scr{J}^\scr{V}_G} X.\]
Such prolongation functors are discussed in detail in \cite[I\S3]{MMSS}
and \cite[I\S2]{MM}. By \cite[I.2.4]{MM}, the unit $\text{Id}\longrightarrow i^*i_*$ of
the adjunction is a natural isomorphism.
We have more concrete descriptions of the functor $i_*$ when $\scr{V}$ consists of a cofinal sequence of representations in some universe $U$. Recall that $\scr{J}_G^\scr{V}(V,V)$ is the orthogonal group $O(V)$ with a disjoint base point.
\begin{lem}\mylabel{describei*}
If $\scr{V}=\{V_i\}\subset \scr{V}'$ is a countable expanding sequence in some $G$-universe $U$, then
\[(i_*X)(V')\cong \scr{J}^{\scr{V}'}_G(V_i,V')\sma_{O(V_i)} X(V_i)\]
where $i$ is the largest natural number such that there is a linear isometry $V_i\longrightarrow V'$.
\end{lem}
\begin{proof}
The forgetful functor $i^*$ is restriction along a functor $\iota\colon
\scr{J}_G^{\scr{V}}\longrightarrow \scr{J}_G^{\scr{V}'}$ and $(i_*X)(V')$ is constructed as the
coequalizer of the pair of parallel maps
\[\xymatrix{\bigvee_{j,k} \scr{J}^{\scr{V}'}_G(V_j,V')\sma_B
\scr{J}^\scr{V}_G(V_k,V_j)\sma_B X(V_k) \ar@<.5ex>[r]\ar@<-.5ex>[r] &
\bigvee_j \scr{J}^{\scr{V}'}_G(V_j,V')\sma_B X(V_j)}\]
given by composition in $\scr{J}_G^{\scr{V}'}$ and by the evaluation maps
associated to the diagram $X$. A cofinality argument that is easily
made precise by use of the explicit description of the category
$\scr{J}^{\scr{V}'}_G$ given in \cite[II.\S4]{MM} shows that the above
coequalizer agrees with the coequalizer of the subdiagram
\[\xymatrix{\scr{J}^{\scr{V}'}_G(V_i,V')\sma_B \scr{J}^\scr{V}_G(V_i,V_i)\sma_B X(V_i)
\ar@<.5ex>[r]\ar@<-.5ex>[r] & \scr{J}^{\scr{V}'}_G(V_i,V')\sma_B X(V_i).}\]
This coequalizer is the space that we have denoted by $\scr{J}^{\scr{V}'}_G(V_i,V')\sma_{O(V_i)} X(V_i)$.
\end{proof}
\begin{rem}
The argument above works in the same way for prespectra. It gives
the conclusion that, for parametrized prespectra $X$ in $G\scr{P}_B^\scr{V}$,
\[(i_*X)(V)\cong \Sigma^{V-V_i}_B X(V_i).\]
\end{rem}
\begin{rem}\mylabel{uniMM}
Assume that $\scr{V}$ and $\scr{V}'$ are closed under finite sums and contain the trivial representation. We can then define the change of universe functors
\[I^\scr{V}_{\scr{V}'}=i_*i'^* \colon G\scr{S}_B^{\scr{V}'} \longrightarrow G\scr{S}_B^\scr{V}\]
where $i\colon \{\mathbb{R}^n\} \subset \scr{V}$ and $i'\colon \{\mathbb{R}^n\} \subset \scr{V}'$. Explicitly
\[(I^\scr{V}_{\scr{V}'} X)(V)\cong \scr{J}_G^\scr{V}(\mathbb{R}^n,V)\sma_{O(n)} X(\mathbb{R}^n).\]
This is the definition given in \cite[V.1.2]{MM}. These change of universe functors $I^\scr{V}_{\scr{V}'}$ are exceptionally well behaved on the point set level
and agree with those we are using when $\scr{V}\subset \scr{V}'$. They are symmetric monoidal equivalences of categories. For collections $\scr{V}$, $\scr{V}'$ and $\scr{V}''$, they satisfy
\[I_{\scr{V}'}^{\scr{V}}\com \Sigma^{\scr{V}'}_B \cong \Sigma^{\scr{V}}_B,\qquad
I_{\scr{V}'}^{\scr{V}} \com I_{\scr{V}''}^{\scr{V}'} \cong I_{\scr{V}''}^{\scr{V}},\qquad
I_{\scr{V}}^{\scr{V}} \cong \text{Id}.\]
Moreover, $I^\scr{V}_{\scr{V}'}$ is continuous and commutes with smash products with
ex-spaces. In particular, it is homotopy preserving and therefore induces equivalences of the classical homotopy categories. Unfortunately, however, the functors $I^\scr{V}_{\scr{V}'}$ are as poorly behaved on the homotopy level as they are
well behaved on the point set level. They do not preserve either level $q$-equivalences or $s$-equivalences in general and the point set level relations above do not descend to the model theoretic homotopy categories that we are interested in. Furthermore, these functors $I^\scr{V}_{\scr{V}'}$ do not exist if $\scr{V}$ is a cofinal expanding sequence. We shall therefore not make much use of them.
\end{rem}
Returning to our full generality, let $i\colon \scr{V}\subset \scr{V}'$. The adjoint pair $(i_*,i^*)$ has good homotopical properties.
\begin{thm}\mylabel{change1}
Let $i\colon \scr{V}\subset \scr{V}'$. Then $i^*$ preserves level $q$-equivalences, level $qf$-fibrations, $s$-fibrations, and $s$-acyclic $s$-fibrations. Therefore $(i_*,i^*)$ is a Quillen adjoint pair in the level $qf$-model structure and in the $s$-model structure. Moreover, $i_*$ on homotopy categories is symmetric monoidal. If $\scr{V}$ is cofinal in $\scr{V}'$, then $i^*$ creates the weak equivalences and $(i_*,i^*)$ is a Quillen equivalence.
\end{thm}
\begin{proof} It is clear from its levelwise definition that $i^*$
preserves level $q$-equi\-va\-lences and level $qf$-fibrations. It follows
that its left adjoint $i_*$ preserves $s$-cofibrations and level acyclic
$s$-cofibrations. This in turn implies that $i^*$ preserves $s$-acyclic
$s$-fibrations, since those are the maps that satisfy the RLP with respect
to the $s$-cofibrations. The levelwise description of $s$-fibrations in
\myref{RLPL} implies that $i^*$ preserves $s$-fibrations. The last statement follows from the definition of homotopy groups and the fact that the unit $\text{id}\longrightarrow i^*i_*$ is an isomorphism. The functor $i_*$
commutes with $\barwedge$ on the point set level, by \cite[I.2.14]{MM},
and this commutation relation descends directly to homotopy categories.
Applying \myref{chvschuni} below to the diagonal map of $B$, it follows
that the derived functor $i_*$ is symmetric monoidal.
\end{proof}
We have constructed the change of universe functors on both the spectrum and prespectrum level and they are compatible with the restriction functors $\mathbb{U}$. However, in order to make use of excellent parametrized prespectra, we must restrict to parametrized prespectra indexed on cofinal sequencess $j\colon \scr{W}\subset \scr{V}$ and $j'\colon \scr{W}'\subset \scr{V}'$ of indexing representations in the given universes $U\subset U'$. But then there need not be an induced inclusion $i\colon \scr{W}\subset \scr{W}'$. We therefore also define change of
universe functors for prespectra indexed on cofinal sequences.
\begin{defn} Let $i\colon \scr{V}\subset \scr{V}'$ and choose cofinal sequences $\scr{W}=\{V_i\}$ and $\scr{W}'=\{V_i'\}$ in $\scr{V}$ and $\scr{V}'$ such that $V_{i+1}=V_i\oplus W_i$ and $V_i'=V_i\oplus Z_i$, where
$Z_{i+1}= Z_{i}\oplus W_i'$ and thus $V_{i+1}' = V_i'\oplus W_i\oplus W_i'$. Define a pair of adjoint functors
\[\xymatrix{G\scr{P}_B^\scr{W} \ar@<.5ex>[r]^{\bar\imath_*} & G\scr{P}_B^{\scr{W}'} \ar@<.5ex>[l]^{\bar\imath^*}}\]
by setting
\[(\bar\imath_*X)(V_i')=\Sigma_B^{Z_i}X(V_i)\qquad\text{and}\qquad
(\bar\imath^*Y)(V_i)=\Omega_B^{Z_i}Y(V_i').\]
The structure maps are induced from the given structure maps in the evident way.
\end{defn}
\begin{prop}\mylabel{chunicomp}
The pair $(\bar\imath_*, \bar\imath^*)$ is a Quillen adjoint pair with respect to both the level $qf$-model structure and the stable model structure.
The following diagram commutes when the vertical arrows point in the same direction.
\[\xymatrix{\text{Ho}\, G\scr{P}^\scr{W}_B \ar@<.5ex>[d]^{\bar\imath_*}
& \text{Ho}\, G\scr{P}^\scr{V}_B \ar@<.5ex>[d]^{i_*}\ar[l]_-{j^*}\\
\text{Ho}\, G\scr{P}_B^{\scr{W}'} \ar@<.5ex>[u]^{\bar\imath^*}
& \text{Ho}\, G\scr{P}_B^{\scr{V}'} \ar@<.5ex>[u]^{i^*}\ar[l]^-{(j')^*}}\]
\end{prop}
\begin{proof}
This is clearly a Quillen adjunction in the level $qf$-model structure, and to show that it is a Quillen adjunction in the stable model structure it therefore suffices to verify the condition of \myref{RLPL}. The homotopy pullback \ref{OMpb} associated to the pair $(V_i,W_i)$ and an $s$-fibration $f\colon X\longrightarrow Y$ is still a homotopy pullback after we apply $\Omega_B^{Z_i}$ to it and displays the required diagram \ref{OMpb} for the map $\bar\imath^*f$. We have that
\[(\bar\imath_*j^*X)(V_i')=\Sigma_B^{Z_i}X(V_i)
\cong \Sigma_B^{V_i'-V_i}X(V_i)=((j')^*i_*X)(V_i')\]
and this point set level isomorphism descends to homotopy categories since the functors $j^*$ and $(j')^*$ preserve all $s$-equivalences. The adjoint structure maps of $X\in G\scr{P}_B^{\scr{V}'}$ induce maps
\[(j^*i^*X)(V_i) = X(V_i)\longrightarrow \Omega_B^{Z_i}X(V_i') = (\bar\imath^*(j')^*X)(V_i).\]
When $X$ is $s$-fibrant, its adjoint structure maps are level $q$-equivalences, and we thus obtain an equivalence $j^*i^*\simeq \bar\imath^*(j')^*$ on homotopy categories.
\end{proof}
On the point-set level, we have the following commutation relations
between change of universe functors and change of base functors.
\begin{lem}\mylabel{pointsetchchuni}
Let $i\colon \scr{V}\subset \scr{V}'$ and let $f\colon A\longrightarrow B$ be a $G$-map. Then $i^*$ commutes up to natural isomorphism with the change of base functors $f_!$, $f^*$, and $f_*$, and $i_*$ commutes up to natural isomorphism with $f_!$ and $f_*$.
\end{lem}
\begin{proof}
The first statement is clear from the levelwise constructions of the base change functors, and the second statement follows by conjugation since $i_*$, $f_!$,
and $f^*$ are left adjoints of $i^*$, $f^*$, and $f_*$.
\end{proof}
The missing relation, $i_*f^*\iso f^*i_*$, would hold with the alternative
point-set level definitions of \myref{uniMM}, where $i^*$ and $i_*$ are inverse equivalences. However, these are point-set level relationships that need not descend to model theoretic homotopy categories. With our preferred definition
of $i_*$ in terms of prolongation, the
following result shows that $i_*f^*\htp f^*i_*$ on homotopy
categories even though we need not have an isomorphism on the
point-set level.
\begin{prop}\mylabel{chvschuni}
Let $i\colon \scr{V}\subset \scr{V}'$ and let $f\colon A\longrightarrow B$ be a $G$-map. Then there are natural equivalences of derived functors
\[i^*f^* \simeq f^*i^*,\quad
i_*f_! \simeq f_!i_*,\quad
i_*f^* \simeq f^*i_*,\quad
i^*f_* \simeq f_*i^*,\quad
i^*f_! \simeq f_!i^*\]
in the relevant homotopy categories.
\end{prop}
\begin{proof}
The first two equivalences are clear since we are commuting Quillen right adjoints and their corresponding Quillen left adjoints. The fourth will follow by adjunction from the third. If $f$ is a homotopy equivalence, then $f^* \htp (f_!)^{-1}$ and in this case the third follows from the second and the fifth from the first. Factoring $f$ as the composite of an $h$-fibration and a homotopy equivalence, we see that the third will hold in general if it holds when $f$ is an $h$-fibration. Similarly, factoring $f$ as the composite of an $h$-cofibration and a homotopy equivalence, we see that the fifth will hold in general if it holds when $f$ is an $h$-cofibration.
Further, for the third equivalence, it suffices to show that $\bar\imath_*f^*\simeq f^*\bar\imath_*$ since \myref{chunicomp} then gives that
\[i_*f^*\simeq i_*j_*j^*f^*\simeq (j')_*\bar\imath_*f^*j^*
\simeq (j')_*f^*\bar\imath^*j^*\simeq f^*(j')_*(j')^*i^*\simeq f^*i^*.\]
Similarly, for the fifth equivalence, it suffices to show that $\bar\imath^*f_!\simeq f_!\bar\imath^*$, for then
\[i^*f_!\simeq i^*(j')_*(j')^*f_!\simeq j_*\bar\imath^*f_!(j')^*
\simeq j_*f_!\bar\imath^*(j')^*\simeq f_!(j')_*(j')^*i^*\simeq f_!i^*.\]
We have reduced the proof of the third equivalence to the situation when $f$ is an $h$-fibration and $i_*$ is replaced by $\bar\imath_*$. The functor $f^*$ preserves excellent prespectra over $B$, but we must apply $T$ to $\bar\imath_*$ before passing to homotopy categories. As in the proof of \myref{Mackeymore}, since $f$ is assumed to be an $h$-fibration
we have a natural homotopy equivalence $\mu\colon Tf^*\longrightarrow f^*T$
in our categories indexed on $\scr{W}$ or on $\scr{W}'$. Therefore
\[T\bar\imath_*f^*\cong Tf^*\bar\imath_*\htp f^*T\bar\imath_*.\]
Similarly, we have reduced the proof of the fifth equivalence to the situation when $f$ is an $h$-cofibration and $i^*$ is replaced by $\bar\imath^*$. Then $f_!$ preserves level $h$-equivalences, and so does $\bar\imath^*$ since it preserves level $q$-equivalences and preserves objects whose total spaces are of the homotopy types of $G$-CW complexes. Since $T$ takes zig-zags of level $h$-equivalences to homotopy equivalences,
\[\xymatrix{Tf_!T\bar\imath^* \ar@{<->}[r]^-{\htp} & Tf_!\bar\imath^*\cong T\bar\imath^*f_! \ar@{<->}[r]^-{\htp} & T\bar\imath^*Tf_!}\]
displays a zig-zag of homotopy equivalences showing that $f_!\bar\imath^*\simeq \bar\imath^*f_!$.
\end{proof}
\section{Restriction to subgroups}
Let $\theta\colon G'\longrightarrow G$ be a homomorphism and let $\theta^*\scr{V}$ be the collection of $G'$-representations $\theta^*V$ for $V\in \scr{V}$, where $\scr{V}$
is our chosen collection of indexing $G$-representations. We have
implicitly used the following result in our earlier results on change
of groups.
\begin{prop}\mylabel{grprestrrQa}
The functor $\theta^*\colon G\scr{S}_B \longrightarrow G'\scr{S}^{\theta^*\scr{V}}_{\theta^*B}$ preserves level $q$-equi\-va\-lences, level $qf$-fi\-bra\-tions, $s$-fi\-bra\-tions, and $s$-equivalences provided that the model structures are defined with respect
to generating sets $\scr{C}_{G}$ and $\scr{C}_{G'}$ of $G$-cell complexes and $G'$-cell complexes such that $\theta_!C = G\times_{G'} C \in \scr{C}_G$ for $C\in\scr{C}_{G'}$.
\end{prop}
\begin{proof} Since $(\theta^*A)^H=A^{\theta^*(H)}$ for a $G$-space $A$ and
a subgroup $H$ of $G'$, this is clear from the definitions of homotopy
groups and from the characterizations of fibrations given in \myref{qffibdef}
and \myref{RLPL}. Note in particular that $\theta^*$ preserves the level
$qf$-fibrant approximations that are used in the definition of the stable
homotopy groups.
\end{proof}
For the remainder of this section fix a subgroup $H$ of $G$ and consider the inclusion $\iota\colon H\subset G$. For an $H$-space $A$, we simplify notation by letting $H\scr{S}_A^{\scr{V}}$ denote the category of $H$-spectra over $A$ indexed on $\iota^*\scr{V}$. Clearly, we then have the restriction of action functor
$$\iota^*\colon G\scr{S}_B^{\scr{V}}\longrightarrow H\scr{S}_{\iota^*B}^{\scr{V}}.$$
For $i\colon \scr{V}\subset \scr{V}'$, we have $\iota^*i^* = i^*\iota^*$ since with
either composite we are just restricting from the representations in $\scr{V}'$
to the representations in $\scr{V}$ and viewing all $G$-spaces in sight as $H$-spaces.
When $\scr{V} = \scr{V}(U)$ for a $G$-universe $U$, there is a quibble here (as was
discussed in \cite[V.10]{MM}). We are using $\iota^*\scr{V}$ as the corresponding indexing collection for $H$. However, if $V$ is an irreducible representation
of $G$, $\iota^*V$ is generally not an irreducible representation of $H$ and we should expand $\iota^*\scr{V}$ to include all representations that
embed up to isomorphism in $\iota^*U$ to fit the definitions into our usual
framework. However, there is a change of universe functor associated to
the inclusion $i\colon \iota^*\scr{V}(U)\subset \scr{V}(\iota^*U)$ that fixes this. The
functor $i^*$ preserves all $s$-equivalences and descends to an equivalence
on homotopy categories. We can and should use these rectifications when restricting to $H$-spectra over $\iota^*B$ for a fixed chosen $H$.
\begin{rem} Consider passage to fibers and recall \myref{FibadQtoo}.
\begin{enumerate}[(i)]
\item Applied to inclusions of orbits, \myref{chvschuni}
implies that the functors $i^*$ for $i\colon \scr{V}\subset \scr{V}'$ are
compatible with passage to fibers, in the sense that
$$(i^*X)_b \iso i^*(X_b)\ \ \text{for}\ \ b\in B,$$
where $i^*$ on the right is the change of universe functor on
$G_b$-spectra.
\item When $\scr{V} = \scr{V}(U)$, we can view the fiber functor
\[(-)_b\colon G\scr{S}_B \longrightarrow G_b\scr{S}\]
as landing in spectra indexed on $\scr{V}(\iota^* U)$, $\iota\colon G_b\longrightarrow G$ by composing with $i_*$ for $i\colon \iota^*\scr{V}(U)\subset \scr{V}(\iota^*U)$. However, these change of universe functors must be used with caution since they are not compatible as $b$ and therefore $G_b$ vary.
\end{enumerate}
\end{rem}
Recall from Propositions \ref{Lchanges2} and \ref{Symmoni} that the equivalence
of categories $(\iota_!, \nu^*\iota^*)$ between $H\scr{S}_A$ and $G\scr{S}_{\iota_!A}$
induces a closed symmetric monoidal equivalence of categories between
$\text{Ho}H\scr{S}_A$ and $\text{Ho}G\scr{S}_{\iota_!A}$. By \myref{LishriekCor2},
we can interpret the restriction
functor $\iota^*\colon \text{Ho}G\scr{S}_B\longrightarrow \text{Ho}H\scr{S}_{\iota^*B}$ as the
composite of base change $\mu^*$ along $\mu\colon \iota_!\iota^*B\longrightarrow B$ and this
equivalence applied to $A = \iota^*B$. The following spectrum level analogue of \myref{changerel} gives compatibility relations between change of
base functors and these results on change of groups.
\begin{prop}\mylabel{substitute}
Let $f\colon A\longrightarrow \iota^*B$ be a map of $H$-spaces and $\tilde{f}\colon
\iota_! A\longrightarrow B$ be its adjoint map of $G$-spaces. Then the following
diagrams commute up to natural isomorphism, where
$\mu\colon \iota_!\iota^*B\longrightarrow B$ and $\nu\colon A\longrightarrow \iota^*\iota_!A$
are the counit and unit of the adjunction $(\iota_!,\iota^*)$.
\[\xymatrix{
G\scr{S}_{\iota_!A} \ar[r]^-{\tilde{f}_!} & G\scr{S}_B \\
H\scr{S}_{A} \ar[r]_-{f_!}\ar[u]^{\iota_!} &
H\scr{S}_{\iota^*B}\ar[u]_{\mu_!\com\iota_!}}
\quad \ \
\xymatrix{
G\scr{S}_B \ar[r]^-{\tilde{f}^*} \ar[d]_{\iota^*}
& G\scr{S}_{\iota_!A} \ar[d]^{{\nu}^*\com\iota^*}\\
H\scr{S}_{\iota^*B} \ar[r]_-{f^*} & H\scr{S}_{A}}\]
These diagrams descend to natural equivalences of composites of
derived functors on homotopy categories.
\end{prop}
\begin{proof}
The point set level diagrams commute by \myref{changerel}, applied
levelwise. The left diagram is one of Quillen left adjoints and the
right diagram is one of Quillen right adjoints, by Propositions
\ref{Qad1too} and \ref{Lchanges2} and \myref{LishriekCor2}.
\end{proof}
We now define a parametrized fixed point functor associated to the inclusion $\iota\colon H\longrightarrow G$. Its target is a category of nonequivariant parametrized spectra. In the next section we will consider a fixed point functor that takes values in a category of parametrized $WH$-spectra, where $WH=NH/H$ is the Weyl group.
Write $G\scr{S}^{\text{triv}}_B$ for $G$-spectra over $B$ indexed on trivial representations. These are ``naive'' parametrized $G$-spectra. As usual, to define fixed point spectra, we must change to the trivial universe before taking fixed points levelwise. Thus let $\scr{V}^G = \{V^G \mid V\in \scr{V}\}$. It is contained in $\scr{V}$ if $\scr{V}=\scr{V}(U)$ for some universe $U$.
\begin{defn}
The \emph{$G$-fixed point functor} $(-)^G\colon G\scr{S}_B\longrightarrow \scr{S}_{B^G}$ is the composite of $i^*$, $i\colon \scr{V}^G\subset \scr{V}$, and levelwise passage to fixed points. For a subgroup $H$ of $G$ the \emph{$H$-fixed point functor}
$(-)^H\colon G\scr{S}_B\longrightarrow \scr{S}_{B^H}$ is the composite of $\iota^*$, $\iota\colon H\subset G$, and $(-)^H$.
\end{defn}
Since the homotopy groups of a level $qf$-fibrant $G$-spectrum $X$ over $B$ are the homotopy groups $\pi_q^H(X_b)$, we see from the nonparametrized analogue \cite[V.3.2]{MM} that these are then the homotopy groups of $X^H$. Recall
in particular that the $s$-fibrant $G$-spectra over $B$ are the $\Omega$-$G$-spectra over $B$, which are level $qf$-fibrant. Therefore, for all subgroups
$H$ of $G$, the homotopy groups of a parametrized $G$-spectrum $X$ are the nonequivariant homotopy groups of the nonequivariant spectra $X^H$, provided
that $(-)^H$ is understood to mean the derived fixed point functor.
On the point-set level, the functor $(-)^G$ is a right adjoint. Thinking of the homomorphism $\varepsilon\colon G\longrightarrow e$ to the trivial group, let $\varepsilon^*\colon \scr{S}_A\longrightarrow G\scr{S}^{\text{triv}}_{\varepsilon^*A}$ be the functor that sends spectra
over a space $A$ to $G$-trivial $G$-spectra over $A$ regarded as a $G$-trivial $G$-space. The following result is immediate by passage to fibers from its nonparametrized special case \cite[V.3.4]{MM}. Let $\scr{A}\!\ell\ell$ denote
the collection of all representations of $G$.
\begin{prop}\mylabel{fixad}
Let $A$ be a space. Let $Y$ be a naive $G$-spectrum over $\varepsilon^*A$ and $X$ be a spectrum over $A$. There is a natural isomorphism
$$G\scr{S}^{\text{triv}}_{\varepsilon^*A}(\varepsilon^*X, Y)\iso \scr{S}_A(X, Y^G).$$
For (genuine) $G$-spectra $Y$ over $\varepsilon^*A$, there is a natural isomorphism
$$G\scr{S}_{\varepsilon^*A}(i_*\varepsilon^*X,Y)\iso \scr{S}_A(X,(i^*Y)^G),$$
where $i\colon \text{triv}\subset \scr{A}\!\ell\ell$.
Both of these adjunctions are given by Quillen adjoint pairs relating the respective level and stable model structures.
\end{prop}
Returning to $G$-spaces $B$ and comparing \myref{FVs} with the proof of \cite[V.3.5-3.6]{MM}, we obtain the following curious results.
\begin{prop}\mylabel{fixcof}
For a representation $V$ and an ex-$G$-space $K$, we have that $(F_V K)^G = *_{B^G}$ unless $G$ acts trivially on $V$, when $(F_VK)^G \iso F_V(K^G)$ as a nonequivariant spectrum over $B^G$. The functor $(-)^G$ preserves $s$-cofibrations, but it does not preserve acyclic $s$-cofibrations.
\end{prop}
\begin{cor}\mylabel{fixsusp}
For ex-$G$-spaces $K$,
$$(\Sigma^{\infty}_BK)^G\iso \Sigma^{\infty}_B(K^G).$$
\end{cor}
This isomorphism of spectra over $B^G$ does \emph{not} descend to the homotopy category $\text{Ho}\, G\scr{S}_{B^G}$. The reader is warned to consult \cite[V\S3]{MM} for the meaning of these results. There is also an analogue of the comparison between $G$-fixed points and smash products in \cite[V.3.8]{MM}, but only when $B=B^G$ and only with good behavior with respect to cofibrant objects when external smash products are used. We shall not state the result formally.
\section{Normal subgroups and quotient groups}
We now turn to quotient homomorphisms and associated orbit and fixed point functors. The material of this section generalizes a number of results from \S2.4, \S7.3, and \S9.5 to the level of parametrized spectra.
Just as we have been using $\iota$ generically for inclusions of subgroups, we shall use $\varepsilon$ generically for quotient homomorphisms. In particular, for an inclusion $\iota\colon H\subset G$, we let $WH = NH/H$, where $NH$ is the normalizer of $H$ in $G$, and we have the quotient homomorphism $\varepsilon\colon NH\longrightarrow WH$. We can study this situation by first restricting from $G$ to $NH$, thus changing the ambient group. Therefore, there is no loss of generality if we focus attention on a normal subgroup $N$ of $G$ with quotient group $J=G/N$, as we do throughout this section.
\begin{defn}\mylabel{defnchN}
Let $G\scr{S}^{\text{$N$-triv}}_B$ be the category of $G$-spectra over $B$ indexed on the $N$-trivial representations of $G$. Regard representations of $J$ as $N$-trivial representations of $G$ by pullback along $\varepsilon\colon G\longrightarrow J$.
For a $J$-space $A$, define
\[\varepsilon^*\colon J\scr{S}_A\longrightarrow G\scr{S}^{\text{$N$-triv}}_{\varepsilon^*A}\]
levelwise by regarding ex-$J$-spaces over $A$ as $N$-trivial $G$-spaces over $\varepsilon^*A$. For a $G$-space $B$, define
\[(-)/N\colon G\scr{S}^{\text{$N$-triv}}_B \longrightarrow J\scr{S}_{B/N}
\qquad\text{and}\qquad
(-)^N\colon G\scr{S}^{\text{$N$-triv}}_B \longrightarrow J\scr{S}_{B^N}\]
by levelwise passage to orbits over $N$ and to $N$-fixed points.
\end{defn}
\begin{lem}\mylabel{fixedptrQa}
The $N$-fixed point functor $(-)^N$ preserves level $q$-equivalences, level $qf$-fibrations, $s$-fibrations, and $s$-equivalences, provided that the model structures are defined with respect to generating sets $\scr{C}_{G}$ and $\scr{C}_{J}$ of $G$-cell complexes and $J$-cell complexes such that $C/N \in \scr{C}_J$ for $C\in\scr{C}_{G}$.
\end{lem}
\begin{proof}
This is a special case of \myref{grprestrrQa}; it also follows directly from the ex-space level analogue in \myref{fixedptrQa0}, the characterization of $s$-fibrations in \myref{RLPL}, and inspection of the definition of the $s$-equivalences.
\end{proof}
\begin{prop}\mylabel{factor} Let $j\colon B^N\longrightarrow B$ be the inclusion
and $p\colon B\longrightarrow B/N$ be the quotient map. Then the following
factorization diagrams commute
\[\xymatrix{G\scr{S}^{\text{$N$-triv}}_B \ar[d]_{p_!}\ar[r]^{(-)/N}
& J\scr{S}_{B/N} \\
G\scr{S}^{\text{$N$-triv}}_{B/N} \ar[ur]_{(-)/N} }
\qquad\text{and}\qquad
\xymatrix{G\scr{S}^{\text{$N$-triv}}_B \ar[d]_{j^*}\ar[r]^{(-)^N}
& J\scr{S}_{B^N} \\
G\scr{S}^{\text{$N$-triv}}_{B^N} \ar[ur]_{(-)^N}}\]
and they descend to natural equivalences on homotopy categories
\[(p_!X)/N \simeq X/N \qquad\text{and}\qquad (j^*X)^N\simeq X^N\]
for $X$ in $\text{Ho}\, G\scr{S}_B^{\text{$N$-triv}}$.
The following adjunction isomorphisms follow.
\begin{enumerate}[(i)]
\item For $Y\in G\scr{S}^{\text{$N$-triv}}_B$ and $X\in J\scr{S}_{B/N}$,
$$J\scr{S}_{B/N}(Y/N,X)\iso G\scr{S}^{\text{$N$-triv}}_B(Y,p^*\varepsilon^*X).$$
\item For $Y\in G\scr{S}^{\text{$N$-triv}}_B$ and $X\in J\scr{S}_{B^N}$,
$$G\scr{S}^{\text{$N$-triv}}_B(j_!\varepsilon^*X,Y)\iso J\scr{S}_{B^N}(X,Y^N).$$
\item For (genuine) $G$-spectra $Y\in G\scr{S}_B$ and $X\in J\scr{S}_{B^N}$,
$$G\scr{S}_B(i_*j_!\varepsilon^*X,Y)\iso J\scr{S}_{B^N}(X,(i^*Y)^N),$$
where $i\colon\text{triv}\subset\scr{A}\!\ell\ell$.
\end{enumerate}
All of these adjunctions are Quillen adjoint pairs with respect to both
the level and the stable model structures and so descend to homotopy categories.
\end{prop}
\begin{proof}
The factorizations follow from the ex-space level analogue \myref{factor0}. The statement about Quillen adjunctions holds since $(-)^N$, $\epsilon^*$ and $i^*$ preserve level $q$-equivalences, level fibrations, $s$-equivalences and level $s$-fibrations, by \myref{fixedptrQa}, \myref{grprestrrQa} and \myref{change1}.
\end{proof}
The behavior of the orbit and fixed point functors with respect to base change is recorded in the following result.
\begin{prop}\mylabel{chvsfixorbit}
Let $f\colon A\longrightarrow B$ be a map of $G$-spaces. Then the following diagrams commute up to natural isomorphism
{\small\[\xymatrix{
G\scr{S}_A^{\text{$N$-triv}} \ar[r]^-{f_!} \ar[d]_{(-)/N}
& G\scr{S}^{\text{$N$-triv}}_B \ar[d]^{(-)/N}\\
J\scr{S}_{A/N} \ar[r]_-{(f/N)_!} & J\scr{S}_{B/N}}
\;\;
\xymatrix{
G\scr{S}^{\text{$N$-triv}}_B \ar[r]^-{f^*} \ar[d]_{(-)^N}
& G\scr{S}^{\text{$N$-triv}}_A \ar[d]^{(-)^N}\\
J\scr{S}_{B^N} \ar[r]_-{(f^N)^*} & J\scr{S}_{A^N}}
\;\;
\xymatrix{
G\scr{S}_{A}^{\text{$N$-triv}} \ar[r]^-{f_!} \ar[d]_{(-)^N}
& G\scr{S}^{\text{$N$-triv}}_B \ar[d]^{(-)^N}\\
J\scr{S}_{A^N} \ar[r]_-{(f^N)_!} & J\scr{S}_{B^N}}\]}
and they descend to the following natural equivalences on homotopy categories
\[(f_!X)/N \simeq (f/N)_! (X/N),
\quad
(f^*X)^N \simeq (f^N)^*(Y^N),
\quad
(f_!X)^N \simeq (f^N)_!(X/N)\]
for $X\in\text{Ho}\, G\scr{S}^\text{$N$-triv}_A$ and $Y\in \text{Ho}\, G\scr{S}^\text{$N$-triv}_B$.
\end{prop}
\begin{proof}
The first statement follows levelwise from the ex-space level analogue \myref{fixorbbase}. The proof that it descends to equivalences on homotopy categories is the same as for the ex-space level analogue \myref{orbfixdescend}.
\end{proof}
Specializing to $N$-free $G$-spaces, we obtain a factorization result that is analogous to those in \myref{factor}, but is less obvious. It is a precursor of the Adams isomorphism.
\begin{prop}\mylabel{ouch}
Let $E$ be an $N$-free $G$-space, let $B = E/N$, and let $p\colon E\longrightarrow B$ be the quotient map. Then the diagram
\[\xymatrix{
G\scr{S}_E^{\text{$N$-triv}} \ar[d]_{p_*} \ar[r]^-{(-)/N} & J\scr{S}_B\\
G\scr{S}_B^{\text{$N$-triv}} \ar[ur]_-{(-)^N}}\]
commutes up to a natural isomorphism, and it descends to a natural equivalence
$$X/N\simeq (p_*X)^N$$
in $G\scr{S}_E^{\text{$N$-triv}}$ for $X\in \text{Ho}\, J\scr{S}_B$. Therefore the left adjoint $(-)/N$ of the functor $p^*\varepsilon^*$ is also its right adjoint.
\end{prop}
\begin{proof}
The point set level result follows levelwise from the ex-space level result \myref{ouch0}. Since it is an isomorphism between a Quillen left adjoint on the left hand side and a composite of Quillen right adjoints on the right hand side, it descends directly to homotopy categories.
\end{proof}
\part{Duality, transfer, and base change isomorphisms}
\chapter{Fiberwise duality and transfer maps}
\section*{Introduction}
We put the foundations of Part III to use in the two chapters of this last part. Unless otherwise stated, we work in the derived homotopy categories, and all functors should be understood in the derived sense. For example, we have the derived fiber functor
$$(-)_b\colon \text{Ho}\, G\scr{S}_B \longrightarrow \text{Ho}\, G_b\scr{S}.$$
Since passage to fibers is a Quillen right adjoint, this means that we
replace $G$-spectra $X$ over $B$ by $s$-fibrant approximations before taking
point-set level fibers. For emphasis, and to make the notation $X_b$ clear
and unambiguous, we may assume that $X$ is $s$-fibrant, but there is no loss
of generality. A map $f$ in $\text{Ho}\, G\scr{S}_B$ is an equivalence if and only if
$f_b$ is an equivalence for all $b\in B$, and that allows us to transer
information back and forth between the parametrized and unparametrized
homotopy categories with impunity. Here we use the word ``equivalence''
to mean an isomorphism in $\text{Ho}\, G\scr{S}_B$, and we use the notation $\simeq$ for this relation. We reserve the symbol $\cong$ to mean an isomorphism on the
point set level.
We have proven that the basic structure enjoyed by the category $G\scr{S}_B$ of parametrized spectra descends coherently to the homotopy category $\text{Ho}\, G\scr{S}_B$. In particular, $\text{Ho}\, G\scr{S}_B$ is closed symmetric monoidal, and
the derived fiber functor is closed symmetric monoidal. In any symmetric monoidal category, we have standard categorical notions of dualizable and invertible objects. In \S\ref{sec:fibdual}, we prove the fiberwise duality theorem, which says that a $G$-spectrum $X$ over $B$ is dualizable or invertible if and only
if each fiber $X_b$ is dualizable or invertible. This allows us to recognize dualizable or invertible $G$-spectra over $B$ when we see them.
In \S\ref{sec:trfr}, we explain how the fiberwise duality theorem leads to a simple and general conceptual definition of trace and transfer maps with good properties. To define the transfer, we regard a Hurewicz fibration $p\colon E\longrightarrow B$ with stably dualizable fibers as a space over $B$. We adjoin a copy of $B$ to obtain a section, and we suspend to obtain a $G$-spectrum over $B$. It is dualizable since its fibers are dualizable, hence it has a transfer map defined by categorical nonsense. Pushing down to $G$-spectra by base change along the map $r\colon B\longrightarrow *$, we obtain the transfer map of $G$-spectra $\Sigma^{\infty}B_+\longrightarrow \Sigma^{\infty}E_+$. This construction is a generalization of various earlier constructions of the transfer \cite{BG1, BG2, CG, Clapp, Waner}, most of which restrict to finite dimensional base spaces and are nonequivariant. An essential point is that the homotopy category of $G$-spectra over $B$ is closed symmetric monoidal with a ``compatible triangulation'', in the sense specified in \cite{Tri}. We defer the proof of the required compatibility relations to \S\ref{sec:comptriang}. This point implies that our traces and transfers satisfy additivity relations as well as the more elementary standard properties.
Some of the classical constructions of the transfer work only for bundles,
but have various properties that are inaccessible to the more general
construction and are important in calculations. These transfers also
admit a perhaps more satisfying construction. Rather than relying on duality
on the level of parametrized spectra, they are obtained by inserting duality
maps for fibers fiberwise into bundles. In the literature, the construction
again usually requires finite dimensional base spaces and is nonequivariant.
We give a general conceptual version of this alternative construction in \S\ref{sec:fibtrfr}.
As a first preliminary, in \S\ref{sec:bdlconstr} we show how to insert parametrized spectra fiberwise into the standard construction of equivariant bundles associated to principal bundles. The general construction is of considerable interest nonequivariantly. The construction on the ex-space
level is easy enough, but even here many of the properties that we describe
seem to be new. The construction is likely to have many further applications. The idea is to generalize the standard construction of the bundle of tangents along the fibers of a bundle by replacing the tangent bundle of the fiber by any spectrum over the fiber. In more detail, we consider $G$-bundles $p\colon E\longrightarrow B$ with fibers $F$. We allow the structure group $\Pi$ and ambient group $G$ to be related by an extension $1 \longrightarrow \Pi\longrightarrow \Gamma\longrightarrow G\longrightarrow 1$, and we take $F$ to be a $\Gamma$-space. The bundle $p$ has an associated principal $(\Pi;\Gamma)$-bundle $\pi\colon P\longrightarrow B$, where $P$ is a $\Pi$-free $\Gamma$-space and $B = P/\Pi$. We show how to construct a $G$-spectrum $P\times_{\Pi}X$ over $E$ from a $\Gamma$-spectrum $X$ over $F$.
As a second preliminary, in \S\ref{sec:PIFree} we develop the theory of
$\Pi$-free parametrized $\Gamma$-spectra. This is a direct generalization
of the nonparametrized theory and is important in many contexts. In
particular, it will play a role in our proof of the Adams isomorphism
in \S\ref{sec:adams}.
The application to transfer maps in \S\ref{sec:fibtrfr} can be described as follows. When $F$ is dualizable, we have a transfer map $\tau\colon S_{\Gamma}\longrightarrow \Sigma^{\infty}_{\Gamma}F_+$ of $\Gamma$-spectra. We insert this into the functor $P\times_{\Pi}(-)$ to obtain a map
\[P\times_{\Pi}\tau\colon P\times_{\Pi} S_{\Gamma}\longrightarrow P\times_{\Pi}\Sigma^{\infty}_{\Gamma}F_+\]
of $G$-spectra over $B$. Again pushing down to a map of $G$-spectra along $r\colon B\longrightarrow *$, we obtain the transfer $G$-map $\Sigma^{\infty}_GB_+\longrightarrow \Sigma^{\infty}_GE_+$. This description hides a subtlety. The construction involves a change of universe functor, and the key point is that this functor is a symmetric monoidal equivalence between categories of parametrized $\Pi$-free $\Gamma$-spectra. This makes it transparent from the naturality of transfer maps with respect to symmetric monoidal functors that the fiberwise transfer map of
a bundle agrees with its transfer map as a Hurewicz fibration.
We assume throughout that all given groups $G$ are compact Lie
groups and all given base $G$-spaces are of the homotopy types
of $G$-CW complexes.
\section{The fiberwise duality theorem}\label{sec:fibdual}
We characterize the dualizable and invertible $G$-spectra over $B$. A recent exposition of the general theory of duality in closed symmetric monoidal
categories appears in \cite{Pic}, to which we refer the reader for discussion
of the relevant categorical definitions and arguments. The following theorem is a substantial generalization of various early results of the same nature about ex-fibrations. These are due, for example, to Becker and Gottlieb \cite[\S4]{BG1}, Clapp \cite[3.5]{Clapp}, and Waner \cite[4.6]{Waner}.
\begin{thm}[The fiberwise duality theorem]\mylabel{bingo}
Let $X$ be an ($s$-fibrant) $G$-spec\-trum over $B$. Then $X$ is dualizable
(respectively, invertible) if and only if $X_b$ is dualizable (respectively, invertible) as a $G_b$-spectrum for each $b\in B$.
\end{thm}
\begin{proof}
By definition, $X$ is dualizable if and only if the natural map
$$\nu\colon D_BX\sma_B X\longrightarrow F_B(X,X)$$
in $\text{Ho}\, G\scr{S}_B$ is an equivalence. Passing to (derived) fibers,
this holds if and only if the resulting map
\[\xymatrix{DX_b\sma X_b \simeq (D_BX\sma_B X)_b \ar[r]^-{\nu_b} & F_B(X,X)_b\simeq F(X_b,X_b)}\]
in $\text{Ho}\, G_b\scr{S}$ is an equivalence for all $b\in B$. By the
categorical coherence observation \myref{coherence}, the latter map is the corresponding natural map $\nu$ in $\text{Ho}\, G_b\scr{S}$. Again by definition, that
map is an equivalence if and only if $X_b$ is dualizable.
Similarly, $X$ is invertible if and only if
the evaluation map
$$\text{ev}\colon D_BX\sma_B X\longrightarrow S_B$$
in $\text{Ho}\, G\scr{S}_B$ is an equivalence. Passing to (derived) fibers,
this holds if and only if the resulting map
\[\xymatrix{DX_b\sma X_b\simeq (D_BX\sma_B X)_b \ar[r]^-{\text{ev}_b} & (S_B)_b\simeq S}\]
in $\text{Ho}\, G_b\scr{S}$ is an equivalence for all $b\in B$. Again by
\myref{coherence}, the latter map
is the evaluation map for $X_b$ in $\text{Ho}\, G_b\scr{S}$, and that map
is an equivalence if and only if $X_b$ is invertible.
\end{proof}
Therefore, to recognize parametrized dualizable and invertible $G$-spectra, it
suffices to recognize nonparametrized dualizable and invertible $G$-spectra.
As we now recall from \cite{FLM}, these are well-understood.
Recall that a $G$-space $X$ is dominated by a $G$-space $Y$ if $X$ is a
retract up to homotopy of $Y$, so that the identity map of $X$ is homotopic
to a composite $X\longrightarrow Y\longrightarrow X$. If $Y$ has the homotopy type of a $G$-CW
complex, then so does $X$. We say that $X$ is finitely dominated
if it is dominated by a finite $G$-CW complex. This does not imply that
$X$ has the homotopy type of a finite $G$-CW complex, even when $X$ and all
of its fixed point spaces $X^H$ are simply connected and therefore, since
they are finitely dominated, homotopy equivalent to finite CW complexes.
For example, a $G$-space $X$ is a $G$-ENR (Euclidean neighborhood retract)\index{Euclidean neighborhood retract} if it can be embedded as a retract of an open subset of some representation $V$. Such open subsets
are triangulable as $G$-CW complexes, so $X$ has the homotopy type of a
$G$-CW complex. A compact $G$-ENR is a retract of a finite $G$-CW complex
and is thus finitely dominated, but it need not have the homotopy type of a finite $G$-CW complex. Non-smooth topological $G$-manifolds give examples
of such non-finite compact $G$-ENRs.
The following result is \cite[2.1]{FLM}.
\begin{thm}\mylabel{dualG}
Up to equivalence, the dualizable $G$-spectra are the $G$-spectra of the
form $\Sigma^{-V}\Sigma^\infty X$ where $X$ is a finitely dominated based
$G$-CW complex and $V$ is a representation of $G$.
\end{thm}
\begin{defn}\mylabel{hrep}
A \emph{generalized homotopy representation} $X$ is a finitely dominated based $G$-CW complex such that, for each subgroup $H$ of $G$, $X^H$ is equivalent to a sphere $S^{n(H)}$. A \emph{stable homotopy representation} is a $G$-spectrum of the form $\Sigma^{-V}\Sigma^{\infty}X$, where $X$ is a generalized homotopy representation and $V$ is a representation of $G$.
\end{defn}
The following result is \cite[0.5]{FLM}.
\begin{thm}\mylabel{inverG}
Up to equivalence, the invertible $G$-spectra are
the stable homotopy representations.
\end{thm}
Combining results, we obtain the following conclusion about ex-$G$-fibrations.
\begin{thm}\mylabel{spaceFDT}
Let $E$ be an ex-$G$-fibration over $B$. If each fiber $E_b$ is a finitely
dominated $G_b$-space, then $\Sigma^{\infty}_BE$ is a dualizable $G$-spectrum
over $B$. If each $E_b$ is a generalized homotopy representation of $G_b$,
then $\Sigma^{\infty}_BE$ is an invertible $G$-spectrum over $B$.
\end{thm}
\begin{proof}
Since the derived suspension spectrum functor commutes
with passage to derived fibers, by \myref{reassuring}, the derived fiber $(\Sigma^{\infty}_BE)_b$ is equivalent to $\Sigma^{\infty}E_b$. The conclusion
follows directly from Theorems \ref{bingo}, \ref{dualG}, and \ref{inverG}.
\end{proof}
In particular, sphere $G$-bundles and, more generally, spherical
$G$-fibrations over $B$, have invertible suspension $G$-spectra over $B$.
\section{Duality and transfer maps}\label{sec:trfr}
Since the stable homotopy category $\text{Ho}\, G\scr{S}_B$ is closed
symmetric mon\-oi\-dal, we have the following generalized trace maps
at our disposal. We state the definition and recall its properties
in full generality, and we then specialize to show how it gives
a simple conceptual definition of the transfer maps associated
to equivariant Hurewicz fibrations.
\begin{defn}\mylabel{tracemap}
Let $\scr{C}$ be any closed symmetric monoidal category with unit object $S$.
For a dualizable object $X$ of $\scr{C}$ with a ``coaction'' map
$\Delta_X\colon X\longrightarrow X\wedge C_X$ for some object $C_X\in\scr{C}$, define the
\emph{trace}\index{trace} $\tau(f)$ of a self map $f$ of $X$ by the diagram
\[\xymatrix{ S\ar[r]^-\eta \ar[d]_{\tau(f)} & X\wedge DX \ar[r]^-\gamma
& DX\wedge X \ar[d]^{Df\wedge \Delta_X} \\
C_X & S\wedge C_X\ar[l]^-\cong & DX\wedge X\wedge C_X.\ar[l]^-{\epsilon\wedge 1}}\]
\end{defn}
\begin{rem}
Such a categorical description of generalized trace maps was first given by Dold
and Puppe \cite{DP}, where they showed that it gives the right framework for trace
maps in algebra, the transfer maps of Becker and Gottlieb \cite{BG1, BG2}, and
the fixed point theory of Dold \cite{Dold}. These early constructions of
transfer maps had finiteness conditions that were first eliminated by Clapp
\cite{Clapp, CP}. Indeed, she gave an early construction of a parametrized
stable homotopy category and proved a precursor of our fiberwise duality theorem.
The equivariant analogue of the attractive space level treatment of Spanier-Whitehead duality given by Dold and Puppe was worked out in \cite{LMS}, and a recent categorical
exposition of duality has been given in \cite{Pic}.
\end{rem}
Two cases are of particular interest. The first is when $C_X=S$ and
$\Delta_X$ is the unit isomorphism. Then $\tau(f)$ is called the
\emph{Lefschetz constant}\index{Lefschetz constant} of $f$ and is denoted by $\chi(f)$; in the special
case when $f=\text{id}$ it is called the \emph{Euler characteristic}\index{Euler characteristic}
of $X$ and denoted by $\chi(X)$. The second is when $C_X=X$. We then
think of $\Delta_X$ as a diagonal map, and $\tau_X=\tau(\text{id})$
is called the \emph{transfer map}\index{transfer map} of $X$.
\begin{rem}
If $C_X$ comes with a ``counit'' map $\xi\colon C_X\longrightarrow S$ such that
the composite
$$\xymatrix{X\ar[r]^-{\Delta} & X\wedge C_X \ar[r]^-{\text{id}\wedge \xi} & X}$$
is the identity, then $\chi(f) = \xi\circ\tau(f)$ by a little diagram chase. The reason for the terminology ``coaction'' and ``counit'' for the maps $\Delta_X$ and $\xi$ is that in many situations $C_X$ is a comonoid and $\Delta_X$ is a coaction of $C_X$
on $X$.
\end{rem}
The following basic properties of the trace are proven in \cite[III\S7]{LMS}
and in \cite{Tri}, where more detailed statements are given. Define a map
$$(f,\alpha)\colon (X,\Delta_X)\longrightarrow(Y,\Delta_Y)$$
to be a pair of maps $f\colon X\longrightarrow Y$ and $\alpha\colon C_X\longrightarrow C_Y$ such
that the following diagram commutes.
\[\xymatrix{X\ar[r]^-{\Delta_X}\ar[d]_f & X\wedge C_X \ar[d]^{f\wedge \alpha}\\
Y\ar[r]_-{\Delta_Y} & Y\wedge C_Y}\]
\begin{prop}\mylabel{traceprop}
The trace satisfies the following properties, where $X$ and $Y$ are dualizable
and $\Delta_X$ and $\Delta_Y$ are given.
\begin{enumerate}[(i)]
\item{\em (Naturality)} If $\scr{C}$ and $\scr{D}$ are closed symmetric monoidal
categories and \linebreak
$F\colon\scr{C}\longrightarrow \scr{D}$ is a lax symmetric monoidal functor
such that $FS_{\scr{C}}\iso S_{\scr{D}}$, then
\[\tau(Ff)=F\tau(f),\]
where $C_{FX} = FC_X$ and $\Delta_{FX}=F\Delta_X$.
\item{\em (Unit property)} If $f$ is a self map of the unit object, then
$\chi(f)=f$.
\item{\em (Fixed point property)}
If $(f,\alpha)$ is a self map of $(X,\Delta_X)$, then
$$\alpha\circ \tau(f)=\tau(f).$$
\item{\em (Invariance under retracts)}
If $X\stackrel{i}{\longrightarrow} Y \stackrel{r}{\longrightarrow} X$ is a retract, $f$ is a self map of $X$, and
$(i,\alpha)$ is a map $(X,\Delta_X)\longrightarrow (Y,\Delta_Y)$, then
$$\alpha\circ\tau(f)=\tau(ifr).$$
\item{\em (Commutation with $\sma$)}
If $f$ and $g$ are self maps of $X$ and $Y$, then
\[\tau(f\wedge g)=\tau(f)\wedge \tau(g),\]
where $\Delta_{X\wedge Y} = (\text{id}\sma\gamma\sma\text{id})\circ (\Delta_X\sma \Delta_Y)$ with
$\gamma$ the transposition.
\item{\em (Commutation with $\vee$)} If $\scr{C}$ is additive
and $h\colon X\vee Y\longrightarrow X\vee Y$ induces $f\colon X\longrightarrow X$ and
$g\colon Y\longrightarrow Y$ by inclusion and retraction, then
\[\tau(h)=\tau(f) + \tau(g),\]
where $C_X = C_Y = C_{X\vee Y}$ and $\Delta_{X\vee Y} = \Delta_X \vee \Delta_Y$.
\item{\em (Anticommutation with suspension)} If $\scr{C}$ is triangulated, then
$$\tau(\Sigma f)=-\tau(f)$$
for all self maps $f$, where $\Delta_{\Sigma X} = \Sigma \Delta_X$.
\end{enumerate}
\end{prop}
In the triangulated context, there is another and very much deeper property.
\begin{thm}[Additivity]\mylabel{addprop} Let $\scr{C}$ be a closed symmetric monoidal
category with a ``compatible triangulation''. Let $X$ and $Y$ be dualizable
and let $\Delta_X$ and $\Delta_Y$ be given, where $C=C_X=C_Y$. Let $(f,\text{id})$ be a
map $(X,\Delta_X)\longrightarrow (Y,\Delta_Y)$ and extend $f$ to a distinguished triangle
$$\xymatrix{X\ar[r]^-f & Y \ar[r]^{g} & Z \ar[r]^-h &\Sigma X.\\}$$
Assume given maps $\phi$ and $\psi$ that make the left square commute in the
first of the following two diagrams.
$$\xymatrix{
X \ar[r]^-f \ar[d]_{\phi} & Y \ar[r]^-g \ar[d]^{\psi}
& Z \ar[r]^-h\ar[d]^{\omega} & \Sigma X \ar[d]^{\Sigma\phi}\\
X \ar[r]_-f & Y \ar[r]_-g & Z \ar[r]_-h & \Sigma X\\}$$
$$\xymatrix{
X \ar[r]^-f\ar[d]_{\Delta_X} & Y \ar[r]^-g \ar[d]^{\Delta_Y}
& Z \ar[r]^-h \ar[d]^{\Delta_Z} & \Sigma X \ar[d]^{\Sigma\Delta_X}\\
X\sma C \ar[r]_-{f\sma\text{id}} & Y\sma C \ar[r]_-{g\sma\text{id}}
& Z\sma C \ar[r]_-{h\sma\text{id}} & \Sigma (X\sma C)\\}$$
Then there are maps $\omega$ and $\Delta_Z$ such that the diagrams commute and
$$\tau(\psi) = \tau(\omega) + \tau(\phi).$$
\end{thm}
The most important case starts with only the distinguished triangle $(f,g,h)$ and concludes with the fundamental additivity relation
$$\chi(Y) = \chi(X) + \chi(Z).$$
The additivity of traces was studied in \cite[III\S7]{LMS} in the equivariant stable homotopy category, but the proof there is incorrect. A thorough investigation of precisely what is needed to prove the additivity of traces is given in \cite{Tri}, where the axioms for a ``compatible triangulation'' are formulated. These axioms hold in all situations previously encountered in algebraic topology and algebraic geometry. However, the model theoretic method of proof described in \cite{Tri} assumes the usual model theoretic compatibilities, such as the pushout-product axiom of \cite{SS}, and these fail to hold in the present context. Since the proof of the following result only makes sense by close comparison with the proof in \cite{Tri}, we shall defer it to \S\ref{sec:comptriang}.
\begin{thm} The category $\text{Ho}\, G\scr{S}_B$ is a closed symmetric monoidal category
with a compatible triangulation.
\end{thm}
With these foundations in place, we can now generalize the
classical construction of transfer maps. The results above specialize
to give more information about them than is to be found in the literature.
If $X$ is a dualizable $G$-spectrum over $B$ with a diagonal map
$\Delta_X\colon X\longrightarrow X\sma_B X$, then we have the transfer map
$\tau_X\colon S_{B}\longrightarrow X$.
We shall apply this to suspension $G$-spectra associated to
$G$-fibrations $p\colon E\longrightarrow B$, but we do not assume that $p$ has a section.
We need some notation. It has been the custom since the beginnings of algebraic
topology to use the same letter $E$ for a bundle and for its underlying total space.
It seems to us that this standard abuse of notation seriously obscures the literature of
parametrized homotopy theory, and for that reason we shall be very pedantic at this
point.
\begin{notn}
For a $G$-space $E$ over $B$, let $(E,p)_+$ denote the
ex-$G$-space $E\amalg B$ over $B$, with section at the disjoint copy of $B$.
The usual notation is $E_+$, but we shall reserve that notation for the union
of the total $G$-space $E$ with a disjoint basepoint. Observe that if $p$ is
a Hurewicz $G$-fibration, then $(E,p)_+$ is an ex-$G$-fibration. Except where
otherwise indicated, we agree to write $r$ for the unique map $B\longrightarrow *$ for
any based $G$-space $B$.
\end{notn}
Recall the desription of the base change functors associated to $r$ from \myref{r!ex}. The spectrum level versions of these functors are central to the deduction of results in classical stable homotopy theory from results in parametrized stable homotopy theory. The following observation is particularly relevant.
\begin{lem}\mylabel{lesstrivial}
For a $G$-map $p\colon E\longrightarrow B$, thought of as a $G$-space over $B$, \[r_!\Sigma^{\infty}_B (E,p)_+\simeq \Sigma^\infty E_+,\]
where $r\colon B\longrightarrow *$. In particular, $r_!S_B \simeq \Sigma^{\infty}B_+$.
\end{lem}
\begin{proof} We have $r_!\Sigma^{\infty}_B\htp \Sigma^{\infty}r_!$. This is a
commutation relation between Quillen left adjoints, and the corresponding
commutation relation for right adjoints holds since
$$r^*\Omega^{\infty}X = B\times X_0 \iso \Omega^{\infty}_B r^* X$$
for a $G$-spectrum $X$. It therefore suffices to show that $r_!(E,p)_+$ is equivalent to $E_+$, where $r_!$ denotes the functor on derived categories.
By \myref{Qad10}, $r_!$ preserves $q$-equivalences between well-sectioned ex-spaces and it follows that $r_!Q(E,p)_+\simeq r_!(E,p)_+\cong E_+$ where the first equivalence is induced by $qf$-cofibrant approximation of $(E,p)_+$.
\end{proof}
To be precise about diagonal maps on the parametrized level,
we consider base change along $\Delta\colon B\longrightarrow B\times B$.
We have the obvious commutative diagram
$$\xymatrix{
E \ar[d]_{p} \ar[r]^-{\Delta} & E\times E \ar[d]^{p\times p}\\
B \ar[r]_-{\Delta} & B\times B.}$$
We consider $E$ as a space over $B\times B$ via this composite. The
diagonal map of $E$ then specifies a natural map
$$\Delta_!((E,p)_+) = (E,\Delta\com p)_+ \longrightarrow (E\times E,p\times p)_+\iso (E,p)_+\barwedge(E,p)_+$$
of ex-spaces over $B\times B$. This is a comparison map between
Quillen left adjoints and therefore descends to a natural map in
$\text{Ho}\, G\scr{K}_{B\times B}$. Its adjoint is a natural map $(E,p)_+\longrightarrow (E,p)_+\sma_B (E,p)_+$ in $\text{Ho}\, G\scr{K}_B$. Apply the (derived) suspension functor $\Sigma^{\infty}_B$ to this map and note that the target is equivalent
to $\Sigma^{\infty}_{B}(E,p)_+\sma_B \Sigma^{\infty}_{B}(E,p)_+$, by \myref{SISISI2}.
This gives the required natural diagonal map
$$\Delta_{(E,p)_+}\colon \Sigma^{\infty}_{B}(E,p)_+
\longrightarrow \Sigma^{\infty}_{B}(E,p)_+\sma_B \Sigma^{\infty}_{B}(E,p)_+$$
in $\text{Ho}\, G\scr{S}_B$.
\begin{defn}[The transfer map]\index{transfer map!for fibrations}\mylabel{fibtransfer}
Let $p\colon E\longrightarrow B$ be a Hurewicz $G$-fibration over $B$ such that each
fiber $E_b$ is homotopy equivalent to a retract of a finite $G_b$-CW-complex.
Then $\Sigma_B^\infty (E,p)_+$ is a dualizable $G$-spectrum over $B$ by \myref{spaceFDT} and we obtain the transfer map
$$\tau_{(E,p)_{+}}\colon S_{B}\longrightarrow \Sigma_B^\infty (E,p)_+$$
in $\text{Ho}\, G\scr{S}_B$. Define the \emph{transfer map} of $E$ to be the map
$$\tau_E = r_!\tau_{(E,p)_{+}}
\colon \Sigma^{\infty} B_+ \iso r_! S_B
\longrightarrow r_!\Sigma_B^\infty (E,p)_+\iso \Sigma^{\infty} E_+$$
in $\text{Ho}\, G\scr{S}$.
\end{defn}
With this definition, all of the standard properties of transfer maps
are direct consequences of the general categorical results
\myref{traceprop} and \myref{addprop} and the properties of $r_!$.
\section{The bundle construction on parametrized spectra}\label{sec:bdlconstr}
The construction of the transfer in the previous section works ``globally'', starting on the parametrized spectrum level. We now give a fiberwise construction of ``stable bundles'' that leads to an alternative fiberwise perspective. However, it is natural to work in greater generality than is needed for the construction of the transfer. The extra generality will be needed in the proof of the Wirthm\"uller isomorphism in \S\ref{sec:fibwirth} and will surely find other applications. The relevant bundle theoretic background was recalled in \S3.2.
Let $\Pi$ be a normal subgroup of a compact Lie group $\Gamma$ such that
$\Gamma/\Pi = G$ and let $q\colon \Gamma\longrightarrow G$ be the quotient homomorphism.
Let $p\colon E\longrightarrow B$ be a $(\Pi;\Gamma)$-bundle with fiber a $\Gamma$-space $F$
and with associated principal $(\Pi;\Gamma)$-bundle $\pi\colon P\longrightarrow B$.
Then $P$ is a $\Pi$-free $\Gamma$-space, $\pi$ is the quotient map to
the orbit $G$-space $B=P/\Pi$, and $p$ is the associated $G$-bundle
$E \iso \T{F}\longrightarrow B$.
To simplify the homotopical analysis, we assume for the rest of this section that $F$ and $P$ are $\Gamma$-CW complexes such that $P$ is $\Pi$-free. We let $E ={\T{F}}$ and $B = \T{*}$. Note that $B$ is a $G$-CW complex. We are thinking of the cases when $F$ is a point or when $F$ is a smooth $\Gamma$-manifold. On the ex-space level, application of $P\times_{\Pi}(-)$ to retracts gives the functor
\[\U{_F} = \T(-)\colon \Gamma\scr{K}_F \longrightarrow G\scr{K}_E.\]\index{bundle construction}
Thus, for an ex-$\Gamma$-space $K$ over $F$, the ex-$G$-space $P\times_{\Pi} K$ over $P\times_{\Pi} F$ has section and projection induced by the section and projection of $K$. Observe that if $F$ is a smooth manifold and $S^{\tau}$ is the sphere bundle obtained by fiberwise one-point compactification of the tangent bundle of $F$, then $\T{S^{\tau}}$ is the $G$-bundle of spherical tangents along the fiber associated to $p$.
We can extend the functor $\U{_F}$ from ex-spaces to ex-spectra. Change of universe must enter since $\Gamma$-spectra are indexed on representations of $\Gamma$ and $G$-spectra are indexed on representations of $G$. We view representations of $G$ as $\Pi$-trivial representations of $\Gamma$. This gives $i\colon q^*\scr{V}_G\longrightarrow \scr{V}_{\Gamma}$. Implicitly applying the functor $i^*$ to $\Gamma$-spectra, {\em we agree to index both $G$-spectra and $\Gamma$-spectra on $\scr{V}_G$ for the rest of the section}. We are interested in $\Gamma$-spectra indexed on a complete universe, and we shall return to this point in the next section. Since $\Pi$ acts trivially on our representations $V$, we have
$$ \U{_F{K}}\sma_E S^V \iso \U{_F{(K\sma_F S^V)}}.$$
Therefore, for a $\Gamma$-spectrum $X$ over $F$, the ex-$G$-spaces $\U{_F{X(V)}}$ over $E$ inherit structure maps from $X$, so that $\U{_FX}$ is a $G$-spectrum over $E$. We have the same definition on the prespectrum level. These functors $\U{_F}$ are exceptionally well-behaved, as the following results show.
\begin{prop}
The functor $\U{_F}\colon \Gamma\scr{S}^{\text{$\Pi$-triv}}_F \longrightarrow G\scr{S}_E$ is both a left and a right Quillen adjoint with respect to the level and stable model structures. Moreover, the functor $\U{_F}\colon \Gamma\scr{P}^{\text{$\Pi$-triv}}_F \longrightarrow G\scr{P}_E$ takes excellent $\Gamma$-prespectra over $F$ to excellent $G$-prespectra over $E = \T{F}$.
\end{prop}
\begin{proof}
Let ${\pi}\colon P\times F\longrightarrow F$ be the projection. Clearly $\U{_F}$ is the composite of ${\pi}^*\colon \Gamma\scr{S}_F \longrightarrow \Gamma\scr{S}_{P\times F}$ and $(-)/\Pi\colon \Gamma\scr{S}_{P\times F}\longrightarrow G\scr{S}_E$. By Propositions \ref{Qad1}, \ref{Qad2}, \ref{Qad1too}, and \ref{Qad2too}, ${\pi}^*$ is both a left and a right Quillen adjoint, provided we use appropriate generating sets in our
definitions of the model structures. By \myref{factor}, the functor $(-)/\Pi$ is a Quillen left adjoint. By \myref{ouch}, it coincides with the right adjoint
$(-)^{\Pi}\com p_*$, where $p$ here is the quotient map $P\times F\longrightarrow P\times_{\Pi}F = E$. Using \myref{rho}, we see that $p\colon E\longrightarrow B$ is a $G$-bundle with CW fibers. Therefore $p_*$ is a Quillen right adjoint by Propositions \ref{Qad2} and \ref{Qad2too}, and $(-)^{\Pi}$ is a Quillen right adjoint by \myref{factor}. The last statement is easily checked from \myref{excel} and \myref{fpOM}.
\end{proof}
We need an observation about the behavior of $\U{_F}$ on fibers.
\begin{lem}\mylabel{tEtB}
Fix $b\in B$. Let $\iota\colon G_b\longrightarrow G$ and $\rho_b\colon G_b\longrightarrow \Gamma$ be the inclusion and the homomorphism of \myref{rho}. Let $b\colon \{b\}\longrightarrow B$ and $i_b\colon E_b\longrightarrow E$ denote the evident inclusions of $G_b$ spaces. The following diagrams commute, and these commutation relations descend to homotopy categories.
\[\xymatrix{
\Gamma\scr{S}_*^{\text{$\Pi$-triv}}\ar[d]_{\U{_{*}}}\ar[r]^-{\rho_b^*} & G_b\scr{S}_{b}\\
G\scr{S}_{B}\ar[r]_{\iota^*} & G_b\scr{S}_B \ar[u]_{b^*}}
\quad \text{and} \quad
\xymatrix{
\Gamma\scr{S}^{\text{$\Pi$-triv}}_F\ar[d]_{\U{_F}}\ar[r]^-{\rho_b^*} & G_b\scr{S}_{E_b} \\
G\scr{S}_E\ar[r]_-{\iota^*} & G_b\scr{S}_E \ar[u]_{i_b^*}}\]
\end{lem}
\begin{proof} On the level of ex-spaces, this is immediate by inspection.
The diagrams extend levelwise to parametrized spectra, and passage to
homotopy categories is clear from the previous result.
\end{proof}
Writing $\Sigma^{\infty}$ for suspension spectra functors indexed on complete universes, we have that $i^*\Sigma^{\infty}_{\Gamma,F}$, where
$i\colon q^*\scr{V}_G\subset \scr{V}_\Gamma$, is the suspension $\Gamma$-spectrum functor indexed on the $\Pi$-trivial $\Gamma$-universe $q^*\scr{V}_G$.
\begin{thm}\mylabel{PSmash}
There is a natural isomorphism of functors
$$\U{_F{i^*\Sigma^{\infty}_{\Gamma,F}}}\iso\Sigma^{\infty}_{G,E}\U{{_F}}\colon
\Gamma \scr{K}_F\longrightarrow G\scr{S}_E,$$
and this isomorphism descends to homotopy categories. The functor
$$\U{_F}\colon \text{Ho}\, \Gamma\scr{S}^{\text{$\Pi$-triv}}_F \longrightarrow \text{Ho}\, G\scr{S}_E$$
is closed symmetric monoidal.
\end{thm}
\begin{proof}
Let $K$ be an ex-$\Gamma$-space over $F$. Since we are indexing on representations $V$ of $G$, we have isomorphisms
\[(\U{_F{i^*\Sigma^{\infty}_{\Gamma,F}K}})(V) = \T{(K\sma_F S^V_F)}
\iso (\T{K})\sma_E S^V = (\Sigma^{\infty}_{G,E}\U{_{F}K})(V).\]
This gives a natural isomorphism of $G$-spectra over $E$, and it descends to homotopy categories since it is a comparison of composites of Quillen left adjoints. Note in particular that $\U{_F}i^*S_{\Gamma,F}$ is isomorphic to $S_{G,E}$. We must show that the functor $\U{_F}$ commutes up to coherent natural isomorphism with smash products and function objects. For ex-$\Gamma$-spaces $K$ and $L$ over $F$, it is easy to check that there is a natural isomorphism
$$ \U{{_F}{(K\sma_F L)}} \longrightarrow \U{{_F}{K}}\sma_E \U{{_F}{L}}$$
of ex-$G$-spaces over $E$. This isomorphism extends levelwise to external smash products (external in the sense of pairs of representations). However, since external pairings (in the sense of pairs of base spaces) do not naturally come into play here, to retain homotopical control it seems simplest to just extend levelwise to handicrafted smash products of $\Gamma$-prespectra; compare \myref{subtlety}. Using excellent prespectra to pass to homotopy categories of prespectra and then using the equivalence $(\mathbb{P},\mathbb{U})$ to pass to homotopy categories of spectra, we obtain the required natural equivalence
$$ \U{{_F}{(X\sma_F Y)}} \longrightarrow \U{{_F}{X}}\sma_E \U{{_F}{Y}}$$
in $\text{Ho}\, G\scr{S}_E$ for $\Gamma$-spectra $X$ and $Y$ over $F$. The
adjoint of the composite
$$\xymatrix{
\U{{_F}{F_F(X,Y)}}\sma_E \U{{_F}{X}}\simeq
\U{{_F}{(F_F(X,Y)\sma_F X)}} \ar[rr]^-{\U{{_F}{(\text{ev})}}}&&
\U{{_F}{Y}}}$$
is a natural map
$$ \U{{_F}{F_F(X,Y)}} \longrightarrow F_E(\U{{_F}{X}},\U{{_F}{Y}})$$
in $\text{Ho}\, G\scr{S}_E$, and we must show that it is an equivalence. This will hold if it holds when restricted to fibers over points of $E$. Since each point is in some $E_b$, it suffices to show that the restriction to each $E_b$ is an equivalence. However, using \myref{tEtB}, we see that the restriction to $E_b$ is the adjoint to the $G_b$-map $\text{ev}\colon F_{E_b}(\rho_b^*X,\rho_b^*Y)\sma_{E_b} \rho_b^*X \longrightarrow \rho_b^*Y$, and is thus the identity map.
\end{proof}
We have the following relations between $\U{_F}$ and base change functors.
\begin{prop}\mylabel{Pfibers}
Consider $r\colon F\longrightarrow *$ and $p = \T{r}\colon E\longrightarrow B$.
For $Y \in \Gamma\scr{S}_*^{\text{$\Pi$-triv}}$ and $X\in \Gamma\scr{S}^{\text{$\Pi$-triv}}_F$,
there are natural isomorphisms
\[p_!\U{_F{X}} \longrightarrow \U{_*{r_!X}}, \quad
\U{_F{r^*Y}}\longrightarrow p^*\U{_*{Y}}, \quad \text{and} \quad
\U{_*{r_*X}}\longrightarrow p_*\U{_F{X}},\]
and these isomorphisms induce natural equivalences on homotopy categories.
\end{prop}
\begin{proof}
We first work on the ex-space level. Let $T$ be a based $G$-space and $K$ be an ex-$G$-space over $F$. Applying the functor $\T{(-)}$ to the maps of retracts that define $r_!K$ and $r^*T$ (see \myref{retract1}), we immediately obtain the first two maps. The first is the natural isomorphism
$$(\T{K})\cup_E B \iso \T{(K/F)}$$
in which the section $F$ is collapsed to a point in $K$ on both sides. The second is the evident natural isomorphism
$$P\times_{\Pi} (F\times T)\iso (P\times_{\Pi} F)\times_B (P\times_{\Pi} T).$$
For the third map, recall that $r_*K =\text{Sec}(F,K)$. The adjoint of
$$\xymatrix{
(\T{\text{Sec}(F,K)})\times_B E
\iso \T{(\text{Sec}(F,K)\times F)} \ar[rr]^-{\T{\text{ev}}} && \T{K}}$$
gives a map
$$ \U{_*{r_*K}} = \T{\text{Sec}(F,K)}\longrightarrow \text{Map}_B(E,\T{K}).$$
Together with the projection of the source to $B$, it induces an isomorphism to $p_*\U{_F{K}}$, which is the pullback along $B\longrightarrow \text{Map}_B(E,E)$ of the projection of the target induced by the projection $P\times_{\Pi}K\longrightarrow P\times_{\Pi}F = E$. Applied levelwise, these point-set level isomorphisms carry over directly to parametrized prespectra and spectra. We must show that they descend to equivalences in homotopy categories. Since \myref{Qad2too} applies to show that both $p_*$ and $r_*$ are Quillen right adjoints (and we have no need to use Brown representability here), the first commutation relation is between composites of left Quillen adjoints, the second is between functors that are both left and right Quillen adjoints, and the third is between Quillen right adjoints, so descent to homotopy categories is immediate.
\end{proof}
\section{$\Pi$-free parametrized $\Gamma$-spectra}\label{sec:PIFree}
We retain the notations of the previous section in this section and the next.
In the next section, we show that the bundle construction on parametrized
spectra leads to a fiberwise generalization of the restriction to bundles
of the trace and transfer maps for fibrations that we described in \S\ref{sec:trfr}. The definition depends on a result that is proven by use
of the theory of $\Pi$-free $\Gamma$-spectra that we present here.
We first recall what it means to say that a $\Gamma$-spectrum $X$ (indexed on any
universe) is $\Pi$-free. Let $\scr{F}(\Pi;\Gamma)$ be the family of subgroups $\Lambda$ of
$\Gamma$ such that $\Lambda\cap \Pi = e$. A $\Gamma$-CW
complex $T$ is $\Pi$-free if and only if the only orbit types $\Gamma/\Lambda$ that appear
in its construction have $\Lambda\in \scr{F}(\Pi;\Gamma)$. We then say that $T$ is an
$\scr{F}(\Pi;\Gamma)$-CW complex. We can make the same definitions for $\Gamma$-CW spectra,
and in general we say that a $\Gamma$-spectrum is $\Pi$-free if it is isomorphic in
$\text{Ho}\, \Gamma\scr{S}$ to an $\scr{F}(\Pi;\Gamma)$-CW spectrum. There is a more conceptual
homotopical reformulation that is the one relevant to the parametrized point of
view and that does not depend on the theory of $\Gamma$-CW spectra.
Let $E(\Pi;\Gamma)$ be the universal $\Pi$-free $\Gamma$-space, so that $E(\Pi;\Gamma)^{\Lambda}$
is contractible if $\Lambda\cap \Pi = e$ and is empty otherwise. We may take $E(\Pi;\Gamma)$
to be an $\scr{F}(\Pi;\Gamma)$-CW complex. Let $B(\Pi;\Gamma) = E(\Pi;\Gamma)/\Pi$ and observe that
$B$ is a $G$-CW complex and therefore also a $\Gamma$-CW complex. We note parenthetically
that the quotient map $p: E(\Pi;\Gamma) \longrightarrow B(\Pi;\Gamma)$ is the universal principal
$(\Pi;\Gamma)$-bundle. That is, pullback along $p$ gives a bijection
\[[X,B(\Pi;\Gamma)]_G\longrightarrow \scr{B}{(\Pi;\Gamma)}(X),\]
where $\scr{B}{(\Pi;\Gamma)}(X)$ denotes the set of equivalence classes of
principal $(\Pi;\Gamma)$-bundles over the $G$-space $X$; see \cite{LM} or
\cite [VII\S2]{EHCT}.
\begin{defn}\mylabel{Nfree} Let $r\colon E(\Pi;\Gamma)\longrightarrow *$ be the projection
and let $\sigma$ be the counit of the (derived) adjunction $(r_!,r^*)$.
A $\Gamma$-spectrum $X$ is said to be \emph{$\Pi$-free} if $\sigma\colon r_!r^* X\longrightarrow X$ is an equivalence.
\end{defn}
The definition should seem reasonable since $r_!r^* T\iso E(\Pi;\Gamma)_+\sma T$
for a $\Gamma$-space $T$. It is equivalent to the original definition in
terms of an equivalence in $\text{Ho}\, G\scr{S}$ to an $\scr{F}(\Pi;\Gamma)$-CW spectrum;
see \cite[II.2.12]{LMS} or \cite[VI\S4]{MM}. The definition generalizes
readily to the parametrized context.
\begin{defn} Let $\pi\colon E(\Pi;\Gamma)\times F\longrightarrow F$ be the
projection and let $\sigma$ be the counit of the (derived) adjunction $(\pi_!,\pi^*)$. An ex-$\Gamma$-space or $\Gamma$-spectrum $X$ over a
$\Gamma$-space $F$ is said to be \emph{$\Pi$-free} if
$\sigma\colon \pi_!\pi^* X \longrightarrow X$ is an equivalence.
\end{defn}
Since the fiber $(\pi_!\pi^* X)_f$ is $E(\Pi;\Gamma)_+\sma X_f$, the definition should seem reasonable. Since equivalences are detected fiberwise, we have
the following results.
\begin{lem}\mylabel{FibPI}
A $\Gamma$-spectrum $X$ over $F$ is $\Pi$-free if and only if each of its fibers $X_f$ is a $(\Pi\cap\Gamma_f)$-free $\Gamma_f$-spectrum.
\end{lem}
\begin{proof}
The fiber of $E(\Pi;\Gamma)\times F\longrightarrow F$ over $f\in F$ is the $\Gamma$-space $E(\Pi;\Gamma)$ with the action restricted along $\iota\colon \Gamma_f\longrightarrow \Gamma$. It is a model of the universal $(\Pi\cap\Gamma_f)$-free $\Gamma_f$-space $E(\Pi\cap\Gamma_f,\Gamma_f)$. Applying $(-)_f$ to the counit $\pi_!\pi^* X \longrightarrow X$ and using \myref{Mackeymore} we obtain the counit $r_!r^* X_f\longrightarrow X_f$ where $r\colon \iota^* E(\Pi;\Gamma)\longrightarrow *$.
\end{proof}
\begin{lem}\mylabel{IsPI} If $P$ is a $\Pi$-free $\Gamma$-space and $X$
is any ex-$\Gamma$-space or $\Gamma$-spectrum over $F$, then $P\times X$
is a $\Pi$-free ex-$\Gamma$-space or $\Gamma$-spectrum over $P\times F$.
\end{lem}
A useful slogan asserts that ``$\Pi$-free $\Gamma$-spectra live in the
$\Pi$-trivial universe''. To explain it, consider the inclusion $i\colon q^*\scr{V}_G\longrightarrow \scr{V}_{\Gamma}$ of the complete $G$-universe $\scr{V}_G$ as the
universe of
$\Pi$-trivial representations in the complete $\Gamma$-universe $\scr{V}_{\Gamma}$.
Then the slogan is given meaning by the following result. In the nonparametrized case $F=*$, it is proven in \cite[II\S2]{LMS} and is
discussed further in \cite[VI\S4]{MM}. Since the parametrized case
presents no complications and the proof is quite easy, we only give a
sketch.
\begin{prop}\mylabel{Nfreeii}
The change of universe adjunction $(i_*,i^*)$ descends to a symmetric monoidal equivalence between the homotopy categories of $\Pi$-free $\Gamma$-spectra over $F$ indexed on $\Pi$-trivial representations of $\Gamma$ on the one hand and indexed on all representations of $\Gamma$ on the other. For $\Pi$-free $\Gamma$-spectra $X$ over $F$ indexed on $\scr{V}_{\Gamma}$, there is a natural equivalence
$i_*(E(\Pi;\Gamma)_+\sma i^*X)\simeq X$.
\end{prop}
\begin{proof}[Sketch Proof] If $\Lambda\cap \Pi = e$, then the quotient map
$q\colon \Gamma\longrightarrow G$ maps $\Lambda$ isomorphically onto a subgroup of $G$. Any
representation $V$ of $\Lambda$ is therefore of the form $q^*W$ for a representation
$W$ of $q(\Lambda)$. It follows that the restrictions to $\Lambda$ of the universes
$\scr{V}_{\Gamma}$ and $q^*\scr{V}_G$ have the same representations. This makes clear that,
on $\Pi$-free $\Gamma$-spectra over $F$, the unit and counit of the adjunction
$(i_*,i^*)$ must be $\scr{F}(\Pi;\Gamma)$-equivalences, in the sense that they are
$\Lambda$-equivalences for any $\Lambda$ in $\scr{F}(\Pi;\Gamma)$. Smashing the unit and counit
with $E(\Pi;\Gamma)_+$, which has trivial fixed point sets for subgroups not in
$\scr{F}(\Pi;\Gamma)$, we obtain natural equivalences, and it follows from
\myref{Nfree} that the unit and counit are themselves equivalences
when applied to $\Pi$-free $\Gamma$-spectra. Alternatively, restricting to
$s$-fibrant $\Gamma$-spectra over $F$, the conclusion follows fiberwise from its
nonparametrized precursor. Since $i_*$ is symmetric monoidal, by \myref{change1},
so is the equivalence. The last statement holds since
\[i_*(E(\Pi;\Gamma)_+\sma i^*X)\htp E(\Pi;\Gamma)_+\sma i_*i^*X\htp X. \qedhere\]
\end{proof}
\section{The fiberwise transfer for $(\Pi;\Gamma)$-bundles}\label{sec:fibtrfr}
We consider a fixed given principal $(\Pi;\Gamma)$-bundle $P$, where
$\Pi$ is a normal subgroup of $\Gamma$ with quotient group $G$ and
quotient map $q\colon \Gamma\longrightarrow G$. We also consider a $\Gamma$-space
$F$ and the associated $(\Pi;\Gamma)$-bundle
$$p\colon E = P\times_{\Pi}F\longrightarrow P\times_{\Pi}* = B.$$
We have the inclusion $i\colon q^*\scr{V}_G\longrightarrow \scr{V}_{\Gamma}$ of the complete
$G$-universe $\scr{V}_G$ as the universe of $\Pi$-trivial representations
in the complete $\Gamma$-universe $\scr{V}_{\Gamma}$.
The change of universe functor $i^*\colon \Gamma\scr{S}_{F}\longrightarrow \Gamma\scr{S}_{F}^{{\Pi}-{\text{triv}}}$ is {\em not}\, symmetric mon\-oid\-al, and it does not preserve dualizable objects. For example, with $F=*$ and $\Pi =e$, the orbit spectrum $i^*\Sigma^{\infty}\Gamma/\Lambda$ is not dualizable if $\Lambda$ is a non-trivial subgroup of $\Gamma$. The bundle theoretic study of transfer maps is based on the following
result, whose proof is based on the theory of $\Pi$-free $\Gamma$-spectra
given in the previous section.
\begin{thm}\mylabel{bomb} The composite functor
$\U{{_F}{i^*}}\colon \text{Ho}\, \Gamma\scr{S}_F\longrightarrow \text{Ho}\, G\scr{S}_E$
is symmetric mon\-oid\-al.
\end{thm}
\begin{proof}
Let $\pi\colon P\times F\longrightarrow F$ be the projection and note that
${\pi}^*X = P\times X$. The functor $\U{_F}$ is the composite of the
symmetric monoidal Quillen left adjoint ${\pi}^*$ and the Quillen left
adjoint $(-)/\Pi$. By \myref{PSmash}, the functor $\U{_F}$ on homotopy
categories is also symmetric monoidal since the $\Gamma$-space $P$ is $\Pi$-free. We observe first that the composite ${\pi}^*i^*$ is symmetric monoidal.
Indeed, for $\Gamma$-spectra $X$ and $Y$ over $F$, we have
\begin{align*}
{\pi}^*i^*(X\sma_FY) & \htp i^*{\pi}^*(X\sma_FY)
&& \text{by \myref{chvschuni}}\\
& \htp i^*({\pi}^*X\sma_{P\times F}{\pi}^*Y)
&& \text{by \myref{Wirthmore}}\\
& \htp i^*{\pi}^*X\sma_{P\times F}i^*{\pi}^*Y
&& \text{by \myref{IsPI} and \myref{Nfreeii}}\\
& \htp {\pi}^*i^*X\sma_{P\times F} {\pi}^*i^*Y
&& \text{by \myref{chvschuni}.}
\end{align*}
It follows directly that $\U{{_F}i^*}$ is symmetric monoidal:
\begin{align*}
\U{{_F}i^*(X\sma_FY)} & = ({\pi}^*i^*(X\sma_FY))/\Pi
&& \text{by definition}\\
& \htp ({\pi}^*i^*X\sma_{P\times F} {\pi}^*i^*Y)/\Pi
&& \text{by the previous display}\\
& \htp ({\pi}^*(i^*X\sma_{F} i^*Y))/\Pi
&& \text{by \myref{Wirthmore}}\\
& = \U{{_F}(i^*X\sma_{F} i^*Y})
&& \text{by definition}\\
& \htp \U{{_F}i^*X}\sma_E \U{{_F}i^*Y}
&& \text{by \myref{PSmash}}. \qedhere
\end{align*}
\end{proof}
Now \myref{traceprop}(i) shows that $\U{{_F}{i^*}}$ commutes with trace maps.
\begin{thm}\mylabel{PtrP}
Let $X\in \text{Ho}\,\Gamma \scr{S}_F$ be dualizable. Then $\bar{P}_F i^*X \in \text{Ho}\, G\scr{S}_{E}$ is dualizable. Suppose given a coaction map $\Delta_X:X\to X\wedge_F C_X$ and a self map $\phi\colon X\longrightarrow X$. Then
\[\tau(\bar{P}_F i^*\phi)\simeq \bar{P}_F i^* \tau(\phi)\colon S_{E}\longrightarrow \U{{_F}{i^*C_X}},\]
where $\bar{P}_F i^*X$ is given the coaction map
$$\bar{P}_Fi^*(\Delta_X):\bar{P}_Fi^* X \longrightarrow
\bar{P}_Fi^* (X \wedge_{F} C_X) \htp \bar{P}_Fi^* X \wedge_{E} \bar{P}_Fi^*C_X.$$
\end{thm}
These trace maps are maps of $G$-spectra over $E$, rather than over $B$.
We can apply $r_!$, $r\colon E\longrightarrow *$, to obtain trace maps of
nonparametrized spectra. This kind of trace map can be viewed as a
fiberwise generalization of the kind of nonparametrized trace map
that is defined bundle theoretically in the literature. To connect
up with the latter, we specialize and change our point of view so as
to arrive at bundle theoretic trace maps over $B$. Specializing further
to transfer maps, we obtain the promised comparison with the transfer
maps of \myref{fibtransfer}.
With these goals in mind, we now focus on the case $F=*$, so that $E$ above becomes $B$, with $p$ the identity map, and our trace maps are parametrized
over $B$. We study our original fixed given $(\Pi;\Gamma)$-bundle $p\colon E\longrightarrow B$ in a different fashion. We rename its fiber $M$ to avoid confusion with respect to the role that space is playing. In the theory above, $F$ was
a base space for paramentrized spectra and there was no need for $F$ to be dualizable. We now consider the case when $M$ is stably dualizable, so that $\Sigma^{\infty}M_+$ is dualizable, and we write $\tau_M$ for the transfer map $S\longrightarrow \Sigma^{\infty}M_+$ in $\Gamma\scr{S}$, as defined in and after \myref{tracemap}. We apply \myref{PtrP} with $F=*$ and $X = \Sigma^{\infty}M_+$ to obtain the following special case. Here we use the diagonal map induced by the diagonal
map of $M$. Observe that, by \myref{PSmash},
$$\U{{_*}{i^*\Sigma^{\infty}M_+}} \simeq \Sigma^{\infty}\U{{_*}{M_+}}
= \Sigma^\infty_B (E,p)_+.$$
\begin{thm}\mylabel{trantran}
Let $M$ be a compact $\Gamma$-ENR and let $p\colon E\longrightarrow B$
be a $(\Pi;\Gamma)$-bundle with fiber $M$ and associated principal
$(\Pi;\Gamma)$-bundle $P$. Let $\phi$ be a self-map of $\Sigma^{\infty}M_+$.
Then
\[\tau(\bar{P}_*i^*\phi)\simeq \bar{P}_*i^*(\tau(\phi)):
S_B\to \Sigma^\infty_B (E,p)_+.\]
Therefore, taking $\phi = \text{id}$ and applying $r_!$, $r\colon B\longrightarrow *$,
\[\tau_E \simeq r_!\bar{P}_*i^*\tau_M\colon \Sigma^{\infty} B_+\longrightarrow \Sigma^{\infty} E_+.\]
\end{thm}
This result gives a clear and precise comparison between
the specialization to bundles of the globally defined transfer map for
Hurewicz fibrations and the fiberwise transfer map for bundles. Effectively,
we have inserted the transfer map for $M_+$ fiberwise into $P\times_{\Pi}(-)$
to obtain an alternative description of the transfer map for the dualizable
$G$-spectrum $\Sigma^{\infty}(E,p)_+$ over $B$.
There is a useful reinterpretation of the
description of transfer maps given by \myref{trantran}. Consider
${\pi}\colon P\longrightarrow *$. Observe that, by \myref{chvsfixorbit},
instead of applying $r_!$,
$r\colon B\longrightarrow *$, to orbit spectra under the action of $\Pi$,
we could first apply ${\pi}_!$ and then pass to orbits. For a
$\Gamma$-spectrum $X$, we have a natural isomorphism
$${\pi}_!{\pi}^*i^*X\iso P_+\sma i^*X$$
and a natural equivalence
$$i_*(P_+\sma i^*X) \htp P_+\sma X.$$
\begin{cor}
let $M$ be a compact $\Gamma$-ENR and let $p\colon E\longrightarrow B$ be a
$(\Pi;\Gamma)$-bundle with fiber $M$ and associated
principal $(\Pi;\Gamma)$-bundle $P$. Then the transfer $\tau_E\colon \Sigma^{\infty} B_+\longrightarrow \Sigma^{\infty} E_+$
is obtained by passage to orbits over $\Pi$ from the map
$$\tilde{\tau} = \text{id}\sma i^*\tau_M
\colon P_+\sma i^*S\longrightarrow P_+\sma i^*\Sigma^{\infty} M_+,$$
and $i_*\tilde{\tau}$ can be identified with
$$\text{id}\sma \tau_M\colon P_+\sma S\longrightarrow P_+\sma \Sigma^{\infty}M_+.$$
\end{cor}
\begin{rem} The corollary gives exactly the transfer map as defined
by Lewis and May \cite[IV.3.1]{LMS}. Working in the nonparametrized
context, they tried in vain to obtain a spectrum level transfer map
for Hurewicz fibrations over general base spaces. The comparison here
also sheds light on the relationship between the two constructions of
Becker and Gottlieb \cite{BG1, BG2}, both of which require finite
dimensional base spaces. The first is bundle theoretic and is easily
seen to be equivalent to the construction in this section by using
Atiyah duality to interpret $\tau_M$ for a $\Gamma$-manifold $M$.
Precisely, by \cite[IV.2.3]{LMS}, if $M$ is embedded in $V$ with normal
bundle $\nu$ and $\tau$ is the tangent bundle of $M$, then the transfer
map $\tau_M$ is homotopic to the map obtained by applying the functor
$\Sigma^{-V}\Sigma^{\infty}$ to the composite of the Pontryagin-Thom map
$S^V\longrightarrow T\nu$ and the map $T\nu\longrightarrow T(\nu\oplus\tau)\iso M_+\sma S^V$
induced by the inclusion $\nu\longrightarrow \nu\oplus \tau$. The second,
which is generalized to the equivariant setting by Waner \cite{Waner},
is fibration theoretic and is easily seen to be equivalent to the
construction of \S\ref{sec:trfr}. Another approach to the comparison is to show
that suitable Hurewicz fibrations are equivalent to bundles,
as is done by Casson and Gottlieb in \cite{CG}.
\end{rem}
\begin{rem} Since our definition coincides with that of \cite[IV.3.1]{LMS},
the properties of the transfer catalogued in \cite[IV\S\S3--7]{LMS}
apply verbatim. Many of these properties generalize directly to the
parametrized trace and transfer maps of \myref{PtrP}. Actually, the
definition of
\cite[IV.3.1]{LMS} works more generally with $P$, or rather $i^*\Sigma^{\infty}P_+$,
replaced by a general $\Pi$-free $\Gamma$-spectrum $\mathbb{P}$ indexed on $\scr{V}_G$. The
constructions here admit similar generalizations. One way to achieve this with
minimal work is to use the case $P=E(\Pi;\Gamma)$ of the construction
already on hand. Thus, for a $\Pi$-free $\Gamma$-spectrum $\mathbb{P}$ over $F$
indexed on $\scr{V}_G$, we can define
$$ \bar{\mathbb{P}}_{F}i^*X = \overline{E(\Pi;\Gamma)}_{F}(\mathbb{P}\sma_F i^*X)$$
and develop parametrized trace and transfer maps from there.
We leave the further development of the theory to the interested reader.
\end{rem}
\section{Sketch proofs of the compatible triangulation axioms}\label{sec:comptriang}
We must explain why $\text{Ho}\, G\scr{S}_B$ is a closed symmetric monoidal
category with a compatible triangulation, in the sense specified in
\cite{Tri}. We have the closed symmetric monoidal structure and the
triangulation, the latter by \myref{yestrian}. We must prove the
compatibility axioms (TC1)--(TC5) of
\cite[\S4]{Tri}. The essential idea is to verify the axioms using external
smash products and function
objects and then pull back along diagonal maps to obtain the conclusions. The axiom
(TC1) only involves suspension, in our case $\Sigma_B$, and is thus easily checked
using \myref{spacesmashpair}. For (TC2), we must show that the functors
$X\sma_B(-)$, $F_B(X,-)$, and $F_B(-,Y)$ preserve distinquished triangles, where
$X$ and $Y$ are $G$-spectra over $B$.
Either model theoretically or by standard topological arguments with cofiber
sequences and fiber sequences, it is easy to see that these conclusions hold with
$\sma_B$ and $F_B$ replaced by the external functors $\barwedge$ and $\bar{F}$.
Since $\Delta^*$ and therefore its right adjoint $\Delta_*$ are exact, the conclusion
internalizes directly. Similarly, the braid axiom (TC3) and additivity axiom (TC4)
hold for $\barwedge$ by the arguments explained in \cite[\S6]{Tri}, and they pull
back along $\Delta^*$ to give these axioms internally in $\text{Ho}\, G\scr{S}_B$.
The braid duality axiom (TC5) is more subtle because it involves simultaneous use
of $\sma_B$ and $F_B$. Externally, we can work over $B\times B$, using $\barwedge$.
Inspecting the argument in \cite[\S7]{Tri}, we see that the only internal homs used
in the verification of the braid duality axiom are duals of the form
$F(-,T)$ for a suitable approximation $T$ of the unit object. In our context, it
turns out that we need to use two analogues of this functor, one to mimic the
proof of (TC5a) given in \cite[pp\,62-64]{Tri} and another to mimic the proof of
(TC5b) given in \cite[pp\,65-67]{Tri}. For the first, let $T\in G\scr{S}_{B\times B}$
be a fibrant model of the derived $\Delta_*S_B$, so that $\bar{F}(X,T)$ is a model for
$DX = F_B(X,S_B)$ in $\text{Ho}\, G\scr{S}_B$. With this replacement for $F(X,T)$,
the cited proof of (TC5a) goes through, first working externally and then
internalizing along $\Delta^*$. The cited proof of (TC5b) relies on a natural
point-set level map
\begin{equation}\label{oldpair}
F(X,T)\sma F(Y,T)\longrightarrow F(X\sma Y,T),
\end{equation}
and this makes no sense in our external context. Working internally, in
$\text{Ho}\, G\scr{S}_B$, we have such a map
\begin{equation}\label{intequ}
F_B(X,S_B)\sma_B F(Y,S_B)\longrightarrow F_B(X\sma_B Y,S_B),
\end{equation}
but we need a point-set level external model for it to carry out the cited argument.
Let $U$ be a fibrant model for $\Delta_*S_{B\times B}$ in
$G\scr{S}_{B\times B\times B\times B}$. Replacing
the functor $D'(-) = F(X,T)$ used in \cite[pp.\, 66-67]{Tri} with the functor
$$D'(-) =\bar{F}(-,U)\colon G\scr{S}_{B\times B} \longrightarrow G\scr{S}_{B\times B},$$
we find that the cited argument goes through verbatim on the external level,
working in the category $G\scr{S}_{B\times B}$, once we construct a natural map
\begin{equation}\label{goodpair}
\bar{F}(X,T)\barwedge \bar{F}(Y,T) \longrightarrow \bar{F}(X\barwedge Y,U)
\end{equation}
in $G\scr{S}_{B\times B}$ to substitute for the pairing (\ref{oldpair}). Starting
from the $(\barwedge,\bar{F})$ adjunction, we obtain an external pairing
\begin{equation}\label{extpair}
\bar{F}(X,T)\barwedge \bar{F}(Y,T)\longrightarrow \bar{F}(X\barwedge Y,T\barwedge T).
\end{equation}
We also have the natural map
\[\xymatrix{
\Delta_*X\barwedge \Delta_* Y \ar[d]^-{\eta} \\
\Delta_*\Delta^*(\Delta_*X\barwedge \Delta_* Y)\ar[d]^{\iso}\\
\Delta_*(\Delta^*\Delta_*X\barwedge \Delta^*\Delta_* Y)\ar[d]^-{\Delta_*(\varepsilon\barwedge\varepsilon)}\\
\Delta_*(X\barwedge Y).}\]
Applying this with $X=Y=S_B$ and using that $S_B\barwedge S_B$ is isomorphic
to $S_{B\times B}$, we obtain a lift $\xi$ in the diagram
$$\xymatrix{
\Delta_* S_B\barwedge \Delta_*S_B \ar[r] \ar[d]
& \Delta_*(S_B\barwedge S_B)\iso \Delta_*S_{B\times B} \ar[r] & U \ar[d]\\
T\barwedge T \ar@{->}[urr]^{\xi} \ar[rr] & & {*}_{B\times B}}$$
Composing $\bar{F}(X\barwedge Y, \xi)$ with the pairing (\ref{extpair}),
we obtain the required pairing (\ref{goodpair}).
Internalization along $\Delta^*$ is then a not altogether trivial exercise which
shows that, on passage to homotopy categories, application of $\Delta^*$ to the
pairing (\ref{goodpair}) gives a model for the pairing (\ref{intequ}).
The latter pairing can be viewed as a map
$$ \Delta^*(\bar{F}(X,T)\barwedge \bar{F}(Y,T))\longrightarrow
\bar{F}(\Delta^*(X\barwedge Y),T),$$
and the essential point of the exercise is to verify that
$\Delta^*\bar{F}(X\barwedge Y,U)$
is equivalent to $\bar{F}(\Delta^*(X\barwedge Y),T)$. Using that
$\Delta^*S_{B\times B} \iso S_{B}$ and looking at represented functors,
we see that a Yoneda lemma argument reduces the verification to the
proof of a derived analogue of (\ref{four}) that is proven in the
same way as \myref{Wirthmore}.
\chapter{The Wirthm\"uller and Adams isomorphisms}
\section*{Introduction}
This chapter consists of variations on a theme. For a $G$-map
$f\colon A\longrightarrow B$, the base change functor $f^*$ from $G$-spectra over
$B$ to $G$-spectra over $A$ has a left adjoint $f_!$ and a right adjoint $f_*$.
We study comparisons between $f_!$ and $f_*$. As preamble, we show in \S16.1 that there is always a natural map $\phi\colon f_!\longrightarrow f_*$ that relates the two adjunctions. It is an equivalence when $f$ is a homotopy equivalence, but not in general. This comparison is largely formal and applies to analogous sheaf theoretic contexts.
In the rest of the chapter, we use our foundations together with formal arguments developed in \cite{FHM} to obtain a simple proof of a general version of the Wirthm\"uller isomorphism and to reprove the Adams isomorphism
as a special case. This material constitutes a considerably simplified
version of work of Po Hu on the same topic \cite{Hu}. We consider $G$-bundles
$p\colon E\longrightarrow B$, as in \S3.2 and \S15.3. We assume that the fiber $M$
is a smooth closed $\Gamma$-manifold; manifolds with boundary work similarly.
The generalized Wirthm\"uller isomorphism computes the relatively mysterious right adjoint $p_*$ of the functor $p^*$ as a suitable shift of the relatively familiar left adjoint $p_!$.
We explain the result in the special case when $E\longrightarrow B$ is $M\longrightarrow *$ in
\S16.2, but we defer the proof to \S16.5. We also show how to relate the Wirthm\"uller
isomorphisms for $M$ and $N$ when $N$ is smoothly embedded in $M$.
When $M = G/H$,
the result specializes under the equivalence between the category of
$G$-spectra over $G/H$ and the category of $H$-spectra to the Wirthm\"uller
isomorphism in the form proven by Lewis and May \cite[II\S6]{LMS}.
As explained in \cite{MayW}, the categorical analysis in \cite{FHM} allows
considerable simplification of that proof. Our proof for general $M$ follows
the same pattern, but it is quite different in detail since the special case
$M=G/H$ has certain simplifying features. For example, when $G$ is finite, that case follows formally from Atiyah duality for $G/H$ and the trivial observation that
$H/K_+$ is an $H$-retract of $G/K_+$ for $K\subset H\subset G$.
In \S16.3, we show that the general case of $G$-bundles $p\colon E\longrightarrow B$ reduces
fiberwise to the special case $M\longrightarrow *$. The proof is an immediate application
of the construction $P\times_{\Pi}(-)$ on parametrized $\Gamma$-spectra that was
studied in \S15.3. This allows
a simple fiberwise construction of the $G$-spectrum over $E$ by which one must shift
$p_!$ to obtain the desired isomorphism. With this construction, it is immediate that
the map of $G$-spectra over $B$ that we wish to prove to be an equivalence coincides
on the fiber over $b$ with a map that we know to be an equivalence by the case
$M\longrightarrow *$. Since equivalences are detected fiberwise, that proves the result.
In turn, we prove in \S16.4 that the Adams isomorphism relating orbit spectra and fixed point spectra that was proven by Lewis and May in \cite[II\S8]{LMS}
is a virtually immediate special case of our generalized Wirthm\"uller isomorphism. These results complete the program originated in \cite{MM} of reproving conceptually all of the basic foundational results that were first proven in a less satisfactory ad hoc way in \cite{LMS}. The pioneering work of Po Hu \cite{Hu} paved the way but, in the absence of adequate foundations, the
bundle construction of \S15.3, and the simplifying framework of \cite{FHM},
her arguments were very long and difficult. Our work recovers variant
versions of all of her results. The basic idea that parametrized $G$-spectra should clarify and simplify the Wirthm\"uller and Adams isomorphisms is due
to Gaunce Lewis \cite{LewisE}.
Again, we assume throughout that all given groups $G$ are compact Lie groups
and all given base $G$-spaces are of the homotopy types of $G$-CW complexes.
\section{A natural comparison map $f_!\longrightarrow f_*$}
The Wirthm\"uller isomorphism that is the subject of the next few sections gives an equivalence between $f_*$ and a shift of $f_!$ for certain equivariant bundles $f$. In the course of our work on that, we came upon a curious natural comparison map $f_!\longrightarrow f_*$ for any map $f$ whatever. We have no current applications for it, but since the relationships among base change functors are so central to the theory and its applications, we shall describe that map in this digressive section. It works just as well on the level of ex-spaces and indeed quite generally in other contexts where one has analogous base change adjunctions.
\begin{thm}\mylabel{Please}
Let $f\colon A\longrightarrow B$ be a $G$-map and let $X$ be a $G$-spectrum over $A$. Let $\varepsilon\colon f^*f_*\longrightarrow\text{Id}$ and $\sigma\colon f_!f^*\longrightarrow \text{Id}$ denote the counits of the adjunctions $(f^*,f_*)$ and $(f_!,f^*)$ relating $\text{Ho}\, G\scr{S}_A$ and $\text{Ho}\, G\scr{S}_B$. There is a natural map $\phi\colon f_!X\longrightarrow f_*X$ in $\text{Ho}\, G\scr{S}_B$ such that the following diagram commutes:
\begin{equation}\label{newdia}
\xymatrix{
& f_!f^*f_*X \ar[dl]_{f_!\varepsilon} \ar[dr]^{\sigma} & \\
f_!X \ar[rr]_-{\phi} & & f_* X.\\}
\end{equation}
\end{thm}
\begin{proof}
Let $K = (K,p,s)$ be an ex-space over $A$. Then $f_!K = K\cup_A B$ and $f^*f_!K = f_!K\times_B A$. Here points $(f(a),a)$ in $B\times_B A$ are identified with points $(s(a),a)$ in $K\times_B A$, and we see that $f^*f_!K$ can be identified with the pullback $K\times_B A$. The projection to $K$ is then a map $\psi\colon f^*f_!K\longrightarrow K$ of ex-spaces over $A$. When $K = f^*L$ for an ex-space $L$ over $B$, $\psi = f^*\sigma\colon f^*f_!f^*L\longrightarrow f^*L$ since $f^*\sigma$ is also given by the projection $f^*L\times_B A\longrightarrow f^*L$. Passing to spectra over $A$ levelwise, we obtain a natural map $\psi\colon f^*f_!X\longrightarrow X$ of spectra over $A$ such that $\psi = f^*\sigma$ when $X = f^*Y$.
To pass to homotopy categories, we take two steps. Factoring $f$ as a composite of a homotopy equivalence and an $h$-fibration, we see that we may assume that
$f$ is either a homotopy equivalence or an $h$-fibration. In the former case, $f_*$ must be inverse to the equivalence $f^*$ and thus equivalent to $f_!$. Here $\varepsilon$ and $\sigma$ are equivalences and we may as well define $\phi$ by the commutativity of (\ref{newdia}). In the latter case, we may work in $G\scr{E}_A$. Since $f$ is an $h$-fibration, we have a natural homotopy equivalence $\mu\colon Tf^*Y\longrightarrow f^*TY$ for $Y\in G\scr{E}_B$. The derived functor $f_!$ is induced by $Tf_!$, and $TX$ is naturally homotopy equivalent to $X$ when $X\in G\scr{E}_A$. The composite
$$\xymatrix{f^*Tf_!X \htp Tf^*f_! X \ar[r]^-{T\psi} & TX\htp X}$$
gives a natural map $\psi\colon f^*f_!X\longrightarrow X$ in $hG\scr{E}_A$. When $X = f^*Y$, we have the commutative naturality diagram
$$\xymatrix{
Tf^*f_!f^*Y \ar[d]_{\mu} \ar[rr]^-{T\psi = Tf^*\sigma} & &Tf^*Y \ar[d]^{\mu}\\
f^*Tf_!f^*Y \ar[rr]_-{f^*T\sigma} & & f^*TY.}$$
The bottom arrow is the derived version of $f^*\sigma$ and the composite around the top is the derived version of $\psi$. Using the equivalences of categories of \S13.5, we obtain a natural map $\psi\colon f^*f_!X \longrightarrow X$ in $\text{Ho}\, G\scr{S}_A$. Let $\eta\colon \text{Id}\longrightarrow f_*f^*$ be the unit of the (derived) adjunction $(f^*,f_*)$ and define $\phi\colon f_!X\longrightarrow f_*X$ to be the adjoint of $\psi$ in $\text{Ho}\, G\scr{S}_A$, so that $\phi = f_*\psi\com \eta$. For $Y\in \text{Ho}\, G\scr{S}_B$, we have $f^*\sigma = \psi\colon f^*f_!f^*Y\longrightarrow f^*Y$. It follows formally that (\ref{newdia}) commutes. Indeed,
$$\varepsilon\com f^*\sigma = \varepsilon\com \psi = \psi\com f^*f_!\varepsilon.$$
The adjoint of $\varepsilon\com f^*\sigma$ is $\sigma$ since
$$f_*(\varepsilon\com f^*\sigma)\com\eta = f_*\varepsilon\com f_*f^*\sigma\com \eta
=f_*\varepsilon\com \eta\com \sigma = \sigma,$$
while the adjoint of $\psi\com f^*f_!\varepsilon$ is $\phi\com f_!\varepsilon$ since
\[f_*(\psi\com f^*f_!\varepsilon)\com \eta = f_*\psi\com f_*f^*f_!\varepsilon\com\eta
=f_*\psi \com\eta\com f_!\varepsilon = \phi\com f_!\varepsilon. \qedhere\]
\end{proof}
\section{The Wirthm\"{u}ller isomorphism for manifolds}\label{sec:mfldwirth}
The classical Wirthm\"{u}ller isomorphism in the equivariant stable homotopy category relates induction and coinduction, the left and right adjoints of the restriction functor from $G$-spectra to $H$-spectra. More precisely, it says that for $H$-spectra $X$, there is a natural equivalence of $G$-spectra \begin{equation}\label{Wirth1} F_H(G_+, X) \simeq G_+\wedge_H (X\wedge S^{-L}), \end{equation} where $L$ is the tangent representation at the identity coset in $G/H$ and $S^{-L}$ is the inverse of the invertible $H$-spectrum $\Sigma^\infty S^L$. Here again, ``equivalence'' means isomorphism in the relevant stable homotopy category and is denoted by $\htp$.
One can also think of this in terms of base change functors. Recall from \myref{changestoo} that the category of $H$-spectra is equivalent to the category of $G$-spectra over $G/H$. The equivalence is given in one direction by applying the functor $G\times_H -$, and in the other by taking the fiber over the identity coset. This equivalence preserves all structure in sight, including the symmetric monoidal and model structures. The map $r\colon G/H\longrightarrow *$ induces a pullback functor $r^*$ from $G$-spectra to $G$-spectra over $G/H$, and it has left and right adjoints $r_!$ and $r_*$. The functor $r^*$ corresponds under the equivalence to the restriction functor and therefore $r_!$ and $r_*$ correspond to the induction and coinduction functors. In this terminology, the Wirthm\"{u}ller isomorphism (\ref{Wirth1}) takes the form
\begin{equation}\label{Wirth1bis}
r_* X \simeq r_!(X\wedge_{G/H} C_{G/H})
\end{equation}
for $G$-spectra $X$ over $G/H$, where $C_{G/H}= \iota_!S^{-L}$, $\iota:H\subset G$ (see \myref{eyeeye}).
We think of $G/H\longrightarrow *$ as the simplest kind of a bundle with a compact manifold as fiber, and we generalize (\ref{Wirth1bis}) to maps $p\colon E\longrightarrow B$ that are equivariant bundles with a smooth closed manifold $M$ as a fiber. We discuss the case $B=*$ in this section and prove the general case in the next. However, it is convenient to begin by describing the form of the map that gives the equivalence in general. For that, we require a $G$-spectrum $C_p$ over $E$ together with an equivalence
\begin{equation}\label{Wobj}
\xymatrix{\alpha_p\colon p_!C_p \ar[r]^-{\htp} & D(p_! S_{E})}
\end{equation}
that identifies the dual of $p_!S_E$.
We call $C_p$,\@bsphack\begingroup \@sanitize\@noteindex{Cp@$C_p$} together with $\alpha_p$, a {\em Wirthm\"uller object}.\index{Wirthmuller object@Wirthm\"uller object}
In \cite{FHM}, Fausk, Hu, and May give a categorical discussion of equivalences of Wirthm\"uller type, including a simplifying formal analysis that describes the minimal amount of information that is needed to prove such a result. In particular, given a Wirthm\"uller object $C_p$, they define a canonical candidate
$$\omega_p\colon p_*X\longrightarrow p_!(X\sma_E C_p)$$
for an equivalence, namely the composite displayed in the commutative diagram
\begin{equation}\label{omega}
\xymatrix{
p_*X \htp p_*X\sma_B D(S_B)\ar[dd]_{\omega_{p}} \ar[rr]^-{\text{id}\sma_BD(\sigma)}
& & p_*X\sma_B D(p_!S_E) \\
& & p_*X\sma_B p_!C_p \ar[u]_{\text{id}\sma_B \alpha_p}^{\htp}\\
p_!(X\sma_E C_p) & & p_!(p^*p_*X\sma_E C_p) \ar[u]_{\htp}\ar[ll]^{p_!(\varepsilon\sma_E\text{id})}.}
\end{equation}
The maps $\sigma\colon p_!S_E\htp p_!p^*S_B \longrightarrow S_B$ and $\varepsilon\colon p_*p^*X\longrightarrow X$
are given by the counits of the adjunctions $(p_!,p^*)$ and $(p^*,p_*)$. The
arrow labelled $\htp$ is an equivalence given by the derived version of the
projection formula (\ref{four}) that is proven in \myref{Wirthmore}.
When $M$ is a smooth closed $G$-manifold and $r$ is the map $\colon M\longrightarrow *$, we write $C_M$ for a Wirth\-m\"uller object $C_r$ and we write $\omega_M$ for $\omega_r$. It is easy to describe $C_M$. Let $\tau$ be the tangent $G$-bundle of $M$. Embed $M$ in a $G$-representation $V$ and let $\nu$ be the normal $G$-bundle of the embedding. By Atiyah duality, the union $M_+$ of $M$ and a
disjoint basepoint is $V$-dual to the Thom $G$-space $T\nu$. A detailed equivariant proof is given in \cite[III\S5]{LMS}, but we require little
beyond the mere statement.
For a $G$-vector bundle $\xi$ over a $G$-space $B$, let $S^{\xi}$ denote the fiberwise one-point compactification of $\xi$, with section given by the points at infinity. This ex-$G$-space over $B$ must not be confused with the Thom complex $T\xi$. The latter is obtained by identifying the section to a point and is precisely $r_!S^{\xi}$, $r\colon B\longrightarrow *$.
\begin{defn}\mylabel{CM}
Define $C_M$ to be the $G$-spectrum $\Sigma_M^{-V}\Sigma^{\infty}_M S^{\nu}$ over $M$.
\end{defn}
\begin{rem}\mylabel{tauinv}
By \myref{spaceFDT}, the suspension $G$-spectrum
$\Sigma^{\infty}_MS^{\tau}$ is invertible. Visibly, $C_M$ is its inverse.
\end{rem}
\begin{lem}\mylabel{starting}
There is an equivalence $\alpha_M\colon r_!C_M \longrightarrow D(r_!S_{M})$,
$r\colon M\longrightarrow *$.
\end{lem}
\begin{proof}
Since $S_{M}(V) = M\times S^V$, $r_!S_{M} = \Sigma^{\infty}M_+$. Since $r_!S^{\nu} = T\nu$ and $r_!$ commutes with shift desuspension functors, $r_!C_M$ is equivalent to $\Sigma^{-V}\Sigma^{\infty}T\nu$. There is a canonical evaluation map $\text{ev}\colon T\nu\wedge M_+\longrightarrow S^V$ of a duality \cite[p.152]{LMS}. Explicitly, using the diagonal of $M$ and the zero section of $\nu$ we obtain an embedding of $M$ in $\nu\times M$ with trivial normal bundle $M\times V$, and $\text{ev}$ is composite of the Pontryagin-Thom map associated to this embedding and the projection $M_+\sma S^V\longrightarrow S^V$. We apply the functor
$\Sigma^{-V}\Sigma^{\infty}$ to obtain
$$\Sigma^{-V}\Sigma^{\infty}T\nu \sma \Sigma^{\infty}M_+ \htp \Sigma^{-V}\Sigma^{\infty}(T\nu\sma M_+) \longrightarrow \Sigma^{-V}\Sigma^{\infty} S^V\htp S.$$
Atiyah duality states that the adjoint of this map is an equivalence from $\Sigma^{-V}\Sigma^{\infty}T\nu$ to $D(M_+)$. This is the required map $\alpha_M$.
\end{proof}
We shall prove the following result in \S\ref{sec:wirthpf}.
\begin{thm}[The Wirthm\"uller isomorphism for manifolds]\mylabel{thmW1}
For $G$-spectra $X$ over $M$ and $r\colon M\longrightarrow *$, the map
$$\omega_M\colon r_*X\longrightarrow r_!(X\sma_M C_M)$$
is a natural equivalence in the homotopy category $\text{Ho}\, G\scr{S}$ of $G$-spectra.
\end{thm}
In an earlier draft of this paper, we thought we could reduce the general case of \myref{thmW1} to the special case $M = G/H$. However, instead of leading to a simplifiction, the argument we had in mind leads to an interesting relative version of the Wirthm\"uller isomorphism. Its starting point is the following observation.
\begin{lem}\mylabel{WMGH}
Let $i\colon N\longrightarrow M$ be an embedding of smooth closed $G$-manifolds and let $\nu_{M,N}$ be the normal bundle of $i$. Then $C_N$ is equivalent to $S^{\nu_{M,N}}\sma_N i^*C_M$.
\end{lem}
\begin{proof}
An embedding of $M$ in a representation $V$ restricts along $i$ to an embedding of $N$ in $V$, and $i^*\nu_M\oplus \nu_{M,N} \iso \nu_{N}$. Commutation relations in \myref{SIfSI} give that
$$i^*\Sigma_M^{-V}\Sigma^{\infty}_M S^{\nu_M}
\htp \Sigma_{N}^{-V}\Sigma^{\infty}_{N}i^*S^{\nu_M}.$$
The conclusion follows after smashing with $S^{\nu_{M,N}}$.
\end{proof}
\begin{cor}[The relative Wirthm\"uller isomorphism]
Let $i\colon N\longrightarrow M$ be a smooth embedding of closed $G$-manifolds. For $G$-spectra $X$ over $N$, there is a natural equivalence
$$\omega_{M,N}\colon r_*i_* X\longrightarrow r_*i_!(X\sma_N S^{\nu_{M,N}}).$$
\end{cor}
\begin{proof}
Here $r\colon M\longrightarrow *$. Write $q = r\com i\colon N\longrightarrow *$. Then $q_*\htp r_*i_*$ and $q_!\htp r_!i_!$. Define $\omega_{M,N}$ by commutativity of the diagram of equivalences
$$\xymatrix{
q_*X \ar[d]_{\htp} \ar[r]^{\omega_N} & q_!(X\sma_N C_N) \ar[d]^{\htp}\\
r_*i_*X \ar[d]_{\omega_{M,N}}
& r_!i_! (X\sma_N S^{\nu_{M,N}}\sma_N i^*C_M) \ar[d]^{\htp}\\
r_*i_!(X\sma_N S^{\nu_{M,N}})\ar[r]_-{\omega_M}
& r_!(i_!(X\sma_N S^{\nu_{M,N}})\sma_M C_M).}$$
Here the derived version of the projection formula (\ref{four}) gives the lower right equivalence.\end{proof}
We explain the strategy that we have not implemented for deducing the Wirth\-m\"uller
isomorphism for $M$ from the Wirthm\"uller isomorphism for orbits.
\begin{rem}
One can use relative Atiyah duality to define an intrinsic map $\alpha_{M,N}$ and use $\alpha_{M,N}$ to define a map $\omega_{M,N}$ directly. One can then obtain the displayed diagram by a chase. If one could prove directly that $\omega_{M,N}$ was an equivalence, then, using the invertibility of $\Sigma^{\infty}_NS^{\nu_{M,N}}$, one could deduce that $\omega_M$ is an equivalence on all $i_!X$ if $\omega_N$ is an equivalence. By \myref{detect}, $\omega_M$ is an isomorphism for all $Y$ if it is an isomorphism for all $Y$ in the detecting set $\scr{D}_M$ of \myref{detecting}. Those $Y$, namely the $S^{n,b}_H$ for $b\in M$ and $H\subset G_b$, are of the form $\tilde{b}_!X$, where $\tilde{b}\colon G/G_b\longrightarrow M$ is the inclusion of the orbit of $b$ and $X$ is a $G$-spectrum over $G/G_b$. Thus the Wirthm\"uller isomorphism for orbits would imply the Wirthm\"uller isomorphism for $M$.
\end{rem}
\section{The fiberwise Wirthm\"uller isomorphism}\label{sec:fibwirth}
As in \S\ref{sec:bdlconstr}, let $G$ be a quotient $\Gamma/\Pi$, where $\Pi$ is a normal subgroup of a compact Lie group $\Gamma$. Let $M$ be a smooth closed $\Gamma$-manifold and let $p\colon E\longrightarrow B$ be a $(\Pi;\Gamma)$-bundle with fiber $M$. This means that $p$ has an associated principal $(\Pi;\Gamma)$-bundle $\pi\colon P\longrightarrow B$ and $p$ is the associated $G$-bundle $E = \T{M}\longrightarrow P/\Pi=B$. We apply the functor $\U{_M}$ to the Wirthm\"uller object $C_M$ to obtain the Wirthm\"uller object $C_p$, and we apply $\U{_M}$ to $\alpha_M$ to obtain the required equivalence $\alpha_p$. This means that the Wirthm\"uller object for $p$ is obtained by inserting the Wirthm\"uller object for $M$ fiberwise into the functor $\T{(-)}$.
\begin{defn}
Define $C_p$ to be the $G$-spectrum $\U{_{M}{i^*C_{M}}}$ over $E$, where $i^*$ is the change of universe functor associated to the inclusion of the $\Pi$-trivial $\Gamma$-universe in the complete $\Gamma$-universe.
\end{defn}
\begin{rem}\mylabel{tauinvtoo}
Recall \myref{tauinv}. By \myref{spaceFDT}, the suspension $G$-spectrum $\Sigma^{\infty}_E(P\times_{\Pi} S^{\tau})$ is invertible. The Wirthm\"uller object $C_p$ is its inverse.
\end{rem}
\begin{lem}\mylabel{startingtoo}
There is an equivalence $\alpha_p\colon p_!C_p \longrightarrow D(p_!S_{E})$.
\end{lem}
\begin{proof}
We define $\alpha_p$ to be the composite
\begin{equation}\label{alf}
\xymatrix{
p_!\U{_M{i^*C_M}} \ar[r] & \U{_{*}{i^*r_!C_M}} \ar[rr]^-{\U{_{*}{i^*\alpha_M}}} & &
\U{_{*}{i^*D(r_!S_M)}} \ar[r] & D(p_!S_{E}).\\}
\end{equation}
The left arrow is given by the first equivalence of \myref{Pfibers} and the last equivalence of \myref{chvschuni}. The middle arrow is an equivalence since $\alpha_M$ is one. The right arrow is the following composite equivalence,
\begin{align*}
\U{_{*}{i^*D(r_!S_M)}} & \htp \U{_{*}{D(i^*r_!S_M)}} \\
& \htp \U{_{*}{D(r_!i^*S_M}}) && \text{by Propositions \ref{chvschuni} and \ref{Nfreeii}}\\
& \htp D(\U{_{*}{r_!i^*S_M}}) && \text{by \myref{PSmash}}\\
& \htp D(p_!\U{_{M}{i^*S_M}}) && \text{by \myref{Pfibers}}\\
& \htp D(p_!S_E) && \text{by \myref{PSmash}}.
\end{align*}
For the first displayed equivalence, $r_!S_M \simeq \Sigma^{\infty}M_+$ by \myref{lesstrivial}, hence
$$D(r_!S_M) \simeq D(\Sigma^{\infty}M_+)\simeq F(M_+,S).$$
For based $\Gamma$-spaces $T$, $i^*F(T,S)\iso F(T,i^*S)$ by inspection. If
$T$ is a based $\Gamma$-CW complex, this is an isomorphism of Quillen right
adjoints and so descends to homotopy categories. Again by inspection,
$$i^*\Sigma^{\infty}\iso \Sigma^{\infty}\colon G\scr{K}_*\longrightarrow G\scr{S}^{\Pi-\text{triv}}.$$
This isomorphism passes to homotopy categories since both sides take
$q$-equi\-va\-lences to level $q$-equi\-va\-lences. Therefore $i^*S\htp S$
and $F(T,i^*S)\htp D(i^*\Sigma^{\infty}T)$.
\end{proof}
\begin{thm}[The fiberwise Wirthm\"uller isomorphism]\mylabel{thmW2}
For $G$-spec\-tra $X$ over $E$, the map
$$\omega_p\colon p_* X \longrightarrow p_!(X\sma_E C_p)$$
is a natural equivalence of $G$-spectra over $B$.
\end{thm}
\begin{proof}
The action of $G_b$ on the fiber $E_b\iso \rho_b^*M$ of $b\in B$ is smooth, hence the Wirthm\"uller isomorphism for manifolds gives the result for $r\colon E_b\longrightarrow *$. We claim that the restriction $\alpha_b$ of $\alpha_p$ to the fiber over $b$ is an equivalence
$$\alpha_{E_b}\colon r_!C_{E_b}\longrightarrow D(r_!S_{E_b})$$
of $G_b$-spectra of the form used to prove \myref{thmW1} for $r$. Indeed, with $p_b = r$, $i\colon E_b\subset E$ and $\iota\colon G_b\subset G$, the derived version of \myref{Johann2} and \myref{tEtB} give that the source of $\alpha_b$ is
$$ (p_!\U{_M{C_M}})_b \htp r_!i^*\iota^*\U{_M{C_M}}\htp r_!\rho_b^*C_M \iso r_!C_{E_b}.$$
For the last isomorphism we must view the representation $V$ of $\Gamma$ that appears in the definition of $C_M$ as a representation of $G_b$ by pullback along $\rho_b$. Similarly, using \myref{reassuring} and the derived version of \myref{Johann2}, the target of $\alpha_b$ is
$$ D(p_!S_{G,E})_b \htp D((p_!S_{G,E})_b) \htp D(r_!i^*\iota^*S_{G,E}) \htp D(r_!S_{G_b,E_b}).$$
In view of the role of $\alpha_M$ in the definition of $\alpha_p$, diagram chases from the definitions show that $\alpha_b$ agrees under these equivalences with the $G_b$-equivalence $\alpha_{E_b}$.
Now, looking at the definition of $\omega_p$ (\ref{omega}), we see that, aside from the equivalence $\alpha_p$, its constituent maps are just counits of adjunctions and derived isomorphisms coming from the closed symmetric monoidal structures. By \myref{reassuring} and the derived versions of commutation relations in \myref{Johann}, these maps restrict on fibers to maps of the same form. Therefore the restriction
$$\omega_b\colon (p_* X)_b \longrightarrow (p_{!}(X\sma_E C_p))_b$$
of $\omega_p$ to the fiber over $b$ can be identified with the map of $G_b$-spectra
$$ \omega_{E_b}\colon r_* X_b \longrightarrow r_!(X_b\sma_{E_b} C_{E_b}).$$
This map is an equivalence of $G_b$-spectra by \myref{thmW1}. Since equivalences of $G$-spectra over $B$ are detected fiberwise, this implies that $\omega_p$ is an equivalence.
\end{proof}
\begin{rem}
When $\Gamma = G\times \Pi$ and only $\Pi$ acts on $M$, one can think of $p\colon E\longrightarrow B$ as a topological $G$-bundle with a reduction of its structural group to a suitably large compact subgroup $\Pi$ of the group of diffeomorphisms of $M$. Our fiberwise Wirthm\"uller isomorphism theorem is a variant of the main theorem, \cite[4.8]{Hu}, of a paper of Po Hu. She worked with $\text{Diff}(M)$ itself as an implicit structure group, without use of an auxiliary group $\Pi$ and without an ambient group $\Gamma$. That bundle theoretic framework leads to formidable complications, hence her arguments are very much more difficult than ours. Her result is both more and less general than the specialization of ours to the case $\Gamma = G\times \Pi$: it allows bundles that might not admit a single compact structure group $\Pi$, but it requires the base spaces to be $G$-CW
complexes with countably many cells. It does not handle more general group extensions.
\end{rem}
\section{The Adams isomorphism}\label{sec:adams}
Let $N$ be a normal subgroup of $G$ and let $\epsilon\colon G\longrightarrow J$ be the quotient by $N$. The conjugation action of $G$ on $N$ induces an action of $G$ on the tangent space of $N$ at the identity element, giving us the adjoint representation $A = A(N;G)$. Let $(i_*,i^*)$ be the change of universe adjunction associated to the inclusion $i\colon q^*\scr{V}_J\longrightarrow \scr{V}_{G}$ of
the complete $J$-universe $\scr{V}_J$ as the universe of $N$-trivial
representations in the complete $G$-universe $\scr{V}_{G}$.
Recall the discussion of $N$-free $G$-spectra from \S\ref{sec:PIFree}, where $\Pi$ and $\Gamma$ played the roles of $N$ and $G$.
\begin{thm}[Adams isomorphism]
For $N$-free $G$-spectra $X$ in $G\scr{S}^{\text{$N$-triv}}$, there is a natural equivalence
\[ X/N\htp (i^*\Sigma^{-A}i_* X)^N \]
in $\text{Ho}\, J\scr{S}^{\text{$N$-triv}}.$
\end{thm}
We shall derive this by applying the fiberwise Wirthm\"{u}ller isomorphism to the quotient $G$-map $p\colon E(N;G) \longrightarrow B(N;G)$, where $E(N;G)$ is the
universal $N$-free $G$-space and $B(N;G) = E(N;G)/N$. To place ourselves in the bundle theoretic context of the previous section, we give another description of $p$, following \cite[II\S7]{EHCT}. It is formal and would similarly identify $p\colon E\longrightarrow E/N$ for any $N$-free $G$-space $E$. Let $\Gamma = G\ltimes N$ be the semi-direct product of $G$ and $N$, where $G$ acts by conjugation on $N$. Write $\Pi$ for the normal subgroup $\{e\}\ltimes N$ of $\Gamma$. We then have an extension
\[1\longrightarrow \Pi \longrightarrow \Gamma \stackrel{\theta}{\longrightarrow} G \longrightarrow 1,\]
where $\theta(g,n)=gn$. Give $N$ the $\Gamma$-action $(g,n)\cdot m = gnmg^{-1}$. Then $N\cong \Gamma/G$ as $\Gamma$-spaces, where we view $G$ as the subgroup $G\ltimes\{e\}$ of $\Gamma$. The composite
\[E(N;G)\cong \theta^*E(N;G)\times_\Pi (\Gamma/G)
\longrightarrow \theta^* E(N;G)\times_\Pi * \cong B(N;G)\]
induced by $\Gamma/G\longrightarrow *$ is $p$. Since $\theta^*E(N;G)$ is a $\Pi$-free $\Gamma$-space, we see that $p$ is a bundle with fiber $\Gamma/G\cong N$ to which the fiberwise Wirthm\"uller isomorphism applies. We must identify the relevant Wirthm\"uller object. We write $r$ for the map $E(N;G)\longrightarrow *$.
\begin{prop}\mylabel{WirthAd}
The Wirthm\"{u}ller object $C_{p}$ is $r^*S^{-A}$.
\end{prop}
\begin{proof}
The tangent bundle of $\Gamma/G\cong N$ is the trivial bundle $N\times A$ \cite[p.\,99]{LMS}. Indeed, let $\Gamma$ act on $A$ via the projection $\epsilon\colon \Gamma\longrightarrow G$, $\varepsilon(n,g)=g$. We obtain a $\Gamma$-trivialization of the tangent bundle of $\Gamma/G$ by sending $(n,a)\in N\times A$ to $d_eL_n(a)$, where $d_eL_n$ is the differential at $e$ of left translation by $n$. It follows that the tangent bundle along the fibers of $p$ is also trivial:
\[\theta^*E(N;G)\times_N(\Gamma/G\times A)
\cong (\theta^*E(N;G)\times_N\Gamma/G))\times A\cong E(N;G)\times A.\]
Thus the spherical bundle of tangents along the fiber is $E(N;G)\times S^A=r^*S^A$, and the inverse of its suspension $G$-spectrum over $E(N;G)$ is $r^*S^{-A}$. In view of \myref{tauinvtoo}, this gives the conclusion.
\end{proof}
\begin{proof}[Proof of the Adams isomorphism]
Let $X\in G\scr{S}^{\text{$N$-triv}}$ be $N$-free. Applying the fiberwise Wirthm\"{u}ller isomorphism to the $G$-spectrum $r^*i_*X$ over $E(N;G)$ and using that $C_p$ is $r^*S^{-A}$, we obtain a natural equivalence
$$ p_*r^*i_* X \htp p_!(r^*i_*X\sma_{E(N;G)} r^*S^{-A})$$
of $G$-spectra over $B(N;G)$. Write $\bar{r}$ for the map $B(N;G)\longrightarrow *$, so that $\bar{r}\com p = r$. Applying the functor $\bar{r}_!((i^*(-))^N)$ to the displayed equivalence, we obtain a natural equivalence
\[\bar{r}_!((i^*p_*r^*i_*X)^N) \htp \bar{r}_!((i^*p_!(r^*i_*X\wedge_{E(N;G)}r^*S^{-A}))^N)\]
in $\text{Ho}\, J\scr{S}^{\text{$N$-triv}}$. We proceed to identify both sides. The source is
\begin{align*}
\bar{r}_!((i^*p_*r^*i_*X)^N)
&\htp \bar{r}_!((p_*r^*i^*i_*X)^N) && \text{by \myref{chvschuni}}\\
&\htp \bar{r}_!((p_*r^*X)^N) && \text{by \myref{Nfreeii}}\\
&\htp \bar{r}_!((p_!r^*X)/N) && \text{by \myref{ouch}}\\
&\htp (\bar{r}_!p_!r^*X)/N && \text{by \myref{chvsfixorbit}}\\
&\htp (r_!r^*X)/N && \text{by functoriality}\\
&\htp X/N. && \text{by \myref{Nfree}}.
\end{align*}
The target is
\begin{align*}
\bar{r}_!((i^*p_!(r^*i_*X &\wedge_{E(N;G)}r^*S^{-A}))^N)\\
&\htp \bar{r}_!((i^*p_!r^*\Sigma^{-A}i_*X)^N) && \text{by \myref{symmonhtp}}\\
&\htp (\bar{r}_!i^*p_!r^*\Sigma^{-A}i_*X)^N && \text{by \myref{chvsfixorbit}}\\
&\htp (i^*\bar{r}_!p_!r^*\Sigma^{-A}i_*X)^N && \text{by Propositions \ref{chvschuni} and \ref{Nfreeii}}\\
&\htp (i^*r_!r^*\Sigma^{-A}i_*X)^N && \text{by functoriality}\\
&\htp (i^*\Sigma^{-A}i_*X)^N && \text{by \myref{Nfree}.}\qedhere
\end{align*}
\end{proof}
\begin{rem}
In outline, the proof just given is essentially that indicated by
Po Hu \cite[pp 81--99]{Hu}. However, her argument, although more conceptual,
is a good deal longer and more complicated than the original
proof in \cite[pp 96--102]{LMS}.
\end{rem}
\section{Proof of the Wirthm\"uller isomorphism for manifolds}\label{sec:wirthpf}
We prove \myref{thmW1} here. Thus consider $r\colon M\longrightarrow *$ for a smooth compact $G$-manifold $M$. With $C_M = \Sigma_M^{-V}\Sigma^{\infty}_M S^{\nu}$, the diagram (\ref{omega}) displays a canonical map
$$\omega = \omega_M\colon r_*(X)\longrightarrow r_!(X\sma_M C_M)$$
of $G$-spectra, where $X$ is a $G$-spectrum over $M$. We must show that $\omega$ is an equivalence. In outline, we follow the pattern of proof explained in \cite{FHM} and illustrated in the case $M = G/H$ in \cite{MayW}, but the details are very different from those applicable in that special case.
We first describe a formal reduction implied by the results of \cite{FHM}. Consider the set $\scr{D}_M$ of detecting objects in $\text{Ho}\, G\scr{S}_M$ that is specified in \myref{detecting}. The objects in $\scr{D}_M$ are compact, by \myref{compact}, and dualizable. We have the analogous detecting set
$\scr{D}_*$ of compact objects in $\text{Ho}\, G\scr{S}$. For $Y$ in $\scr{D}_*$, $r^*Y$ is dualizable and it follows formally, by \cite[2.1.3(d)]{HPS}, that $r^*Y$ is compact (in the sense of \myref{compact}). Therefore $r_*$, as well as $r_!$, preserves coproducts \cite[7.4]{FHM}. This verifies the hypotheses of the
formal Wirthm\"uller isomorphism theorem, \cite[8.1]{FHM}, and that result
shows that $\omega$ will be an equivalence for all $G$-spectra $X$ over $M$ if
it is an equivalence for those $X$ in $\scr{D}_M$.
Such $X$ are of the form $S^{n,m}_H=\widetilde{m}_!\iota_!S^n_H$, where $n\in \mathbb{Z}$, $m\in M$, $H\subset G_m$, and $\iota$ is the inclusion of $G_m$ in $G$. By commutation with suspension, we can assume that $n\geq 0$. Then $X$ is of the form $\Sigma^{\infty}_MK$ for an ex-$G$-space $K$ over $M$, and $X$ can be any such $G$-spectrum over $M$ in the rest of the proof. By \cite[6.3]{FHM}, it suffices to construct a map ${\xi_X} \colon r^*r_!(X\wedge_M C_M)\longrightarrow X$ such that certain diagrams commute. To be precise, let $\sigma$ and $\zeta$ be the counit and unit of the $(r_!,r^*)$ adjunction, note that $r^*S\iso S_M$, and define maps $\tau = \tau_S$ and $\xi = \xi_{S_M}$ by commutativity of the diagrams
\begin{equation}\label{diagdis}
\xymatrix{S\ar[r]^-{\tau} \ar[d]_{\simeq} & r_!C_M\ar[d]^{\alpha_M}\\
DS \ar[r]_-{D\sigma} & Dr_!r^*S}
\quad\text{and}\quad
\xymatrix{r^*r_!C_M\ar[rr]^-{\xi}\ar[d]_{r^*\alpha_M} & & S_M \ar[d]^\simeq\\
r^*Dr_!S_M \ar[r]_-{\simeq} & Dr^*r_!S_M\ar[r]_-{D\zeta} & DS_M}
\end{equation}
Then define $\tau_Y$ for a general $G$-spectrum $Y$ to be the composite
\begin{equation}\label{tau}
\xymatrix{
\tau_Y\colon Y\simeq Y\wedge S \ar[r]^-{\text{id}\wedge \tau} & Y\wedge r_!C_M
\htp r_!(r^*Y\wedge_M C_M)\\}
\end{equation}
and define $\xi_{r^*Y}$ for the $G$-spectrum $r^*Y$ over $M$ to be the composite
\begin{equation}\label{xi}
\xymatrix{
\xi_{r^*Y}\colon r^*r_!(r^*Y\wedge_M C_M)\simeq
r^*Y \wedge_M r^*r_!C_M \ar[r]^-{\text{id}\wedge \xi} & r^*Y\wedge_M S_M
\simeq r^*Y.\\}
\end{equation}
Here the equivalences are given by the derived versions of (\ref{one})
and the projection formula (\ref{four}). With these notations, we shall
prove the following result.
\begin{prop}\mylabel{prop:spec}
For $X=\Sigma^\infty_M K$, there is a map
$$\xi_X\colon r^*r_!(X\wedge_M C_M)\longrightarrow X$$
such that the composite
\begin{equation}\label{eq:trieq}
\xymatrix{
r_!(X\wedge_M C_M) \ar[d]^-{\tau_{r_!(X\sma_M C_M)}} \\
r_!(r^*r_!(X\wedge_M C_M)\sma_M C_M)
\ar[d]^-{r_!(\xi_X\sma_M\text{id})}\\
r_!(X\wedge_M C_M)\\}
\end{equation}
is the identity map (in $\text{Ho}\, G\scr{S}$)
and, for any map $\theta\colon r^*Y \longrightarrow X$ of $G$-spectra over $M$, the
following diagram commutes in $\text{Ho}\, G\scr{S}_M$.
\begin{equation}\label{eq:nat}
\xymatrix{r^*r_!( r^*Y\wedge_M C_M)\ar[r]^-{\xi_{r^*Y}}
\ar[d]_{r^*r_!(\theta\wedge \text{id})} & r^*Y \ar[d]^\theta\\
r^*r_!(X\wedge_M C_M) \ar[r]_-{\xi_X} & X}
\end{equation}
\end{prop}
This will complete the proof of the theorem by the cited reduction
from \cite{FHM}.
\begin{cor} For $X$ in $\scr{D}_M$, $\omega_M\colon r_*(X)\longrightarrow r_!(X\sma_M C_M)$
is an equivalence with inverse the adjoint of $\xi_X$.
\end{cor}
\begin{proof} Taking $Y$ to be $r_*X$ and $\theta$ to be the counit
of the $(r^*,r_*)$ adjunction in (\ref{eq:nat}), the conclusion
is a direct application of \cite[6.3]{FHM}.
\end{proof}
Thus it suffices to prove \myref{prop:spec}. We shall construct the map
$\xi_X$ and prove that it satisfies the stated properties by reducing to
space level considerations. We begin with a space level description
of the maps $\tau$ and $\xi$ displayed in (\ref{diagdis}),
and we need some space level notations.
\begin{notns}\mylabel{notns}
Recall that $\nu$ denotes the normal bundle of $M$ and that we have the
duality map $\text{ev}\colon T\nu\wedge M_+\longrightarrow S^V$ specified in the
proof of \myref{starting}. Also, recall that
\[r_!K = K/s(M), \quad r^*T = T_M = M\times T,
\quad\text{and}\quad
r_*K = \text{Sec}(M,K)\]
for any based $G$-space $T$ and any ex-$G$-space $(K,p,s)$ over $M$.
In particular,
$$r_!S^{\nu} = T\nu, \quad r_!S^0_M = M_+, \quad \text{and}\quad
r_*r^*T \iso F(M_+,T).$$
Therefore the adjoint $\widetilde{\text{ev}}\colon T\nu\longrightarrow F(M_+,S^V)$
is a map $r_!S^{\nu}\longrightarrow r_*S^V_M$.
Let $t\colon S^V\longrightarrow r_!S^\nu$ be the Pontryagin-Thom
construction and $k$ be the composite
$$\xymatrix{k\colon r^*r_!S^\nu \ar[r]^-{r^*\tilde{\text{ev}}}
& r^*r_*S^V_M \ar[r]^-{\varepsilon} & S^V_M,\\}$$
where $\varepsilon\colon r^*r_*\longrightarrow \text{id}$ is the counit of the adjunction
$(r^*,r_*)$; note that, in general, $\varepsilon$ is just the evaluation map
$M\times \text{Sec}(M,K)\longrightarrow K$.
\end{notns}
Recall from Propositions \ref{SIfSI} and \ref{SISISI2} that we can commute
suspension spectrum functors past smash products and base change functors.
\begin{lem}\mylabel{lemma:units} With these definitions of $t$ and $k$,
\[\tau\simeq \Sigma^{-V}\Sigma^\infty t\colon S\iso \Sigma^{-V}\Sigma^\infty S^V
\longrightarrow \Sigma^{-V}\Sigma^\infty r_!S^\nu \htp r_!C_M\]
and
\[\xi \simeq \Sigma^{-V}_M\Sigma^\infty_M k\colon
r^*r_!C_M\htp \Sigma^{-V}_M\Sigma^\infty_M r^*r_!S^{\nu} \longrightarrow
\Sigma^{-V}_M\Sigma^\infty_M S^V_M \htp S_M.\]
\end{lem}
\begin{proof} By \cite[III.5.2]{LMS},
the dual of $t$ is the projection $\delta\colon M_+\longrightarrow S^0$.
This means that the following diagram is stably homotopy commutative.
$$\xymatrix{
S^V\sma M_+ \ar[r]^-{t\sma\text{id}} \ar[d]_{\text{id}\sma \delta}
& T\nu\sma M_+ \ar[d]^{\text{ev}}\\
S^V\sma S^0 \ar@{=}[r] & S^V\\}$$
Here $\delta\colon M_+ =r_!r^*S^0\longrightarrow S^0$ is the counit of the space level
adjunction $(r_!,r^*)$, and we can identify $\Sigma^{\infty}\delta$ with the
counit $\sigma\colon \Sigma^{\infty}M_+\iso r_!r^*S\longrightarrow S$ of the spectrum level
adjunction $(r_!,r^*)$. Applying $\Sigma^{-V}\Sigma^\infty$ to the
diagram and passing to adjoints, the right vertical arrow becomes
$$\alpha_M\colon r_!C_M \htp \Sigma^{-V}\Sigma^{\infty}T\nu\longrightarrow D(M_+) = Dr_!r^*S,$$
by the proof of \myref{starting}. Comparing the resulting diagram with
the diagram that defines $\tau$, we conclude that
$\tau\simeq \Sigma^{-V}\Sigma^\infty t$.
For the identification of $\xi$, we consider the
composite equivalence $r^*r_!C_M\htp Dr^*r_!S_M$ in the diagram
that defines $\xi$ to be an identification of the dual of
$r^*r_!S_M$. To identify the dual of $\zeta$
modulo that identification, we observe that the following diagram is
commutative.
$$\xymatrix{
r^*r_!S^{\nu}\sma_M S^0_M \ar[rr]^-{k\sma\text{id}}
\ar[d]_{\text{id}\sma\zeta}
& & S^V_M\sma_M S^0_M \ar@{=}[d] \\
r^*r_!S^{\nu}\sma_M r^*r_!S^0_M \ar[r]_{\cong}
& r^*(r_!S^{\nu}\sma r_!S^0_M) \ar[r]_-{r^*\text{ev}} & r^*S^V\\}$$
Indeed, recalling that $k = \varepsilon\circ r^*\tilde{\text{ev}}$ and rewriting
the diagram in more familiar notation, it becomes
$$\xymatrix{
M\times T\nu \ar[r]^-{\text{id}\times \tilde{\text{ev}}} \ar[d]_-{\text{id}\times \zeta}
& M\times \text{Sec}(M,M\times S^V) \ar[d]^{\varepsilon}\\
M\times (T\nu\sma M_+) \ar[r]_-{\text{id}\times \text{ev}}
& M\times S^V,\\}$$
and both composites send $(m,x)$ to $(m,\text{ev}(x\sma m))$.
Applying $\Sigma^{-V}\Sigma^{\infty}$ to the first diagram and
comparing with the definition of $\xi$, we conclude that
$\xi \simeq \Sigma^{-V}_M\Sigma^\infty_M k$.
\end{proof}
The following space level result will imply \myref{prop:spec}.
\begin{prop}\mylabel{prop:space}
Let $K$ be a $G$-space over $M$. There is a natural map
$$u_K\colon r^*r_!(K\wedge_M S^\nu) \longrightarrow \Sigma^V_M K$$
in $\text{Ho}\, G\scr{K}_M$ which satisfies the following properties.
\begin{enumerate}[(i)]
\item When $K=S^0_M$,
$u_K\simeq k\colon r^*r_!S^{\nu}\longrightarrow S^V_M$.
\item For a based $G$-space $T$, the following diagram commutes in
$\text{Ho}\, G\scr{K}_M$.
\[\xymatrix{ r^*r_!(T_M\wedge_M K\wedge_M S^\nu) \ar[r]^\simeq
\ar[d]_{_{u_{T_M\sma_M K}}} & T_M \wedge_M r^*r_!(K\wedge_M S^\nu)
\ar[d]^{\text{id}\wedge u_K}\\
\Sigma^V_M(T_M\wedge_M K) \ar[r]^\cong & T_M \wedge_M \Sigma^V_M K}\]
Here the top equivalence is given by (\ref{one}) and (\ref{four}) in
$\text{Ho}\, G\scr{K}_M$.
\item The following diagram commutes in $\text{Ho}\, G\scr{K}_*$.
\[\xymatrix{r_!(K\wedge_M S^\nu)\wedge S^V
\ar[rr]^-{\text{id}\wedge t}\ar[d]_\simeq
&& r_!(K\wedge_M S^\nu)\wedge r_! S^\nu \ar[d]^\simeq \\
r_!(\Sigma^V_M K\wedge_M S^\nu)
&& r_!(r^*r_!(K\wedge_M S^\nu)\wedge_M S^\nu) \ar[ll]^-{r_!(u_K\wedge \text{id})}}\]
Here the right vertical equivalence is given by (\ref{four}) in $\text{Ho}\, G\scr{K}_M$.
The left vertical equivalence is the composite
\[r_!(K\wedge_M S^\nu)\wedge S^V
\simeq r_!(K\wedge_M S^\nu\wedge_M S^V_M)
\simeq r_!(K\wedge_M S^V_M\wedge_M S^\nu),\]
where the second equivalence is obtained by moving the copy of $S^\tau$
from $S^V_M\cong S^\tau\wedge_M S^\nu$ and amalgamating it with the displayed
copy of $S^\nu$.
\end{enumerate}
\end{prop}
\begin{proof}[Proof of \myref{prop:spec}]
Let $X=\Sigma^\infty_M K$. Define $\xi_X$ to be the map
\[\xymatrix{
r^*r_!(X\wedge_M C_M)
\simeq \Sigma^{-V}_M \Sigma^\infty_M r^*r_!(K\wedge_M S^\nu)
\ar[rr]^-{\Sigma^{-V}_M\Sigma^\infty_M u_K} &&
\Sigma^{-V}_M \Sigma^\infty_M \Sigma^V_M K\iso X.}\]
Using \myref{prop:space}(ii), we see that
$\xi_{\Sigma^V_MX}$ can be identified with $\Sigma^V_M\xi_X$,
which in turn can be identified with $\Sigma^{\infty}_Mu_K$.
To show that the diagram (\ref{eq:nat}) commutes, it suffices to
show that the diagram obtained from it by applying $\Sigma^V_M$
commutes. We have just identified the lower horizontal arrow of
the resulting diagram in space level terms. Similarly, the
definition (\ref{xi}) of its upper horizontal arrow, together with
\myref{lemma:units} and \myref{prop:space}(i),
identifies its upper horizontal arrow, with $Y$ serving as a dummy
variable. More explicitly, using the projection formula
(\ref{four}), we see that the diagram can be rewritten as
\[\xymatrix{r^*r_!(r^*Y\wedge_M S^\nu) \ar[r]^\simeq
\ar[d]_{r^*r_!(\theta\wedge \text{id})} & r^*Y \wedge_M r^*r_! S^\nu
\ar[r]^-{\text{id}\wedge u_{S^0_M}} & r^*Y\wedge_M S^V_M \ar[d]^{\theta\wedge \text{id}}\\
r^*r_!(X\wedge_M S^\nu)\ar[rr]_{\Sigma^\infty_M u_K} && X\wedge_M S^V_M}\]
Consider the dummy variable $Y$ levelwise. We see from the case
$K=S^0_M$ of \myref{prop:space}(ii) that, at level $V$,
the top row is the map $u_{r^*Y(V)}$. Therefore the diagram commutes levelwise
by the naturality of $u$.
To prove that the composite (\ref{eq:trieq}) is the identity map, we apply
$\Sigma^V$ to it. To abbreviate notation, write $Y = r_!(X\wedge_M C_M)$ and
consider the following diagram.
\[\xymatrix{
\Sigma^V Y \ar[rrr]^-{\Sigma^V(\text{\text{id}}\sma\tau)} \ar@{=}[d]
\ar[drr]^-{\Sigma^V\tau_{Y}} & & & \Sigma^V(Y\sma r_!C_M)
\ar[dl]^{\htp} \ar[d]^{\htp}\\
\Sigma^V Y \ar[d]_{\htp} & &\Sigma^V r_!(r^*Y\sma_M C_M)
\ar[ll]_-{\Sigma^Vr_!(\xi_X\sma_M\text{id})} \ar[dr]^{\htp}
& Y\wedge r_! S^\nu \ar[d]^\simeq\\
r_!(X\sma_M S^{\nu}) & & &
r_!(r^*Y\wedge_M S^\nu)
\ar[lll]^-{r_!(\xi_X\wedge_M \text{id})}
}\]
We must prove that the triangle at the upper left commutes. The
arrows marked $\htp$ are given by (\ref{four}) and the evident
equivalence $\Sigma^V_MC_M\htp \Sigma^{\infty}S^{\nu}$. The upper
triangle commutes by the definition of $\tau_Y$ in terms of $\tau$,
the bottom trapezoid commutes by naturality, and the triangle at
the right commutes by inspection of projection formula isomorphisms.
Thus it suffices to prove that traversal of the perimeter gives a
commutative diagram, and this will hold for $X$ if it holds for
$\Sigma^V_MX$. In that case, we see from \myref{lemma:units},
\myref{prop:space}(ii), and a diagram chase that the
perimeter agrees with the diagram that is obtained by applying the
suspension spectrum functor to the diagram in
\myref{prop:space}(iii).
\end{proof}
The proof of \myref{prop:space} is based on the following construction
of a natural map
\[w_K\colon r^*r_!K \longrightarrow \tilde{K}\sma_M S^\tau\]
that will give rise to the required map $u_K$. Here $\tilde{K}$ is a
suitably ``fattened up'' version of $K$.
\begin{con} As in \cite[11.5]{MilS}, identify the tangent bundle of $M$
with the normal bundle of the diagonal embedding $M\longrightarrow M\times M$.
Let $U$ be a tubular neighborhood of the diagonal. Let $\text{pr}_1$ and
$\text{pr}_2$ be the projections $M\times M\longrightarrow M$ and let
$\pi_i\colon U\longrightarrow M$ be their restrictions to $U$. For an ex-space
$(K,p_K,s_K)$ over $M$, consider the following diagram of
retracts, where $\Delta$ is the diagonal and $\iota$ is the inclusion
of $U$ in $M\times M$.
Note that $\pi_i = \text{pr}_i \com \iota$ and define
$\tilde{K} = (\pi_1)_!\pi_2^* K$.
\[\xymatrix@=.4cm{M \ar[rr]^\Delta\ar[dd]\ar@{=}[drrr]
&& U \ar[rr]^{\iota}\ar[dd]\ar[dr]^(.6){\pi_1}
&& M\times M \ar[rr]^-{\text{pr}_2}\ar@{-}[d]\ar[dr]^(.6){\text{pr}_1}
&& M\ar[dr]^(.6)r \ar@{-}[d]\\
&&& M\ar@{=}[rr]\ar[dd] &\ar[d]& M \ar[rr]^(.3){r}\ar[dd] &\ar[d]& {*} \ar[dd]\\
K \ar[rr]\ar[dd] && \pi_2^* K \ar@{-}[r]\ar[dd]\ar[dr]
&\ar[r]& M\times K \ar@{-}[r]\ar[dd]\ar[dr] &\ar[r]& K\ar[dr]\ar@{-}[d]\\
&&& \tilde{K}\ar[dd] && r^*r_!K \ar[rr]\ar[dd] &\ar[d]& r_!K\ar[dd]\\
M \ar[rr]\ar@{=}[drrr] && U \ar@{-}[r]\ar[dr] &\ar[r]
& M\times M \ar@{-}[r]\ar[dr] &\ar[r]& M\ar[dr]\\
&&& M\ar@{=}[rr] && M \ar[rr] && {*}}\]
The floor and ceiling of the diagram are identical.
The back wall is formed by pulling $K$ back along the maps of base spaces and
then the front wall is obtained from the back wall by applying lower shriek
functors. Here we have used the canonical isomorphism
${\text{pr}_1}_!(M\times K)={\text{pr}_1}_!\text{pr}_2^*K\iso r^*r_!K$
associated to the pullback square on the right side of the floor. Since
the $\pi_i$ are homotopy equivalences, the maps
\begin{equation}\label{kappak}
K\longrightarrow \pi_2^*K\longrightarrow \tilde K
\end{equation}
at the left are equivalences when $K$ is $q$-cofibrant and $q$-fibrant, and
we denote the displayed composite equivalence as $\kappa$.
To get a better feeling for the spaces in the diagram, we make the
following schematic picture.
\[\begin{xy}
0;(50,0) **@{-};(60,20) **@{-};(60,60) **@{-};(10,60) **@{-};(0,40)
**@{-};(50,40) **@{-};(60,60) **@{-}
,(50,40);(50,0) **@{-},(0,40);(0,0) **@{-};(10,20) **@{--};(10,60)
**@{--},(10,20);(60,20) **@{--}
,(5,0);(58.5,17) **@{.},(1.5,3);(55,20) **@{.}
,(5,40);(58.5,57) **@{.},(1.5,43);(55,60) **@{.} ,(24,6);(26.4,10.8)
**@{-};(26.4,50.8) **@{-};(24,46) **@{-};(24,6) **@{-}
,(55,30) *{K},(34,30) *{\pi^*_2K|_{U_m}}
,(35,51) *{U},(21,47) *{U_m}
,(0,-5);(50,-5)**@{-};(60,15)**@{-};(10,15)**@{-};(0,-5)**@{-};(60,15)
**@{-}
,(38,5) *{\Delta},(30,-3) *{M},(58,4) *{M}
\end{xy}\]
The cube represents $M\times K$ with $M$ in the horizontal direction
and $K$ in the other two directions. It is sitting over $M\times M$
with vertical projection and we think of the top face as being the
image of the section. We can view $\pi_2^* K$ as the subspace of
$M\times K$ sitting over the neighborhood $U$ of the diagonal in
$M\times M$. Passing to the front face of the diagram, the fibers of
$\tilde{K}$ over points $m$ in $M$ are the slices $\pi_2^*K|_{U_m}$ of
$\pi_2^* K$ as displayed with basepoint obtained by identifying the
fiber $U_m=\pi_1^{-1}(m)$ over $m$ in the bundle $U$ to a single
point. We therefore think of $\tilde{K}$ as a fattening of the space
$K$ that sits over the diagonal in $M\times M$.
For a sub $G$-space $A$ of a $G$-space $K$ over $M$, not necessarily sectioned,
passage to fiberwise quotients gives an ex-$G$-space $K/\!_M A$ over $M$ with
total space $K\cup_A M$. Given two such pairs $(K,A)$ and $(L,B)$, we obtain
a product pair by setting
\[(K,A)\times_M (L,B)=(K\times_M L, K\times_M B \cup A\times_M L).\]
Its fiberwise quotient is the ex-$G$-space $(K/\!_M A)\wedge_M (L/\!_M B)$ over $M$.
The pair $(M\times M, M\times M-U)$ is a model for the Thom
complex $T\tau$, and we can identify $T\tau$ with the
quotient space $(M\times M)/(M\times M-U)$. More relevantly
for us, the fiberwise quotient $(M\times M)/\!_M(M\times M-U)$
is a model for $S^{\tau}$. View $M\times K=\text{pr}_2^* K$,
$M\times M$, and $U$ as $G$-spaces over $M$ via projection to the
first factor and embed $U$ in $M\times K$ by sending $(m,n)$ to
$(m,s_K(n))$. We have the diagonal map
\[M\times K\longrightarrow (M\times K, U) \times_M (M\times M, M\times M - U)\]
of $G$-spaces over $M$ that sends $(m,k)$ to $((m,k), (m,p_K(k)))$ for $m\in M$
and $k\in K$. It induces the top map in the following diagram in $G\scr{K}_M$.
\[\xymatrix{ M\times K \ar[r]\ar[d]\ar@{->}[dr]
& ((M\times K)/\!_M U)\wedge_M {S^\tau}\\
r^*r_!K \ar@{->}[r]_{w_K} & \tilde{K}\wedge_M {S^\tau} \ar[u]}\]
Here $(M\times K)/\!_M U$ is obtained from $M\times K$ by identifying all points
of the form $(m,s_K(n))$ such that $(m,n)\in U_m$ to a single basepoint in the
fiber over $m$. It therefore contains $\tilde K$ as a subspace, and this gives
the right vertical inclusion. The image of the top arrow lands in the
image of the right vertical arrow since if $(m,p_K(k))$ is not in $U_m$,
then $(m,k)$ maps to the basepoint in $S^{\tau}$ and therefore to the
basepoint in $(M\times K)/\!_M U\sma_M S^{\tau}$. This gives the diagonal arrow.
Note that $r^*r_!K = M\times (K/s_K(M))$ and the left vertical arrow is the
obvious quotient map. Since the diagonal arrow maps $(m,x)$ with $x\in s_K(M)$
to the base point of the fiber over $m$ in $\tilde K\wedge_M {S^\tau}$, it is
constant on the fibers of the left vertical arrow. It therefore factors through
a map $w_K$. Explicitly, $w_K$ is specified by
\begin{equation}\label{doubleu}
w_K(m,[x])=\begin{cases}
\; [m,x]\wedge[m,p_K(x)] &\text{if $(m,p_K(x))\in U_m$,}\\
\; * & \text{otherwise,}
\end{cases}
\end{equation}
where $m\in M$, $x\in K$, and the square brackets denote equivalence classes.
\end{con}
\begin{proof}[Proof of \myref{prop:space}]
Here we are working in homotopy categories, and we may assume that $K$
is $qf$-fibrant and $qf$-cofibrant. Let $L=K\wedge_M S^\nu$.
We define the map $u_K$ in $\text{Ho}\, G\scr{K}_M$ by the natural zig-zag
\[\xymatrix{
r^*r_! L \ar[r]^-{w_{L}}
& \tilde{L}\wedge_M {S^\tau}
&& L \wedge_M S^\tau \ar[ll]_-{\kappa\sma_M\text{id}} \ar[r]^-{\mu}
& \Sigma^V_M K}\]
of arrows in $G\scr{K}_M$, where $\mu$ is induced by the isomorphism
\[\xymatrix{
S^\nu\wedge_MS^\tau \ar[r]^-{\text{twist}}
& S^\tau\wedge_MS^\nu \ar[r]^-{\cong}
& S^V_M}\]
and $\kappa\colon L\longrightarrow \tilde L$ is an equivalence as in ({kappak}).
\noindent
\emph{Proof of (i)}
We must show that $u_{S^0_M}\simeq k$ in $\text{Ho}\, G\scr{K}_M$. Using our zig-zag definition of $u_K$, we see that it suffices to show that the composite
\[\xymatrix{
M\times T\nu \ar[r]^-{k} & M\times S^V
\ar[r]^-{\mu^{-1}} & S^\nu\wedge_M S^\tau
\ar[r]^-{\kappa\wedge\text{id}} & \widetilde{S^\nu}\wedge_M S^\tau}\]
is homotopic to the map $w_{S^\nu}$ defined in (\ref{doubleu}). As noted in the proof of \myref{lemma:units}, $k(m,[v])=(m,\text{ev}([v]\wedge m))$, where $v\in S^\nu$ and brackets denote equivalence classes. Recall that the map $\text{ev}$ depends on a choice of a tubular neighborhood of $M$ in $\nu\times M$ (as in the proof of \myref{starting}). We use the obvious choice
\[\{(v,m)\, |\, v\in \nu,\ \, m\in M, \ \, \text{and}\ \ (p_\nu(x),m)\in U\}.\]
Under our identification of the normal bundle of $\Delta:M\to M\times M$ and thus of its tubular neighborhood $U$ with $\tau$, this tubular neighborhood is identified with $M\times V\cong \nu\oplus \tau$. When not in the section, we can view points $[m,n]\in S^\tau=(M\times M)/_M(M\times M-U)$ as vectors $(m,n)$ in the tangent space $U_m\cong\tau_m\subset V_m\cong V$ of $M$ at $m$. We then have that $(m,\text{ev}([v]\wedge m)) = (m,(p_\nu(v),m)+v)$. To identify this point in the image of $\mu$, let $u\in\nu_m$ be such that $(p_\nu(v),m)+v = (m,p_\nu(v))+u$ in $V$ and note that $u$ depends continuously on $m$ and $v$. Since $\mu(u\wedge [m,p_\nu(v)])=(m,(m,p_\nu(v))+ u)$, the composite displayed above is given by
\[(\kappa\wedge_M \text{id})\mu^{-1}k(m,[v])=
\begin{cases
\; [m,u]\wedge [m,p_\nu(v)] & \text{if $(m,p_\nu(v))\in U_m$,}\\
\; * & \text{otherwise.}
\end{cases}\]
A linear homotopy in the fibers of $\nu$ shows that this map is
homotopic to $w_{S^\nu}$.
\noindent
\emph{Proof of (ii)}
Inspection of the construction of $w$ gives the following naturality diagram for based $G$-spaces $T$ and ex-$G$-spaces $K$ over $M$.
\[\xymatrix{ r^*r_!(T_M\wedge_M K) \ar[r]^\simeq\ar[d]_{w}
& T_M \wedge_M r^*r_!K \ar[d]^{\text{id}\wedge w}\\
(\widetilde{T_M\wedge_M K}) \wedge_M {S^\tau}\ar[r]^\simeq
& T_M \wedge_M \tilde{K}\wedge_M {S^\tau} }.\]
Here, using $r^*r_!\iso (\pi_1)_!\pi_2^*$, the bottom equivalence is the following application of the projection formula.
\begin{align*}
(\pi_1)_!\pi_2^*(r^*T\wedge_M K)
&\simeq (\pi_1)_!(\pi_2^*r^*T\wedge_M \pi_2^*K)\\
&\simeq (\pi_1)_!(\pi_1^*r^*T\wedge_M \pi_2^*K)\\
&\simeq r^*T\wedge_M (\pi_1)_!\pi_2^*K
\end{align*}
This use of the projection formula is compatible with its use for $r_!$ to obtain the equivalence of the top row. Analogous naturality diagrams for the other two maps in the definition of $u_K$ give the conclusion.
\noindent
\emph{Proof of (iii)}
Again let $L=K\wedge_M S^\nu$. Expanding the diagram in the statement of (iii) in terms of the definition of $u_K$, we must prove that the following diagram commutes in $\text{Ho}\, G\scr{K}_*$, where the equivalences here are the vertical arrows of the diagram in (iii).
\[\xymatrix@=.75cm{
r_!(\Sigma^V_MK\wedge_MS^{\nu})
&& r_!L \wedge S^V \ar[rr]^-{\text{id}\wedge t} \ar[ll]_-{\simeq}
&& r_!L \wedge r_! S^\nu \ar[d]^\simeq\\
r_!(L\wedge_M {S^\tau} \wedge_M S^\nu) \ar[u]^{r_!\mu}
\ar[rr]_-{r_!(\kappa\wedge\text{id})}
&& r_!(\tilde L\wedge_M {S^\tau} \wedge_M S^\nu)
&& r_!(r^*r_!L\wedge_M S^\nu) \ar[ll]^-{r_!(w_L\wedge \text{id})}}\]
We chase the diagram starting in $r_!(L\wedge_M {S^\tau} \wedge_M S^\nu)$ and mapping to $r_!(\tilde L\wedge_M {S^\tau} \wedge_M S^\nu)$. Let $x = k\sma w \in L = K\sma_M S^{\nu}$, $u\in S^\tau$, and $v\in S^\nu$ be points in fibers over a given $m\in M$. Using square brackets to denote passage to quotient spaces (the lower shriek functors), we see that $r_!(\kappa\sma\text{id})$ sends $[x\sma u\sma v]$ to $[[m,x]\sma u\sma v]$. The definitions of $\mu$ and of the top left equivalence (which is the left vertical equivalence in the diagram of the statement) are arranged in such a way that the composite of $\mu$ and the inverse of the equivalence sends $[x\sma u\sma v]$ to $[x\sma(u+v)]$. Let $t(u+v)=[z]$, $z\in S^\nu$, and let $n=p_\nu(z)$. Chasing $[x\sma u\sma v]$ around the top of the diagram, when we do not arrive at the basepoint we arrive at the point $[[n,x]\sma [n,m]\sma z]$, where $[n,m]$ is an element of $U\cong \tau$. We can identify the target space with $r_!(\tilde{L}) \sma S^V$ using the identification of $S^\tau \sma S^\nu$ with $M \times S^V$ and the projection formula. Then our two maps are homotopic by a homotopy $h$ that can be written in the form
\[h([x\wedge u\wedge v],s)=[m+s[m,n],x]\wedge(u+v+2s[n,m]).\]
Here $m+s[m,n]$ denotes a point on the path from $m$ to $n$ in $M$ that is the image under the exponential map of the line segment from $0$ to $[m,n]$ in the tangent space at $m$. The Thom map takes $u+v$ in $S^V$ (which in $M \times S^V$ is based at $m$) to the point $z$ in $S^\nu$ based at $n$. In $S^V$, we have $u+v = [m,n]+z$. Since $[n,m]= -[m,n]$ under the identification of $\tau$ with $U$ (as in \cite[11.5]{MilS}), we see that the homotopy ends at the composite around the top of the diagram, and it clearly begins at $r_!(\kappa\sma\text{id})$.
\end{proof}
\backmatter
|
{
"timestamp": "2004-11-30T14:37:42",
"yymm": "0411",
"arxiv_id": "math/0411656",
"language": "en",
"url": "https://arxiv.org/abs/math/0411656"
}
|
\section{Pointing and Apparent Star Altitudes}\label{Sec.lateral}
\subsection{Basics: Flat Earth Model}\label{sec.flatearth}
The standard model of an interferometric setup
and delay line correction for a star at the true zenith angle $z$ is shown
in Fig.\ \ref{DelModl.ps}:
The optical path difference (OPD) $D$ shows up once in the vacuum above
telescope 1, and is added for telescope 2 on the ground at some local index of
refraction.
The atmosphere is horizontally homogeneous and the earth flat; therefore
no correction is needed for the ray path curvature induced by
any vertical gradient of the index of refraction through the atmosphere, because
these two paths match each other at each height above ground.
\begin{figure}[h]
\plotone{DelModl.eps}
\caption{
The standard model of delay correction and recovery
for a star at zenith angle $z$
observed by two telescopes that look through horizontal layers
of the stratified atmosphere.
$D=b\sin z=P \tan z$.
}
\label{DelModl.ps}
\end{figure}
The pointing difference $R$ between the apparent and actual altitude
of a star that is induced by the refraction of the earth atmosphere
is in a simple model of a flat earth \citep{Green1985,FilippenkoPASP94}
\begin{equation}
R\approx (n_0-1)\tan z_0,
\label{eq.Rofnflat}
\end{equation}
where $n_0$ is the index of refraction on the ground, $z_0$
the apparent zenith distance on the ground, and
\begin{equation}
R=z-z_0 >0
\end{equation}
obtained in radian.
This implies chromatic effects \citep{Livengood,LanghansAN324,ColavitaPASP116,RoePASP114}
and dependencies on the atmospheric model,
commonly summarized under the label ``transversal atmospheric dispersion'' in
Astronomy.
There is a rainbow effect in Eq.\ (\ref{eq.Rofnflat}): as $n_0$
is a function of the wavelength, $R$ becomes dispersive,
too: between wavelengths of $2$ $\mu$m and $2.4$ $\mu$m,
we get a difference of $\Delta n_0\approx 1.05\cdot 10^{-7}$ at 2600 m above
sea level,
which translates
into a spectral smear of $\Delta R\approx 22$ mas $\cdot\tan z_0$.
\label{sec.rainbow}
Eq.\ (\ref{eq.Rofnflat})
suggests that the relative error in the star's altitude definition
is close to the relative error in the dielectric function and susceptibility
at the telescope site.
In addition, Snell's law of refraction states that product $n\sin z$ between
the height-dependent index of refraction $n$ and the sine of the angle of
refraction at that height remains constant along the path
\cite[(4.2)]{Green1985}.
Therefore the optical path delay is $D=b\sin z=bn_0\sin z_0$; it changes
as the astronomical object changes position in $z$, or, supposed $b$
is fixed, according to atmospheric parameters accessible at the ground level.
The benefit of this analysis is that both, the pointing correction $R$
in Eq.\ (\ref{eq.Rofnflat}) and the OPD measured on the ground, are functions
of the index of refraction at the telescope site, not functionals
of the entire layered atmosphere.
The theme of this paper are corrections to these statements considering
a non-turbulent atmosphere covering an earth surface of constant, but
non-negligible curvature.
The rest of Sec.\ \ref{Sec.lateral} shortly describes the standard theory
of refraction and defines two different baseline lengths.
Sec.\ \ref{sec.zeni} concentrates on the integral formulation
of the OPD calculation through the atmosphere:
geometries with constrained azimuths suffice to introduce all
relevant concepts; general star positions are then reduced
to the constrained case.
\subsection{Spherical Earth: Geometry}\label{sec.sphergeo}
Things are more complicated if we start to look at the more realistic
model of a spherical earth. A telescope distance of $b=100$ m
on an earth of radius $\rho=6368$ km leads to a pointing mismatch of purely
geometric origin
of about $b/\rho\approx 3.2$ arcsec (Fig.\ \ref{PointRho.ps}).
\begin{figure}[htb]
\plotone{PointRho.eps}
\caption{
Two telescopes placed on the earth of radius $\rho$ at a distance
$b$ looking at the same star experience local zenith angles that differ
by $Z\approx b/\rho$ rad.
}
\label{PointRho.ps}
\end{figure}
The baseline $b$ is the distance between the telescope locations on the
earth, the length of the straight secant drawn in
Figs.\ \ref{PointRho.ps} and \ref{Base.ps},
\begin{eqnarray}
b&=&\rho\sqrt{2(1-\cos Z)} \label{eq.bofZ} \\
&=&2\rho\sin\frac{Z}{2} \\
&\approx& \rho\left(Z-\frac{Z^3}{24}+\frac{Z^5}{1920}\ldots \right).
\label{eq.bforz}
\end{eqnarray}
The inversion of this series reads
\begin{equation}
Z\approx \frac{b}{\rho}
+\frac{1}{24}\left(\frac{b}{\rho}\right)^3
+\frac{3}{640}\left(\frac{b}{\rho}\right)^5\ldots.
\label{eq.zforb}
\end{equation}
The angle approximation $Z\approx b/\rho$ is an estimate to $b/\rho =2\sin (Z/2)$,
a limit of a baseline so short that it does not matter whether it is
measured along a straight line (as drawn in Fig.\ \ref{PointRho.ps})
or along the circular perimeter.
The relative error in this approximation
is $\approx Z^2/24$, or $\approx 10^{-11}$ for the example of
$b=100$ m.
Pointing/guiding is a functionality of the individual telescopes: the existence
of a nonzero pointing difference is absorbed in the telescope
operation, and any delay originating from there is to first order recovered
in the tracking.
We hereby explicitly discard any ``separation'' term of Gubler and Tytler
eminent from their consideration of single telescopes
\citep{GublerPASP110}, and acknowledge that both telescopes
are ``mispointing'' at the same time.
This cross-eyed geometry indirectly transforms into a contribution to
the delay that may be understood---and calculated to lowest order---on
the basis of what is said in Sec.\ \ref{sec.flatearth}, but has no
parallel with single telescopes as long as their diameters are much
smaller than the earth radius: The example of $3.2$ arcsec
from above translates into an additional angle of
refraction of
$\Delta R\approx \Delta z_0(1+\tan^2z_0)(n_0-1)>\Delta z_0(n_0-1)\approx
Z (n_0-1)\approx 650$ $\mu$as at 2600 m above sea level,
which is the first derivative of Eq.\ (\ref{eq.Rofnflat}) w.r.t.\ $z_0$.
Dropping the term $\tan^2z_0$ here means this is a lower estimate
in the limit of stars at the zenith.
The effect on the delay could be understood in the
``standard model of delay line correction,'' where the two rays of
the star that will eventually hit the two telescopes
``generate'' a phase difference in the vacuum (undone later on
in the delay line tunnel) as they hit the top layer of the earth
atmosphere with a path difference $D$
(Fig.\ \ref{DelModl.ps}). The curvature correction means
that this top layer ``bends back'' a bit more for the telescope further away
from the star, which slightly increases the angle of incidence on the
atmosphere.
\begin{figure}[h]
\plotone{Base.eps}
\caption{
Interferometry with two telescopes located on a non-planar surface
needs to be aware of the vagueness of the definition of a baseline $b$.
}
\label{Base.ps}
\end{figure}
Fig.\ \ref{Base.ps} shows that the only obvious definition of the
baseline length $b$ is on earth.
Three alternatives have been marked with question
marks in the figure:
\begin{enumerate}
\item
Simple outward projection along some
``common'' zenith direction of both telescopes is futile, because
it would only scale $b$ with some arbitrary distance from
the earth center.
\item
Some sliding definition along the curved
rays fails because there are no fix points or right angles for reference.
\item
The baseline cannot be rigorously defined
far outside the atmosphere. Actually only the projection $P$
of the two telescopes remains well defined as the rays can be considered
parallel, and so remains some direction angle $\alpha$ relative to some
geostationary coordinate system.
One could try to use the b? in Fig.\ \ref{Base.ps} as some line parallel to
the earth-based $b$ anchored at $P$, but needs to stay aware of the
fact that this replaces the zenith angle $z$ in Fig.\ (\ref{DelModl.ps})
by the angle $\beta=\alpha-\phi$, where $\alpha$ is the polar coordinate
of the star and $\phi$ the mean polar coordinate of the telescopes
in the geocentric coordinate system.
\end{enumerate}
In a formal way, I define an effective baseline length $b^*$
above the atmosphere via
\begin{equation}
b^{*2} \equiv P^2+D^2
\label{eq.beff}
\end{equation}
which turns out to be longer than $b$ because the earth's atmosphere represents
a gradient lens \citep{HuiAJ572}. This effective baseline is a function of the baseline $b$
and of the star zenith angle, and a functional of the atmospheric refraction
$n(r)$.
\subsection{Spherical Earth: Atmospheric Layers}\label{sec.spherlay}
The delay line modification described in the previous paragraph is
of purely geometrical nature and is closely related to the discussion
of differences between two light rays, but there are more pointing
``corrections'' that already show up in the single ray case, as discussed
next.
Eq.\ (\ref{eq.Rofnflat}) is based on the assumption that the gradient
of the refractive index $n$ along the ray path is parallel to a global, constant
zenith vector. As the gradient and the zenith vector change in direction
along the path through a spherically symmetric atmosphere, and as we assume
that $n$ becomes a function of the radial distance to the spherical surface of
the earth, the equation becomes more accurately
\cite[(4.19)]{Green1985}\citep{ThomasJHUAPL17,AuerAJ119,NenerJOSA20,NoerdlingerJPRS54,Tannousarxiv01}
\begin{equation}
R=\rho n_0\sin z_0\int_1^{n_0}\frac{dn}{n(r^2n^2-\rho ^2n_0^2\sin^2z_0)^{1/2}},
\label{eq.RofnInt}
\end{equation}
an integral over the refractive index, to start above the atmosphere
($n=1$) and to end at the telescope position ($n=n_0$), and where
$r\ge \rho$ is the distance to the earth center.
Its Taylor expansion is
commonly written as an expansion in powers of $\tan z_0$ \citep{StonePASP108},
the observed quantity,
\begin{eqnarray}
R&=&\rho n_0\tan z_0\int_1^{n_0}\frac{dn}{n^2r} \nonumber \\
&&+\frac{\rho n_0\tan^3 z_0}{2} \int_1^{n_0}\frac{\rho^2n_0^2-r^2n^2}{n^4r^3}dn \nonumber \\
&& +\frac{3\rho n_0\tan^5 z_0}{8} \int_1^{n_0}
\frac{(\rho^2n_0^2-r^2n^2)^2}{n^6r^5}dn
+\ldots.
\end{eqnarray}
The major new aspect is that the pointing direction becomes a functional
of the height spectrum of the index $n$, and the angle of arrival becomes
site-dependent \citep{ConanJOSA17}.
Accurate modeling of $n(r)$ is not within the scope of this treatise here
and unambitiously discussed in App.\ \ref{sec.nofr}\@.
Tables and graphs to follow use an exponential depletion of the susceptibility
\begin{equation}
\chi=\epsilon-1=n^2-1
\label{eq.chiofn}
\end{equation}
to the vacuum of the universe with a scale height $K$,
\begin{equation}
\chi(r)=\chi_0e^{-(r-\rho )/K}.
\label{eq.chiofr}
\end{equation}
The explicit parameters are
\begin{equation}
\chi_0=4\cdot 10^{-4},\quad \rho=6380 \mathrm{ km},\quad K=10 \mathrm{ km},
\label{eq.chiofrPar}
\end{equation}
unless otherwise noted, representing a prototypical K-band value at 2600 m
above sea level \citep{MatharAO43}.
Within this model, Fig.\ \ref{Rdiff.ps} shows the change in $R$ introduced by
switching from the flat earth model,
\begin{equation}
R \stackrel{\rho\to \infty}{\longrightarrow}
n_0\sin z_0\int_1^{n_0}\frac{dn}{n(n^2-n_0^2\sin^2z_0)^{1/2}},\label{eq.Rflat}
\end{equation}
to the spherical model of the atmosphere.
\begin{figure}[h]
\plotone{Rdiff.eps}
\caption{
Solid line: The relative error in the refraction angle $R$, in percent, made
by switching from the spherical
model, Eq.\ (\ref{eq.RofnInt}), to the limit of the
flat earth, Eq.\ (\ref{eq.Rflat}). Dashed line: The relative error by
switching from the spherical model to the approximation (\ref{eq.Rofnflat}).
The {\em absolute} errors approach 0 as $z\to 0$.
}
\label{Rdiff.ps}
\end{figure}
\section{Accumulated Optical Path Lengths for Spherical Geometry}\label{sec.zeni}
\subsection{Planar, Overhead Geometry}\label{sec.zeni2D}
\subsubsection{Single Star}
The optical path length along the curved ray trajectory is the line integral
of the refractive index over the geometric path, similar to
Eq.\ (\ref{eq.RofnInt})
\begin{equation}
L=\int_{n=1}^{n_0}n \frac{dr}{\cos \psi},
\label{eq.Daux}
\end{equation}
where $dr/\cos \psi$ is the length of the diagonal path element in
Fig.\ \ref{Psi.ps}.
\begin{figure}[hbt]
\plotone{Psi.eps}
\caption{The integrals (\ref{eq.RofnInt}) and (\ref{eq.Daux}) accumulate
the change in the geocentric zenith angle $\psi(r)$ and the optical
path length $n/\cos\psi$, respectively, along the light path, applying
Snell's law $r n \sin\psi(r)=$ const at each differential layer.
}
\label{Psi.ps}
\end{figure}
$\psi$ the local zenith angle of the star, and the $r$ the distance
from the earth center
\cite[Fig. 4.4]{Green1985}.
This quantity includes the geometric path length up to the star and is
infinite in our applications.
We assume a maximum height $H$ of the atmosphere, $n_{|r>\rho+H}=1$, with
the option to look at the limit $H\to\infty$ if the density has no
clear ceiling, like in the models of Eq.\ (\ref{eq.chiofr}).
We define the impact parameters $I_1$ and $I_2$ of the rays,
$0\le I_2\le I_1\le \rho$, which were their
smallest distances to the earth center if they would pass by along geometric
straight lines without any diffraction---as used in atomic collision theory.
\begin{equation}
P=I_1-I_2
\label{eq.PofIs}
\end{equation}
is the projected baseline, measured above the atmosphere.
Sec.\ \ref{sec.zeni2D} considers the case
in which the star,
the two telescopes (the baseline), and the earth center
are coplanar: Fig.\ \ref{DelModl2.ps}\@.
Comments on the general configuration of an unconstrained star azimuth follow in
Sec.\ \ref{sec.zeni3D}.
\begin{figure}[hbt]
\plotone{DelModl2.eps}
\caption{
Sketch of the impact parameters $I_i$, the auxiliary geocentric
zenith angles $\psi^{(i)}_H$, the true zenith angles
$z_i$, the angles of refraction
$R_i$, and the projected baseline $P$. The two light rays enter
from the right, hit the upper atmosphere at a height $H$ where indicated by the
dotted quarter-circle, and eventually the telescopes $T_i$.
}
\label{DelModl2.ps}
\end{figure}
The difference between the integrals
Eq.\ (\ref{eq.Daux})
\begin{eqnarray}
&&D=L_1-L_2\\
&&\!=\!\int\limits_{r=\rho}^{\sqrt{(\rho+H)^2+I_1^2-I_2^2}}
\!\!\!\!
\frac{n\, dr}{\cos \psi^{(1)}}
-\int\limits_{r=\rho}^{\rho+H}\!\frac{n\, dr}{\cos \psi^{(2)}},
\end{eqnarray}
is the OPD for two telescopes at a common baseline $b$, with different
apparent and different true zenith angles (individual pointing), but looking
at the same star.
The additional straight line segment for the ray to telescope 1 through vacuum
before it reaches the altitude $H$ is
\begin{eqnarray}
D_v &\equiv& \sqrt{(\rho+H)^2-I_2^2}-\sqrt{(\rho+H)^2-I_1^2} \nonumber \\
&=& (\rho+H)[\cos \psi_H^{(2)}-\cos \psi_H^{(1)}].
\label{eq.Dv}
\end{eqnarray}
The Taylor series of this term up to third order in $P$ is
\begin{equation}
D_v\approx P\tan \psi_H^{(2)}+\frac{1}{2I_2}
\frac{\tan \psi_H^{(2)}}{\cos^2 \psi_H^{(2)}}P^2
+\frac{1}{2I_2^2}
\frac{\tan^3 \psi_H^{(2)}}{\cos^2 \psi_H^{(2)}}P^3.
\label{eq.taylDv}
\end{equation}
We separate this
piece from the integral for the telescope 1,
\begin{equation}
D =D_v+\int_{r=\rho}^{\rho+H} n\Big[\frac{1}{\cos\psi^{(1)}} \nonumber\\
-\frac{1}{\cos\psi^{(2)}}\Big]dr,
\end{equation}
where the major difference w.r.t.\ Fig.\ \ref{DelModl.ps} is that this
integral does not vanish, because the atmosphere is now hit at two different
angles $\psi_H^{(1)}\neq \psi_H^{(2)}$.
The integral would also not vanish, if the factor $n$ would be dropped to
deduce the {\em geometric} path difference of the curved beams.
The constance of the product $rn\sin\psi$ along each curved trajectory
\cite[(4.16)]{Green1985},
\begin{eqnarray}
rn\sin\psi^{(1)} &=&(\rho+H)\sin \psi_H^{(1)}=I_1 ; \\
rn\sin\psi^{(2)} &=&(\rho+H)\sin \psi_H^{(2)}=I_2 ,
\label{eq.Snell}
\end{eqnarray}
is inserted into the previous equation,
\begin{eqnarray}
D =D_v&+&\int\limits_{r=\rho}^{\rho+H} rn^2\Big[
\frac{1}{\sqrt{r^2n^2-(\rho+H)^2\sin^2\psi_H^{(1)}}}\nonumber\\
&&-\frac{1}{\sqrt{r^2n^2-(\rho+H)^2\sin^2\psi_H^{(2)}}}\Big]dr.
\label{eq.DIntf}
\end{eqnarray}
The term in square brackets allows another Taylor expansion
\begin{eqnarray}
&&\frac{1}{\sqrt{r^2n^2-I_1^2}}-\frac{1}{\sqrt{r^2n^2-I_2^2}}
\approx \nonumber \\
&& \frac{I_2}{(r^2n^2-I_2^2)^{3/2}}P
+\frac{1}{2}\frac{r^2n^2+2I_2^2}{(r^2n^2-I_2^2)^{5/2}}P^2 \nonumber \\
&&+\frac{1}{2}\frac{3r^2n^2+2I_2^2}{(r^2n^2-I_2^2)^{7/2}}I_2P^3.
\label{eq.TaylIntf}
\end{eqnarray}
All correction terms of the spherical geometry in
Eqs.\ (\ref{eq.taylDv})--(\ref{eq.TaylIntf}) have a positive sign.
The contribution of the term of $O(P^2)$ amounts to $\approx 2$ mm,
and of the term of $O(P^3)$ to $\approx 60$ nm as the zenith angle approaches
$60$ deg at $b=100$ m.
A consistency check of Eq.\ (\ref{eq.TaylIntf}) is that its contribution of the
first, linear Taylor order to the integral in Eq.\ (\ref{eq.DIntf}) becomes
\begin{eqnarray}
&&
\int_{r=\rho}^{\rho+H} rn^2\frac{I_2P}{(r^2n^2-I_2^2)^{3/2}}dr \nonumber\\
&&\stackrel{n\to 1}{\longrightarrow}-P\tan \psi_H^{(2)}+P\frac{I_2}{\sqrt{\rho^2-I_2^2}}
\end{eqnarray}
in the vacuum limit, such that the term $P \tan\psi_H^{(2)}$ cancels
the first term in Eq.\ (\ref{eq.taylDv}), and only the term $P\tan z$ depending
on the true topocentric zenith angle remains, equivalent to
Fig.\ \ref{DelModl.ps}.
The influence of the spherical geometry on the atmospheric path length
difference is demonstrated in Fig.\ \ref{Dds.ps}.
\begin{figure}[h]
\plotone{Dds.eps}
\caption{
The difference between the full integral (\ref{eq.DIntf}) of the
delay $D$ and the approximation $D\approx b\sin \bar z$, which one would
derive from Fig.\ \ref{DelModl.ps} using the mean true zenith
angle $\bar z \equiv (z_1+z_2)/2$, for baselines between
25 and 200 m. A convergence test to the limit $H\to\infty$
is indicated for $b=200$ m.
}
\label{Dds.ps}
\end{figure}
\subsubsection{Astrometry (two true zenith distances)}
Up to here, an accurate computation of the delay $D$ by integration over the
atmosphere layers is not competitive against measuring the equivalent value
on the ground (Fig.\ \ref{DelModl.ps}), since the gas densities on the
ground along the beam path---input to calculation of the refractive
indexes---are accessible to sensors and much better known
than the remote, high-flying layers of air.
As we turn to the astrometric task of completing the right-angled triangle
formed by $D$, $P$ and $b^*$ in Fig.\ \ref{Base.ps}---with the aim
of precise determination of either of the base angles at $b^*$---,
the ``baseline calibration'' emerges as an additional focus.
This means determination of $b^*$, the image of $b$. The following sections
consider the atmospheric lensing correction $b^*-b$ to a, in principle,
rock-solid and accessible ground baseline $b$.
The computational strategy is to derive $P$ and $D$, then to use
Eq.\ (\ref{eq.beff}).
Within this framework, the geometric baseline $b$ is defined joining
the ``ends of the paths through the atmosphere,'' and is assumed to
be the same
vector for both stars in the case of astrometry. We do not ask the
question whether the corresponding terminal points $T_1$ and $T_2$ are that
well defined for real telescope optics with chromatic, azimuth
dependent foci and trusses that bent under the load of their own weight
or the wind.
The formal solution to the problem of tracing four beams
(two stars, separated by an angle $\tau$ and labeled $P$ for ``primary''
and $S$ for ``secondary,'' to two telescopes) for a given $Z$ in
Eq.\ (\ref{eq.zforb}) is then given by computation of $D_\textrm{P}$ for the
primary at some $Z$ (some baseline $b$, see Eq.\ (\ref{eq.bforz})),
computation of $D_\textrm{S}$ for the secondary at the
same $\Delta z=Z=z_{1\textrm{S}}-z_{2\textrm{S}}
=z_{1\textrm{P}}-z_{2\textrm{P}}$
(since the secondary must be caught by the same two telescopes)
but slightly different true zenith angles
$z_{1\textrm{S}}= z_{1\textrm{P}}+\tau$,
$z_{2\textrm{S}}= z_{2\textrm{P}}+\tau$.
It should be noted that the earth-bound baseline is {\em not\/} tilted here---
there is no lifting of one telescope and sinking of the other
to acquire the secondary---, and $b$ refers
to the geometrical distance between two foci that define a
common reference for both stars.
Any residual effects
of the spherical atmosphere and/or atmospheric layering are then
caused by the second derivative of $L(z)$ (Fig.\ \ref{Lofz.ps}).
\begin{figure}[hbt]
\plotone{Lofz.eps}
\caption{A difference in the delays $D_\textrm{P}$ and $D_{\textrm{S}}$
measured for the primary star and the secondary star is caused by
the nonlinearity of the path length $L$, Eq.\ (\ref{eq.Daux}), as a function
of the true zenith position.
}
\label{Lofz.ps}
\end{figure}
From (\ref{eq.DIntf}) with (\ref{eq.Snell}), omitting a constant $\rho+H$,
\begin{eqnarray}
L=&& -\sqrt{(\rho+H)^2-\rho^2n_0^2\sin^2 z_0} \nonumber \\
&&+\int\limits_{r=\rho}^{\rho+H}\!\!
\frac{rn^2}{(r^2n^2-\rho^2n_0^2\sin^2 z_0)^{1/2}}dr, \label{eq.LInt}
\end{eqnarray}
where we insert $z_0=z-R$.
For a spherical earth without atmosphere, the vacuum limit,
\begin{equation}
R\stackrel{n\to 1}{\longrightarrow} 0,\quad n_0\longrightarrow 1,\quad
\end{equation}
\begin{eqnarray}
L& \longrightarrow& \rho(1-\cos z)\nonumber \\
&=& \frac{\rho}{2}\tan^2z
-\frac{3}{8}\rho\tan^4z+\frac{5}{16}\rho\tan^6z\ldots,
\end{eqnarray}
\begin{equation}
D\longrightarrow \rho(\cos z_2-\cos z_1).
\label{eq.Dvac}
\end{equation}
This suggests we define an effective true zenith angle $\bar z$ via
\begin{equation}
D\equiv b\sin \bar z=P\tan \bar z,
\end{equation}
and use Eqs.\ (\ref{eq.bforz}) and (\ref{eq.Dvac}) to prove that this equals
the mean,
\begin{equation}
\bar z=\frac{z_1+ z_2}{2}.
\label{eq.zbar}
\end{equation}
Since the astrometric signature is hidden in the second derivative of $L(z)$,
the computationally most appealing expansion of $R$ and $L$ is
a Taylor series around some reference value of $z$:
\begin{equation}
R_{|z+x}\equiv \xi_0+\xi_1x+\xi_2x^2+\xi_3x^3+\ldots, \qquad \xi_0 = R_{|z}.
\label{eq.rtayl}
\end{equation}
We define refractivity integrals
as a short-cut to the notation, covering
Eq.\ (\ref{eq.RofnInt}) as a special case:
\begin{eqnarray}
R_j& \equiv& I^{2j+1}\int_1^{n_0}\frac{dn}{n(r^2n^2-I^2)^{j+1/2}}, \label{eq.RofnInt2}\\
I& \equiv& \rho n_0\sin z_0, \qquad j=0,1,2,\ldots
\end{eqnarray}
A stable numerical scheme for these integrals is proposed
in App.\ \ref{sec.Rnum}\@.
Insertion of the series (\ref{eq.rtayl}) into the l.h.s.\ of
(\ref{eq.RofnInt}) and into the arguments $z_0=z-R$ of the sines at the r.h.s.\
yields the expansion coefficients
\begin{eqnarray}
\xi_0&=&R_0, \\
\xi_1 &=& \frac{R_0+R_1}{\tan z_0}/ \left( 1+\frac{R_0+R_1}{\tan z_0} \right), \\
\xi_2&=&\frac{(\xi_1-1)^3}{2}
\left[R_0+R_1-3\frac{R_1+R_2}{\tan^2z_0}\right] \nonumber \\
&=& \frac{[3R_{12}-R_{01}\tan^2 z_0]\tan z_0}{2 \hat R_{01}^3}, \\
\xi_3 &=& -(\xi_1-1)^2
\Big[
-(R_0+R_1)\xi_2
+3\frac{R_1+R_2}{\tan^2z_0}\xi_2 \nonumber \\
&&\quad +\frac{(\xi_1-1)^2}{\tan z_0}
\Big\{
\frac{1}{6}R_0
+\frac{5}{3}R_1-3\frac{R_2}{\tan^2z_0} \nonumber \\
&&\quad -\frac{1}{2}\frac{R_1}{\tan^2 z_0}
-\frac{5}{2}\frac{R_3}{\tan^2 z_0}+\frac{3}{2}R_2
\Big\}
\Big] \\
&=& -\frac{\tan z_0}{6\hat R_{01}^5}\Big[
15(R_1-R_3)\hat R_{01} \nonumber \\
&& +9R_{12}(3R_{12}-2\hat R_{01}-\bar R_{01}\tan^2 z_0) \nonumber \\
&& +R_{01}(\hat R_{01}+3R_{01}\tan^2z_0)\tan^2z_0
\Big] ,
\end{eqnarray}
with the doubly-indexed shorthands
\begin{eqnarray}
R_{ij}&\equiv& R_i+R_j,\\
\hat R_{ij}&\equiv& R_i+R_j+\tan z_0,\\
\bar R_{ij}&\equiv& R_i+R_j-\tan z_0. \label{eq.Renddef}
\end{eqnarray}
In Eqs.\ (\ref{eq.RofnInt2})--(\ref{eq.Renddef}) and App.\ \ref{sec.Rnum},
the subscripts of $R$ are the exponential $j$ of the
definition (\ref{eq.RofnInt2}); elsewhere they indicate the
telescope number/site.
\begin{figure}[hbt]
\plotone{Rtayl.eps}
\caption{
The expansion coefficients $\xi_j$ of Eq.\ (\ref{eq.rtayl}) as a function
of zenith angle $z$ for $j=0,\ldots,3$ and the atmospheric model
(\ref{eq.chiofr})--(\ref{eq.chiofrPar}).
$R$ is an odd function of $z$, so the even-indexed $\xi_j$ approach 0 for
$z\to 0$. $\xi_1$ approaches $n_0-1$, see Eq.\ (\ref{eq.Rofnflat}).}
\label{Rtayl.ps}
\end{figure}
$x$ in Eq.\ (\ref{eq.rtayl}) is of the order of $b/(2\rho)$ if the reference
azimuth $z$
is chosen close to the middle between the telescopes, and therefore
not larger than $1.6\cdot 10^{-5}$ rad for $b<200$ m. Because the $\xi_j$ are
approximately of the same magnitude (Fig.\ \ref{Rtayl.ps}), collecting the
terms up to $j=3$ ought establish a relative accuracy of
$\approx 5\cdot 10^{-14}$ in the angle of refraction.
The expansion
\begin{equation}
L_{|z+x} = L_{|z}+\sum_{i=1,2,3,\ldots} l_i x^i, \label{eq.Lofl}
\end{equation}
proceeds via insertion of Eq.\ (\ref{eq.rtayl}) into the sines of
Eq.\ (\ref{eq.LInt}), and employs an auxiliary set of integrals
\begin{eqnarray}
v_i\equiv I^{2i}\Big[ &&
\frac{1}{[(\rho +H)^2-I^2]^{i-1/2}} \nonumber \\
&&+(2i-1)
\int_{r=\rho}^{\rho+H} \frac{rn^2}{(r^2n^2-I^2)^{i+1/2}}dr\Big] \nonumber \\
&& \stackrel{n\to 1, I\to \rho\sin z}{\longrightarrow} \rho\tan^{2i}z\cos z
.
\end{eqnarray}
\begin{eqnarray}
l_1 &=& \frac{1-\xi_1}{\tan z_0}v_1
\stackrel{n\to 1, I\to \rho\sin z}{\longrightarrow} \rho\sin z,
\\
l_2 &=& -\frac{1}{2}v_1
\left[(1-\frac{1}{\tan^2 z_0})(1-\xi_1)^2+\frac{2\xi_2}{\tan z_0}\right] \nonumber \\
&& +\frac{1}{2}v_2\frac{(1-\xi_1)^2}{\tan^2 z_0} \\
&& \stackrel{n\to 1, I\to \rho\sin z}{\longrightarrow} \frac{1}{2}\rho\cos z, \label{eq.l2vac}
\\
l_3 &=&
\frac{1}{3}v_1(1-\xi_1)
\left[-2\frac{(1-\xi_1)^2}{\tan z_0}+3\xi_2(1-\frac{1}{\tan^2 z_0})\right] \nonumber \\
&& -\frac{1}{2}v_2\frac{1-\xi_1}{\tan z_0}
\left[(1-\frac{1}{\tan^2 z_0})(1-\xi_1)^2+\frac{2\xi_2}{\tan z_0}\right] \nonumber \\
&&\qquad +\frac{1}{2}v_3\frac{(1-\xi_1)^3}{\tan^3 z_0}
\stackrel{n\to 1, I\to \rho\sin z}{\longrightarrow} -\frac{1}{6}\rho\sin z, \nonumber
\\
l_4 &&
\stackrel{n\to 1, I\to \rho\sin z}{\longrightarrow} -\frac{1}{24}\rho\cos z .
\end{eqnarray}
\begin{figure}[hbt]
\plotone{Ltayl.eps}
\caption{
The differences between the Taylor expansion coefficients $l_j$ of
Eq.\ (\ref{eq.Lofl}) and their vacuum values as a function
of zenith angle $z$ for $j=1,\ldots,3$, parametrized through the exponential
model (\ref{eq.chiofr})--(\ref{eq.chiofrPar}).
$L$ is an even function of $z$, so the odd-indexed $l_j$ approach 0 for
$z\to 0$. The vacuum limits, Eq.\ (\ref{eq.Dvac}), are $l_1\to \rho\sin z$,
$l_2\to \rho(\cos z)/2$, $l_3\to -\rho(\sin z)/6$.
}
\label{Ltayl.ps}
\end{figure}
If the $l_i$ are calculated at the mean position $\bar z=(z_1+z_2)/2$,
the delay for a single sky position becomes
\begin{eqnarray}
D &\approx&
l_1\left(z_1-z_2\right)
+l_2\left(z_1^2-z_2^2\right) \nonumber \\
&& +l_3\left(z_1^3-z_2^3\right)
+l_4\left(z_1^4-z_2^4\right)+\cdots \\
&\approx&
l_1\Delta z
+l_3 (\Delta z)^3/4 + l_5(\Delta z)^5/16 \nonumber \\
&& +l_7(\Delta z)^7/64+\cdots \label{eq.Dofl}
\end{eqnarray}
If the $l_i$ are calculated at the mean position $\bar z=(z_{1\textrm{P}}+z_{2\textrm{P}})/2$,
the differential delay is to lowest orders in $\tau$
\begin{eqnarray}
\!\!\!&&\!\!\!\!\!D_\textrm{S}-D_\textrm{P} \nonumber \\
\!\!\!&&\!\!\!\!\!=\Delta z\tau \Big[
2 l_2
+3\tau l_3
+\left((\Delta z)^2+4\tau^2\right)l_4 \nonumber \\
&& +\left(\frac{5}{2}(\Delta z)^2\tau+5\tau^3\right)l_5 \nonumber \\
&&+ \left(\frac{3}{8}(\Delta z)^4+5(\Delta z)^2\tau^2+6\tau^4\right)l_6+\cdots
\Big].
\label{eq.DssDps}
\end{eqnarray}
The leading term $2\Delta z \tau l_2$ contains
\begin{itemize}
\item
the familiar geometric ``vacuum'' contribution
$\Delta z \rho \tau \cos z\approx b\tau\cos z$; see Eqs.\ (\ref{eq.zforb})
and (\ref{eq.l2vac}),
\item
an atmospheric correction to $l_2$ of the order of 1--10 m/rad$^2$
(Fig.\ \ref{Ltayl.ps}).
It adds some tens of nanometers to the differential delay, if
$\tau <$ 1 arcmin $=3\cdot 10^{-4}$ rad, and $\Delta z < 3\cdot 10^{-5}$ rad
($b < 200$ m).
There is no equivalent contribution of this kind in planar earth models
like
Fig.\ \ref{DelModl.ps}.
\end{itemize}
With Eqs.\ (\ref{eq.PofIs}) and (\ref{eq.rtayl}),
\begin{eqnarray}
I_1 &=& \rho n_0 \sin(z_1-R_1) \\
&\approx&\rho n_0 \sin\Big(\bar z -\xi_0+(1-\xi_1)\frac{\Delta z}{2}\nonumber \\
&&\qquad -\xi_2\frac{(\Delta z)^2}{4}-\xi_3\frac{(\Delta z)^3}{8}-\ldots\Big) , \label{eq.I1ofDz}\\
I_2 &=& \rho n_0 \sin(z_2-R_2) \\
&\approx&\rho n_0 \sin\Big(\bar z -\xi_0+(\xi_1-1)\frac{\Delta z}{2}-\xi_2\frac{(\Delta z)^2}{4}\nonumber\\
&&\qquad +\xi_3\frac{(\Delta z)^3}{8}-\ldots\Big), \label{eq.I2ofDz}
\end{eqnarray}
we may expand $P^2$ in a power series of $\Delta z$,
\begin{eqnarray}
P^2\!\!\!&\approx&\!\!\!\left\{\rho n_0 \cos \bar z_0(1-\xi_1)\right\}^2(\Delta z)^2 \nonumber \\
&& +\frac{1}{12} \rho^2 n_0^2\cos \bar z_0 (1-\xi_1) \nonumber \\
&& \times \left\{
[(\xi_1-1)^3-6\xi_3]\cos \bar z_0+6\xi_2[1-\xi_1]\sin \bar z_0
\right\} \nonumber \\
&&\times (\Delta z)^4 +\ldots
\label{eq.Pofdeltaz}
\end{eqnarray}
We do the same for $D^2$ via Eq.\ (\ref{eq.Dofl}), and eventually
combine these power series of $\Delta z$ at the r.h.s.\
of Eq.\ (\ref{eq.beff}),
\begin{eqnarray}
&&\!\! b^{*2}\approx\Big\{(1-\xi_1)^2\rho^2n_0^2\cos^2 \bar z_0+l_1^2\Big\}
(\Delta z)^2 \nonumber\\
&&+ \bigg\{\frac{\rho^2 n_0^2(1-\xi_1)\cos \bar z_0}{12}
\Big([(\xi_1-1)^3-6\xi_3]\cos \bar z_0 \nonumber \\
&&\qquad +6\xi_2[1-\xi_1]\sin \bar z_0\Big)
+\frac{l_1l_3}{2}\bigg\} (\Delta z)^4 \nonumber\\
&&\quad+\ldots \label{eq.bstarlen}
\end{eqnarray}
which turns into Eq.\ (\ref{eq.bforz}) in the vacuum limit. Examples
of this ``baseline magnification'' introduced by the atmosphere are
shown in Fig.\ \ref{Beff.ps}; the effect becomes larger if the transition
into the free space is smoothed by choosing a large cut-off height $H$.
\begin{figure}[h]
\plotone{Beff.eps}
\caption{
The difference $b^*-b$ between the length of the effective baseline
above the atmosphere and the geometric baseline on the ground according
to Eq.\ (\ref{eq.bstarlen}) as a function of $z_1$,
parametrized through the model (\ref{eq.chiofr})--(\ref{eq.chiofrPar}),
which is cut off at $H=20$, 40 or 80 km.
}
\label{Beff.ps}
\end{figure}
\subsection{3D Geometry}\label{sec.zeni3D}
\subsubsection{Geographic Coordinates}
The previous section dealt with the case where the telescopes, the star and
the earth center are coplanar. In the general case, the direction of the star
and the direction of the second telescope do not share the same azimuth $A$.
Let the telescopes have geographical latitudes $\Phi_i$
and longitudes $\lambda_i$ in a geocentric spherical coordinate system:
\begin{equation}
{\bf r}_i=\rho \left(\begin{array}{c}
\cos \lambda_i \cos\Phi_i \\
\sin \lambda_i \cos\Phi_i \\
\sin\Phi_i \\
\end{array}\right), \quad (i=1,2).
\end{equation}
For the Very Large Telescope Interferometer in Northern Chile $\Phi\approx -0.4298$ rad
and $\lambda\approx -1.229$ rad, for instance.
The baseline angle in Eq.\ (\ref{eq.bofZ}) becomes
\begin{equation}
\cos Z = \sin\Phi_1\sin\Phi_2+\cos\Phi_1\cos\Phi_2\cos(\Delta\lambda),
\label{eq.b3d}
\end{equation}
where
$\Delta \lambda \equiv \lambda_1-\lambda_2$.
To transform Cartesian coordinates from the local alt-az-system of telescope $i$
(with the Cartesian coordinate $z$ pointing to the zenith, $x$ horizontally
tangentially to the earth toward north and the local horizon as indicated by
the dotted line in Fig.\ \ref{PointRho.ps}, $y$ horizontally toward west)
to the geocentric system (with $z$ pointing from the earth center to the north pole, $x$
from the center to the equator south of Greenwich, $y$ from the center
to the equator 1000 km west of Sumatra) we translate the coordinates
into a tilted system originating from the earth center, then (de)rotate them:
\begin{eqnarray}
{\bf r}_c&=&\left(
\begin{array}{ccc}
-\sin \Phi_i \cos \lambda_i & \sin\lambda_i & \cos\Phi_i\cos\lambda_i \\
-\sin \Phi_i \sin \lambda_i & -\cos\lambda_i & \cos\Phi_i\sin\lambda_i \\
\cos \Phi_i & 0 & \sin\Phi_i \\
\end{array}
\right) \nonumber \\
&&\quad \cdot \left ({\bf r}_i +\left( \begin{array}{c} 0 \\ 0 \\ \rho \end{array} \right) \right),
\quad i=1,2.
\label{eq.rgeo}
\end{eqnarray}
The inverse operation with the inverse matrix (which equals the
transpose matrix) is
\begin{eqnarray}
{\bf r}_i\!\!\!&=&\!\!\!\left(
\begin{array}{ccc}
-\sin \Phi_i \cos \lambda_i & -\sin\Phi_i\sin\lambda_i & \cos\Phi_i \\
\sin \lambda_i & -\cos\lambda_i & 0 \\
\cos \Phi_i\cos\lambda_i & \cos\Phi_i\sin\lambda_i & \sin\Phi_i \\
\end{array}
\right) \cdot {\bf r}_c \nonumber \\
&&\quad -\left( \begin{array}{c} 0 \\ 0 \\ \rho \end{array} \right) .
\label{eq.rgeoinv}
\end{eqnarray}
The product of two such operations with indexes 1 and 2 converts the two
alt-az systems:
starting in Eq.\ (\ref{eq.rgeo}) with ${\bf r}_2=0$, computing ${\bf r}_c$,
and inserting this into (\ref{eq.rgeoinv}) for $i=1$ shows
that the origin of coordinates of telescope 2 is located at
\begin{eqnarray}
&&\!\!\!{\bf b}_{12}= \rho\cdot \nonumber \\
&&\!\!\!\left(
\begin{array}{c}
-\sin \Phi_1 \cos \Phi_2 \cos(\Delta\lambda)+\cos\Phi_1\sin\Phi_2 \\
\cos\Phi_2 \sin(\Delta\lambda) \\
\cos \Phi_1\cos\Phi_2 \cos(\Delta\lambda) +\sin\Phi_1\sin\Phi_2 -1\\
\end{array}
\right)
\nonumber
\end{eqnarray}
seen from the origin of telescope 1\@.
The length of this vector is $b=|{\bf b}_{12}|$
of Eq.\ (\ref{eq.bofZ}), using Eq.\ (\ref{eq.b3d});
the third coordinate is negative since the
second telescope lies below the horizon of the first telescope (and vice versa),
as illustrated in Fig.\ \ref{PointRho.ps}.
\subsubsection{Vacuum limit}
If the atmosphere is absent, the star direction is defined as
\begin{equation}
{\bf s}_i =\left(
\begin{array}{c}
\cos A_i \sin z_i \\
\sin A_i \sin z_i \\
\cos z_i \\
\end{array}
\right),\qquad i=1,2, \label{eq.stars}
\end{equation}
in terms of the true local azimuth $A_i$ and true zenith angle $z_i$ in
the $i$th telescope
coordinate system. $A$ is counted positive
starting from N to W\@. [This azimuth convention is the one
of \cite[\S II]{Smart}; the alternative convention of
\citep{Taff,Karttunen} is obtained with
the replacement $A_i\rightarrow \pi-A_i$.]
The cosine of the angle
between the star and the baseline in Fig.\ \ref{PointRho.ps} is
\begin{equation}
\cos \varphi_1 = {\bf s}_1 \cdot {\bf b}_{12}/b , \label{eq.blen}
\end{equation}
where ${\bf s}_1 \cdot {\bf b}_{12}$ is known as the geometric optical path delay.
If one swaps the indexes $1$ and $2$, the cosine switches its sign, because
in this parallax-free situation the angle between star and baseline is the
$180 ^\circ$-complement of the angle relative to the other telescope:
\begin{equation}
\cos \varphi_2 = {\bf s}_2 \cdot {\bf b}_{21}/b = - \cos\varphi_1.
\end{equation}
This may be verified with the standard coordinate transformations
between the hour angles $h_i$ and right ascension $\delta$
for $i=1,2$,
\begin{equation}
\cos\delta \sin h_i = \sin z_i \sin A_i,
\end{equation}
\begin{equation}
\sin z_i \cos A_i = \cos \Phi_i \sin\delta -\sin\Phi_i\cos\delta \cos h_i,
\label{eq.sinzcosA}
\end{equation}
\begin{equation}
\cos z_i = \sin \Phi_i \sin\delta +\cos\Phi_i\cos\delta \cos h_i,
\label{eq.zofRA} \label{eq.cosz}
\end{equation}
\begin{equation}
h_1-h_2 =\lambda_1-\lambda_2
\end{equation}
So if the atmosphere is absent, this angle
relates to $P=|\sin \varphi_i|b$ ($i=1,2$) as shown in Fig.\ \ref{Base.ps}.
The mean and difference in the true zenith angles remain defined as
in Eqs.\ (\ref{eq.zbar}) and
\begin{equation}
\Delta z \equiv z_1-z_2,
\end{equation}
and can be retrieved from the geographical coordinates (\ref{eq.sinzcosA})--(\ref{eq.cosz})
and \cite[(4.3.34)-(4.3.37)]{AS}.
\subsubsection{Ray tracing}
There is one distinguished geocentric coordinate system for
the case of a single star, shown in Fig.\ \ref{DelModl3.ps}, in
which the direction vectors of the incoming light above the atmosphere
have only a component along the polar axis.
The side view of this geometry reduces to Fig.\ \ref{DelModl2.ps} if
the positions
1, 1v, 2 and 2v lie on the same projected straight line.
(In the following, ``projected'' means projected onto a plane
perpendicular to the ray propagation above the atmosphere.)
\begin{figure}[hbt]
\plotone{DelModl3.eps}
\caption{A top view of the earth coordinates as seen from the incoming two rays
in which both ${\bf s}_i$ are perpendicular to the plane of the drawing.
If the atmosphere were absent, the first ray would see telescope 1 at
the position 1v. The atmospheric refraction pulls rays toward
the earth center, which means it has to
relocate its impact at the top of the atmosphere to the actual image
position 1 above the atmosphere
to end up at telescope 1 on the ground. The same effect for ray 2 means
the projected baseline vector $P$ above the atmosphere from 1 to 2 is
both longer and tilted by $\Delta \eta$ compared to the vacuum case from
1v to 2v.
}
\label{DelModl3.ps}
\end{figure}
The strategy to transform the star position to the projected baseline
(vector) $P$ is (i) to calculate the triangle formed by the vacuum baseline
and the earth center seen by the incoming light, and (ii) stretch this
radially outward to include the effect of the two difference refraction
angles $R_i$:
the two true zenith angles $z_i$ for both telescopes
are assumed to be given via Eq.\ (\ref{eq.zofRA}). The impact parameters
$I_i$ of the rays
for the vacuum case are $\rho \sin z_i$ and they equal the lengths of the
(projected) station vectors
from the earth center to the points 1v and 2v in Fig.\ \ref{DelModl3.ps}\@.
The vacuum baseline in Fig.\ \ref{DelModl3.ps} stretches from
1v to 2v, which has the projected length $b|\sin \varphi_i|$ as written
down in Eq.\ (\ref{eq.blen}).
The projected baseline aperture angle $\sigma$ in the triangle with
side lengths
$\rho \sin z_i$ and $b|\sin \varphi_i|$ relates to the impact parameters
and projected vacuum baseline by planar trigonometry \cite[4.3.148]{AS},
\begin{equation}
\sin^2 z_1+\sin^2 z_2 = 2 \sin z_1 \sin z_2 \cos \sigma
+ \frac{b^2}{\rho^2}\sin^2 \varphi_1.
\end{equation}
$\sigma$ is the projection of $Z$ defined in Eq.\ (\ref{eq.b3d}).
The equation mingles the true zenith angles from the two telescope's point
of view with the angles $\varphi_i$, which represent the star distance from the
baseline direction. One may reduce this to the geographic
coordinates by multiplying
${\bf r}_1+{\bf b}_{12}={\bf r}_2$ with ${\bf s}_1$ to get
\begin{eqnarray}
\cos\sigma&=&\frac{\cos Z-\cos z_1\cos z_2}{\sin z_1\sin z_2}\\
&=& 1-\frac{\cos(z_1-z_2)-\cos Z}{\sin z_1\sin z_2}.
\end{eqnarray}
$\cos\sigma$ and $\cos Z$ are close to unity in contemporary
optical interferometry,
so the actual implementation avoids the use of the cosine
in favor of the haversine,
\begin{eqnarray}
&&\mathop{\mathrm{hav}}\nolimits\sigma=
\frac{\sin\frac{Z+\Delta z}{2} \sin\frac{Z-\Delta z}{2}}{\sin z_1\sin z_2} \\
&&=
\frac{Z^2-(\Delta z)^2}{4\sin^2\bar z}
-\left(\frac{1}{16}-\frac{1}{48}\sin^2\bar z\right)
\left(\frac{\Delta z}{\sin\bar z}\right)^4 \nonumber \\
&&\, -\frac{1}{48}\left(\frac{Z}{\sin\bar z}\right)^4
+\frac{1}{16}\frac{Z^2 (\Delta z)^2}{\sin^4\bar z} \nonumber \\
&&\,
-\left(45-30\sin^2\bar z+2\sin^4\bar z\right)
\frac{(\Delta z)^6}{2880 \sin^6\bar z} \nonumber \\
&&\, +\left(3-\sin^2 \bar z\right)\frac{(\Delta z)^4Z^2}{192 \sin^6\bar z}
-\frac{(\Delta z)^2Z^4}{192 \sin^4\bar z} \nonumber \\
&&\, +\frac{Z^6}{1440 \sin^2\bar z}+\ldots , \label{eq.havsigma}
\end{eqnarray}
to protect against cancellation of significant digits,
and returns from there to the sine, if needed,
\begin{eqnarray}
&&\sin\sigma = 2\sqrt{\mathop{\mathrm{hav}}\nolimits \sigma}\sqrt{1-\mathop{\mathrm{hav}}\nolimits\sigma} \\
&&= 2\sqrt{\mathop{\mathrm{hav}}\nolimits \sigma}\left[1-\sum_{j=1}^\infty \frac{(2j-3)!!}{j!}
\left(\frac{\mathop{\mathrm{hav}}\nolimits \sigma}{2}\right)^j \right] \nonumber \\
&&=\sqrt{\mathop{\mathrm{hav}}\nolimits\sigma}\left(2-\mathop{\mathrm{hav}}\nolimits\sigma-\frac{\mathop{\mathrm{hav}}\nolimits^2\sigma}{4}
-\frac{\mathop{\mathrm{hav}}\nolimits^3\sigma}{8}
\cdots\right). \nonumber
\end{eqnarray}
The special coplanar case of Sec.\ \ref{sec.zeni2D} is included
as $\Delta z=Z$ and $\sigma=0$ or $\pi$.
(On some Linux systems, where the {\tt cosl} library function
does not support the full accuracy of the 96 bit {\tt long double}
number representation, it makes sense to switch to alternative
high-precision implementations of the cosine \citep{Schonfelder}.)
Two calculations for the actual impact parameters
$I_i=\rho n_0 \sin z_0^{(i)}$ starting from
the given $z_i$ would be done as in the preceding, ``aligned'' geometry of
Sec.\ \ref{sec.zeni2D}, and these inserted into
\begin{equation}
I_1^2+I_2^2 = 2 I_1 I_2 \cos \sigma + P^2
\end{equation}
to calculate the projected baseline $P$ and to generalize Eq.\ (\ref{eq.PofIs}).
$P^2=(I_1-I_2)^2+4I_1I_2\mathop{\mathrm{hav}}\nolimits\sigma$ comprises the terms of
Eq.\ (\ref{eq.Pofdeltaz}) of the constrained geometry augmented by
\begin{equation}
4I_1I_2\mathop{\mathrm{hav}}\nolimits \sigma
=\left(\rho n_0\sin \bar z_0\right)^2
\frac{Z^2-(\Delta z)^2}{\sin ^2\bar z}
+\ldots
\end{equation}
as read from Eq.\ (\ref{eq.havsigma}) in combination with
Eqs.\ (\ref{eq.I1ofDz})--(\ref{eq.I2ofDz}).
To calculate the path difference $D=L_1-L_2$ between the two rays above the
atmosphere, Eq.\ (\ref{eq.Dv}) remains valid, but in general a Taylor
expansion akin to (\ref{eq.taylDv}) does not exist,
because $P\ge I_i\sin\sigma$ ($i=1,2$) excludes small $P$ at arbitrary $\sigma$.
Eq.\ (\ref{eq.rtayl}) remains in use to expand the refraction angle in a
neighborhood of $\bar z$, and so do Eqs.\ (\ref{eq.Lofl})--(\ref{eq.Dofl})
that depend only on zenith distances but not on azimuths.
\subsubsection{Baseline Rotation}
The rotation angle between the baseline $b$ (projected on
a plane perpendicular to the star direction) and $P$ is $\Delta \eta\equiv\eta_P-\eta_b$
where $\eta_b$ and $\eta_P$ are the angles from $b$ to $I_2$ and
$P$ to $I_2$ respectively, and given by
\begin{eqnarray}
\frac{I_2}{I_1}&=&\cos\sigma +\sin \sigma \cot \eta_P, \\
\frac{\sin z_2}{\sin z_1}&=&\cos\sigma +\sin \sigma \cot \eta_b.
\end{eqnarray}
Numerical examples are presented in Figs.\ \ref{ProtP.ps}--\ref{Protz.ps}:
By symmetry, the rotation effect vanishes if the star azimuth is along
the baseline or perpendicular to it.
The angle $\Delta \eta$ may be about five times larger than the
interferometric resolution of a 200 m baseline in the K-band:
the interferometric fringes appear slightly rotated on the detector,
and the true $(u,v)$ coordinate is found by rotating the ``apparent''
vector by $-\Delta\eta$.
\begin{figure}[h]
\plotone{ProtP.eps}
\caption{The baseline rotation $\Delta \eta$ introduced in
Fig.\ \ref{DelModl3.ps} for a baseline $b=200$m, for 6 different azimuth
angles $A_1$ measured from $T_1$ toward the baseline, and for $\sin z_1$
changing in equidistant steps of $0.05$ from $0.05$ to $0.85$.
}
\label{ProtP.ps}
\end{figure}
\begin{figure}[h]
\plotone{Protz.eps}
\caption{An alternative view on the six lines of Fig.\ \ref{ProtP.ps} with the abscissa
switched from the projected baseline length to the azimuth angle.
Each small cross has a counterpart in Fig.\ \ref{ProtP.ps}. This graph would
not change visibly choosing a baseline of $b=100$ m.
}
\label{Protz.ps}
\end{figure}
There is no intrinsically new aspect compared to the analysis of Sec.\ \ref{sec.zeni2D}:
The rotation of the baseline vector $P$ relative to the projection of $b$
would again be absorbed into the pointing direction of the two telescopes.
With only one (scalar) observable, which is the differential delay, one cannot
measure both the astrometric angle (distance) between
two stars and the positioning angle of the S relative to the P
at the same time.
\subsubsection{Astrometric Case}
Eq.\ (\ref{eq.DssDps}) obviously looses its meaning
because the secondary star now has two degrees of freedom
and can no longer be positioned by a single angle $\tau$.
The details follow
by writing down Eq.\ (\ref{eq.cosz}) for both telescopes and both stars.
If the star distances $\tau_h$ and $\tau_\delta$ are defined as
\begin{equation}
h_{\mathrm S}\equiv h_{\mathrm P}+\tau_h,\qquad
\delta_{\mathrm S}\equiv \delta_{\mathrm P}+\tau_\delta,\qquad
\end{equation}
the expansion of Eq.\ (\ref{eq.cosz}) yields
for $z_{i\mathrm S}\equiv z_{i\mathrm P}+\tau_{iz}$
to lowest order in $\tau_h$ and $\tau_\delta$:
\begin{eqnarray}
\tau_{iz}&=&\frac{\cos\Phi_i \sin\delta_{\mathrm P}\cos h_{\mathrm P}
-\sin\Phi_i\cos\delta_{\mathrm P}}{\sin z_{i\mathrm P}}\tau_\delta
\nonumber \\
&& +\frac{\cos\Phi_i \cos\delta_{\mathrm P}\sin h_{\mathrm P}}
{\sin z_{i\mathrm P}}\tau_h+\ldots
\end{eqnarray}
The changes $\tau_{iA}$ in the azimuths are not detailed here, since
the refraction is determined by the zenith angles; they are
expected to ensure that the associated change in the star direction
(\ref{eq.stars}) forms an acute angle to the baseline to maintain the
interferometric resolution.
The symmetry suggested in Fig.\ \ref{Lofz.ps} will generally be broken:
$\Delta z_\textrm{P}\neq \Delta z_\textrm{S}$.
Restarting from Eq.\ (\ref{eq.Dofl}), the differential delay is expanded in
powers of the doubly differential $\Delta \tau_z$:
\begin{eqnarray}
&&D_\textrm{S}-D_\textrm{P} \nonumber \\
&& =\Delta z_{\mathrm P}\bar\tau_z \Big[
2 l_2
+3\bar\tau_z l_3
+\left(
(\Delta z_{\mathrm P})^2+ 4\bar\tau_z^2\right)l_4 \nonumber \\
&& \qquad\qquad +\cdots \Big]+O(\Delta \tau_z) ,
\end{eqnarray}
with $\bar \tau_z\equiv (\tau_{1z}+\tau_{2z})/2$ and
$\Delta \tau_z\equiv \tau_{1z}-\tau_{2z}$, where
the $l_j$ are again evaluated at the mean primary zenith,
$(z_{1\mathrm P}+z_{2\mathrm P})/2$.
\section{Summary}
The optical path length integral of star light passing through the
atmosphere can be handled with numerical and analytical methods known from
treatments of the more familiar refractivity integral. Subtraction of two
of these computes the optical path difference, and renormalizes the
optical path lengths to start from a common plane tangential to the
earth's upper atmosphere, which at the same time defines the projected baseline
in this plane perpendicular to the two rays. Definition of a right triangle
above the atmosphere with the projected baseline and the path difference
as two catheti defines a hypotenuse, which is an effective baseline that
is longer than and rotated relative to the geometric baseline between
the receiving telescopes on earth.
|
{
"timestamp": "2004-11-14T21:46:21",
"yymm": "0411",
"arxiv_id": "astro-ph/0411384",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411384"
}
|
\section{Introduction}
The equation of state of strongly interacting matter at finite
temperature $T$ and small chemical potential $\mu$ has become accessible
fairly recently through first principle lattice QCD
calculations~\cite{All03,Csi04,Fod02,Fod03}. Apart from solving QCD on the
lattice, there exist analytical approaches such as resummed HTL
scheme, $\Phi$ functional approach etc. (cf.~\cite{Ris04} for a
recent survey). Ab initio approaches~\cite{Bla99,Bla02} are
restricted in describing lattice data on temperatures
$T\gtrsim 2.5T_{\rm c}$, where $T_{\rm c}$ is the transition
temperature of deconfinement and chiral symmetry restoration. In contrast,
phenomenological models with adjustable
parameters~\cite{Pes94,Pes00,Lev98} cover the region $T\gtrsim T_{\rm
c}$. Here, we present new developments of our quasi-particle model
(QPM)~\cite{Pes94,Pes00}.
In~\sref{sec:model} the QPM is reviewed. In~\sref{sec:coeffs} the
model is confronted with recent lattice QCD data. Motivating our
model,~\sref{sec:QCD}
briefly illustrates a chain of approximations within a $\Phi$ funtional
approach. The results are summarized in~\sref{sec:conclusions}.
\section{Quasi-particle model}
\label{sec:model}
The model is based on the idea that the quark-gluon fluid can be
expressed in terms of quasi-particles. The pressure of $N_q$
light (q), strange (s) quarks and gluons (g) reads
\begin{equation}
\label{equ:pres}
\fl
p(T,\mu)=\sum_{i\,=\,q,s,g} p_i(T,\mu) - B(T,\mu) .
\end{equation}
$p_i$ are thermodynamic standard expressions in which $T$
and $\mu$ dependent self-energies $\Pi_i$ enter. $B(T,\mu)$, together
with the stationarity condition $\delta p/\delta m_i^2=0$~\cite{Gor95},
ensures thermodynamic self-consistency (cf.~\cite{Pes00,Pes02} for
details). Thus, the entropy density $s=\partial p/\partial
T=\sum_{i\,=\,q,s,g} s_i$ explicitly reads
\begin{equation}
\label{equ:entr}
\fl
s_i=\frac{d_i}{2\pi^2T}\int_0^\infty \rmd
k\,k^2\left\{\frac{\left(\frac{4}{3}k^2 +
m_i^2\right)}{\omega_i(k)}\left[f_i^+(k) + f_i^-(k)\right] -
\mu\left[f_i^+(k)-f_i^-(k)\right]\right\}
\end{equation}
with statistical distribution functions $f_i^\pm (k)=(\exp
[(\omega_i(k)\mp\mu_i)/T]+S_i)^{-1}$, $S_q=S_s=1$, $S_g=-1$,
$\mu_q=\mu$, $\mu_s=\mu_g=0$ and degeneracies $d_q=6N_q$, $d_s=6$ and
$d_g=8$.
In the thermodynamically relevant region of momenta $k\sim
T$, $\mu$, the quasi-particle dispersion relations are approximated by
the asymptotic mass shell expressions near the light cone
$\omega^2_i(k)=k^2 + m_i^2$ with $m_i^2=\Pi_i(k;T,\mu) +
(x_iT)^2$. The self-energies $\Pi_i$ of the quasi-particle excitations are
approximated by their 1-loop expressions at hard
momenta~\cite{Pes00} neglecting imaginary parts, and $x_iT$
represent the quark masses used on the
lattice~\cite{All03}. Replacing the running coupling in the
self-energies by an effective coupling, $G^2(T,\mu)$, non-perturbative
effects are thought to be taken into
account~\cite{Pes02}. Imposing thermodynamic consistency onto $p$, a flow equation
for $G^2(T,\mu)$ follows~\cite{Pes00}
\begin{equation}
\label{equ:flow}
\fl
a_\mu\frac{\partial G^2}{\partial\mu} + a_T\frac{\partial
G^2}{\partial T} = b
\end{equation}
which can be solved as Cauchy problem by knowing $G^2$ on an arbitrary curve
$T(\mu)$. One convenient choice is parametrizing $G^2(T(\mu=0))$
appropriately (cf.~\cite{Blu04}) such that $p$ and $s$ can be computed at
non-vanishing $\mu$ from~(\ref{equ:pres},\ref{equ:entr}).
\section{Expansion coefficients}
\label{sec:coeffs}
Apart from~(\ref{equ:pres}), the pressure can be decomposed into a
Taylor series
\begin{equation}
\label{equ:series}
\fl
\frac{p(T,\mu)}{T^4}=\frac{p(T,\mu=0)}{T^4} + \frac{\Delta
p(T,\mu)}{T^4} = \sum_{n=0}^\infty
c_{2n}(T)\left(\frac{\mu}{T}\right)^{2n}
\end{equation}
with $c_0(T)=p(T,\mu=0)/T^4$ and vanishing $c_k$ for odd $k$. The
expansion coefficients $c_k$ have been subject of recent lattice
evaluations~\cite{All03} by computing derivatives of the thermodynamic
potential. They follow from~(\ref{equ:pres}) as
$c_k(T)=\left.\partial^k p/\partial\mu^k\right|_{\mu=0}T^{k-4}/k!$ in
our model and read
\begin{eqnarray}
\label{equ:coeff2}
\fl
c_2 & = \frac{3N_q}{\pi^2T^3}\int_0^\infty \rmd k\, k^2
\frac{\rme^{\omega_q/T}}{\left(\rme^{\omega_q/T}+1\right)^2} \, , \\
\label{equ:coeff4}
\fl
c_4 & = \frac{N_q}{4\pi^2T^3}\int_0^\infty \rmd k\, k^2
\frac{\rme^{\omega_q/T}}{\left(\rme^{\omega_q/T}+1\right)^4}
\left\{\rme^{2\omega_q/T} - 4\rme^{\omega_q/T} +1
-\frac{T}{\omega_q}\left[\rme^{2\omega_q/T}-1\right]A_2\right\}\,,
\end{eqnarray}
where $A_2=(G^2/\pi^2+3x_q\sqrt{G^2/(6\pi^4)}+
[\frac{3}{2}x_qT^2/\sqrt{6G^2}+\frac{1}{2}T^2]\partial^2
G^2/\partial\mu^2)|_{\mu=0}$, $\omega_q$ is taken at
$\mu=0$ and $\partial^2G^2/\partial\mu^2$ follows from
differentiating~(\ref{equ:flow}).
In the left panel of~\Fref{fig:coeffsexcess}, the QPM results of $c_{2,4}$
calculated from~(\ref{equ:coeff2},\ref{equ:coeff4}) are compared with
lattice QCD results~\cite{All03} for the two-flavour case,
i.e. $N_q=2$. Using $x_q=0.4$ and $x_g=0$
as in~\cite{All03} and setting $T_{\rm c}(\mu=0)\equiv T_0=170$ MeV,
$G^2(T(\mu=0))$ is adjusted to describe $c_2(T)$. Since
$c_4$ in~(\ref{equ:coeff4}) only depends on $G^2$ and its derivatives
at $\mu=0$, no further assumptions enter into the evaluation once the
parametrization is fixed. A fairly good agreement is found for
$c_4(T)$. Note, in particular, that the pronounced peak structure of
$c_4$ at $T_0$ solely originates from the term including $\left.\partial^2
G^2/\partial\mu^2\right|_{\mu=0}$.
\begin{figure}[h]
\begin{center}
\includegraphics[width=.5\textwidth]{c2_22_1.eps}
\includegraphics[width=.48\textwidth]{delta51.eps}
\caption{\label{fig:coeffsexcess} Left: Expansion coefficients $c_2(T)$
(squares) and $c_4(T)$ (circles). Data from~\cite{All03}. Full
and dashed curves represent, respectively, corresponding QPM
results~(\ref{equ:coeff2},\ref{equ:coeff4}). Right: Scaled excess
pressure $\Delta p$ from~\cite{All03} (symbols)
and QPM results (full lines) from truncating~(\ref{equ:series}) at
order $(\mu/T)^4$ for different
$\mu/T_0=0.2,0.4,0.6,0.8,1.0$ (lower to upper curve). Dashed
lines depict corresponding full QPM results.}
\end{center}
\end{figure}
In~\cite{All03}, the excess pressure $\Delta p$ has been
calculated as the truncation of the expansion~(\ref{equ:series})
including the order $(\mu/T)^4$. The right panel
of~\Fref{fig:coeffsexcess} exhibits the
comparison of $\Delta p$ calculated by employing only $c_{2,4}$
in~(\ref{equ:series}) (full lines) with the lattice data for different
$\mu/T_0$. An impressively good agreement with the data
is observed for small values of $\mu$ in which case $c_4$ is of less
importance. Similarly, $\Delta p$ can be evaluated as infinite series
from the QPM by combining~(\ref{equ:pres}) and~(\ref{equ:series})
(dashed lines). These full results
differ noticeably from the truncated results only for $T\approx
T_0$. It should be noted that the model is successfully applied to
describing the equation of state with strange quarks~\cite{Sza03}.
\section{Contact with QCD}
\label{sec:QCD}
Motivating the strong assumptions made in formulating the QPM
expressions~(\ref{equ:pres},\ref{equ:entr}), a chain of reasonable
approximations starting from QCD would be of desire. The thermodynamic
potential $\Omega=-pV$ in ghost free gauge reads~\cite{Bla99}
\begin{equation}
\fl
\Omega[D,S] = T\left\{\frac{1}{2}{\Tr}\left[\ln D^{-1}-\Pi
D\right] - {\Tr}\left[\ln S^{-1}-\Sigma S\right]\right\} +
T\Phi[D,S]
\end{equation}
with dressed propagators $D$ and $S$ of bosons and fermions and
corresponding exact self-energies $\Pi$ and $\Sigma$ from Dyson's
equations. The functional $\Phi$ is given by the sum over all 2
particle irreducible skeleton diagrams and $\Pi$, $\Sigma$ follow from
truncating dressed propagator lines in these
diagrams~\cite{Bla99}. The sum over Matsubara frequencies in the trace
$\Tr$ is transformed into an appropriate contour integral in the
complex energy plane. Computing the entropy density $s=-\partial
(\Omega/V)/\partial T$, an ultra-violet finite expression
\begin{eqnarray}
\label{equ:phientropy}
\nonumber
\fl
s = &-{\rm tr}\int\frac{d^4k}{(2\pi)^4}\frac{\partial
n(\omega)}{\partial T} [{\rm Im}\ln D^{-1}-{\rm Im}\,\Pi\,{\rm Re}
D] \\
\fl
&- 2{\rm\, tr}\int\frac{d^4k}{(2\pi)^4}\frac{\partial
f(\omega)}{\partial T} [{\rm Im}\ln S^{-1}-{\rm Im}\,\Sigma\,{\rm
Re} S] + s'
\end{eqnarray}
is derived. After truncating $\Phi$ at 2 loop order, $s'=0$ is
found. Lost gauge invariance gets restored by employing hard thermal
loop expressions for $\Pi$ and $\Sigma$ which show the correct limiting
behaviour for $k\sim T$, $\mu$. Performing the remaining trace tr over discrete
indices in~(\ref{equ:phientropy}), quantum numbers of quarks and gluons are
recovered. Furthermore, the exponentially suppressed longitudinal
gluon modes and the plasmino branch can be neglected. In addition,
neglecting imaginary parts in $\Pi$ and $\Sigma$ as well as Landau
damping and approximating self-energies and dispersion relations
suitably, expression~(\ref{equ:entr}) for the entropy density $s$ is
recovered.
\section{Conclusion}
\label{sec:conclusions}
The quasi-particle model has been reviewed and successfully compared
with recent lattice data of the expansion coefficients $c_{2,4}$ and
the excess pressure at finite temperature and chemical
potential. Briefly, a chain of approximations within the $\Phi$
functional scheme starting from QCD has been summarized which leads to
our model.
\ack
Inspiring discussions with A~Peshier are gratefully acknowledged. The
work is supported by BMBF, GSI and EU-I3HP.
\Bibliography{10}
\bibitem{All03} Allton C R \etal 2003 {\it\PR}D {\bf 68} 014507
\bibitem{Csi04} Csikor F \etal 2004 {\it J. High Energy Phys.}
JHEP05(2004)046; 2004 {\it Prog. Theor. Phys. Suppl.} {\bf 153} 93--105
\bibitem{Fod02} Fodor Z and Katz S D 2002 {\it\PL} B {\bf 534} 87--92
\bibitem{Fod03} Fodor Z \etal 2003 {\it\PL} B {\bf 568} 73--7
\bibitem{Ris04} Rischke D H 2004 {\it Prog. Part. Nucl. Phys.} {\bf
52} 197--296
\bibitem{Bla99} Blaizot J P \etal 1999 {\it\PL} B {\bf 470} 181--8
\bibitem{Bla02} Blaizot J P \etal 2002 {\it\NP} A {\bf 698} 404--7
\bibitem{Pes94} Peshier A \etal 1994 {\it\PL} B {\bf 337} 235--9; 1996
{\it\PR} D {\bf 54} 2399--402
\bibitem{Pes00} Peshier A \etal 2000 {\it\PR} C {\bf 61} 045203
\bibitem{Lev98} Levai P and Heinz U 1998 {\it\PR} C {\bf 57} 1879--90
\nonum Schneider R A and Weise W 2001 {\it\PR} C {\bf 64} 055201
\nonum Letessier J and Rafelski J 2003 {\it\PR} C {\bf 67} 031902
\nonum Rebhan A and Romatschke P 2003 {\it\PR} D {\bf 68} 025022
\nonum Thaler M A \etal 2004 {\it\PR} C {\bf 69} 035210
\bibitem{Gor95} Gorenstein M I and Yang S N 1995 {\it\PR} D {\bf 52}
5206--12
\bibitem{Pes02} Peshier A \etal 2002 {\it\PR} D {\bf 66} 094003
\bibitem{Blu04} Bluhm M \etal 2004 {\it Preprint} hep-ph/0402252; 2004
{\it Preprint} hep-ph/0411106
\bibitem{Sza03} Szabo K K and Toth A I 2003 {\it J. High Energy Phys.}
JHEP06(2003)008
\endbib
\end{document}
|
{
"timestamp": "2005-01-24T16:45:48",
"yymm": "0411",
"arxiv_id": "hep-ph/0411319",
"language": "en",
"url": "https://arxiv.org/abs/hep-ph/0411319"
}
|
\section{Introduction}
\label{intro}
The determination of the equation of state (EOS) at or above nuclear
densities is a long sought goal in high energy astrophysics. In this
respect direct observations of neutron stars (NSs) may
provide invaluable insight into several key issues of fundamental
physics, like quantum chromodynamics/electrodynamics, superfluidity and
superconductivity, that could not be otherwise tested under laboratory
conditions. An ideal way to place tight constraints on the EOS is
the simultaneous mass and radius measurement of NSs (see
e.g. \citealt{lapra2001}). Precise mass determinations have been
obtained for NSs in binaries, especially in radio pulsar systems
(e.g. \citealt{tho99}). Simultaneous mass and radius measurements are
presently available for a few X-ray binaries, although masses
derived from gravitational redshift of spectral lines are still
uncertain in many cases (e.g. \citealt{c2002}). Mass estimates can be
obtained also from quasi-periodic oscillation
measurements (\citealt{bul2000}), but results based on this, as well as other
approaches, are still quite model dependent. The study of glitches
observed in radio pulsars, and recently
in anomalous X-ray pulsars (AXPs) as well
(e.g. \citealt{hor2004}; \citealt{osso2003}),
promise further insight
into the understanding of the internal structure of NSs, as well as
future observations of neutrino and gravitational wave emission
from neutron star sources.
For the time being, however, X-ray observations of NSs are,
and will still be for some time,
central for our understanding of the star interior. Isolated
neutron stars that emit at X-ray energies as they
cool are particularly promising in this respect. Their thermal radiation
directly comes from the star surface, carrying information on the
physical conditions of the emitting matter, in particular on the star
surface temperature. Thermal X-ray emission has been detected from about 20
isolated NSs so far, including normal radio pulsars, central
compact objects in supernova remnants (CCOs in SNRs),
radioquiet NSs, AXPs and soft $\gamma$-repeaters
(see e.g. \citealt{paza03}; \citealt{kaspi04}; \citealt{hab04} for reviews).
Significant progress in the understanding of NS thermal evolution has
been made in recent years and cooling curves have been computed
by several groups (see e.g. \citealt{kam};
\citealt{tsuruta}; \citealt{page}; \citealt{bgv2004} and references therein).
The present work is based on NS cooling calculations performed in
the {\it Nuclear medium cooling scenario} (\citealt{bgv2004}) which differs
from the other abovementioned approaches in a consistent inclusion of medium
effects. Processes of internal heating (\citealt{tsuruta}) are not
included.
Since the cooling history crucially depends on the assumed physical
conditions inside the star, comparison with observations may rule out
some models in favor of others.
To study this we will vary assumptions on the nuclear pairing
gaps, the relation between crust and surface temperature as well as
presence or absence of pion condensation.
A customary way of testing predictions of
cooling calculations is to construct a temperature vs. age
($\mathrm T$-$\mathrm t$ for short) plot for
the largest sample of sources. Despite its wide application
and undisputed usefulness, this test has a number of limitations.
In this paper we suggest use of the $\mathrm{Log\, N}$-$\mathrm{Log\, S}$
distribution of close-by NSs as an additional probe for NS cooling models.
The idea is based on the comparison of present
observational data with NS population synthesis calculations in which cooling
curves are one of the ingredients. Our approach extends the actual
calculation of
observational properties of the population of close-by young NSs
that has already been developed by
\cite{p03,p04} (hereafter Paper I and II)
by including nuclear medium effects in the cooling code and by investigating
the contribution of massive
progenitors in the Gould Belt to the local NS population
in more details.
The paper is organized as follows. In section \ref{tests} we discuss
the advantages and limitations of the two methods
in extracting information from observational data.
Section \ref{popsynth} presents our population synthesis model in more
detail, especially concerning the choice of the set of cooling curves
which will be used in our calculation.
We present our results and discuss them in section \ref{results}.
Section \ref{concl} contains our conclusions.
\section{The two tests}
\label{tests}
In this section we briefly discuss and compare the capabilities of
the conventional $\mathrm T$-$\mathrm t$ test and of the
$\mathrm{Log\, N}$-$\mathrm{Log\, S}$ test we propose here.
Our main conclusion is that the two approaches should be used together
as their advantages and disadvantages are mostly complementary to each other
especially in the case of available samples of observed objects.
\subsection{The $\mathrm T$-$\mathrm t$ test}
The $\mathrm T$-$\mathrm t$ test is the most appropriate one
to compare results of thermal evolution calculations with observations.
An important advantage of this test is that there are no additional theoretical
uncertainties except those connected with the cooling model: theoretical
cooling
curves do not depend on unknown (or poorly known) astrophysical parameters but
only on the input physics of the star interior (for a discussion of
cooling processes and references to earlier papers, see e.g. \citealt{yak99}
and \citealt{bgv2004}). Still, there are some well known drawbacks to this
method.
For sources associated with radiopulsars or CCOs, the star age is usually estimated from
the spindown age $P/\dot P$, or inferred from the age of the
supernova remnant.
To which extent these determinations are indeed representative
of the neutron star age is still uncertain.
The situation is even worse for {\it ROSAT\/} isolated neutron stars which are
not associated with a SNR and are with no exception radio-silent.
Current age estimates for the two brightest objects in this class
(RX J1856.5-3754 and RX~J0720.4-3125), based on dynamical considerations,
should be regarded only as guesses.
The $\mathrm T$-$\mathrm t$ test is not very sensitive to objects with ages
$\ga 10^5$ yrs. There are two reasons for this.
First, there are only a few sources older than
this value to which the test can be applied.
Then, cooling curves sharply drop at the photon cooling
stage. Shifts between different cooling curves are comparable with data error
bars. So, it is difficult to discriminate between models that differ
mainly in this respect.
Cooling calculations provide the star temperature at the core-crust boundary
and the actual surface temperature is then obtained applying a bridging
formula (e.g. \citealt{t1979}; \citealt{y2004}).
Detailed modeling of heat transport in the highly magnetized envelope
indicates that the surface temperature may be influenced by several effects,
among which the magnetic field distribution inside the star is very important
(\citealt{gkp2004}).
The temperature is derived from a spectral fit to the data and, as such,
depends on the assumed emission model for the surface.
Different models (blackbody, H/heavy elements atmosphere with/without magnetic
field, solid surface) give values which can differ by a factor of a few.
Recently, it has been suggested that the Temperature-Age test should be
refined by the introduction of a {\it brightness constraint} (BC),
which demands that cooling curves for objects within a mass interval
anticipated as typical should not appear significantly brighter than the
brightest of already observed objects for a given age (\citealt{g2005},
hereafter G05).
It has been shown in this work that some of the cooling models discussed in
BGV which would violate the BC become acceptable when the crustal
properties are modified.
These modifications do not affect old stars with ages $\ga 10^5$ yrs for
which the $\mathrm{Log\, N}$-$\mathrm{Log\, S}$ test is relevant.
Due to our lack of knowledge of the mass distribution
and atmospheric properties of cooling compact stars, we must be cautious in
applying the strong BC.
Therefore we refer in this paper to results of the $\mathrm T$-$\mathrm t$
test with and without BC.
Finally, the data sample is non-uniform: there are sources of different types,
and, more important, the sample is neither flux, nor volume limited,
and is a strongly selected one, containing objects for
which age and temperature estimates
are available\footnote{Note that
there are some young cooling NSs for which their age is not known,
in particular those not observed as active radiopulsars.
These sources
can not be included in the sample of the Temperature-Age test.}.
Clearly these sources are not representative of any real population of NSs.
\subsection{The $\mathrm{Log\, N}$-$\mathrm{Log\, S}$ test}
The $\mathrm{Log\, N}$-$\mathrm{Log\, S}$ distribution is a widely used tool
in many branches of astronomy.
For isolated NSs such an approach has been already
used by \cite{nt99} and \cite{p00b} to probe the origin of isolated NSs in the
solar proximity.
The main conclusion of these investigations is that the observed
$\mathrm{Log\, N}$-$\mathrm{Log\, S}$
cannot be easily explained assuming that the local population of NSs
originated only in the Galactic disc.
As shown in Paper I and II\nocite{p03,p04}, accounting for massive
progenitors in the Gould Belt reconciles theoretical predictions with the data.
One immediate advantage of the $\mathrm{Log\, N}$-$\mathrm{Log\, S}$ test is
that, at variance with the $\mathrm T$-$\mathrm t$ test, no degree of
arbitrariness is introduced when observational data are analyzed: both the
fluxes and (of course) the number of sources are well measured.
In addition, this approach is a ``global'' one.
In our scenario it would not be possible to explain some particular sources
by invoking slight changes in the cooling physics.
Once the parameters of the model other than those related to the cooling
process are fixed (see section~\ref{popsynth}), a particular cooling curve
either fits the population as a whole or not. Furthermore, the
$\mathrm{Log\, N}$-$\mathrm{Log\, S}$ sample is a uniform one, i.e. objects
are flux (and probably volume) limited, and no
strong selection criteria are introduced.
For the $\mathrm{Log\, N}$-$\mathrm{Log\, S}$ test
the only necessary observational piece of information is the ROSAT
count rate. The method can
be applied to objects with unknown ages. This makes it possible
to include, for example, all the ROSAT X-ray dim NSs,
and 3EG~J1835+5918 (the Geminga twin)
in the testing sample. The $\mathrm{Log\, N}$-$\mathrm{Log\, S}$
test is mostly sensitive to NSs older than $\sim 10^5$~yrs.
Older sources dominate in number,
and in the solar proximity there are about a dozen of them in comparison
to very few with $t\la 10^5$ years.\footnote{Of course, the
$\mathrm{Log\, N}$-$\mathrm{Log\, S}$ test can be sensitive to young objects
if another uniform sample is used. For example, it is very important to make
a population synthesis of sources in SN remnants.
Then, as we mentioned above,
differences between cooling curves for large ages are tiny. However, in the
$\mathrm{Log\, N}$-$\mathrm{Log\, S}$ approach we, in some sense, integrate
these small differences along the curves, so their impact on the final
$\mathrm{Log\, N}$-$\mathrm{Log\, S}$ distribution is significant.
}
Nevertheless there are significant limitations too.
One source of uncertainty is our incomplete knowledge of some important
ingredients of the population synthesis model. These are discussed in more
detail in the next section and concern the spatial distribution of the NS
progenitors, the NS mass and velocity spectrum, and their emission properties.
However, all these issues, to some extent, can be addressed by considering
different cases believed to cover
the entire range of acceptable scenarios. In addition, there could be unknown
correlations among some of the quantities we use to parametrize our model, so
that they should not be treated as independent ones. Examples of such possible
correlations are those between the star kick and the internal structure
(because of quark deconfinement, see \citealt{bp04}),
and between the star mass and the magnetic field
(because of fallback, see \citealt{petal02, hws04}).
A more severe problem arises in connection with the low statistics of the
sample, since there are only about 20 thermally emitting NSs known to date.
This implies that the bright end of the $\mathrm{Log\, N}$-$\mathrm{Log\, S}$
relation comprises very few objects so that it is difficult to account for
statistical fluctuations. We do not know much
either about the properties of very faint sources, i.e. the dim end of the
$\mathrm{Log\, N}$-$\mathrm{Log\, S}$ distribution.
\section{The population synthesis model}
\label{popsynth}
The main physical ingredients that enter our population synthesis model
are:
\begin{itemize}
\item the initial NS spatial distribution;
\item the kick velocity distribution;
\item the NS mass spectrum;
\item the cooling curves;
\item the surface emission;
\item the interstellar absorption.
\end{itemize}
The calculation of the NS spatial evolution as they move in the Galactic
gravitational potential follows that presented in
Papers I and II \nocite{p03, p04}. The same
treatment of the interstellar absorption is retained and the kick distribution
is that proposed by \cite{acc2002}. We do not account for atmospheric
reprocessing of thermal radiation, and assume that the emitted spectrum is a
pure blackbody.
Although this is clearly an oversimplification, it is a reasonable
starting assumption and will serve for our, mainly illustrative, purposes.
A more detailed description of surface emission may be easily accommodated in
our model later on. For the time being, we perform our calculations for nine
different sets of cooling curves among those discussed in \cite{bgv2004},
hereafter BGV, to which we refer for all details.
This issue, together with the initial spatial distribution of NSs and their
mass spectrum, is further discussed below.
\subsection{The initial NS spatial distribution}
Following the results of previous investigations (paper~I\nocite{p03}), we
take as an established fact that the population of nearby NSs is
genetically related to the Gould Belt.
The contribution of the Belt has dominated the production of compact
remnants in the solar proximity over the past $\sim 30$~Myrs
(see \citealt{p97} for a detailed description of the Belt structure).
About two thirds of massive stars in the $\sim 600$~pc around
the Sun belong to the Belt (\citealt{torra}). Since they are bright objects,
it is possible
to track their positions exactly for example by using HIPPARCOS data.
However, here we use a simplified distribution of the progenitor stars, as
discussed in Papers I and II\nocite{p03, p04}.
NSs are born in the Belt, for which a simple ring-shaped
structure is assumed, and in the Galactic disc. The outer belt radius is a
model parameter
and two values, 300~pc and 500~pc, were used, the second being an
upper limit. The supernova rate was taken from \cite{g2000}, and appears
in agreement with the historical rate estimated by \cite{tammann}.
\subsection{The NS mass spectrum}
\label{mass-spect}
Since cooling curves are strongly dependent on the star mass,
the mass spectrum is one of the most important ingredients and,
unfortunately, one of the lesser known.
We cannot rely on the mass measurements in binary radio pulsars
(e.g. \citealt{tho99}) because they refer to a ``twice selected''
population
(i.e. selection effects due to evolution in a binary can be
important together with possible conditions necessary for radio
pulsar formation).
Probably not all NSs
go through the active radio pulsar stage (e.g. \citealt{gv00}), and
the properties of NSs in binaries may be different to those
of isolated objects (e.g. \citealt{plp2003}).
Note that the local NS mass spectrum can be different
to the global NS mass spectrum in the Galaxy. Even stronger deviations can
be expected between the mass spectra of local NSs and of those sources
usually used for the $\mathrm T$-$\mathrm t$ plot.
As the population of NSs in $\sim 1$ kpc around the Sun may be slightly
different from the average galactic population, we estimated the
mass spectrum for these
objects directly (see Paper II \nocite{p04} for more details).
The basic idea is to use HIPPARCOS data on massive stars around the Sun
in conjunction with the calculations by \cite{whw02}. Knowing the mass
distribution
of progenitors through their spectral classes, we use a fit to a plot from
\cite{whw02}
in order to obtain the NS mass from the mass of the progenitor.
We use eight mass bins centered at
$\mathrm{M/M}_{\odot}=$ 1.1, 1.25, 1.32, 1.4, 1.48, 1.6, 1.7, 1.76.
The adopted mass spectrum is shown in Fig.~\ref{fig:mass}.
The lower limit for the NS mass is still an open question.
\cite{tww1996} suggested that there are no NSs with ${\mathrm M}\la 1.27\,
\mathrm{M_{\sun}}$, although their conclusion is not definite
(see also \citealt{whw02}).
For this reason we decide to use also a truncated mass spectrum, in
which the first bin is suppressed and all objects originally contained there
are added to the second one. Each bin corresponds to one of the calculated
cooling curves.
According to the mass spectrum, each curve has a statistical weight of
$31.75\%$, $25.75\%$, $11\%$, $28.125\%$, $0.875\%$, $1.125\%$, $0.75\%$,
and $0.625\%$.
For the truncated one the weights of the first two bins
are replaced by 0.0 and $57.5\%$ respectively.
We note that sampling also relatively low masses is important
since low-mass NSs seem to be required to interpret data on
the $\mathrm T$-$\mathrm t$ plot (BGV).
\begin{figure}[t]
\vbox{\psfig{figure=massbins_nocut.ps,width=\hsize}}
\caption[]{The adopted mass spectrum, binned over eight intervals of
different widths.
}
\label{fig:mass}
\end{figure}
\begin{table*}[h]
\begin{tabular}{|c|c|c|c|c|c|c|c|c}
\hline
Model&Reference &$\pi$ Cond& Gaps & Crust&\multicolumn{2}{c|}{T -- t} &
Log N -- Log S\\
& && & &wo BC&w BC & \\
\hline
\hline
I & BGV, Fig. 21 & Yes & A & C & Yes & Yes & Yes\\
II & BGV, Fig. 13 & No & B & D & Yes & No & No\\
III & BGV, Fig. 15 & Yes & B & C & Yes & No & No \\
IV & BGV, Fig. 12 & No & B & C & Yes & No & No\\
V & BGV, Fig. 16 & Yes & B & D & Yes & No & No\\
VI & BGV, Fig. 14 & No & B & E & Yes & Yes & No\\
VII & BGV, Fig. 18 & Yes & B'& C & Yes & No & No\\
VIII& BGV, Fig. 19 & Yes & B''&C & Yes& Yes & Yes\\
IX & BGV, Fig. 20 & No & A & C & Yes & Yes & Yes \\
X & G05, Fig. 2 & Yes & B & E & Yes & Yes & No\\
XI & G05, Fig. 2 & Yes & B'& E & Yes & Yes & No\\
\hline
\end{tabular}
\vspace{5mm}
\caption{Properties of the selected cooling curves: A - gaps from
\cite{tt2004}, $^3{\mathrm P}_2$ neutron gap suppressed by 0.1; B
- gaps from \cite{y2004}, $^3{\mathrm P}_2$ neutron gap suppressed
by 0.1; B' - same as for B and $^1{\mathrm P}_0$ proton gap
suppressed by 0.5; B'' - same as for B, $^1{\mathrm P}_0$ proton
gap suppressed by 0.2 and $^1{\mathrm P}_0$ neutron gap suppressed
by 0.5; C - $T_{\rm s} - T_{\rm in}$ relation fit from BGV; D -
$T_{\rm s} - T_{\rm in}$ relation by \cite{t1979}; E - $T_{\rm s}
- T_{\rm in}$ relation from \cite{y2004} and $\eta = 4\times
10^{-6}$. {The last three entries indicate whether the
model complies or not with the tests: temperature-age without
(wo) or with (w) the additional brightness constraint and
$\mathrm{Log\, N}$-$\mathrm{Log\, S}$
(see text and also BGV and G05 for more details)}.
\label{tab:models}}
\end{table*}
\subsection{Cooling curves}
\label{cooling}
In their recent paper \cite{bgv2004} presented sixteen sets of cooling curves.
Each set contains models for several values of the star mass while different
sets refer to different assumptions on heat transport in the crust and on the
physical processes in the NS interior.
Five of these models are unable to reproduce the observed
temperature-age plot and will not be considered further.
From the remaining eleven sets, all of which give results not in contradiction
with observations
(see, however, the discussion in sec. \ref{results}), we select nine
representative models for our population synthesis calculations
(models I - IX in Table \ref{tab:models}).
We add two models (X and XI) from the recent analysis in G05,
which correspond to models III and VII, respectively, when calculated
with different crustal properties.
All of them have superfluid nuclear matter and medium modifications of the
neutrino processes.
They differ in the assumptions about the superfluid gaps, the presence/absence
of a pion condensate and the properties of the neutron star crust.
The latter governs the relationship between the temperature
of outermost core layer ($\mathrm{T_{in}}$) to that of the star surface
($\mathrm{T_{s}}$).
The main characteristics of the selected models are summarized in
Table \ref{tab:models}. All of the eleven models satisfy the temperature - age
test according to BGV, whereas only six of them fulfill the additional
brightness constraint introduced in G05\nocite{g2005}. {The last
column anticipates the results discussed in the following section
and shows if the model complies with the
$\mathrm{Log\, N}$-$\mathrm{Log\, S}$ test}.
\section{Results and discussion}
\label{results}
\begin{figure*}
\hbox{\psfig{figure=temp21.ps,width=9cm}
\psfig{figure=lnls1t_13_n.ps,width=9cm}}
\caption[]{ Model I. Left: cooling curves for (from top to bottom)
1.1, 1.25, 1.32, 1.4, 1.48, 1.6, 1.7, 1.76 $\mathrm{M}_\odot$. Right: the
corresponding $\mathrm{Log\, N}$-$\mathrm{Log\, S}$ distribution for
$\mathrm{R_{belt}}=300$~pc and non-truncated mass spectrum (full line)
and 500~pc and truncated mass spectrum (dotted line). See text
for details.
\label{fig:m1}}
\end{figure*}
\begin{figure*}
\hbox{\psfig{figure=temp15.ps,width=9cm}
\psfig{figure=lnls6t_18_22_n.ps,width=9cm}}
\caption[]{Same as in Fig. \ref{fig:m1} for Model III. The dashed line
refers to a calculation in which the full (non-truncated)
mass spectrum was used and $\mathrm{R_{belt}}$ was assumed to be $500$~pc.
Model X should produce nearly the same $\mathrm{Log\, N}$-$\mathrm{Log\, S}$
distribution as it differs only in the type of the crust.
\label{fig:m3}}
\end{figure*}
\begin{figure*}
\hbox{\psfig{figure=temp14.ps,width=9cm}
\psfig{figure=lnls9t_15_20_n.ps,width=9cm}}
\caption[]{Same as in Fig. \ref{fig:m3} for Model VI.
{Model IV gives quite similar results since the change of the crust model
from C to E does not affect the log N - log S distribution}.
\label{fig:m6}}
\end{figure*}
\begin{figure*}
\hbox{\psfig{figure=temp18.ps,width=9cm}
\psfig{figure=lnls10t_14_21_n.ps,width=9cm}}
\caption[]{Same as in Fig. \ref{fig:m3} for Model VII.
Model XI should produce nearly the same $\mathrm{Log\, N}$-$\mathrm{Log\, S}$
distribution as it differs only in the type of the crust.
\label{fig:m7}}
\end{figure*}
\begin{figure*}
\hbox{\psfig{figure=temp19.ps,width=9cm}
\psfig{figure=lnls11t_12_n.ps,width=9cm}}
\caption[]{Same as in Fig. \ref{fig:m1} for Model VIII.
\label{fig:m8}}
\end{figure*}
\begin{figure*}
\hbox{\psfig{figure=temp20.ps,width=9cm}
\psfig{figure=lnls25_26_27_n.ps,width=9cm}}
\caption[]{Same as in Fig. \ref{fig:m3} for Model IX.
\label{fig:m9}}
\end{figure*}
In each run we calculate 5000 individual tracks for the spatial evolution of a
single star with a time step of $10^4$~yrs.
Each track is applied to all eight (or seven for the truncated spectrum)
masses, and the thermal evolution is followed by the correspondig cooling
curve. Results are then collected according to the statistical
weight of each mass bin.
Results are summarized in Table \ref{tab:models} and
Figs.~\ref{fig:m1}-\ref{fig:m9}, which refer to a selected
sub-sample of the cooling curve sets listed in
that Table.
In the left panels of Figs.~\ref{fig:m1}-\ref{fig:m9}, the corresponding
cooling curves for the various masses are shown.
Results are plotted for ages $> 10^4$ yrs and temperatures above
$10^5$~K. The right panels illustrate the $\mathrm{Log\, N}$-$\mathrm{Log\, S}$
distributions computed for the same sets of cooling curves.
All models have been calculated using both the full mass spectrum and the
truncated one, although results for the latter are not shown in all cases
(see the following discussion).
Two values of the outer radius of the Gould Belt have been used to test the
dependence of our calculation
on the assumed geometry, $R_\mathrm{belt}=300$ pc and 500 pc.
Theoretical distributions are superimposed on the observed
$\mathrm{Log\, N}$-$\mathrm{Log\, S}$
for isolated NSs. The data points are derived from the sample
of thirteen sources listed in Paper~I\nocite{p03}.
Error bars correspond to Poisson statistics and are plotted to
illustrate the statistical significance of
the points. We note, however, that there can be more
unidentified sources, especially at fluxes below 0.1 cts~s$^{-1}$ (see
\citealt{rut2003} for a recent discussion). In this respect, the fact that the
two points at the lowest fluxes lie below the general trend of the observed
$\mathrm{Log\, N}$-$\mathrm{Log\, S}$ is not surprising.
In addition, the observational
upper limit derived by \cite{rut2003} on the number of fainter sources
is also shown (marked as BSC in the figures).
Bounds on the total number of sources with flux $> 0.2$~cts~s$^{-1}$ have
been also presented
by \cite{s99}, on the basis of the {\it ROSAT\/} Bright Sources
Catalogue (BSC).
The comparison of the predicted and observed
$\mathrm{Log\, N}$-$\mathrm{Log\, S}$
distributions in Figs.~\ref{fig:m1}-\ref{fig:m9} indicates that at most three
cooling models (model I and possibly models VIII and IX)
are in agreement with the data.
All the others substantially overpredict the observed number of sources
at all fluxes, even though they comply with the $\mathrm T$-$\mathrm t$ test.
This latter statement deserves further comment.
BGV did not reject models II--VII on the basis of
the fact that the cooling curves cover, for the assumed mass range, the entire
region in the $\mathrm T$-$\mathrm t$ plane where the observed sources lie.
This approach is sound, and it represents the only possible option
to discriminate among different cooling scenarios at the zero level, i.e
without introducing additional information.
Clearly, if some assumptions on the NS mass distribution are
made, the $\mathrm T$-$\mathrm t$ test can be used to exclude some further
models.
If the same mass distribution discussed in Sec. \ref{mass-spect} is applied to
BGV sets of cooling curves, one is immediately led to discard models II--V
and VII because they predict quite high temperatures for low mass NSs
($\mathrm M\la 1.3\mathrm M_\odot$)
which, according to our mass spectrum, are very abundant.
Although such hot objects would be
detectable even to large distances they are not actually observed
(this issue is further discussed in G05, see also below).
However, care must be taken in using such an argument.
Our mass spectrum is meant to be representative
of the local population of
isolated NSs, and its application to the very biased and limited
sample of objects which can be placed on the
$\mathrm T$-$\mathrm t$ plots is uncertain.
Nevertheless, one can reverse the argument by saying that the two tests,
{\em when provided with the same amount of information},
yield results that are broadly consistent, as they should.
Still, we note that a strict interpretation of the $\mathrm{Log\, N}$-
$\mathrm{Log\, S}$ test
results in the exclusion of two further models (VI and VIII) and that even
in the most optimistic case model VI is rejected.
The $\mathrm{Log\, N}$-$\mathrm{Log\, S}$ test is not equally sensitive
to changes in the three main groups of parameters in Table \ref{tab:models}
(presence or absence of the pion condensate, gaps, and type of the crust).
Mostly the test reacts to changes in the gap parameters.
Conversely, changes in the type of the crust are not very important as here
we consider a sample of relatively old NSs (nevertheless uniform samples
of younger NSs can also be studied).
Variations in the crust properties are discussed in detail in G05, where the
{ brightness constraint\/} was introduced.
G05\nocite{g2005} shows that models II--V and VII can be ruled out when the
absence of very bright young NSs is considered as a constraint.
The fact that the $\mathrm{Log\, N}$-$\mathrm{Log\, S}$ test leads to the
same conclusion is however a completely independent result, since objects
from different age ranges are used in the two approaches.
As mentioned earlier, the $\mathrm{Log\, N}$-$\mathrm{Log\, S}$ test applied
to close-by NSs is not sensitive to changes in the crustal properties.
This implies that if a cooling model with
a given crust is rejected then its variants with other types of crust
(at least from the set considered here) can be ruled out, too.
For example, additional models X and XI (see Table \ref{tab:models}) do not
satisfy the test because their {\it twins} (models III and VII, respectively)
do not (see Figs. \ref{fig:m3} and \ref{fig:m7}).
In G05\nocite{g2005} it has been shown that it is possible to fulfil the
BC by changing the crust.
For example, model III with crust E can fit the data.
However, on the basis of the $\mathrm{Log\, N}$-$\mathrm{Log\, S}$ test this
model is ruled out.
This is because the latter test mainly ``feels'' the changes in gaps. And
if they are ``wrong'' then changes in the crust properties are of no avail.
Jumping to the conclusion that the present analysis can provide direct
information on the physical state of star interior would be premature.
Our adopted scenario contains a number of uncertainties, as discussed in
sections~\ref{tests} and \ref{popsynth}.
Nevertheless, we believe that the case presented here convincingly shows
that by combining theoretical cooling curves with populations synthesis
calculations one has the potential to discriminate between competing cooling
scenarios.
The main outcome of this investigation is not that models
I, and possibly VIII and IX, fit the data while others do not.
This may be the result of our starting assumptions.
What matters is the fact that, within the same set of assumptions,
different cooling models
produce different results when compared with observations.
Current limitations of this approach are due to our present incomplete
knowledge of some key issues, chiefly the NS mass spectrum, their surface
emission properties and initial spatial distribution.
We attempted to account for some of these uncertainties
in our model by considering different configurations which should bracket the
true behavior. We note that in all the six cases for which no agreement
has been found between theoretical and observed distributions,
this is largely independent of the assumptions we introduced. Using different
values of $R_{\mathrm{belt}}$ or taking variants of the mass spectrum does not help
in reconciling predictions with the data. However, we caution that other
effects, like those introduced by the proper inclusion of an atmospheric
model, may be important. This, and other issues, will be the subject of
future work.
In all the cases we examined (see again Figs.~\ref{fig:m1}-\ref{fig:m9}),
with the possible exception of model VIII, the capability of our test to
discriminate between different cooling scenarios seems to be quite robust.
The three models (I, VIII and IX) which can reproduce the observed
$\mathrm{Log\, N}$-$\mathrm{Log\, S}$ for our choice of parameters have also
been considered by BGV as the most realistic ones.
Among models I--IX which according to BGV are not in contradiction
with the Temperature--age test,
the three
mentioned above are the theoretically most appealing ones since either
the superfluid gaps were calculated with
the same nucleon-nucleon interaction that formed the basis for the equation
of state and thus the structure of the neutron star configurations
(models I and IX) or the gaps were modelled such as to mimic these results
(model VIII).
\section{Conclusions}\label{concl}
In this paper we suggest adopting the $\mathrm{Log\, N}$-$\mathrm{Log\, S}$
test as an useful addition to the standard $\mathrm T$-$\mathrm t$ test in
probing neutron star cooling models.
To illustrate the capabilities of the proposed approach,
we applied it to nine sets of cooling curves from \cite{bgv2004}.
Out of sixteen sets described in that paper
and two additional ones taken from \cite{g2005}, these eleven produce
results that are not in immediate contradiction with the
$\mathrm T$-$\mathrm t$ test.
The application of the $\mathrm{Log\, N}$-$\mathrm{Log\, S}$ test rules out
at least eight out of eleven investigated cooling models,
resulting in just three models able to pass both tests.
Requiring that the tested cooling models should fulfill in addition
the more stringent bightness constraint of G05, there are still six
models left out of which only 50\% pass the
$\mathrm{Log\, N}$-$\mathrm{Log\, S}$ test.
One of the most challenging questions for the application of the test
suggested in this paper is to use it next for a possible discrimination
between purely hadronic compact star cooling scenarios and hybrid ones for
stars having a color superconducting quark core \cite{gbv2005} which have
already successfully passed the T-t test.
Our conclusion is that the $\mathrm{Log\, N}$-$\mathrm{Log\, S}$ may
therefore become a powerful strategy in uncovering the properties
of dense nuclear matter under the extreme conditions in neutron star
interiors.
\begin{acknowledgements}
We thank D.~Yakovlev and other members of the St-Petersburg group
for many helpful discussions. We are indebted to D.~Voskresensky
and M.~Prokhorov for their constant interest and support.
Comments by the referee were very useful and helped to improve the paper.
S.P. acknowleges a postdoctoral fellowship from the University of Padova where
most of this work was carried out and a fellowship from
the ``Dynasty'' Foundation. The work of S.P. was partly supported through
RFBR grant 03-02-16068.
H.G. is grateful to the Department of
Physics, University of Padova, for hospitality and acknowledges financial
support from the Virtual Institute ``Dense hadronic matter and QCD phase
transition'' of the Helmholtz Association (grant No. VH-VI-049) and from
Deutsche Forschungsgemeinschaft (grant No. 436 ARM 17/4/05).
The work was partially supported by the Italian
Ministry for Education, University and Research under grant PRIN-2002-027245.
\end{acknowledgements}
|
{
"timestamp": "2005-10-29T08:17:21",
"yymm": "0411",
"arxiv_id": "astro-ph/0411618",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411618"
}
|
\section{Hamiltonian and the Intermediate Hamiltonian}
Consider a network $\Omega=\Omega_0 \cup \omega_1\cup\dots$
on $(x,z)$ plane combined of a circular quantum well
$\Omega_0$ and three straight semi-infinite quantum
wires $\omega_s,\,\, s= 1,2,3$ of constant width $\delta$
attached to the well $\Omega_0$ such that the orthogonal bottom
sections $\gamma_{_{s}}$ of the wires $\omega_s$ are parts of
the piece-wise smooth boundary $\partial \Omega_{0}$ of the well
$\Omega_{0}$ . We consider scattering of electrons in the network
in presence of a strong electric field directed orthogonally
to the $(x,z)$ plane. The wave-function $\Psi$ of the electron
is presented by spinor $(\psi_1,\,\,\psi_2)$, and the
spin-orbital interaction is taken in form of Rashba Hamiltonian
\cite{Rashba} : a cross product - of the vector $\sigma$ of
Pauli matrices and the vector $p$ of momentum of electron :
\begin{equation}
\label{spinorb} H_{_R}= \alpha\left[\sigma,\,\, p \right],
\end{equation}
where $\alpha$ is an absolute constant. In presence of a strong
electric field ${\cal E} = |{\cal E}| e_{_{y}} $ directed
orthogonally to the $(x,z)$ plane the corresponding
Schr\"{o}dinger equation on the well is
\begin{equation}
\label{Schredinger} L u = -\frac{\hbar^2}{2 m*}\bigtriangleup u +
V u +
\alpha[\sigma_z \ p_x - \sigma_x \ p_z] u
\end{equation}
where $\alpha[\sigma_z \ p_x - \sigma_x \ p_z] = \left(
H_{_R}\right)_{_y}$ is the $y$-component of the cross product
$\alpha\left[\sigma,\,\, p \right]$, and the linear potential $ V
: = |{\cal E}| y $ on the well is defined by the macroscopic
electric field ${\cal E}$. The above Rashba Hamiltonian is
non-self-adjoint in the space $L_2 (\Omega)$ but the whole
Schr\"{o}dinger operator (\ref{Schredinger}), is a self-adjoint
operator in $L_2 (\Omega)$ on the domain of sufficiently smooth
functions with appropriate conditions imposed
on their boundary values, see (\ref{boundcondG},\ref{boundcondR}). On the
wires the potential is constant $V = V_{\infty}$ , but we assume
that the Schr\"{o}dinger equation on the wires $\omega_{_{s}} =
\left\{ 0< \eta_{_{s}}< \delta,\, 0< \xi_{_{s}} <\infty\right\}$
contains an anisotropic tensor of effective mass:
\begin{equation}
\label{Schrwires} l u = -\frac{\hbar^2}{2
m^{\parallel}}\,\frac{d^2 u}{d \xi ^2} - \frac{\hbar^2}{2
m^{\bot}}\,\frac{d^2 u}{d \eta ^2} + V_{\infty} u ,
\end{equation}
and the width of wires is constant and equal to $\delta$. We
neglect the spin-orbital interaction in the wires. On the sum
$\Gamma = \sum_{s=1}^3 \gamma_s$ of bottom sections of the wires,
separating the wires from the well, we impose proper matching
boundary conditions on the boundary of the well:
\begin{equation}
\label{boundcondG}
\frac{\hbar^2}{2 m^*}\,\,\,\frac{\partial
u}{\partial n} - \frac{i \alpha}{2} \left[ \sigma,\, n\right]u
\bigg|_{\partial \Omega_{_{0}} \backslash \Gamma} = 0,
\end{equation}
\begin{equation}
\label{boundcondR}
\frac{\hbar^2}{2 m^*}\,\,
\frac{\partial
u}{\partial n} - \frac{\hbar^2}{2
m^{\parallel}}\,\,\,\frac{\partial u_s}{\partial n} - \frac{i
\alpha}{2} \left[ \sigma,\, n\right]u \bigg|_{\gamma_s} = 0,
\end{equation}
They define a self-adjoint operator ${\cal L}$ on $L_2
(\Omega)$ which plays a role of the Hamiltonian of the
electron on the network. Following the pattern of \cite{boston}
we consider the scattering problem for ${\cal L}$ on the network
$\Omega$ and calculate the transmission coefficients from the
input wire $\omega_1$ to the terminals $\omega_2,\,\, \omega_3$
across the well and estimate the quantum conductance on resonance
energy, based on Landauer formula,see \cite{Landauer70, Buttiker85}.
\par
Denote by $e_l = \sqrt{\frac{2}{\sqrt{2 m^{^{\bot}}}\delta}}
\sin \frac{\pi l \eta}{\sqrt{2 m^{^{\bot}}}\delta},\,\,
l=1,2,3,\,\,\dots,\, 0<\eta<\delta$ the eigenfunctions of the
cross-sections of the wires and assume that the Fermi level
in the wires lies in the middle of the first spectral band in
the wires $E_{_F} = V_{\infty} + \frac{5}{2} \frac{\hbar^2}{2
m^{\bot}}\frac{\pi^2}{\delta^2}$. Denote by $P_+ $ the
orthogonal projection onto the linear hull of the entrance
vectors $\bigvee_{_{s}} e_{_{1,s}} = E_{_{+}} $ of the open first
channel, and by $P_{-}$ the complementary projection in $L_2
(\Gamma) \,\, : \,\,I = P_{+} + P_{-}$. We define the {\it
Intermediate Hamiltonian} $\hat{\cal L}$ by the same
Schr\"{o}dinger differential expressions (\ref{Schredinger},
\ref{Schrwires}) but replacing the matching conditions
(\ref{boundcondR}) by the ``chopping-off'' boundary conditions in open channels:
\begin{equation}
\label{match+} P_+ u_s\big|_{_{\gamma_s}} = 0,\, \,s=1,2,3.
\end{equation}
and the matching conditions in closed channels :
\begin{equation}
\label{match-}
P_- [u_0 - u_s]\,\,\big|_{_{\gamma_s}} = 0,\,\,
\frac{\hbar^2}{2 m^*}\,\,\,P_-\frac{\partial
u_0}{\partial n} - \frac{\hbar^2}{2
m^{\parallel}}\,\,\,P_-\frac{\partial u_s}{\partial n} - \frac{i
\alpha}{2} \left[ \sigma,\, n\right] P_- u_0 \bigg|_{\gamma_s} =
0.
\end{equation}
Note that the boundary term $\frac{i \alpha}{2} \left[
\sigma,\, n\right]$ arising from the Rashba Hamiltonian commutes
with projection onto the entrance vectors of the channels. The
operator $ {\hat{\cal L}}$ on the network defined by the above
differential expressions (\ref{Schredinger}, \ref{Schrwires}) with
the boundary conditions (\ref{match-},\ref{match+}) and the
Meixner conditions at the inner angles of the domain is
self-adjoint. The operator $\hat{\cal L}$ is split as an
orthogonal sum $\hat{\cal L} = {\bf l}_{_{1}} \oplus L_{_{R}}$ of
the operator ${\bf l}_{_{1}}$ on the open channel in the wires and the operator
$L_{_{R}}$ acting in the orthogonal complement of the open channels.
This operator plays a role of an intermediate operator . We will derive
an explicit formula for the scattering matrix in terms of spectral data
of $L_{_R}$, see (\ref{Smatrix}). The spectrum of the part
${\bf l}_{_1}$ of $\hat{\cal L}_{_{R}}$ in the open channels is
just a semi-axis $V_{\infty}+ \frac{\pi^2
\hbar^{^2}}{2m^{\bot}\delta^2} < \lambda < \infty.$ The
absolutely-continuous spectrum of the part ${L}_{_{R}}$ of
the operator ${ L}_{_{R}} = \hat{\cal L}\ominus {\bf
l}_{_1}$ on the orthogonal complement of the open channels
consists of a countable family of branches
$\cup_{l=2}^{\infty}\left[\left.\frac{l^2 \hbar^{^2} \pi^2}{2
\mu^{\bot}\delta^2} +
V_{\infty},\,\infty \right.\right)$.
\vskip0.5cm
\section{Scattering matrix}
Denote by $G_{_{R}} $ the Green function of the the operator
${L}_{_{R}}$. The solution ${\bf u}$ of the Dirichlet problem for the
former equation with the data $\left\{ u_{_{1}},\, u_{_{2}},\, u_{_{2}}
\right\}: = u_{_{\gamma}}\in E_+$ on $\Gamma$ can be presented as
\[
u (x)= - \int_{\Gamma}\left( \frac{\hbar^2}{2 \mu^*}\frac{\partial
G_{_R} (x,s)}{\partial n_{_{s}}} - i\alpha [\sigma,
n_{_{s}}]G_{_R} (x,s) \right) u_{_{\gamma}} (s) d\Gamma := {\cal
P} u_{_{\gamma}}.
\]
We match $u$ with the
Scattering Ansatz
${\bf u} = e^{-i p \xi} \nu + e^{i p \xi} S \nu,\, \nu \in E_+,\,\,
p = \frac{\sqrt{2 \mu^{^{\parallel}}}}{\hbar} \,\,\sqrt{\lambda -
V_{_{\infty}} - \frac{\pi^{^{2}}}{2 \mu^{^{\bot}} \,
\delta^{^{2}}}}$ in the first channel :
\[
\frac{\hbar^2}{2 m^*}
\,\,\,P_+\frac{\partial u}{\partial n} -
\frac{i
\alpha}{2}
\left[ \sigma,\, n\right] P_+ u \bigg|_{\gamma_s} =
\frac{\hbar^2}{2 m^{\bot}}\,\,\,P_+\frac{\partial {u}_{_{s}}}{\partial
n}.
\]
Taking into account the continuity $[u-{ u}_{_{s}}]
\bigg|_{_{\gamma_{_{s}}}} = 0$ and denoting by ${\cal D}_{_R}$
the boundary differential operation $ {\cal D}^{^R}_{_{x}} u =
\left( \frac{\hbar^2}{2 \mu^*}\frac{\partial u }{\partial n} -
i\alpha [\sigma, n] u \right) $ and by $\Lambda_R (\eta,\eta')$
the generalized kernel of the corresponding Dirichlet-to-Neumann
map (DN-map) $\Lambda_R $, see \cite{SU2}, of the operator
${L}_{_R} $ on $\Gamma$ : $ \Lambda_{_R} (\eta, \eta')= -
{\cal D}^{^R}_{_{x}}\,\, {\cal D}^{^R}_{_{x'}} \,\, G_{_R}
(x,\,x')$, with $x|_{_{\Gamma}} = \eta,\, x'|_{_{\Gamma}} =
\eta'$, we calculate the DN map $P_{_{1}}\Lambda_R P_{_{1}}: =
\Lambda_R^{^1}$ framed by the orthogonal projections onto the
entrance vectors $e_{_{1,s}}$ of the first channel :
$P_{_{1}} = \sum_{_{s = 1}}^{^3} e_{_{1,s}}\rangle \,\,\langle e_{_{1,s}}$.
This gives the following formula for the scattering matrix
:
\begin{equation}
\label{Smatrix} S = - \frac{\Lambda_R^{^1} + ip I}{\Lambda_R^{^1}
- ip I}.
\end{equation}
The transport properties of the filter for given temperature $T$ are
defined via averaging of the corresponding transmission coefficients
$S_{_{s1}}$ over Fermi distribution on the essential spectral interval,
$\Delta_{_{T}} = \left\{E_{_{F}} - \kappa T <\lambda< E_{_{F}} +
\kappa T\right\} $ , see for instance \cite{Aver_Xu01}.
We may obtain a reasonably good approximation for the
Scattering matrix, substituting for $\Lambda_{_R} (x,y)$ the
corresponding spectral sum over all eigenvalues $\lambda_m$ of
the operator $\hat{L}_{_R}$ on the essential spectral interval, if
the interval
does not overlap with the continuous spectrum of $L_{_{R}}$:
\begin{equation}
\label{essential}
\Lambda_{_R}^{^1} (\eta, \eta') \approx
-\sum_{\Delta_{_{T}}} \,\,\frac{P_{_{1}}{\cal D}^{^R}_{x}\,\,
\varphi_m (\eta)\rangle \,\,\langle \,
P_{_{1}}{\cal D}^{^R}_{x'}\,\, \varphi_m (\eta')}{\lambda_{_m} - \lambda
},\,\,\,\,\, \eta, \eta' \in \Gamma.
\end{equation}
The formulae (\ref{Smatrix},\ref{essential}) show that the
eigenvalues and eigenfunctions of the intermediate operator define
the structure of the Scattering Matrix on $\Delta_{_{T}}$ and the transport
properties of the spin-filter based on the quantum well. Recovering
necessary spectral data of the intermediate operator with the
non-standard boundary conditions
(\ref{match+},\ref{match-}) cannot be done with existing commercial software
and needs creation of special programs.
\par
Assume
that there exist a resonance eigenvalue $\lambda_{_0}$ of the
operator $\hat{L}_{_R}$ which is equal to the Fermi -level in the
wires $\lambda_{_0} = E_{_F}$. Then on a (small) part of the
essential spectral interval defined by the temperature we may
substitute the DN-map by the resonance term only.:
\[
\Lambda_{_R}^{^1} (\eta,\eta') \approx \Lambda_{_{essential}}^{^1}
(\eta,\eta') - \frac{P_{_{+}}{\cal D}^{^R}_{_x}\,\, \varphi_0
(\eta)\rangle \,\,\langle \,
P_{_{+}}{\cal D}^{^R}_{_{x'}}\,\, \varphi_0 (\eta')}{\lambda_{_0} - \lambda }
\]
The residue of the resonance polar term is proportional to the
projection onto the one-dimensional subspace spanned by the
portion
$P_{_{+}}{\cal D}^{^R}_{{x}}\,\, \varphi_0 (\eta)\,\bigg|_{_{\gamma_s}} = \phi_{_{s}}$
of the resonance eigenfunction in the entrance subspace of the
wire $\omega_{_{s}}$:
\[
\Lambda^{^{1}}_{_{R}} (\eta,\, \eta') \approx
\frac{P_{_{1}}{\cal D}^{^R}_{_{x}}\,\, \varphi_0 (\eta)\rangle \,\,\langle
\,
P_{_{1}}{\cal D}^{^R}_{_{x'}}\,\, \varphi_0 (\eta)}{\lambda - \lambda_{_0}}
= |{\bf \phi}|^{^{2}} \left\{
P_{_{\phi}}\right\}_{_{s,\, s'}},
\]
where $\phi = \oplus\sum_{_{s=1}}^3 \, \phi_{_{s}}$ ,
$|\phi|^{^{2}} = \sum_{_{s=1}}^{^{3}} |\phi_{_{s}}|^{^{2}}$.
This gives the
one-pole approximation for the scattering matrix :
\begin{equation}
\label{onepole}
S (\lambda) \approx S_{_{approx}} (\lambda) = P^{^{\bot}}_{_{\phi}} -
\frac{|\phi|^{^2} + i p (\lambda - \lambda_{_0})} {|\phi|^{^2} - i
p (\lambda - \lambda_{_0})} \,\, P_{_{\phi}},
\end{equation}
where $P^{^{\bot}}_{_{\phi}} = I - P_{_{\phi}} $. The last
formula implies the following expressions for transmission
coefficients at the resonance energy $\lambda_0 = E_{_{F}}$
for low temperature:
\[
S_{_{s1}} (\lambda_0) \approx ( S_{_{approx}})_{_{s1}} (\lambda_0)= - 2
\frac{{\bf \phi}_{_{s}}\rangle \, \langle
\,{\bf \phi}_{_{1}}} {|{\bf \phi }|^{^2}}.
\]
Hence the transmission coefficient from the input wire
$\omega_{_1}$ to the wire $\omega_{_s}$ is represented by the
$2\times 2$ matrix constructed as a product of spinors, and the
spin-dependent current is calculated based on the relevant
version of the Landauer formula, see \cite{Buttiker85}. In \cite{MP01,boston}
the resonance eigenfunctions are scalar and real.
Then the transmission coefficients are defined by
the integrals
\[
\int_{_{\gamma_{_{s}}}} e_{_{\eta}}\, \frac{\partial \varphi_{_{0}} }{\partial n}(\eta) d\eta\, \int_{_{\gamma_{_{1}}}} e_{_{\eta}}\, \frac{\partial \varphi_{_{0}} }{\partial n}(\eta) d\eta.
\]
The selectivity of the devices in \cite{MP01,boston} is guaranteed by the
presence of the zeros of $ \frac{\partial \varphi_{_{0}} }{\partial n}$ in
the middle of the bottom section $\gamma_{_{s}}$ of the wire $s$.
In our case the intergrands in the corresponding integrals are spinors,
and the transmission coefficients are presented by matrices . Using the
superscripts $\pm $ for components of spinors with spin $\pm \, 1/2$
we write down the formulae for transmission coefficients , for instance:
\begin{equation}
-\frac{1}{2} |\phi|^{^2} S^{^{++}}_{_{s1}} (\lambda_{_{0}}) =
\int_{_{\gamma_{_{s}}}} e_{_{1}} (\eta)\, \left( {\cal D}_{x}^{^R}\,\, \varphi_0 \right)^{^{+}}\bigg|_{_{\gamma_{_{s}}}}
(\eta) d\eta\,
\int_{_{\gamma_{_{1}}}} e_{_{\eta}}\, \left( {\cal D}_{x}^{^R}\,\, \varphi_0 \right)^{^{+}}\bigg|_{_{\gamma_{_{1}}}}(\eta) d\eta.
\end{equation}
The magnitude of the transmission coefficients defines the selectivity
of the spin-filter. But recovering of details of the parameter regimes of the switch
now is more complicated task, that in \cite{MP01,boston} , since
the matrices are complex and the selectivity is not defined by zeros
of one real function. Nevertheless, if the geometry of the well and the
positions of the contact zones $\gamma_{_s}$ are chosen such
that, for given electric field , the matrix elements $S^{^{++}}_{_{s1}} (\lambda_{_{0}}),\,\,S^{^{+-}}_{_{s1}} (\lambda_{_{0}})$ bigger than
$S^{^{-+}}_{_{s1}} (\lambda_{_{0}})\,\,\,S^{^{--}}_{_{s1}} (\lambda_{_{0}})$, then
the electrons with the spin up prevail in the exit wire $s$, for non-polarized
incoming flow in the wire $1$.
Based on the above formula (\ref{onepole}) one
can calculate also the position of the resonance in the complex
plane ( the pole of the Scattering Matrix), which essentially defines the speed of switching.
Based on the one-pole approximation one can construct a solvable model , \cite{Kurasov}
of the spin filter, in form of a quantum graph with the resonance boundary
conditions at the node.
\vskip0.3cm
{\small
|
{
"timestamp": "2004-11-12T22:45:39",
"yymm": "0411",
"arxiv_id": "cond-mat/0411354",
"language": "en",
"url": "https://arxiv.org/abs/cond-mat/0411354"
}
|
\section{Introduction}
The lightest neutralino state, $\tilde \chi^0_1$, is often assumed
to be the Lightest Supersymmetric Particle (LSP) \cite{LSP}.
As such it is also a candidate for the origin of Dark Matter
(DM) \cite{DMLSP}. This assumption has of course to be verified though, by analyzing the
results or constraints reached by experiments trying to detect Dark Matter
through direct or indirect methods \cite{DMobs, DMann}.
However, even in the minimal MSSM version of the SUSY models,
the large parameter space induces great uncertainties in the neutralino
properties. So to check the consistency of the DM idea, it is essential
to establish the neutralino properties through production
at high energy hadron and lepton colliders.
The first such possibility of neutralino production, will probably be through cascades at
the CERN LHC \cite{SUSYsearches1, SUSYsearches2}. But precious additional independent information
from LHC could also be obtained by studying the smaller signals of the
direct ($\tilde \chi^0_i \tilde \chi^0_j$) pair production, as well as the production
in association with other sparticles in processes as $(\tilde \chi^0_i \tilde g)$,
($\tilde \chi^0_i \tilde q_{L,R}$) or ($\tilde \chi^0_i \tilde \chi^{\pm}_j$). When
the LC collider, will finally be built, a wealth of additional information will become
accessible \cite{gamma-gamma}.
Studies of the pure QCD effects to these channels at LO
and NLO have already appeared \cite{Spiratalk, Beenakker}.
A recent summary can be found in
\cite{Plehn}, where the results of a NLO QCD computation
are presented for various processes including neutralino production in association with
a gluino, squark, slepton, chargino, or another neutralino.
The overall conclusion of these computations is that at the LHC range, the pure
QCD soft and collinear
corrections always {\it increase} the LO cross section by an amount which,
depending on the subprocess c.m. energy and the masses of the particles involved, lies
in the range of 10\% to 40\%.
Also important at LHC though, turn out to be the
leading and subleading 1-loop logarithmic (LL) electroweak (EW) corrections.
Particularly for processes characterized by
non-vanishing Born contributions, such effects show a largely
universal structure with the leading $\ln^2(\hat s)$-terms solely
determined by the couplings of the known gauge bosons $(W^\pm,Z,\gamma)$ to the
external particles of the process; which in turn are fixed completely by their quantum numbers.
The situation is different for the subleading single-$\ln (\hat s)$ terms though, which
depend on the couplings and masses of all
virtual particles, gauge or non-gauge, shaping up the underlying dynamics
\cite{BRV, Denner, qqLHC, Beenakker}.
Thus, depending on whether SUSY is "near by" with all
MSSM sparticles below the TeV range,
or some of the sparticles are very heavy,
or even that the pure simple SM model
stays correct till very high scales, will only affect the subleading single $\ln(\hat s)$-terms
\cite{BRV, qqLHC, gam-gam-VV}.
The most striking characteristic when comparing these EW corrections
to the aforementioned pure QCD ones, is that
they are of roughly similar magnitude, but have {\it opposite sign}
\cite{BRV, Denner, qqLHC, gam-gam-VV}.
Particularly for the subprocess $q\bar q\to\tilde \chi^0_i \tilde \chi^0_j$ contributing
to the neutralino-pair production, these effects have been studied in
\cite{pp-2chi0}, where the calculation of the pure 1-loop
process $gg\to\tilde \chi^0_i \tilde \chi^0_j$ was also included. If the masses
of the squarks of the 1rst and 2nd family turn out to be very heavy,
it might happen at LHC, that the $gg\to\tilde \chi^0_i \tilde \chi^0_j$ contribution
is comparable to that of the LO process
$q\bar q\to\tilde \chi^0_i \tilde \chi^0_j$, particularly at low invariant masses where
the gluon flux is very large.\\
Additional information on neutralinos in a hadron collider could be obtained
from the single neutralino production triggered by the subprocesses:
\begin{equation}
~~q\bar q\to \tilde \chi^0_i \tilde g~~,~~ gg\to \tilde \chi^0_i \tilde g~~,~~
g q\to \tilde \chi^0_i \tilde q_{L,R}~~,~~
q\bar q'\to \tilde \chi^0_i \tilde \chi^{\pm}_j~~,~~
\label{processes}
\end{equation}
where the indices $(i,j)$ now enumerate the neutralino and chargino respectively.
The aim of the present paper is to study the physical consequences of these
subprocesses using the same procedure as in \cite{pp-2chi0}. Since different particles
are involved in each of them, their combined study is sensitive to different aspects of the
underlying model.
For the first, third and fourth of the subprocesses in (\ref{processes}), this model sensitivity
arises already at the Born level mainly caused by the
(gaugino-higgsino) mixing matrices
\footnote{The notation of \cite{Rosiek} is used here.}
multiplying the basic gaugino and higgsino couplings.
The relevant diagrams are shown
in Figs.\ref{qq-gluino-fig}-\ref{qq-chargino-fig} .
This model sensitivity is further enhanced when including also
the leading logarithmic part of the 1-loop corrections, calculated by
following the procedure of \cite{BRV}.
Thus, rather simple expressions for the amplitudes
of these processes are reached, which apart
from being very sensitive to the physical dynamics,
should also be quite adequate for LHC energies and accuracies.
Further model sensitivity is induced in the case of $\tilde \chi^0_i \tilde g$ production,
by the contribution of the genuine 1-loop subprocess $gg\to \tilde \chi^0_i \tilde g$.
The generic form of the relevant diagrams is shown in Fig.\ref{gg-gluino-fig}.
On the basis of these, a numerical Fortran code called PLATONgluino
is released, calculating $\mathrm d\sigma(gg\to \tilde \chi^0_i \tilde g)/d\hat t $
for any set of real $\mu$ and MSSM soft breaking parameters
at the electroweak scale \cite{Platon}.
To explore the actual physical situation that might be realized within the SUSY approach,
typical MSSM benchmark models with real parameters are used \cite{Snowmass, Arnowitt, CDG}.
The LHC cross sections for proton proton collisions are then computed
by convoluting the $q\bar q$, $gg$, $qg$
subprocess cross sections,
with the corresponding quark
and gluon distribution functions taken from \cite{MRST} . As in \cite{pp-2chi0},
invariant mass and angular distributions are constructed,
illustrations of which are given below.
The results obtained in this paper should be useful for precise
applications at LHC taking into account decay branching ratios
and final state identifications. We will come back to this point
in the conclusion.
The organization of the paper is the following. In Section 2.1 and Appendix A.1, the
general form of the Born amplitudes for $q\bar q\to \tilde \chi^0_i \tilde g $ are given,
together with the 1-loop LL EW
and\footnote{The SUSY-QCD corrections describe a
special part of the complete QCD correction, intimately related to
the SUSY dynamics. The fact that we consider them together with the EW corrections,
rather than the pure QCD ones, is a matter of choice.
See \cite{qqLHC} for its exact definition.} SUSY QCD corrections to them,
as well as the explicit Born expressions for the helicity amplitudes.
The corresponding results for $q g \to \chi^0_i \tilde q_{L,R}$ and
$q\bar q'\to \tilde \chi^0_i \tilde \chi^{\pm}_j$ are given in Sections 2.2 and 2.3, and Appendices
A.2 and A.3 respectively; while in Section 3, the 1-loop process
$gg\to \tilde \chi^0_i \tilde g$ is discussed. Finally, in Section 4 we discuss our results,
and Section 5 presents the Conclusions.\\
\section{The processes $q\bar q\to \tilde \chi^0_i \tilde g$,~
$g q\to \tilde \chi^0_i \tilde q_{L,R}$,~
$q\bar q'\to \tilde \chi^0_i \tilde \chi^{\pm}_j$.}
The momenta, energies and masses in these subprocesses,
as well as in $gg \to \tilde \chi^0_i \tilde g$ of Section 3,
are defined as
\begin{equation}
a(q_1)~ b(q_2)~ \to ~A(p_1, E_1, m_i)~ B(p_2, E_2, m_j)~~, \label{prosses-kinematics}
\end{equation}
where the masses of the incoming particles are neglected.
Denoting by $(p,~\theta)$ the final state c.m. momentum
and scattering angle, we have
\begin{eqnarray}
&& \hat s= (p_1+p_2)^2=(q_1+q_2)^2 ~~, \nonumber \\
&& \hat t= (q_2-p_2)^2=(p_1-q_1)^2 ~~, \nonumber \\
&& \hat u= (q_2-p_1)^2=(p_2-q_1)^2 ~~, \nonumber \\
&& p=\frac{1}{2\sqrt{\hat s}} \Big\{ [\hat s-(m_i+m_j)^2][\hat s-(m_i-m_j)^2] \Big\}^{1/2}~~,
\nonumber \\
&& \beta=\frac{2 p}{\sqrt{\hat s}} ~~~, ~~
E_1=\frac{\hat s +m_i^2-m_j^2}{2\sqrt{\hat s}}~~~,~~~
E_2=\frac{\hat s +m_j^2-m_i^2}{2\sqrt{\hat s}}~~~, \nonumber \\
&& q_1=\frac{\sqrt{\hat s}}{2}(1, 0,0,1) ~~~,~~~ q_2= \frac{\sqrt{\hat s}}{2}(1, 0,0,-1) ~, \nonumber \\
&& p_1= (E_1, p \sin\theta, 0, p \cos\theta) ~~,~~
p_2= (E_2, -p \sin\theta, 0, -p \cos\theta) ~~. \label{kinematics}
\end{eqnarray}
The common characteristic of the subprocesses of the present Section,
is that they all receive
non-vanishing Born contributions determined by the diagrams in
Figs.\ref{qq-gluino-fig}-\ref{qq-chargino-fig}.
Since we neglect initial masses,
the only needed vertices for calculating the diagrams for the first two processes,
are those given by the
neutral gaugino-quark-squark couplings
\begin{equation}
A^{0L}_i(\tilde{u}_L)=-~{e\over 3\sqrt{2}s_Wc_W}
(Z^N_{1i}s_W+3Z^N_{2i}c_W)~~,~~
A^{0L}_i(\tilde{d}_L)=-~{e\over 3\sqrt{2}s_Wc_W}
(Z^N_{1i}s_W-3Z^N_{2i}c_W)~,\label{chi0-Lsquarks}
\end{equation}
\begin{equation}
A^{0R}_i(\tilde{u}_R)=~{2e\sqrt{2}\over 3c_W}
Z^{N*}_{1i}~~,~~
A^{0R}_i(\tilde{d}_R)=-~{e\sqrt{2}\over 3c_W}
Z^{N*}_{1i}~, \label{chi0-Rsquarks}
\end{equation}
and the corresponding chargino-$q\tilde q'_{L,R}$-ones
\begin{equation}
A^{cL}_j(\tilde u_L)=-\frac{e}{s_W}Z^+_{1j}~~,~~
A^{cL}_j(\tilde d_L)=-\frac{e}{s_W}Z^-_{1j}~~.
\label{chip-Lsquarks}
\end{equation}
The notation of \cite{Rosiek} is used for the neutralino and chargino mixing matrices, and
$i$ and $j$ in (\ref{chi0-Lsquarks}, \ref{chi0-Rsquarks}) and (\ref{chip-Lsquarks}),
denote the neutralino and chargino index respectively.\par
For the third process $q\bar q'\to \tilde \chi^0_i \tilde \chi^{\pm}_j$,
determined by the three Born diagrams of Figs.\ref{qq-chargino-fig}a,b,c
one needs in addition the $W$-chargino-neutralino
couplings\footnote{We use the same notation as
in \cite{pp-2chi0, Rosiek}.}
\begin{equation}
O^{WL}_{ji}=Z^N_{2i}Z^{+*}_{1j}-{1\over\sqrt{2}}Z^N_{4i}Z^{+*}_{2j}
~~~,~~~ O^{WR}_{ji}=Z^{N*}_{2i}Z^{-}_{1j}+{1\over\sqrt{2}}Z^{N*}_{3i}Z^{-}_{2j}~~.
\label{OWLRji}
\end{equation}
\subsection{The process $q\bar q\to \tilde \chi^0_i \tilde g$ to the LL 1-loop EW order.}
Writing this process in more detail as
\begin{equation}
q_{c_1}(q_1,\lambda_1)~ \bar q_{ c_2}(q_2,\lambda_2)~
\to ~\tilde \chi^0_i(p_1,\tau_1)~ \tilde g_l(p_2, \tau_2)~~,
\label{process-gluino}
\end{equation}
we denote by $(c_1, c_2)$ the color indices
for $(q,~\bar q)$ respectively, and by $l$ the color index of
$\tilde g$. The helicity amplitude is then written as
$F_{\lambda_1\lambda_2;\tau_1\tau_2}$,
with color indices suppressed and $\lambda_1,\lambda_2,\tau_1,\tau_2$
denoting the helicities. The mass $m_j$ in (\ref{kinematics}) now describes
the gluino mass.
The Born level contributions to this amplitude arising from the two diagrams,
in Fig.\ref{qq-gluino-fig}a,b are
\begin{eqnarray}
F^{\rm B (a)}&=&- {g_s\sqrt{2}A^{0L}_i(\tilde q_L)
\over \hat t-m^2_{\tilde q_L}} \Big ({\lambda^l\over2}\Big )_{c_2c_1}
[\bar v(q_2)P_Ru^c(\tilde g)]
[\bar u(\chi^0_i)P_Lu(q_1)]\nonumber\\
&& + {g_s\sqrt{2}A^{0R}_i(\tilde q_R)
\over \hat t-m^2_{\tilde q_R}} \Big ({\lambda^l\over2}\Big )_{c_2c_1}
[\bar v( q_2)P_Lu^c(\tilde g)] [\bar u(\chi^0_i)P_Ru(q_1)] ~~, \nonumber \\
F^{\rm B (b)}&=& {g_s\sqrt{2}A^{0L*}_i(\tilde q_L)
\over \hat u-m^2_{\tilde q_L}}
\Big ({\lambda^l\over2}\Big )_{c_2c_1}
[\bar v(q_2)P_Ru^c(\chi^0_i)]
[\bar u(\tilde g)P_Lu(q_1)]\nonumber\\
&&- {g_s\sqrt{2}A^{0R*}_i(\tilde q_R)
\over \hat u-m^2_{\tilde q_R}}
\Big ({\lambda^l\over2}\Big )_{c_2c_1}
[\bar v(q_2)P_Lu^c(\chi^0_i)]
[\bar u(\tilde g)P_Ru(q_1)]~. \label{Born-amp-gluino}
\end{eqnarray}
\noindent
where (\ref{chi0-Lsquarks},\ref{chi0-Rsquarks})
have been used and $g_s$ denotes the QCD coupling.
The explicit expressions of the Born helicity amplitudes
$F^B_{\lambda_1\lambda_2,\tau_1\tau_2}$
are given in (\ref{Born-amp-gluino-app}). To get full helicity amplitudes containing also
the 1-loop LL EW and SUSY QCD contributions, the corrections
in (\ref{universal-amp-gluino-cor})
and (\ref{angular-amp-gluino-cor}) should be added.
The differential cross section is then obtained as
\begin{equation}
{d\sigma (q\bar q\to \tilde \chi^0_i \tilde g)
\over d \cos\theta}={\beta \over 1152 \pi s}
\sum_{\rm col, ~spins}|F_{\lambda_1\lambda_2,\tau_1\tau_2}|^2 ~~.
\label{dsigma-qq-gluino}
\end{equation}
At asymptotic energies, much larger than all masses,
both the dominant amplitudes (see (\ref{qq-gluino-amp-asym})), and the
differential cross sections simplify considerably. \\
\subsection{The process $ q g\to \tilde \chi^0_i \tilde q_{L,R}$ to the LL
1-loop EW order}
Writing this process as
\begin{equation}
q_{c_1}(q_1,\lambda_1) g_l(q_2,\mu_2)
\to \tilde \chi^0_i(p_1,\tau_1) \tilde q_{Ic_2}(p_2)~~, \label{prosses-squark}
\end{equation}
we denote by $(c_1,~c_2, ~l)$ the color indices
for $(q,~\tilde q_I,~ g )$ respectively, while $(I=L,~R) $ determines the
type of the produced squark of the first or second family.
The helicities of initial quark and gluon, as well as the
helicity of the final neutralino, are
respectively described by $\lambda_1, ~\mu_2, ~\tau_1$. Correspondingly, the polarization
vector of the initial gluon is denoted as $\epsilon_2$, and the
full helicity amplitudes for the process is written as
$F_{\lambda_1 \mu_2;\tau_1}$.
As before, the kinematics are fixed by (\ref{prosses-kinematics}, \ref{kinematics}),
with $m_j$ now describing the squark mass.
The two relevant Born diagrams are shown in Fig.\ref{qg-squark-fig}.
Since the incoming quark is massless, the squark specification by the index
$(I=L,~R) $, is uniquely associated with the quark helicity being $( \lambda_1=-1/2, ~+1/2)$
respectively; this property remaining true at 1-loop LL level also.
With the momenta and helicities defined by (\ref{prosses-squark}), the contributions
from the diagrams in Figs.\ref{qg-squark-fig}a,b may then be written as
\begin{eqnarray}
F^{\rm B(a)} &= & {g_sA^{0I}_i(\tilde q_I )\over \hat s}
\Big ({\lambda^l\over2}\Big )_{c_2c_1}
[\bar u(\chi^0_i)P_I (\rlap/ q_1+\rlap / q_2)) \rlap / \epsilon_2 u(q_1)] ~~, \nonumber \\
F^{\rm B(b)}& = & {g_sA^{0I}_i(\tilde q_I)\over \hat t-m^2_{\tilde q_I} }
\Big ({\lambda^l\over2}\Big )_{c_2c_1}
[\bar u(\chi^0_i)P_I u(q)](2\epsilon_2 .p_2 )~~.\label{Born-amp-squark}
\end{eqnarray}
The resulting Born helicity amplitudes appear in
(\ref{Born-amp-squark1-app}, \ref{Born-amp-squark2-app}), while their asymptotic expressions
are given in (\ref{Born-amp-asym-squark2}). The universal and angular parts of the
LL EW and SUSY corrections to these amplitudes are respectively given in
(\ref{universal-amp-cor-squark}, \ref{angular-amp-cor-squark}).
After averaging over spins and colors, the cross sections are obtained from these amplitudes
by
\begin{equation}
{d\sigma(q g\to \tilde \chi^0_i \tilde q_I)\over d \cos\theta}
= {\beta\over3072\pi s} \sum_{col,spins}|F_{\lambda_1,\mu_2,\tau_1}|^2 ~~.
\label{dsigma-qg-squark}
\end{equation}
The asymptotic expressions of the amplitudes including all LL EW and SUSY QCD corrections
appear in (\ref{amp-asym-squark}). \\
\subsection{The process $q\bar q'\to \tilde \chi^0_i \tilde \chi^\pm_j$
to the LL 1-loop EW order.}
The two contributing processes in this case, namely $u\bar d \to \tilde \chi^0_i+\tilde \chi^+_j$ and
$d\bar u \to \tilde \chi^0_i+\tilde \chi^-_j$, should give equal differential cross sections,
because of the CP invariance valid for real soft MSSM breaking and $\mu$ parameters:
\begin{equation}
\frac{\mathrm d\sigma (u\bar d \to \tilde \chi^0_i+\tilde \chi^+_j)}{\mathrm d\cos\theta}=
\frac{\mathrm d\sigma (d\bar u \to \tilde \chi^0_i+\tilde \chi^-_j)}{\mathrm d\cos\theta}~~. \label{CP-chargino}
\end{equation}
We therefore concentrate on
\begin{equation}
u(q_1,\lambda_1) \bar d(q_2,\lambda_2)~ \to \tilde \chi^0_i(p_1,\tau_i) ~ \tilde \chi^+_j(p_2, \tau_j)~~,
\label{prosses-chi0p}
\end{equation}
where the helicities and momenta are defined, so that (\ref{kinematics}) keeps describing the
kinematics with $m_j $ now being the chargino mass.
The helicity amplitudes are denoted as
$F_{\lambda_1 \lambda_2;\tau_i\tau_j}$.
The Born level contributions arise
from the three diagrams in Figs.\ref{qq-chargino-fig}a,b,c caused respectively
by the exchanges of a $W^+$ in the s-channel, a $\tilde u_L$-squark in the $t$-channel, and
a $\tilde d_L$-squark in the $u$-channel, and suitably analyzed as
\begin{equation}
F^{ijB}_{\lambda_1 \lambda_2;\tau_i\tau_j}=
S^{ijB}_{\lambda_1 \lambda_2;\tau_i\tau_j}
+T^{ijB}_{\lambda_1 \lambda_2;\tau_i\tau_j}
+U^{ijB}_{\lambda_1 \lambda_2;\tau_i\tau_j} ~~ ,\label{Born-amp-chi0p}
\end{equation}
where the indices $(i,j )$ refer to the neutralino and chargino respectively.
Note that, since we neglect quark masses, there are no R-squark exchange
contributions.
Defining the momenta and helicities as in (\ref{prosses-chi0p}), and
using (\ref{OWLRji}),
the contributions from the three diagrams in Figs.\ref{qq-chargino-fig}a,b,c
to the Born helicity amplitudes appear in (\ref{chi0p-Born-s}, \ref{chi0p-Born-t},
\ref{chi0p-Born-u}) respectively.
At asymptotic energies, only $F^{ij}_{-+-+}$ and $F^{ij}_{-++-}$ retain a non-vanishing
Born contribution appearing in (\ref{chi0p-Born-asym1},\ref{chi0p-Born-asym2});
while the associated EW universal, SUSY QCD, RG and angular LL corrections are
shown respectively in (\ref{Univ-chi0p-mpmp}, \ref{Univ-chi0p-mppm}),
(\ref{SQCD-chi0p}), (\ref{RG-chi0p}) and (\ref{ang-chi0p-mpmp},\ref{ang-chi0p-mppm}).
The spin and color averaged differential cross section is calculated from
\begin{equation}
{d\sigma (u \bar d\to \tilde \chi^0_i \tilde \chi^+_j)\over d \cos\theta}
={d\sigma (d \bar u\to \tilde \chi^0_i \tilde \chi^-_j)\over d \cos\theta}
={\beta \over 384 \pi s}
\sum_{\rm spins}|F^{ij}_{\lambda_1\lambda_2,\tau_1\tau_2}|^2 ~~. \label{dsigma-chi0p}
\end{equation}\\
\section{The one loop process $gg\to \tilde \chi^0_i \tilde{g}$ }
The momenta, helicities and color indices $(a_1,a_2,a_3)$ of the particles participating
in this process, together with the polarization vectors of the gluons,
are defined though
\begin{equation}
g_{a_1}(q_1, \epsilon_1(\mu_1))+ g_{a_2}(q_2, \epsilon_2(\mu_2))
\to
\tilde \chi^0_i(p_1, \lambda_1)~+~\tilde g_{a_3}(p_2, \lambda_2)
~~ . \label{process-gg-gluino}
\end{equation}
The kinematics is defined in (\ref{kinematics}), with $m_i$ denoting the neutralino mass and
$m_j$ the mass of the gluino. The helicity amplitude of the process denoted as
$F_{\mu_1\mu_2; \lambda_1\lambda_2}^{a_1a_2; a3}(\theta)$,
satisfies
\begin{equation}
F_{\mu_1\mu_2; \lambda_1\lambda_2}^{a_1a_2; a3}(\theta)
=(-1)^{\lambda_1-\lambda_2} F_{\mu_2\mu_1; \lambda_1\lambda_2}^{a_2a_1; a3}(\pi-\theta)
~~, \label{gg-Bose}
\end{equation}
because of Bose symmetry among the initial gluons.
This process first appears at the 1-loop level, driven by the diagrams generically shown in
Fig.\ref{gg-gluino-fig}. These consist of three types of box diagrams named (B1, B2, B3),
which are of exactly the same form as those met in neutralino pair production
in an $LC_{\gamma \gamma}$ collider, or in the calculation of
the reverse process of dark matter annihilation
to photons \cite{gamma-gamma, DMann}. In addition to them, there are
three types of s-channel triangular diagrams
(s1, s2, s3), and two types of t-channel triangles.
These diagrams have been calculated exactly and the results were
used to construct the FORTRAN code named PLATONgluino which,
after averaging over all spins and colors, calculates
\[
\frac{\mathrm d\sigma (g g \to \tilde \chi^0_i \tilde g) }{\mathrm d\hat t} ~~~{\rm in } ~~~ fb/TeV^2
\]
for any value of the subprocess c.m. scattering angle $\theta$
given in radians, and any set of real MSSM parameters at the electroweak scale. As with
other related codes we have constructed, PLATONgluino may be obtained from
\cite{Platon}.
\section{Results}
In this Section we discuss the LHC predictions for the direct production of
a single neutralino associated with a gluino, squark or chargino,
according to the four subprocesses presented
in Sections 2 and 3. The predictions are valid for any
MSSM model with real SUSY parameters. An exploration
of the possible results has been made, using typical benchmark models
\cite{Snowmass, Arnowitt, CDG}. These benchmarks have also
been used in other recent neutralino explorations,
and their sole purpose is to help identifying the physical parameters
mainly affecting the neutralino production at LHC \cite{DMann, gamma-gamma, pp-2chi0}.
For the parton distribution functions inside
the proton, we use the MRST2003c package \cite{MRST}
at the factorization scale
\begin{equation}
Q=\frac{E_{Ti}+E_{Tj}}{4}~~. \label{Q-scale}
\end{equation}
A complete summary of the relevant parton formulae and kinematics
may be found in Appendix B of \cite{pp-2chi0}.
As observables we use the invariant mass distribution
$d\sigma/d\hat s$ of the aforementioned subprocesses, and the
c.m. angular distribution $d\sigma/d\chi$ defined {\it e.g. } in
eqs.(B.39,B.43) of \cite{pp-2chi0}. The $\chi$-variable is always taken
to describe the particle accompanying the neutralino
in the subprocess and is defined in terms of c.m. variables
\begin{eqnarray}
\chi_j\equiv e^{2y_j^*}&= & {1 - {p^*\over E^*_j} \cos\theta^*
\over 1 +{p^*\over E^*_j} \cos\theta^*} ~~ , \label{chi}
\end{eqnarray}
\noindent
so that our treatment covers both, the LSP $\tilde \chi^0_1$ case, as well
as the case of a heavier $\tilde \chi^0_i$.
The transverse momentum distribution is not shown in any detail here,
since it presents the same features as the mass distribution;
a similar situation has already been noticed for $\tilde \chi^0_i\tilde \chi^0_j$ production
\cite{pp-2chi0}. Depending on the experiment of course,
such distributions may also be useful. \\
Extensive sensitivity of the single neutralino production processes
to the SUSY MSSM parameters, is observed.
This is caused mainly by the dependence of the neutralino couplings
on the percentage of their gaugino (Bino and Wino)
or higgsino components, through the $Z^N_{ji}$ mixing matrices
\cite{Rosiek}. The four processes in (\ref{processes}) react differently to
this percentage, as the first three are mainly controlled
by the gaugino components\footnote{Some dependence on the higgsino component appears
also for the subprocess $gg\to \tilde \chi^0_i \tilde g$, caused by
$(\tilde t, \tilde b)$-loop contributions. It should be remembered though that
the contribution of this subprocess, being of higher order,
is generally suppressed.}, whereas the fourth process
depends both, on the gaugino and on the higgsino
components.
As we are especially interested in the structure of the
LSP, supposedly the lightest $\tilde \chi^0_1$, which, depending on
the benchmark, can predominantly be either Bino, or Wino, or
higgsino, this explains the large sensitivity to the
chosen benchmark.\par
The single neutralino production processes
are also very sensitive to the masses of the exchanged squarks.
In the gluino case, the relative importance of the one loop process $gg\to\tilde \chi^0_i\tilde{g}$,
is also strongly depending on the squark masses.
For light squarks, this gives cross sections which are about
a hundred times smaller than the ones from the $q\bar q$ process.
But if the squarks of the first two generations become heavy, while those of the third
remain rather light, it may turn out that kinematical regions exist where the
1-loop subprocess $gg\to\tilde \chi^0_i\tilde{g}$ is appreciable, compared to the
Born-level subprocess $q \bar q \to\tilde \chi^0_i\tilde{g}$; so that it
cannot be ignored. Comparing with the $gg\to\tilde \chi^0_i\tilde \chi^0_k$ treatment of \cite{pp-2chi0},
we should note that the $gg\to\tilde \chi^0_i \tilde{g}$ process cannot be enhanced
by resonance effects like those enhancing $\tilde \chi^0_i\tilde \chi^0_k$ production. \par
For what concerns the electroweak radiative corrections to the three
Born processes, the computations of the leading and subleading
logarithmic contributions show a
negative effect, regularly increasing with the invariant mass,
which is of the order of (10-20)\% at the TeV range, as expected from \cite{BRV}.
This effect is comparable, but of opposite sign, to the analogous
QCD correction\cite{Spiratalk,Beenakker,Plehn},
and it should also be taken into account
for a precise analysis.\\
On the basis of our explorations, we present
in Figs.\ref{SPS1a-mass-fig}-\ref{ADfg9a-mass-chi-fig} below,
illustrations for four different
typical cases\footnote{In all cases we have used ${\rm Suspect2\_3}$ to calculate
the various masses \cite{suspect}.}:\\
\noindent
1) a gaugino (Bino)-type model for $\tilde \chi^0_1$, with light squark masses, SPS1a
\cite{Snowmass},\\
1a) a Bino-type model with heavy squark masses\footnote{SPS1aa is constructed
from SPS1a of \cite{Snowmass}, by simply putting the high scale SUSY breaking soft
sfermion masses of the first and second generations, at 5000 GeV.}, SPS1aa, \\
2) a higgsino type model for $\tilde \chi^0_1,~\tilde \chi^\pm_1$,
with light squark masses\footnote{This model is
extracted from Fig.9 of \cite{Arnowitt}. It is characterized by the high
scale values: $m_{1/2}=420 ~GeV$,
$A_0=420~ GeV$, $\tan\beta=40$, and $m_0=600~ GeV$ for all scalar masses except
$m_{H_u}=600 \sqrt{2} ~GeV$. To preserve the predominantly higgsino nature of $\tilde \chi^0_1$,
$m_t=174 ~GeV$ should be used here, as this was the case when the model was constructed.},
AD(fg9) \cite{Arnowitt},\\
2a) a related higgsino type model, but
with heavy squark masses\footnote{It is constructed from the EW
scale masses of AD(fg9), by only changing
the sfermion soft SUSY breaking masses of the first two
generations, which are now put at 5000 GeV.}, AD(fg9a) \cite{Arnowitt},\\
\noindent
discussed in turn below:
\begin{itemize}
\item Let us first consider
the $\tilde \chi^0_1$ gaugino-type model (SPS1a). This model gives invariant mass distributions
for the three Born processes in (\ref{processes}),
which are largely observable in the 1 TeV range; {\it i.e. } cross sections of
about 100fb for the first two cases, but only 10fb or less
for the third one. It also gives an invariant mass distribution for the
$ q g\to \tilde \chi^0_i \tilde q_{L,R}$ channel, which
is more important at low masses, due to the behavior of the gluon
distribution function. The angular distribution
for $q\bar q \to \tilde \chi^0 \tilde g$, described at Born
level by the t- and u- channel squark exchanges indicated in
Fig.\ref{qq-gluino-fig}, flattens out
at large $\chi_{\tilde g}$ in this model.
It is amusing to remark that a very similar behavior is also expected in the
universal m-SUGRA type model which has been identified by
\cite{Ellis}, as "a best" description of all
present particle and cosmological constraints \cite{DMpresent}.
\item Comparing SPS1aa to SPS1a, one notices
a reduction of the invariant mass distribution in
the 1 TeV range, by more than an order of magnitude for
$\tilde \chi^0_i \tilde g$,
by somewhat less than an order of magnitude for $\tilde \chi^0_i\tilde \chi^\pm_1$;
and, obviously, a complete suppression of the $\tilde \chi^0_i \tilde q$ production.
Moving from SPS1a to SPS1aa, there appears also a change in
the $\chi$-distribution, which becomes steeper for $\tilde \chi^0_i \tilde g$,
but remains roughly similar for $\tilde \chi^0_i\tilde \chi^\pm_1$.
\item We next turn to the higgsino type model AD(fg9)
for the three channels studied here. Comparing them to the gaugino SPS1a and SPS1aa
models, we find that the predicted cross sections for
the $\tilde \chi^0_i \tilde g$ and $\tilde \chi^0_i \tilde q$ channels, are much smaller now,
since they are controlled essentially by the
gaugino components; (see Figs.\ref{ADfg9-mass-fig},\ref{ADfg9a-mass-chi-fig}).
On the contrary, the cross section for the $\tilde \chi^0_i\tilde \chi^\pm_1$ process
is larger, because of the presence of the
s-channel W exchange diagram involving higgsino
components; (see Fig.\ref{qq-chargino-fig}). The $\chi$-distribution
becomes also flatter, for the same reason. These can be seen by
comparing Figs.\ref{ADfg9-mass-fig}c,\ref{ADfg9-chi-fig}c and \ref{ADfg9a-mass-chi-fig}b,d
with Figs.\ref{SPS1a-mass-fig}c,\ref{SPS1a-chi-fig}c and \ref{SPS1aa-mass-chi-fig}b,d
respectively.
\item Finally, we compare the results of the models
AD(fg9) and AD(fg9a), in both of which there is a large
higgsino component to $\tilde \chi^0_1$, as well as to $\tilde \chi^0_2$.
In AD(fg9a) the squark contribution, coupled through the
gaugino component, is further reduced compared to AD(fg9), leading
to an even smaller prediction
for the $\tilde \chi^0_i \tilde g$; compare Figs.\ref{ADfg9-mass-fig}a,\ref{ADfg9-chi-fig}a
with \ref{ADfg9a-mass-chi-fig}a,c. On the other hand,
for the $\tilde \chi^0_1\tilde \chi^\pm_1$ and $\tilde \chi^0_2\tilde \chi^\pm_1$ channels, the cross
sections are comparable, since they both receive a large contribution from
the higgsino component coupled to the intermediate $W$ boson,
which is not affected by the change in the squark mass;
compare Figs.\ref{ADfg9-mass-fig}c,\ref{ADfg9-chi-fig}c
with \ref{ADfg9a-mass-chi-fig}b,d.
\end{itemize}
Finally we comment on the difference between the magnitude of
the $\tilde \chi^0_1$ production cross section (in which we are mainly
interested in), and the $\tilde \chi^0_2$ one;
$\tilde \chi^0_2$ production is generally more copious than the
$\tilde \chi^0_1$ one, becoming progressively more pronounced
as we go from the $\tilde \chi^0_i \tilde g$ channel, to $\tilde \chi^0_i \tilde q$, and eventually to
the $\tilde \chi^0_i\tilde \chi^\pm_1$ channel, where it reaches a factor 10 in the SPS1a model.
This factor is even larger in the SPS1aa model; of order 100.
These differences are due to the
$Z_{ji}$ mixing matrix elements appearing in the Born amplitudes, which control the
Bino, Wino and higgsino components of the neutralinos.\\
\section{General Conclusion on $\tilde \chi^0_i $ production}
In this paper we have analyzed the single
neutralino production processes
$\tilde \chi^0_i \tilde g$, $\tilde \chi^0_i \tilde g$ and
$\tilde \chi^0_i \tilde \chi^{\pm}_j$ at LHC.\par
The complete set of helicity amplitudes
for the Born terms of the subprocesses
$q\bar q\to \tilde \chi^0_i \tilde g$, $g q\to \tilde \chi^0_i \tilde q_{L,R}$,
$q\bar q'\to \tilde \chi^0_i \tilde \chi^{\pm}_j$ has been written down,
together with the leading and subleading logarithmic electroweak
corrections to them. Compact analytic expressions are presented,
which are applicable to any MSSM model with real parameters.
We have also included the complete one loop calculation of the
subprocess $gg\to \tilde \chi^0_i \tilde g$, for which a numerical code called PLATONgluino
is released \cite{Platon}.\par
The pure QCD corrections, which have already been
given in previous papers \cite{Beenakker, Plehn}, have not been reexamined.
But we have emphasized that, contrary may be to naive expectations,
the leading logarithmic EW and QCD corrections at the LHC energies have
similar magnitudes, but opposite sign.
Thus, they should both be taken into account in analyzing the experimental
data. \par
The single neutralino production processes have been found to be mainly sensitive on
two physical sets of quantities; namely the amount of gaugino and higgsino components
of the neutralinos, and the scale of the soft breaking parameters for the
squarks of the first and second generations. To emphasize this, a set of
illustrations for LHC invariant mass and angular distributions have been presented
which indeed show this sensitivity. These were based on
four "benchmark" models, but more were explored in our actual runnings.\par
This physics output should of course be joint to the one that can be obtained
from the $\tilde \chi^0_i \tilde \chi^0_j$ production studied previously \cite{pp-2chi0}.
For that purpose we have added
Figs.\ref{SPS1a-chi0chi0-fig},\ref{SPS1aa-chi0chi0-fig}, which show the
invariant mass and $\chi$ distributions for
$\tilde \chi^0_2 \tilde \chi^0_1$ and $\tilde \chi^0_2 \tilde \chi^0_2$ production, in the same
SPS1a and the SPS1aa models used for the single neutralino
case\footnote{Analogous results have been shown in Figs.11,12 of \cite{pp-2chi0},
where though, $\chi$ is defined as the inverse of the present one.}.
In going from SPS1a to SPS1aa, one sees a
reduction of the Born contribution, rather similar to what happens
in the $\chi^0_{1,2}\tilde{g}$ case; but one also sees that the
relative role of the one loop $gg$ process in SPS1aa, is more important for
$\tilde \chi^0_2 \tilde \chi^0_1$ production, then for $\chi^0_{1,2}\tilde{g}$.
So the neutralino pair production channel has its own
typical features.\par
Summarizing, we have observed that the channels
$\tilde \chi^0_i \tilde g$, $ \tilde \chi^0_i \tilde g$,
$\tilde \chi^0_i \tilde \chi^{\pm}_j$ and $\tilde \chi^0_i \tilde \chi^0_j$
present an important sensitivity to the neutralino structure;
particularly to the relative magnitude of its gaugino
and higgsino components.
They also present a considerable
sensitivity to the MSSM mass spectrum
for the gluino, squarks, charginos and Higgses;
the later being able to lead
to possible resonance effects.\par
We conclude by emphasizing that the results obtained
in this paper should be completed by detail
experimental studies dedicated for LHC. Observables should then be
constructed addressing neutralino, gluino and squark
decay channels to various numbers of jets and leptons.
Such observables should also
reflect, at some important level, the sensitivity to the neutralino
properties\footnote{If discovered at LHC or
a Linear Collider, it would also be the first time
that physical quantities highly sensitive to the Majorana
nature of a particle reach such a high observability.}, that we have
observed at the level of the basic processes.
We hope that their measurement will be able to
confirm or infirm the possibility
that the neutralino is an important component
of the Dark Matter of the Universe.\\
\noindent
{\large\bf{Acknowledgement}}\\
\noindent
G.J.G. gratefully acknowledges also the support by the European Union RTN contract
MRTN-CT-2004-503369, and the hospitality extended to him by the CERN Theory
Division during the later part of this work.
\newpage
|
{
"timestamp": "2005-03-29T09:57:26",
"yymm": "0411",
"arxiv_id": "hep-ph/0411366",
"language": "en",
"url": "https://arxiv.org/abs/hep-ph/0411366"
}
|
\section{Introduction}\label{sec.1}
Non-relativistic strings (NR ) in flat transverse space with a
non-trivial spectrum were introduced in
\cite{Gomis:2000bd}\cite{Danielsson:2000gi},
in the static gauge they become free theories
\cite{Garcia:2002fa}. The actions of NR strings and branes
can be constructed as Wess-Zumino
terms \cite{Brugues:2004an}
of the underlying Galilei groups using the method of
non-linear realizations for space-time groups \cite{Ivanov:1975zq}
\cite{Gauntlett:1990nk}.
NR superstring theories in flat transverse space were recently
analyzed in \cite{Gomis:2004pw}. The action was obtained by performing a
suitable non relativistic limit of Green-Schwarz type IIA string. These
NR superstring theories have gauge diffeomorphism and kappa symmetry.
We hope the study of NR string theory could be usefull to find a new sector of
the theory where the AdS/CFT correspondence can be tested.
In this paper we consider NR bosonic string in a curved transverse space
in terms of a Polyakov type action.
In the static gauge case, we
derive a BPS bound for the energy.
This theory is conformal at quantum level
if the transverse metric is Ricci flat.
We study rotating solutions of the classical equations of motion
for the flat case and
for the singular conifold \cite{Candelas:1989js}. For the case of
the conifold the solutions we found
have three angular momenta. The relation among the energy and
angular momenta is of non-relativistic type $E\sim J^2$ where
$J$ represents the angular momenta.
We also analyze the supersymmetric properties of the bosonic
solutions we have found by studying the conditions imposed by
the kappa symmetry and supersymmetry of the NR superstring
\cite{Gomis:2004pw}.
The organization of the paper is as follows. In section 2 we
introduce Polyakov form NR string and derive a BPS bound for the
energy. In section 3 we find rotating solutions of NR string in
flat space time. In section 4 we consider solutions in the case of
a singular conifold. In section 5 we study the supersymmetric
properties of our solutions. We also give some conclusions in
section 6.
\section{Non-relativistic strings in transverse curved backgrounds
and BPS bound}
\label{sec.2}
We study a non-relativistic string in
a $d$-dimensional space-time, with coordinates $x^\mu$,
$\mu=0,1$, along the string and transverse
coordinates $X^a$, $a=2,\ldots,d-1$.
The $d$-dimensional metric is
\begin{eqnarray}
\label{metrica}
g_{MN}=\left(\begin{array}{cc}\eta_{\mu\nu}&0\\0&
G_{ab}(X^a)\end{array}\right).
\end{eqnarray}
The $\eta_{\mu\nu}$ is the
$2$-dimensional Minkowski metric with signature $(-,+)$ and
$G_{ab}(X^a)$ is a metric on some $(d-2)$-dimensional Riemannian manifold
${\cal M}$.
The action of NR string is given by
\begin{equation}\label{azionephi}
S = - {1\over2}T \int d^{2}\sigma \sqrt{-\det g}\, g^{ij}
\partial_i X^a \partial_j X^b G_{ab}
\end{equation}
where $g^{ij}$ is the inverse of the two dimensional metric
\begin{equation}
g_{ij} = \partial_i x^\mu \partial_j x^\nu\eta_{\mu\nu},
\end{equation}
the worldsheet coordinates are $\tau$ and $\sigma$,
$\sigma^i= (\tau, \sigma), i=1,2$.
The action is
invariant under 2d diffeomorphism of the world sheet.
In order to have a consistent non relativistic string theory at quantum level
we have to
consider the coordinate $x^1$ toroidally compactified
\cite{Gomis:2000bd}\cite{Danielsson:2000gi}, i.e.:
\begin{equation}
x^1\backsim x^1 + 2\pi R.
\end{equation}
We now analyze the dynamics of non relativistic strings at
classical level by using
the Hamiltonian formalism. Let us indicate by $p_\mu$ the
canonical momenta associated to the longitudinal coordinates $x^\mu$
and $P_a$ the transverse momenta. As a consequence of the
gauge symmetry of the action (\ref{azionephi}), the canonical
variables satisfy the following two primary first class constraints:
\begin{eqnarray}\label{constraints0}
V_0&=&p_\mu\;\vep^{\mu\rho}\eta_{\rho\nu}\;{x'}^\nu+
\frac12\left(\frac{P_aP_b}{T} G^{ab}(X)+{X'}^a{X'}^b TG_{ab}(X)\right)
\sim 0,
\\\label{constraints1}
V_1&=&p_\mu {x'}^\mu+P_a{X'}^a\sim 0.
\end{eqnarray}
The Dirac hamiltonian is:
\begin{equation}\label{hamdir}
H_{D}=\int d\sigma
\left[
\lambda_{0} V_0+\lambda_{1} V_1\right]
\end{equation}
where $\lambda_{0},\lambda_{1}$
are arbitrary functions of the world sheet
coordinates $\sigma^i$.
If we write these arbitrary functions in terms of an auxiliary two
dimensional metric $\gamma_{ij}$,
$\lambda_{0}=\frac{\sqrt{-\gamma}}{\gamma_{11}},
\lambda_{1}=\frac{\gamma_{01}}{\gamma_{11}}$ we will find a gauge
invariant Polyakov's formulation of this non-relativistic string.
In fact if we consider the first order action \begin{equation} {\tilde
S}_P=\int d^2\sigma \left[ p_\mu {\dot x}^\mu+P_a{\dot X}^a
\right]-\int d \tau H_D \end{equation} and we eliminated the momenta $P_a$ we
get \begin{equation} S_P=\int d^2\sigma \Bigg[-\frac{T}2 \sqrt{-\gamma}
\gamma^{ij}\partial_i X^a\partial_j X^b G_{ab}+ p_\mu(\dot
x^\mu-\frac{\sqrt{-\gamma}}{\gamma_{11}} \vep^{\mu\rho}\
\eta_{\rho\nu}\;{x'}^\nu-
\frac{\gamma_{01}}{\gamma_{11}}{x'}^\mu)\Bigg]. \label{polyakov}
\end{equation} This action is invariant under 2d diffeomorphism\footnote{We
acknowledge Kiyoshi Kamimura for useful discussions on this
point.} \begin{eqnarray}
\delta x^\mu&=& \xi^k\partial_k x^\mu\\
\delta p_\mu&=& \partial_k(\xi^kp_\mu)+ p_\mu(-{\dot
\xi}^0+{\xi'}^0
\frac{\gamma_{01}}{\gamma_{11}})+p^\nu\vep_{\nu\mu}{\xi'}^0
\frac{\sqrt{-\gamma}}{\gamma_{11}}\\
\delta X^a&=& \xi^k\partial_k X^a\\
\delta\gamma_{ij}&=& {\cal L}_\xi\gamma_{ij}
\end{eqnarray}
where ${\cal L}_\xi$ is the Lie derivative along $\xi$. The action is
also invariant under
gauge
Weyl transformations of the auxiliary metric $\gamma_{ij}$.
If we choose the conformal gauge $ \sqrt{-\gamma}
\gamma^{ij}=\eta^{ij}$, the action becomes
\begin{equation} S_P=\int
d^2\sigma \left[-\frac{T}2 \eta^{ij}\partial_i
X^a\partial_j X^b G_{ab}+
p_\mu(\dot x^\mu-
\vep^{\mu\rho}\ \eta_{\rho\nu}\;{x'}^\nu)
\right]. \label{polyakov1}
\end{equation}
The equations of motion in the
conformal gauge are
\begin{eqnarray} \label{eqmocur1}
0&=& \eta^{ij}\partial_i\partial_j X^a+\Gamma^a_{bc}
\eta^{ij}\partial_i X^b \partial_j X^c\\
0&=& \dot x^\mu-
\vep^{\mu\rho}\ \eta_{\rho\nu}\;{x'}^\nu\label{eqmocur1a}\\
0&=& \dot p_\mu- \vep^{\nu\rho}\ \eta_{\rho\mu}\;{p'}_\nu \end{eqnarray}
where $\Gamma^a_{bc}$ are the Christoffel symbols of the metric
$G_{ab}$. From \bref{eqmocur1a} we deduce that the longitudinal
coordinates verify the relations $\dot x x'=0$ and ${\dot
x}^2+{x'}^2=0$.
The boundary conditions are the same as the ones considered for the flat space
case \cite{Garcia:2002fa}, i.e. for a closed string
\begin{equation}
\label{chiuse} x^\mu(\tau,\sigma+2\pi)=x^\mu(\tau,\sigma)+
2\pi nR\delta^\alpha_1 \qquad
X^a(\tau,\sigma+2\pi)=X^a(\tau,\sigma)
\end{equation}
where $n\in \mathbb{Z}$ is the winding number of the string.\\
For an open string
\begin{eqnarray} \label{aperte}\nonumber
x'^0|_{\sigma=\pi}=x'^0|_{\sigma=0}=0\qquad
x^1(\sigma=0)=0 \qquad x^1(\sigma=\pi)=2\pi nR \\
X'^a_N|_{\sigma=\pi}=X'^a_N|_{\sigma=0}=0 \qquad
\dot{X}^b_D|_{\sigma=\pi}=\dot{X}^b_D|_{\sigma=0}=0,
\end{eqnarray}
also in this case $n\in \mathbb{Z}$ .
If we introduce the coordinates
$\gamma=X^0+X^1,\bar\gamma=X^0-X^1, \beta=p_0+p_1,
\bar\beta=p_0-p_1$ the first order part of the action
\bref{polyakov1} is the conformal (2,2) $\beta\gamma$ system
introduced in \cite{Gomis:2000bd}. Therefore our NR string will
be conformal invariant at quantum level if the metric $G_{ab}$ is
Ricci flat \cite{Callan:1985ia} and the spacetime dimension is 26
\cite{Gomis:2000bd} like for the ordinary bosonic relativistic
string.
To solve the classical equations of motion is convenient to work in the
static gauge. We fix this gauge by imposing two gauge fixing
constraints:
\begin{eqnarray}\label{statclos}
\Phi_0\equiv x^0-K\tau=0 \qquad \Phi_1\equiv x^1-K\sigma=0
\end{eqnarray}
where $K$ is a constant. These constraints make the constraints
\bref{constraints0} \bref{constraints1} second class, they are stationary
if $\lambda_{0}=1, \lambda_{1}=0$. Therefore the static
gauge is included in the conformal gauge. In the static gauge the
transverse degrees of freedom, $(X^a, P_a)$, are the
independent degrees of freedom of the NR string.
The dynamics is given by \bref{eqmocur1}.
The momenta ${\cal P}_1$ along the string,
in the static gauge, using the constraints \bref{constraints1}
is given by
\begin{equation}\label{momenta1}
{\cal P}_1=-\frac{1}{K}\int d\sigma\left(P_a{ X'}^a \right)
\end{equation}
and the energy is
\begin{equation}
\label{energia} E=\frac{1}{K}\int
d\sigma\left(\frac{(P)^2}{2T}+ \frac{T(X')^2}{2}\right)
\end{equation}
where we have used the constraint \bref{constraints0}. Note that
the energy can be written as \begin{equation}\label{relb} E=\int d\sigma\frac
{(P^a\pm TX'^a)^2}{2TK}\pm {\cal P}_1 =\int d\sigma\frac {T(\dot
X^a\pm X'^a)^2}{2K}\pm {\cal P}_1.\end{equation}
Therefore ${\cal P}_1$ is the BPS bound
\begin{equation}
\label{bound} E\ge \pm {\cal P}_1\ge
0,
\end{equation}
the bound is saturated by the BPS configurations \begin{equation}\label{bpseq}
\dot X^a\pm X'^a=0. \end{equation}
The configurations that satisfy the BPS equation \bref{bpseq}
verify also the second order equations of
motion \bref{eqmocur1}.
\section{Classical Solutions for NR strings in flat space time}
We now analyze non relativistic strings in flat transverse space,
i.e. we consider $G_{ab}(X^a)=\delta_{ab}$. The equations of
motion (\ref{eqmocur1}) simplify to:
\begin{equation}
\label{onda} \ddot{X}^a=X''^a.
\end{equation}
Here we are interested in solutions of (\ref{onda}) describing
rotating strings, i.e. solutions with angular momentum different
from zero. The component of the angular momentum perpendicular to
the $X^a-X^b$ plane is:
\begin{equation} \label{mom}
J^{ab}=\int d\sigma\left(X^aP^b-X^bP^a\right).
\end{equation}
\subsection{Closed Strings}
We now consider solutions of the equations (\ref{onda}) satisfying
boundary conditions for closed strings (\ref{chiuse}). To satisfy
the boundary conditions for the longitudinal coordinates we have
to fix $K=nR$ in the static gauge constraints (\ref{statclos}).
The simplest solution describing a rotating closed string is:
\begin{eqnarray}\nonumber
x^0=nR\tau \qquad
X^2=A\sin(\omega\sigma)\sin\left(\omega\tau\right)\\\label{soluzione1}
x^1=nR\sigma\qquad
X^3=A\sin(\omega\sigma)\cos\left(\omega\tau\right)
\end{eqnarray}
where $\omega\in\mathbb{Z}$ and $A$ is a dimensional constant,
$[A]=L$; these conventions are valid also for the following
solutions. The (\ref{soluzione1}) describes a circular string with
angular momentum perpendicular to the $X^2-X^3$ plane (figure
\ref{f12}).
\begin{figure}
\centerline{\includegraphics[width=10cm]{figura12x.eps}}
\caption{String described by solution (\ref{soluzione1}) when
$\omega=1$}\label{f12}
\end{figure}
The velocity modulus of the points of the string is:
\begin{eqnarray}
v^2=\left(\frac{\partial X^2}{\partial
x^0}\right)^2+\left(\frac{\partial X^3}{\partial
x^0}\right)^2=\left(\frac{\omega
A}{nR}\right)^2\left(\sin(\omega\sigma)\right)^2\qquad
v_{max}=\frac{A\omega}{nR} \qquad\label{speed1}.
\end{eqnarray}
Like for the relativistic case the turning points are the faster
points, but in this case the maximum velocity is not limited by
the velocity of light, which in natural units is $v=1$.
This is a typical feature of non
relativistic theory. By using (\ref{energia}) and (\ref{mom}) we
find:
\begin{equation}
\label{rels} E=\frac{J^2}{TA^2\pi nR}.
\end{equation}
Interpreting $\frac{1}{2}TA^2\pi nR$ like the inertial
momentum of the string respect to the rotation axis, the
(\ref{rels}) is the dispersion relation for a classical rotating
system.
The momentum along the string ${\cal P}_1$ vanishes for this configuration.
Since the energy of this configuration \bref{rels} is different from zero
this configuration does not saturate the bound of the energy \bref{bound}
and therefore we expect it will be non-supersymmetric if we embed this
NR string in a supersymmetric theory.
A generalization of this solution with more than two spikes in the
transverse space is given by
\begin{eqnarray}
\nonumber x^0=nR\tau\qquad
X^2=B\left(\cos((z-1)(\tau+\sigma))+(z-1)\cos(\tau-\sigma)\right)
\\\label{spike} x^1=nR\sigma\qquad
X^3=B\left(\sin((z-1)(\tau+\sigma))+(z-1)\sin(\tau-\sigma)\right),
\end{eqnarray}
where z is an integer and corresponds to the number of spikes of the strings.
When $z=2$, we recover the solution \bref{soluzione1} with
$A=2B$ and $\omega=1$.
These solutions are
the NR version of the solutions studied in \cite{Kruczenski:2004wg}.
The spikes are the fastest point of the strings, the modulus of their velocity
is
\begin{eqnarray}
v=\frac{2B(z-1)}{nR}\label{spikevel}\end{eqnarray}
i.e. a non-limited quantity. Using (\ref{energia}) and (\ref{mom}) we
find:
\begin{equation}
E=\frac{J^2}{TB^2\pi nRz^2}.
\label{spikeene}\end{equation}
The inertial momentum $\frac{1}{2}TB^2\pi nRz^2$ is proportional to the square
of the number of spikes. This is in agreement with the fact that increasing the
number of spikes, the string is on average more distant from the rotational
axis.
For these solutions also ${\cal P}_1=0$ and the BPS bound is not saturated.
\vspace{5mm}
We now analyze a solution describing a string that is moving in a
$s$-dimensional sphere embedded in the transverse space. The
transverse part is described by a $(s+1)$-dimensional vector of
solutions \cite{Hoppe:1987vv}:
\begin{eqnarray}
\label{hoppe}
\overrightarrow{X}(\tau,\sigma)=D(\tau)\overrightarrow{m}(\sigma)
\end{eqnarray}
where:
\begin{eqnarray}
\overrightarrow{m}(\sigma)=(A_1\sin(\omega_1\sigma),A_1\cos(\omega_1\sigma),
A_2\sin(\omega_2\sigma),A_2\cos(\omega_2\sigma),\ldots)
\\\nonumber\\D(\tau)=\left(\begin{array}{ccccc}\sin\left(\omega_1\tau\right)&
\cos\left(\omega_1\tau\right)&0&0&\ldots\\\\
\cos\left(\omega_1\tau\right)&-\sin\left(\omega_1\tau\right)&0&0&\ldots\\\\
0&0&\sin\left(\omega_2\tau\right)&\cos\left(\omega_2\tau\right)&\ldots\\\\
0&0&\cos\left(\omega_2\tau\right)&-\sin\left(\omega_2\tau\right)&\ldots\\\\
\vdots & \vdots &\vdots & \vdots &\ddots
\end{array}\right).
\end{eqnarray}
This solution verify the BPS equation \bref{bpseq},
$\dot{\vec X}+ \vec X'=0$,
and therefore
satisfy the equations of motion
(\ref{onda}) with the boundary
conditions (\ref{chiuse}). This solution describe
a string on a $s$-dimensional sphere since
\begin{eqnarray}
\label{sferas}\overrightarrow{X}(\tau,\sigma)^2=\textrm{constant}.
\end{eqnarray}
The simplest solution of this kind, considering also the
longitudinal part, is:
\begin{eqnarray}\label{susysol}
\nonumber x^0=nR\tau\qquad
X^2=A\left(\sin(\omega\sigma)\sin\left(\omega\tau\right)+\cos(\omega\sigma)\cos\left(\omega\tau\right)\right)
\\\label{semhoppe} x^1=nR\sigma\qquad
X^3=A\left(-\cos(\omega\sigma)\sin\left(\omega\tau\right)+\sin(\omega\sigma)\cos\left(\omega\tau\right)\right).
\end{eqnarray}
The solution (\ref{semhoppe}) describes a circular string
with angular momentum perpendicular to the plane $X^2$-$X^3$,
(figure \ref{f34}).
\begin{figure}
\center{\includegraphics[width=10cm]{figura34x.eps}}
\caption{String described by solution (\ref{semhoppe}) when
$\omega=1$}\label{f34}
\end{figure}
The velocity modulus is the same for all the points of the string:
\begin{equation}
v=\frac{A\omega}{nR},
\label{speed2}
\end{equation}
also in this case there is not a limit for the velocity. From
(\ref{energia}) and (\ref{mom}) we have:
\begin{equation} \label{rel}
E=\frac{J^2}{2TA^2\pi nR}.
\end{equation}
For this solution the inertial momentum is $TA^2\pi nR$ and is
bigger than that one of the solution \bref{soluzione1}. This is
in agreement with the classical intuition because now the string
is on an average more distant from the rotation axis.
For this solution the momenta along the string is given by \begin{equation}
{\cal P}_1=\frac{J^2}{2TA^2\pi nR} \end{equation} and coincides with the
energy of the solution because this configuration verifies the BPS
equation and therefore it saturates the BPS bound. We will see
that this configuration is ${1\over 4}$ BPS supersymmetric when we
embed the bosonic NR string in the NR superstring
\cite{Gomis:2004pw}.
Relativistic closed string on spheres with all points moving at
the same velocity in modulus, was considered in
\cite{Hoppe:1987vv} .
They have at least
two component of the angular momentum different from zero. However
they are different of our NR solutions among other things
because they are not supersymmetric.
\subsection{Open strings}
We now turn to open strings, i.e. solutions of the equations
(\ref{onda}) satisfying boundary conditions (\ref{aperte}). To
satisfy the boundary conditions for the longitudinal coordinates
we have to fix $K=2nR$ in the static gauge constraints
(\ref{statclos}). We consider a solution with all transverse
coordinates satisfying Neumann boundary conditions:
\begin{eqnarray}
\nonumber x^0=2nR\tau \qquad
X^2=A\cos(\omega\sigma)\sin\left(\omega\tau\right)\\\label{san}
x^1=2nR\sigma\qquad
X^3=A\cos(\omega\sigma)\cos\left(\omega\tau\right).
\end{eqnarray}
This is an open string with angular momentum perpendicular to the
$X^2-X^3$ plane (figure \ref{an}).
\begin{figure}
\centerline{\includegraphics[width=10cm]{apertanx.eps}}
\caption{String described by solution (\ref{san}) when
$\omega=1$}\label{an}
\end{figure}
The string extrema are the faster points and also in this case
the velocity modulus $v_{max}=\frac{A\omega}{2nR}$ in not limited.
The relation between energy and angular momentum is
\begin{equation}
\label{rela} E=\frac{J^2}{TA^2\pi nR},
\end{equation}
i.e. the same relation that we have obtained for the first kind of closed string.
For relativistic strings there are different energy-angular momentum relations for
open and closed strings.
\section{Classical Solutions for NR strings in curved space time}
We now analyze the dynamics of strings propagating in a curved
transverse space. Since the transverse metric should be Ricci flat
in order to have a consistent NR string at quantum level, here we
consider the case of a singular conifold with metric
$ds^2=dr^2+r^2 ds^2_{T^{1,1}}$. The $T^{1,1}$ is the conifold
base, i.e. the 5-dimensional space
$T^{1,1}$\cite{Candelas:1989js},
\begin{eqnarray} \label{(T)}\nonumber
ds^2_{(T^{1,1})}=\qquad\qquad\qquad\qquad\qquad\qquad\\=\left(\frac{1}{9}(d\psi+\cos\theta_1
d\phi_1+\cos\theta_2
d\phi_2)^2+\frac{1}{6}(d\theta_1^2+\sin^2\theta_1
d\phi_1^2+d\theta_1^2+ \sin^2\theta_2 d\phi_2^2)\right)\qquad
\end{eqnarray}
where the coordinates
$X^a$ are renamed $r, \psi$, $\phi_1$,
$\phi_2$, $\theta_1$, $\theta_2$. This metric does not depend on $\psi$,
$\phi_1$, $\phi_2$ and the action is invariant under
translations of these angles. The conserved quantities
associated to these symmetries are the angular momenta:
\begin{eqnarray}
\label{momcurv}\nonumber J_i=\int^{2\pi}_{0}d\sigma
P_{\hat a}=T\int^{2\pi}_{0}d\sigma\dot{X^{\hat b}}
G_{\hat a \hat b}^{(T^{1,1})}\qquad
\hat a, \hat b=1,2,3 \\\qquad(X^1=\psi\qquad X^2=\phi_1\qquad
X^3=\phi_2)\qquad
\end{eqnarray}
We now consider solutions of the equations of motion
(\ref{eqmocur1}) describing closed strings, i.e. solutions
satisfying the boundary conditions (\ref{chiuse}).
\subsection{Point like strings}
We write three different solutions describing strings that in the
transverse space are point particle periodically moving.
Considering also the longitudinal part, these are circular
extended strings. For these solutions the energy is obtained by
(\ref{energia}) and the angular momentum by (\ref{momcurv}). All
of them have $r=C$, where $C$ is a constant:
\begin{itemize}
\item One $\tau$-dependent transverse coordinate:
\begin{eqnarray}
\label{punto1}
x^0=nR\tau\qquad x^1=nR\sigma\qquad
\psi=\nu nR\tau \qquad\phi_1=\phi_2= \theta_1=\theta_2=0
\qquad\end{eqnarray}
\begin{eqnarray}
\label{mompunt} J_\psi=J_{\phi_1}=J_{\phi_2}=TnR2\pi\nu
\frac{C^2}{9} \qquad\qquad E=nRT\pi\nu^2\frac{C^2}{9}
\end{eqnarray}
\item Two $\tau$-dependent transverse coordinates:
\begin{eqnarray} \label{punto2}
x^0=nR\tau\qquad x^1=nR\sigma\qquad
\psi=\phi_1=\nu nR\tau \qquad \theta_1=\theta_2=\phi_2=0\qquad
\end{eqnarray}
\begin{eqnarray}
\label{mompunt} J_\psi=J_{\phi_1}=J_{\phi_2}=TnR2\pi\nu
\frac{2}{9}C^2 \qquad\qquad E=nRT\pi\nu^2\frac{4}{9}C^2
\end{eqnarray}
\item Three $\tau$-dependent transverse coordinates:
\begin{eqnarray}
\label{punto3}
x^0=nR\tau\qquad x^1=nR\sigma\qquad
\psi=\phi_1=\phi_2=\nu nR\tau \qquad \theta_1=\theta_2=0\qquad
\end{eqnarray}
\begin{eqnarray}
\label{mompunt} J_\psi=J_{\phi_1}=J_{\phi_2}=TnR2\pi\nu
\frac{C^2}{3}\qquad\qquad E=nRT\pi\nu^2C^2
\end{eqnarray}
\end{itemize}
For the three point like solutions, the relation between energy
and angular momentum is the same, i.e.:
\begin{equation} \label{enpunt}
E=3\frac{J_\psi^2}{4\pi nRTC^2}+3\frac{J_{\phi_1}^2}{4\pi
nRTC^2}+3\frac{J_{\phi_2}^2}{4\pi nRTC^2}.
\end{equation}
For all these point like solutions the momenta ${\cal P}_1=0$.
\subsection{Extended string}
We now consider a string that is extended also in the transverse space and like
\bref{semhoppe} is moving along its own extension:
\begin{eqnarray}\label{3.10like}
x^0=nR\tau\qquad x^1=nR\sigma\qquad\phi_1=\nu
nR\tau-\omega\sigma\qquad\psi=\phi_2=\theta_1=\theta_2=0\qquad
\end{eqnarray}
where $\omega\in\mathbb{N}$. This is a closed
string because
$\phi_1\in(0,2\pi)$.
>From (\ref{momcurv}) we obtain:
\begin{eqnarray}
J_\psi=J_{\phi_1}=J_{\phi_2}=TnR2\pi\nu \frac{C^2}{9}.
\end{eqnarray}
Using (\ref{energia}) we have:
\begin{eqnarray}\label{en3.10like}
E=nRT\pi\nu^2\frac{C^2}{9}+\pi\frac{C^2}{9}\frac{T}{nR}\omega^2
\end{eqnarray}
and finally we find
\begin{equation} \label{enext1}
E=3\frac{J_\psi^2}{4\pi nRTC^2}+3\frac{J_{\phi_1}^2}{4\pi
nRTC^2}+3\frac{J_{\phi_2}^2}{4\pi
nRTC^2}+\pi\frac{C^2}{9}\frac{T}{nR}\omega^2.\qquad\end{equation}
For this solution \begin{equation}\label{P3.10like} {\cal
P}_1=T\nu\omega2\pi\frac{C^2}{9}.\end{equation}
The BPS equation \bref{bpseq} is satisfied when $\nu=\frac{\omega}{nR}$; in fact, in this case from
\bref{en3.10like} and \bref{P3.10like} we have $E={\cal P}_1$.
\section{Supersymmetry properties of the solutions}
In \cite{Gomis:2004pw} it has been considered a
supersymmetric extension of the bosonic
string \bref{azionephi} in flat space time.
The NR superstring has diffeomorphism and kappa invariance in 10 dimensions.
The action in the static gauge and with half of the fermions
set to zero has the following form
\begin{equation}\label{susyaction}
S= -T\;\int d^2\sigma \left[ \frac{1}2\;\eta} \newcommand{\Th}{\Theta^{ij} \partial} \def\lrpa{\partial^\leftrightarrow_i
X \cdot \partial} \def\lrpa{\partial^\leftrightarrow_j X \;+\;2
i\; {\overline\theta}_+\Gamma^i\partial} \def\lrpa{\partial^\leftrightarrow_i\theta} \newcommand{\vth}{\vartheta_+\right]
\label{Lag22gf2}
\end{equation}
where $\theta_+$ is a $+1$ eigenspinor of $\Gamma_*=
\Gamma_0\Gamma_1\Gamma_{11}$ with 16 components and
$\bar\theta_+$ is the conjugate
spinor.
Note that this lagrangian }%\def\lag{Lagrangian is quadratic in the bosonic and
fermionic variables.
The action \bref{Lag22gf2} is invariant under the supersymmetry
transformations
\begin{equation}
\D\theta} \newcommand{\vth}{\vartheta_+= \epsilon} \newcommand{\vep}{\varepsilon_+\;+\;\frac{1}{2}\Gamma_\mu\partial_i x^\mu\partial} \def\lrpa{\partial^\leftrightarrow^i X^a
\Gamma_a \epsilon} \newcommand{\vep}{\varepsilon_-,\quad \quad
\D X^a\;=\;2i{\overline\theta}_+\Gamma^a\epsilon} \newcommand{\vep}{\varepsilon_-
\label{ressusyst}
\end{equation}
where $\epsilon} \newcommand{\vep}{\varepsilon_+$ and $\epsilon} \newcommand{\vep}{\varepsilon_-$ are 16 components constant spinors.
The bosonic supersymmetric configurations of the action \bref{susyaction}
should verify
\begin{equation}
\epsilon} \newcommand{\vep}{\varepsilon_+\;=\;-\frac{1}{2} \Gamma_\mu\partial_i x^\mu\partial} \def\lrpa{\partial^\leftrightarrow^i x^a\Gamma_a \epsilon} \newcommand{\vep}{\varepsilon_-,
\label{susycond}
\end{equation}
with $ \epsilon} \newcommand{\vep}{\varepsilon_\pm$ constants spinors. When
this relation is verified for some non-vanishing
components of $\epsilon} \newcommand{\vep}{\varepsilon_\pm$ the configuration preserves some of the 32
supersymmetries of the action \bref{Lag22gf2}.
The vacuum configuration of the string
\begin{equation}\label{vacuum}
x^0=K\tau, \quad \quad x^1=K\sigma, \quad \quad \;X^a= {\rm constant}
\end{equation}
is supersymmetric if $\epsilon} \newcommand{\vep}{\varepsilon_+\;=\;0$ which implies
\begin{equation}
\Gamma^0\Gamma^1\Gamma_{11} \epsilon} \newcommand{\vep}{\varepsilon=\epsilon} \newcommand{\vep}{\varepsilon.
\label{gammacond2}
\end{equation}
Therefore the string is a $\frac12$ BPS configuration.
Note that the vacuum solution \bref{vacuum} is a solution of both
the relativistic and NR string.
There are other possible bosonic supersymmetric configurations
with non-constant transverse coordinates. If we write
$X^a=X^a(\tau,\sigma)$, we can have a solution of \bref{susycond} if
\begin{equation}\label{susyconn}
\partial_\tau X^b=\pm\partial_\sigma X^b,\quad\quad {\rm and}\quad\quad
\Gamma_0\epsilon} \newcommand{\vep}{\varepsilon_-=\pm\Gamma_1\epsilon} \newcommand{\vep}{\varepsilon_-.
\label{quarters2}\end{equation}
Together with the condition $\epsilon} \newcommand{\vep}{\varepsilon_+=0$ the susy parameter $\epsilon} \newcommand{\vep}{\varepsilon$
should satisfy \begin{equation} \mp\Gamma^0\Gamma^1\epsilon} \newcommand{\vep}{\varepsilon=\epsilon} \newcommand{\vep}{\varepsilon. \label{quarters}
\end{equation}
This configuration is a ${1\over4}$ BPS configuration. It
represents a wave propagating, with the velocity of light, along a
string with arbitrary profile in the transverse directions.
The supersymmetric (BPS) condition \bref{susyconn} was obtained previously as
a condition to saturate the energy bound \bref{bound}.
Now we can study the symmetry properties of the solutions we have
found. The solutions \bref{soluzione1} and \bref{spike} are not
supersymmetric because the condition \bref{susycond} can not be
verified. Instead the solution \bref{susysol} verifies the
supersymmetry condition if the spinors verify $\epsilon} \newcommand{\vep}{\varepsilon_+\;=\;0$ and
$\Gamma_0\epsilon} \newcommand{\vep}{\varepsilon_-=\pm\Gamma_1\epsilon} \newcommand{\vep}{\varepsilon_-$. It is therefore a ${1\over4}$
BPS configuration. For the solutions in flat space time that we
have considered, we observe that if the BPS energy bound \bref{bound} is verified the
solution is supersymmetric.
In the case of a transverse curved manifold there is not yet a
general discussion about the supersymmetry properties of a
NR superstring in these backgrounds. Therefore for
the solutions we have found in transverse curved background
we make the ansatz that the solutions
that saturate the energy bound \bref{bound} will be supersymmetric.
\section{Conclusions}
In this paper we have considered NR string theory in curved
transverse space. Our starting point has been the Nambu-Goto action for
NR strings
in a curved transverse space \bref{azionephi}.
We have then rewritten the action in a
Polyakov like form \bref{polyakov}
and we have derived a BPS bound for the energy in the static gauge.
We have seen that the theory is
conformal at quantum level if the transverse space is Ricci flat.
For this reason we have considered the case of a singular conifold
\cite{Candelas:1989js}.
We also have found classical rotating solutions of NR
strings. The solutions have
typical features of a non relativistic theory. In
particular, the velocity of the strings has not any limit
\bref{speed1},\bref{spikevel},\bref{speed2} and the energy is proportional to
squared angular momentum
\bref{rels},\bref{spikeene},\bref{rel},\bref{rela},\bref{enpunt},\bref{enext1}.
For the flat space case, the BPS equation is obtained as condition to saturate
the energy bound \bref{bpseq} or as a supersymmetry condition
\bref{susyconn}.
Some of the solutions are ${1\over4}$ BPS supersymmetric and represent
a wave with the velocity of light propagating along the string.
These NR string have a energy spectrum of non-relativistic theories
but keep relativistic properties along the longitudinal directions.
\section{Acknowledgements}
We are grateful to Roberto Emparan, Jaume Gomis, Kiyoshi Kamimura,
Tomas Ort\'{\i}n, Josep Maria Pons,
Paul Townsend and Toine Van Proeyen for useful discussions and
comments.
This work is supported in part by the European Community's Human
Potential Programme under contract MRTN-CT-2004-005104 `Constituents,
fundamental forces and symmetries of the universe'.
The work of F.P. is supported in part by the Federal Office for
Scientific, Technical and Cultural Affairs through the "Interuniversity
Attraction Poles Programme -- Belgian Science Policy" P5/27.
|
{
"timestamp": "2005-07-28T14:54:40",
"yymm": "0411",
"arxiv_id": "hep-th/0411195",
"language": "en",
"url": "https://arxiv.org/abs/hep-th/0411195"
}
|
\section{Introduction and motivation}
\label{sec:int}
The $K^+N$ interaction below the pion-production threshold is fairly weak and
featureless, as anticipated from an `exotic' channel corresponding to quark
content $qqqq{\bar s}$, where $q$ denotes a light nonstrange quark.
This merit has motivated past suggestions to probe nuclear in-medium effects
by studying scattering and reaction processes with $K^+$ beams below 800
MeV/c; see Ref.\cite{DWa82} for an early review.
Limited total cross-section data \cite{BGK68} on carbon, and elastic
and inelastic differential cross section data \cite{MBC82} on carbon and
calcium, drew theoretical attention already in the 1980s to the insufficiency
of the impulse-approximation $t\rho$ form of the $K^+$ - nucleus optical
potential, where $t$ is the free-space $K^+N$ $t$ matrix, particularly
with respect to its reaction content (`reactivity' below). In order to
account for the increased reactivity in $K^+$ - nucleus interactions,
Siegel {\it et al.}\cite{SKG85} suggested that nucleons `swell' in the
nuclear medium, primarily by increasing the dominant hard-core $S_{11}$
phase shift. Brown {\it et al.}\cite{BDS88} suggested that the extra
reactivity was due to the reduced in-medium masses of exchanged vector
mesons, and this was subsequently worked out in detail in Ref.\cite{CLa96}.
Another source for increased reactivity in $K^+$ - nucleus interactions
was discussed in the 1990s and is due to meson exchange-current effects
\cite{JKo92,GNO95}.
Some further experimental progress was made during the early 1990s,
consisting mostly of measuring attenuation cross sections in $K^+$
transmission experiments at the BNL-AGS on deuterium and several other
nuclear targets in the momentum range $p_{\rm lab} = 450 - 740$ MeV/c
\cite{KAA92,SWA93,WAA94,FGW97} and of measuring $K^+$ quasielastic
scattering on several targets at 705 MeV/c \cite{KPS95}. New measurements
of $K^+$ elastic and inelastic differential cross sections on C and Ca at
715 MeV/c were reported in Ref.\cite{MBB96} and analyzed in Ref.\cite{CSP97},
and self-consistent final values of $K^+$ integral (reaction and total) cross
sections on $^6$Li, $^{12}$C, $^{28}$Si and $^{40}$Ca were published in
Refs.\cite{FGM97a,FGM97b}. By the late 1990s, experimentation in
$K^+$ - nuclear physics has subsided, and with it died out also theoretical
interest. The subject was reviewed last time in HYP97,
concluding that ``every experiment of $K^+$ mesons with complex nuclei finds
cross sections larger than those predicted" \cite{Pet98}; and based on
analyses of $K^+$ - nuclear integral cross sections \cite{FGM97a,FGM97b},
it was concluded that ``at present theory misses some unconventional in-medium
effects" \cite{Gal98}.
The $\Theta^+$(1540) exotic baryon \cite{PDG04} provides a new mode of
reactivity to $K^+$ - nuclear interactions.
The $\Theta^+$ couples directly to nucleons,
$\Theta^+ \rightarrow K^+n,K^0p$, but its small width of order 1 MeV or
less \cite{CTr04,Gib04,SHK04} indicates that this coupling is weak. The
$K^+n$ phase-shift input to the $t\rho$ optical potential in the vicinity
of the $\Theta^+$ mass (corresponding to $p_{\rm lab} \sim 440$ MeV/c)
is derived from $K^+d$ scattering data across the $\Theta^+$ resonance
energy. Therefore, whatever the $KN\Theta$ coupling is, its effect is
already included at least implicitly in the $t\rho$ impulse approximation.
A deuteron target also allows for a two-body production reaction
$K^+d \rightarrow \Theta^+p$ with threshold at $p_{\rm lab} \sim 400$
MeV/c \cite{Ash04}, but the available $K^+d$ scattering data do not give
any evidence for the opening of such a production channel.
It is plausible that {\it denser} nuclear targets would help
to fuse the $K^+$'s $\bar s$ quark together with two
$(ud)_{S=0,I=0}$ diquarks, each of which belongs to a distinct nucleon,
into the $(ud)(ud)\bar s$ configuration according to the Jaffe-Wilczek
model of $\Theta^+$ \cite{JWi03}, and similarly for the diquark-triquark
Karliner-Lipkin model \cite{KLi03}. Therefore, one should look for traces
of $K^+$ {\it absorption} on two nucleons, $K^+nN \rightarrow \Theta^+N$,
in normal nuclei. In the present work we show that traces of such two-body
$K^+$ absorption may be identified in $K^+$ - nucleus dynamics and help
resolve the long-standing anomalies in the $K^+$ - nucleus integral cross
sections. We determine the $K^+$ absorption cross section on nuclei at
$p_{\rm lab}=488$ MeV/c which is the closest momentum to the $\Theta^+$
rest mass, where good data are available.
Our results provide the first concrete demonstration that nuclear targets
are potentially useful in the study of exotic baryons.
\section{Methodology, Results and Discussion}
\label{sec:res}
In the calculations presented below the Klein Gordon equation is solved,
using the simplest possible $t\rho$ form for the optical potential
\begin{equation}
\label{equ:Vopt}
2 \varepsilon^{(A)}_{\rm red} V_{\rm opt}(r) = -4\pi F_A b_0 \rho(r) ~~,
\end{equation}
where $\varepsilon^{(A)}_{\rm red}$ is the reduced energy in the cm
system, $F_A$ is a kinematical factor resulting from the transformation
of amplitudes between the $KN$ and the $K^+$ - nucleus cm systems and
$b_0$ is the (complex) value of the isospin-averaged $KN$ scattering
amplitude in the forward direction. The Coulomb potential due to the
charge distribution of the nucleus is included.
This form of the potential takes into account $1/A$ corrections,
an important issue when handling as light a nucleus as $^6$Li.
Using this approach Friedman {\it et al.}\cite{FGM97a} showed
that no {\it effective} value for $b_0$ could be found that fits
satisfactorily the reaction and total cross sections derived from the
BNL-AGS transmission measurements at $p_{\rm lab} = 488, 531, 656, 714$
MeV/c on $^6$Li, $^{12}$C, $^{28}$Si, $^{40}$Ca. This is demonstrated
in the upper part of Fig. \ref{kplusfig1} for the reaction cross sections
per nucleon $\sigma_R/A$ at 488 MeV/c, where the calculated cross sections
using a best-fit $t\rho$ optical potential (dashed line) are compared
with the experimental values listed in Ref. \cite{FGM97b}.
The best-fit values of
Re$b_0$ and Im$b_0$ which specify this $t\rho$ potential are given in the
first row of Table \ref{tab:FGa04} where Im$b_0$ represents $10-15\%$
increase with respect to the free-space value. The $\chi ^2$/N of this
density-independent fit is very high. Its failure is due to the
impossibility to reconcile the $^6$Li data (which for the total cross
sections are consistent with the $K^+d$ `elementary' cross sections)
with the data on the other, denser nuclei, as is clearly exhibited in
Fig. \ref{kplusfig1} for the best-fit $t\rho$ dashed line.
If $^6$Li were removed out of the data base, then it would have become
possible to fit reasonably well the rest of the nuclei, but the rise
in Im$b_0$ would then be substantially higher than that for the $t\rho$
potential used here. Such fits, excluding $^6$Li, are less successful at
the higher energies.
\begin{figure}[t]
\centerline{\includegraphics[height=6.8cm]{kplusfig1.eps}}
\caption{Data and calculations for $K^+$ reaction cross sections per
nucleon ($\sigma_R/A$) at $p_{\rm lab}=488$ MeV/c are shown in the
upper part. Calculated $K^+$ absorption cross sections per nucleon
($\sigma_{\rm abs}/A$) are shown in the lower part, see text.}
\label{kplusfig1}
\end{figure}
To incorporate $K^+nN \rightarrow \Theta^+N$ two-nucleon absorption
into the impulse-approximation motivated $V_{\rm opt}(r)$,
Eq.(\ref{equ:Vopt}), we add a $\rho^2 (r)$ piece, as successfully
practised in pionic atoms to account for $\pi^-$ absorption on two
nucleons:
\begin{equation}
\label{equ:DD1}
b_0~ \rho (r) \rightarrow b_0~ \rho (r)~+~B~ \rho^2 (r)~,
\end{equation}
where the parameter $B$ reperesents the effect of $K^+$ nuclear absorption
into exotic $S=+1$ baryonic channels. Using this potential we have repeated
fits to all 32 data points for the reaction and total cross sections.
As there were correlations between the parameters of Re$V_{\rm opt}$
we subsequently kept the parameter Re$b_0$ fixed at its free kaon-nucleon
value. The results are summarized in Table \ref{tab:FGa04} (marked as
`Eq.(\ref{equ:DD1})') and the substantial improvement compared to the
$t\rho$ potential is self evident. However, the fits at the higher
momenta are not as successful as the fit at 488 MeV/c, suggesting that
one needs a more effective way to distinguish between $^6$Li and the denser
nuclear targets. In fact, it was shown {\it empirically} \cite{FGM97a,FGM97b}
that the average nuclear density
${\bar \rho}=\frac{1}{A}\int\rho^2d{\bf r}$ provides such discrimination
and is instrumental in achieving good agreement with experiment. Therefore,
we replace Eq.(\ref{equ:DD1}) by the simplest ansatz
\begin{equation}
\label{equ:DD2}
b_0~ \rho (r) \rightarrow b_0~ \rho (r)~+~B~ {\bar \rho}~\rho (r)~.
\end{equation}
The added piece is a functional of the density which to lowest order
reduces to a $\rho^2$ form. To justify it theoretically would require
to understand quantitatively the production mechanisms of the $\Theta^+$
pentaquark in dense nuclear matter. The resulting fits are shown in
Table \ref{tab:FGa04}, marked as `Eq.(\ref{equ:DD2})'.
The superiority of this ${\bar \rho} \rho$ version compared to the
$\rho ^2$ one is very clearly observed.
This conclusion remains valid also when the data for the elastic scattering
of 715 MeV/c $K^+$ by $^6$Li and $^{12}$C \cite{MBB96} are included
in the analysis. The calculated reaction cross sections at 488 MeV/c,
using Eq.(\ref{equ:DD2}), are shown by the solid line marked
$t\rho + \Delta V_{\rm opt}$ in the upper part of Fig. \ref{kplusfig1},
where $\Delta V_{\rm opt}$ is the added piece of $V_{\rm opt}$ due to
a nonzero value of $B$. Clearly, it is a very good fit.
Inspecting the results in Table \ref{tab:FGa04} one notes that
the splitting of Im$V_{\rm opt}$ into its two reactive components Im$b_0$
and Im$B$ appears well determined by the data at all energies, with very
accurate values of Im$b_0$ thus derived. These values of Im$b_0$ are close
to, but somewhat below the corresponding free-space values. The two-nucleon
absorption coefficient Im$B$ rises slowly with energy as appropriate to the
increased phase space available to the underlying two-nucleon absorption
process $K^+nN \rightarrow \Theta^+N$. Its values in this energy range
are roughly independent of the form of $\Delta V_{\rm opt}$, the more
conservative Eq.(\ref{equ:DD1}) or the more effective Eq.(\ref{equ:DD2}),
used to derive these values from the data.
Regarding Re$V_{\rm opt}$, and recalling that
${\bar \rho} \sim 0.1~ {\rm fm}^{-3}$ for the dense nuclear targets,
it is clear that Re$V_{\rm opt} \sim 0$ at the two higher momenta,
illustrating the inadequacy of the $t\rho$ model in which the best-fit
solution flips from repulsion at the two lower momenta (consistently
with the impulse approximation) to attraction of a similar order of
magnitude at the two higher momenta.
We note that our $K^+nN \rightarrow \Theta^+N$ absorption reaction
is related to the mechanism proposed recently in Ref. \cite{CLM04}
as causing strong $\Theta^+$ - nuclear attraction, based on $K\pi$ two-meson
cloud contributions to the self energy of $\Theta^+$ in nuclear matter.
However, it would appear difficult to reconcile as strong
$\Theta^+$ - nuclear attraction as proposed there with the magnitude of
Re$B$ derived in the present work.
\begin{table}
\caption{Fits to the eight $K^+$-nuclear integral cross sections at four
laboratory momenta $p_{\rm lab}$ (in MeV/c), using different potentials.}
\label{tab:FGa04}
\begin{tabular}{ccccccc}
\hline \hline
$p_{\rm lab}$&$V_{\rm opt}$&Re$b_0$(fm)&Im$b_0$(fm)&Re$B$(fm$^4$)&
Im$B$(fm$^4$)&$\chi ^2$/N \\ \hline
488&$t\rho$ &$-$0.205(27)&0.173(7)& & & 18.2 \\
&Eq.(\ref{equ:DD1}) &$-$0.178&0.120(6)&0.80(34)&0.92(8) &1.60 \\
&Eq.(\ref{equ:DD2}) &$-$0.178&0.126(4)&0.19(11)&0.67(6)&0.69 \\
& & & & & & \\
531&$t\rho$ &$-$0.198(41)&0.203(10)& & & 63.4 \\
&Eq.(\ref{equ:DD1}) &$-$0.172&0.157(13)&1.80(35)&0.68(24) &7.38 \\
&Eq.(\ref{equ:DD2}) &$-$0.172&0.144(7)&0.50(28)&0.82(9)&6.06 \\
& & & & & & \\
656&$t\rho$ &0.168(34)&0.250(11)& & & 38.2 \\
&Eq.(\ref{equ:DD1}) &$-$0.165&0.205(17)&1.75(55)&0.85(31) &8.78 \\
&Eq.(\ref{equ:DD2}) &$-$0.165&0.203(5)&2.16(19)&0.78(8)&1.42 \\
& & & & & & \\
714&$t\rho$ &0.176(39)&0.275(13)& & & 51.0 \\
&Eq.(\ref{equ:DD1}) &$-$0.161&0.221(21)&1.57(83)&1.04(40) &11.4 \\
&Eq.(\ref{equ:DD2}) &$-$0.161&0.218(8)&1.75(41)&0.97(12)&2.40 \\
\hline \hline
\end{tabular}
\end{table}
The $K^+$ nuclear absorption cross section $\sigma_{\rm abs}^{(K^+)}$
due to the availability of $\Theta^+$ - nuclear final states is driven
by Im($\Delta V_{\rm opt}$). We approximate it by
\begin{equation}
\label{equ:abs}
\sigma_{\rm abs}^{(K^+)} \sim -~ {\frac{2}{\hbar v}}
\int {\rm Im} (\Delta V_{\rm opt}(r))~
|\Psi^{(+)}({\bf r})|^2~d{\bf r} ~~,
\end{equation}
where the distorted waves $\Psi^{(+)}$ are evaluated
in two different ways in order to assess the theoretical uncertainty.
Calculated absorption cross sections {\it per target nucleon} at
$p_{\rm lab}=488$ MeV/c are shown in the lower part of Fig. \ref{kplusfig1}
for the fit using Eq. (\ref{equ:DD2}) for $V_{\rm opt}$ in Table
\ref{tab:FGa04}. The triangles are for the case where $\Delta V_{\rm opt}$
does not enter the evaluation of the distorted waves $\Psi^{(+)}$,
as appropriate to the DWIA approximation, and the solid circles are
for the case where $\Psi^{(+)}$ are the fully distorted waves, including
the effect of $\Delta V_{\rm opt}(r)$.
The error bars plotted are due to the uncertainty in the
parameter Im$B$. It is seen that these calculated absorption cross sections,
for the relatively dense targets of C, Si and Ca, are proportional to the
mass number $A$, and the cross section per target nucleon due to
Im$B \neq 0$ is estimated as close to 3.5 mb. Although the less successful
Eq.(\ref{equ:DD1}) gives cross sections larger by $40\%$, this value
should be regarded an upper limit, since the best-fit density-dependent
potentials of Refs.\cite{FGM97a,FGM97b} yield values smaller than 3.5 mb
by a similar amount.
The experience gained from studying $\pi$-nuclear absorption \cite{ASc86}
leads to the conclusion that $\sigma_{\rm abs}(K^+NN)$ is smaller than the
extrapolation of $\sigma_{\rm abs}^{(K^+)}/A$ in Fig. \ref{kplusfig1} to
$A=1$, and since the $KN$ interaction is weaker than the $\pi N$ interaction
one expects a reduction of roughly $50\%$, so that
$\sigma_{\rm abs}(K^+NN) \sim 1 - 2$ mb.
We note in Fig.\ref{kplusfig1} the considerably smaller absorption cross
section per nucleon calculated for $^6$Li which, considering its low density,
suggests a cross section of order fraction of millibarn for
$K^+ d \rightarrow \Theta^+ p$, well below the order 1 mb which as Gibbs has
argued recently \cite{Gib04} could indicate traces of $\Theta^+$ in $K^+ d$
total cross sections near $p_{\rm lab} \sim 440$ MeV/c. To be definite,
we assume that $\sigma(K^+ d \rightarrow \Theta^+ p) \sim 0.1~-~0.5$ mb.
It is worth noting that this {\it two-nucleon} cross section is considerably
larger than what a {\it one-nucleon} production process
$K^+ n \rightarrow \Theta^+$ would induce on a deuteron target.
Such a one-step process at the $\Theta^+$ mass ($p_{\rm lab}=440$ MeV/c)
\begin{equation}
\label{equ:Theta}
K^+ ~n \rightarrow \Theta^+ ~, ~~~ \Theta^+ ~N \rightarrow \Theta^+ ~N
\end{equation}
may be compared to pion absorption near the (3,3) resonance energy
\cite{ASc86}, where the primary two-nucleon absorption mechanism is
through a direct $\Delta$ production
\begin{equation}
\label{equ:Delta}
\pi^+ ~p \rightarrow \Delta^{++}~, ~~~ \Delta^{++}~n \rightarrow p~p ~,
\end{equation}
with $\sigma(\pi^+ d \rightarrow pp) \sim 12.5$ mb.
Scaling this value by the ratio of coupling constants squared
$g^2_{KN\Theta}/g^2_{\pi N \Delta} \sim 2.5 \times 10^{-3}$, assuming
$J^{\pi}(\Theta^+) = {(\frac{1}{2})}^+$ and
$\Gamma (\Theta^+ \rightarrow K N) \sim 1$ MeV, we estimate a cross section
level of 0.03 mb for the one-step production process at the $\Theta^+$
resonance energy. [Assuming $J^{\pi}(\Theta^+) = {(\frac{1}{2})}^-$, the
one-step production cross section is lower by at least another order of
magnitude.] The one-step cross section affordable by the neutron Fermi
motion at $p_{\rm lab}=488$ MeV/c would be considerably smaller than this
estimate \cite{Lip04}. In contrast, the two-nucleon reaction need not
involve the suppressed $KN\Theta$ coupling and its cross section,
$\sigma(K^+ d \rightarrow \Theta^+ p) \sim 0.1~-~0.5$ mb at
$p_{\rm lab}=488$ MeV/c, should depend smoothly on energy.
\section{Conclusions}
\label{sec:conc}
In conclusion, we have demonstrated a very
good agreement between experiment and calculation for all the available
integral cross-section data by adding to the $t\rho$ optical potential
a density-dependent term which simulates absorption channels.
We have identified these absorption channels, as exhibited by the
anomalous reactivity established systematically in past phenomenological
analyses of $K^+$ - nuclear interactions, with the $\Theta^+$ production
reaction $K^+ nN \rightarrow \Theta^+ N$ with threshold at
$p_{\rm lab} \sim 400$ MeV/c. The analysis of these
data is consistent with an upper limit of about 3.5 mb on the $K^+$
absorption cross section per nucleon, for $\Theta^+$ production on the
denser nuclei of C, Si, Ca, and indicates a sub-millibarn cross section
for $\Theta^+$ production on deuterium.
We urge experimenters to look for the $K^+ d \rightarrow \Theta^+ p$
two-body production reaction \cite{Ash04} which requires experimental
accuracies of 0.1 mb in cross section measurements. Given the magnitude
of the $K^+$ nuclear absorption cross sections, as derived in the present
work, we also urge doing ($K^+,p$) experiments on nuclear targets. This
reaction which has a `magic momentum' about $p_{\rm lab} \sim 600$ MeV/c,
where the $\Theta^+$ is produced at rest, is particularly suited to study
bound or continuum states in {\it hyponuclei} \cite{Gol82}. It might prove
more useful than the large momentum transfer ($K^+,\pi^+$) reaction
proposed in this context \cite{NHO04}.
Undoubtedly, precise $K^+$ - nuclear scattering and reaction data would be
extremely useful to obtain further, more direct evidence for the presence
of the $\Theta^+$ exotic baryon and its effects in the nuclear medium.
In particular, $K^+d$ and $K^+$ - nuclear data in the range
$p_{\rm lab} \sim 300-500$ MeV/c would be very helpful to study the onset
of strange-pentaquark dynamics in the nuclear medium.
\begin{acknowledgments}
This work was supported in part by the Israel Science Foundation grant 131/01.
\end{acknowledgments}
|
{
"timestamp": "2005-01-21T13:14:03",
"yymm": "0411",
"arxiv_id": "nucl-th/0411052",
"language": "en",
"url": "https://arxiv.org/abs/nucl-th/0411052"
}
|
\section{INTRODUCTION}
The measured polarization fractions in the $B\to\phi K^*$ decays
have exhibited an anomaly. Let $R_{L,\parallel,\perp}$ denote the
longitudinal, parallel, and perpendicular polarization fractions
of a $B\to VV$ mode, respectively. It is well known from a
helicity argument that these fractions for light vector mesons $V$
follow the naive counting rules,
\begin{eqnarray}
R_L\sim 1-O(m_V^2/m_B^2)\;,\;\;\;\;R_\parallel \sim R_\perp\sim
O(m_V^2/m_B^2)\;, \label{nai}
\end{eqnarray}
if the emission topology of diagrams dominates, where $m_B$
($m_V$) is the $B$ ($V$) meson mass. That is, Eq.~(\ref{nai})
holds for tree-dominated modes, such as $B \to \rho \rho$, whose
longitudinal polarization fraction has been observed to be
$R_L\sim 1$ \cite{BelleRhopRho0,BaBarRhopRho0}. For
penguin-dominated modes, Eq.~(\ref{nai}) could be modified by
annihilation contributions to some extent \cite{CKL2,Li04}.
However, the $B \to \phi K^{*}$ polarization fractions shown in
Table~\ref{tab:tab1} were found to dramatically differ from the
naive counting rules, and have been considered as a puzzle.
\begin{table}[ht]
\begin{center}
\begin{tabular}{c c c c}\hline \hline
Mode&Pol. Fraction&Belle&Babar \\
\hline
$B^+\to\phi K^{*+}$&$R_L$&$0.49\pm 0.13\pm 0.05$ \cite{Zhang04}&$0.46\pm0.12\pm0.03$ \cite{BaBarRhopRho0}\\
&$R_\perp$&$0.12^{+0.11}_{-0.08}\pm 0.03$ \cite{Zhang04}&\\
$B^0\to\phi K^{*0}$&$R_L$&$0.52\pm 0.07\pm 0.05$ \cite{Zhang04}
&$0.52\pm0.05\pm0.02$ \cite{Bar017}\\
&$R_\perp$&$0.30\pm 0.07\pm 0.03$ \cite{Zhang04
&$0.22\pm0.05\pm0.02$ \cite{Bar017}\\
\hline \hline
Mode&Pol. Fraction&Belle&Babar\\
\hline
$B^+\to\rho^0 K^{*+}$&$R_L$&&$0.96^{+0.04}_{-0.15}\pm0.04$ \cite{BaBarRhopRho0}\\
$B^+\to\rho^+ K^{*0}$&$R_L$&$0.50\pm 0.19^{+0.05}_{-0.07}$ \cite{Bel102}&$0.79\pm 0.08\pm0.04\pm 0.02$ \cite{Bar093}\\
\hline \hline
\end{tabular}
\end{center}
\caption{Polarization fractions in the penguin-dominated $B\to VV$
decays. }\label{tab:tab1}
\end{table}
In this work we shall investigate all the $B\to VV$ polarization
data carefully, and understand more the above puzzle. We show that
the $B\to VV$ modes can be classified into four categories: the
first category can be easily understood by means of kinematics in
the heavy-quark limit. Take the $B^0\to (D_s^{*+}, D^{*+},
\rho^+)D^{*-}$ modes as examples. Under the naive factorization
assumption (FA) \cite{BSW}, QCD dynamics in different helicity
amplitudes is absorbed into the universal Isgur-Wise (IW) function
\cite{IW}. The polarization fractions are then completely
determined by the kinematic factors, leading to the observation
that $R_L$ increases from 0.5 to 0.9, when the vector mesons from
the weak vertex change through $D_s^{*}$, $D^{*}$ and $\rho$. We
shall examine subleading effects by deriving the perturbative QCD
(PQCD) \cite{KLS,LUY} factorization formulas for the $B\to
D_{(s)}^*D^*$ decays up to next-to-leading power in
$m_{D_{(s)}^*}/m_B$, $m_{D_{(s)}^*}$ being the $D_{(s)}^*$ meson
mass. To this level of accuracy, the various form factors deviate
from the IW function, and the nonfactorizable contributions
appear. It will be demonstrated that the simple kinematic
estimates are robust under these subleading corrections. As a
byproduct, we observe that $R_L\sim 1$ for $B^0\to {\bar D}^{*0}
D^{*0}$, governed by nonfactorizable $W$-exchange amplitudes,
differs from $R_L\sim 0.5$ for other $B\to D^* D^*$ modes. This
PQCD prediction can be confronted with data in the future.
The second category is understandable via kinematics in the
large-energy limit, which consists of tree-dominated $B$ meson
decays into light vector mesons, such as $B\to (\rho,
\omega)\rho$. Their helicity amplitudes can be expressed in terms
of various heavy-to-light transition form factors under FA, which
are related to each other by the large-energy symmetry relations.
It turns out that only two universal form factors associated with
the longitudinally and transversely polarized final states are
relevant. If these two form factors do not differ much, the
polarization fractions will be completely determined by the
kinematic factors, leading to $R_L\sim 1$ consistent with
Eq.~(\ref{nai}). The same argument applies to the $B^+\to
(D_s^{*+}, D^{*+})\rho^0$ decays with the $D_s^{*+}, D^{*+}$
mesons being emitted from the weak vertex, whose $R_L\sim 0.7$ are
predicted, and can be compared with the future data. We then
examine subleading corrections to the large-energy symmetry
relations by including the two-parton twist-4 contribution. Adding
this piece to the longitudinal polarization amplitude makes
complete the next-to-leading-power analysis at two-parton level,
since the transverse polarization amplitudes are of
next-to-leading power by themselves. It will be demonstrated that
this subleading effect is also negligible.
The third category contains penguin-dominated modes, such as
$B\to\rho (\omega) K^*$, which can exhibit a sizable deviation
from the naive counting rules in Eq.~(\ref{nai}). As shown in
\cite{CKL2}, the annihilation amplitudes associated with the
$(S-P)(S+P)$ operators follow the counting rules,
\begin{eqnarray}
R_L\sim R_\parallel \sim R_\perp\;. \label{mod}
\end{eqnarray}
The PQCD analysis has indicated that the penguin annihilation
contribution, together with nonfactorizable corrections, bring the
longitudinal polarization fraction in a pure-penguin mode from
$R_L\sim 0.9$ down to $0.75$ \cite{CKL2}. Therefore, the
$B^+\to\rho^+K^{*0}$ polarization data in Table~\ref{tab:tab1} can
be well accommodated within the Standard Model, showing no
anomaly. The longitudinal polarization fraction of another mode
$B^+\to\rho^0K^{*+}$ remains as $R_L\sim 0.9$ because of the
interference between the penguin amplitude and an additional tree
amplitude. From the viewpoint of PQCD, the $B\to\omega K^*$ decays
do not differ much from $B\to\rho^0 K^*$, and are expected to show
similar $R_L$. This prediction can be tested by the future data.
The fourth category, involving the $B\to\phi K^*$ decays, is the
abnormal one. These decays occur mainly through the penguin
operators, but their $R_L$ are as small as 0.5, much lower than
0.75 expected from PQCD. The mechanism proposed in the literature
to explain the $B\to\phi K^*$ polarization data includes new
physics \cite{G03}, the annihilation contribution \cite{AK} in the
framework based on the QCD-improved factorization (QCDF)
\cite{BBNS}, the charming penguin \cite{BPRS}, the rescattering
effect \cite{CDP,LLNS,CCS}, and the $b\to sg$ transition (the
magnetic penguin) \cite{HN}. We shall comment on these proposals:
the annihilation contribution has to be parameterized in QCDF, and
varying free parameters to fit the data can not be conclusive. The
charming penguin strategy, demanding many free parameters, does
not help understand dynamics. The rescattering effect is based on
a model-dependent analysis \cite{W,Ligeti04}, and constrained by
the $B\to\rho K^{*}$ data. The magnetic penguin is suppressed by
the $G$-parity, and not sufficient to reduce $R_L$ down to 0.5.
Therefore, none of these proposals is satisfactory \cite{Li04}.
However, we are not concluding that the $B\to\phi K^*$
polarization data signal new physics, since the complicated QCD
dynamics in $B\to VV$ modes has not yet been fully explored.
In Sec.~II we study the kinematic effects on the polarizations of
tree-dominated decays using FA in the heavy-quark or large-energy
limit. The next-to-leading-power corrections to the $B\to
D_{(s)}^*D^*$ decays, and the two-parton twist-4 corrections to
the $B\to\rho\rho$ decays are calculated in Sec.~III. We comment
on the proposals for explaining the abnormal $B\to\phi K^*$ data
in Sec.~IV. Section V is the conclusion.
\section{NAIVE FACTORIZATION}
In this section we demonstrate that the polarization fractions of
tree-dominated $B\to VV$ decays can be simply understood by means
of kinematics in the heavy-quark or large-energy limit.
\subsection{Heavy-quark Limit}
We first investigate the polarizations in the decays $B^0\to
(D_s^{*+}, D^{*+}, \rho^+)D^{*-}$ in the heavy quark limit. These
modes are color-allowed with the $D_s^{*+}, D^{*+}, \rho^+$ mesons
emitted from the weak vertex, respectively, and FA is supposed to
work well. Take the $B^0\to D_s^{*+} D^{*-}$ decay as an example.
The $B$ meson momentum $P_1$, the $D^*$ meson momentum $P_2$, and
the $D_s^*$ meson momentum $P_3$ are chosen, in the light-cone
coordinates, as
\begin{eqnarray}
P_1&=&\frac{m_B}{\sqrt{2}}(1,1,0_T)\;,\nonumber\\
P_2&=&\frac{m_B}{\sqrt{2}}(r_2\eta^+,r_2\eta^-,{\bf 0}_T)\;,\nonumber\\
P_3&=&\frac{m_B}{\sqrt{2}}(1-r_2\eta^+,1-r_2\eta^-,{\bf 0}_T)\;,
\label{pal}
\end{eqnarray}
where the factors $\eta^\pm$ are defined by
$\eta^\pm=\eta\pm\sqrt{\eta^2-1}$ with $\eta=v_1\cdot v_2$ being
the velocity transfer, $v_1\equiv P_1/m_B$ and $v_2\equiv
P_2/m_{D^*}$ the $B$ meson and $D^*$ meson velocities,
respectively, and $r_2=m_{D^*}/m_B$ the mass ratio. To extract the
helicity amplitudes, the following parametrization for the
longitudinal polarization vectors is useful:
\begin{eqnarray}
\epsilon_2(L)&=&v_2-\frac{m_{D^*}}{P_2\cdot
n_-}n_-=\frac{1}{\sqrt{2}}(\eta^+,-\eta^-,{\bf 0}_T)\;,\nonumber\\
\epsilon_3(L)&=&v_3-\frac{m_{D_s^*}}{P_3\cdot
n_+}n_+=\frac{1}{\sqrt{2}r_3}\left(-\frac{r_3^2}{1-r_2\eta^-},1-r_2\eta^-,{\bf
0}_T\right)\;,\label{lpo}
\end{eqnarray}
with the $D_s^*$ meson velocity $v_3\equiv P_3/m_{D_s^*}$, the
mass ratio $r_3=m_{D_s^*}/m_B$, and the null vectors
$n_+=(1,0,{\bf 0}_T)$ and $n_-=(0,1,{\bf 0}_T)$, which satisfy the
normalization $\epsilon_2^2(L)=\epsilon_3^2(L)=-1$ and the
orthogonality $\epsilon_2(L)\cdot P_2=\epsilon_3(L)\cdot P_3=0$.
For the transverse polarization vectors, we simply choose
\begin{eqnarray}
\epsilon_2(T) =(0,0,{\bf 1}_T)\;,\;\;\;\; \epsilon_3(T) =(0,0,{\bf
1}_T)\;.\label{tpo}
\end{eqnarray}
The relevant effective weak Hamiltonian is given by
\begin{eqnarray}
{\cal H}_{\rm eff} = {G_F\over\sqrt{2}}\, V_{cb}V_{cs}^*
\Big[C_1(\mu)O_1(\mu)+C_2(\mu)O_2(\mu)\Big]\;,
\end{eqnarray}
where $V$'s are the Cabibbo-Kobayashi-Maskawa (CKM) matrix
elements, $C_1$ and $C_2$ the Wilson coefficients, and
\begin{eqnarray}
O_1= (\bar sb)_{V-A}(\bar cc)_{V-A}\;,\qquad\qquad O_2= (\bar
cb)_{V-A}(\bar sc)_{V-A}\;,
\end{eqnarray}
the four-fermion operators with the definition $(\bar
q_1q_2)_{V-A}\equiv \bar q_1\gamma_\mu(1- \gamma_5)q_2$. The
contributions from $O_1$ and $O_2$ can be combined, and the
resultant coefficient appears as $a_1= C_2 + C_1/ N_c$, $N_c$
being the number of colors.
The $B^0\to D_s^{*+} D^{*-}$ decay amplitude in FA is expressed as
\begin{eqnarray}
{\cal M}^{(\sigma)}&=&\langle D_s^{*+}(P_3,\epsilon_3^*)
D^{*-}(P_2,\epsilon_2^*) | {\cal H}_{\rm eff} | B^0(P_1)
\rangle\;,\nonumber\\
&=& \displaystyle{G_F \over \sqrt{2}}V_{cb}^*V_{cs} a_1 \langle
D_s^{*+}(P_3,\epsilon_3^*)|{\bar c}\gamma_\mu(1-\gamma_5) s|0
\rangle\nonumber\\
& &\times\langle D^{*-}(P_2,\epsilon_2^*)|{\bar
b}\gamma^\mu(1-\gamma_5) c|B^0(P_1) \rangle\;, \label{nfa}
\end{eqnarray}
where the superscript $\sigma$ denotes a possible final helicity
state. The first matrix element defines the $D_s^*$ meson decay
constant,
\begin{eqnarray}
\langle D_s^{*+}(P_3,\epsilon_3^*)|{\bar c}\gamma_\mu(1-\gamma_5)
s|0 \rangle=m_{D_s^*}f_{D_s^*}\epsilon_{3\mu}^*\;.\label{dcn}
\end{eqnarray}
The matrix elements for the $B\to D^*$ transitions are
parameterized as
\begin{eqnarray}
\langle D^{*-}(P_2,\epsilon_2^*)|{\bar b}\gamma^\mu
c|B^0(P_1)\rangle&=&
i\sqrt{m_Bm_{D^*}}\xi_V(\eta)\epsilon^{\mu\nu\alpha\beta}
\epsilon^*_{2\nu} v_{2\alpha}v_{1\beta}\;,\nonumber\\
\langle D^{*-}(P_2,\epsilon_2^*)|{\bar b}\gamma^\mu\gamma_5
c|B^0(P_1) \rangle&=&\sqrt{m_Bm_{D^*}}
\left[{\xi_{A1}}(\eta)(\eta+1)\epsilon^{*\mu}_2-{\xi_{A2}}(\eta)
\epsilon_2^*\cdot v_1v_1^\mu\right.\nonumber\\
& &\left.-{\xi_{A3}}(\eta)\epsilon_2^*\cdot v_1 v_2^\mu \right]
\;, \label{bds}
\end{eqnarray}
where the form factors $\xi_{A_1}$, $\xi_{A_2}$, $\xi_{A_3}$, and
$\xi_V$ satisfy the relations in the heavy-quark limit,
\begin{equation}
\xi_V=\xi_{A_1}=\xi_{A_3}=\xi,\;\;\;\; \xi_{A_2}=0\;, \label{iwr}
\end{equation}
with $\xi$ being the IW function \cite{IW}.
The $B^0\to D_s^{*+} D^{*-}$ decay rate is given by
\begin{equation}
\Gamma =\frac{P_c}{8\pi m^{2}_{B} }\sum_\sigma
{\cal M}^{(\sigma)\dagger }{\cal M^{(\sigma)}}\;,
\label{dr1}
\end{equation}
where $P_c\equiv |P_{2z}|=|P_{3z}|=m_Br_2\sqrt{\eta^2-1}$
is the momentum of either of the vector mesons. The amplitude
$\cal M^{(\sigma)}$ is decomposed into
\begin{eqnarray}
{\cal M}^{(\sigma)}
=\left(m_{B}^{2}{\cal M}_{L}, m_{B}^{2}{\cal M}_{N}
\epsilon^{*}_{2}(T)\cdot\epsilon^{*}_{3}(T), i{\cal
M}_{T}\epsilon^{\alpha \beta\gamma \rho}
\epsilon^{*}_{2\alpha}\epsilon^{*}_{3\beta} P_{2\gamma }P_{3\rho
}\right)\;,
\end{eqnarray}
where the first term corresponds to the configuration with both
the vector mesons being longitudinally polarized, and the second
(third) term to the two configurations with both the vector mesons
being transversely polarized in the parallel (perpendicular)
directions. The helicity amplitudes are then defined as,
\begin{eqnarray}
A_{L}&=&-G m^{2}_{B}{\cal M}_{L}, \nonumber\\
A_{\parallel}&=&G \sqrt{2}m^{2}_{B}{\cal M}_{N}, \nonumber \\
A_{\perp}&=&G m_{D_s^*} m_{D^*} \sqrt{2[(v_2\cdot v_3)^{2}-1]}
{\cal M }_{T}\;, \label{ase}
\end{eqnarray}
with the normalization factor $G=\sqrt{P_c/(8\pi m^2_{B}\Gamma)}$,
which satisfy the relation,
\begin{eqnarray}
|A_{L}|^2+|A_{\parallel}|^2+|A_{\perp}|^2=1\;.
\end{eqnarray}
It is easy to read the helicity amplitudes off
Eqs.~(\ref{nfa})-(\ref{bds}),
\begin{eqnarray}
A_{L}&\propto&
\epsilon_2^*(L)\cdot \epsilon_3^*(L)(\eta+1)\xi_{A_1}
-\epsilon_2^*(L)\cdot v_1\left[\epsilon_3^*(L)\cdot v_1\xi_{A_2}
+\epsilon_3^*(L)\cdot v_2\xi_{A_3}\right]
\;, \nonumber\\
A_{\parallel}&\propto&- \sqrt{2}
(\eta+1)\xi_{A_1}\;, \nonumber \\
A_{\perp}&\propto&-
r_3\sqrt{2[(v_2\cdot v_3)^{2}-1]}\xi_V\;. \label{ase1}
\end{eqnarray}
In the heavy-quark limit, i.e., applying Eq.~(\ref{iwr}), all the
helicity amplitudes depend on a single IW function $\xi$, which
absorbs QCD dynamics.
Simply inserting $m_B=5.28$ GeV, $m_{D_s^*}=2.11$ GeV, and
$m_{D^*}= 2.01$ GeV into the kinematic factors, we obtain
\begin{eqnarray}
R_L\sim 0.52\;,\;\;\;\; R_{\parallel}\sim 0.43\;,\;\;\;\;
R_{\perp}\sim 0.05\;. \label{arr}
\end{eqnarray}
Equation (\ref{ase1}) is applicable to the $B^0\to D^{*+}D^{*-}$
and $B^0\to \rho^{+}D^{*-}$ decays by substituting $m_{D^*}$ and
$m_\rho=0.77$ GeV for $m_{D_s^*}$, leading to
\begin{eqnarray}
& &R_L\sim 0.54\;,\;\;\;\; R_{\parallel}\sim 0.41\;,\;\;\;\;
R_{\perp}\sim 0.05\;, \label{ard}\\
& &R_L\sim 0.88\;,\;\;\;\; R_{\parallel}\sim 0.10\;,\;\;\;\;
R_{\perp}\sim 0.02\;, \label{ar}
\end{eqnarray}
respectively. All the above results are consistent with the
observed values listed in Table~\ref{tab:tab2}, and with the
predictions in \cite{Ros90}. The estimated polarization fractions
for the decay $B^-\to K^{*-}D^{*0}$ are close to Eq.~(\ref{ar})
and consistent with the data. For the $B^+\to \rho^{+}\bar D^{*0}$
decay, the internal-$W$ emission amplitudes involving the
$B\to\rho$ form factors are suppressed by the vanishing
coefficient $a_2=C_1+C_2/N_c$. Hence, this mode is similar to
$B^0\to \rho^{+} D^{*-}$, and the result in Eq.~(\ref{ar}) applies
as shown in Table~\ref{tab:tab2}. It is easy to find that the
longitudinal polarization fraction increases as the mass of the
vector meson from the weak vertex decreases.
\begin{table}[ht]
\begin{center}
\begin{tabular}{c c c c}\hline \hline
Mod
& Pol. Fraction& Data\\
\hline
$B^0\to D_s^{*+} D^{*-}$&$R_L
&$0.52\pm0.05$ \cite{PDG}\\
$B^0\to D^{*+} D^{*-}$ &$R_L
&$0.57\pm0.08\pm0.02$ \cite{Belich}\\
&$R_\perp$ &$0.19\pm0.08\pm0.01$ \cite{Belich}\\
&$R_\perp$ & $0.063\pm 0.055\pm 0.009$ \cite{PDG,BAR0306}\\
$B^0\to \rho^{+} D^{*-}$&$R_L
&$0.885\pm0.016\pm0.012$ \cite{PDG}\\
$B^+\to \rho^{+} {\bar D}^{*0}$&$R_L
&$0.892\pm0.018\pm0.016$ \cite{PDG}\\
$B^-\to K^{*-} D^{*0}$ &$R_L
&$0.86\pm0.06\pm0.03$ \cite{BAR0308}\\
\hline \hline
Mode& Pol. Fraction &Belle &Babar\\
\hline
$B^+\to\rho^+ \rho^0$&$R_L$&$0.95\pm0.11\pm0.02$ \cite{BelleRhopRho0}&$0.97^{+0.03}_{-0.07}\pm0.04$ \cite{BaBarRhopRho0}\\
$B^0\to\rho^+ \rho^-$&$R_L$&&$0.99 \pm 0.03^{+0.04}_{-0.03}$ \cite{Aubert:2003xc}\\
$B^+\to \rho^+\omega$&$R_L$&&$0.88^{+0.12}_{-0.15}\pm 0.03$ \cite{gritsan}\\
\hline \hline
\end{tabular}
\end{center}
\caption{Polarization fractions in the tree-dominated $B\to VV$
decays.}\label{tab:tab2}
\end{table}
\subsection{Large-energy Limit}
We then show that the polarization fractions in the $B\to
(\rho,\omega)\rho$ decays can be understood by means of kinematics
in the large-energy limit. The parametrizations of the momenta in
Eq.~(\ref{pal}) and of the polarization vectors in
Eqs.~(\ref{lpo}) and (\ref{tpo}) hold here, with the mass ratio
$r_2$ for the vector meson from the $B$ meson transition and $r_3$
for the vector meson emitted from the weak vertex. The transition
form factors associated with a $B\to V$ transition are defined via
the matrix elements,
\begin{eqnarray}
\langle V(P_2,\epsilon_2^\ast)| \bar b \gamma^\mu q |B(P_1)
\rangle& =&
\frac{2iV(q^2)}{m_B+m_V} \epsilon^{\mu\nu\rho\sigma}
\epsilon_{2\nu}^{\ast} P_{2\rho} P_{1\sigma},
\label{V}\\
\langle V(P_2,\epsilon_2^\ast)|\bar b \gamma^\mu\gamma_5 q |
B(P_1) \rangle &=&
2m_VA_0(q^2)\frac{\epsilon_2^\ast\cdot q}{q^2}q^\mu +
(m_B+m_V)A_1(q^2)\left(\epsilon_2^{\ast\mu}-
\frac{\epsilon_2^\ast\cdot q}{q^2}q^\mu\right)
\nonumber\\
&& -A_2(q^2)\frac{\epsilon_2^\ast\cdot q}{m_B+m_V}
\left(P_1^\mu+P_2^{\mu} -\frac{m_B^2-m_V^2}{q^2}q^\mu\right),
\end{eqnarray}
with the momentum $q=P_1-P_2$. The form factors $V$, $A_1$,
and $A_2$ satisfy the symmetry relations in the large-energy
limit,
\begin{eqnarray}
&&\frac{m_B}{m_B+m_V} V(q^2) = \frac{m_B+m_V}{2 E} A_1(q^2)
= \xi_\perp(E)\;,
\label{rho1}\\
& &\frac{m_B+m_V}{2 E}A_1(q^2) - \frac{m_B-m_V}{m_B}A_2(q^2)
=\frac{m_V}{E} \xi_\parallel(E)\;, \label{rho2}
\end{eqnarray}
where $E$ is the $V$ meson energy, and the function
$\xi_\parallel(E)$ ($\xi_\perp(E)$) is associated with the
transition into a longitudinally (transversely) polarized $V$
meson. Our definition of $\xi_\parallel$ differs from those in
\cite{BF,Ch}, such that we have $\xi_\perp\approx \xi_\parallel$,
when not distinguishing the longitudinal and transverse
polarizations.
Consider the $B^+\to \rho^{+}\rho^0$ mode as an example, which is
dominated by a tree contribution (assuming that the electroweak
penguin contribution is negligible). The explicit expression of
the relevant effective weak Hamiltonian will not be shown here. In
FA, the decay amplitude is written as
\begin{eqnarray}
\langle \rho^+ \rho^0 | {\cal H}_{\rm eff}|B^+ \rangle &=&
\displaystyle{G_F \over \sqrt{2}}V_{ub}^*V_{ud} \left[a_1 \langle
\rho^+(P_3,\epsilon_3^*)|{\bar u}\gamma_\mu(1-\gamma_5) d|0
\rangle\langle \rho^0(P_2,\epsilon_2^*)|{\bar
b}\gamma^\mu(1-\gamma_5) u|B^+(P_1)
\rangle\right.\nonumber\\
& &\left.+ a_2 \langle \rho^0(P_2,\epsilon_2^*)|{\bar
u}\gamma_\mu(1-\gamma_5) u|0 \rangle\langle
\rho^+(P_3,\epsilon_3^*)|{\bar b}\gamma^\mu(1-\gamma_5) d|B^+(P_1)
\rangle\right]\;,\label{amr}
\end{eqnarray}
where the two terms can be combined, leading to the helicity
amplitudes,
\begin{eqnarray}
A_{L}&\propto&
(1+r_2)\epsilon_2^*(L)\cdot
\epsilon_3^*(L)\left[A_1 -\frac{2\epsilon_2^*\cdot P_3P_2\cdot
\epsilon^*_3}{m_B^2(1+r_2)^2\epsilon_2^*\cdot
\epsilon_3^*}A_2\right]\;, \nonumber\\
A_{\parallel}&\propto&-\sqrt{2}
(1+r_2)A_1\;, \nonumber \\
A_{\perp}&\propto&-
\frac{2r_2r_3}{1+r_2}\sqrt{2[(v_2\cdot v_3)^{2}-1]} V\;.
\label{ase1r}
\end{eqnarray}
The relative phases among the helicity amplitudes are
$\phi_\parallel=\phi_\perp=\pi$ under FA in our convention. In the
large-energy limit, i.e., employing Eqs.~(\ref{rho1}) and
(\ref{rho2}), Eq.~(\ref{ase1r}) becomes
\begin{eqnarray}
A_{L}&\propto& 2
r_2\epsilon_2^*(L)\cdot\epsilon_3^*(L) \xi_\parallel\;, \nonumber\\
A_{\parallel}&\propto&
-2\sqrt{2}
r_2\eta\,\xi_\perp\;,\nonumber\\
A_{\perp}&\propto&- 2
r_2r_3\sqrt{2[(v_2\cdot v_3)^{2}-1]}\, \xi_\perp\;, \label{ase2r}
\end{eqnarray}
implying that QCD dynamics has been absorbed into the two
functions $\xi_\parallel$ and $\xi_\perp$. For a rough estimate,
adopting $\xi_\parallel\approx (f_\rho/f_\rho^T)\xi_\perp$,
$f_\rho=200$ MeV and $f_\rho^T=160$ MeV \cite{BBKT}, the $B^+\to
\rho^{+}\rho^0$ polarization fractions are given by,
\begin{eqnarray}
R_L\sim 0.95\;,\;\;\;\; R_{\parallel}\sim 0.03\;,\;\;\;\;
R_{\perp}\sim 0.02\;, \label{ase3r}
\end{eqnarray}
consistent with the data in Table~\ref{tab:tab2}. The polarization
fractions of other tree-dominated $B\to VV$ modes, including
$\rho^+\rho^-$ and $\rho^+\omega$, can be explained in a similar
way.
We generalize Eq.~(\ref{ase2r}) to the $B^+\to (D_s^{*+},
D^{*+})\rho^0$ modes, which are mainly governed by the $B\to\rho$
form factors, with
the masses $m_{D_s^{*}}$, $m_{D^{*}}$ being substituted for
$m_\rho$, respectively. Their polarization fractions are predicted
to be,
\begin{eqnarray}
& &D_s^{*+}\rho^0:\;\;\; R_L\sim 0.70\;,\;\;\;\;
R_{\parallel}\sim 0.16\;,\;\;\;\;R_{\perp}\sim 0.14\;,\nonumber\\
& &D^{*+}\rho^0:\;\;\; R_L\sim 0.72\;,\;\;\;\; R_{\parallel}\sim
0.15\;,\;\;\;\;R_{\perp}\sim 0.13\;, \label{asr3}
\end{eqnarray}
which can be compared with the data in the future.
\section{SUBLEADING CORRECTIONS}
Away from the heavy quark limit, the form factors in
Eq.~(\ref{bds}) deviate from the IW function. Beyond FA,
nonfactorizable contributions appear. Both corrections, being
subleading \cite{TLS2,KKLL}, can be calculated more reliably in
the PQCD approach based on $k_T$ factorization theorem
\cite{BS,LS} due to the stronger end-point suppression in the
former \cite{TLS2} and to the soft cancellation between a pair of
nonfactorizable diagrams in the latter \cite{CKL}. In this section
we shall examine whether the simple estimates made in the previous
section are robust under these subleading corrections. The $k_T$
factorization theorem for the $B\to D^{*}$ form factors in the
large-recoil region of the $D^{*}$ meson can be proved following
the procedure in \cite{NL}, which are expressed as the convolution
of hard kernels with the $B$ and $D^{*}$ meson wave functions in
both the momentum fractions $x$ and the transverse momenta $k_T$
of partons. A hard kernel, being infrared-finite, is calculable in
perturbation theory. The $B$ and $D^{*}$ meson wave functions,
collecting the soft dynamics in the decays, are not calculable but
universal. After including the parton $k_T$, the end-point
singularities, which usually break QCDF at subleading level, do
not appear, and PQCD factorization formulas are well-defined. This
formalism has been applied to the semileptonic decay $B\to
D^{(*)}l\nu$ \cite{TLS2}, and the nonleptonic decays $B\to
D^{(*)}\pi(\rho)$ \cite{KKLL,WYL} and $B\to D_s^{(*)}D_s^{(*)}$
\cite{LLX04} successfully.
\subsection{$B^0\to D_s^{*+} D^{*-}$}
\begin{figure}[t]
\centerline{
\includegraphics[width=15cm]{fig_dsd_1.eps}
} \caption{Lowest-order diagrams for the $B^0\to D_s^{*+}D^{*-}$
decay.} \label{fig1}
\end{figure}
For the $B^0\to D_s^{*+} D^{*-}$ mode, we shall neglect the
penguin contribution, which is suppressed by the Wilson
coefficients $C_{3-6}\sim 0.05 a_1$. According to the lowest-order
external-$W$ emission diagrams in Fig.~\ref{fig1}, the decay
amplitudes for different final helicity states are expressed as
\begin{eqnarray}
{\cal
M^{(\sigma)}}&=&\frac{G_F}{\sqrt{2}}V^*_{cb}V_{cs}\left[m_B^2\,
(f_{D_s^*}\, {\cal F}_{L}+{\cal M}_{L}),\; m_B^2
\epsilon^{*}_{2}(T)\cdot\epsilon^{*}_{3}(T)\, (f_{D_s^*}\, {\cal
F}_{N}+{\cal M}_{N}),\right.
\nonumber\\
& &\left. i\, \epsilon^{\alpha \beta\gamma \rho}
\epsilon^{*}_{2\alpha}\epsilon^{*}_{3\beta} P_{2\gamma }P_{3\rho
}\, (f_{D_s^*}\, {\cal F}_{T}+{\cal M}_{T})\right]\;, \label{M1}
\end{eqnarray}
where ${\cal F}_{L,N,T}$ come from the factorizable diagrams,
Figs.~\ref{fig1}(a) and \ref{fig1}(b), and ${\cal M}_{L,N,T}$ from
the nonfactorizable diagrams, Figs.~\ref{fig1}(c) and
\ref{fig1}(d).
We first compute each factorizable amplitude in terms of the
``form factors'' as an expansion in $r_2$,
\begin{eqnarray}
\xi_i&=&\xi
+\xi_i^{({\rm NL})}\;,\;\;i=A_1, A_3, V\;,\nonumber\\
\xi_{A2}&=&\xi_{A2}^{({\rm NL})}\;, \label{xi_form}
\end{eqnarray}
where the superscript NL denotes the next-to-leading-power
corrections. Equation (\ref{xi_form}) is equivalent to the
heavy-quark expansion of the heavy-heavy currents in $1/m_b$ and
in $1/m_c$ \cite{N1}, $m_b$ ($m_c$) being the $b$ ($c$) quark
mass. We refer the explicit expressions of the PQCD factorization
formulas for $\xi$ and $\xi_i^{({\rm NL})}$ to \cite{TLS2}, which
have absorbed the Wilson coefficient $a_1$ in the current
analysis.
In terms of Eq.~(\ref{xi_form}), the factorizable amplitudes are
written as
\begin{eqnarray}
{\cal F}_{L}
&=&
\sqrt{r_2}r_3\,
\left\{
\epsilon_2^*(L)\cdot \epsilon_3^*(L)(\eta+1)\,
\left( \xi + \xi_{A1}^{({\rm NL})}\right)
\right.\nonumber\\
& &\left.\ \ \ \ \ \ \ \ \
- \epsilon_2^*(L)\cdot v_1 \,
\left[ \epsilon_3^*(L)\cdot v_1 \xi_{A2}^{({\rm NL})}
+ \epsilon_3^*(L)\cdot v_2
\left( \xi + \xi_{A3}^{({\rm NL})}\right)
\right]
\right\}\;,
\nonumber\\
{\cal F}_{N}
&=&
\sqrt{r_2}r_3\, (\eta+1) \,
\left( \xi + \xi_{A1}^{({\rm NL})}\right)\;,
\nonumber\\
{\cal F}_{T}
&=&
\frac{r_3}{\sqrt{r_2}}\,
\left( \xi + \xi_V^{({\rm NL})}\right)\;.
\label{xi_sum}
\end{eqnarray}
A numerical analysis gives
\begin{eqnarray}
\xi_{A_1}^{\rm (NL)}\sim -0.02\, \xi\;,\;\;\;\; \xi_{A_2}^{\rm
(NL)}\sim -0.19\, \xi\;,\;\;\;\; \xi_{A_3,V}^{\rm (NL)}\sim
-0.05\, \xi\;. \label{ff1}
\end{eqnarray}
Inserting the deviation from the IW function into
Eq.~(\ref{ase1}), the polarization fractions are modified only
slightly:
\begin{eqnarray}
R_L\sim 0.54\;,\;\;\;\; R_{\parallel}\sim 0.41\;,\;\;\;\;
R_{\perp}\sim 0.05\;. \label{ard1}
\end{eqnarray}
The largest next-to-leading-power correction comes from
$\xi_{A_2}^{\rm (NL)}$, which is, however, suppressed by the
factor $r_2$.
Next we calculate the subleading corrections from the
nonfactorizable diagrams in Figs.~\ref{fig1}(c) and \ref{fig1}(d).
For simplicity, we shall expand all the kinematical factors and
the polarization amplitudes up to next-to-leading power in
$r_{2,3}$.
The factorization formulas are given by
\begin{eqnarray}
{\cal M}_{L}
&=&
16\pi C_F \sqrt{2N_c}\,m_B^2
\int_0^1 dx_1dx_2dx_3
\int_0^\infty b_1db_1\, b_3db_3\,
\phi_B(x_1,b_1)\, \phi_{D^*}(x_2)\, \phi_{D_s^*}(x_3)
\nonumber\\
& &\times\,
\left[\left(x_3-x_1+r_2x_2\right)\,
E_b(t_b^{(1)})\, h_b^{(1)}(x_1,x_2,x_3,b_1,b_3)
\right.\nonumber \\
& & \left. \ \ \ \ \ \ \
- \left( 1+x_2-x_3-x_1-x_2r_2 \right)\,
E_b(t_b^{(2)})\, h_b^{(2)}(x_1,x_2,x_3,b_1,b_3)
\right]
\;, \label{mL}
\\
{\cal M}_{N}
&=&
16\pi C_F \sqrt{2N_c}\,m_B^2
\int_0^1 dx_1dx_2dx_3
\int_0^\infty b_1db_1\, b_3db_3\,
\phi_B(x_1,b_1)\, \phi_{D^*}(x_2)\, \phi_{D_s^*}(x_3)
\nonumber\\
& &\times\,
\left[ x_3\,r_3\,
E_b(t_b^{(1)})\, h_b^{(1)}(x_1,x_2,x_3,b_1,b_3)
\right.\nonumber \\
& & \left. \ \ \ \ \ \ \
- \left( 1+x_3\right)\,r_3\,
E_b(t_b^{(2)})\, h_b^{(2)}(x_1,x_2,x_3,b_1,b_3)
\right]
\;, \label{mN}
\\
{\cal M}_{T}
&=&
2{\cal M}_{N}
\;. \label{mT}
\end{eqnarray}
Radiative corrections to the meson wave functions generate the
double logarithms $\alpha_s\ln^2 k_T$ from the overlap of
collinear and soft enhancements, whose Sudakov resummation has
been studied in \cite{LY1}. The Sudakov factors from $k_T$
resummation for the $B$ meson, the $D^*$ meson and the $D_s^*$
meson are given, according to \cite{LL04}, by
\begin{eqnarray}
\exp[-S_{B}(\mu)]&=&\exp\left[-s(k_1^-,b_1)
-\frac{5}{3}\int_{1/b_1}^\mu
\frac{d{\bar\mu}}{\bar\mu}\gamma(\alpha_s({\bar\mu}))\right]\;,
\nonumber\\
\exp[-S_{D^*}(\mu)]&=&\exp\left[-s(k_2^+,b_2)
-\frac{5}{3}\int_{1/b_2}^\mu
\frac{d{\bar\mu}}{\bar\mu}\gamma(\alpha_s({\bar\mu}))\right]\;,
\nonumber\\
\exp[-S_{D_s^*}(\mu)]&=&\exp\left[-s(k_3^-,b_3)
-\frac{5}{3}\int_{1/b_3}^\mu
\frac{d{\bar\mu}}{\bar\mu}\gamma(\alpha_s({\bar\mu}))\right]\;,
\label{ktd}
\end{eqnarray}
respectively, with the quark anomalous dimension
$\gamma=-\alpha_s/\pi$. The momenta $k_1^-=x_1P_1^-$,
$k_2^+=x_2P_2^+$, and $k_3^-=x_3P_3^-$, carried by the light
valence quarks in the $B$, $D^*$, and $D_s^*$ mesons,
respectively, define the momentum fractions $x$. The impact
parameters $b_1$, $b_2$, and $b_3$ are conjugate to the transverse
momenta carried by the light valence quarks in the $B$, $D^*$, and
$D_s^*$ mesons, respectively. For the explicit expression of the
Sudakov exponent $s$, refer to \cite{KLS}. Note that the
coefficient $5/3$ of the quark anomalous dimension in
Eq.~(\ref{ktd}) differs from 2 for a light meson. The reason is
that the rescaled heavy-quark field adopted in the definition of a
heavy-meson wave function has a self-energy correction different
from that of the full heavy-quark field \cite{LL04}. The evolution
factors are then given by
\begin{eqnarray}
E_b(t) &=& \alpha_s(t)\frac{C_1(t)}{N_c}
\exp\left[-S(t)|_{b_2=b_1}\right]\;,
\end{eqnarray}
with the Sudakov exponent $S=S_B+S_{D^*}+S_{D^*_s}$.
The functions $h^{(j)}_b$, $j=1$ and $2$, are written as
\begin{eqnarray}
h^{(j)}_b
&=&
\left[\theta(b_1-b_3)
K_0\left(Bm_B b_1\right)
I_0\left(Bm_Bb_3\right)
\right. \nonumber \\
& &\quad \left.
+ \theta(b_3-b_1)
K_0\left(Bm_B b_3\right)
I_0\left(Bm_B b_1\right)
\right]
\nonumber \\[1mm]
& & \times
\left(
\begin{array}{cc}
K_{0}\left(B_{j}m_Bb_{3}\right) & \mbox{for $B^2_{j} \geq 0$}
\\[1mm]
\frac{i\pi}{2} H_{0}^{(1)}\left(\sqrt{|B_{j}^2|}m_Bb_{3}\right) &
\mbox{for $B^2_{j} \leq 0$}
\end{array} \right)\;,
\end{eqnarray}
with the variables
\begin{eqnarray}
B^{2} &=& x_1 x_2 r_2\eta^+\;,
\nonumber \\
B_{1}^{2} &=& x_1 x_2 r_2\eta^+
- x_2 x_3 \left(r_2\eta^+ - r_2^2\right)\;,
\nonumber \\
B_{2}^{2} &=& x_1 x_2r_2\eta^+
- x_2 (1-x_3)\left(r_2\eta^+-r_2^2\right)
+ (x_1+x_3)\left(1-r_2\eta^+\right)
+ x_3\left(r_2^2-r_2\eta^-\right)\;.
\label{mis}
\end{eqnarray}
The scales $t_b^{(j)}$ are chosen as
\begin{eqnarray}
t_b^{(j)} &=&
{\rm max}\left(Bm_B,\sqrt{|B_j^2|}m_B,1/b_1,1/b_3\right)\;.
\end{eqnarray}
The $B$ and $D^*$ meson wave functions involved in
Eqs.~(\ref{mL})-(\ref{mT}) are also referred to \cite{TLS2}:
\begin{eqnarray}
\phi_B(x,b)&=&N_Bx^2(1-x)^2
\exp\left[-\frac{1}{2}\left(\frac{xm_B}{\omega_B}\right)^2
-\frac{\omega_B^2 b^2}{2}\right]\;, \label{os}\\
\phi_{D^*}(x)&=&\frac{3f_{D^*}}{\sqrt{2N_c}}x(1-x)
\left[1+C_{D^*}(1-2x)\right]\;.
\end{eqnarray}
The $D_s^*$ meson wave function $\phi_{D_s^*}$ will be assumed to
have the same functional form as $\phi_{D^*}$. We do not consider
the $b$ dependence of $\phi_{D^*}$ in the large recoil region of
the $D^*$ meson \cite{TLS2}. The Gaussian form of $\phi_B$ was
motivated by the oscillator model in \cite{BW}. The shape
parameter $\omega_B=0.40$ GeV comes from \cite{TLS}, and
$C_{D^*}=1.04$ is determined from the IW function $\xi(\eta=1.3)=
0.7$. The normalization constant $N_B$ is related to the decay
constant $f_B$ through
\begin{eqnarray}
\int dx\phi_B(x,b=0)=\frac{f_B}{2\sqrt{2N_c}}\;.
\end{eqnarray}
We shall not distinguish the $D^*$ meson wave functions associated
with the longitudinal and transverse polarizations. There are
various models of the $B$ meson wave functions available in the
literature \cite{LL04}. It has been confirmed that the model in
Eq.~(\ref{os}) and the model derived in \cite{KQT} with a
different functional form lead to similar numerical results for
the $B\to\pi$ form factor \cite{WY}.
Taking into account only the nonfactorizable corrections, the
polarization fractions become
\begin{eqnarray}
R_L\sim 0.54\;,\;\;\;\; R_{\parallel}\sim 0.40\;,\;\;\;\;
R_{\perp}\sim 0.06\;. \label{ardn1}
\end{eqnarray}
Including both the subleading factorizable and nonfactorizable
corrections, we have
\begin{eqnarray}
R_L\sim 0.56\;,\;\;\;\; R_{\parallel}\sim 0.39\;,\;\;\;\;
R_{\perp}\sim 0.06\;, \label{ardt1}
\end{eqnarray}
which are still close to the values in Eq.~(\ref{arr}). Hence, the
simple kinematic estimate made in the heavy-quark limit is very
reliable.
At last, we compute the $B^0\to D_s^{*+}D^{*-}$ branching ratio,
considering only the leading contribution. Employing the CKM
matrix elements $V_{cb}\, =\, 0.0412$, $V_{cs}\, =\, 0.996$, and
$V_{cd}\, =\, -0.224$, the
quark masses $m_b\, =\, 4.8 \; {\rm GeV}$ and $m_t\, =\, 174.3 \;
{\rm GeV}$, the meson decay constants $f_{B}\, =\, 200\; {\rm
MeV}$, $f_{D^*}\,=\, 230\; {\rm MeV}$, and $f_{D_s^*}\, =\, 240
\;{\rm MeV}$,
the lifetimes $\tau_{B^0}\, =\, 1.542\times 10^{-12}\;{\rm sec}$
and $\tau_{B^\pm}\, =\, 1.674\times 10^{-12} \;{\rm sec}$, and the
Fermi constant $G_F\, =\, 1.16639\times 10^{-5}\;{\rm GeV}^{-2}$,
we predict
\begin{eqnarray}
B(B^0\to D_s^{*+} D^{*-})= \left(2.6^{+1.1}_{-0.8}\right) \%
\;,\label{brds}
\end{eqnarray}
which is consistent with the recent measurement $(1.85\pm 0.09\pm
0.16)\%$ \cite{Bar040}. The theoretical uncertainty in
Eq.~(\ref{brds}) arises from the allowed range of the shape
parameter $\omega_B=(0.40\pm 0.04)$ GeV \cite{TLS} in the $B$ meson
wave function. Note that the polarization fractions are
insensitive to this overall source of uncertainty.
\subsection{$B\to D^*D^*$}
\begin{figure}[t]
\centerline{
\includegraphics[width=15cm]{dpdm.eps}
} \caption{$W$-exchange diagrams for the $B^0\to D^{*+}D^{*-}$
decay.} \label{fig_dpdm}
\end{figure}
For the $B^0\to D^{*+}D^{*-}$ mode, there exists an additional
contribution from the $W$-exchange topology shown in
Fig.~\ref{fig_dpdm} compared to $B^0\to D_s^{*+}D^{*-}$. The
factorizable $W$-exchange diagrams, Figs.~\ref{fig_dpdm}(a) and
\ref{fig_dpdm}(b), vanish exactly because of the helicity
suppression. An explicit evaluation shows that the nonfactorizable
$W$-exchange diagrams, Figs.~\ref{fig_dpdm}(c) and
\ref{fig_dpdm}(d), contribute only 2\% of the external-$W$
emission ones in Fig.~\ref{fig1}. Therefore, the $W$-exchange
topology is even less important than the penguin one. Note that
the $W$-exchange contribution is as important as the
nonfactorizable contribution for the $B\to D^{(*)}\pi(\rho)$
decays \cite{KKLL}, to which the helicity suppression does not
apply. The $B^0\to D^{*+}D^{*-}$ factorization formulas are then
similar to those of $B^0\to D_s^{*+}D^{*-}$ but with the
appropriate replacements of the $D_s^*$ meson mass, decay
constant, and wave function by the $D^*$ meson ones, respectively.
Similarly, if neglecting the $W$-exchange and penguin
contributions, the $B^+\to D^{*+}{\bar D}^{*0}$ decay amplitudes
will be the same as of $B^0\to D^{*+}D^{*-}$.
The numerical results are summarized below. Including both the
subleading factorizable and nonfactorizable corrections, we obtain
the polarization fractions of the $B^0\to D^{*+}D^{*-}$ and
$B^+\to D^{*+}{\bar D}^{*0}$ modes,
\begin{eqnarray}
R_L\sim 0.56\;,\;\;\;\;
R_{\parallel}\sim 0.38\;,\;\;\;\;
R_{\perp}\sim 0.06\;,
\label{ard2}
\end{eqnarray}
which are also close to the simple estimate in Eq.~(\ref{ard}).
Using the same input parameters for the quark masses, the $B$
meson lifetimes,..., we predict the branching ratio,
\begin{eqnarray}
B(B^0\to D^{*+} D^{*-})= \left(1.2\,{}^{+0.5}_{-0.3}\right)\times
10^{-3} \;,\label{bdd}
\end{eqnarray}
which is consistent with the updated measurement $B(B^0\to D^{*+}
D^{*-})=(8.1\pm 0.8\pm 0.1)\times 10^{-4}$ \cite{Belich}. The
branching ratio $B(B^+\to D^{*+} D^{*0})$ can be obtained simply
by changing the $B$ meson lifetime, whose value is close to that
in Eq.~(\ref{bdd}).
\begin{figure}[t]
\centerline{
\includegraphics[width=15cm]{d0d0.eps}
} \caption{Lowest-order diagrams for the $B^0\to \bar
D^{*0}D^{*0}$
decay.}
\label{fig_d0d0}
\end{figure}
Since only the $W$-exchange topology in
Figs.~\ref{fig_d0d0}(a)-(d) and the penguin annihilation topology
in Figs.~\ref{fig_d0d0}(e)-(h) contribute, the $B^0\to {\bar
D}^{*0}D^{*0}$ decay must have a tiny branching ratio. For both
topologies, the factorizable amplitudes diminish because of the
helicity suppression (only the penguin annihilation from the
$(S-P)(S+P)$ operators survives, which do not exist in this mode).
Hence, they are not exhibited in Fig.~\ref{fig_d0d0}. With the
penguin amplitudes being down by the Wilson coefficients, we shall
calculate only the tree contribution. By measuring the $B^0\to
{\bar D}^{*0}D^{*0}$ mode, we learn how the nonfactorizable
$W$-exchange contribution governs the polarization fractions. It
will be shown that the polarization fractions in this decay differ
dramatically from those in the $B^0 \to D^{*+}D^{*-}$ and $B^+ \to
D^{*+}\bar D^{*0}$ decays. On the other hand, it has been proposed
\cite{ADL04} to extract the weak phase $\phi_3$ from the $B\to
D^{(*)}D^{(*)}$, $D_s^{(*)}D^{(*)}$ measurements.
The amplitudes from Figs.~\ref{fig_d0d0}(a) and \ref{fig_d0d0}(b)
are given by
\begin{eqnarray}
{\cal M}_{L}
&=&
16\pi C_F \sqrt{2N_c}\,m_B^2
\int_0^1 dx_1dx_2dx_3
\int_0^\infty b_1db_1\, b_2db_2\,
\phi_B(x_1,b_1)\, \phi_{D^*}(x_2)\, \phi_{D^*}(x_3)
\nonumber\\
& &\times\,
\left[ \left( x_3-x_1 \right)\,
E_f(t_f^{(1)})\, h_f^{(1)}(x_1,x_2,x_3,b_1,b_2)
\right.\nonumber \\
& & \left. \ \ \ \ \ \ \
-x_2\,
E_f(t_f^{(2)})\, h_f^{(2)}(x_1,x_2,x_3,b_1,b_2)
\right]\;, \label{excmL}
\\
{\cal M}_{N,T} &=& O\left(r_i^2\right) \;, \label{excmNT}
\end{eqnarray}
and those from Figs.~\ref{fig_d0d0}(c) and \ref{fig_d0d0}(d) are
\begin{eqnarray}
{\cal M}'_L
&=&
16\pi C_F \sqrt{2N_c}\,m_B^2
\int_0^1 dx_1dx_2dx_3
\int_0^\infty b_1db_1\, b_2db_2\,
\phi_B(x_1,b_1)\, \phi_{D^*}(x_2)\, \phi_{D^*}(x_3)
\nonumber\\
& &\times\,
\left[ \left( 1-x_2 \right) \,
E_f(t_{f}^{'(1)})\, h_{f}^{'(1)}(x_1,x_2,x_3,b_1,b_2)
\right.\nonumber \\
& & \left. \ \ \ \ \ \ \
- \left( 1- x_3 + x_1 \right)\,
E_f(t_{f}^{'(2)})\, h_{f}^{'(2)}(x_1,x_2,x_3,b_1,b_2)
\right]
\;, \label{excmL2}
\\
{\cal M}'_{N,T} &=& O\left(r_i^2\right) \;. \label{excmNT2}
\end{eqnarray}
It is observed that the transverse polarization amplitudes vanish
in the current next-to-leading-power accuracy. The evolution
factors are
\begin{eqnarray}
E_f(t) &=& \alpha_s(t)\frac{C_2(t)}{N_c}
\exp\left[-S(t)|_{b_3=b_2}\right]\;.
\end{eqnarray}
The hard functions $h^{(\prime)(j)}_{f}$, $j=1$ and $2$, are
written as
\begin{eqnarray}
h^{(\prime)(j)}_{f}
&=&
\frac{i\pi}{2}
\left[\theta(b_1-b_2)
H_0^{(1)}\left(F^{(\prime)}m_B b_1\right)
J_0\left(F^{(\prime)}m_Bb_2\right)
\right. \nonumber \\
& &\quad \left.
+ \theta(b_2-b_1)
H_0^{(1)}\left(F^{(\prime)}m_B b_2\right)
J_0\left(F^{(\prime)}m_B b_1\right)
\right]
\nonumber \\[1mm]
& & \times
\left(
\begin{array}{cc}
K_{0}\left(F^{(\prime)}_{j}m_Bb_{1}\right) &
\mbox{for $F^{(\prime)2}_{j} \geq 0$} \\[1mm]
\frac{i\pi}{2} H_{0}^{(1)}
\left(\sqrt{|F^{(\prime)2}_{j}|}m_Bb_{1}\right) &
\mbox{for $F^{(\prime)2}_{j} \leq 0$}
\end{array} \right)\;,
\end{eqnarray}
with the variables
\begin{eqnarray}
F^{2} &=& x_2 r_2\eta^+ x_3(1-r_2\eta^-)\;,
\nonumber \\
F_{1}^{2} &=& x_2 r_2\eta^+
\left[ x_1 - x_3(1-r_2\eta^-) \right]\;,
\nonumber \\
F_{2}^{2} &=& 1+(1-x_2r_2\eta^+)
\left[ x_1 - 1 + x_3(1-r_2\eta^-) \right]\;,
\nonumber \\
F^{\prime\, 2} &=&
(1-x_2r_2\eta^+)
\left[ 1 - x_3(1-r_2\eta^-) \right]\;,
\nonumber \\
F_{1}^{\prime\, 2} &=&
(1-x_2r_2\eta^+)
\left[ x_1 -1 + x_3(1-r_2\eta^-) \right]\;,
\nonumber \\
F_{2}^{\prime\, 2} &=&
1+ x_2r_2\eta^+
\left[ x_1 - x_3(1-r_2\eta^-) \right]
\;.
\end{eqnarray}
The variables $F^\prime$'s can be obtained from $F$'s by
interchanging $x_2r_2\eta^+$ and $1-x_2r_2\eta^+$, and
$x_3(1-r_2\eta^-)$ and $1 - x_3(1-r_2\eta^-)$.
The hard scales $t_{f}^{(\prime)(j)}$ are chosen as
\begin{eqnarray}
t_{f}^{(\prime)(j)}
&=& {\rm max}
\left(F^{(\prime)}m_B,\sqrt{|F^{(\prime)2}_j|}m_B,1/b_1,1/b_2\right)\;.
\end{eqnarray}
The longitudinal polarization fraction dominates:
\begin{eqnarray}
R_L\sim 1\;,\;\;\;\; R_{\parallel}\sim R_{\perp}\sim {\rm
few}\%\;, \label{ard3}
\end{eqnarray}
which differs very much from that in Eq.~(\ref{ard2}). Therefore,
the comparison of the theoretical prediction with the future data
can test the PQCD approach. We also predict the branching ratio,
\begin{eqnarray}
B(B^0\to {\bar D}^{*0}D^{*0}) =
\left(8.9^{+1.4}_{-1.1}\right)\times 10^{-5} \;,
\end{eqnarray}
which can also be compared with the future data.
\subsection{$B\to\rho\rho$}
In this subsection we examine whether the simple estimate of the
polarization fractions in the tree-dominated $B$ meson decays into
two light vector mesons is robust under subleading corrections.
Similar to the previous subsection, the $O(m_V/m_B)$ terms should
be included into the factorizable amplitudes at this level of
accuracy. At the same time, the two-parton twist-4 contribution
appears, since the linear end-point singularity involved in
collinear factorization theorem modifies the power behavior from
$O(m_V^2/m_B^2)$ into $O(m_V/m_B)$ \cite{TLS}. The inclusion of
these two corrections makes complete the next-to-leading-power
analysis at the two-parton level. The nonfactorizable amplitudes
have been known to be small due to the strong cancellation between
a pair of nonfactorizable diagrams \cite{KLS,LUY}. The
annihilation amplitudes for tree-dominated modes are also
negligible due to helicity suppression \cite{CKL}. Hence, we shall
not consider the two-parton twist-4 correction to these two
subleading contributions. Below we analyze the $B\to\rho\rho$
longitudinal polarization amplitude as an example.
The two-parton $\rho$ meson distribution amplitudes up to twist 4
are defined by the following expansion \cite{BBKT},
\begin{eqnarray}
\langle \rho^-(P_2,\epsilon_{2L}^*)|\bar d(z)_ju(0)_l|0\rangle
&=&\frac{1}{\sqrt{2N_c}}\int_0^1 dx e^{ixp_2\cdot z}\Bigg[\not
p_2\phi_{\rho}(x) +m_{\rho}(\not n_+\not n_--1)
\phi_{\rho}^{t}(x)\nonumber\\
& &\;\;\;\;\;\;+m_{\rho}
I\phi_{\rho}^s(x)-\frac{m_{\rho}^2}{2p_2\cdot n_-}\not
n_-\phi_{\rho}^{g}(x)\Bigg]_{lj}\;, \label{klpf}
\end{eqnarray}
where the new vector $p_2$ contains only the plus (large)
component of $P_2$. The distribution amplitude $\phi_\rho$ is of
twist 2 (leading twist), $\phi_{\rho}^{t}$ and $\phi_{\rho}^{s}$
of twist 3, and $\phi_\rho^g$ of twist 4. The twist-3 distribution
amplitudes in fact give leading-power contribution due to the
similar modification from the end-point singularity. The explicit
expressions of the above $\rho$ meson distribution amplitudes are
referred to \cite{TLS}, and $\phi_\rho^g$ is given by \cite{BB98}
\begin{eqnarray}
\phi_\rho^g(x)=\frac{f_\rho}{2\sqrt{2N_c}}\left[1-1.62C_2^{1/2}(2x-1)
-0.41C_4^{1/2}(2x-1)\right]\;,
\end{eqnarray}
with the Gegenbauer polynomials,
\begin{eqnarray}
& &C_2^{1/2}(t)=\frac{1}{2}(3t^2-1)\;,\;\;\;
C_4^{1/2}(t)=\frac{1}{8}(35 t^4 -30 t^2 +3)\;.
\end{eqnarray}
The longitudinal factorizable amplitude in the $B\to\rho\rho$
decays is written, up to twist 4, as,
\begin{eqnarray}
{\cal F}_{L} &=& 8 \pi C_F m_B^2 \int_0^1 {\rm d}x_1 {\rm d}x_2
\int_0^{\infty} b_1{\rm d}b_1\, b_2{\rm d}b_2\, \phi_B(x_1,b_1)
\nonumber \\
& &\hspace{5mm}\times\, \bigg\{ \left[ \left( (1+x_2)(1-r_2^2)
-(1+2x_2)r_2^2 \right)\,\phi_{\rho}(x_2)
\right. \nonumber\\
& & \left.\hspace{13mm} + r_2(1-2x_2) \left(
\phi_{\rho}^s(x_2)+\phi_{\rho}^t(x_2) \right) \right]
E_{e}(t^{(1)}_e) h_{e}(x_1,x_2,b_1,b_2)
\nonumber\\
& & + r_2 \left[ 2\, \phi_{\rho}^s(x_2) + r_2\,
\phi_{\rho}^{g}(x_2) \right] E_{e}(t^{(2)}_e)
h_{e}(x_2,x_1,b_2,b_1) \bigg\} \;,
\end{eqnarray}
where the first (second) term containing $E_{e}(t^{(1)}_e)$
$[E_{e}(t^{(2)}_e)]$ comes from the lowest-order diagram similar
to Fig.~\ref{fig1}(a) [Fig.~\ref{fig1}(b)], but with the $D_s^*$
and $D^*$ mesons being replaced by the $\rho$ mesons. The
evolution factor is
\begin{eqnarray}
E_e(t)=\alpha_s(t)\,a_1(t)\,\exp[-S_B(t)-S_{\rho}(t)]
\;,\label{eet}
\end{eqnarray}
with the Sudakov factor from the $k_T$ resummation,
\begin{eqnarray}
\exp[-S_{\rho}(\mu)]=\exp\left[-s(k_2^+,b_2)-s(P_2^+-k_2^+,b_2)
-2\int_{1/b_2}^\mu
\frac{d{\bar\mu}}{\bar\mu}\gamma(\alpha_s({\bar\mu}))\right]\;,
\label{srho}
\end{eqnarray}
and the hard scales,
\begin{eqnarray}
t^{(1)}_e&=&{\rm max}(\sqrt{x_2}m_B,1/b_1,1/b_2)\;, \;\;
t^{(2)}_e={\rm max}(\sqrt{x_1}m_B,1/b_1,1/b_2)\;.
\end{eqnarray}
The hard function is given by
\begin{eqnarray}
h_e(x_1,x_2,b_1,b_2) &=& S_t(x_2)\,
K_{0}\left(\sqrt{x_1x_2}m_Bb_1\right)
\left[\theta(b_1-b_2)K_0\left(\sqrt{x_2}m_B
b_1\right)I_0\left(\sqrt{x_2}m_Bb_2\right)\right.
\nonumber \\
& & \hspace{33mm}
\left.+\theta(b_2-b_1)K_0\left(\sqrt{x_2}m_Bb_2\right)
I_0\left(\sqrt{x_2}m_Bb_1\right)\right]\;.
\end{eqnarray}
The Sudakov factor $S_t(x)$ arises from the threshold resummation
of the double logarithms $\alpha_s\ln^2 x$, which are produced by
the radiative corrections to the hard kernels. Its expression
\cite{UL},
\begin{eqnarray}
S_t(x)=\frac{2^{1+2c}\Gamma(3/2+c)}{\sqrt{\pi}\Gamma(1+c)}
[x(1-x)]^c\;,
\end{eqnarray}
with the constant $c\sim 0.3$, provides further suppression in the
end-point region of $x\to 0$, and improves the perturbative
calculation. Since we have performed the leading-logarithm
resummation so far, only the behavior of $S_t(x)$ at small $x$ is
reliable. The parametrization in the large $x$ region, also
vanishing, was proposed for convenience \cite{UL}.
The numerical results are listed in Table~\ref{t4}, where diagram
(a) [diagram (b)] refers to the lowest-order diagram with the hard
gluon being on the $B$ ($\rho$) meson side. It indicates that the
next-to-leading-power corrections decrease (increase) the
leading-power contribution from diagram (a) [diagram (b)]
slightly. Considering the net effect, these corrections are indeed
negligible.
\begin{table}[tp]
\begin{center}
\begin{tabular}{cccc}
\hline\hline & Diagram (a) & Diagram (b) & sum
\\
\hline
Leading-power & $0.330$ & $0.088$ & $0.418$ \\
Plus next-to-leading power & $0.324$ & $0.096$ & $0.420$ \\
\hline\hline
\end{tabular}
\caption{Contributions to ${\cal F}_{L}$ up to next-to-leading
power. \label{t4}}
\end{center}
\end{table}
\section{COMMENTS ON PLAUSIBLE EXPLANATIONS}
We now comment on the mechanism proposed in the literature to
explain the abnormal polarization fractions of the $B\to \phi K^*$
decays. It will be argued that these proposals involve many free
parameters, can not account for the polarizations of all $B\to VV$
modes simultaneously, or are too small to achieve the purpose.
When discussing the annihilation effect on penguin-dominated
decays, we found that the $B\to\rho K^*$ polarization data can be
understood, which form the third category mentioned in the
Introduction.
\subsection{Annihilation Contribution}
If the power-suppressed annihilation contribution from the
$(S-P)(S+P)$ penguin operators is enhanced by some mechanism, one
could have the different counting rules as shown in
Eq.~(\ref{mod}), and it might be possible to reach $R_L\sim 0.5$.
This is the strategy adopted in \cite{AK}, whose analysis was
performed in the QCDF approach \cite{BBNS}. In QCDF an
annihilation amplitude is not calculable due to the end-point
singularity, and has to be formulated in terms of several free
parameters, such as $\rho_A$. In the current case, different
$\rho_A$ have been introduced for the longitudinal and transverse
polarization amplitudes in order to fit the data. These parameters
greatly reduce the predictive power of QCDF. Because varying free
parameters to explain the data can not be conclusive, we shall
estimate the annihilation contribution in the PQCD approach,
viewing that the PQCD predictions for the penguin annihilation are
consistent with the measured direct CP asymmetries in $B^0\to
K^+\pi^-$, $\pi^+\pi^-$ \cite{KLS,LUY}. Such a calculation for a
pure-penguin $VV$ mode has been performed in \cite{CKL2}. As shown
in Table~\ref{tab5}, both the penguin annihilation and
nonfactorizable contributions help reduce $R_L$. However, the
combined effect is still not sufficient to lower the fractions
$R_L$ of the $B\to\phi K^*$ decays down to around 0.5.
Note that our predicted relative strong phases among $A_{L}$,
$A_{\parallel}$, and $A_{\perp}$ are consistent with the $B\to\phi
K^{*0}$ data:
\begin{eqnarray}
& &\phi_{\parallel}=2.21\pm 0.22\pm 0.05\,
(rad.)\;,\;\;\;\;\phi_{\perp}=2.42\pm 0.21 \pm
0.06\,(rad)\,\cite{Zhang04} \;,\nonumber\\
& &\phi_{\parallel}=2.34^{+0.23}_{-0.20}\pm 0.05\,
(rad.)\;,\;\;\;\;\phi_{\perp}=2.47\pm 0.25\pm
0.05\,(rad)\,\cite{Bar017}\;.
\end{eqnarray}
Table~\ref{tab5} also implies that $R_L$ can decrease down to 0.75
for a pure-penguin $VV$ mode, after taking into account the
penguin annihilation and nonfactorizable contributions. Since the
$B^+\to \rho^+K^{*0}$ decay is a pure-penguin process, and the
$\rho$ meson mass is not very different from the $\phi$ meson
mass, the above PQCD analysis applies. Hence, we expect the
longitudinal fraction $R_L\sim 0.75$ for the $B^+\to \rho^+K^{*0}$
decay, which is consistent with the Babar measurement, but a bit
larger than the Belle measurement. Due to the large uncertainty of
the Belle data, there is in fact no discrepancy.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{cccccc}
\hline\hline
Mode & $ |A_{L}|^{2}$ & $ |A_{\parallel}|^{2}$ & $
|A_{\perp}|^{2}$ & $\phi_{\parallel}(rad.)$ & $\phi_{\perp}(rad.)$
\\ \hline
$\phi K^{*0}$(I) & $0.923$ & $0.040$ & $0.035$ &
$\pi$ & $\pi$\\
\hspace{0.7cm}(II)& $0.860$ & $0.072$ & $0.063$ & $3.30$
& $3.33$ \\
\hspace{0.7cm}(III) & $0.833$ & $0.089$ & $0.078$ &
$2.37$ & $2.34$ \\
\hspace{0.7cm}(IV) & { $0.750$} & $0.135$ & $0.115$ & $2.55$ & $2.54$ \\
\hline $\phi K^{*+}$(I) & $0.923$ & $0.040$ & $0.035$ &
$\pi$ & $\pi$ \\
\hspace{0.7cm}(II) &$0.860$ & $0.072$ & $0.063$ & $3.30$ & $3.33$
\\
\hspace{0.7cm}(III) &$0.830$ & $0.094$ & $0.075$ &
$2.37$ & $2.34$ \\
\hspace{0.7cm}(IV) &{ $0.748$} & $0.133$ & $0.111$ & $2.55$ & $2.54$ \\
\hline\hline
\end{tabular}
\end{center}
\caption{(I) Without nonfactorizable and annihilation
contributions, (II) add only nonfactorizable contribution, (III)
add only annihilation contribution, (IV) add both nonfactorizable
and annihilation contributions.}\label{tab5}
\end{table}
We then come to another mode $B^+\to \rho^0K^{*+}$, to which the
tree operators contribute. Adopt the Babar measurement $R_L\sim
0.8$ for the $B^+\to \rho^+K^{*0}$ decay, and assume that the tree
contribution, which obeys the counting rules in Eq.~(\ref{nai}),
affects only the longitudinal fraction. Hence, we simply add the
color-allowed and color-suppressed tree amplitudes $T+C\sim 0.6
\exp(-90^oi)P$, which was extracted from the $B\to K\pi$ data
\cite{Charng}, to the $B^+\to \rho^+K^{*0}$ longitudinal
polarization amplitude $P$. The phase $-90^o$ has included the
weak phase $\phi_3\sim 60^o$. Without an explicit computation, we
derive the polarization fractions for the $B^+\to \rho^0K^{*+}$
decay,
\begin{eqnarray}
R_L\sim 0.86\;,\;\;\;\;R_\parallel\sim R_\perp\sim 0.07\;,
\end{eqnarray}
which are consistent with the data within $1\sigma$. We emphasize
that we did not attempt a rigorous calculation of the $B\to \rho
K^*$ decays here, which deserves a separate paper. In the
perturbation theories, such as PQCD and QCDF, the factorizable
$B\to \pi K$ amplitudes and the factorizable $B\to \rho K^*$
amplitudes with longitudinal polarizations are very similar. Small
differences arise only from the meson masses and the distribution
amplitudes. Therefore, the estimation using the three amplitudes
from the $B\to \pi K$ modes, which have been available in the
literature, makes sense.
The analysis and the result of the modes $B\to \omega K^*$ should
be similar to those of $B\to \rho^0 K^*$. The explicit PQCD
analysis of the $B\to\rho (\omega) K^*$ polarizations will be
performed elsewhere. In conclusion, it is not difficult to
accommodate the polarization data of the third category, the $B\to
\rho K^{*}$ decays, within the Standard Model by means of the
penguin annihilation and nonfactorizable contributions. It is also
interesting to propose that the measurement of the $B\to\omega
K^*$ polarizations can test the PQCD approach.
\subsection{Charming Penguin}
A charming penguin arises from the nonperturbative dynamics
involved in a charm quark loop \cite{charming}. It is not
calculable, has to be parameterized as a free parameter, and could
be as large as a leading contribution. Recently, it has been
introduced into soft-collinear effective theory (SCET) in order to
account for the large $B\to\pi^0\pi^0$ branching ratio
\cite{BPRS}. The inputs of the measured CP asymmetries
$S_{\pi\pi}$ and $A_{\pi\pi}$ demand a complex charming penguin,
leading to a large
penguin-over-tree ratio $|P/T|\sim 0.7$ \cite{BPRS}. With this
$P/T$ from the data, a large branching ratio
$B(B^0\to\pi^0\pi^0)\sim 1.9\times 10^{-6}$ was obtained. SCET
does not attempt to explain why $|P/T|$ is so large, even though
the Standard Model calculations based on PQCD and QCDF give
$|P/T|=0.23$-0.29. Another concern is that the $B\to\pi$ form
factor from the data fitting is as small as 0.17 in the presence
of the large charming penguin, in conflict with the values 0.28
from lattice QCD \cite{DB02} and from light-cone sum rules
\cite{KR,PB3}.
It has been also proposed that the charming penguin may be large
enough to modify the counting rules in Eq.~(\ref{nai}), and to
explain the abnormal $B\to\phi K^*$ polarization data \cite{BPRS}.
However, one also requires different free parameters for the
different helicity amplitudes in order to lower the longitudinal
polarization fraction, and to enhance the transverse polarization
fractions. In this sense, SCET is similar to QCDF \cite{AK}, where
different $\rho_A$ were introduced for the different helicity
amplitudes. One needs different parameters for different modes
too, such as $B\to\rho K^*$ and $B\to\phi K^*$. Our comment on
SCET is then the same as on QCDF: the explanation by introducing
as many parameters as necessary is always plausible, but can not
be conclusive. We point out that the current SCET formalism is
only of leading power: the chirally enhanced terms, proportional
to $m_0/m_B$, have been dropped, and the annihilation (or
$W$-exchange) amplitudes have not yet been formulated. We
speculate that if the annihilation amplitude is included into
SCET, the charming penguin may not be so essential. On the other
hand, the charm-loop correction is well-behaved in perturbation
theory without any infrared singularity, which has been known as
the Bander-Silverman-Soni mechanism \cite{BSS}, implying that its
nonperturbative piece is unlikely to be large. Besides, the
light-cone-sum-rule analysis has supported a small charming
penguin \cite{KMM}.
We have taken this chance to investigate the charm-quark loop
correction to the $B\to\phi K^*$ polarization fractions in the
PQCD approach. The gluon invariant mass attaching the charm-quark
loop can be defined unambiguously as
\begin{eqnarray}
q^2=(1-x_2)x_3m_B^2-|{\bf k}_{2T}-{\bf k}_{3T}|^2\;,
\end{eqnarray}
with $x_2$ and $k_{2T}$ ($x_3$ and $k_{3T}$) being the momentum
fraction and the transverse momentum in the $K^*$ ($\phi$) meson,
respectively. It turns out that this effect increases $R_L$ by
about 5\%, and is negligible. It also decreases the relative
strong phases $\phi_\parallel$ and $\phi_\perp$ a bit.
\subsection{Rescattering Effect}
It has been proposed to explain the $B\to\phi K^*$ polarization
data through the rescattering effect \cite{CDP,LLNS,CCS},
\begin{eqnarray}
B \to D_s^{(*)} D^{(*)} \to \phi K^*\;.
\end{eqnarray}
The motivation is that the longitudinal polarization fraction of
the intermediate states $D_s^{*} D^{*}$, as low as 0.5, might
propagate into the final state $\phi K^*$. First, the massive $B$
meson can decay into $\phi K^*$ through many intermediate states.
The analysis in \cite{CDP,LLNS,CCS} was restricted to only a few
channels, and likely to be model-dependent \cite{Ligeti04}. The
truncation of the higher intermediate states in this kind of
analyses has been criticized \cite{W}. Second, if this mechanism
works for the $B\to\phi K^*$ modes, it will also work for
$B\to\rho K^*$, which involve the same intermediate states. As
obtained in \cite{CCS}, $R_L$ of both the $B^+\to\rho^+ K^{*0}$
and $B^+\to\rho^0 K^{*+}$ decays are as low as 0.6. This
observation is expected, since the additional tree amplitudes in
the latter can not change $R_L$ very much. However, the data in
Table~\ref{tab:tab1} indicate $R_L\sim 0.96$ for the $B^+\to\rho^0
K^{*+}$ decay. In other words, the $B^+\to\rho^0 K^{*+}$
polarization data, obeying the naive counting rules, have strongly
constrained the rescattering effect.
Third, the $D_s^*D$ and $D_s D^*$ intermediate states,
contributing to the $P$-wave component, could affect the
perpendicular polarization of the $B\to\phi K^*$ decays.
Unfortunately, there exists a strong cancellation among these two
channels due to the CP and SU(3) (CPS) symmetries \cite{CCS}. The
$D_s^* D^*$ intermediate state survives the CPS symmetry, which,
however, exhibits a vanishing $R_\perp$ as in Eq.~(\ref{arr}).
Therefore, the rescattering effect leads to the pattern,
\begin{eqnarray}
R_L\sim R_\parallel \gg R_\perp\;, \label{res}
\end{eqnarray}
contrary to the observed approximate equality $R_\parallel\approx
R_\perp$.
Furthermore, it has been known that the $B\to KK$ decays are
sensitive to rescattering effects. The $B\to KK$ branching ratios
measured recently well agree with the PQCD predictions \cite{CL00}
as shown in Table~\ref{tab3}, leaving very limited room for the
rescattering effect. Note that no theoretical errors were
presented in \cite{CL00}, since the detailed investigation of
uncertainties in the PQCD approach was available only after
Ref.~\cite{TLS}. Roughly speaking, the theoretical errors on PQCD
predictions for branching ratios of two-body charmless $B$ meson
decays are about 30\%. Viewing the contradiction of
Eq.~(\ref{res}) to the $B\to\phi K^*$ polarization data, and the
constraints from the measured $B^+\to\rho^0 K^{*+}$ polarizations
and from the measured $B\to KK$ branching ratios, we intend to
conclude that the rescattering effect is not a satisfactory
resolution to the polarization puzzle.
\begin{table}[ht]
\begin{center}
\begin{tabular}{c c c}\hline \hline
Branching Ratio&PQCD& Babar \cite{Bar080}\\
\hline $B(B^+\to K^+K^0)$&$1.65\times
10^{-6}$&$(1.45^{+0.53}_{-0.46}\pm
0.11)\times 10^{-6}$\\
$B(B^0\to K^0{\bar K}^0)$&$1.75\times 10^{-6}$&$(1.19^{+0.40}_{-0.35}\pm 0.13)\times 10^{-6}$\\
\hline \hline
\end{tabular}
\end{center}
\caption{PQCD predictions for the CP-averaged $B\to KK$ branching
ratios and the data. }\label{tab3}
\end{table}
\subsection{Magnetic Penguin}
\begin{figure}[t]
\centerline{
\includegraphics[width=13cm]{fig_dsd_4.eps}
} \caption{Some diagrams from the magnetic penguin which
contribute to the transverse polarization fraction of the
$B\to\phi K^*$ decays.} \label{fig4}
\end{figure}
If the $B\to\rho K^*$ data can be understood in the Standard Model
by means of the penguin annihilation and nonfactorizable
contributions, and only the $B\to\phi K^*$ decays exhibit an
anomaly, it is natural to look for a unique mechanism for the
latter. Such a mechanism, the $b\to sg$ transition, has been
proposed in \cite{HN}. The novel idea is that the transversely
polarized gluon from the transition propagates into the $\phi$
meson, enhancing the transverse polarization amplitudes. The
relevant matrix element was then parameterized in terms of a
dimensionless free parameter $\kappa$. Assuming this parameter to
be $\kappa\sim -0.25$, the authors of \cite{HN} claimed that the
$B\to\phi K^*$ polarization data could be accommodated within the
Standard Model. Similarly, varying a free parameter to fit the
data can not be conclusive, and a reliable estimate of the
$\kappa$ value is necessary. As pointed out in \cite{HN}, the same
mechanism also contributes to the $B\to\omega K^*$ decays, and
small $R_L\sim 0.5$ have been predicted. Therefore, the
measurement of the $B\to\omega K^*$ polarizations will impose a
stringent test on this proposal in the future. Note that PQCD
postulates, contrary to \cite{HN}, that $R_L$ of the $B\to\omega
K^*$ decays are as large as those of $B\to\rho^0 K^*$.
Besides the above experimental discrimination, we shall estimate
the order of magnitude of $\kappa$ in the framework of FA,
following the method in \cite{CL0307}. The weak effective
Hamiltonian contains the $b\to sg$ transition,
\begin{eqnarray}
-\frac{G_F}{\sqrt{2}}V_{ts}^*V_{tb}C_{8g}O_{8g}\;,
\end{eqnarray}
with the magnetic penguin operator,
\begin{eqnarray}
O_{8g}=\frac{g}{8\pi^2}m_b{\bar
s}_i\sigma_{\mu\nu}(1+\gamma_5)T_{ij}^aG^{a\mu\nu}b_j\;,
\end{eqnarray}
$i$, $j$ being the color indices. The picture described in
\cite{HN} is displayed in Fig.~\ref{fig4}: one or more collinear
gluons, emitted from the $B\to K^*$ form factor, produce the $s$
and $\bar s$ quarks in the color-octet state. They, together with
the transversely polarized gluon from the $b\to sg$ transition,
fragment into the color-singlet transversely polarized $\phi$
meson. According to this picture, we introduce three-parton
distribution amplitudes to absorb the nonperturbative dynamics
associated with the $\phi$ meson. Another diagram, in which the
transversely polarized gluon produces the $s$ and $\bar s$ quarks
and the collinear gluon flows into the $\phi$ meson directly, does
not contribute, since the transverse polarization of the $\phi$
meson is mainly carried by its gluonic parton.
We first argue that the leading picture in Fig.~\ref{fig4}(a),
where the collinear gluon attaches the $s$ quark from the $b\to
sg$ transition, diminishes due to the $G$-parity. The
corresponding three-parton distribution amplitudes are defined via
the matrix elements \cite{BBKT},
\begin{eqnarray}
& &\langle \phi(P_3,\epsilon^*_3(T))|\bar
s(-z)gG_{\mu\nu}(vz)\gamma_\alpha
s(z)|0\rangle\nonumber\\
& &=-iP_{3\alpha}\left[P_{3\mu}\epsilon^*_{3\nu}(T)-
\epsilon^*_{3\mu}(T)P_{3\nu}\right] f_{3\phi}^V {\tilde
V}(v,P_3\cdot z)\;,\label{mat1}\\
& &\langle\phi(P_3,\epsilon^*_3(T))|\bar s(-z)g\tilde
G_{\mu\nu}(vz)\gamma_\alpha\gamma_5
s(z)|0\rangle\nonumber\\
& &=P_{3\alpha}\left[P_{3\nu}\epsilon^*_{3\mu}(T)
-P_{3\mu}\epsilon^*_{3\nu}(T)\right] f_{3\phi}^A {\tilde
A}(v,P_3\cdot z)\;,\label{mat2}
\end{eqnarray}
with the dual gluon field strength tensor $\tilde
G_{\mu\nu}=\epsilon_{\mu\nu\rho\sigma}G^{\rho\sigma}/2$. Other
three-parton distribution amplitudes, irrelevant to the discussion
below, are not quoted here. The Fourier transformation of the
distribution amplitude $\tilde V$ gives
\begin{eqnarray}
{\tilde V}(v,P_3\cdot z)=\int [dx]\exp[iP_3\cdot z(x_{\bar
s}-x_s+vx_g)]V(x_s,x_{\bar s},x_g)\;,\label{four}
\end{eqnarray}
with $x_s$, $x_{\bar s}$, and $x_g$ being the momentum fractions
carried by the $s$ quark, the ${\bar s}$ quark, and the gluon,
respectively, and the integration measure,
\begin{eqnarray}
\int [dx]\equiv \int_0^1 dx_{\bar s}\int_0^1 dx_s\int_0^1
dx_g\delta\left(1-\sum_i x_i\right)\;.
\end{eqnarray}
The Fourier transformation of $\tilde A$ is defined in the same
way. The three-parton twist-3 vector meson distribution amplitudes
have been also studied in \cite{IH}. The asymptotic models of $V$
and $A$ have been parameterized as \cite{BBKT,CZ84,SVZ}
\begin{eqnarray}
V(x_s,x_{\bar s},x_g)&=&5040(x_s-x_{\bar s})x_sx_{\bar
s}x_g^2\;,\label{vda}\\
A(x_s,x_{\bar s},x_g)&=&360x_sx_{\bar
s}x_g^2\left[1+\omega_{1,0}^A\frac{1}{2}(7x_g-3)\right] \;,
\end{eqnarray}
with the shape parameter $\omega_{1,0}^A=-2.1$. The antisymmetry
(symmetry) of $V$ ($A$) between the exchange of $x_s$ and $x_{\bar
s}$ is a consequence of the $G$-parity transformation in the SU(3)
limit \cite{BBKT}. The constant $f_{3\phi}^V$ is chosen such that
$V$ is normalized according to
\begin{eqnarray}
\int [dx](x_s-x_{\bar s})V(x_s,x_{\bar s},x_g)=1\;.\label{vdan}
\end{eqnarray}
We factorize the matrix element in Eq.~(\ref{mat1}) and the $B\to
K^*$ transition form factor out of Fig.~\ref{fig4}(a). The
remaining part is the hard kernel, which must be symmetric under
the exchange of $x_s$ and $x_{\bar s}$. Therefore,
Fig.~\ref{fig4}(a), written as the convolution of the symmetric
hard kernel with the antisymmetric three-parton distribution
amplitude $V$, vanishes. The diagram with the collinear gluon
attaching the $b$ quark does not contribute, because its hard
kernel is also symmetric. No matter how many infrared gluons are
involved in the $s$-$\bar s$ quark pair production, there is no
contribution for the same reason. If factorizing the matrix
element in Eq.~(\ref{mat2}) out of Fig.~\ref{fig4}(a), it is easy
to find that the corresponding hard kernel vanishes, because the
$s$-$\bar s$ quark pair does not form an axial-vector current.
Therefore, we conclude that the leading diagram does not
contribute to the transverse polarization amplitudes of the
$B\to\phi K^*$ decays.
To survive the above suppressions, one has to consider subleading
diagrams such as Fig.~\ref{fig4}(b), in which both the $s$ quark
and the gluon from $b\to sg$ flow into the $\phi$ meson, or such
as Fig.~\ref{fig4}(c), in which one more infrared gluon fragments
into the $\phi$ meson. For Fig.~\ref{fig4}(b), an extra hard gluon
is necessary for producing the $s$-$\bar s$ quark pair, such that
the price to pay is the $\alpha_s$ suppression. For
Fig.~\ref{fig4}(c), four-parton distribution amplitudes are
involved, whose contribution is power-suppressed. We shall show
that the order of magnitude of $\kappa$ from Fig.~\ref{fig4}(b)
is, unfortunately, as small as 0.01, far away from $\kappa\sim
-0.25$ required by the data. For simplicity, we analyze the
contribution from the three-parton distribution amplitude $V$, and
our conclusion applies to that from $A$. Due to the lack of the
information of the four-parton twist-4 distribution amplitudes, we
can not estimate the contribution from Fig.~\ref{fig4}(c) in a
reliable way. However, it is of higher power in $m_\phi/m_B$, and
unlikely to be huge.
Insert the Fierz identity,
\begin{eqnarray}
I_{ij}I_{lk}=\frac{1}{4}(\gamma_\alpha)_{ik}(\gamma^\alpha)_{lj}
+\cdots\;,
\end{eqnarray}
where the irrelevant terms have been suppressed, and the identity
for color matrices,
\begin{eqnarray}
I_{ij}I_{lk}=2(T^b)_{ik}(T^b)_{lj} +\frac{1}{N_c}I_{ik}I_{lj}\;,
\end{eqnarray}
to change the fermion and color flows of the outgoing $s$ and
$\bar s$ quarks, respectively. Figure \ref{fig4}(b), where the
additional hard gluon attaches the $s$ quark going into the $K^*$
meson, is then factorized into
\begin{eqnarray}
{\cal M}&=&-\frac{G_F}{\sqrt{2}}V_{ts}^*V_{tb}
\frac{g^2}{8\pi^2}m_b C_{8g}\int[dx]{\rm IFT}\langle
\phi(P_3,\epsilon^*_3(T))|\bar
s(-z)gG^{a\mu\nu}(vz)T^b\gamma_\alpha
s(z)|0\rangle\nonumber\\
& & \times\frac{1}{4}tr\left[\cdots\gamma^\lambda\gamma^\alpha
\gamma_\lambda\frac{\not P_1-x_g\not P_3}{(P_1-x_gP_3)^2}
\sigma_{\mu\nu}(1+\gamma_5)\cdots\right] \frac{
2tr(T^cT^bT^cT^a)}{(P_2+x_{\bar s}P_3)^2}\;,
\end{eqnarray}
where IFT means the inverse Fourier transformation. The indices
$\lambda$ denote the hard gluon vertices, and the dots in the
trace represent the Feynman rules associated with the $B\to K^*$
form factor.
Employing $2tr(T^cT^bT^cT^a)=-\delta^{ab}/(2N_c)$,
Eq.~(\ref{mat1}) and Eq.~(\ref{four}), the above expression
becomes
\begin{eqnarray}
{\cal M}&=&-\frac{G_F}{\sqrt{2}}V_{ts}^*V_{tb}
\frac{g^2}{8\pi^2N_c}m_b C_{8g}f_{3\phi}^V\int[dx]V(x_s,x_{\bar
s},x_g)
\nonumber\\
& & \times\frac{1}{4}tr\left[\cdots\gamma^\lambda\not P_3
\gamma_\lambda\frac{\not P_1-x_g\not P_3}{(P_1-x_gP_3)^2}
i\sigma_{\mu\nu}(1+\gamma_5)\cdots\right]
\frac{P_3^{\mu}\epsilon^{*\nu}_3(T)}{(P_2+x_{\bar s}P_3)^2}\;.
\end{eqnarray}
Neglecting the light meson masses, assuming $m_b\approx m_B$, and
working out the product of the Dirac matrices in the trace, we
derive
\begin{eqnarray}
{\cal M}&=&-\frac{G_F}{\sqrt{2}}V_{ts}^*V_{tb}
\frac{\alpha_s}{4\pi N_c}C_{8g}\frac{f_{3\phi}^V}{m_B}
\int[dx]\frac{V(x_s,x_{\bar s},x_g)}{x_{\bar
s}(x_s+x_{\bar s})}\nonumber\\
& &\times \langle K^{*-}(P_2,\epsilon^{*}_2(T))|\bar
si\sigma_{\mu\nu}(1+\gamma_5)b|B^-(P_1)\rangle
P_3^\nu\epsilon_3^{*\mu}(T) \;.
\end{eqnarray}
Comparing the above expression with Eq.~(4) in \cite{HN}, the
parameter $\kappa$ is given by
\begin{eqnarray}
\kappa=\frac{\alpha_s}{4\pi N_c}\zeta_{3\phi}^V
\int[dx]\frac{V(x_s,x_{\bar s},x_g)}{x_{\bar s}(x_s+x_{\bar
s})}\;,
\end{eqnarray}
with $\zeta_{3\phi}^V\equiv f_{3\phi}^V/(f_\phi m_\phi)$. For the
values $\alpha_s=0.4$ and $\zeta_{3\phi}^V=0.013$ \cite{BBKT}, and
the model distribution amplitude in Eq.~(\ref{vda}), we obtain
$\kappa\approx 0.004$. Other diagrams with the hard gluon
attaching the $b$ quark and the transversely polarized gluon can
be analyzed in a similar way, and the results are of the same
order of magnitude.
Adding these contributions leads to
\begin{eqnarray}
\kappa\approx 0.01\;.
\end{eqnarray}
Hence, we intend to conclude that the magnetic penguin is not
sufficient to resolve the $B\to\phi K^*$ puzzle.
\section{CONCLUSION}
In this paper we have investigated most of the $B\to VV$ modes
carefully. Our observation is that the $B\to VV$ modes can be
classified into four categories. For the tree-dominated decays,
the polarization fractions are basically determined by kinematics.
The upper part of Table~\ref{tab:tab2} can be understood by
kinematics in the heavy-quark limit. The longitudinal polarization
fractions follow the mass hierarchy among the $D_s^{*}$, $D^{*}$
and $\rho$ ($K^*$) mesons. The lower part of Table~\ref{tab:tab2}
can be understood by kinematics in the large-energy limit. We
always have $R_L\sim 1$ for the decays into two light vector
mesons. It has been found that the above simple kinematic
estimates in the heavy-quark and large-energy limits are robust
under subleading corrections. For this part, we have analyzed the
next-to-leading-power corrections to the universal IW function,
the nonfactorizable contributions, the two-parton twist-4
contributions, and part of next-to-leading-order contributions
from the charm-quark loop, all of which are negligible. That is,
QCD dynamics plays only a minor role for the polarizations of the
tree-dominated decays.
As a byproduct, we have predicted the longitudinal polarization
fractions of the $B^+\to (D_s^{*+}, D^{*+})\rho^0$ modes in the
large-energy limit using FA, and found $R_L\sim 0.7$. We have also
calculated the polarization fractions of the $B^0\to {\bar
D}^{*0}D^{*0}$ decay explicitly in the PQCD approach based on
$k_T$ factorization theorem. The result $R_L\sim 1$ is quite
different from $R_L$ of other $B\to D^*D^*$ modes, since it is
dominated by the nonfactorizable $W$-exchange topology. The above
predictions can be confronted with the future data.
For the penguin-dominated modes, the polarization fractions can
deviate from the naive counting rules based on kinematics, because
of the important annihilation contribution from the $(S-P)(S+P)$
operators. This mechanism explains the third category listed in
the lower part of Table~\ref{tab:tab1}: $R_L$ can decrease to 0.75
for the $B^+\to\rho^+ K^{*0}$ mode. Adding the tree contribution,
$R_L$ of the $B^+\to\rho^0 K^{*+}$ decay can go up to about 0.9.
We have postulated from the viewpoint of PQCD that the $B\to\omega
K^*$ decays also belong to the third category, and should show
$R_L$ similar to those of $B\to\rho^0 K^*$. All the above three
categories can be accommodated within the Standard Model. Only the
fourth category, the $B\to\phi K^*$ decays, can not. They are
dominated by the penguin contribution, but their $R_L\sim 0.5$ are
much lower than 0.75. We have carefully analyzed the various
mechanism proposed in the literature to resolve this anomaly, and
concluded that none of them is satisfactory. Therefore, the
$B\to\phi K^*$ polarization data remain as a puzzle. However, we
emphasize that we are not claiming a signal of new physics, since
the complicated QCD dynamics in the $B\to VV$ decays has not yet
been fully explored. For example, a smaller $B\to K^*$ form factor
$A_0$ could decrease $R_L$ significantly \cite{L0411}.
\vskip 1.0cm We thank C. Bauer, I.I. Bigi, P. Chang, C.H. Chen,
K.F. Chen, H.Y. Cheng, C.K. Chua, W.S. Hou, Y.Y. Keum, Z. Ligeti,
M. Nagashima, D. Pirjol, A.I. Sanda, and I. Stewart for useful
discussions. This work was supported by the National Science
Council of R.O.C. under Grant No. NSC-93-2112-M-001-014, by the
Taipei Branch of the National Center for Theoretical Sciences of
R.O.C., and by the Grants-in-aid from the Ministry of Education,
Culture, Sports, Science and Technology, Japan under Grant No.
14046201. HNL acknowledges the hospitality of Department of
Physics, Tohoku University, where this work was initiated.
|
{
"timestamp": "2005-04-01T11:38:27",
"yymm": "0411",
"arxiv_id": "hep-ph/0411146",
"language": "en",
"url": "https://arxiv.org/abs/hep-ph/0411146"
}
|
\section{Introduction}
\begin{figure*}
\centering
\includegraphics[width=10 cm]{fig1.ps}
\caption{Radio contours of the cluster A2255
overlaid on the Rosat X-ray image (colors).
The radio image is at 1.4 GHz and has a FWHM of
15$^{\prime\prime}$ $\times$ 15$^{\prime\prime}$ (uniform weighting).
The sensitivity (1$\sigma$) is 16 $\mu$Jy/beam and the dynamic range
is $\simeq$ 6300.
Contour levels are: 0.05 0.1 0.2 0.4 0.8 1.6 3.2 6.4 12.8 25.6 51.2 mJy/beam.
No primary beam correction has been applied to the image.
}
\label{fig1}
\end{figure*}
An ever increasing number of galaxy clusters exhibit
large-scale, diffuse, steep-spectrum
synchrotron sources associated with the intracluster medium.
These radio sources have been generally
classified as radio halos or relics depending on their
morphology and location.
Radio halos are located at the cluster center
and are characterized by a
regular shape and extremely low surface brightness.
Relics are similar, but are found at the cluster periphery
and in general have an elongated shape.
In some clusters, both a central halo and a peripheral
relic are present.
While relics are usually strongly polarized,
no significant polarization has been detected so far in radio halos.
In the Coma cluster, the upper limit to the halo fractional polarization
is $\sim$ 10\% at 1400 MHz (Feretti \& Giovannini 1998).
Good upper limits ($<$ 5\%) have been placed for the powerful
radio halos of the galaxy clusters A2219, A2163 and $1E0657-57$
(Bacchi et al. 2003, Feretti et al. 2001, Liang et al. 2000).
In addition to the analysis of the wide, diffuse synchrotron sources,
cluster magnetic fields can be constrained
through the detection
of non-thermal emission of inverse
Compton origin in the hard X-ray wavelengths or
by studying Faraday rotation measure (RM) of polarized radio galaxies.
It is known that these different
methods of analysis give somewhat discrepant results for the magnetic field
strength
(see e.g. reviews by Carilli \& Taylor 2002,
Govoni \& Feretti 2004, and references therein).
The knowledge of the magnetic field structure may be the key issue
to understand the origin of this discrepancy.
Recently, En{\ss}lin \& Vogt (2003), Vogt \& En{\ss}lin (2003),
Murgia et al. (2004), showed that the RM
of radio galaxies can be used to infer
not only the cluster magnetic field strength,
but also the power spectrum of the cluster magnetic
field fluctuations.
Murgia et al. (2004) derived the magnetic field power spectrum
of a sample of clusters for which good RM data of cluster
radio galaxies were
available. They found that A2255 appears as one of the clusters with a
very steep intracluster magnetic field power spectra.
Moreover they pointed out that
morphology and polarization information of radio halos may
provide important constraints on the
power spectrum of the magnetic field fluctuations
on large scales.
In particular, their simulations showed that if the intracluster magnetic
field fluctuates up to scales of some hundred kpc,
then steep magnetic field power spectra
may give rise to detectable polarized filaments.
A2255 is a nearby (z=0.0806, Struble \& Rood 1999),
rich cluster which shows signs of
undergoing a merger event.
It is characterized by the presence of
a diffuse radio halo source at the cluster center,
a relic source at the cluster periphery, and several
embedded head-tail radio galaxies (Jaffe \& Rudnick 1979,
Harris et al. 1980, Burns et al. 1995, Feretti et al. 1997).
We observed A2255 with the purposes
of detecting polarized emission
from the radio halo and obtaining information on the degree of
ordering of the cluster magnetic field.
In this letter we report the results of this high resolution and
sensitivity observation.
Throughout this paper we assume a $\Lambda$CDM cosmology with
$H_0$ = 71 km s$^{-1}$Mpc$^{-1}$,
$\Omega_m$ = 0.3, and $\Omega_{\Lambda}$ = 0.7.
At the distance of A2255, 1$^{\prime\prime}$~ corresponds to 1.5 kpc.
\section{Radio Data}
\begin{figure*}
\centering
\includegraphics[width= 8.8 cm]{fig2a.ps}
\includegraphics[width= 8.8 cm]{fig2b.ps}
\caption{Total intensity radio contours of A2255 at 1.4 GHz with a
FWHM of 25$^{\prime\prime}$$\times$25$^{\prime\prime}$ (natural weighting).
The sensitivity (1$\sigma$) is $\simeq$ 24 $\mu$Jy/beam and the dynamic
range is $\simeq$ 6000.
Contour levels are: 0.07 0.14 0.28 0.56 4.48 mJy/beam.
No primary beam correction has been applied.
Left: the contours of the total intensity
are overlaid on the linearly
polarized intensity (grey-scale). The sensitivity (1$\sigma$)
of the U and Q images is $\simeq$ 11 $\mu$Jy/beam.
The grey-scale shows the polarized flux from 50-150 $\mu$Jy/beam.
Right: the contours of the total intensity
are overlaid on the polarization vectors.
The vector orientation represents the projected E-field
(not corrected for the contribution of the galactic rotation)
while their length is proportional to the fractional polarization
(1$^{\prime}$=50\%).
All pixels with a fractional polarization less than 2$\sigma$,
or with an error in the
polarization angle greater than 10$^{\circ}$, have been blanked.
}
\label{fig2}
\end{figure*}
We observed A2255
with the Very Large Array
(VLA) in C configuration on 2004 April 19,
for a total integration time of $\simeq$11 hours on source.
This full-synthesis produced excellent ($u$,$v$) coverage.
A bandwidth of 25 MHz was used for each of the two IF channels centered at
1465 MHz and 1415 MHz.
The observing frequencies were selected in order to
avoid interferences.
Faraday rotation effects
between the two channels should be small for RMs $<$100 rad m$^{-2}$.
The choice of the bandwidth was selected to reduce bandwidth smearing
effects.
Calibration and imaging were performed with the
NRAO\footnote{The National Radio Astronomy Observatory (NRAO)
is a facility of the National Science Foundation, operated under
cooperative agreement by Associated Universities, Inc.}
Astronomical Image Processing System
(AIPS), following the standard procedure:
Fourier-Transform, Clean and
Restore. Self-calibration was applied to remove residual
phase variations.
Images in the Stokes parameters I, Q and U were made,
using uniform and natural weighting.
The images of the polarized intensity (corrected for the
positive bias), the
fractional polarization and the position angle of polarization
were derived from the I, Q and U images.
The final images, shown here, were convolved with a
circular Gaussian with a FWHM of 15$^{\prime\prime}$~ and 25$^{\prime\prime}$~ respectively.
\section{Results}
A full resolution (FWHM=15$^{\prime\prime}$) image of A2255,
including several radio galaxies, the central halo
and the peripheral relic, is shown in Fig. 1 (contours).
The radio emission is overlaid on the Rosat X-ray image (Feretti et al. 1997)
to show the gas distribution of the cluster.
The diffuse extended radio sources are resolved and
at this high resolution only their brighter regions are visible.
The relic (label R)
appears as a long, straight filament of about 12$^{\prime}$~
in length located 10$^{\prime}$~ to the North-East
from the cluster center and elongated in the
South-East North-West direction.
The halo shows a complex morphology.
The most prominent features detected at this resolution
are straight filaments (labels F1, F2, F3),
each about 6$^{\prime}$~ in length and 2$^{\prime}$~ in width,
nearly perpendicular to each other.
A bridge of low brightness emission (label B),
seems to connect the halo and the relic on the northeast side.
The filaments F1 and F3 are almost parallel to the relic,
while the bridge B is nearly parallel to the filament F2.
In Fig. 2 (contours) a
lower resolution (FWHM=25$^{\prime\prime}$)
image of the cluster radio emission is shown.
Owing to the higher signal to noise ratio,
the low brightness regions of the
diffuse sources are easily visible.
On the left, the contours of the total intensity
are overlaid on the linear
polarized intensity (grey-scale).
On the right, the contours of the total intensity
are overlaid on the polarization vectors.
The vector orientation shows the projected E-field and their
length is proportional to the fractional polarization
(1$^{\prime}$=50\%).
In the figure, all pixels
with a fractional polarization less than 2$\sigma$
or with an error in the polarization angle greater than 10$^{\circ}$~
were blanked.
The polarized emission at 15$^{\prime\prime}$~ resolution (not shown here)
displays similar results.
The most important result is that the bright filaments of the
halo appear strongly polarized at levels of $\simeq$ 20$-$40\%
($\simeq$4$\sigma$$-$8$\sigma$ detections).
Regions of ordered magnetic field of $\sim$400 kpc in size
can be observed.
In the rest of the cluster we don't detect significant
polarized emission except in the brighter regions of the relic
where the fractional polarization is in the
range $\simeq$15$-$30\% ($\simeq$3$\sigma$$-$7$\sigma$ detections).
The upper limit (2$\sigma$) to the fractional polarization in the
fainter regions of the halo (i.e. where the average total intensity emission
is about 0.15 mJy/beam), is $\simeq$ 15\%.
The galactic RM in the direction of A2255
is expected to be about $-6$ rad m$^{-2}$, based on the average
of the RM galactic contribution published by
Simard-Normandin et al. (1981) for sources near the cluster.
Therefore, even if no Faraday rotation occurs within the cluster,
the position angle of the E-field observed at 1.4\,GHz
is rotated by $\sim$15$^{\circ}$~ counter-clockwise
with respect to the intrinsic (at $\lambda$=0) orientation.
The electric polarization vectors of the relic tend to
be roughly perpendicular to the relic elongation
indicating aligned magnetic field structures within it,
while the electric polarization vectors of the halo seems roughly
parallel to the filaments.
Fig. 3 shows the total intensity
image at 25$^{\prime\prime}$~ resolution,
with the discrete sources
subtracted.
The discrete sources were identified by making an image
using long spacings, then their components were subtracted
directly in the {\it uv}-plane (AIPS task UVSUB).
To estimate the flux density of the cluster diffuse emission
the primary beam correction was applied to the image in Fig. 3
(AIPS task PBCOR).
The halo has a total flux density of $\simeq$ 56$\pm$3 mJy,
the relic $\simeq$ 23$\pm$1 mJy,
and their connecting bridge $\simeq$ 6$\pm$0.5 mJy.
The three filaments F1, F2, F3 have flux densities of
9$\pm$0.5, 3$\pm$0.5, and 5$\pm$0.5mJy respectively, indicating
a total flux $\simeq$30\% of the flux of the entire halo.
\section{Discussion}
The absence of a significant polarization in halos
has been interpreted as the result of two concurrent effects:
internal Faraday rotation and beam depolarization.
The thermal intracluster gas is mixed with the relativistic
plasma, thus due to internal Faraday rotation,
significant depolarization may occur within the radio halos.
Moreover as a consequence of their
extremely low surface brightness,
radio halos have been studied so
far at low spatial resolution. This could result
in a significant decrease
of the observed fractional polarization, if the cluster
magnetic field is tangled on scales smaller than the beam.
Murgia et al. (2004) showed that if the outer
scale of the magnetic field fluctuations extends up to
some hundred kpc, and if the power spectrum
\footnote{$\rm{|B_k|^2\propto k^{-n}}$ where n
is the index of the power spectrum of the magnetic field fluctuations.
The power spectrum
is expressed as a vectorial form in $k$-space.}
of the cluster magnetic field is relatively steep (n$\geq$3)
there could be a chance
of detecting filamentary polarized emission in the halo.
The deep and high resolution radio observation
of A2255 presented here confirms their prediction.
The radio halo of A2255 shows, for the first time,
filaments of strong polarized emission.
Moreover, the distribution of the polarization angles
indicates that the magnetic field of this cluster
fluctuates up to scales of about 400 kpc in size.
The detection of polarized emission in a synchrotron halo in A2255 opens up
new questions regarding its nature, origin, and connection
with the history of merging.
The halo filaments could result from a
compression wave, which enhances and aligns disordered
magnetic fields.
Most turbulence theories involve the processes by which the energy
is injected into a medium at large spatial scales and than converted into
motions at smaller and smaller spatial scales until reaching
scales at which it is dissipated.
What we have detected in A2255 may be
the injection in the intracluster medium of energy on large scales,
produced for example by a shock during a cluster merger.
However both the radio morphology of the filaments nearly perpendicular
to each other, and the electric polarization vectors running
roughly parallel to the elongation of the radio halo filaments
(indicating magnetic field structures perpendicular within them)
are quite unusual and difficult to explain in this framework.
Spatial spectral index information, in conjunction with high resolution
cluster X-ray and temperature images, will test
electron re-acceleration models (e.g. turbulence, shock)
responsible for the halo filaments formation and could help to determine whether
these structures are produced by shock waves resulting from
a cluster merger.
Another important issue is to understand why the halo filaments are so
strongly polarized.
They could be cluster foreground structures in which
the Faraday rotation is negligible.
RM information, obtained at well separated frequencies,
will be of great importance to evaluate this possibility.
In this case they should have a low rotation measure.
Finally it is not clear whether these filamentary polarized structures,
are typical features of clusters or if
A2255 is a peculiar case.
One can devise a scenario in which all clusters have
magnetic fields fluctuating both on
small (visible thorough RM data) and large scales
(visible thorough radio halos). But only for those
clusters for which the power spectrum of the magnetic field
fluctuations is steep enough will these polarized
filamentary structures be detectable.
Future observations on other clusters containing radio halos,
and selected on the basis of their magnetic field power spectrum,
are necessary to test these ideas.
\begin{figure}
\centering
\includegraphics[width= 8.5 cm]{fig3.ps}
\caption{
The 1.4 GHz image of the cluster radio emission
after subtraction of the discrete sources.
The FWHM is 25$^{\prime\prime}$ $\times$ 25$^{\prime\prime}$ (natural weighting).
Contour levels are: 0.07 0.14 0.28 0.56 1.12 2.24 4.48 8.96 19.92 35.84
71.68 mJy/beam.
The grey scale flux is $30-450$ $\mu$Jy/beam.
No primary beam correction has been applied to this image.
}
\label{fig3}
\end{figure}
|
{
"timestamp": "2004-11-26T15:47:30",
"yymm": "0411",
"arxiv_id": "astro-ph/0411720",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411720"
}
|
\section{Introduction}
It has been well recognized that
heavy-ion collisions at energies around the Coulomb barrier
are strongly affected by the internal structure of
colliding nuclei \cite{DHRS98,BT98}.
The couplings of the
relative motion to the
intrinsic degrees of freedom (such as collective
inelastic excitations of the colliding nuclei and/or transfer
processes) results in a single potential barrier being replaced
by a number of distributed barriers.
It is now well known that a barrier
distribution can be extracted experimentally
from the fusion excitation function
$\sigma_{\rm fus}(E)$ by taking the second derivative of the product
$E\sigma_{\rm fus}(E)$ with respect to the center-of-mass energy $E$, that is,
$d^2(E\sigma_{\rm fus})/dE^2$ \cite{RSS91}.
The extracted fusion barrier distributions have been
found to be very sensitive to the structure of the colliding nuclei
\cite{DHRS98,L95},
and thus the barrier distribution method has opened up
the possibility of
exploiting the heavy-ion fusion reaction as a ``quantum tunneling
microscope'' in order to investigate both the static and
dynamical properties of atomic nuclei.
The same barrier distribution interpretation can be applied to
the scattering process as well.
In particular, it was suggested in Ref. \cite{ARN88}
that the same information as the fusion cross
section may be obtained from the cross section for quasi-elastic
scattering (a sum of elastic, inelastic, and transfer cross sections)
at large angles.
Timmers {\it et al.}
proposed to use the first derivative of the ratio of the
quasi-elastic cross section $\sigma_{\rm qel}$
to the Rutherford cross section $\sigma_R$ with
respect to energy, $-d (d\sigma_{\rm qel}/d\sigma_R)/dE$, as an
alternative
representation of the barrier distribution \cite{TLD95}.
Their experimental data have revealed
that the quasi-elastic barrier distribution is indeed similar to that
for fusion, although the former may be somewhat smeared and thus
less sensitive to nuclear structure effects
(see also Refs.\cite{PKP02,MSS03,SMO02} for recent measurements).
As an example, we show in Fig. 1 a comparison between the
fusion and the quasi-elastic barrier distributions for the
$^{16}$O + $^{154}$Sm system \cite{HR04}.
\begin{figure}
\includegraphics[scale=0.4,clip]{fig1}
\caption{
(a) The fusion barrier distribution
for the $^{16}$O + $^{154}$Sm reaction.
The solid line is obtained with the
orientation-integrated formula with $\beta_2=0.306$ and $\beta_4$=
0.05.
The dashed lines indicate the contributions from the six
individual eigenbarriers. These lines are
obtained by using a
Woods-Saxon potential with a surface diffuseness parameter $a$ of 0.65
fm. The dotted line is the fusion barrier distribution calculated with
a potential which has $a$ = 1.05 fm.
(b) Same as Fig. 1(a), but for the quasi-elastic barrier
distribution.
(c) Comparison between
the barrier distribution for fusion (solid line)
and that for quasi-elastic scattering (dashed line).
These functions are both normalized to unit area in the energy interval
between 50 and 70 MeV.}
\end{figure}
In this contribution, we
undertake a detailed discussion of the
properties of the quasi-elastic barrier distribution \cite{HR04},
which are less known than the fusion counterpart.
We shall discuss possible advantagges
for its exploitation, putting a particular emphasis on future
experiments with radioactive beams.
\section{Quasi-elastic barrier distributions}
Let us first discuss
heavy-ion reactions between inert nuclei.
The classical fusion cross section is given by,
\begin{equation}
\sigma^{cl}_{\rm fus}(E)=\pi
R_b^2\left(1-\frac{B}{E}\right)\,\theta(E-B),
\end{equation}
where $R_b$ and $B$ are the barrier position and the barrier height,
respectively.
From this expression, it is clear that the first derivative of
$E\sigma^{cl}_{\rm fus}$ is proportional to the classical
penetrability for a 1-dimensional barrier of height $B$ or
eqivalently the s-wave penetrability,
\begin{equation}
\frac{d}{dE}[E\sigma^{cl}_{\rm fus}(E)]=\pi R_b^2\,\theta(E-B)
=\pi R_b^2\,P_{cl}(E),
\end{equation}
and the second derivative to a delta function,
\begin{equation}
\frac{d^2}{dE^2}[E\sigma^{cl}_{\rm fus}(E)]=\pi R_b^2\,\delta(E-B).
\label{clfus}
\end{equation}
In quantum mechanics,
the tunneling effect smears the delta function in Eq. (\ref{clfus}).
If we define the fusion test function as
\begin{equation}
G_{\rm fus}(E)=\frac{1}{\pi R_b^2}\frac{d^2}{dE^2}
[E\sigma_{\rm fus}(E)],
\end{equation}
this function has the following properties:
i) it is symmetric around $E=B$, ii) it is centered on $E=B$, iii)
its integral over $E$ is unity, and iv) it has a relatively narrow width
of around $\hbar\Omega\ln(3+\sqrt{8})/\pi \sim 0.56 \hbar\Omega$,
where $\hbar\Omega$ is the curvature of the Coulomb barrier.
We next ask ourselves
the question of how best to define a similar test function
for a scattering problem.
In the pure classical approach, in the limit of a strong Coulomb field,
the differential cross sections for elastic scattering at $\theta=\pi$
is given by,
\begin{equation}
\sigma_{\rm el}^{cl}(E,\pi)=\sigma_R(E,\pi)\,\theta(B-E),
\end{equation}
where $\sigma_R(E,\pi)$ is the Rutherford cross section.
Thus, the ratio $\sigma_{\rm el}^{cl}(E,\pi)/\sigma_R(E,\pi)$ is the
classical reflection probability $R(E)$ ($=1-P(E)$), and
the appropriate test function for
scattering is \cite{TLD95},
\begin{equation}
G_{\rm qel}(E)=-\frac{dR(E)}{dE}
=-\frac{d}{dE}\left(\frac{\sigma_{\rm el}(E,\pi)}{\sigma_R(E,\pi)}\right).
\label{qeltest}
\end{equation}
In realistic systems, due to the effect of nuclear
distortion, the differential cross section deviates from the
Rutherford cross section even at energies below the barrier.
Using the semi-classical perturbation theory,
we have derived
a semi-classical formula for the backward scattering
which takes into account the nuclear effect to the leading order \cite{HR04}.
The result for a scattering angle $\theta$
reads,
\begin{equation}
\frac{\sigma_{\rm el}(E,\theta)}{\sigma_R(E,\theta)}
=\alpha(E,\lambda_c)\cdot |S(E,\lambda_c)|^2,
\label{ratio}
\end{equation}
where
$S(E,\lambda_c)$ is the total (Coulomb + nuclear)
$S$-matrix at energy $E$ and angular momentum
$\lambda_c = \eta\cot(\theta/2)$, with $\eta$ being the usual Sommerfeld
parameter.
Note that
$|S(E,\lambda_c)|^2$ is nothing but the reflection probability of the
Coulomb barrier, $R(E)$. For $\theta=\pi$, $\lambda_c$ is zero, and
$|S(E,\lambda_c=0)|^2$ is given by
\begin{equation}
|S(E,\lambda_c=0)|^2 = R(E) =
\frac{\exp\left[-\frac{2\pi}{\hbar\Omega}(E-B)\right]}
{1+\exp\left[-\frac{2\pi}{\hbar\Omega}(E-B)\right]}
\end{equation}
in the parabolic approximation.
$\alpha(E,\lambda_c)$ in Eq. (\ref{ratio}) is given by
\begin{eqnarray}
\alpha(E,\lambda_c)&=&1+\frac{V_N(r_c)}{ka}\,
\frac{\sqrt{2a\pi k\eta}}{E}\,\\
&\times&
\left[1-\frac{r_c}{Z_PZ_Te^2}\cdot
2V_N(r_c)
\left(\frac{r_c}{a}-1\right)\right],
\end{eqnarray}
where $k=\sqrt{2\mu E/\hbar^2}$, with $\mu$ being the reduced mass for
the colliding system. The nuclear potential $V_N(r_c)$ is evaluated at
the Coulomb turning point $r_c=(\eta+\sqrt{\eta^2+\lambda_c^2})/k$,
and $a$ is the diffuseness parameter in the nuclear potential.
\begin{figure}
\includegraphics[scale=0.4,clip]{fig2}
\caption{
The ratio of elastic scattering to the Rutherford cross section at
$\theta=\pi$ (upper panel) and the quasi-elastic test
function $G_{\rm qel}(E)=-d/dE (\sigma_{\rm el}/\sigma_R)$ (lower panel)
for the $^{16}$O + $^{144}$Sm reaction.
}
\end{figure}
Figure 2 shows an example of the
excitation function of the cross sections
and the corresponding quasi-elastic test function, $G_{\rm qel}$
at $\theta=\pi$ for the $^{16}$O + $^{144}$Sm reaction.
Because of the nuclear distortion factor $\alpha(E,\lambda_c)$,
the quasi-elastic test function behaves a little less simply
than that for fusion.
Nevertheless, the quasi-elastic test function
$G_{\rm qel}(E)$ behaves rather similarly to the fusion test function
$G_{\rm fus}(E)$.
In particular,
both functions have a similar, relatively narrow, width,
and their integral over $E$ is unity.
We may thus consider that the quasi-elastic test function is an excellent
analogue of the one for fusion, and we exploit this fact in
studying barrier structures in heavy-ion scattering.
In the presence of the channel couplings,
the fusion and the quasi-elastic
cross sections may be given as a weighted sum of the cross sections
for uncoupled eigenchannels,
\begin{eqnarray}
\sigma_{\rm fus}(E)&=&\sum_\alpha w_\alpha
\sigma_{\rm fus}^{(\alpha)}(E), \label{crossfus}\\
\sigma_{\rm qel}(E,\theta)&=&\sum_\alpha w_\alpha
\sigma_{\rm el}^{(\alpha)}(E,\theta), \label{crossqel}
\end{eqnarray}
where $\sigma_{\rm fus}^{(\alpha)}(E)$ and
$\sigma_{\rm el}^{(\alpha)}(E,\theta)$
are the fusion and the elastic cross sections for a potential
in the eigenchannel $\alpha$.
These equations
immediately lead to
the expressions for the barrier distribution in terms of the test
functions,
\begin{eqnarray}
D_{\rm fus}(E)&=&\frac{d^2}{dE^2}[E\sigma_{\rm fus}(E)]=
\sum_\alpha w_\alpha
\pi R_{b,\alpha}^2\,G_{\rm fus}^{(\alpha)}(E),
\label{weightedsum}
\\
D_{\rm qel}(E)&=&
-\frac{d}{dE}\left(\frac{\sigma_{\rm qel}(E,\pi)}{\sigma_R(E,\pi)}\right)
=
\sum_\alpha w_\alpha
G_{\rm qel}^{(\alpha)}(E).
\end{eqnarray}
\section{Advantages over fusion barrier distributions}
There are certain attractive experimental advantages to
measuring the quasi-elastic
cross section $\sigma_{\rm qel}$ rather than the fusion cross sections
$\sigma_{\rm fus}$ to extract a representation of
the barrier distribution.
These are: i) less accuracy is required in the data for taking the
first derivative rather than the second derivative, ii) whereas
measuring the fusion cross section requires specialized recoil
separators (electrostatic deflector/velocity filter) usually of low
acceptance and efficiency, the measurement of
$\sigma_{\rm qel}$ needs only very simple charged-particle detectors,
not necessarily possessing good resolution either in energy or in
charge, and iii) several effective energies can be measured at a
single-beam energy, since, in the semi-classical approximation, each
scattering angle corresponds to scattering at a certain angular
momentum, and the cross section can be scaled in energy by taking
into account the centrifugal correction.
Estimating the centrifugal potential at the Coulomb turning
point $r_c$, the effective energy may be expressed as \cite{TLD95}
\begin{equation}
E_{\rm eff}\sim E
-\frac{\lambda_c^2\hbar^2}{2\mu r_c^2}
=2E\frac{\sin(\theta/2)}{1+\sin(\theta/2)}.
\label{Eeff}
\end{equation}
Therefore, one expects that the function
$-d/dE (\sigma_{\rm el}/\sigma_R)$
evaluated
at an angle $\theta$ will correspond to the quasi-elastic test function
(\ref{qeltest})
at the effective energy given by eq. (\ref{Eeff}).
This last point not only improves the efficiency of the experiment,
but also allows the use of a cyclotron accelerator where the
relatively small energy steps required for barrier
distribution experiments cannot be obtained from the
machine itself \cite{PKP02}.
Moreover, these advantages all point to greater ease of measurement with
low-intensity exotic beams, which will be discussed in the
next section.
\begin{figure}
\includegraphics[scale=0.4,clip]{fig3}
\caption{
Comparison of the
ratio $\sigma_{\rm el}/\sigma_R$ (upper panel) and its energy
derivative $-d/dE (\sigma_{\rm el}/\sigma_R)$ (lower panel)
evaluated at two
different scattering angles.
}
\end{figure}
In order to check the scaling property of the quasi-elastic test function
with respect to the angular momentum,
Fig. 3 compares the functions
$\sigma_{\rm el}/\sigma_R$ (upper panel) and
$-d/dE (\sigma_{\rm el}/\sigma_R)$ (lower panel)
obtained at two different scattering angles.
The solid line is evaluated at $\theta=\pi$, while the dotted line at
$\theta=160^{\rm o}$. The dashed line is the same as the dotted line, but
shifted in energy by $E_{\rm eff}-E$.
Evidently, the scaling does work well, both at
energies below and above the Coulomb barrier, although
it becomes less good
as the scattering angle decreases \cite{HR04}.
\section{Quasi-elastic scattering with radioactive beams}
Low-energy radioactive beams have become increasingly
available in recent years,
and heavy-ion fusion reactions involving neutron-rich nuclei
have been performed for a few systems
\cite{SYW04,LSG03,RSC04}.
New generation facilities have been under
construction at several laboratories, and many more
reaction measurements
with exotic beams at low energies will be performed in the near future.
Although it would still be difficult to perform high-precision
measurements of fusion cross sections with radioactive beams,
the measurement of the quasi-elastic barrier distribution, which
can be obtained much more easily than the fusion
counterpart as we discussed in the previous section, may be feasible.
Since the quasi-elastic barrier distribution contains similar
information as the fusion barrier distribution,
the quasi-elastic measurements at backward angles
may open up a
novel way to probe the structure of exotic neutron-rich nuclei.
\begin{figure}
\includegraphics[scale=0.4,clip]{fig4}
\caption{
The excitation function for quasi-elastic scattering
(upper panel) and the quasi-elastic barrier distribution
(lower panel) for the
$^{32}$Mg + $^{208}$Pb reaction around the Coulomb barrier.
The solid and the dashed lines are the results of coupled-channels
calculations which assume that $^{32}$Mg is a rotational and a
vibrational nucleus, respectively.
The single octupole-phonon excitation in
$^{208}$Pb is also included in the calculations. }
\end{figure}
In order to demonstrate the usefulness of the study of the
quasi-elastic barrier
distribution with radioactive beams, we take as an example the
reaction $^{32}$Mg and $^{208}$Pb, where the quadrupole
collectivity of the neutron-rich $^{32}$Mg remains to be clarified
experimentally.
Fig. 4 shows the excitation function of the quasi-elastic scattering
(upper panel) and the quasi-elastic barrier distribution
(lower panel) for this system.
The solid and dashed lines are results of coupled-channels
calculations where $^{32}$Mg is assumed to be a rotational or
a vibrational nucleus, respectively.
We include the quadrupole excitations in $^{32}$Mg up to the second
member (that is, the first 4$^+$ state in the rotational band for the
rotational coupling, or the double phonon state for the vibrational
coupling). In addition, we include the single octupole phonon
excitation at 2.615 MeV in $^{208}$Pb.
We use a version of the computer code {\tt CCFULL} \cite{HRK99} in order
to integrate the
coupled-channels equations.
One clearly sees well separated peaks in the
quasi-elastic barrier distribution both for the rotational and for the
vibrational couplings. Moreover, the two lines are considerably
different at energies around and above the Coulomb barrier, although
the two results are rather similar below the
barrier.
We can thus expect that the
quasi-elastic barrier distribution can indeed be utilized
to discriminate between the rotational and the
vibrational nature of the quadrupole collectivity in $^{32}$Mg,
although these results might be somewhat perturbed by other
effects which are not considered in the present calculations, such as
double octupole-phonon excitations in the target, transfer
processes or hexadecapole deformations.
We mention that the distorted-wave Born approximation (DWBA) yields
identical results for both
rotational and vibrational couplings (to first order).
In order to discriminate whether the transitions are vibration-like
or rotation-like, at least second-step processes (reorientation
and/or couplings to higher members) are necessary.
The coupling effect plays a more important role in low-energy
reactions than at high and intermediate energies. Therefore,
we expect that quasi-elastic scattering around the
Coulomb barrier will provide a useful means
to allow
the detailed study of the structure of neutron-rich nuclei in
the near future.
\begin{acknowledgments}
This work was supported by the Grant-in-Aid for Scientific Research,
Contract No. 16740139,
from the Japan Society for the Promotions of
Science.
\end{acknowledgments}
|
{
"timestamp": "2004-11-15T02:16:12",
"yymm": "0411",
"arxiv_id": "nucl-th/0411055",
"language": "en",
"url": "https://arxiv.org/abs/nucl-th/0411055"
}
|
\section*{References}}{}
\renewcommand{\cite}[1]{{\sf[##1]}}
\renewcommand{\bibitem}[1]{\par\noindent{\sf[##1]}}
}
\def\thesection.\arabic{equation}{\thesection.\arabic{equation}}
\preprint{hep-th/0411256}
\title{Quantum Geometries of $A_{2}$}
\author{Girma Hailu\thanks{hailu@physics.harvard.edu}
\\
\small\sl Jefferson Laboratory of Physics\\
\small\sl Harvard University \\
\small\sl Cambridge, MA 02138 }
\abstract{We solve $\mathcal{N}=1$ supersymmetric $A_{2}$ type
$U(N)\times U(N)$ matrix models obtained by deforming
$\mathcal{N}=2$ with symmetric tree level superpotentials of any
degree exactly in the planar limit. These theories can be
geometrically engineered from string theories by wrapping D-branes
over Calabi-Yau threefolds and we construct the corresponding exact
quantum geometries.}
\starttext
\setcounter{equation}{0}
\section{Introduction\label{intr}}
A class of supersymmetric gauge theories with tree level
superpotentials can be geometrically engineered from type $IIA$ and
type $IIB$ string theories by wrapping D-branes over various cycles
of Calabi-Yau threefolds. See \cite{GV-1,V-1,KKLM-1,CIV-1,CFIKV} for
instance. The quantum Calabi-Yau geometries can be studied using the
related geometrically engineered gauge theories. Constructing the
exact quantum geometries associated to product gauge group theories
with general tree level superpotentials is a highly nontrivial
problem. More recently, important connections between matrix models
and supersymmetric gauge theories have been found by Dijkgraaf and
Vafa. \cite{DV-1a,DV-1,DV-1b} It was also found by expansion
\cite{DV-1} that the quantum Calabi-Yau geometry that engineers
$A_{2}$ could be expressed in terms of two polynomials. In this
note, we will give analytic proof and find these polynomials by
solving the matrix models for $\mathcal{N}=1$ supersymmetric $A_{2}$
type $U(N)\times U(N)$ gauge theories with symmetric tree level
superpotentials of any degree in the planar limit and thus construct
the corresponding exact quantum Calabi-Yau geometries.
First let us start with a general $\mathcal{N}=2$ supersymmetric
$\prod_{i}U(N_i)$ gauge theory with link chiral superfields $Q_{ij}$
and $Q_{ji}=Q_{ij}^{\dagger}$ transforming as $(\Box,\,\bar{\Box})$
and $(\bar{\Box},\,\Box)$ respectively under $U(N_{i})\times
U(N_{j})$ and the corresponding matrix model. Consider the tree
level superpotential
\begin{equation} W_{\mathrm{tree}}(\Phi,Q)=\sum_{i,j}s_{ij}\mathrm{Tr }\,
Q_{ij}\Phi_{j}Q_{ji}+\sum_{i}\mathrm{Tr }\,
W_{i}(\Phi_{i}),\label{eq:n2-1}\end{equation} where $\Phi_{i}$ is
the scalar chiral superfield associated with $U(N_i)$ and the
indices are ordered such that $s_{ij}=-s_{ji}=1$ with $j>i$ when the
$i^{th}$ and $j^{th}$ gauge groups are linked and $s_{ij}=0$
otherwise. The first term in (\ref{eq:n2-1}) comes from the
superpotential of $\mathcal{N}=2$ with bifundamental
hypermultiplets. The second term is a polynomial in each $\Phi_{i}$
and it will contain quadratic mass terms which break $\mathcal{N}=2$
down to $N=1$. This theory can be geometrically engineered from type
IIB string theory with D3, D5 and D7 branes wrapped over various
cycles of Calabi-Yau threefolds and also from type IIA string theory
with D6 branes wrapped over Calabi-Yau threefolds.
\cite{GV-1,V-1,KKLM-1,CIV-1,CFIKV} The partition function is defined
as
\begin{equation}
Z=\frac{1}{Z_{0}}\int
\prod_{i}d\Phi_{i}\prod_{i<j}dQ_{ij}dQ_{ji}\,e^{-\frac{1}{g_{s}}
W_{\mathrm{tree}}(\Phi,Q)}.\label{zmoose}
\end{equation}
and normalized such that in terms of the eigenvalues
$\lambda_{i,1},\cdots,\lambda_{i,I}, \cdots\lambda_{iN_{i}}$ of
$\Phi_{i}$ it becomes
\begin{equation}
Z=\int\prod_{i,I}d\lambda_{i,I}\mathrm{exp}(-S_{\mathrm{eff}}),
\label{eq:zpq-2}
\end{equation}
with the effective action \cite{K-1}
\begin{equation}
S_{\mathrm{eff}}=\frac{1}{g_{s}}\sum_{i,I}W_{i}(\lambda_{i,I})
-2\sum_{i,I<J}\mathrm{log}|\lambda_{i,I}-\lambda_{i,J}|
+\sum_{i<j,I,J}|s_{ij}|\mathrm{log}|\lambda_{i,I}
-\lambda_{j,J}|.\label{eq:zpq-3}
\end{equation}
Note that the small letter index $i$ denotes the $i^{\mathrm{th}}$
gauge group and the upper letter index $I$ denotes eigenvalues. The
equations of motion are obtained by minimizing (\ref{eq:zpq-3}) with
$\lambda_{i,I}$,
\begin{equation}
W'_{i}(\lambda_{i,I})-2g_{s}\sum_{J\ne
I}\frac{1}{\lambda_{i,I}-\lambda_{i,J}}+g_{s}\sum_{j,J}|s_{ij}|
\frac{1}{\lambda_{i,I}-\lambda_{j,J}}=0.\label{eq:zpq-4}
\end{equation}
Let us introduce the resolvents,
\begin{equation}
w_{i}(x)=\frac{1}{N_{i}}\sum_{I=1}^{N_{i}}\frac{1} {\lambda_{i,I}-x},\label{eq:zpq-5}
\end{equation}
where $x$ is complex. Note that $w_{i}(x)$ obey the asymptotic
larger $x$ behavior
\begin{equation}
w_i(x)\to -\frac{1}{x}.\label{wasymp}
\end{equation}
The eignevalues are distributed on the real
axis of $x$. We will consider the case in which the gauge symmetry
in the low energy theory is unbroken in this note. This corresponds
to the case in which each set of eignevalues $\lambda_{i,I}$ is
separately distributed on a single interval $[a_{i},\,b_{i}]$. The
equations of motion (\ref{eq:zpq-4}) expressed in terms of the
resolvents give
\begin{equation}
S_{i}(w_{i}(x+i0)+w_{i}(x-i0))-\sum_{j}|s_{ij}|S_{j}w_{j}(x)+W'_{i}(x)=0
\label{resqe}
\end{equation}
for $x\in [a_{i},\,b_{i}]$ where
\begin{equation}
S_{i}\equiv g_{s}N_{i}.\label{sing}
\end{equation}
Following Dijkgraaf and Vafa \cite{DV-1a,DV-1,DV-1b}, $S_i$ will be
identified with the glueball superfields defined in terms of the
gauge chiral superfields $W_{i\alpha}$ associated to the confining
$SU(N_{i})$ subgroup of $U(N_{i})$ as
\begin{equation}
S_{i}=-\frac{1}{32\pi^2}\mathrm{Tr\,}W_{i}^{\alpha}W_{i\alpha}.\label{eq:rev4-2}
\end{equation}
In the large $N$ limit, the eigenvalues are continuously
distributed and each resolvent $w_i(x)$ can be written as a sum a
regular function $w_{ir}(x)$ which is a particular solution of
(\ref{resqe}) and another function $w_{is}(x)$ which contains the
singular part of $w_{i}(x)$,
\begin{equation}
w_{i}(x)=w_{ir}(x)+w_{is}(x). \label{resrs}
\end{equation}
We can think of this as a substitution for $w_{i}(x)$ in terms of
$w_{ir}(x)+w_{is}(x)$ where $w_{ir}(x)$ satisfies the regular
equation (\ref{zpq4s}) below and we will then solve for $w_{is}(x)$
such that the asymptotic behavior (\ref{wasymp}) is satisfied. We
will find that $w_{is}(x)$ is the singular part of the resolvent.
Putting (\ref{resrs}) in (\ref{resqe}) and setting
\begin{equation}
2S_{i}w_{ir}(x)-\sum_{j}|s_{ij}|S_{j}w_{jr}(x)+W'_{i}(x)=0,\label{zpq4s}
\end{equation}
we obtain
\begin{equation}
S_{i}(w_{is}(x+i0)+w_{is}(x-i0))-\sum_{j}|s_{ij}|S_{j}w_{js}(x)=0,\label{resqes}
\end{equation}
for $x$ in the branch cut $[a_i,\,b_i]$ of $w_{is}(x)$. In the large
$N$ limit, the eigenvalues become continuous and we introduce the
eigenvalue densities
\begin{equation}
\rho_{i}(\lambda)=\frac{1}{N_{i}}\sum_{I}
\delta(\lambda-\lambda_{i,I}),\label{eq:mrho-1}
\end{equation}
normalized such that $\int \rho_{i}(\lambda)d\lambda=1$, and
(\ref{eq:zpq-5}) becomes
\begin{equation} w_{i}(x)=\int\frac{\rho_{i}
(\lambda)d\lambda}{\lambda-x}.\label{wixr}
\end{equation}
Once $w_i(x)$ are found, (\ref{wixr}) can be inverted to determine
$\rho_{i}(\lambda)$ and
\begin{equation}
\rho_{i}(\lambda)=\frac{1}{2\pi i}(w_i(\lambda+i0)-w_i(\lambda-i0)).
\label{wiri}
\end{equation}
The multi-matrix planar free energy can be conveniently written as
\begin{equation} \mathcal{F}_{0} =
\frac{1}{2}\sum_{i}S_{i}\int d\lambda\rho_{i}
(\lambda)W_{i}(\lambda)-\frac{1}{2}\sum_{i,j}C_{ij}S_{i}S_{j}\int
d\lambda\rho_{i} (\lambda)\mathrm{log}|\lambda|\,,\label{f0rs}
\end{equation}
where $C_{ij}=2\delta_{ij}-|s_{ij}|$ is the Cartan matrix. We will
not do free energy calculations in this note. The reason we have
added this last paragraph is because we find the free energy given
by (\ref{f0rs}) in terms of single integrals simpler and useful for
doing calculations and we have not seen it in the literature on
multi-matrix models. The derivation is given in Appendix
\ref{appfii}.
\setcounter{equation}{0}
\section{Quantum geometries of $A_{2}$}\label{sqga2}
In this section we will explicitly construct the quantum Calabi-Yau
geometries associated to $\mathcal{N}=1$ supersymmetric $A_{2}$ type
$U(N)\times U(N)$ gauge theories obtained by deforming
$\mathcal{N}=2$ with symmetric tree level superpotentials of any
degree and the gauge symmetry unbroken in the low energy theory. In
the low energy theory, the $U(1)$ subgroup of each $U(N)$ decouples
and the $SU(N)$ subgroup confines. The most general asymptotically
free product gauge theories of the type discussed in Section
\ref{intr} for the confining $\Pi_{i} SU(N_{i})$ subgroup with
$\mathcal{N}=2$ supersymmetry and link chiral superfields in the
bifundamental representation are constrained to be only of $A-D-E$
type Dynkin diagrams. See \cite{KMV} for instance. The reason is
that the condition of asymptotic freedom for the $i^\mathrm{th}$
gauge group can be written as $(2\delta_{ij}-\sum_{j\neq
i}|s_{ij}|)N_{j}>0$ and this results in the constraint that all
eigenvalues of the connectivity matrix $|s_{ij}|$ need to be less
that $2$ in order for the theory to be asymptotically free. Thus
$(2\delta_{ij}-|s_{ij}|)$ is the Cartan matrix of $A-D-E$ type
Dynkin diagrams and the most general asymptotically free such
$\mathcal{N}=2$ product gauge theories with link chiral superfields
in the bifundamental representation are of $A-D-E$ type. When the
eignevalues of the connectivity matrix also contain $2$, the beta
function vanishes and theory is conformal and the diagram is that of
affine $\hat{A}-\hat{D}-\hat{E}$ type.
Our interest is $\mathcal{N}=2$ supersymmetric $A_2$ type
$U(N)\times U(N)$ gauge theory deformed to $\mathcal{N}=1$ by
symmetric tree level superpotentials with the gauge symmetry
preserved in the low energy theory. This corresponds to two separate
cuts for the resolvents associated to each gauge group in the matrix
model. It follows from (\ref{wiri}) that each branch cut is a square
root branch cut. The regular parts of the resolvents $w_{1r}(x)$ and
$w_{2r}(x)$ are solutions of the following equations which follow
from (\ref{zpq4s}) with $i,j$ running over $1,2$,
\begin{equation}
2S_{1}w_{1r}(x)-S_{2}w_{2r}(x)+W'_{1}(x)=0\,, \quad
2S_{2}w_{2r}(x)-S_{1}w_{1r}(x)+W'_{2}(x)=0. \label{a2-19}
\end{equation}
The solutions are
\begin{equation}
w_{1r}(x)=-\frac{1}{3S_{1}}\Bigl(2W'_{1}(x)+W'_{2}(x)\Bigl)\,,\quad
w_{2r}(x)=-\frac{1}{3S_{2}}\Bigl(2W'_{2}(x)+W'_{1}(x)\Bigl)\label{a2-19a}
\end{equation}
Now the tree level superpotential (\ref{eq:n2-1}) becomes
\begin{equation} W_{\mathrm{tree}}(\Phi,Q)=\mathrm{Tr }\,
Q_{12}\Phi_{2}Q_{21}-\mathrm{Tr }\, Q_{21}\Phi_{1}Q_{12}+\mathrm{Tr
}\, W_{1}(\Phi_{1})+\mathrm{Tr }\, W_{2}(\Phi_{2}).\label{eq:n2-a2}
\end{equation}
The classical equations of motion are
\begin{eqnarray}
Q_{12}\Phi_{2}-\Phi_{1}Q_{12}=0,\quad
Q_{21}\Phi_{1}-\Phi_{2}Q_{21}=0,\nonumber \\
-Q_{12}Q_{21}+\frac{\partial{W_{1}(\Phi_{1})}}{\partial{\Phi_{1}}}=0,\quad
Q_{21}Q_{12}+\frac{\partial{W_{2}(\Phi_{2})}}{\partial{\Phi_{2}}}=0.\label{a2class-1}
\end{eqnarray}
Combining these equations, we can write
\begin{eqnarray}
(X-S_{1}w_{1r})(X+S_{1}w_{1r}-S_{2} w_{2r})=0,\quad \quad (X+S_{2}w_{2r})=0,\nonumber\\
(Y-S_{2}w_{2r})(Y-S_{1}w_{1r}+S_{2}w_{2r})=0,\quad \quad
(Y+S_{1}w_{1r})=0, \label{a2class-2}
\end{eqnarray}
where $X=-Q_{21}Q_{12}-S_{1}w_{1r}+S_{2} w_{2r}$ and
$Y=Q_{12}Q_{21}+S_{1}w_{1r}-S_{2} w_{2r}$. The singular classical
spectral curve can be written in terms of a complex variable $y$ as
\begin{equation}
(y+S_{1}w_{1r}(x))(y-S_{2}w_{2r}(x))(y-S_{1}w_{1r}(x)+S_{2}
w_{2r}(x))=0. \label{class-sc1}
\end{equation}
The corresponding classical Calabi-Yau geometry is the singular
threefold,
\begin{equation}
uv+(y+S_{1}w_{1r}(x))(y-S_{2}w_{2r}(x))(y-S_{1}w_{1r}(x)+S_{2}
w_{2r}(x))=0, \label{class-cy1}
\end{equation}
which describes the $A_2$ fibration over the $x$ plane, where $u$,
$v$ and $y$ are complex coordinates. At the quantum level, the
classical singularities are resolved and the spectral curve that
describes the quantum resolution of the geometry is that of the
resolved threefold and it should be given by (\ref{class-sc1}) with
the classical values of the resolvents replaced by the singular
parts of the quantum resolvents,
\begin{eqnarray}
&&(y+S_{1}w_{1s}(x)))(y-S_{2}w_{2s}(x))(y-S_{1}w_{1s}(x)+S_{2}
w_{2s}(x))\nonumber\\
&&=(y-S_{1}w_{1r}(x)+S_{1}w_{1}(x)))(y+S_{2}w_{2r}(x)-S_{2}w_{2}(x))\nonumber\\
&& (y+S_{1}w_{1r}(x)-S_{2} w_{2r}(x)-S_{1}w_{1}(x)+S_{2}
w_{2}(x))=0. \label{quantum-sc1}
\end{eqnarray}
Putting the decomposition given by (\ref{resrs}) in
(\ref{quantum-sc1}) and using the classical solution given by
(\ref{a2-19a}) gives
\begin{equation}
(y+S_{1}w_{1r}(x))(y-S_{2}w_{2r}(x))(y-S_{1}w_{1r}(x)+S_{2}
w_{2r}(x))-f(x)\,y-g(x)=0, \label{quantum-sc1b}
\end{equation}
where
\begin{eqnarray}
f(x)&=&S_{1}^{2}w_{1s}(x)^{2}+S_{2}^{2}w_{2s}(x)^{2}
-S_{1}S_{2}w_{1s}(x)w_{2s}(x)\nonumber\\
&&-\frac{1}{3}(W_{1}'(x)^2+
W_{2}'(x)^2+W_{1}'(x)W_{2}'(x))\label{quantum-f}
\end{eqnarray}
and
\begin{eqnarray}
g(x)&=&S_{1}^2 S_{2} w_{1s}(x)^{2}w_{2s}(x)-S_{1} S_{2}^{2}
w_{1s}(x)
w_{2s}(x)^{2}\nonumber\\
&&-\frac{1}{27}(2W_{1}'(x)^{3}-2W_{2}'(x)^{3}
+3W_{1}'(x)^{2}W_{2}'(x)-3W_{1}'(x)W_{2}'(x)^{2}).\label{quantum-g}
\end{eqnarray}
The relation between Calabi-Yau geometries and matrix models was
first found in \cite{DV-1a,DV-1,DV-1b} and the general classical and
quantum curves as in the form (\ref{class-sc1}) and
(\ref{quantum-sc1b}) in terms of two general polynomial functions
$f(x)$ and $g(x)$ was given in \cite{DV-1}. These polynomials
describe the resolution of the singularities in the quantum theory
and constructing them for general tree level superpotential is a
highly nontrivial problem. Here we will construct the exact
polynomials $f(x)$ and $g(x)$ that describe the quantum resolved
geometry for $U(N)\times U(N)$ with symmetric tree level
superpotentials $W_{1}(x)$ and $W_{2}(x)=W_{1}(-x)$ of any degree
with the gauge symmetry preserved in the low energy theory.
The singular parts of the resolvents satisfy (\ref{resqes}) which
for the case of $U(N_1)\times U(N_2)$ becomes
\begin{equation}
S_{1}(w_{1s}(x+i0)+w_{1s}(x-i0))-S_{2}w_{2s}(x)=0\quad \mathrm{for}\, x\in [a_1,\,
b_1]\label{a2-1}
\end{equation}
and
\begin{equation}
S_{2}(w_{2s}(x+i0)+w_{2s}(x-i0))-S_{1}w_{1s}(x)=0\quad \mathrm{for}\, x\in [a_2,\,
b_2].\label{a2-2}
\end{equation}
The resolvents satisfy two independent equations, one quadratic and
the other cubic,
\begin{eqnarray}
S_{1}^{2}w_{1}(x)^{2}+S_{2}^{2}w_{2}(x)^{2}
-S_{1}S_{2}w_{1}(x)w_{2}(x)+S_{1}W_{1}'(x)w_{1}(x)\nonumber \\
+S_{2}W_{2}'(x)w_{2}(x)
+f_{1}(x) +f_{2}(x)=0\label{w12quad}
\end{eqnarray}
and
\begin{eqnarray}
S_{1}^2 S_{2} w_{1}(x)^{2}w_{2}(x)-S_{1} S_{2}^{2} w_{1}(x)
w_{2}(x)^{2}+S_{1}^{2} w_{1}(x)^2W_{1}'(x)+S_{1}w_{1}(x)W_{1}'(x)^2
+f_{1}(x)W_{1}'(x)\nonumber\\
-g_{1}(x)-S_{2}^{2}w_{2}(x)^2
W_{2}'(x)-S_{2}w_{2}(x)W_{2}'(x)^2-f_{2}(x)W_{2}'(x)+g_{2}(x)=0,
\label{w12cub}
\end{eqnarray}
where
\begin{equation}
f_{i}(x)\equiv\frac{S_{i}}{N_{i}}\sum_{I}\frac{W_{i}'(x)
-W_{i}'(\lambda_{iI})}{x-\lambda_{i,I}}\label{aq11}
\end{equation}
and
\begin{equation}
g_{1}(x)\equiv\frac{S_{1}S_{2}}{N_{1}N_{2}}\sum_{I,J}\frac{W_{1}'(x)
-W_{1}'(\lambda_{1I})}{(\lambda_{1,I}-\lambda_{2,J})(x-\lambda_{1,I})},\quad
g_{2}(x)\equiv\frac{S_{1}S_{2}}{N_{1}N_{2}}\sum_{I,J}\frac{W_{2}'(x)
-W_{2}'(\lambda_{2I})}{(\lambda_{2,I}-\lambda_{1,J})(x-\lambda_{2,I})}.
\label{aq11b}
\end{equation}
are polynomials. The quadratic equation for the $O(n)$ matrix model
was obtained in \cite{EKZ} and the quadratic and cubic equations
(\ref{w12quad}) and (\ref{w12cub}) for the $A_2$ model were obtained
in \cite{L-1,KLLRNSW,NSY-1,CT-1}. We have also given a derivation in
Appendix \ref{appqce}. The most general independent equations that
$w_{1s}(x)$ and $w_{2s}(x)$ satisfy are at most cubic in either
$w_{1s}(x)$ and $w_{2s}(x)$ or their combinations and there are
three Riemann sheets with one cut for $w_{1s}(x)$ joining the first
and the second sheets and a second cut for $w_{2s}(x)$ joining the
second and third sheets.
In the large $N$ limit, $w_{i}(x)$ can be expressed as in
(\ref{resrs}) which with (\ref{a2-19a}) in (\ref{w12quad}) and
(\ref{w12cub}) gives
\begin{equation}
S_{1}^{2}w_{1s}(x)^{2}
+S_{2}^{2}w_{2s}(x)^{2}-S_{1}S_{2}w_{1s}(x)w_{2s}(x)=3p(x)\label{wb12a}
\end{equation}
and
\begin{equation}
S_{1}^{2}S_{2}w_{1s}(x)^{2}w_{2s}(x)
-S_{1}S_{2}^{2}w_{1s}(x)w_{2s}(x)^{2} =2q(x),\label{wb12b}
\end{equation}
where
\begin{equation}
p(x)=\frac{1}{9}\Bigl(W_{1}'(x)^2+W_{2}'^2+W_{1}(x)'W_{2}'(x)
-3f_{1}(x)-3f_{2}(x)\Bigr)\label{ppol}
\end{equation}
and
\begin{eqnarray}
q(x)&=&\frac{4}{27}((W'_{1}(x)^{3}-W'_{2}(x)^{3})
+\frac{1}{18}(W'_{1}(x)^{2}W'_{2}(x)-W'_{1}(x)W'_{2}(x)^{2})\nonumber\\
&&-\frac{1}{2}(W'_{1}(x)f_{1}(x)-W'_{2}(x)f_{2}(x))
-((W'_{1}(x)-W'_{2}(x))p(x) +\frac{1}{2}(g_{1}(x)-g_{2}(x)).
\label{qpol}
\end{eqnarray}
Using (\ref{wb12a})-(\ref{qpol}) in (\ref{quantum-f}) and
(\ref{quantum-g}) we have
\begin{equation}
f(x)=3p(x)-\frac{1}{3}(W_{1}'(x)^2+
W_{2}'(x)^2+W_{1}'(x)W_{2}'(x))\label{quantum-f1}
\end{equation}
and
\begin{equation}
g(x)=2q(x)-\frac{1}{27}(2W_{1}'(x)^{3}-2W_{2}'(x)^{3}
+3W_{1}'(x)^{2}W_{2}'(x)-3W_{1}'(x)W_{2}'(x)^{2}).\label{quantum-g1}
\end{equation}
Note that $p(x)$ is a polynomial of degree $2n_{1}$ or $2n_{2}$ and
$q(x)$ is a polynomial of degree $3n_{1}$ or $3n_{2}$ depending on
whether $n_{1}>n_{2}$ or not. Combining (\ref{wb12a}) and
(\ref{wb12b}) gives
\begin{equation}
S_{1}^{3}w_{1s}(x)^{3}-3p(x)S_{1}w_{1s}(x)=
-S_{2}^{3}w_{2s}(x)^{3}+3p(x)S_{2}w_{2s}(x)=2q(x),\label{wb12}
\end{equation}
Our interest is the case in which the two gauge groups are the same
and the potentials $W_{1}(x)$ and $W_{2}(x)$ have the same degree
$n+1$ with $n_{1}=n_{2}=n$ and are symmetric about the origin such
that $W_{2}(x)=W_{1}(-x)$. We will set $N_1=N_2=N$ from now on and
the gauge symmetry is $U(N)\times U(N)$ and it will be preserved in
the low energy theory. Thus we also have $S_1=S_2=g_{s}N\equiv S$.
This corresponds to two separate cuts in $w_{1}(x)$ and $w_{2}(x)$
associated to each gauge group in the matrix model. We will set up
the potentials $W_1(x)$ and $W_2(x)$ such that the branch cuts of
$w_{1}(x)$ and $w_{2}(x)$ will be symmetrically on the positive and
the negative real axis of $x$ respectively and we have
$\lambda_{1,I}>0$ and $\lambda_{2,J}<0$. The equations that
$Sw_{1s}(x)$ and $-Sw_{2s}(x)$ satisfy are similar to that of the
$O(n)$ matrix model investigated in \cite{EKZ} and we will use
techniques developed in \cite{EKZ} to solve the $U(N)\times U(N)$
matrix model. We will impose appropriate boundary conditions that
produce the desired properties described above. First we start with
one of the solutions to the cubic equation (\ref{wb12}) for
$w_{1s}(x)$,
\begin{equation}
w_{1s}(x)=\frac{1}{S}\Bigl(e^{-2\pi i/3} w_{s+}(x) +e^{2\pi
i/3}w_{s-}(x)\Bigr),\label{w1bs}
\end{equation}
where
\begin{equation}
w_{s\pm}(x)=\Bigl(q(x)\mp
\sqrt{q(x)^2-p(x)^3}\Bigr)^{1/3}.\label{wpmg}
\end{equation}
It follows from (\ref{wpmg}) that
\begin{equation}
p(x)=w_{s+}(x)w_{s-}(x), \label{pwpwm}
\end{equation}
\begin{equation}
q(x)=\frac{1}{2}\Bigl(w_{s+}(x)^{3}+w_{s-}(x)^{3}\Bigl),
\label{qwqwm}
\end{equation}
and
\begin{equation}
\sqrt{q(x)^2-p(x)^3}=\frac{1}{2}\Bigl(w_{s+}(x)^{3}
-w_{s-}(x)^{3}\Bigl).\label{sqdw}
\end{equation}
The second resolvent $w_{2s}(x)$ also follows as one of the three
solutions to the cubic equation in (\ref{wb12}) with appropriate
boundary conditions to be imposed,
\begin{equation}
w_{2s}(x)=\frac{1}{S}\Bigl(e^{-\pi i/3} w_{s+}(x) +e^{\pi
i/3}w_{s-}(x)\Bigr).\label{w2bs}
\end{equation}
The third solution to the cubic equation is a linear combination of
the two resolvents.
The square root branch cuts in $w_{1s}(x)$ and $w_{2s}(x)$ come from
$\sqrt{q(x)^2-p(x)^3}$. In order to fulfill the constraint that the
two branch cuts be symmetric, we need to have $a_2=-b_1$ and
$b_2=-a_1$ so that $w_{1s}(x)$ will have its branch cut on
$x\in[a\,,b]$ and $w_{2s}(x)$ on $x\in[-b\,,-a]$ with $b>a>0$. This
will be achieved by imposing the following symmetries which are
extensions of the symmetries imposed in \cite{EKZ} for the $O(n)$
matrix model,
\begin{equation}
w_{s\pm}(x-i0)=e^{\pm 2\pi i/3}w_{s\mp}(x+i0)\quad \mathrm{for} \,
x\in [a,\,b], \label{w1barbc}
\end{equation}
\begin{equation}
w_{s\pm}(x-i0)=e^{\pm 4\pi i/3}w_{s\mp}(x+i0)\quad \mathrm{for} \,
x\in [-b,\,-a] \label{w2barbc}
\end{equation}
\begin{equation}
w_{s+}(x)=w_{s-}(-x). \label{wpmconst}
\end{equation}
It then follows from (\ref{w1bs}), (\ref{wpmg}), (\ref{w2bs}) and
(\ref{wpmconst}) that $w_{2s}(x)=-w_{1s}(-x)$. Note also that
combining (\ref{wpmconst}) with (\ref{sqdw}) implies that
$q(0)^2=p(0)^3$. The main reason for the choice of the symmetries
(\ref{w1barbc}) - (\ref{wpmconst}) on $w_{1s}(x)$ and $w_{2s}(x)$
given in (\ref{w1bs}) and (\ref{w2bs}) is that we have for
$x\in[a\,,b]$,
\begin{equation}
w_{1s}(x-i0)=\frac{1}{S}(w_{s+}(x+i0)+w_{s-}(x+i0)),\quad\quad
w_{2s}(x-i0)=w_{2s}(x+i0), \label{w1sabb}
\end{equation}
and for $x\in[-b\,,-a]$,
\begin{equation}
w_{2s}(x-i0)=-\frac{1}{S}(w_{s+}(x+i0)+w_{s-}(x+i0)),\quad\quad
w_{1s}(x-i0)=w_{1s}(x+i0). \label{w2smabb}
\end{equation}
Thus $w_{1s}(x)$ has a branch cut across $x\in[a\,,b]$ and no
discontinuity across $x\in[-b\,,-a]$. On the other hand, $w_{2s}(x)$
has a branch cut across $x\in[-b\,,-a]$ and no branch cut across
$x\in[a\,,b]$. This is exactly what we wanted.
It follows from (\ref{w1bs}), (\ref{w2bs}) and (\ref{wpmconst}) that
\begin{equation}
w_{s\pm}(x)=-\frac{iS}{\sqrt{3}}\Bigl(e^{-2\pi i/3} w_{1s}(\pm x) -
e^{2\pi i/3}w_{1s}(\mp x)\Bigr)=\frac{iS}{\sqrt{3}}\Bigl(e^{-\pi
i/3} w_{2s}(\pm x) - e^{\pi i/3}w_{2s}(\mp x)\Bigr).\label{w1ps1s}
\end{equation}
The asymptotic behavior of $w_{i}(x)$ given by (\ref{wasymp})
combined with (\ref{resrs}) and (\ref{w1ps1s}) gives the asymptotic
large $x$ behaviors
\begin{equation}
w_{\pm}(x)\to\mp\frac{iS}{\sqrt{3}x},\label{w1ps1sasy}
\end{equation}
where
\begin{equation}
w_{\pm}(x)=w_{r\pm}(x)+w_{s\pm}(x)\label{wpmrsa}
\end{equation}
and $w_{r\pm}(x)$ are given by the same expressions given in
(\ref{w1ps1s}) for $w_{s\pm}(x)$ with $w_{is}(\pm x)$ replaced by
$w_{ir}(\pm x)$. Noting that $Sw_{r\pm}(x)$ are independent of $S$,
let us define
\begin{equation}
\Omega_{\pm}(x)\equiv\frac{\partial
\,(Sw_{\pm})}{\partial{S}}=\frac{\partial
\,(Sw_{s\pm})}{\partial{S}}.\label{omegadef}
\end{equation}
It is then convenient to decompose $w_{s\pm}(x)$ as
\begin{equation}
w_{s\pm}(x)=\Omega_{\pm}(x)h_{\pm}(x),\label{w1spmdec}
\end{equation}
with $\Omega_{\pm}(x)$ obeying the same boundary conditions
(\ref{w1barbc}) - (\ref{wpmconst}) as $w_{s\pm}(x)$ and having the
same large $x$ asymptotic behaviors given in (\ref{w1ps1sasy}) for
$w_{\pm}(x)$. Note that because $\Omega_{\pm}(x)$ are obtained by
taking first derivative of $Sw_{s\pm}(x)$ which have square root
branch cuts, $\Omega_{\pm}(x)$ must have simple poles at the branch
points. Moreover, because $\Omega_{\pm}(x)$ obey the boundary
conditions given in (\ref{w1barbc}) - (\ref{wpmconst}) with the
asymptotic behaviors given by (\ref{w1ps1sasy}),
$\Omega_{+}(x)\Omega_{-}(x)$ is even in $x$ and with the simple
poles at $\pm a$ and $\pm b$, we can write it in its most general
form as
\begin{equation}
\Omega_{+}(x)\Omega_{-}(x)=\frac{S^2}{3}\frac{x^2-e^2}
{(x^2-a^2)(x^2-b^2)}\,,
\
\label{wpmprop1}
\end{equation}
where $e$ is a constant and $\Omega_{+}(x)\Omega_{-}(x)$ has two
zeros at $x=\pm e$. We will choose $x=+e$ to be a zero of
$\Omega_{+}(x)$ and $x=-e$ to be a zero of $\Omega_{-}(x)$.
Following \cite{EKZ}, it is convenient to define functions that will
simplify our notations,
\begin{equation}
g_{\pm}(x)=\frac{\sqrt{(x^2-a^2)(x^2-b^2)}\pm\frac{x}{e}
\sqrt{(e^2-a^2)(e^2-b^2)}} {x^2-e^2}.\label{gpm1}
\end{equation}
The functions $\Omega_{\pm}(x)$ that satisfy the above properties
can then be written in their most general forms as
\begin{equation}
\Omega_{\pm}(x)=-\frac{i}{\sqrt{(x^2-a^2)(x^2-b^2)}}
\Bigl((x^2-e^2)(cg_{\mp}(x)\pm d x)\Bigl)^{1/3}, \label{wpgs}
\end{equation}
where $c$ and $d$ are constants. Putting (\ref{wpgs}) in
(\ref{wpmprop1}), we obtain
\begin{eqnarray}
\frac{a^2 b^2}{e^2}c^{2}-c^2 x^2-d^2 e^2
x^2-2\frac{cd}{e}\sqrt{(e^2-a^2)(e^2-b^2)}\,x^2 +d^2
x^4-\frac{1}{27}S^6(x^2-e^2)^2=0. \label{abeeqn}
\end{eqnarray}
The constants $d$, $c$ and $e$ are expressed in terms of $a$ and $b$
using (\ref{abeeqn}) at $x=\infty$, $0$ and $e$ which give
\begin{equation}
d=\frac{1}{3\sqrt{3}} S^{3}\,,\label{cdpol3}
\end{equation}
\begin{equation}
c=-\frac{2}{3\sqrt{3}}S^3
\frac{\sqrt{(e^2-a^2)(e^2-b^2)}}{e-a^2b^2/e^3},\label{cdpol4}
\end{equation}
and
\begin{equation}
e^{4}+2ab\sqrt{(e^2-a^2)(e^2-b^2)}-a^2b^2=0, \label{cdpol2}
\end{equation}
where the appropriate signs are chosen such that the desired
asymptotic behaviors are produced.
It also follows from (\ref{w1spmdec}) and the constraint that
$w_{s\pm}(x)$ and $\Omega_{\pm}(x)$ satisfy the same boundary
conditions that
\begin{equation}
h_{\pm}(x-i0)= h_{\mp}(x+i0)\quad \mathrm{for} \, x\in [a,\,b]\,
\mathrm{and}\,x\in [-b,\,-a]. \label{hpmarasy}
\end{equation}
Thus $h_{+}(x)+h_{-}(x)$ is regular while $h_{+}(x)-h_{-}(x)$ has
square root branch cuts across $x\in [a,\,b]$ and $x\in [-b,\,-a]$.
Because $w_{\pm}(x)$ and $\Omega_{\pm}(x)$ have the asymptotic
behaviors given in (\ref{w1ps1sasy}) and $w_{s\pm}(x)$ have the
asymptotic behavior $\sim x^{n}$, $h_{\pm}(x)$ must have the large
$x$ behavior $\sim x^{n+1}$. We can write $h_{\pm}(x)$ that
satisfies these constraints in the most general form as
\begin{equation}
h_{\pm}(x)=\sqrt{(x^2-a^2)(x^2-b^2)} \Bigl(A(x^2)g_{\pm}(x) \pm
xB(x^2)\Bigr),\label{hgenf}
\end{equation}
where $A(x^2)$ and $B(x^2)$ are even polynomials of degree at most
$n-2$ and $n-4$ respectively if $n$ is even and each of degree at
most $n-3$ if $n$ is odd. We then obtain $w_{s\pm}(x)$ using
(\ref{wpgs}) and (\ref{hgenf}) with (\ref{cdpol3}) - (\ref{cdpol2})
in (\ref{w1spmdec}),
\begin{eqnarray}
w_{s\pm}(x)=-i\frac{1}{\sqrt{3}}S\Bigl((x^2-e^2)(\frac{e^3}{ab}
g_{\mp}(x)\pm x)\Bigl)^{1/3} \Bigl(A(x^2)g_{\pm}(x) \pm
xB(x^2)\Bigr), \label{wispm-2}
\end{eqnarray}
where $A(x^2)$, $B(x^2)$ and the constants $a$, $b$ and $e$ are
calculated putting (\ref{a2-19a}) and (\ref{wispm-2}) in
(\ref{wpmrsa}), making use of the asymptotic behaviors
(\ref{w1ps1sasy}), and using the constraint given by (\ref{cdpol2})
for any given tree level superpotentials $W_1(x)$ and $W_2(x)$
symmetric about the origin and the resolvents having separate cuts.
With the resolvents completely determined in terms of the input
parameters of the theory, we have solved the matrix model in the
planar limit.
The polynomials $p(x)$ and $q(x)$ follow from (\ref{wispm-2}) in
(\ref{pwpwm}) and (\ref{qwqwm}), see Appendix \ref{pqpolyn} for more
details,
\begin{equation}
p(x)=\frac{S^2}{3}\Bigl((x^2-\frac{a^2 b^2}{e^2})A(x^2)^2+
(\frac{e^3}{ab}-\frac{ab}{e}) x^2 A(x^2)B(x^2) -(x^4-e^2 x^2)
B(x^2)^2\Bigr) \label{pxpol-1}
\end{equation}
and
\begin{eqnarray}
q(x)&=&\frac{i}{6\sqrt{3}}S^{3}\Bigl[ \Bigl((3\frac{a^2
b^2}{e}-e^3)x^2+2\frac{a^3 b^3}{e^3}\Bigl)A(x^2)^3 +
3\Bigl(2x^4-(\frac{a^2b^2}{e^2}+e^2)x^2\Bigl)A(x^2)^2 B(x^2)
\nonumber\\&&+ 3\Bigl((\frac{e^3}{ab}+\frac{a^2b^2}{e})x^4-2abex^2
\Bigl)A(x^2) B(x^2)^2 + \Bigl(2x^6+(\frac{e^6}{a^2 b^2}-3e^2)x^4
\Bigl)B(x^2)^3 \Bigr]. \label{qxpol-1}
\end{eqnarray}
With $p(x)$ and $q(x)$ in hand, we have found the explicit forms of
the quantum resolution functions putting (\ref{pxpol-1}),
(\ref{qxpol-1}) and (\ref{a2-19a}) in (\ref{quantum-f1}) and
(\ref{quantum-g1}).
The final result for the spectral curve that describes quantum
geometry follows from (\ref{a2-19a}), (\ref{quantum-sc1b}) -
(\ref{quantum-g}), (\ref{pxpol-1}) and (\ref{qxpol-1}),
\begin{eqnarray}
&&(y-\frac{1}{3}(2W'_1(x)+W'_2(x))(y
+\frac{1}{3}(W'_1(x)+2W'_2(x))\nonumber\\
&&(y+\frac{1}{3}(W'_1(x)-W'_2(x))-f(x)\,y-g(x)=0,
\label{quantum-geomf}
\end{eqnarray}
where $f(x)$ and $g(x)$ are given by
\begin{eqnarray}
f(x)&=& S^2 \Bigl((x^2-\frac{a^2 b^2}{e^2})A(x^2)^2+
(\frac{e^3}{ab}-\frac{ab}{e}) x^2 A(x^2)B(x^2) -(x^4-e^2 x^2)
B(x^2)^2\Bigr) \nonumber\\&&-\frac{1}{3}(W_{1}'(x)^2+
W_{2}'(x)^2+W_{1}'(x)W_{2}'(x)),\label{quantum-ff1}
\end{eqnarray}
\begin{eqnarray}
g(x)&=& \frac{i}{3\sqrt{3}}S^{3}\Bigl[ \Bigl((3\frac{a^2
b^2}{e}-e^3)x^2+2\frac{a^3 b^3}{e^3}\Bigl)A(x^2)^3 +
3\Bigl(2x^4-(\frac{a^2b^2}{e^2}+e^2)x^2\Bigl)A(x^2)^2 B(x^2)
\nonumber\\&&+ 3 \Bigl((\frac{e^3}{ab}+\frac{a^2b^2}{e})x^4-2abex^2
\Bigl)A(x^2) B(x^2)^2 + \Bigl(2x^6+(\frac{e^6}{a^2 b^2}-3e^2)x^4
\Bigl)B(x^2)^3
\Bigr]\nonumber\\&&-\frac{1}{27}(2W_{1}'(x)^{3}-2W_{2}'(x)^{3}
+3W_{1}'(x)^{2}W_{2}'(x)-3W_{1}'(x)W_{2}'(x)^{2}).\label{quantum-gf1}
\end{eqnarray}
The even polynomials $A(x^2)$ and $B(x^2)$ and the constants $a$,
$b$ and $e$ in (\ref{quantum-ff1}) and (\ref{quantum-gf1}) are
completely determined for any given symmetric tree level
superpotentials such that the branch cuts of $w_1(x)$ and $w_2(x)$
are disconnected and on opposite sides of the origin making use of
the relation given by (\ref{cdpol2}) and the asymptotic behaviors of
$w_{\pm}(x)$ given by (\ref{w1ps1sasy}).
Let us now apply our results to the simple case of symmetric
quadratic potentials,
\begin{equation}
W_{1}(x)=\frac{1}{2}m x^{2}-\alpha x \quad \mathrm{and} \quad
W_{2}(x)=\frac{1}{2}m x^{2}+\alpha x\label{a2-18}
\end{equation}
where $m$ and $\alpha$ are constants such that $w_{1}(x)$ and
$w_{2}(x)$ have non overlapping branch cuts so that the gauge
symmetry is unbroken in the low energy theory. The regular parts of
the resolvents $w_{1r}(x)$ and $w_{2r}(x)$ follow from (\ref{a2-18})
in (\ref{a2-19a}),
\begin{equation}
w_{1r}(x)= -\frac{1}{3S}\Bigl(3m x-\alpha \Bigr),\quad\quad
w_{2r}(x)= -\frac{1}{3S}\Bigl(3m x+\alpha \Bigr). \label{a2-19aa}
\end{equation}
Now (\ref{a2-19aa}) in (\ref{w1ps1s}) with $w_{is}$ and $w_{s\pm}$
replaced by $w_{ir}$ and $w_{r\pm}$ give
\begin{equation}
w_{r\pm}(x)= \mp\frac{i}{\sqrt{3}}m x-\frac{1}{3}\alpha.
\label{w1rpab}
\end{equation}
In this case, the asymptotic behaviors of $w_{\pm}(x)$ require that
$B(x^2)=0$ and $A(x^2)=A$, where $A$ is a constant, which with
(\ref{wispm-2}) and (\ref{w1rpab}) in (\ref{wpmrsa}) give
\begin{eqnarray}
w_{\pm}(x)=\mp\frac{i}{\sqrt{3}}m x-\frac{1}{3}\alpha
-i\frac{1}{\sqrt{3}}A S\Bigl((x^2-e^2)(\frac{e^3}{ab}g_{\mp}(x)\pm
x)\Bigl)^{1/3} g_{\pm}(x). \label{wplusall}
\end{eqnarray}
Note that there are a total of four unknown parameters $A$, $a$, $b$
and $e$. Demanding that $w_{\pm}(x)$ in (\ref{wplusall}) obey the
asymptotic large $x$ limits given by (\ref{w1ps1sasy}) gives three
equations and we have one additional constraint among $a$, $b$ and
$e$ given by (\ref{cdpol2}). The asymptotic limits (\ref{w1ps1sasy})
on (\ref{wplusall}) give the following relations
\begin{equation}
A=-\frac{m}{S}, \label{w1pc1}
\end{equation}
\begin{equation}
e=-i\sqrt{3}\frac{ m}{\alpha}a b, \label{w1pc2}
\end{equation}
and
\begin{equation}
m(a^2+b^2)+4\frac{m^3}{\alpha^2}a^2 b^2-6\frac{m^7}{\alpha^6}a^4 b^4
=2S. \label{w1pc3}
\end{equation}
Combining (\ref{cdpol2}) and (\ref{w1pc3}), we obtain a simple
expression for the sum of the squares of the locations of the branch
points,
\begin{equation}
a^2+b^2=18\frac{S}{m}+2\frac{\alpha^2}{m^2}. \label{w1pc4}
\end{equation}
Explicit expressions for the locations of the branch points $\pm a$
and $\pm b$ and the constant parameter $e$ are given in Appendix
\ref{appbp}. The functions $f(x)$ and $g(x)$ that parameterize the
quantum resolution of the classical Calabi-Yau singularities are
also given in Appendix \ref{appqrq}. The constants $a$, $b$ and $e$
are all completely determined in terms of the parameters of the
theory $m$, $\alpha$ and $S$ through (\ref{aquadf}), (\ref{bquadf})
and (\ref{equadf}). Note also that $a$, $b$ and $e$ have nice
relations. The constants $a$ and $b$ are real for real $S$, $m$ and
$\alpha$ as we demanded and the magnitudes of $a$ and $b$ become
larger as the critical points of the potentials $x=\pm \alpha/m$ get
further away from the origin. Moreover, $e$ is pure imaginary and
nonzero for $a\ne 0$. Our solution describes a theory in which the
gauge symmetry in the low energy theory is preserved and $\alpha/m$
is such that the two branch cuts, one from $w_{1}(x)$ and the other
from $w_{2}(x)$ are disconnected with $b>a>0$. As the parameters of
the theory $\alpha$, $m$ and $S$ are varied, the locations of the
branch points move on the real axis of $x$ and this is related to a
movement of $e$ on the imaginary axis of $x$.
\section{Conclusion}\label{concl}
In conclusion, matrix models in combination with supersymmetric
gauge theories provide very powerful tools that allow us to study
exact nonperturbative physics for systems involving quite general
tree level superpotentials where symmetries and holomorphy alone are
not enough. On the other hand, a class of supersymmetric gauge
theories with tree level superpotentials can be geometrically
engineered in type $IIA$ and type $IIB$ string theories by wrapping
D-branes over various cycles of Calabi-Yau threefolds. The
singularities in the classical Calabi-Yau geometry are resolved by
quantum effects. In this note we have used the combined power of
supersymmetry and matrix models to construct the exact quantum
Calabi-Yau geometries associated to $\mathcal{N}=1$ supersymmetric
$A_{2}$ type $U(N)\times U(N)$ gauge theories with quite general
symmetric tree level polynomial superpotentials of any degree. Even
though our interest in this note was the construction of the quantum
geometries, our exact results could be used to compute the free
energies in the planar limit and the exact nonperturbative dynamical
superpotentials of $A_2$.
\section*{Acknowledgements}
Part of this research was supported by the Department of Energy
under grant number DE-FG02-91ER40654.
|
{
"timestamp": "2005-01-24T18:38:25",
"yymm": "0411",
"arxiv_id": "hep-th/0411256",
"language": "en",
"url": "https://arxiv.org/abs/hep-th/0411256"
}
|
\section{Introduction}
\label{sec:intro}
In hierarchical galaxy formation models, based on the cold dark matter (CDM) paradigm,
very small dwarf galaxies are excellent candidates for earliest formed galaxies. Numerical
simulations based on these models make definitive predictions about the average density
and shape of the dark matter distribution in such galaxies. These dwarf
systems are typically dark matter dominated, even in the innermost regions,
unlike bright galaxies where the stellar population is dynamically important.
Dwarf galaxies could hence provide an unique opportunity to compare observations
with the CDM simulations.
According to the numerical simulations, mass-density distribution
in the inner parts of the simulated dark matter halos could be well
described by a cusp i.e. r$^{-1}$ power law. This cusp in the density
distribution manifests itself as a steeply rising rotation curve in the
inner regions of galaxies. However, this prediction from CDM simulations
is found to disagree with the observed rotation curves of several
dwarf galaxies (e.g. deBlok \& Bosma 2002); the data indicate a constant
density core dark matter distribution. Apart from the shape of the dark
matter halos, numerical simulations also predict an anti-correlation
between the characteristic density and the virial mass of the dark matter
halos. In hierarchical scenarios, the low mass halos form at the earlier times, when the
background was higher, hence dwarf galaxies should have larger dark
matter densities.
All these predictions of the numerical simulations were however untested
in the faintest dwarf galaxies, as it was widely believed that very faint dwarf
irregular galaxies do not show any systematic rotational motions.
From a systematic study of the kinematics of a sample of dwarfs, C\^{o}t\'{e}, Carignan
\& Freeman (2000) found that normal rotation is seen only in galaxies
brighter than -14 mag, while fainter dwarfs have disturbed kinematics.
This result is consistent with the earlier findings of Lo, Sargent \& Young (1993), who
from a study of a sample of dwarfs (with M$_{B\rm} \sim -9$ to M$_{B\rm} \sim -15$)
found that very faint dwarf irregulars have chaotic velocity fields. However, most of the
previous studies were done with a coarser velocity resolution and modest sensitivities.
Inorder to study the kinematics of very faint dwarf irregular galaxies, we obtained
high velocity resolution and high sensitivity GMRT HI 21 cm-line observations of a sample
of galaxies with M$_B>-$12.5.
The GMRT has a hybrid configuration which simultaneously
provides both high angular resolution ($\sim$ 2$''$ if one uses baselines between the arm antennas)
as well as sensitivity to extended emission (from baselines between the antennas in the central
square). This unique hybrid configuration of the GMRT makes it an excellent facility
to study such galaxies. The velocity resolution used for our observations was $\sim1.6$ kms$^{-1}$
and the typical integration time on each source was $\sim 16~-$18 hrs, which gave a typical rms
of $\sim$ 1.0 mJy/Beam per channel. The galaxies in our sample have typical HI masses $\sim$ 10
$^{6-8}$ M$_\odot$.
We present here the results obtained from our GMRT observations.
\section{Results of The GMRT Observations}
\label{sec:result}
Figs.1[B],3[B] show the velocity fields for some of the galaxies in our sample.
Our GMRT observations clearly show that the earlier conclusions about the kinematics
of faintest galaxies were a consequence of observational bias. Since most of the
earlier studies of faint dwarf galaxies were done with coarser
velocity resolutions ($\sim 6 - 7$ kms$^{-1}$) and modest sensitivities,
hence, such observations could not detect systematic gradients, which
are typically $\sim 10-15$ kms$^{-1}$, in the velocity fields of such faint
dwarfs. On the other hand, our high velocity resolution and high
sensitivity observations found large scale systematic velocity gradients
in even the faintest galaxy in our sample i.e. DDO210. The velocity field for
DDO210, differs significantly from the velocity field derived earlier by
Lo et al.(1993). The pattern seen in the GMRT velocity field of DDO210 (fig.~\ref{fig:camb_mom}[B]) is,
to zeroth order, consistent with that expected from rotation, on the other
hand, the velocity field derived by Lo et al.(1993) based on a coarser velocity
resolution of $\sim$ 6 kms$^{-1}$, is indeed ``chaotic". This difference in the
observed kinematics suggests that high velocity resolution and high sensitivity
is crucial in determining the systematic gradients in the velocity fields of
faint galaxies. For some of the galaxies in our sample this large scale
systematic gradients could be modeled as systematic rotation, hence allowing
us to derive the rotation curve for those galaxies and to determine the structure
of their dark matter halos through mass modelling. The rest of this paper discusses
the results obtained from the detailed mass modelling of two of our
sample galaxies viz. Camelopardalis B (Cam B;M$_B\sim-$10.9) and DDO210 (M$_B\sim-$10.6).
\begin{figure}[h!]
\rotatebox{-90}{\epsfig{file=GS75F1.PS,width=2.5in, height=5.5in}}
\caption{{\bf{[A]}}Integrated HI emission map of Cam B at
$40^{''} \times 38^{''}$ resolution overlayed on the Digitised Sky Survey
Image. The contour levels are $3.7, 8.8, 19.1, 24.3, 29.4, 34.6, 39.8, 44.9, 50.1,
55.2, 60.4, 65.5~\&~ 70.7\times 10^{19}$ atoms~cm$^{-2}$.
{\bf{[B]}}The HI velocity field of Cam~B at $24^{''}\times 22^{''}$
resolution. The contours are in steps of 1~kms$^{-1}$ and
range from 70.0~kms$^{-1}$ (the extreme North East contour)
to 84.0~kms$^{-1}$ (the extreme South West contour).
}
\label{fig:camb_mom}
\end{figure}
\subsection{Camelopardalis B}
\label{ssec:camb}
Fig.~\ref{fig:camb_mom}[A] shows the integrated HI column density image of Cam B at 40$''\times 38''$
resolution overlayed on the optical DSS image. The HI mass obtained from the integrated profile
(taking the distance to the galaxy to be 2.2 Mpc) is 5.3$\pm$0.5 $\times10^6$M$_\odot$ and the M$_{HI}$/L$_B$
ratio is found to be 1.4 in solar units.
The velocity field derived from the moment analysis of 24$''\times 22''$ resolution data is shown
in the Fig.~\ref{fig:camb_mom}[B]. The velocity field is regular and the isovelocity contours are approximately
parallel, this is a signature of rigid body rotation. The kinematical major axis of the galaxy has a position
angle $\sim$ 215$^\circ$, i.e. it is well aligned with the major axis of both the HI distribution and the optical
emission.
\begin{figure}[h!]
\rotatebox{-90}{\epsfig{file=GS75F2.PS,width=3.0in,height=5.5in}}
\caption{{\bf{[A]}}The derived rotation curve for Cam B (dashes)
and the rotation curve after applying
the asymmetric drift correction (dash dots).
{\bf{[B]}}Mass models for Cam B using the corrected rotation curve.
The points are the observed data. The total mass of gaseous disk (dashed line)
is $6.6\times10^6 M_\odot$ (after scaling the total HI mass by a factor of 1.25, to include
the contibution of primordial Helium). The stellar disk (short dash dot line) has
$\Upsilon_V=0.2$, giving a stellar mass of $0.7 \times10^6 M_\odot$. The
best fit total rotation curve for the constant density halo model is shown as
a solid line, while the contribution of the halo itself is shown as a
long dash dot line (the halo density is $\rho_0=13.7\times10^{-3}
M_\odot$ pc$^{-3}$). The best fit total rotation curve for an NFW type halo
(for $c=1.0$ and $\Upsilon_V=0.0$) is shown as a dotted line.
}
\label{fig:camb_vrot}
\end{figure}
The rotation curve for Cam B was derived from the velocity fields at 40$''\times 38''$, 24$''\times 22''$
and 16$''\times 14''$ resolution using the tilted ring method (see Begum, Chengalur and Hopp 2003 for details). The
derived hybrid rotation curve is shown as a dashed curve in Fig.~\ref{fig:camb_vrot}[A]. We find that the peak
inclination corrected rotational velocity for Cam B ($\sim$ 7.0 kms$^{-1}$) is comparable to the observed HI
dispersion i.e. V$_{\rm{max}}/\sigma_{\rm{HI}}\sim1.0$. This implies that the random motions provide significant
dynamical support to the system. In other words, the observed rotational velocities underestimate the total
dynamical mass in the galaxy due to a significant pressure support of the HI gas. Hence,
the observed rotation velocities were corrected for this pressure support, using the usual
``asymmetric drift" correction, before constructing mass models (see Begum et al (2003) for details).
The dot-dashed curve in Fig.~\ref{fig:camb_vrot}[A] shows the ``asymmetric drift" corrected rotation curve.
Using the ``asymmetric drift" corrected curve, mass models for Cam B were derived.
The best fit mass model using a modified isothermal halo is shown
in Fig.~\ref{fig:camb_vrot}[B]. Also shown in the figure is
the best fit total rotation curve for an NFW type halo.
As can be seen, the kinematics of Cam~B is well fit with
a modified isothermal halo while an NFW halo provides
a poor fit to the data.
The ``asymmetric drift" corrected rotation curve for Cam B is
rising till the last measured point (Fig.~\ref{fig:camb_vrot}[B]), hence the core
radius of the isothermal halo could not be constrained
from the data. The best fit model for a constant density halo
gives central halo density ($\rho_0$) of 12.0$\times 10^{-3}~M_
\odot$~pc$^{-3}$.
The derived $\rho_0$ is relatively insensitive to the assumed
mass-to-light ratio of the stellar disk ($\Upsilon_V$). We found that
by changing $\Upsilon_V$ from a value of 0 (minimum disk fit)
to a value of 2.0 (maximum disk fit), $\rho_0$ changes by $<$20\%.
From the last measured point of the observed rotation curve, a total dynamical mass of
1.1$\times 10^8 M_\odot$ is derived, i.e. at the last measured point
more than 90\% of the mass of Cam~B is dark. Futher, the dominance of the
dark matter halo together with the linear shape of the rotation
curve (after correction for ``asymmetric drift'') means that one
cannot obtain a good fit to the rotation curve using an NFW halo
regardless of the assumed $\Upsilon_V$.
\begin{figure}[h!]
\rotatebox{0}{\epsfig{file=GS75F3.PS,width=5.5in}}
\caption{{\bf{[A]}} The optical DSS image of DDO210 (greyscales) with
the GMRT 44$^{''}\times37^{''}$ resolution integrated HI
emission map (contours) overlayed.
The contour levels are $0.7, 10.2, 19.8, 29.3, 38.3, 45.5, 57.1,
67.5, 77.1, 86.7, 96.2, 105.7, 121.4~ \& ~124.8 \times 10^{19}$ atoms~cm$^{-2}$.
{\bf{[B]}}The HI velocity field of DDO210 at 29$^{''}\times 23^{''}$
resolution. The contours are in steps of 1~kms$^{-1}$ and
range from $-$145.0~kms$^{-1}$ to $-$133.0~kms$^{-1}$.
}
\label{fig:ddo210_mom}
\end{figure}
\begin{figure}[h!]
\rotatebox{-90}{\epsfig{file=GS75F4.PS,width=3.0in,height=5.5in}}
\caption{{\bf{[A]}}The hybrid rotation curve (dashes) and the rotation curve
after applying the asymmetric drift correction (dots).
{\bf{[B]}} Mass models for DDO210 using the corrected rotation curve.
The points are the observed data. The total mass of gaseous
disk (dashed line) is $3.6\times10^6 M_\odot$ (including the contibution
of primodial He).The stellar
disk (short dash dot line) has
$\Upsilon_B=3.4$, giving a stellar mass of $9.2 \times10^6 M_\odot$.
The best fit total rotation curve for the constant density
halo model is shown as
a solid line, while the contribution of the halo itself
is shown as a long dash dot line (the halo density is
$\rho_0=29~\times10^{-3} M_\odot$ pc$^{-3}$).
The best fit total rotation curve for an NFW type halo,
using $\Upsilon_B=0.5$, c=5.0 and $v_{200}$=38.0~km s$^{-1}$
is shown as a dotted line. See text for more details.
}
\label{fig:ddo210_vrot}
\end{figure}
\subsection{DDO210}
\label{ssec:camb}
DDO210 is the faintest known gas rich dwarf galaxy in our local group.
Karachentsev et al.(2002), based on the HST observations of the I magnitude of
the tip of the red giant branch, estimated the distance to this galaxy to
be 950$\pm$50 kpc.
Fig.~\ref{fig:ddo210_mom}[A] shows the integrated HI column density image of
DDO210 at 44$''\times 37''$
resolution overlayed on the optical DSS image. The HI isodensity contours are
elongated in eastern and southern half of the galaxy, indicating a density enhancement
in these directions.The HI mass obtained from the integrated profile
(taking the distance to the galaxy to be 1.0 Mpc) is 2.8$\pm$0.3 $\times10^6$M$_\odot$
and the M$_{HI}$/L$_B$ ratio is found to be 1.0 in solar units.
The velocity field of DDO210 derived from the moment analysis of 29$''\times 23''$
resolution data cube is shown in Fig.~\ref{fig:ddo210_mom}[B]. The velocity field is
regular and a systematic velocity gradient is seen across the galaxy.
Fig.~\ref{fig:ddo210_vrot}[A] shows the derived rotation curve for DDO210 and the
``asymmetric drift" corrected rotation curve. We find that the ``asymmetric
drift" corrected rotation curve of DDO210 can be well fit with a
modified isothermal halo (with a central density $\rho_0 \sim 29\times10^
{-3}$ $M_\odot$ pc$^{-3}$). In the case of the NFW halo, we find that
there are a range of parameters which provide acceptable fits e.g. (v$_{200}\sim$ 20 kms$^{-1}$,
c $\sim$ 10) to (v$_{200}\sim$ 500 kms$^{-1}$, c$\sim$ 0.001); the halo parameters
could not be uniquely determined from the fit. However, the best fit value of the
concentration parameter c, at any given v$_{200}$ was found to be consistently smaller
than the value predicted by numerical simulations for the $\Lambda$CDM universe (Bullock et al. 2001).
Fig.~\ref{fig:ddo210_vrot}[B] shows the best fit mass models for DDO210 (see Begum \& Chengalur 2004 for details).
\begin{figure}[h!]
\rotatebox{0}{\epsfig{file=GS75F5.PS,width=4.0in}}
\caption{Scatter plots of the central halo density against the
circular velocity and the absolute blue magnitude . The data (empty squares)
are from Verheijen~(1997), Broeils~(1992),
C\^{o}t\'{e} et al.~(2000) and Swaters~(1999). The filled
squares are the medians of the binned data, and the straight
lines are the best fits to the data. Cam~B and DDO210 are
shown as crosses.
}
\label{fig:dens}
\end{figure}
\section{Discussion}
\label{sec:result}
Fig.~\ref{fig:dens} shows the core density of isothermal halo against circular
velocity and absolute blue magnitude for a sample of galaxies, spanning a range of
magnitudes from M$_B\sim-10.0$ mag to M$_B\sim-23.0$ mag. Cam B and DDO210 are also
shown in the figure, lying at the low luminosity end of the sample.
As can be seen in the figure,
there is a trend of increasing halo density with a decrease in circular velocity
and absolute magnitude, shown by a best fit line to the data (solid line),
although the correlation is very weak and noisy. Further, as a guide to an
eye, we have also binned the data and plotted the median value (solid points).
Binned data also shows a similar trend. Such a tread is expected in
hierarchical structure formations scenario (e.g. Navarro,
Frenk \& White 1997).
|
{
"timestamp": "2004-11-24T14:54:32",
"yymm": "0411",
"arxiv_id": "astro-ph/0411664",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411664"
}
|
\section{Equivalence of the SCET and QCD factorization formulae}
We first note that the coefficient function of the spectator-scattering term
can be represented as a convolution $T^{\rm II}=C^{\rm II}\star J$ of hard and
hard-collinear coefficient functions. The formula quoted in
\cite{Bauer:2004tj} follows from (\ref{ff}) by rewriting
\begin{equation}\label{rearrange}
T^{\rm II}\star\Phi_B\star\Phi_\pi\star\Phi_\pi
= C^{\rm II}\star\zeta_J^{B\pi}\star\Phi_\pi \,,
\end{equation}
with $\zeta_J^{B\pi}$ defined as $J\star\Phi_B\star\Phi_\pi$, and $C^{\rm II}$
defined by the decomposition of $T^{\rm II}$ shown above. In addition, in
\cite{Bauer:2004tj} the SCET form factor $\zeta^{B\pi}$ rather than the
physical QCD form factor $F^{B\to\pi}$ is used. As discussed in
\cite{Beneke:2000ry}, this implies another rearrangement of this type.
In general SCET provides a powerful tool to simplify
factorization proofs (a task that has not yet been completed for
the case of $B\to\pi\pi$ considered here), but the resulting QCD
factorization formulae can also be obtained using traditional factorization
methods, as is frequently done in practice.
The authors of \cite{Bauer:2004tj} entertain the possibility that the
hard-collinear scale may be non-perturbative, and hence they choose not to
factorize $\zeta_J^{B\pi}$ into $J\star\Phi_B\star\Phi_\pi$ as indicated
above. This is a logical possibility in the QCD factorization approach.
However, we show below that it is not supported by theoretical calculations.
We disagree with \cite{Bauer:2004tj} on the statement that
$\zeta_J^{B\pi}\ll F^{B\to\pi}$ is a prediction of QCD factorization, whereas
the SCET treatment suggests $\zeta_J^{B\pi}\sim F^{B\to\pi}$. The
factorization formula states that both terms in (\ref{ff}) are of the same
order in $1/m_b$ power counting and that $T^{\rm I}$ and $T^{\rm II}$ start at
$O(\alpha_s^0)$ and $O(\alpha_s(\sqrt{m_b\Lambda}))$, respectively. However,
it does not predict the relative size of the two terms, and as we explain
below, the numerical value of $\zeta_J^{B\pi}$ depends on several
hadronic input parameters, which are rather uncertain at present.
The authors of \cite{Bauer:2004tj} point out that the hard-collinear kernel
$J$ is universal, so that only a single function $\zeta_J^{B\pi}$ appears in
(\ref{rearrange}), which is the same that appears in the factorization of the
$B\to\pi$ form factors. This fact is important for phenomenology when one opts
to treat the hard-collinear scale as non-perturbative, but does not by itself
represent a conceptual difference between the formulae given in
\cite{Beneke:1999br,Beneke:2000ry} and \cite{Bauer:2004tj}. Furthermore, the
usefulness of the universality of $J$ is limited to the approximation where
one neglects radiative corrections to the hard-scattering kernels
$C^{\rm II}$, since only then does the function $\zeta_J^{B\pi}$ reduce to a
single number \cite{Beneke:2003pa,Hill:2004if}. In other words, a
phenomenological treatment of the hard-collinear scale as non-perturbative
relies on the approximation that the kernels are restricted to their
tree-level approximations, whereas one of the key features of QCD
factorization (as opposed to naive factorization) is that one can consistently
include radiative corrections.
\section{Charm-penguin loops}
In \cite{Bauer:2004tj} it is claimed that there is a possible exception to
factorization from diagrams with charm-quark loops (see
Figure~\ref{fig:penguin}). The argument is based on the observation that when
the gluon virtuality is near the $c\bar c$ threshold, $q^2\approx 4 m_c^2$,
the non-relativistic scales $m_c v$ and $m_c v^2$ become important. Since
$m_c v^2\approx\Lambda$ numerically, this appears to introduce a sensitivity
to non-perturbative scales without power-suppression in $1/m_b$. The authors
of \cite{Bauer:2004tj} suggest that these diagrams do not factorize, in the
sense that the long-distance physics at leading-power in the $1/m_b$ expansion
cannot be factored into form factors or light-cone distribution amplitudes. We
emphasize that the question of non-factorizable $c\bar c$ effects at leading
order in the heavy-quark expansion is a different issue than the one raised in
\cite{Ciuchini:2001gv}, where it is speculated that power corrections to the
QCD charm-penguin amplitudes may be numerically large.
\begin{figure}
\begin{center}
\epsfig{file=cpeng2.eps,width=6.0cm}
\end{center}
\vspace{-0.4cm}
\caption{\label{fig:penguin}
Charm-penguin loop.}
\end{figure}
Factorization statements, be they derived
diagrammatically or with soft-collinear effective theory, always concern
properties of the amplitude in certain asymptotic limits, here an expansion in
$1/m_b$, independent of the actual size of the expansion parameter. It is
therefore important to clearly distinguish the issue of factorization at
leading order in the $1/m_b$ expansion from the question of whether there are
non-factorizable contributions which are formally power-suppressed, but which
nevertheless may be numerically significant. In the following discussion, we
focus on the question of factorization in the formal heavy-quark limit.
The most intuitive way of understanding why the threshold region does not
require special treatment is based on quark-hadron duality
\cite{Beneke:2000ry}. The integration over the gluon virtuality in the range
$0\le q^2\le m_b^2$ is weighted by the pion distribution amplitude, which is
smooth over the entire integration region. This provides the necessary
smearing of the loop amplitude, which ensures that the result is given by a
simple partonic calculation up to power corrections, in complete analogy with
the standard justification of the partonic interpretation of inclusive
heavy-meson decays, or cross sections in general. The smearing also ensures
that one can apply the same power-counting arguments that demonstrate
factorization of other diagrams to charm-penguin diagrams with no need to
single out the threshold region. In particular, in the non-relativistic
situation implied in \cite{Bauer:2004tj} there is no need to sum Coulomb
ladder diagrams, since these do not result in large perturbative corrections
after the integration over $q^2$.
Even without invoking the duality argument, the fact that the charm threshold
region comprises only a parametrically small portion of the entire integration
implies a phase-space suppression. This fact has been neglected in the
argument of \cite{Bauer:2004tj}. More precisely, writing $q^2=\bar x m_b^2$,
where $\bar x$ denotes the longitudinal momentum fraction of the anti-quark in
one of the pions, this region is $\Delta\bar x\sim v^2\,(m_c/m_b)^2$ or
$\Delta\bar x\sim\Lambda m_c/m_b^2$, whichever is larger. In order to study
the question of whether long-distance $c\bar c$ loop effects are of leading
order or not, it is necessary to decide how the limit $m_b\to\infty$ is to be
taken. If we define the heavy-quark limit by $m_b\to\infty$ with $m_c$ fixed,
one may distinguish several possibilities such as $m_c\sim\Lambda$,
$m_c v^2\sim\Lambda$, or $m_c v^2\gg\Lambda$. While the physics of the
threshold region is very different for all these cases, they share the common
feature that, for the purposes of power counting, the charm quark can be
considered to be a light quark, and the suppression of long-distance
$c\bar{c}$ effects has the same origin as that for the corresponding diagrams
with light-quark loops, which implies $\Delta\bar x\sim 1/m_b^2$. In addition,
since in this region $\bar x\sim m_c^2/m_b^2$, there is a further suppression
due to the end-point behavior of the pion distribution amplitude (which
vanishes linearly as $\bar x\to 0$).
If we define the heavy-quark limit as $m_{b,c}\to\infty$ with the ratio
$m_c/m_b$ fixed, then there is no power suppression due to the phase-space or
end-point behavior of the distribution amplitudes, but the threshold region is
perturbative, up to a small non-perturbative contribution of order
$v^2\cdot v^2\,(\Lambda/(m_c v^2))^4\sim(\Lambda/m_b)^4$
\cite{Voloshin:1979uv}.
We conclude that charm-penguin diagrams factorize at leading power in $1/m_b$.
The argument for factorization remains valid also for more complicated
higher-order penguin graphs whenever the threshold region is phase-space (and
end-point) suppressed or the charm quark is heavy so that perturbation theory
is applicable.
\section{Validity of perturbation theory at the hard-collinear scale}
In applying the QCD factorization formula to phenomenology the authors of
\cite{Bauer:2004tj} treat the hard-collinear scale $\sqrt{m_b\Lambda}$ as
non-perturbative, and hence the quantity $\zeta_J^{B\pi}$ as an unknown
phenomenological function. This is justified {\em a posteriori\/} following
the result of a phenomenological fit (on which we comment below). This line of
argument ignores the fact that perturbation theory at scales of order
$\sqrt{m_b\Lambda}\sim m_c$ has been used successfully in many important
applications in $B$-physics, including all determinations of $|V_{ub}|$ from
inclusive $B$-decays, and studies of the hadronic decay rate of the $\tau$
lepton. As already mentioned, a serious drawback of treating the
hard-collinear scale as non-perturbative is that it renders the factorization
approach unpredictive beyond the tree approximation, because only the integral
over $\zeta_J^{B\pi}$ and not its functional form can be extracted from
measurements.
A systematic way to address the question of the perturbativity of the
hard-collinear scale is to calculate higher-order corrections in $\alpha_s$ to
the jet function $J$ in the product $T^{\rm II}=C^{\rm II}\star J$, defined as
the Wilson coefficient function arising in the matching of certain (type-B)
SCET$_{\rm I}$
current operators onto four-quark operators of SCET$_{\rm II}$. The
next-to-leading order terms have been computed recently and were found to be
small \cite{Hill:2004if}. Specifically, for the case of light pseudoscalar
mesons and an asymptotic light-cone distribution amplitude
$\Phi_\pi(x)=6x(1-x)$, one finds that the convolution integrals over the jet
function give rise to the series
\begin{equation}\label{nlocorrection}
\frac{\alpha_s(\mu_i)}{\lambda_B}
\left[ 1 + {\alpha_s(\mu_i)\over\pi} \left( \frac{\langle L^2\rangle}{3}
- 1.31 \langle L\rangle + 1.00 \right) + \dots \right] ,
\end{equation}
where $\mu_i\sim\sqrt{m_b\Lambda}$ is the hard-collinear scale, $\lambda_B$ is
the first inverse moment of the $B$-meson distribution amplitude
$\Phi_B(\omega,\mu_i)$, $L=\ln(m_b\omega/\mu_i^2)$, and $\langle\dots\rangle$
denotes an average over $\Phi_B(\omega,\mu_i)$ with measure $d\omega/\omega$.
While the precise form of the $B$-meson distribution amplitude is unknown,
the fact that $\omega\sim\Lambda$ ensures that $L$ cannot be large, giving a
small coefficient to the next-to-leading term. For example, using the results
of \cite{Braun:2003wx} for the moments $\langle L^2\rangle$,
$\langle L\rangle$ the coefficient of $\alpha_s/\pi$ in (\ref{nlocorrection})
is $2.2\pm 0.6$ for $\mu_i^2=0.5\,\mbox{GeV}\,m_b$. There is thus no evidence
that perturbation theory cannot be applied at the hard-collinear scale. We
also note in this context that the power corrections from the hard-collinear
scale are $1/m_b$ suppressed (and not $1/\sqrt{m_b}$) just as those from the
hard scale.
Since the perturbative corrections to the jet function are well behaved, the
quantity $\zeta_J^{B\pi}$ can be factorized and expressed in terms of
convolution integrals over light-cone distribution amplitudes. The question of
the numerical value of $\zeta_J^{B\pi}$, and whether it is a small
contribution to the physical form factor $F^{B\to\pi}$, rests on the
properties of these amplitudes, as well as on other parameters such as the
strange-quark mass. At leading order in perturbation theory, we obtain for
$\zeta_J^{B\pi}$ the result
\begin{equation}
\zeta_J^{B\pi} = \frac{3\pi\alpha_s C_F}{N_c^2}\,
\frac{f_B f_\pi}{M_B\lambda_B}\left(\langle \bar y^{-1}\rangle_\pi
+ r_\chi^\pi X_H\right)
\end{equation}
where the notations of \cite{Beneke:2003zv} have been used. Taking the default
values and uncertainties of the input parameters from this reference, and
adding errors in quadrature, yields $\zeta_J^{B\pi}=0.016\mbox{--}0.064$,
which is small compared with typical values $F^{B\to\pi}=0.24\mbox{--}0.30$.
($F^{B\to\pi}=\zeta^{B\pi}+\zeta_J^{B\pi}$ when hard matching corrections are
neglected.) Taking some correlated parameter variations so as to reproduce the
data on $B\to\pi\pi$ decays, scenario S2 of \cite{Beneke:2003zv} yields the
somewhat increased value $\zeta_J^{B\pi}=0.080$. To obtain significantly
larger results would require a very small value of the hadronic parameter
$\lambda_B$. While this is a logical possibility, a recent QCD sum-rule
calculation of $\lambda_B$ gives a value around $0.45\,$GeV
\cite{Braun:2003wx}, which is in fact somewhat larger than the estimate
adopted in \cite{Beneke:1999br,Beneke:2000ry}.
These estimates are to be compared to the fit result
$\zeta_J^{B\pi}=0.11\pm 0.03$ obtained by the authors of \cite{Bauer:2004tj},
who also find that the bulk of the $B\to\pi$ form factor comes from
$\zeta_J^{B\pi}$, giving the very small result $F^{B\to\pi}=0.17\pm 0.02$.
This picture contradicts the QCD sum rules for heavy-to-light form factors, in
which $\zeta_J^{B\pi}$ must be associated with a radiative correction
\cite{Bagan:1997bp}. The preference of the data for a smaller $B\to\pi$ form
factor together with an increase of the hard-spectator scattering contribution
to the color-suppressed tree amplitude $a_2$ has already been discussed in
\cite{Beneke:2003zv}, which however did not arrive at a similarly extreme
conclusion. This discrepancy can be traced to a few omissions in the
calculation of \cite{Bauer:2004tj}, each of which has a minor effect: the
absence of radiative corrections, the absence of phases in tree amplitudes,
the absence of the scalar up-penguin amplitude in $T_c$, the use of asymptotic
wave functions, and finally, a larger value of $|V_{ub}|$. When these effects
are taken into account and combined with the most recent experimental data,
one finds a significantly smaller value of $\zeta_J^{B\pi}$ and a larger value
of $F^{B\to\pi}$, in qualitative agreement with theoretical expectations.
\section{The QCD penguin amplitude}
We now turn to the discussion of the phenomenological analysis of the
$B\to\pi\pi$ data performed in \cite{Bauer:2004tj}. Our principal criticism in
addition to what has already been described concerns the evaluation of the QCD
penguin amplitude. It may be written as
\begin{equation}
P = a_4 + r_\chi^\pi a_6 + \beta_3 \approx -0.09 \,,
\end{equation}
where $a_4\approx -0.023\,[\alpha_s^0]-0.002\,[\alpha_s^1]$ represents the
vector penguin contribution,
$r_\chi^\pi a_6\approx -0.038\,[\alpha_s^0]-0.014\,[\alpha_s^1]$ the
$1/m_b$ suppressed scalar penguin contribution, and $\beta_3\approx -0.011$ a
power suppressed and rather uncertain penguin annihilation term. (The numbers
are based on the analysis in \cite{Beneke:2003zv}. Without errors they should
be taken only for illustration purposes. In particular, we neglected all
phases, since they are unimportant for the following discussion.)
In the calculation of \cite{Bauer:2004tj} the lowest order ($\alpha_s^0$) and
some of the $\alpha_s$ contributions to $a_4$ (those included in the
phenomenological parameters for hard scattering and charm penguins) are taken
into account. The term $r_\chi a_6+\beta_3$ is dropped, because it is
power-suppressed. Now while it is true that the factorization properties of
power-suppressed contributions in general, and scalar penguin contributions in
particular, have not yet been investigated to all orders in perturbation
theory, the large tree-level contribution to $r_\chi a_6$ suggests that if one
neglects power corrections entirely (as done in \cite{Bauer:2004tj}) one is
certain to obtain a poor approximation. We emphasize that this is unrelated to
the charm-quark loops discussed above, which appear only in the small
$\alpha_s$ corrections. By dropping the scalar penguin amplitude, the authors
of \cite{Bauer:2004tj} are forced to erroneously assign the QCD penguin
amplitude almost entirely to the charm-quark loops.
There is considerable phenomenological evidence that the scalar penguin
amplitude is in approximate agreement with our theoretical expectations. The
suppression of the pseudoscalar-vector and vector-pseudoscalar penguin
amplitudes relative to the pseudoscalar-pseudo\-scalar penguin amplitude
\cite{Beneke:2003zv}, as well as the pattern of drastically different
branching fractions for the decay modes $B\to \eta^{(\prime)} K^{(*)}$
\cite{Beneke:2002jn}, can be attributed directly to the different size and
sign of the $r_\chi a_6$ term relative to $a_4$ in the QCD penguin amplitude.
We are unaware of any other theoretical framework that can explain these
facts. From such studies of penguin dominated $B$-decays we are therefore led
to the conclusion that there is little room for extra contributions to the QCD
penguin amplitude.
\begin{acknowledgments}
We are grateful to Iain Stewart for useful discussions. M.B.\ and M.N.\ would
like to thank the KITP, Santa Barbara, for hospitality during the preparation
of this note. The work of M.B.\ is supported in part by the DFG
Sonderforschungsbereich/Transregio 9 ``Computer-gest\"utzte Theoretische
Teilchenphysik''. The research of M.N.\ is supported by the National Science
Foundation under Grant PHY-0355005, and by the Department of Energy under
Grant DE-FG02-90ER40542. The work of C.T.S.\ is supported by PPARC Grant
PPA/G/0/2002/00468.
\end{acknowledgments}
|
{
"timestamp": "2004-11-12T16:53:00",
"yymm": "0411",
"arxiv_id": "hep-ph/0411171",
"language": "en",
"url": "https://arxiv.org/abs/hep-ph/0411171"
}
|
\section{Introduction}
A central goal of the GAIA mission is to teach us how the Galaxy functions
and how it was assembled. We can only claim to understand the structure of
the Galaxy when we have a dynamical model galaxy that reproduces the
data. Therefore the construction of a satisfactory dynamical model is in a
sense a primary goal of the GAIA mission, for this model will encapsulate
the understanding of galactic structure that we have gleaned from GAIA.
Preliminary working models that are precursors of the final model will also
be essential tools as we endeavour to make astrophysical sense of the GAIA
catalogue. Consequently, before launch we need to develop a model-building
capability, and with it produce dynamical models that reflect fairly fully
our current state of knowledge.
\section{Current status of Galaxy modelling}
The modern era of Galaxy models started in 1980, when the first version of
the Bahcall-Soneira model appeared \citep{BahcallS1}. This model broke new
ground by assuming that the Galaxy is built up of components like those seen
in external galaxies. Earlier work had centred on attempts to infer
three-dimensional stellar densities by directly inverting the observed star
counts. However, the solutions to the star-count equations are excessively
sensitive to errors in the assumed obscuration and the measured magnitudes,
so in practice it is essential to use the assumption that our Galaxy is
similar to external galaxies to choose between the infinity of statistically
equivalent solutions to the star-count equations. Bahcall \& Soneira showed
that a model inspired by data for external galaxies that had only a dozen or
so free parameters could reproduce the available star counts.
\cite{BahcallS1} did not consider kinematic data, but \cite{CaldwellO}
updated the classical work on mass models by fitting largely kinematic data
to a mass model that comprised a series of components like those seen in
external galaxies. These data included the Oort constants, the tangent-velocity
curve, the escape velocity at the Sun and the surface density of the disk
near the Sun.
\cite{BienaymeRC} were the first to fit both kinematic and star-count data
to a model of the Galaxy that was inspired by observations of external
galaxies. They broke the disk down into seven sub-populations by age. Then
they assumed that motion perpendicular to the plane is perfectly decoupled
from motion within the plane, and further assumed that as regards vertical
motion, each subpopulation is an isothermal component, with the velocity
dispersion determined by the observationally determined age-velocity
dispersion relation of disk stars. Each sub-population was assumed to form
a disk of given functional form, and the thickness of the disk was
determined from the approximate formula
$\rho(R,z)/\rho(R,0)=\exp\{[\Phi(R,0)-\Phi(R,z)]/\sigma^2\}$, where $\Phi$
is an estimate of the overall Galactic potential. Once the thicknesses of
the sub-disks have been determined, the mass of the bulge and the parameters
of the dark halo were adjusted to ensure continued satisfaction of the
constraints on the rotation curve $v_c(R)$. Then the overall potential is
recalculated, and the disk thicknesses were redetermined in the new
potential. This cycle was continued until changes between iterations were
small. The procedure was repeated several times, each time with a different
dark-matter disk arbitrarily superposed on the observed stellar disks. The
geometry and mass of this disk were fixed during the interations of the
potential. Star counts were used to discriminate between these dark-matter
disks; it turned out that the best fit to the star counts was obtained with
negligible mass in the dark-matter disk. Although in its essentials the
current `Besan\c on model' \citep{Robin03} is unchanged from the original
one, many refinements and extensions to have been made. In particular, the
current model fits near IR star counts and predicts proper motions and
radial velocities. It has a triaxial bulge and a warped, flaring disk. Its
big weakness is the assumption of constant velocity dispersions and
streaming velocities in the bulge and the stellar halo, and the neglect of
the non-axisymmetric component of the Galaxy's gravitational field.
A consensus that ours is a barred galaxy formed in the early 1990s
\citep{BlitzS1,Binneyetal} and models of the bulge/bar started to appear soon
after. \cite{BinneyGS} and \cite{Freudenreich} modelled the luminosity
density that is implied by the IR data from the COBE mission, while
\cite{Zhao1} and \cite{Hafneretal} used extensions of Schwarzschild's (1979)
modelling technique to produce dynamical models of the bar that predicted
proper motions in addition to being compatible with the COBE data. There was
an urgent need for such models to understand the data produced by searches
for microlensing events in fields near the Galactic centre. The interplay
between these data and Galaxy models makes rather a confusing story because
it has proved hard to estimate the errors on the optical depth to
microlensing in a given field.
The recent work of the Basel group \citep{Bissantz0,Bissantz1,Bissantz2} and the
microlensing collaborations \citep{Eros,popowski} seems at last to have
produced a reasonably coherent picture. \cite{Bissantz1} fit a model to
structures that are seen in the $(l,v)$ diagrams that one constructs from
spectral-line observations of HI and CO. The model is based on
hydrodynamical simulations of the flow of gas in the gravitational potential
of a density model that was fitted to the COBE data
\citep{Bissantz0}. They show that structures observed in the $(l,v)$ plane
can be reproduced if three conditions are fulfilled: (a) the pattern speed
of the bar is assigned a value that is consistent with the one obtained by
\cite{DehnenPattern} from local stellar kinematics; (b) there are four spiral arms
(two weak, two strong) and they rotate at a much lower pattern speed; (c)
virtually all the mass inside the Sun is assigned to the stars rather than a
dark halo.
\cite{Bissantz2} go on to construct a stellar-dynamical model that
reproduces the luminosity density inferred by \citep{Bissantz0}. The model,
which has no free parameters, reproduces both (a) the stellar kinematics in
windows on the bulge, and (b) the microlensing event timescale distribution
determined by the MACHO collaboration \citep{Alcocketal}. The magnitude of
the microlensing optical depth towards bulge fields is still controversial,
but the latest results agree extremely well with the values predicted by
Bissantz \& Gerhard: in units of $10^{-6}$, the EROS collaboration report
optical depth $\tau_6=0.94\pm0.3$ at $(l,b)=(2.5^\circ,-4^\circ)$
\citep{Eros} while Bissantz \& Gerhard predicted $\tau_6=1.2$ at this
location; the MACHO collaboration report $\tau_6=2.17^{+0.47}_{-0.38}$ at
$(l,b)=(1.5^\circ,-2.68^\circ)$ \citep{popowski}, while Bissantz \& Gerhard
predicted $\tau_6=2.4$ at this location.
Thus there is now a body of evidence to suggest that the Galaxy's mass is
dominated by stars that can be traced by IR light rather than by invisible
objects such as WIMPS, and that dynamical galaxy models can successfully
integrate data from the entire spectrum of observational probes of the Milky
Way.
\section{Where do we go from here?}
Since 1980 there has been a steady increase in the extent to which Galaxy
models are dynamical. A model must predict stellar velocities if it is to
confront proper-motion and radial velocity data, or predict microlensing
timescale distributions, and it needs to predict the time-dependent,
non-axisymmetric gravitatinal potential in order to confront spectra-line
data for HI and CO. Some progress can be made by adopting characteristic
velocity dispersions for different stellar populations, but this is a very
poor expedient for several reasons. (a) Without a dynamical model, we do not
know how the orientation of the velocity ellipsoid changes from place to
place. (b) It is not expected that any population will have Gaussian velocity
distributions, and a dynamical model is needed to predict how the
distributions depart from Gaussianity. (c) An arbitrarily chosen set of
velocity distributions at different locations for a given component are
guaranteed to be dynamically inconsistent. Therefore it is imperative that
we move to fully dynamical galaxy models. The question is simply, what
technology is most promising in this connection?
\subsection{Schwarzschild modelling}
The market for models of external galaxies is currently dominated by models
of the type pioneered by \cite{Schwarzschild}. One guesses the galactic
potential and calculates a few thousand judiciously chosen orbits in it,
keeping a record of how each orbit contributes to the observables, such as
the space density, surface brightness, mean-streaming velocity, or velocity
dispersion at a grid of points that covers the galaxy. Then one uses linear
or quadratic programming to find non-negative weights $w_i$ for each orbit
in the library such that the observations are well fitted by a model in
which a fraction $w_i$ of the total mass is on the $i$th orbit.
Schwarzschild's technique has been used to construct spherical, axisymmetric
and triaxial galaxy models that fit a variety of observational constraints.
Thus it is a tried-and-tested technology of great flexibility.
It does have significant drawbacks, however. First the choice of initial
conditions from which to calculate orbits is at once important and obscure,
especially when the potential has a complex geometry, as the Galactic
potential has. Second, different investigators will choose different
initial conditions and therefore obtain different orbits even when using the
same potential. So there is no straightforward way of comparing the
distributon functions of their models. Third, the method is computationally
very intensive because large numbers of phase-space locations have to be
stored for each of orbit. Finally, predictions of the model are subject to
discreteness noise that is larger than one might naively suppose because
orbital densities tend to be cusped (and formally singular) at their edges
and there is no natural procedure for smoothing out these singularities.
\subsection{Torus modelling}
In Oxford over a number of years we developed a technique in which orbits
are not obtained as the time sequence that results from integration of the
equations of motion, but as images under a canonical map of an orbital torus
of the isochrone potential. Each orbit is specified by its actions $\b J$
and is represented by the coefficients $S_{\b n}(\b J')$ that define the
function $S(\b J',\theta)=\b J'\cdot\theta+\sum_{\b n}S_\b n\e^{\i\b
n\cdot\theta}$ that generates the map. Once the $S_\b n$ have been
determined, analytic expressions are available for $\b x(\theta)$ and $\b
v(\theta)$, so one can readily determine the velocity at which the orbit
would pass through any given location. Since orbits are labelled by actions,
which define a true mapping of phase space, it is straightforward to
construct an orbit library by systematically sampling phase space at the
nodes of a regular grid of actions $\b J'$. Moreover, a good approximation
to an arbitrary orbit can be obtained by interpolating the $S_\b n(\b J')$.
If the orbit library is generated by torus mapping, it is easy to determine
the distributon function from the weights. When the orbit weigts are
normalized such that $\sum_iw_i=1$, and the distribution function is
normalized such that $\int{\rm d}}\def\e{{\rm e}}\def\i{{\rm i}^3\b x{\rm d}}\def\e{{\rm e}}\def\i{{\rm i}^3\b v\,f=1$, then
\[
f(\b J)={1\over(2\pi)^3}\sum_iw_i\delta^{(3)}(\b J-\b J_i).
\]
If the action-space gid is regular with spacing $\Delta$, we can obtain an
equivalent smoothed distribution function by replacing $\delta^{(3)}(\b J-\b J_i)$
by $\Delta^{-3}$ if $\b J$ lies within a cube of side $\Delta$ centred on
$\b J_i$, and zero otherwise. Different modellers can easily compare their
smoothed distribution functions. Finally, with torus mapping many fewer
numbers need to be stored for each orbit -- just the $S_\b n$ rather than
thousands of phase-space locations $(\b x,\b v)$.
The drawbacks of torus mapping are these. First, it requires complex
special-purpose software, whereas orbit integration is trivial. Second, it
has to date only been demonstrated for systems that have two degrees of
freedom, such as an axisymmetric potential \citep{McGillB}, or a planar bar
\citep{kaasalainen1}. Finally, orbits are in a fictitious integrable
Hamiltonian \citep{kaasalainenB} rather than in the, probably non-integrable,
potential of interest. I return to this point below.
\subsection{Syer--Tremaine modelling}
In both the Schwarzschild and torus modelling strategies one starts by
calculating an orbit library, and the weights of orbits are determined only
after this step is complete. \cite{SyerT} suggested an alternative stratey,
in which the weights are determined simultaneously with the integration of
the orbits. Combining these two steps reduces the large overhead involved in
storing large numbers of phase-space coordinates for individual orbits.
Moreover, with the Syer--Tremaine technique the potential does not have to
be fixed, but can be allowed to evolve in time, for example through the usual
self-consistency condition of an N-body simulation.
To describe the Syer--Tremaine algorithm we need to define some notation.
Let $\b z\equiv(\b x,\b v)$ denote an arbitrary point in phase space. Then
each observable $y_\alpha$ is defined by a kernel $K_\alpha(\b z)$ through
\[
y_\alpha=\int{\rm d}}\def\e{{\rm e}}\def\i{{\rm i}^6\b z\,f(\b z)K_\alpha(\b z).
\]
For example, if $Y_\alpha$ is the density at some point $\b x_\alpha$, then
$K_\alpha(\b x,\b v)$ would be $\delta^{(3)}(\b x-\b x_\alpha)$. In an orbit model
we take $f$ to be of the form $f(\b z)=\sum_iw_i\delta^{(6)}(\b z-\b z_i)$ and the
integral in the last equation becomes a sum over orbits:
\[\label{discy}
y_\alpha=\sum_iw_iK_\alpha(\b z_i).
\]
If we simultaneously integrate a large number of orbits in a common
potential $\Phi$ (which might be the time-dependent potential that is
obtained by assigning each particle a mass $w_iM$), then through equation
(\ref{discy}) each observable becomes a function of time. Let $Y_\alpha$ be
the required value of this observable, then Syer \& Tremaine adjust the
value of the weight of the $i$th orbit at a rate
\[
{{\rm d}}\def\e{{\rm e}}\def\i{{\rm i} w_i\over{\rm d}}\def\e{{\rm e}}\def\i{{\rm i} t}=- w_i\sum_\alpha{K_\alpha[\b z_i(t)]\over Z_\alpha}
\left({y_\alpha\over Y_\alpha}-1\right).
\]
Here the positive numbers $Z_\alpha$ are chosen judiciously to stress the
importance of satisfying particular constraints, and can be increased to
slow the rate at which the weights are adjusted. The numerator $K_\alpha[\b
z_i(t)]$ ensures that a discrepancy between $y_\alpha(t)$ and $Y_\alpha$
impacts $w_i$ only in so far as the orbit contributes to $y_\alpha$. The right
side starts with a minus sign to ensure that $w_i$ is decreased if
$y_\alpha>Y_\alpha$ and the orbit tends to increase $y_\alpha$.
\cite{Bissantz2} have recently demonstrated the value of the Syer \& Tremaine
algorithm by using it to construct a dynamical model of the inner Galaxy in
the pre-determined potential of \cite{Bissantz0}.
N-body simulations have been enormously important for the development of our
understanding of galactic dynamics. To date they have been of rather limited
use in modelling specific galaxies, because the structure of an N-body model
has been determined in an obscure way by the initial conditions from which
it is started. In fact, a major motivation for developing other modelling
techniques has been the requirement for initial conditions that will lead to
N-body models that have a specified structure \citep[e.g][]{KuijkenD}.
Nothwithstanding this difficulty, \cite{Fux} was able to find an N-body
model that qualitatively fits observations of the inner Galaxy. It will be
interesting to see whether the Syer--Tremaine algorithm can be used to
refine a model like that of Fux until it matches all observational
constraints.
\section{Hierarchical modelling}
When trying to understand something that is complex, it is best to proceed
through a hierarchy of abstractions: first we paint a broad-bruish picture
that ignores many details. Then we look at areas in which our first picture
clearly conflicts with reality, and understand the reasons for this
conflict. Armed with this understanding, we refine our model to eliminate
these conflicts. Then we turn to the most important remaining areas of
disagreement between our model and reality, and so on. The process
terminates when we feel that we have nothing new or important to learn from
residual mismatches between theory and measurement.
This logic is nicely illustrated by the dynamics of the solar system. We
start from the model in which all planets move on Kepler ellipses around the
Sun. Then we consider the effect on planets such as the Earth of Jupiter's
gravitational field. To this point we have probably assumed that all bodies
lie in the ecliptic, and now we might consider the non-zero inclinations of
orbits. One by one we introduce disturbances caused by the masses of the
other planets. Then we might introduce corrections to the equations of motion from
general relativity, followed by consideration of effects that arise because
planets and moons are not point particles, but spinning non-spherical
bodies. As we proceed through this hierarchy of models, our orbits will
proceed from periodic, to quasi-periodic to chaotic. Models that we
ultimately reject as oversimplified will reveal structure that was
previously unsuspected, such as bands of unoccupied semi-major axes in the
asteroid belt. The chaos that we will ultimately have to confront will be
understood in terms of resonances between the orbits we considered in the
previous level of abstraction.
The impact of Hipparcos on our understanding of the dynamics of the solar
neighbourhood gives us a flavour of the complexity we will have to confront
in the GAIA catalogue. When the density of stars in the $(U,V)$ plane was
determined \citep{DehnenStruct,Fux}, it was found to be remarkably lumpy,
and the lumps contained old stars as well as young, so they could not be
just dissolving associations, as the classical interpretation of star
streams supposed. Now that the radial velocities of the Hipparcos survey
stars are available, it has become clear that the Hyades-Pleiades and Sirius
moving groups are very heterogeous as regards age \citep{Famaey}. Evidently
these structures do not reflect the patchy nature of star formation, but
have a dynamical origin. They are probably generated by transient spiral
structure \citep{Desimone}, so they reflect departures of the Galaxy from
both axisymmetry and time-independence. Such structures will be most readily
understod by perturbing a steady-state, axisymmetric Galaxy model.
A model based on torus mapping is uniquely well suited to such a study because
its orbits are inherently quasi-periodic structures with known
angle-action coordinates. Consequently, we have everything we need to use
the powerful techniques of canonical perturbation theory.
\begin{figure}[ht]
\begin{center}
\leavevmode
\centerline{\psfig{file=binney_fig1a.ps,width=\hsize}}
\centerline{\psfig{file=binney_fig1b.ps,width=\hsize}}
\end{center}
\caption{Using perturbation theory to model a resonant family of orbits.
Dots show consequents obtained by numerical integration. The curves in the
top panel
show resonant orbits obtained by applying perturbation theory to
orbits obtained by torus mapping. The curves in the lower panel show three
of these orbits, one through the centre of the resonant region and one on
each side. From \cite{KaasalainenT}.}
\label{fig1}
\end{figure}
\begin{figure}[ht]
\begin{center}
\leavevmode
\centerline{\psfig{file=binney_fig2a.ps,width=\hsize}}
\centerline{\psfig{file=binney_fig2b.ps,width=\hsize}}
\end{center}
\caption{Obtaining resonant orbits by direct torus mapping. The lower
panel shows a surface of section that is largely taken up by several
powerful resonances. Three non-resonant orbits obtained by torus mapping
are shown, one through the middle and one on each side of the largest
resonant family. The uper panel shows several orbits of the resonant
family that are obtained by directly mapping isochrone orbits. Notice that
the chaotic region is nicely contained between two of these mapped orbits.
From \cite{KaasalainenT}.}
\label{fig2}
\end{figure}
Even in the absence of departures from axisymmetry or time-variation in the
potential, resonances between the three characteristic frequencies of a
quasi-periodic orbit can deform the orbital structure from that encountered
in analytically integrable potentials. Important examples of this phenomenon
are encountered in the dynamics of triaxial elliptical galaxies, where
resonant `boxlets' almost entirely replace box orbits when the potential is
realistically cuspy \citep{MerrittF}, and in the dynamics of disk galaxies,
where the 1:1 resonance between radial and vertical oscillations probably
trapped significant numbers of thick-disk stars as the mass of the thin disk
built up \citep{SridharT}. \cite{kaasalainen3} has shown that such families
of resonant orbits may be very successfully modelled by applying
perturbation theory to orbits obtained by torus mapping. If the resonant
family is exceptionally large, one may prefer to obtain its orbits directly
by torus mapping \citep{kaasalainen2} rather than through perturbation
theory. Figures \ref{fig1} and \ref{fig2} show examples of each approach to
a resonant family. Both figures show surfaces of section for motion in a
planar bar. In Figure \ref{fig1} a relatively weak resonance is successfuly
handled through perturbation theory, while in Figure \ref{fig2} a more
powerful resonance that induces significant chaos is handled by directly
mapping isochrone orbits into the resonant region. These examples
demonstrate that if we obtain orbits by torus mapping, we will be able to discover what
the Galaxy would look like in the absence of any particular resonant family
or chaotic region, so we will be able to ascribe particular features in the
data to particular resonances and chaotic zones. This facility will make the
modelling process more instructive than it would be if we adopted a simple
orbit-based technique.
\section{Confronting the data}
A dynamical model Galaxy will consist of a gravitational potential $\Phi(\b
x)$ together with distribution functions $f_\alpha(\b J)$ for each of
several stellar populations. Each distribution function may be represented by
a set of orbital weights $w_i$, and the populations will consist of
probability distributions in mass $m$, metallicity $Z$ and age $\tau$ that a
star picked from the population has the specified characteristics. Thus a
Galaxy model will contain an extremely large number of parameters, and
fitting these to the data will be a formidable task.
Since so much of the Galaxy will be hidden from GAIA by dust, interpretation
of the GAIA catalogue will require a knowledge of the three-dimensional
distribution of dust. Such a model can be developed by the classical method
of comparing measured colours with the intrinsic colours of stars of known
spectral type and distance. At large distances from the Sun, even GAIA's
small parallax errors will give rise to significantly uncertain distances,
and these uncertainties will be an important limitation on the reliability
of any dust model that one builds in this way.
Dynamical modelling offers the opportunity to refine our dust model because
Newton's laws of motion enable us to predict the luminosity density in
obscured regions from the densities and velocities that we see elsewhere,
and hence to detect obscuration without using colour data.
Moreover, they require that the luminosity distributions of hot components
are intrinsically smooth, so fluctuations in the star counts of these
populations at high spatial frequencies must arise from small scale
structure in the obscuring dust. Therefore, we should solve for the
distribution of dust at the same time as we are solving for the potential and
the orbital weights.
In principle one would like to fit a Galaxy model to the data by predicting
from the model the probability density $P(\alpha,\ldots,v)$ of detecting a
star at given values of the catalogue variables, such as celestial
coordinates $\alpha$, parallax $\varpi$, and proper motons $\mu$, and then
evaluating the likelihood $L=\prod_i P_i$, where the product runs over stars
in the catalogue and
\begin{eqnarray}
P_i=\int{\rm d}}\def\e{{\rm e}}\def\i{{\rm i}^2\alpha\,{\rm d}}\def\e{{\rm e}}\def\i{{\rm i}\varpi\,{\rm d}}\def\e{{\rm e}}\def\i{{\rm i}^2\mu\,{\rm d}}\def\e{{\rm e}}\def\i{{\rm i} v\,
{\e^{-(\alpha-\alpha_i)^2/2\sigma^2_\alpha}\over2\pi\sigma^2_\alpha}
\nonumber\\
\times\cdots\times
{\e^{-(v-v_i)^2/2\sigma^2_v}\over(2\pi\sigma^2_v)^{1/2}}\,P(\alpha,\ldots,v)
\end{eqnarray}
with $\alpha_i\ldots v_i$ the measured values and
$\sigma_\alpha\ldots\sigma_v$ the associated uncertainties. Unfortunately,
it is likely to prove difficult to obtain the required probability density
$P$ from an orbit-based model, and we will be obliged to compare the real
catalogue to a pseudo-catalogue derived from the current model. Moreover,
standard optimization algorithms are unlikely to find the global maximum in
$L$ without significant astrophysical input from the modeller. In any event,
evaluating $P_i$ for each of $\sim10^9$ observed stars is a formidable
computational problem. Devising efficient ways of fitting models to the
data clearly requires much more thought.
\subsection{Adaptive dynamics}
Fine structure in the Galaxy's phase space may provide crucial assistance in
fitting a model to the data. Two inevitable sources of fine structure are
(a) resonances, and (b) tidal streams. Resonances will sometimes be marked
by a sharp increase in the density of stars, as a consequence of resonant
trapping, while other resonances show a deficit of stars. Suppose the data
seem to require an enhanced density of stars at some point in action space
and you suspect that the enhancement is caused by a particular resonance. By
virtue of errors in the adopted potential $\Phi$, the frequencies will not
actually be in resonance at the centre of the enhancement. By appropriate
modification of $\Phi$ it will be straightforward to bring the frequencies
into resonance. By reducing the errors in the estimated actions of orbits, a
successful update of $\Phi$ will probably enhance the overdensity around the
resonance. In fact, one might use the visibility of density enhancements to
adjust $\Phi$ very much as the visibility of stellar images is used with
adaptive optics to configure the telescope optics.
A tidal stream is a population of stars that are on very similar orbits --
the actions of the stars are narrowly distributed around the actions of the
orbit on which the dwarf galaxy or globular cluster was captured.
Consequently, in action space a tidal stream has higher contrast than it does
in real space, where the stars' diverging angle variables gradually spread
the stars over the sky. Errors in $\Phi$ will tend to disperse a tidal
stream in action space, so again $\Phi$ can be tuned by making the tidal stream as
sharp a feature as possible.
\section{Conclusions}
Dynamical Galaxy models have a central role to play in attaining GAIA's core
goal of determining the structure and unravelling the history of the Milky
Way.
Even though people have been building Galaxy models for over half a
century, we are still only beginning to construct fully dynamical models,
and we are very far from being able to build multi-component dynamical
models of the type that the GAIA will require.
At least three potentially viable Galaxy-modelling technologies can be
identified. One has been extensively used to model external galaxies, one
has the distinction of having been used to build the currently leading
Galaxy model, while the third technology is the least developed but
potentially the most powerful. At this point we would be wise to pursue all
three technologies.
Once constructed, a model needs to be confronted with the data. On account
of the important roles in this confrontation that will be played by
obscuration and parallax errors, there is no doubt in my mind that we need
to project the models into the space of GAIA's catalogue variables $(\alpha,
\varpi,\ldots)$. This projection is simple in principle, but will be
computationally intensive in practice.
The third and final task is to change the model to make it fit the data
better. This task is going to be extremely hard, and it is not clear at this
point what strategy we should adopt when addressing it. It seems possible
that features in the action-space density of stars associated with
resonances and tidal streams will help us to home in on the correct
potential.
There is much to do and it is time we started doing it if we want to have a
reasonably complete box of tools in hand when the first data arrive in
2012--2013. The overall task is almost certainly too large for a single
institution to complete on its own, and the final galaxy-modelling machinery
ought to be at the disposal of the wider community than the dynamics
community since it will be required to evaluate the observable implications
of any change in the characteristics or kinematics of stars or interstellar
matter throughout the Galaxy. Therefore, we should approach the problem
of building Galaxy models as an aspect of infrastructure work for GAIA,
rather than mere science exploitation.
I hope that in the course of the next year interested parties will enter
into discussions about how we might divide up the work, and define interface
standards that will enable the efforts of different groups to be combined in
different combinations. It is to be hoped that these discussions lead before
long to successful applications to funding bodies for the resources that
will be required to put the necessary infrastructure in place by 2012.
|
{
"timestamp": "2004-11-09T16:43:12",
"yymm": "0411",
"arxiv_id": "astro-ph/0411229",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411229"
}
|
\section{Introduction}
One of the most important prediction of general
relativity is gravitational radiation. Coalescing neutron star
binaries are considered among the strongest and most likely sources of
gravitational waves to be seen by VIRGO/LIGO interferometers
\cite{Kalogera04, Belczynski02}. Due to the emission of
gravitational radiation, binary neutron stars decrease their orbital
separation and finally merge. Gravitational waves emitted during the
last few orbits of inspiral could yield important informations about
the equation of state (EOS) of dense matter
\cite{Faber02,TanigG03,Oechslin04,Bejger04}. With accurate templates
of gravitational waves from coalescing binary compact stars, it may be
possible to extract information about physics of neutron stars from
signals observed by the interferometers and to solve one of the
central but also most complex problem of physics --- the problem of the
absolute ground state of matter at high densities. It is still an open
question whether the core of a neutron star consists mainly of
superfluid neutrons or exotic matter like strange quark matter, pions
or kaons condensates (see e.g. Ref.~\cite{Haens03} for a recent
review). The possibility of the existence of quark
matter dates back to the early seventies. Bodmer \cite{Bodmer71}
remarked that matter consisting of deconfined up, down and strange
quarks could be the absolute ground state of matter at zero pressure
and temperature. If this is true then objects made of such matter,
the so-called {\em strange stars}, could exist
\cite{Witten84,HaensZS86,AlcocFO86}.
Strange quark stars are currently considered as a
possible alternative to neutron stars as compact objects (see
e.g. \cite{Weber04, Madsen99, Gondek03} and references
therein).
The evolution of a binary system of compact objects is entirely driven
by gravitational radiation and can be roughly divided into three
phases : point-like inspiral, hydrodynamical inspiral and merger. The
first phase corresponds to large orbital separation (much larger than
the neutron star radius) and can be treated analytically
using the post-Newtonian (PN) approximation to general relativity
(see Ref.~\cite{Blanc02b} for a review). In the second phase the orbital separation
becomes only a few times larger than the radius of the star, so
the effects of tidal deformation, finite size and hydrodynamics
play an important role. In this phase, since the shrinking
time of the orbital radius due to the emission of gravitational waves
is still larger than the orbital period, it is possible to approximate
the state as quasi-equilibrium
\cite{BaumgCSST97,BonazGM99}.
The final phase of the evolution is the merger of the two objects,
which occur dynamically
\cite{ShibaU00,ShibaU01,ShibaTU03,OoharN99}.
Note that quasi-equilibrium computations from the second phase
provide valuable initial data for the merger
\cite{ShibaU00,ShibaTU03,Oechslin04,FaberGR04}.
Almost all studies of the final phase of the inspiral
of close binary neutron star systems employ a simplified EOS of
dense matter, namely a polytropic EOS
\cite{BonazGM99,MarroMW99,UryuE00,UryuSE00,GourGTMB01,Faber02,
TanigG02b,TanigG03,FaberGR04,MarronDSB04}. There are only two
exceptions: (i) Oechslin et al. have used a pure nuclear matter EOS,
based on a relativistic mean field model and a `hybrid' EOS
with a phase transition to quark matter at high density \cite{Oechslin04};
(ii) Bejger et al. have computed quasi-equilibrium sequences based
on three nuclear matter EOS \cite{Bejger04}.
In this article we present results on the hydrodynamical phase of
inspiraling binary strange
quark stars described by MIT bag model. The calculations are performed
in the framework of {\em Isenberg-Wilson-Mathews} approximation to
general relativity (see Ref. \cite{BaumgS03} for a review). We
consider binary systems consisting of two identical stars. We choose
the gravitational mass of each star to be $1.35 \, M_\odot$ in infinite
separation in order to be consistent with recent population synthesis
calculations \cite{BulikGB04} and with the current set of
well-measured neutron star masses in relativistic binary radio
pulsars \cite{Lorim01,Burga03}.
We compare the evolution of a strange star binary system
with a neutron star binary in order to find any characteristic
features in the gravitational waveform that will help to distinguish between
strange stars
and neutron stars. We consider two limiting cases of velocity flow in stellar
interior: the irrotational and the synchronized case in order to
exhibit the differences between these two extreme states. The
irrotational case is more realistic since the viscosity of neutron
star matter (or strange star matter) is far too low to ensure
synchronization during the late stage of the inspiral
\cite{BildsC92,Kocha92}.
Due to the finite density at the surface of bare strange stars,
we had to introduce a treatment of the boundary condition for the
velocity potential (in the irrotational case) different from that
of neutron stars, where the density vanishes at the stellar surface.
The paper is organized in the following way: Sec. II is a brief
summary of the assumptions upon which this work is based, Sec. III is
devoted to the description of the EOS used to describe strange stars
and neutron stars. In Sec. IV we briefly describe the basic equations
for quasi-equilibrium and derive the boundary condition required for
solving the fluid equation of irrotational flow with finite surface
density, which is relevant for strange stars. In Sec. V we present the
numerical results for corotating and irrotational strange stars
binaries and compare their quasistationary evolution with that of
neutron stars, as well as with that of post-Newtonian point-masses.
Section VI contains the final discussion. Throughout the paper, we
use geometrized units, for which $G=c=1$, where $G$ and $c$ denote the
gravitational constant and speed of light respectively.
\section{Assumptions}
The first assumption regards the matter stress-energy tensor
$\w{T}$, which we
assume to have the {\bf perfect fluid} form:
\begin{equation}
\w{T} = (e+p)\w{u}\otimes\w{u} +p \, \w{g},
\end{equation}
where $e$, $p$, $\w{u}$ and $\w{g}$ are respectively the fluid proper energy
density, the fluid pressure, the fluid 4-velocity, and the spacetime metric.
This is a very good approximation for neutron star matter or strange star
matter.
The last orbits of inspiraling binary compact stars can be studied in
the {\bf quasi-equilibrium} approximation. Under this assumption the
evolution of a system is approximated by a sequence of exactly
circular orbits. This assumption results from the fact that the time
evolution of an orbit is still much larger than the orbital period and that
the gravitational radiation circularizes an orbit of a binary system.
This implies a continuous spacetime symmetry, called {\em helical
symmetry} \cite{BonazGM97,FriedUS02} represented by the Killing
vector: \begin{equation} \w{\ell} = {\partial \over \partial t} +\Omega {\partial \over \partial \varphi}, \end{equation}
where $\Omega$ is the orbital angular velocity and $\partial/\partial t$ and
$\partial/\partial \varphi$ are the natural frame vectors associated with the time
coordinate $t$ and the azimuthal coordinate $\varphi$ of an asymptotic
inertial observer.
One can then introduce the {\em shift vector} $\w{B}$ of co-orbiting
coordinates by means of the orthogonal decomposition of $\w {\ell}$ with
respect to the $\Sigma_t$ foliation of the standard 3+1 formalism:
\begin{equation} \label{e:helicoidal_n} \w {\ell} = N \, \w{n} - \w{B} , \end{equation}
where $\w{n}$ is the unit future directed vector normal to
$\Sigma_t$, $N$ is called the {\em lapse function} and
$\w{n}\cdot\w{B}=0$.
We also assume that the spatial part of the metric
(i.e. the metric induced by $\w{g}$ on each hypersurface $\Sigma_t$)
is conformally flat,
which corresponds to the {\bf Isenberg-Wilson-Mathews} (IWM)
approximation to general relativity \cite{Isenb78,IsenbN80,WilsoM89}
(see Ref. \cite{FriedUS02} for a discussion). Thanks to this
approximation we have to solve only five of the ten Einstein
equations. In the IWM approximation, the spacetime metric takes the
form: \begin{equation} ds^2 =-(N^2 -B_i B^i) dt^2 -2B_i \, dt \, dx^i +A^2 f_{ij}
dx^i dx^j,
\label{eq:metric}
\end{equation}
where $A$ is some conformal factor,
$f_{ij}$ the flat spatial metric and Latin indices run in $\{1,2,3\}$
(spatial indices).
The comparison between the IWM results presented here and the
non-conformally flat ones will be performed in a future article
\cite{UryuL04}.
The fourth assumption concerns the fluid motion inside each star. We
consider two limiting cases: {\bf synchronized} (also called
corotating) motion and {\bf irrotational} flow (assuming that
the fluid has zero vorticity in the inertial frame).
The latter state is more realistic.
We consider only {\bf equal-mass} binaries consisting of identical
stars with gravitational masses $M_1=M_2=1.35\ M_{\odot}$ measured in
infinite separation. The main reason for choosing these particular
masses is that five out of six observed binary radio pulsars have
mass ratio close to unity and gravitational masses of each star $\sim
1.3-1.4 M_{\odot}$ \cite{Lorim01,Burga03}. In addition
population synthesis calculations \cite{BulikGB04} have shown that a
significant fraction of the observed binary neutron stars in
gravitational waves will contain stars with equal masses $\sim 1.4\
M_{\odot}$ and systems consisting of a low and a high mass neutron
star.
\section{The equation of state and stellar models}
\begin{figure}
\vskip 0.8cm
\includegraphics[width=0.45\textwidth]{MRSSPOLI.eps}
\caption{Gravitational mass $M$ versus areal radius $R$
for sequences of
static strange quark stars described by the simplest MIT bag model
(solid line) and neutron stars described by polytropic EOS with
$\gamma =2.5$ and $\kappa=0.0093\ m_0 \, n_{\rm nuc}^{1.5}$
(dashed line). The two sequences
are crossing at the point $M=1.35\ M_\odot$
and $R=10.677 {\ \rm km}$ (marked by a circle). }\label{f:MRSSPOLI}
\end{figure}
It has been shown \cite{Faber02,TanigG03}
that the evolution of equal-mass binary neutron
stars depend mainly on the compactness parameter $M/R$,
where $M$ and $R$ are the gravitational mass
measured by an observer at infinity for a single isolated neutron star
and the stellar radius respectively. It is therefore interesting to
check if the properties of inspiraling strange stars can be predicted
by studying binaries consisting of polytropic neutron stars having the
same mass and the same compactness parameter. Therefore we perform
calculations for two different equations of state of dense matter:
a strange quark matter EOS and a polytropic EOS.
Typically, strange stars are modeled \cite{AlcocFO86, HaensZS86} with an
equation of state based on the MIT-bag model of
quark matter, in which quark confinement is described by an energy
term proportional to the volume \cite{FahriJ84}. The equation
of state is given by the simple formula
\begin{equation} \label{e:sseos}
p=a(\rho-\rho_0),
\end{equation}
\begin{equation} \label{sseosn} n(p) = n_0\cdot\left[1+{1+a\over
a}{p\over\rho_0}\right]^{1/(1+a)}, \end{equation}
where $n$ is the baryon
density and $a,\ \rho_0 , n_0$ are some constants depending on the 3 parameters of the model
(the bag constant $B$, the mass of the strange quarks $m_{\rm s}$ and the
strenght of the QCD coupling constant $\alpha$). In general this
equation corresponds to a self-bound matter with mass density $\rho_0$
and baryon density $n_0$ at zero pressure and with a fixed sound
velocity ($\sqrt{a}$) at all pressures. It was shown that different
strange quark models can be approximated very well by Eqs
(\ref{e:sseos}) and (\ref{sseosn}) \cite{Gondek00, Zdunik00}.
In the numerical calculations reported in the present paper we
describe strange quark matter using the simplest MIT bag model (with
massless and non-interacting quarks), for which the formula
(\ref{e:sseos}) is exact. We choose the value of the bag constant to
be $B=60\ {\rm MeV\ fm^{-3}}$. For this model we have $a=1/3$,
$\rho_0=4.2785\times 10^{14}\ {\rm g\ cm}^{-3}$ and ${n_0}=0.28665\
{\rm fm^{-3}}$. In general for the MIT bag model the density of
strange quark matter at zero pressure is in the range $\sim 3 \times
10^{14}-6.5 \times 10^{14}\ {\rm g\ cm^{-3}}$ and $a$ between 0.289
and 1/3 (for $0 \le \alpha \le 0.6 \ {\rm and}\ 0 \le m_{\rm s} \le 250 \ {\rm
MeV}$) \cite{Zdunik00}. The higher value of $a$ and of
$\rho_0$ the higher compactness parameter of a star with fixed
gravitational mass.
Up to now, the majority of calculations of the hydrodynamical inspiral
phase \cite{BonazGM99,MarroMW99,UryuE00,UryuSE00,GourGTMB01,Faber02,
TanigG02b,TanigG03,FaberGR04,MarronDSB04}
and all calculations of the merger phase
\cite{ShibaU00,ShibaU01,ShibaTU03,OoharN99}
have been performed for
binary systems containing neutron stars described by a polytropic
EOS:
\begin{equation}
p=\kappa n^{\gamma},
\label{eeospolyp}
\end{equation} where $\kappa$ and $\gamma$ coefficients are some constant numbers:
$\kappa$ represents the overall compressibility of matter
while $\gamma$ measures the stiffness of the EOS.
The total energy density is related to the baryon density by
\begin{equation}\label{eeospolye}
e(n) = {\kappa \over \gamma-1} n^\gamma + \mu_0 \, n \ ,
\end{equation}
where $\mu_0$ is the chemical potential at zero pressure.
In order to compare results for strange stars with those for neutron
stars, we determine the values of $\kappa$ and $\gamma$ which yield to
the same radius for the gravitational mass $M = 1.35\, M_{\odot}$ as
that obtained for a static strange star. It was shown \cite{Bejger04}
that the properties of inspiraling neutron stars described by
realistic EOS can be, in a good approximation, predicted by studying
binaries with assumed polytropic EOSs with $\gamma=2$ or 2.5. For a
1.35 $M_{\odot}$ strange star we have a high value of compactness
parameter $M/R=0.1867$ so we have choosen $\gamma = 2.5$, for which we found
$\kappa=0.00937 \ m_0 \, n_{\rm nuc}^{1.5}$, with the rest mass of relativistic particles $m_0:=1.66\times
10^{-27} {\rm\ kg}$ and $n_{\rm nuc} = 0.1 {\ \rm fm}^{-3}$.
\begin{figure}
\vskip0.8cm
\includegraphics[width=0.46\textwidth]{rhoR.eps}\vskip 0.9cm
\includegraphics[width=0.45\textwidth]{Gamma_R.eps}
\caption{Mass density (top panel) and the adiabatic index
$\gamma $ (bottom panel) versus the radial coordinate $r$ for a
static strange quark star (solid line) and a polytropic neutron star
(dashed line), having both a gravitational mass of $1.35\ M_\odot$ and
an areal radius $R=10.677 {\ \rm km}$ (resulting in the compactness
parameter $M/R = 0.1867$). The vertical dotted line corresponds to
the stellar surface.}
\label{f:rho_R}
\end{figure}
In Fig. \ref{f:MRSSPOLI} we present the mass-radius relation for a
sequence of static stars described by the simplest MIT bag model
(solid line) and the polytropic EOS (dashed line) parametrized by
central density. For small mass strange stars $M\sim R^3$ since
density is almost constant inside a star $\sim \rho_0$.
In the top panel of Fig. \ref{f:rho_R} we show the mass
density distribution inside the
strange star (solid line) and the neutron star described by polytropic
EOS (dashed line) having gravitational mass 1.35 $M_\odot$ and areal
radius $10.667 {\rm \ km}$
(the configurations corresponding to the crossing point on
Fig. \ref{f:MRSSPOLI}). The huge density jump at the surface of the strange
star corresponds to $\rho_0 = 4B$. The value of density at the surface
describes strongly or weakly bound strange matter, which in each case
must be absolutely stable with respect to ${}^{56}{\rm Fe}$.
An important quantity relevant for evolution of binary compact stars
is the adiabatic index: \begin{equation} \gamma= {\rm d}\ln p/{\rm d}\ln n . \end{equation}
We assume that matter is catalized so the adiabatic index can be
calculated directly from EOS (see Refs.~\cite{GourgHG95} and
\cite{GondeHZ97} for discussion on different kind of adiabatic indices
and corresponding timescales). Note that for the polytropic EOS given
by Eq.~(\ref{eeospolyp}) the index $\gamma$ coincides with the
adiabatic index of a relativistic isentropic fluid. Dependence of the
adiabatic index $\gamma$ on stellar radii for both EOS is shown in the
bottom pannel of Fig.~\ref{f:rho_R}. The adiabatic index of strange
matter is qualitatively different from the adiabatic index for
polytropic EOS or for realistic EOS. The values of $\gamma$ in the
outer layers of strange stars are very large and for $\rho\rightarrow
\rho_0$ we have $\gamma=a+\rho/(\rho-\rho_0)\rightarrow \infty $. The
EOS of neutron stars for densities lower than $\sim 10^{14} \ {\rm g\,
cm}^{-3}$ (the crust) is well established \cite{Haens03}. In the
outer crust of an ordinary neutron star the pressure is dominated by
the ultra-relativistic electron gas, so we have $\gamma=4/3$. The
values of the local adiabatic index in the inner crust of a neutron
star depends strongly on density and varies from $\gamma\simeq 0.5$
near the neutron drip point to $\gamma\simeq 1.6$ in the bottom layers
near the crust-core interface.
In our calculations we use equation of state in the form:
\begin{equation}
n=n(H),\ \ \ \ \ e=e(H),\ \ \ \ \ p=p(H),
\end{equation} where H is
pseudo-enthalpy (the log-enthalpy) defined by:
\begin{equation} \label{eeospolyh}
H(n) := \ln \left( {e+p \over n E_0}\right),
\end{equation}
where the energy per unit baryon number is $E_0=m_0$ for a polytropic EOS,
and $E_0=\rho_0 /n_0= 837.26 {\ \rm MeV}$ for strange quark model described
above.
For our model of strange quark matter we have:
\begin{equation}
\rho =\rho_0(3{\rm e}^{4H} + 1)/4, \ \ \ p = \rho_0({\rm e}^{4H} -1)/4~,\ \ \ \ n = n_0{\rm e}^{3H}~
\end{equation}
\section{Equations to be solved}
We refer the reader to Ref.~\cite{GourgGTMB01} for the derivation
of the equations describing quasi-equilibrium binary stars within
the IWM approximation to general relativity.
After recalling these equations, we mainly
concentrate on the equation for the velocity potential of
irrotational flows. Actually this equation has a different structure
for strange stars than for neutron stars. This results from the
non-vanishing of the density at the stellar surfaces of strange
stars (cf. the top panel in Fig.~\ref{f:rho_R}).
\subsection{The gravitational field equations}
The gravitational field equations have been obtained within the 3+1
decomposition of the Einstein's equations \cite{York79,Cook00}, taking
into account the helical symmetry of spacetime.
The trace of the spatial part of the Einstein equations
combined with the Hamiltonian constraint results in two equations:
\begin{eqnarray}
\underline\Delta \nu &=& 4\pi A^2 (E+S) + A^2 K_{ij} K^{ij}
- \overline\nabla_i \nu \overline\nabla^i \beta, \\
\underline\Delta \beta &=& 4\pi A^2 S +{3 \over 4} A^2 K_{ij} K^{ij} \nonumber\\
&& -{1 \over 2} (\overline\nabla_i \nu \overline\nabla^i \nu +\overline\nabla_i \beta \overline\nabla^i \beta),
\end{eqnarray}
where $\overline\nabla_i$ stands for the covariant derivative associated with
the flat 3-metric $f_{ij}$ and $\underline\Delta := \overline\nabla^i \overline\nabla_i$ for the
associated Laplacian operator.
The quantities $\nu$ and $\beta$ are defined by
$\nu := \ln N$ and $\beta := \ln (AN)$, and
$K_{ij}$ denotes the extrinsic curvature tensor of the $t={\rm const}$
hypersurfaces.
$E$ and $S$ are respectively the matter energy density and the trace of the
stress tensor, both as measured by the observer whose 4-velocity is $n^\mu$
{\em (Eulerian observer)}:
\begin{eqnarray}
E &:=& T_{\mu \nu} n^{\mu} n^{\nu} , \\
S &:=& A^2 f^{ij} T_{ij}.
\end{eqnarray}
In addition, we have also to solve the momentum constraint, which writes
\begin{eqnarray}
\underline\Delta N^i +{1 \over 3} \overline\nabla^i (\overline\nabla_j N^j) &=& -16\pi N A^2 (E+p) U^i \nonumber\\
&& +2 N A^2 K^{ij} \overline\nabla_j (3\beta -4\nu),
\end{eqnarray}
where $N^i:=B^i +\Omega (\partial/\partial \varphi)^i$ denotes the shift vector of
nonrotating coordinates, and $U^i$ is the fluid 3-velocity.
\subsection{The fluid equations}
Apart from the gravitational field equations, we have to solve the
fluid equations. The equations governing the quasi-equilibrium state
are the relativistic Euler equation and the equation of baryon number conservation.
Both cases of irrotational and synchronized motions admit a first
integral of the relativistic Euler equation:
\begin{equation} H + \nu - \ln \Gamma_0 + \ln \Gamma = {\rm
const.},
\end{equation}
where $\Gamma_0$ is the Lorentz factor between
the co-orbiting observer and
the Eulerian observer and $\Gamma$ is the Lorentz factor between the fluid
and the co-orbiting observers ($\Gamma=1$ for synchronized binaries).
For a synchronized
motion, the equation of baryon number conservation is trivially
satisfied, whereas for an irrotational flow, it is written as
\begin{eqnarray}
\label{eq:vel_pot} \zeta H \underline\Delta \Psi &+& \overline\nabla^i H \overline\nabla_i \Psi \nonumber\\
&=& A^2 h \Gamma_{\rm n} U^i_0 \overline\nabla_i H + \zeta H \nonumber\\
&& \times [\overline\nabla^i \Psi \overline\nabla_i
(H-\beta) +A^2 h U^i_0 \overline\nabla_i \Gamma_{\rm n}],
\end{eqnarray}
where $\Psi$ is the velocity potential, $h := \exp(H)$, $\zeta$ the
thermodynamical coefficient:
\begin{equation} \label{e:zeta_h}
\zeta := {d \ln H \over d \ln n},
\end{equation}
and
$\Gamma_{\rm n}$ denotes the Lorentz factor between the fluid and the
Eulerian observer
and $U^i_0$ is the orbital 3-velocity with respect to the Eulerian
observers: \begin{equation} U^i_0 = -{B^i \over N}. \label{eq:ov_wrt_euler} \end{equation}
The fluid 3-velocity $U^i$ with respect to the Eulerian observer is equal to
$U^i_0$ for synchronized binary systems, whereas
\begin{equation} \label{e:u-euler}
U^i = {1 \over A^2 \Gamma_{\rm n} h} \overline\nabla^i \Psi \label{eq:iv_wrt_corot}
\end{equation}
for irrotational ones.
\subsection{Boundary condition for the velocity potential}
The method of solving the elliptic equation (\ref{eq:vel_pot}) for the
velocity potential is different for neutron stars and strange stars.
In the case of neutron stars, the coefficient $\zeta H$ in front of
the Laplacian vanishes at the surface of the star so
Eq. (\ref{eq:vel_pot}) is not merely a Poisson type equation for
$\Psi$. It therefore deserves a special treatment (see Appendix B in
\cite{GourGTMB01} for a discussion). In the case of strange stars,
the coefficient $\zeta H = 1/3$ in whole star so we have to deal with
a usual Poisson equation and consequently we have to impose a boundary
condition for the velocity potential at the stellar surface.
We can define the surface of the star by $n_{| \rm{surf}} = n_0 =
\rm{constant}$.
The surface of the fluid ball is obviously Lie-dragged along the fluid
4-velocity vector $\w{u}$, so that this last condition gives
\begin{equation} \label{e:lie1}
\left. (\pounds _{\w{u}} n) \right| _{\rm{surf}} = 0,
\end{equation}
where $\pounds _{\w{u}}$ is the Lie derivative along the vector field
$\w{u}$.
Let us decompose $\w{u}$ in a part along the helical Killing vector
$\w {\ell}$ and a part $\w{S}$ parrallel to the hypersurface $\Sigma_t$:
\begin{equation} \label{e:decomposition}
\w{u} = \lambda (\w {\ell} + \w{S}) .
\end{equation}
The condition (\ref{e:lie1}) is then equivalent to
\begin{equation} \label{e:lie2}
\left. \lambda ( \pounds _{\w {\ell}} n
+ \pounds _{\w{S}} n ) \right| _{\rm{surf}} = 0.
\end{equation}
Now, if the fluid flows obeys to the helical symmetry
$\pounds _{\w {\ell}} n = 0$; inserting this relation
into Eq.~(\ref{e:lie2}) leads to
$\left. (\pounds _{\w{S}} n ) \right| _{\rm{surf}} = 0$
or equivalently (since $\w{S}$ is a spatial vector):
\begin{equation} \label{e:condition1}
\left. (S^i \overline\nabla_i n) \right| _{\rm{surf}} = 0.
\end{equation}
Now, let us express $\w{S}$ in terms of the spatial vectors
$\w{U}$ and $\w{B}$. First, Eq. ~(\ref{e:decomposition}) implies
$\w{n} \cdot \w{u} = \lambda \w{n} \cdot \w {\ell}$.
Secondly, the fluid motion $\w{u}$ can be described by the orthogonal
decomposition $\w{u} = \Gamma_{\rm n} (\w{n} + \w{U})$ which yields
$\w{n} \cdot \w{u} = - \Gamma_n$.
Finally, from Eq. ~(\ref{e:helicoidal_n}), we have
$\w{n} \cdot \w {\ell} = - N$ so that the factor $\lambda$ can be expressed as
$\lambda = \Gamma_n / N$
and Eq. ~(\ref{e:decomposition}) becomes
\begin{equation} \label{e:decomposition2}
\w{u} = {\Gamma_n \over N} (\w {\ell} + \w{S}).
\end{equation}
Now, combining Eq. ~(\ref{e:decomposition2})
and Eq. ~(\ref{e:helicoidal_n}), we have
\begin{equation} \label{e:decomposition3}
\w{u} = \Gamma_n \left[ \w{n} + {1 \over N} (\w{S} - \w{B}) \right] .
\end{equation}
Comparing with the orthogonal decomposition
$\w{u} = \Gamma_{\rm n} (\w{n} + \w{U})$, we deduce that
$\w{S} = N \w{U} + \w{B}.$
Inserting this relation into Eq.~(\ref{e:condition1}) leads to the boundary
condition
\begin{equation}
\left. (N U^i \overline\nabla_i n + B^i \overline\nabla_i n) \right| _{\rm{surf}} = 0.
\end{equation}
Now, using Eq. ~(\ref{e:u-euler}), we obtain a Neumann-like boundary condition
for $\Psi$:
\begin{equation} \label{e:condition2}
\left. (\overline\nabla^i n \overline\nabla_i \Psi) \right| _{\rm{surf}}
= - \left. \left({\Gamma_n h A^2 \over N} B^i \overline\nabla_i n\right)
\right| _{\rm{surf}} .
\end{equation}
Considering the elliptic equation (\ref{eq:vel_pot}) for $\Psi$ we
see that the boundary condition we have obtained is consistent with the
case $n = 0$ (or equivalently $\zeta H = 0$) at the stellar surface since,
from Eq. ~(\ref{e:zeta_h}), $\overline\nabla^i H = {\zeta H \over n} \overline\nabla^i n$.
\section{Numerical results}
\subsection{The method}
The resolution of the above nonlinear elliptic equations is
performed thanks to a numerical code based on multidomain spectral
methods and constructed upon the {\sc Lorene} C++ library \cite{Lorene}.
The detailed description of the whole algorithm,
as well as numerous tests of the code can be found in \cite{GourgGTMB01}.
Additional tests have been presented in Sec.~3 of \cite{TanigG03}.
The code has
already been used successfully for calculating the final phase of
inspiral of binary neutron stars described by polytropic EOS
\cite{BonazGM99,TanigGB01,TanigG02a,TanigG02b,TanigG03} and realistic EOS
\cite{Bejger04}.
It is worth to stress that the adaptation of the
domains (numerical grids) to the stellar surface (surface-fitted coordinates)
used in this code is particulary usefull here,
due to the strong discontinuity of the density field at the surface of strange
stars (cf. the top panel in Fig.~\ref{f:rho_R}). Adapting the grids to the stellar surface
allows to avoid the severe Gibbs phenomenon that such a discontinuity
would necessary generate when performing polynomial expansions of the
fields \cite{BonazGM98}.
The hydrodynamical part of the code has been amended for the
present purpose, namely to solve Eq.~(\ref{eq:vel_pot}) for the
velocity potential $\Psi$ subject to the boundary condition
(\ref{e:condition2}). Let us recall that in the original version
of the code, the treatment of Eq.~(\ref{eq:vel_pot}) was different
due to the vanishing of the density field at the stellar surface
(see Appendix~B of Ref.~\cite{GourgGTMB01}).
We have used one numerical domain for each star and
3 (resp. 4) domains for the space around
them for a small (resp. large) separation. In each domain, the number
of collocation points of the spectral method is chosen to be $N_r
\times N_{\theta} \times N_{\varphi} = 25 \times 17 \times 16$, where
$N_r$, $N_{\theta}$, and $N_{\varphi}$ denote the number of
collocation points ($=$ number of polynomials used in the spectral
method) in
the radial, polar, and azimuthal directions respectively. The
accuracy of the computed relativistic models has been estimated using a
relativistic generalization of the virial theorem
\cite{FriedUS02} (see also Sec.~III.A of Ref. \cite{TanigG03}). The
virial relative error is a few times $10^{-5}$ for the closest
configurations.
\subsection{Evolutionary sequences}
For each EOS we construct an {\em evolutionary sequence}, i.e. a
sequence of quasi-equilibrium configurations
with fixed baryon mass and decreasing separation. Such a sequence is
expected to approximate pretty well the true evolution of binary
neutron stars, which is entirely driven by the reaction to
gravitational radiation and hence occur at fixed baryon number and
fluid circulation.
For a given rotational state we calculate evolutionary sequences of
binary system composed of two identical neutron stars or two identical
strange stars. The evolution of inspiraling corotating (irrotational)
binaries is shown in Fig. \ref{f:EJ_cor} (Fig. \ref{f:EJ_irr}).
Fig. \ref{f:EJ_cor} and upper panel of Fig. \ref{f:EJ_irr}
show the binding energy $E_{\rm bind}$ versus frequency of
gravitational waves $f_{\rm GW}$ and lower panel of Fig. \ref{f:EJ_irr}
show the total
angular momentum of the systems as a function of $f_{\rm GW}$. The
binding energy is defined as the difference between the actual ADM
mass of the system, $M_{\rm ADM}$, and the ADM mass at infinite
separation ($2.7 \, M_{\odot}$ in our case). The frequency of
gravitational waves is twice the orbital frequency, since it
corresponds to the frequency of the dominant part $l=2,\; m=\pm 2$.
Solid and dashed lines denote quasi-equilibrium sequences of strange
quark stars binaries and neutron stars binaries respectively. Dotted
lines in Fig. \ref{f:EJ_cor} and Fig. \ref{f:EJ_irr} correspond to the
3rd PN approximation for point masses derived by \cite{Blanc02}.
Finally in Fig. \ref{f:Ebind_f_all} we compare our results with third
order post-Newtonian results for point-mass particles obtained in the
effective one body approach by Damour et al. 2000 \cite{DamourJS},
Damour et al. 2002 \cite{DamouGG02} and in the standard nonresummed
post-Newtonian framework by Blanchet 2002 \cite{Blanc02}.
\begin{figure}{}
\vskip 0.5cm
\includegraphics[angle=0,width=0.45\textwidth]{Ebincor.eps}
\caption{ Binding energy as a function of gravitational wave frequency
along evolutionary sequences of corotating binaries. The solid line
denotes strange quark stars, the dashed one neutron stars with
polytropic EOS, and the dotted one point-mass binaries in the 3PN
approximation \cite{Blanc02}. The diamonds locate the minimum of the
curves, corresponding to the innermost stable circular orbit;
configurations to the right of the diamond are securaly
unstable. }\label{f:EJ_cor}
\end{figure}
A turning point of $E_{\rm bind}$ along an evolutionary sequence
indicates an orbital instability \cite{FriedUS02}.
This instability originates both from relativistic effects (the well-known
$r=6M$ last stable orbit of Schwarzschild metric) and hydrodynamical
effects (for instance, such an instability exists for sufficiently
stiff EOS in the Newtonian regime, see e.g. \cite{TanigGB01} and
references therein). It is secular for
synchronized systems and dynamical for irrotational ones.
In the case where no turning point of $E_{\rm bind}$ occurs along the
sequence, the mass-shedding limit (Roche lobe overflow) marks the
end of the inspiral phase of the binary system, since recent dynamical
calculations for $\gamma = 2$ polytrope have shown that the time to
coalescence was shorter than one orbital period for configurations at
the mass-shedding limit \cite{ShibaU01, MarronDSB04}. Thus the
physical inspiral of binary compact stars terminates by either the
orbital instability (turning point of $E_{\rm bind}$) or the
mass-shedding limit. In both cases, this defines the {\em innermost
stable circular orbit (ISCO)}. The orbital frequency at the ISCO is a
potentially observable parameter by the gravitational wave detectors,
and thus a very interesting quantity.
\subsection{Corotating binaries}
Quasi-equilibrium sequences of equal mass corotating binary neutron
stars and strange stars are presented in Fig. \ref{f:EJ_cor}. For
both sequences we find a minimum of the binding energy.
In the present rotation state, this locates
a secular instability \cite{FriedUS02}.
The important difference between neutron stars and
strange stars is the frequency at which this
instability appears. Indeed, there is a difference of more than 100~Hz:
1020~Hz for strange stars and 1140~Hz for neutron stars. The binding
energy is the total energy of gravitational waves emitted by the
system: a corotating binary strange star system emits less energy
in gravitational
waves and loses less angular momentum before the ISCO than a binary neutron
star one with the same mass and compaction parameter in infinite
separation.
Comparison of our numerical results with 3rd order PN
calculations reveals a good agreement for small frequencies (large
separations) (see Fig. \ref{f:EJ_cor} and
\ref{f:Ebind_f_all}). The deviation from PN curves
at higher frequencies (smaller separation) is due to hydrodynamical
effects, which are not taken into account in the PN approach.
\subsection{Irrotational binaries}
\begin{figure}{}
\vskip 0.5cm
\includegraphics[angle=0,width=0.45\textwidth]{Ebind_f_irr.eps} \qquad \\
\vskip 1.2cm
\includegraphics[angle=0,width=0.45\textwidth]{J_f_irr.eps}
\caption{Binding energy (top panel) and angular momentum (bottom
panel) as a function of gravitational wave frequency along
evolutionary sequences of irrotational binaries. The solid line denotes
strange quark stars, the dashed one polytropic neutron stars,
and the dotted one point-mass binaries in the 3PN approximation \cite{Blanc02}.
The diamonds correspond to dynamical
orbital instability (the ISCO).}
\label{f:EJ_irr}
\end{figure}
\begin{figure}{}
\vskip 0.5cm
\includegraphics[angle=-90,width=0.35\textwidth]{vel_ISCO_cor.eps}\qquad
\vskip 0.5cm
\includegraphics[angle=-90,width=0.35\textwidth]{vel_ISCO_ref.eps}
\caption{Internal velocity fields of irrotational strange quark stars binaries
at the ISCO. {\em upper panel:} velocity $\w{U}$ in the orbital plane
with respect to the ``inertial'' frame (Eulerian observer);
{\em lower panel:} velocity field with respect to the
corotating frame.
The thick solid lines denote the surface of each star. }\label{f:psi0}
\end{figure}
\begin{figure}{}
\vskip 1cm
\includegraphics[angle=0,width=0.45\textwidth]{a1a2.eps}
\caption{Coordinate ``radius'' (half the coordinate size of a star in
fixed direction) versus coordinate separation for
irrotational quasi-equilibrium sequences of binary strange stars
(solid line) and neutron stars (dashed line). Upper lines correspond
to equatorial radius $R_x$ (radius along the x axis going through
the centers of stars in a binary system) and lower lines are polar
radius $R_z$ (radius along the rotation axis).}
\label{f:a1a2}
\end{figure}
In Fig. \ref{f:EJ_irr} we present the evolution of the
binding energy and angular momentum
for irrotational sequences of binary neutron stars and strange stars.
We also verify that these sequences are in a good agreement with PN
calculations for large separations.
We note important differences in the
evolution of binary systems consisting of strange stars or
neutron stars.
The strange star sequence shows a minimum of the
binding energy at $f_{\rm GW}\simeq 1390{\rm\ Hz}$,
which locates a dynamical instability \cite{FriedUS02}
and thus defines the ISCO.
The minimum of $E_{\rm bind}$ coincides with the minimum of
total angular momentum $J$. This is in accordance with the ``first
law of binary relativistic star thermodynamics''
within the IWM approximation
as derived by Friedman, Uryu and Shibata \cite{FriedUS02}
and which states that, along an evolutionary sequence,
\begin{equation}
\delta M_{\rm ADM} = \Omega \delta J.
\end{equation}
The surface of strange stars at the ISCO is smooth
(see Fig. \ref{f:psi0}).
On the contrary the neutron star sequence does not present any
turning point of $E_{\rm bind}$, so that the ISCO
in this case corresponds to the mass-shedding limit
(final point on the dashed curves in Fig.~\ref{f:EJ_irr}).
The gravitational wave frequency at the ISCO
is much lower for neutron star binaries than for strange star binaries.
As already mentioned the adiabatic index in the outer layers of a
compact star in a binary system plays a crucial role in its evolution,
especially in setting the mass-shedding limit. Although the crust of
a $1.35\, M_\odot$ neutron star contains only a few percent of the
stellar mass, this region is easily deformed under the action of the
tidal forces resulting from the gravitational field produced by the
companion star. The end of inspiral phase of binary stars strongly
depends on the stiffness of matter in this region. It has been shown
that the turning-point orbital instability for irrotational polytropic
neutron stars binaries can be found only if $\gamma \ge 2.5$ and if
the compaction parameter is smaller than certain value
(\cite{TanigG03}, \cite{UryuSE00}). In fact, as shown in Fig. 31 of
paper \cite{TanigG03}, they didn't find ISCO for irrotational binary
neutron stars with $\gamma = 2.5$ or $\gamma = 3$ for compaction
parameter as high as $M/R = 0.187$.
In Fig.~\ref{f:a1a2} we present the evolution of two different
stellar radii: the {\em equatorial radius} $R_x$, defined as half the
diameter in the direction of the companion and the {\em polar
radius}, defined as half the diameter parallel to the rotation axis.
For spherical stars $R_x=R_z$. We see that at the end of the inspiral
phase, neutron stars are, for the same separation, more oblate (more
deformed) than strange stars. Binary neutron stars reach the
mass-shedding limit (the point at which they start to exchange matter -
a cusp form at the stellar surface in the direction of the
companion) at coordinate separation $d\sim 25 {\rm km}$. We don't see
any cusps for strange stars even for distances slightly smaller that
the distance corresponding to the ISCO $\sim 23.5\ {\rm km}$.
It is worth to remind here the results on
rapidly rotating strange stars and neutron stars. The Keplerian
limit is obtained for higher oblatness (more deformed stars),
measured for example by the ratio of polar and equatorial radius, in
the case of strange stars than in the case of neutron stars
\cite{Cook94, Sterg99,Gondek00, ZduniHGG00, AmsteBGK00, Gondek01}.
The differences in the evolution of binary (or rotating)
strange stars and neutron stars stem from the fact that strange stars
are principally bound by another force than gravitation: the
strong interaction between quarks.
As already mentioned the frequency of gravitational waves is one of
potentially observable parameters by the gravitational wave detectors.
We can see from Fig. \ref{f:Ebind_f_all} that the 3rd PN
approximations for point masses derived by different authors are
giving ISCO at very high frequencies of gravitational waves $> 2\ {\rm
kHz}$. Since in the hydrodynamical phase of inspiral the effect of a
finite size of the star (e.g. tidal forces) is very important we see
deviation of our numerical results from point-masses calculations. The
frequency of gravitational waves at the ISCO strongly depends on
equation of state and the rotational state. For irrotational equal
mass (of 1.35 $M_{\odot}$ at infinite separation) strange stars
binaries described by the simple MIT bag model this frequency is $\sim
1400 {\rm Hz}$ and for neutron stars binaries described by four
different realistic EOS it is between $ 800{\rm Hz}$ and $1230{\rm
Hz}$ \cite{Oechslin04, Bejger04}.
\begin{figure*}{}
\vskip 0.5cm
\includegraphics[angle=0,width=0.75\textwidth ]{Ebind_f_all.eps} \qquad
\caption{Binding energy versus frequency of gravitational waves along
evolutionary sequences of corotational (thick dashed lines) and
irrotational (thick solid lines) equal mass (of $1.35\ M_{\odot}$)
strange stars and polytropic neutron stars binaries compared with
analytical results at the 3rd post-Newtonian order for point-masses by
Damour et al. 2000 \cite{DamourJS}, Damour et al. 2002
\cite{DamouGG02} and Blanchet 2002 \cite{Blanc02} }
\label{f:Ebind_f_all}
\end{figure*}
\section{Summary and discussion}
We have computed evolutionary sequences of irrotational and corotating
binary strange stars by keeping the baryon mass constant to a value
that corresponds to individual gravitational masses of $1.35\,
M_\odot$ at infinite separation. The last orbits of inspiraling
binary strange stars have been studied in the quasi-equilibrium
approximation and in the framework of Isenberg-Wilson-Mathews
approximation of general relativity. In order to calculate
hydrodynamical phase of inspiraling irrotational strange stars
binaries, i.e. assuming that the fluid has zero vorticity in the
inertial frame, we found the boundary condition for the velocity
potential. This boundary condition is valid for both the case of
non-vanishing (e.g. self-bound matter) and vanishing density at the
stellar surface (neutron star matter). In our calculations strange
stars are built by strange quark matter described by
the simplest MIT bag model (assuming massless and non-interacting
quarks).
We have located the end of each quasi-equilibrium sequence (ISCO),
which corresponds to some orbital instabilities (the dynamical
instability for irrotational case or the secular one for synchronized
case) and determined the frequency of gravitational waves at this
point. This characteristic frequency yields important information
about the equation of state of compact stars and is one of the
potentially observable parameters by the gravitational wave
detectors. In addition, the obtained configurations provide valuable
initial conditions for the merger phase. We found the frequency of
gravitational waves at the ISCO to be $\sim 1400\ {\rm Hz}$ for
irrotational strange star binaries and $\sim 1000\ {\rm Hz}$ for
synchronized case. The irrotational case is more realistic since the
viscosity of strange star matter is far too low to ensure
synchronization during the late stage of the inspiral. For
irrotational equal mass (of 1.35 $M_{\odot}$) neutron star binaries
described by realistic EOS \cite{Oechslin04, Bejger04} the frequency
of gravitational waves at the ISCO is between $ 800{\rm Hz}$ and
$1230{\rm Hz}$, much lower than for a binary strange quark star built
of self-bound strange quark matter. We have considered only strange
quark stars described by the simple MIT bag model with massless and
non-interacting quarks. In order to be able to interpret future
gravitational-wave observations correctly it is necessary to perform
calculations for different strange star EOS parameters (taking also
into account the existence of a thin crust) and for large sample of
neutron stars described by realistic equations of state. For some MIT
bag model parameters one is able to obtain less compact stars than
considered in the present paper. In this case the frequency of
gravitational waves at the end of inspiral phase will be lower than
obtained by us. It should be also taken into account that stars in a
binary system can have different masses \cite{BulikGB04}. The case
of binary stars (with equal masses and different masses) described by
different strange quark matter models will be presented in a separate
paper \cite{GondeL05}.
We have shown the differences in the inspiral phase between strange quark
stars and neutron stars described by polytropic equation of state
having the same gravitational mass and radius in the infinite
separation. It was already shown by Bejger et al. 2005
\cite{Bejger04} that the frequency of gravitational waves at the end
point of inspiraling neutron stars described by several realistic EOS
without exotic phases (such as meson condensates or quark matter) can
be predicted, in a good approximation, by studying binaries with
assumed polytropic EOSs with $\gamma=2$ or 2.5. For realistic EOS
and polytropes with $\gamma \le 2.5$ \cite{UryuSE00, TanigG03} a
quasi-equilibrium irrotational sequence terminates by mass-shedding
limit (where a cusp on the stellar surface develops).
We found that it wasn't the case for inspiraling strange star binaries
which are self-bound objects having very large adiabatic index in the
outer layer. For both synchronized and irrotational configurations, we
could always find a turning point of binding energy along an
evolutionary sequence of strange quark stars, which defines an orbital
instability and thus marks the ISCO in this case. In the irrotational
case for the same separation strange stars are less deformed than
polytropic neutron stars and for the same ratio of coordinate radius
$R_x/R_z$ their surfaces are more smooth. A cusp doesn't appear on
the surface of a strange star in a binary system even for separation
corresponding to orbital instability. The frequency of gravitational
waves at the end of inspiral phase is higher by 300 Hz for the strange
star binary system than for the polytropic neutron star binaries.
The differences in the evolution of binary (or rotating) strange stars
and neutron stars stem from the fact that strange stars are
principally bound by an additional force, strong interaction between
quarks.
\acknowledgements We thank our anonymous referee for helpful comments.
Partially supported by the KBN grants 5P03D.017.21 and
PBZ-KBN-054/P03/2001; by ``Ayudas para movilidad de Profesores de
Universidad e Invesigadores espanoles y extranjeros'' from the Spanish
MEC; by the ``Bourses de recherche 2004 de la Ville de Paris'' and by
the Associated European Laboratory Astro-PF (Astrophysics
Poland-France).
|
{
"timestamp": "2005-03-21T11:15:26",
"yymm": "0411",
"arxiv_id": "gr-qc/0411127",
"language": "en",
"url": "https://arxiv.org/abs/gr-qc/0411127"
}
|
\section{INTRODUCTION}
The Carina nebula (NGC~3372) is a giant H~{\sc ii} region rich in
complex structure that has just recently been recognized as an
important region of ongoing active star formation. Early surveys of
the central part of the nebula in molecular lines and far-infrared
(IR) continuum suggested that neutral gas and dust is mostly evacuated
from the core of the nebula and that it lacks significant current star
formation (Harvey et al.\ 1979; Ghosh et al.\ 1988; de Grauww et al.\
1981). It was assumed that radiation and stellar winds from hot
massive stars in Carina are just clearing away the last vestiges of
their natal molecular cloud.
However, recent studies at IR wavelengths have dramatically altered
this view. Megeath et al.\ (1996) found an embedded near-IR source at
the edge of a dark cloud near $\eta$ Car. A large-scale thermal-IR
survey with the {\it Midcourse Space Experiment} ({\it MSX}) revealed
dozens of compact IR sources that were suggested as potential sites of
ongoing and possibly triggered star formation (Smith et al.\ 2000).
Many of these are found in the southern part of the nebula at the
heads of giant dust pillars pointing back toward the massive stars in
the core of the nebula (Smith et al.\ 2000; Rathborne et al.\ 2004).
One southern pillar contains a luminous Class~I protostar that drives
the parsec-scale HH666 outflow (Smith et al.\ 2004a), and several
other dust pillars may contain embedded IR sources and even star
clusters seen in 2MASS data (Rathborne et al.\ 2004). Thus, the dust
pillars in the Carina nebula are akin to the famous pillars in M16,
which contain embedded near-IR sources and may be sites of triggered
star formation (Sugitani et al.\ 2002; Thompson et al.\ 2002;
McCaughrean \& Andersen 2002). With this ongoing star formation in
the vicinity of some of the hottest and most massive stars known in
the Galaxy (Walborn 1995; Walborn et al.\ 2002), the Carina nebula may
also be an analog of two-stage starburst regions like 30~Dor or
NGC~604. These types of regions are useful for studying a second
generation of stars, whose formation may have been triggered by
feedback from the first generation. At a distance of $\sim$2.3 kpc
(Walborn 1995), the Carina nebula provides a laboratory to study this
phenomenon in exquisite detail compared to extragalactic examples.
An object we refer to as the ``Treasure Chest'' (a dust pillar
associated with the star CPD~$-$59$\arcdeg$2661) is located in the southern part of the
Carina nebula (see Figure 1). The H$\alpha$ nebulosity surrounding
CPD~$-$59$\arcdeg$2661\ was first noted by Thackeray (1950), and its visual-wavelength
emission was studied in greater detail by Walsh (1984; see also Herbst
1975; van den Bergh \& Herbst 1975). Walsh (1984) noted that the
spectral type of CPD~$-$59$\arcdeg$2661\ was O9.5 V, and that the compact nebula
appeared to be expanding at $\sim$25 km s$^{-1}$. The nebula and
CPD~$-$59$\arcdeg$2661\ are coincident with G287.84-0.82, one of the brightest of the
`South Pillars' identified from thermal-IR emission by Smith et al.\
(2000). Rathborne et al.\ (2004) noted that G287.84-0.82 appeared to
harbor a probable young star cluster seen in 2MASS data (first noted
by Dutra \& Bica 2001), and showed that the IR spectral energy
distribution implied a total integrated luminosity of $\sim$10$^{6.6}$
L$_{\odot}$. The spectral type of CPD~$-$59$\arcdeg$2661, the embedded cluster, and the
high IR luminosity imply that the Treasure Chest is a site of recent
{\it massive} star formation. In this paper we take a closer look at
the emission from the dust pillar around CPD~$-$59$\arcdeg$2661, the young luminous star
cluster embedded within it, and the relationship between them.
\section{OBSERVATIONS AND DATA REDUCTION}
\subsection{Narrow-band Optical Images}
We obtained narrowband images of the southern Carina nebula on 2001
December 18 using the 8192$\times$8192 pixel imager MOSAIC2 mounted at
the prime focus of the Cerro Tololo Interamerican Observatory (CTIO)
4m Blanco telescope. This camera has a pixel scale of
$\sim$0$\farcs$27 and provides a 35$\farcm$4 field of view, only a
small portion of which is discussed here. The seeing during the
observations was about 0$\farcs$8. We used narrowband interference
filters ($\Delta\lambda \; \approx \; 80$ \AA) centered on [O~{\sc
iii}] $\lambda$5007, H$\alpha$ (also transmitting [N~{\sc ii}]
$\lambda$6548 and $\lambda$6583), and [S~{\sc ii}]
$\lambda\lambda$6717,6731. In each filter, we took several individual
exposures with slight positional offsets to correct for gaps in the
CCD array, and to correct for detector artifacts. Total exposure
times and other details are listed in Table 1. We reduced the data in
the standard fashion with the {\sc mscred} package in
IRAF,\footnote{IRAF is distributed by the National Optical Astronomy
Observatories, which are operated by the Association of Universities
for Research in Astronomy, Inc., under cooperative agreement with the
National Science Foundation.} and absolute sky coordinates were
computed with reference to US Naval Observatory catalog stars.
Emission-line images were flux calibrated with reference to {\it
Hubble Space Telescope} images of the Keyhole
Nebula\footnotemark\footnotetext{See {\url
http://oposite.stsci.edu/pubinfo/pr/2000/06/} and Smith et al.\
(2004b).}, obtained from the {\it HST} archive (the position observed
by {\it HST} was included in the MOSAIC2 field of view). Appropriate
corrections were made for the H$\alpha$ filter, which is wider than
the {\it HST}/WFPC2 F656N filter that mostly excludes the [N~{\sc ii}]
lines, based on line intensities in the H~{\sc ii} region near the
Keyhole (Smith et al.\ 2004b). Figure 2 shows a 3-color composite
image of CPD~$-$59$\arcdeg$2661\ and its surroundings made from optical CTIO/MOSAIC2
data.
\subsection{Near-Infrared Images}
Infrared images of the dust pillar and cluster surrounding CPD~$-$59$\arcdeg$2661\ were
obtained on 2003 March 11 using SOFI, the facility near-IR imager and
spectrograph mounted on the New Technology Telesope (NTT) of the
European Southern Observatory (ESO) at La Silla, Chile. SOFI uses a
1024$\times$1024 pixel Hawaii HgCdTe array, with a pixel scale of
0$\farcs$288 and a field of view of roughly 4$\farcm$9. Under
photometric skies with 0$\farcs$7 seeing, we obtained images of the
Treasure Chest in the $J$, $H$, and $K_S$ broadband filters, as well
as narrowband filters isolating Pa$\beta$ at 1.28 $\micron$, [Fe~{\sc
ii}] at 1.64 $\micron$, and H$_2$ 1-0 S(1) at 2.12 $\micron$. In each
filter, multiple frames were taken at many different positions
(dithering). Total exposure times and other details are listed in
Table 1. A sky position roughly 2$\arcdeg$ south of the nebula was
also observed to characterize the sky emission. The observations were
reduced using standard IR data reduction procedures in {\tt IRAF}. To
flux calibrate the $J$, $H$, and $K_S$ images, we chose several field
stars included in our images that were near CPD~$-$59$\arcdeg$2661\ and reasonably well
isolated, and we adopted their fluxes listed in the 2MASS point source
catalog.\footnotemark\footnotetext{{\url
http://irsa.ipac.caltech.edu/2mass.html}} To flux calibrate the
narrowband images, we used the same field stars and interpolated their
continuum flux to the filter wavelength; calibration of the narrow
filters has a roughly 20\% uncertainty in absolute flux. Figure 3
shows a 3-color composite image of CPD~$-$59$\arcdeg$2661\ and its surroundings in
near-IR emission lines made from NTT/SOFI data, and Figure 4 shows the
same for the $J$, $H$, and $K_S$ broadband filters.
\subsection{Optical Spectroscopy}
Low-resolution ($R \ \sim \ 700-1600$) spectra from 3600 to 9700 \AA \
were obtained on 2002 March 1 and 2 using the RC Spectrograph on the
CTIO 1.5-m telescope. Long-slit spectra were obtained with the
1$\farcs$5-wide slit aperture oriented at P.A.$\approx$60$\arcdeg$
offset about 1$\arcsec$ south of CPD~$-$59$\arcdeg$2661\ as shown in Figure 5. The pixel
scale in the spatial direction was 1$\farcs$3. Spectra were obtained
on two separate nights in two different wavelength ranges (blue,
3600-7100 \AA, and red, 6250-9700 \AA), with total exposure times and
other details listed in Table 1. Sky subtraction was accomplished by
observing a blank sky position roughly 2$\fdg$5 south. Sky conditions
were mostly photometric, although a few thin transient clouds were
present on the night of March 2 when the blue spectrum was
obtained. Flux calibration and telluric absorption correction were
accomplished using similar observations of the standard stars LTT-3218
and LTT-2415.
Although the long-slit spectra were sky subtracted, the spectrum of
our target is contaminated by bright emission from the background
Carina nebula H~{\sc ii} region. This background was subtracted using
a fit to several positions in the background sky on either side of the
target (the several background positions are labeled ``B'' in Figure
5). From the resulting background-subtracted long-slit spectra, we
made an extracted one-dimensional (1-D) spectrum from a segment of the
slit that sampled emission from the compact H~{\sc ii} region
surrounding CPD~$-$59$\arcdeg$2661\ (see Figure 5). The blue and red wavelength ranges
of these extracted 1-D spectra were merged to form a single 3600-9700
\AA \ spectrum, with a common dispersion of 2 \AA \ pixel$^{-1}$; the
average of the two was taken in the region of the spectrum near
H$\alpha$ where the blue and red spectra overlapped. A small
correction of about +5\% was made to the absolute flux of the blue
spectrum so that it matched the red in the overlapping region; this
difference was probably due to the very thin transient clouds present
on the second night when the blue spectrum was obtained. The final
flux-calibrated spectrum is shown in Figure 6. Uncertainty in the
absolute flux calibration is roughly $\pm$10\%, but our analysis of
the spectrum below relies on relative line fluxes, where uncertainty
in the reddening and measurement errors (for faint lines) dominate the
results.
Observed intensities of many relevant lines are listed in Table 2,
relative to H$\beta$=100. Uncertainties in these line intensities
vary depending on the strength of the line and the measurement method.
The integrated fluxes of isolated emission lines were measured; for
these, brighter lines with observed intensities greater than 10 in
Table 2 typically have measurement errors of a few percent, and weaker
lines may have uncertainties of $\pm$10 to 15\%. The uncertainties
increase somewhat at the blue edge of the spectrum. Blended pairs or
groups of lines were measured by fitting Gaussian profiles. For
brighter blended lines like H$\alpha$+[N~{\sc ii}] and [S~{\sc ii}],
the measurement uncertainty is typically 5 to 10\%. Obviously, errors
will be on the high end for faint lines adjacent to bright lines, and
errors will be on the low end for the brightest lines in a pair or
group, or lines in a pair with comparable intensity. Table 2 also
lists dereddened line intensities. The reddening used to correct the
observed line intensities was determined by comparing observed
strengths of Hydrogen lines to the Case B values calculated by Hummer
\& Storey (1987). As shown in Figure 7, the observed Balmer and
Paschen decrements suggest a value for $E(B-V)$ of roughly
0.65$\pm$0.04, using the reddening law of Cardelli, Clayton, \& Mathis
(1989) with $R_V = A_V \div E(B-V) \ \approx \ 4.8$, which is
appropriate for local extinction from dust clouds around the Keyhole
nebula (Smith 1987; Smith 2002). This is close to the value of
$E(B-V)$=0.6 derived by Walsh (1984) for the star CPD~$-$59$\arcdeg$2661. Table 3 lists
representative physical quantities like electron density and
temperature derived from a standard deductive nebular analysis of the
usual line ratios (e.g., Osterbrock 1989). These are useful to guide
photoionization models described below in \S 4. The ``model'' line
intensities listed in the last column of Table 2 will be discussed in
\S 4.
\section{IMAGES OF THE NEBULOSITY}
At visual wavelengths (Figure 2), CPD~$-$59$\arcdeg$2661\ is found roughly at the center
of diffuse emission-line nebulosity extending to radii of
15-20$\arcsec$ (roughly 0.2 pc), as noted by Thackeray (1950) and
Walsh (1984). Filamentary structure is also seen farther from the
star, especially in [S~{\sc ii}], which appears to outline part of a
dark dust pillar seen as a silhouette in [O~{\sc iii}] and H$\alpha$
in Figure 2.
Images at near-IR wavelengths that penetrate the dust screen clearly
suggest a more interesting morphology. Instead of a single star
surrounded by diffuse nebulosity, Figures 3 and 4 show that CPD~$-$59$\arcdeg$2661\ is a
member of a rich cluster of stars in a cavity embedded inside the head
of an externally-illuminated dust pillar. The wall of the cavity
(seen best in H$_2$ 2.122 $\micron$ emission in Fig.\ 3) has a radius
of roughly 25$\arcsec$ (0.3 pc), which is larger than the nebulosity
seen at visual wavelengths. We refer to this cluster and the
surrounding nebulosity as the ``Treasure Chest'', because the
morphology is reminiscent of an opened container with sparkling riches
inside.\footnotemark\footnotetext{One usually expects to find a
treasure chest at the bottom of the sea in a sunken pirate ship.
Following this vein, we point out a molecular globule roughly
2$\arcmin$ to the west/southwest of the Treasure Chest, with a
morphology that begs to be called the ``Sea Horse'' nebula.
Additionally, many tadpole-shaped globules are seen throughout the
Carina nebula (Smith et al.\ 2003).} The embedded cluster and cavity
inside the head of a dust pillar are reminiscent of the ``Mount St.\
Helens'' pillar in 30 Doradus (Walborn 2001), but perhaps at a
somewhat earlier phase just before it has removed its summit. The
presence of a compact cluster implies ongoing {\it massive} star
formation (consistent with the O9.5 V spectral type of CPD~$-$59$\arcdeg$2661; Walsh
1984), as compared to the previously documented cases of intermediate-
and low-mass star formation in Carina (Smith et al.\ 2004a, Megeath et
al.\ 1996).
The multiwavelength emission-line structure of the outer edges of the
dust pillar is consistent with an externally-ionized photoevaporative
flow from the surface of a dense molecular cloud (see the discussion
of the Finger globule in the northern part of the Carina nebula; Smith
et al.\ 2004b). Limb-brightened [S~{\sc ii}] emission is seen at the
edge of the cloud (the ionization front) with more extended H$\alpha$
and [O~{\sc iii}] at larger distances in the ionized evaporative flow.
A thin layer of H$_2$ emission is seen behind the ionization front,
indicating a dusty and geometrically thin photodissociation region
reaching column densities of $n_H\gg$10$^{22}$ cm$^{-2}$. The eastern
side of the dust pillar has a remarkably straight edge pointing back
toward $\eta$ Carinae and the Tr16 cluster in the heart of the Carina
nebula (see Fig.\ 1), much like the globule associated with HH~666
(Smith et al.\ 2004a), suggesting that $\eta$ Car or other massive
stars in Tr16 were responsible for shaping the Treasure Chest.
Figure 2 indicates severely non-uniform extinction across the field of
view; both on large spatial scales associated with the foreground of
the Carina nebula, and on smaller scales associated with compact
clumps within the dust pillar, superposed on the compact H~{\sc ii}
region around CPD~$-$59$\arcdeg$2661. Thus, the value of $E(B-V)$=0.65 that we derived
earlier (Table 3 and Fig.\ 7) indicates a representative average,
while the true reddening may vary strongly with position.
The visual-wavelength emission lines apparently trace only a part of
the ionized cavity that is beginning to break out of its surrounding
cocoon. Walsh (1984) observed line splitting of $\sim$25 km s$^{-1}$
at the position of the nebula, indicating expansion at roughly $\pm$12
km s$^{-1}$ (near the sound speed). However, emission-line images
show no clear evidence for outflow activity like highly-collimated
bipolar Herbig-Haro jets (Reipurth \& Bally 2001) or wider-angle
bipolar outflows that would likely be seen in shock tracers such as
[S~{\sc ii}], [Fe~{\sc ii}], or H$_2$ emission. Nothing like the
dramatic outflow from the BN/KL region of Orion (Salas et al.\ 1999;
Schild et al.\ 1997; Genzel \& Stutzki 1989; Shuping et al.\ 2004) is
seen here, even though the inferred luminosity of $\sim$10$^{6.6}$
L$_{\odot}$ (Rathborne et al.\ 2004) is significantly higher than
BN/KL.
The expansion speed of $\pm$12 km s$^{-1}$ observed by Walsh (1984)
implies an age for the cavity (radius of $\sim$25$\arcsec$ or 0.3 pc)
of a few $\times$10$^4$ yr. This cavity size is also much smaller
than a typical Str\"{o}mgrem sphere for an O9.5 V star and an ambient
density of $\sim$500 cm$^{-3}$ (Table 3), which would be almost 1 pc.
The young dynamic age and the small size of the cavity compared to the
expected Str\"{o}mgren sphere radius suggest that the cluster around
CPD~$-$59$\arcdeg$2661\ is extrememly young, and that the cavity is just in the initial
stages of expansion. Alternatively, the H~{\sc ii} region may be
dusty, and grains may absorb a large fraction of the Lyman continuum
luminosity. In any case, the compact H~{\sc ii} region is probably
caught in the early phases of breaking out of and destroying the head
of the surrounding dust pillar. Thus, the H~{\sc ii} region is not in
photoionization equilibrium, which may affect the analysis of the
spectrum in the next section. Judging by the morphology in Figures 3
and 4, the expansion of the cavity seems to be encountering less
resistance toward the southwest, while a significant reservior of
molecular material impedes its expansion toward the northeast (see
also Rathborne et al.\ 2004). Note that many of the reddened stars
that are presumably cluster members are also located toward the north
and northeast of CPD~$-$59$\arcdeg$2661\ near the edge of the cavity (Fig.\ 4).
Finally, we note that the bright star located $\sim$30$\arcsec$
northeast of CPD~$-$59$\arcdeg$2661\ (at $\alpha_{2000}$=10$^{\rm h}$45$^{\rm
m}$57$\fs$3, $\delta_{2000}$=$-$59$\arcdeg$56$\arcmin$43$\arcsec$) is
probably not a member of the embedded cluster. This star (Hen
3-485=Wra 15-642) is a Be star and is therefore somewhat evolved, and
so it is probably too old to be associated with the extremely young
cluster around CPD~$-$59$\arcdeg$2661. Instead, Massey \& Johnson (1993) considered it
to be a member of Tr16. It also appears somewhat disconnected from
the bright nebulosity around CPD~$-$59$\arcdeg$2661, and may be in the foreground of the
Treasure Chest, while still within the confines of the larger Carina
nebula.
\section{SPECTROSCOPIC ANALYSIS OF THE H~{\sc ii} REGION}
To understand the dereddened spectrum of the nebula around CPD~$-$59$\arcdeg$2661\
quantitatively, we employed the spectral synthesis code {\sc cloudy}
(Ferland 1996), using simplified assumptions about the geometry and
other factors gleaned from the analysis above. We approximated the
nebula as a 0.5 pc sphere filled with a density\footnote{This density
comes from the electron density in Table 3, derived from the relative
intensities of the [S~{\sc ii}] $\lambda\lambda$6717,6731 lines
observed in the nebula.} $n_H$=500 cm$^{-3}$, and with the density
rising slightly to 600 cm$^{-3}$ at the outer edge. We used {\sc
cloudy}'s standard H~{\sc ii} region abundances and dust content
(similar to the Orion nebula; see Ferland 1996). A range of
properties for the ionizing source was tested; the best results were
found using a blackbody with T=31,500~K and L=8$\times$10$^{4}$
L$_{\odot}$. This is typical for a main-sequence O 9.5 V star, in
agreement with the observed spectral type of CPD~$-$59$\arcdeg$2661\ (Walsh 1984). To
adequately match the observed spectrum, however, we needed to adjust
this blackbody by extinguishing roughly half the hydrogen and helium
ionizing photons. This provided much better results than simply using
a cooler blackbody, for example, which has a different spectral energy
distribution in the UV. The rationale for the depletion of ionizing
photons in the model might be that the nebula around CPD~$-$59$\arcdeg$2661\ is not in
equilibrium, while {\sc cloudy} is an equilibrium code. The cavity's
observed size in Figure 3 is smaller than a Str\"{o}mgren sphere for
an O 9.5 V star, and it is observed to be expanding (Walsh et al.\
1984). In that case, one can interpret the deficit of ionizing
photons as ionizations not balanced by recombinations as the nebula
increases in size (another way to approximate the observed spectrum
was to simply use a larger radius than observed). With these input
assumptions, the model produced a fair approximation of the observed
line intensities, and ionizing photons were used-up near the model
nebula's outer boundary as the gas became fully molecular, as expected
for the observed cavity.
However, some discrepancies between the simple model and the observed
spectrum were difficult to reconcile without modifying the chemical
abundances. Important cooling lines of S and O in the visual spectrum
were too weak, and N lines were too strong. Therefore, we lowered the
abundance of N by 20\%. This adjustment allowed the [N~{\sc ii}]
lines to be fit satisfactorally, and also increased the strength of
[S~{\sc ii}], [S~{\sc iii}], and [O~{\sc ii}] lines as they took on
the additional burden of cooling the
nebula.\footnotemark\footnotetext{The Ne abundance was also increased
to account for some additional cooling in order to match the observed
electron temperatures in Table 3; however, this {\it ad hoc}
adjustment could be any additional source of cooling and is in no way
a true indicator of an enhanced Ne abundance.} The final spectrum
with these slightly modified abundances matched the observed spectrum
quite well, and several important lines predicted by the model are
listed in Table 2 for comparison with the dereddened line intensities.
H and He line intensities are reproduced well; in fact, the perfect
agreement of H$\alpha$ suggests that our reddening value of
$E(B-V)$=0.65 is correct. Important forbidden cooling lines and
temperature/density diagnostics are also well matched, with the
exceptions noted below.
There are still two unsatisfying discrepancies between our model
predictions in Table 2 and the dereddened spectrum of the nebula
around CPD~$-$59$\arcdeg$2661. First, the model underpredicted the strength of [O~{\sc
ii}] $\lambda\lambda$3726,3729 by 14\%. Other lines in the spectrum
were over- or under-predicted by similar amounts in our model, but
[O~{\sc ii}] $\lambda\lambda$3726,3729 is one of the strongest lines
in the nebula and an important coolant, so the poor agreement is
bothersome. Simply increasing the oxygen abundance would not fix this
problem, because the increased cooling would dramtically affect the
rest of the spectrum, and it would cause additional problems because
the intensities of the red [O~{\sc ii}] lines are matched quite well
by the current model. One possible explanation is that our observed
spectrum has relatively large calibration uncertainty at the blue end
of the spectrum, which is enough to account for [O~{\sc ii}]
$\lambda\lambda$3726,3729. A second (and even more severe)
discrepancy between our model and the observed/dereddened spectrum of
CPD~$-$59$\arcdeg$2661\ is that our model underpredicts the [O~{\sc iii}] lines by about
a factor of two. We could not fix this discrepancy in a satisfactory
way with adjustments to the model; increasing the effective
temperature of the ionizing source enough to match the [O~{\sc iii}]
intensities, for example, would dramatically increase the strengths of
He~{\sc i} and [S~{\sc iii}] lines as well (some of which are already
too strong). The only potential solution to this underestimate of the
model [O~{\sc iii}] lines is that the observed spectrum of the nebula
around CPD~$-$59$\arcdeg$2661\ may be contaminated by emission from gas along the line
of sight in the Carina nebula outside the dust pillar. Although we
tried to carefully subtract the background nebular emission, the local
extinction is patchy (Fig.\ 2). The background Carina nebula has
extremely strong [O~{\sc iii}] lines (e.g., Smith \& Morse 2004; Smith
et al.\ 2004b), and even a small amount of contamination might account
for the discrepancy.
In summary, with the caveats that 1) we needed to make a minor
adjustment to the N abundance, 2) we needed to extinguish some of the
ionizing photons to account for the non-equilibrium state of the
nebula, and 3) we needed to invoke some contamination of the [O~{\sc
iii}] lines from the surrounding Carina nebula, we were able to match
the observed spectrum of the nebula around CPD~$-$59$\arcdeg$2661\ in a satisfactory way
with a simple geometry that was consistent with the observed
morphology in images, and using an ionizing source consistent with an
O 9.5 V star, which is the observed spectral type of CPD~$-$59$\arcdeg$2661\ itself
(Walsh 1984). This proves that CPD~$-$59$\arcdeg$2661\ is indeed the dominant ionizing
source of the Treasure Chest and is likely a member of the embedded
star cluster seen in IR continuum images, as discussed below in \S 5.
\section{STELLAR CONTENT\label{stellar}}
To conduct a preliminary stellar census of the cluster associated with
the Treasure Chest, we present an analysis of the $JHK$ photometry of
point sources in our broad-band near-IR images (\S 2.2). We begin by
describing the identification of all stellar sources, including
completeness limits, and our measurement of their $JHK$ magnitudes and
colors. We then place the stars on an $H-K$ vs.\ $K$ color-magnitude
diagram (CMD) and compare their placement with appropriate
pre--main-sequence (PMS) isochrones to derive a likely cluster age.
Next, we study $J-H$ and $H-K$ colors to determine the fraction of
stellar sources with IR excesses indicative of circumstellar
disks. Finally, we present a K-band luminosity function (KLF) for the
cluster which will serve as the basis for a follow-up study of the
cluster's initial mass function.
\subsection{$JHK$ photometry\label{photometry}}
We performed standard point-spread-function (PSF) photometry on our
$JHK_S$ images using the IRAF {\sc daophot} package. Stellar point
sources were identified with {\sc daofind} using a signal-to-noise
ratio of 5. An empirical, spatially variable, model PSF was
constructed separately for each image using $\approx 20$ bright and
relatively isolated stars. To construct the cleanest model PSF
possible, we disregarded stars at the very center of the images, where
the nebular emission is strongest. At the same time, it was necessary
to exclude PSF stars near the edges of the field of view because the
PSF is highly spatially variable and it is in the center of the images
where the PSF is best and where the stars of greatest interest
reside. Thus, we carefully selected PSF stars just at the periphery of
the strong nebular background.
Owing to the strong and spatially variable nature of the nebular
background emission, we performed the PSF photometry in an iterative
fashion with the aim of accurately modeling and subtracting this
background. First, we subtracted all stellar sources identified by
{\sc daofind}, and then interpolated the background over the positions
of the subtracted sources, to produce a model of the nebular
background alone. This background was then subtracted from the
original image, and the PSF photometry performed anew on this
background-subtracted image, with the background in the PSF fitting
now fixed at zero. We were not able to derive photometry for CPD~$-$59$\arcdeg$2661\
itself, since this bright star was saturated in our images.
Stellar fluxes derived from the PSF fitting were converted to $JHK$
magnitudes on the CIT system using the photometric zero-points and
color terms determined by the 2MASS\footnote{See
\url{http://www.ipac.caltech.edu/2mass/releases/allsky/doc/sec6\_4b.html}.}
project. From inspection of histograms of the $JHK$ magnitudes so
derived, we find that we are sensitive to $JHK$ magnitudes of 20.4,
19.2, and 18.1, respectively. To determine our completeness limits in
$JHK$, we used the IRAF package {\sc artdata} to add artificial stars
of varying brightness to the original images. We successfully
recovered 90\% of these artificial stars at $JHK$ magnitudes of 19.3,
18.2, and 17.2, respectively. These then define our 90\% completeness
limits.
\subsection{Cluster boundary and ``background" region\label{boundary}}
In order to distinguish the properties of the young stars in the
Treasure Chest from those of the young stellar population likely
associated with the surrounding star-forming region, we defined a
cluster boundary by tracing the edge of the nebular emission visible
in our narrow-band images (\S 3; Fig.\ 3). This marks the boundary of
the surrounding elephant trunk that harbors the embedded cluster.
We also defined a circular annulus around this cluster boundary as a
``background" region for comparison. To be sure, this region does not
sample a truly background stellar population, in the sense that this
region is likely to be dominated by young stars in the Carina
star-forming region, not just foreground and background field
stars. But defining the background region in this way will allow us to
characterize the stellar population in the Treasure Chest cluster
separately from that of other young stars in the immediate vicinity of
the cluster. In the analysis that follows, we will differentiate
between stars in the ``cluster" and ``background" regions so
defined.
\subsection{Stellar number density\label{density}}
From our $K_S$ image we can determine the surface number density of
stellar point sources in the Treasure Chest cluster. At a distance of
2.3 kpc, our image scale of 0$\farcs$27 pixel$^{-1}$ corresponds to
0.003 pc pixel$^{-1}$. We have measured the surface number density by
counting stars in boxes 33$\times$33 pixels, corresponding to
0.1$\times$0.1 pc. In the background annulus region, we detect an
average stellar surface number density of 250 pc$^{-2}$, while in the
cluster region itself we detect a maximum of 1020 pc$^{-2}$ in the
northern part of the cluster.
This is almost certainly a lower limit to the true cluster
density, as even in our $K_S$ image we can see clear traces of
extinction, and in our analysis below we find stars whose colors
indicate significant extinction, $A_V > 40$ (see \S \ref{cmd}).
Deeper images at 2 $\mu$m will presumably detect even more
sources, although at our current completeness limit of
$K \approx 17.2$ (\S \ref{photometry}), the KLF for the cluster
region already blends into the KLF of the background region
(see \S \ref{klf}).
Nonetheless, we can say here that the maximum stellar surface number
density in the Treasure Chest is at least $\approx 770$ pc$^{-2}$
(i.e., from 1020--250, as noted above), which is comparable to that
found in other rich young clusters, such as IC348 (Lada \& Lada 1995),
though probably not as high as that found in NGC~2024 (Haisch et al.\
2000) and the Trapezium (Hillenbrand \& Hartmann 1998).
\subsection{Color-magnitude diagram\label{cmd}}
Within the cluster boundary defined above, we detected 172 point
sources in our $J$ image, 194 in our $H$ image, and 199 in our $K_S$
image. Of these, 156 are common to all three images, and 183 are
detected in both the $H$ and $K_S$ images.
The $H-K$ vs.\ $K$ CMD for stars in the cluster and background regions
is shown in Figure 8. Stars located within the cluster region are
displayed as filled points. Dot-dashed lines represent isochrones for
stars with masses from 0.02 M$_\odot$ to 3.0 M$_\odot$, at ages of 0.1
Myr, 1 Myr, and 100 Myr, from the PMS models of D'Antona \& Mazzitelli
(1997), assuming a cluster distance of 2.3 kpc. The isochrones have
been transformed from the $T_{\rm eff}/L$ plane to the $(H-K)/K$ plane
using the relationship between $T_{\rm eff}$ and $HK$ bolometric
corrections compiled by Muench et al.\ (2002). Dashed lines are
reddening vectors for stars with masses of 2.5 M$_\odot$, 0.3
M$_\odot$, 0.08 M$_\odot$, and 0.05 M$_\odot$, and extinctions $A_V$
of 70, 40, 20, and 10 mag, respectively. These extinction vectors
assume the reddening law of Bessell \& Brett (1988) and a ratio of
total-to-selective extinction, $R_V$, of 4.8 (see Smith 2002). Dotted
lines represent our sensitivity and completeness limits. Crosses
along the right side of the figure represent typical observational
error bars at various $K$ magnitudes. Our photometry is evidently
complete for a star at the hydrogen-burning limit (0.08 M$_\odot$), at
an age of 0.1 Myr, seen through $A_V = 20$ mag of extinction.
The CMD of the Treasure Chest cluster exhibits several interesting
features that provide some insight into the nature of its stellar
population. First, the majority of sources in the cluster region
(filled circles) show very red $H-K$ colors, consistent with a young
population of stars still embedded in significant quantities of
intervening dust. Indeed, we find stars with extinctions as large as
$A_V \sim 50$. This population of highly reddened stars exists in the
background annulus region as well (open circles), indicating the
presence of significant numbers of young stars in the surrounding
star-forming region.
Comparison with the PMS isochrones of D'Antona \& Mazzitelli (1997)
provides further indications of extreme stellar youth. For stellar
magnitudes down to $K \sim 16$, the cluster stars (filled circles)
display a marked blue ``edge" that is well traced by the 0.1 Myr
isochrone, with a marked decrease in the number of stars blueward of
that isochrone. For $K > 16$, the 0.1 Myr isochrone begins to merge
in $H-K$ color with the 1 Myr isochrone, which may be more
representative of the young stellar population associated with the
surrounding star-forming region. At these faint magnitudes,
particularly at $K > 16.5$, we also begin to see a significant number
of stars in both the cluster and background regions with blue $H-K$
colors indicative of main-sequence and giant field stars.
\subsection{Color-color diagram\label{ccd}}
In Figure 9 we show a $J-H$ vs.\ $H-K$ color-color diagram for the
stars that we detected in all three passbands. The upper panel shows
stars within the cluster boundary defined above, while the lower panel
shows stars within the background annulus. Solid lines represent the
colors of main-sequence stars and giants from Bessell \& Brett (1988),
transformed to the CIT system. Dashed lines represent reddening
vectors emanating from the extrema of the main-sequence and giant
colors. Stars within these reddening vectors are consistent with
reddening due to intervening interstellar dust, although small amounts
of reddening due to circumstellar material cannot be ruled out.
In both panels, red points represent stars redward of the 1 Myr
isochrone in the CMD (Fig.\ 8), whereas green points represent stars
blueward of this isochrone, and which are therefore unlikely to be
young stars associated with the Treasure Chest or the surrounding
star-forming region. Indeed, with few exceptions the green points
follow the main-sequence and giant color relations closely.
The dash-dotted line is the locus of classical T~Tauri stars from
Meyer et al.\ (1997). Stars on or above this line and to the right of
the dashed lines are those with IR excesses indicative of massive
circumstellar disks.
Both the cluster and background regions show evidence for stars with
disks. However, such stars are found in higher proportion in the
cluster region. In the cluster region, 67\% of likely cluster members
(red points) show evidence for circumstellar disks; in the background
region this fraction is 44\%.
The large disk fraction found for the cluster region is almost
certainly a lower limit to the true disk fraction, since $JHK$
photometry is not the most sensitive tracer of disks. For example, in
NGC~2024 (age $<1$ Myr) Haisch et al.\ (2000) found a disk fraction of
$\sim 60\%$ from analysis of its $JHK$ excess fraction, but found a
much higher disk fraction of $\sim 90\%$ when they included $L$-band
photometry in the analysis. That the disk fraction we find in the
Treasure Chest is larger than that found in NGC~2024 via $JHK$
photometry suggests that the true disk fraction in this cluster may
prove to be among the highest yet seen for a young cluster, and
further corroborates the extremely young age from our analysis of the
CMD (\S \ref{cmd}).
\subsection{$K$-band luminosity function\label{klf}}
One of our principal aims in studying the Treasure Chest is ultimately
to measure the mass spectrum of an extremely young cluster that may
have been triggered by an earlier episode of nearby, massive star
formation. One of the primary means for determining the initial mass
function (IMF) of a young cluster is by analysis of its KLF (Muench et
al.\ 2002).
The KLF of the Treasure Chest cluster is shown in Figure 10,
which includes only stars redward of the 0.1 Myr isochrone in the CMD.
The dashed histogram shows the corresponding KLF for stars in the
background annulus region, scaled to the same spatial area as the
cluster region.
The KLF shows a clear excess of stars over the background for all $K$
down to our completeness limit. We can place a lower limit on the
number of members in the Treasure Chest cluster by summing over the
cluster KLF after subtracting the background KLF.
We find a lower limit of 69 stars down to our completeness limit.
For $K \gtrsim 17$, the cluster KLF merges with the KLF of the
surrounding star-forming region. Thus, while deeper imaging of the
cluster may identify additional stars, it may prove difficult to
statistically separate members of the Treasure Chest cluster from
members of the surrounding nebula for $K \gtrsim 17$ ($M < 0.05$
M$_\odot$ with $A_V = 10$ at an age of $\lesssim 1$ Myr).
\section{SUMMARY AND CONCLUSIONS}
We have undertaken a detailed observational analysis of a dust pillar
in the Carina nebula called the Treasure Chest, as well as its
associated embedded star cluster. Narrowband images,
visual-wavelength spectra, and broadband near-IR photometry point
toward the following main conclusions, all of which provide
independent evidence of extreme youth:
1. Emission-line images of the Treasure Chest reveal an embedded star
cluster occupying a cavity inside the head of an
externally-ionized dust pillar, and part of the embedded compact
H~{\sc ii} region appears to be breaking out of the dust pillar
into the surrounding giant H~{\sc ii} region. The dust pillar
points toward $\eta$ Carinae and other stars in the Tr16 cluster,
raising suspicion that the birth of this young cluster was
triggered by feedback from nearby massive stars.
2. The visual-wavelength spectrum of ionized gas from the cavity that
is breaking through the dust cocoon is ionized by a late O-type
star, consistent with the spectral type of O~9.5~V derived for the
central star CPD~$-$59$\arcdeg$2661\ by Walsh (1984).
3. Analysis of the color-magnitude diagram of the embedded cluster
suggests that it has an age $\la$0.1 Myr, which is in reasonable
agreement with the dynamical age of the expanding nebular cavity
(a few times 10$^4$ yr).
4. Stellar photometry also reveals several members of the cluster
that are highly embedded, with extinction values as high as
$A_V\sim$50. This is much higher than the average extinction for
the ionized gas or CPD~$-$59$\arcdeg$2661\ itself, suggesting that CPD~$-$59$\arcdeg$2661\ and its
compact H~{\sc ii} region are breaking out of the dust pillar on
the side facing the Earth, while star formation in the cloud
may be continuing on the far side.
5. Two-thirds of the cluster members show strong IR excess emission
indicative of cirumstellar disks, but the fraction may be much
higher if longer-wavelength data are considered. Thus, the disk
fraction for the Treasure Chest cluster may be among the highest
yet seen for any young clusters. Some field stars outside the
cluster (but still within the Carina nebula) also show IR excess
indicative of young disks, but the fraction is lower than in the
cluster itself.
We have also provided a preliminary K-band luminosity function, which
suggests that new data with higher spatial resolution and sensitivity
will allow us to accurately measure the cluster's mass function. The
mass function of this particular cluster will be of great interest,
because it is a ``second-generation'' cluster at the periphery of a
giant H~{\sc ii} region.
\acknowledgements \scriptsize
Support for N.S.\ was provided by NASA through grant HF-01166.01A from
the Space Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy, Inc., under
NASA contract NAS~5-26555. NOAO funded N.S.'s travel to Chile and
accommodations while at CTIO. Additional support was provided by NSF
grant AST 98-19820 and NASA grants NCC2-1052 and NAG-12279 to the
University of Colorado.
|
{
"timestamp": "2004-11-08T06:47:35",
"yymm": "0411",
"arxiv_id": "astro-ph/0411178",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411178"
}
|
\section{\@startsection{section}{1}{\z@}{3.5ex plus 1ex minus
.2ex}{2.3ex plus .2ex}{\large\bf}}
\defAppendix \Alph{section}{\arabic{section}}
\def\Alph{section}.\arabic{subsection}{\arabic{section}.\arabic{subsection}}
\def\arabic{subsubsection}{\arabic{subsubsection}}
\def
|
{
"timestamp": "2005-01-31T23:23:22",
"yymm": "0411",
"arxiv_id": "hep-th/0411074",
"language": "en",
"url": "https://arxiv.org/abs/hep-th/0411074"
}
|
\section{Introduction}
Modern methods of estimating the age of the Galactic thin
disk\footnote{All references to the \emph{Galactic disk} must be
regarded, in this work, as references to the \emph{thin} disk,
unless otherwise specified.}, like dating the oldest open clusters
by isochrone fitting and whites dwarfs by cooling sequences, rely
heavily on stellar evolution calculations. Nucleocosmochronology
is a dating method that makes use of only a few results of main
sequence stellar evolution models, therefore allowing a
quasi-independent verification of the afore-mentioned techniques.
Nucleocosmochronology employs the abundances of radioactive
nuclides to determine timescales for astrophysical objects and
events. The Th/Eu chronometer was first proposed by
\citet{pagel89} as a way of assessing the age of the Galactic
disk. Its main advantages are that \element[][232]Th has a
14.05~Gyr half-life, i.e., of the order of magnitude of the age
being assessed, and that Eu provides a satisfactory element for
comparison, being produced almost exclusively \citep[97\%,
according to\space][]{burrisetal00} by the same nucleosynthetic
process that produces all Th, the rapid neutron-capture process
(r-process). Its main disadvantage lies in the difficulty one
encounters when trying to determine stellar Th abundances. This
difficulty arises from the very low equivalent width (EW) of the
\ion{Th}{ii} line used, and from the fact that it is severely
blended with other much stronger lines.
The first to suggest the presence of a \ion{Th}{ii} line in the
solar spectrum, located at 4019.137~\AA, were
\citet{sitterly&king43}. \citet{severny58} was the first to
measure the EW of this line, deriving an upper limit for the solar
thorium abundance. The first confirmed Th detection and abundance
determination in a star other than the Sun was accomplished by
\citet{cowleyetal75} in HR~465, an Ap~star. Until today, only two
works available in the literature have presented Th abundances for
samples of Galactic disk stars: \citet{dasilvaetal90} and
\citeauthor{morelletal92} (1992, MKB92). Of these, only
\citeauthor{dasilvaetal90} present a chronological analysis.
This project aims at resuming investigation of Galactic disk
dating using [Th/Eu] abundance ratios, a theme absent from the
literature since 1990.\footnote{In this paper we obey the
following customary spectroscopic notations: absolute abundance
$\log\varepsilon(\mbox{A})\equiv\log_{10}(N_{\mathrm{A}}/N_{\mathrm{H}})+12.0$,
and abundance ratio
$\mbox{[A/B]}\equiv\log_{10}(N_{\mathrm{A}}/N_{\mathrm{B}})_{\mathrm{star}}
-\log_{10}(N_{\mathrm{A}}/N_{\mathrm{B}})_{\mbox{\scriptsize\sun}}$,
where $N_{\mathrm{A}}$ and $N_{\mathrm{B}}$ are the abundances of
elements A and B, respectively, in atoms~cm$^{-3}$.} In this
paper, Part~I of a series, we carried out the preliminary,
observationally-oriented steps required. First, an appropriate
stellar sample had to be elected. Selection considered the
suitability of the sample to the task of Galactic \emph{disk}
dating, with criteria like metallicity and spectral type range. In
stars with spectral types and luminosity classes adequate to our
study, which are F5--K2 dwarfs and subgiants (see
Sect.~\ref{sec:selection_criteria}), there are only one Th and one
Eu line adequate for abundance determinations. This means that
spectral synthesis techniques had to be called upon so as to
obtain acceptably accurate results. As prerequisites to the
synthesis, we determined accurate atmospheric parameters and
chemical abundances of the elements that contaminate the Th and Eu
spectral regions (Ti, V, Cr, Mn, Co, Ni, Ce, Nd, and Sm). Eu and
Th abundances were determined for all sample stars by spectral
synthesis, using our atmospheric parameters and abundances. The
[Th/Eu] abundance ratios thus obtained were used, in the second
paper of this series \citep[\space Paper~II]{delpelosoetal05b}, to
determine the age of the Galactic disk.
\section{Sample selection, observations and data reduction}
The stellar sample was defined using the Hipparcos catalogue
\citep{hipparcos}, upon which a series of selection criteria were
applied. All objects in the final sample were observed with the
Fiber-fed Extended Range Optical Spectrograph \citep[FEROS;\
][]{kauferetal99} fed by the 1.52~m European Southern Observatory
(ESO) telescope, in the ESO-Observat\'orio Nacional, Brazil,
agreement. Spectra were also obtained with a coud\'e spectrograph
fed by the 1.60~m telescope of the Observat\'orio do Pico dos Dias
(OPD), LNA/MCT, Brazil, and with the Coud\'e \'Echelle
Spectrometer (CES) fiber-fed by ESO's 3.60~m telescope and Coud\'e
Auxiliary Telescope (CAT). The selection criteria are related in
detail below, followed by the description of the observations and
their reduction.
\subsection{Selection criteria}
\label{sec:selection_criteria}
Multiple criteria were applied to the Hipparcos catalogue,
composed of 118\,218 objects. Initially, we eliminated objects
with parallaxes lower than 0.010\arcsec, in order to ensure the
minimum uncertainty in the derived bolometric magnitudes (22\,982
objects left). Only stars with declination $\delta~\le~+20\degr$
were kept, to allow the observations to be carried out in the
southern hemisphere (15\,898 objects left). Stars fainter than
visual magnitude $V~=~7.0$ were removed, so that high resolution,
high S/N ratio spectra could be obtained with relatively short
total exposures (i.e., less than 3 hours in total) on small and
medium sized telescopes (3272 objects left).
The subsequent criteria aimed at constructing the most suitable
sample for Galactic disk nucleocosmochronology. Spectral types
were restricted to the F5--K2 range because these stars have
life-times comparable to the age of the Galaxy (1744~objects
left). Only luminosity classes IV and V were allowed, to avoid
stars whose photospheric abundances had been altered by dredge-up
episodes, and to minimise non-LTE effects (925 objects left).
Since spectral analysis would be differential, we limited the
sample to stars with atmospheric parameters similar to the Sun by
selecting objects with a (B$-$V) color index in the interval
[+0.45, +0.82], so that their effective temperatures fell in the
range $T_{\mathrm{eff}\mbox{\scriptsize\sun}}\pm400$ (744 objects
left). As the ultimate objective of this work is the determination
of the age of the Galactic \emph{disk}, we eliminated stars with
[Fe/H]$<-$1.00 (252~objects left); for this purpose, we employed
average metallicities from the catalogue of
\citet{cayreldestrobeletal01}.
Stars listed as double in the Bright Star \citep{brightstar} or
Hipparcos catalogues were rejected. As a last criterium, we only
kept stars whose masses could be determined with the Geneva sets
of evolutionary tracks (\citealt{schalleretal92},
\citealt{charbonneletal93},
\citealt{schaereretal93b,schaereretal93}, and
\citealt{charbonneletal96}, hereafter collectively referred to as
Gen92/96). For this purpose, stars were required to be located
between tracks in at least two of the HR diagrams constructed for
different metallicities. This left 157 objects in the sample.
To reach a suitably sized sample, we kept only the brightest stars
to end up with 2 to 4 objects per 0.25~dex metallicity bin in the
interval $-1.00\leq\mbox{[Fe/H]}\leq+0.50$, and a total of 20
dwarfs and subgiants of F5 to G8 spectral type
(Table~\ref{tab:sample}).
\begin{table*} \caption[]{Selected stellar sample.}
\label{tab:sample}
\begin{tabular}{ l r r l c c r @{.} l r @{.} l c }
\hline \hline HD & HR & HIP & Name & R.A. & DEC &
\multicolumn{2}{c}{Parallax} & \multicolumn{2}{c}{V} &
Spectral type\\
& & & & 2000.0 & 2000.0 & \multicolumn{2}{c}{(mas)} &\multicolumn{2}{c}{} &and\\
& & & & (h m s) & (d m s) & \multicolumn{2}{c}{}&\multicolumn{2}{c}{}& luminosity class\\
\hline 2151 & 98 & 2021 & $\beta$ Hyi & 00 25 45 & $-$77 15 15 & 133&78 & 2&80 & G1 IV\\
9562 & 448 & 7276 &--& 01 33 43 & $-$07 01 31 & 33&71 & 5&76 & G3 V\\
16\,417 & 772 & 12\,186 & $\lambda^2$ For & 02 36 59 & $-$34 34 41
& 39&16 &
5&78 & G1 V\\
20\,766 & 1006 & 15\,330 & $\zeta^1$ Ret & 03 17 46 & $-$62 34 31
& 82&51 &
5&54 & G3--5 V\\
20\,807 & 1010 & 15\,371 & $\zeta^2$ Ret & 03 18 13 & $-$62 30 23
& 82&79 &
5&24 & G2 V\\
22\,484 & 1101 & 16\,852 & 10 Tau & 03 36 52 & +00 24 06 & 72&89 & 4&28 & F8 V\\
22\,879 &--& 17\,417 &--& 03 40 22 & $-$03 13 01 & 41&07 & 6&74 & F7--8 V\\
30\,562 & 1536 & 22\,336 &--& 04 48 36 & $-$05 40 27 & 37&73 & 5&77 & G5 V\\
43\,947 &--& 30\,067 &--& 06 19 40 & +16 00 48 & 36&32 & 6&63 & F8 V\\
52\,298 &--& 33\,495 &--& 06 57 45 & $-$52 38 55 & 27&38 & 6&94 & F5--6 V\\
59\,984 & 2883 & 36\,640 &--& 07 32 06 & $-$08 52 53 & 33&40 & 5&93 & G5--8 V\\
63\,077 & 3018 & 37\,853 & 171 Pup & 07 45 35 & $-$34 10 21 &
65&79 & 5&37 & G0
V\\
76\,932 & 3578 & 44\,075 &--& 08 58 44 & $-$16 07 58 & 46&90 &
5&86 & F7--8
IV--V\\
102\,365 & 4523 & 57\,443 &--& 11 46 31 & $-$40 30 01 & 108&23 & 4&91 & G3--5 V\\
128\,620 & 5459 & 71\,683 & $\alpha$ Cen A & 14 39 37 & $-$60 50
02 & 742&12 &
$-$0&01 & G2 V\\
131\,117 & 5542 & 72\,772 &--& 14 52 33 & $-$30 34 38 & 24&99 &
6&29 & G0--1
V\\
160\,691 & 6585 & 86\,796 & $\mu$ Ara & 17 44 09 & $-$51 50 03 &
65&46 & 5&15 &
G3 IV--V\\
196\,378 & 7875 & 101\,983 & $\phi^2$ Pav & 20 40 03 & $-$60 32 56
& 41&33 &
5&12 & F7 V\\
199\,288 &--& 103\,458 &--& 20 57 40 & $-$44 07 46 & 46&26 & 6&52 & G0 V\\
203\,608 & 8181 & 105\,858 & $\gamma$ Pav & 21 26 27 & $-$65 21 58
& 108&50 & 4&22
& F7 V\\
\hline
\end{tabular}
{References: Coordinates: SIMBAD (FK5 system); parallaxes:
Hipparcos catalogue \citep{hipparcos}; visual magnitudes: Bright
Star Catalogue \citep{brightstar} for stars with an HR~number and
SIMBAD for those without; spectral types and luminosity classes:
Michigan Catalogue of HD~Stars
\citep{michigancatalog1,michigancatalog2,michigancatalog3,michigancatalog4,michigancatalog5}
for all stars, with the exception of \object{HD~182\,572} and
\object{HD~196\,755} (Bright Star Catalogue), and
\object{HD~43\,047} \citep{fehrenbach61}.}
\end{table*}
\subsection{Checks of sample contamination}
\label{sec:kinematic}
Our sample is restricted to objects closer than 40~pc. It is a
very localised sample, considering that the Galactic thin disk has
a 290~pc scale height \citep{buseretal98}. This greatly reduces
the probability of halo stars contaminating our sample, because
the local density of halo stars is less than 0.05\%
\citep{buseretal98}. In order to reduce even further the
probability of contamination, we restricted our sample to
relatively high metallicities ($\mbox{[Fe/H]}\ge-1.00$). However,
these criteria are not perfect, because there exists an
intersection between the metallicity distributions of disk and
halo stars, i.e., there exists a non-negligible, albeit small,
number of halo stars with $\mbox{[Fe/H]}\ge-1.00$. As an
additional check of contamination, we performed a kinematic
analysis of the objects.
We developed a code to calculate the U, V, and W spatial velocity
components. This code is based on the \citet{johnson&soderblom87}
formulation and makes use of the \citet{perrymanetal98} coordinate
transformation matrix. It calculates the velocity components
relative to the Sun, and then converts them to the local standard
of rest (LSR) using the velocity of the Sun taken from
\citet{dehnen&binney98}. We adopted a right-handed coordinate
system, in which U, V, and W are positive towards the Galactic
center, the Galactic rotation and the Galactic north pole,
respectively. The code input parameters were taken from the
Hipparcos catalogue, with the exception of the radial velocities,
which were determined by us when Doppler velocity corrections were
applied to the FEROS spectra. Four of our objects
(\object{HD~22\,484}, \object{HD~22\,879}, \object{HD~76\,932},
and \object{HD~182\,572}) have radial velocities determined with
the CORAVEL spectrometers \citep{udryetal99}, which have high
precision and accuracy, and there is excellent agreement between
these and our results (linear correlation with dispersion
$\sigma=0.05$~km~s$^{-1}$). Mark that \object{HD~182\,572},
although initially selected, was not included in the final sample,
because we could not obtain Th spectra for this object;
nonetheless, we used it for comparison with \citet{udryetal99}
because we had its FEROS spectrum and its derived radial velocity
($-100.8$~km~s$^{-1}$). Distances were calculated directly from
the trigonometric parallaxes; extinction was not corrected for, as
it can be considered negligible due to the proximity of our
objects (less than 40~pc). Our results are contained in
Table~\ref{tab:kinematic_data}, along with the radial velocities
used in the calculations.
\begin{table}
\caption[]{Radial velocities (RV) and spatial velocity components
(U, V, and W) of the sample stars, in a right-handed Galactic
system and relative to the LSR. All values are in km~s$^{-1}$.}
\label{tab:kinematic_data}
\begin{tabular}{ l
r @{.} l r @{.} l r @{.} l r @{.} l } \hline \hline
HD & \multicolumn{2}{c}{RV} & \multicolumn{2}{c}{U} & \multicolumn{2}{c}{V} & \multicolumn{2}{c}{W}\\
\hline
2151 & +23 & 7 & $-$50 & 4 $\pm$1.2 & $-$41 & 9 $\pm$0.9 & $-$24 & 5 $\pm$0.4\\
9562 & $-$14 & 0 & +1 & 4 $\pm$0.5 & $-$21 & 2 $\pm$0.8 & +20 & 2 $\pm$0.4\\
16\,417 & +11 & 8 & +32 & 1 $\pm$0.5 & $-$18 & 4 $\pm$0.7 & $-$2 & 0 $\pm$0.4\\
20\,766 & +12 & 8 & $-$60 & 5 $\pm$0.9 & $-$42 & 1 $\pm$0.9 & +22 & 8 $\pm$0.6\\
20\,807 & +12 & 1 & $-$60 & 0 $\pm$0.8 & $-$41 & 3 $\pm$0.9 & +23 & 2 $\pm$0.6\\
22\,484 & +28 & 1 & +11 & 2 $\pm$0.4 & $-$10 & 0 $\pm$0.6 & $-$34 & 8 $\pm$0.5\\
22\,879 & +120 & 7 & $-$99 & 7 $\pm$0.6 & $-$80 & 5 $\pm$1.6 & $-$37 & 7 $\pm$0.9\\
30\,562 & +77 & 3 & $-$42 & 1 $\pm$0.5 & $-$67 & 5 $\pm$1.3 & $-$13 & 9 $\pm$0.6\\
43\,947 & +41 & 6 & $-$29 & 8 $\pm$2.0 & $-$6 & 1 $\pm$0.8 & +4 & 6 $\pm$0.4\\
52\,298 & +4 & 0 & +70 & 9 $\pm$1.2 & +1 & 4 $\pm$0.6 & $-$15 & 2 $\pm$0.6\\
59\,984 & +55 & 4 & +9 & 1 $\pm$0.8 & $-$74 & 8 $\pm$1.0 & $-$21 & 8 $\pm$0.8\\
63\,077 & +106 & 2 & $-$137 & 1 $\pm$1.0 & $-$55 & 6 $\pm$0.8 & +45 & 9 $\pm$0.5\\
76\,932 & +119 & 7 & $-$38 & 1 $\pm$0.4 & $-$84 & 9 $\pm$0.7 & +76 & 7 $\pm$0.8\\
102\,365 & +17 & 4 & $-$49 & 7 $\pm$0.7 & $-$33 & 7 $\pm$0.7 & +12 & 5 $\pm$0.4\\
128\,620 & $-$23 & 9 & $-$21 & 1 $\pm$0.7 & +7 & 3 $\pm$0.9 & +19 & 8 $\pm$0.4\\
131\,117 & $-$28 & 6 & $-$50 & 2 $\pm$1.5 & $-$29 & 6 $\pm$1.8 & +17 & 4 $\pm$1.0\\
160\,691 & $-$8 & 9 & $-$3 & 6 $\pm$0.4 & $-$3 & 2 $\pm$0.6 & +3 & 2 $\pm$0.4\\
196\,378 & $-$31 & 5 & $-$55 & 2 $\pm$1.2 & $-$42 & 6 $\pm$1.2 & +5 & 9 $\pm$1.0\\
199\,288 & $-$7 & 4 & +32 & 7 $\pm$0.8 & $-$96 & 4 $\pm$1.9 & +51 & 8 $\pm$1.0\\
203\,608 & $-$29 & 4 & $-$2 & 7 $\pm$0.4 & +48 & 9 $\pm$0.6 & +12 & 6 $\pm$0.4\\
\hline
\end{tabular}
\end{table}
It is not possible to discriminate the disk from the halo
population based on kinematic or metallicity criteria,
independently. Nonetheless, this discrimination can be achieved
using these criteria \emph{simultaneously}. \citet{schusteretal93}
demonstrated that halo stars whose metallicities are higher than
[Fe/H]~=~$-$1.00 have V velocity component lower than disk stars
of the same metallicity, and that it is possible to separate these
populations with a line in the V vs. [Fe/H] diagram. Analyzing
such diagram constructed for our sample stars
(Fig.~\ref{fig:kinematics}), we can see that all of them are
located far above the cut off line, indicating a highly remote
probability of one of them belonging to the halo. Note that the
metallicities used for the graph were taken from papers
individually chosen among the spectroscopic determinations that
compose the \citet{cayreldestrobeletal01} catalogue, i.e., they
were not determined by us. Even though they could have been wrong,
these determinations would have to have been \emph{overestimated}
by at least 0.52~dex for a star to cross the line, which is very
unlikely. After having determined our own metallicities
(Table~\ref{tab:adopted_atmospheric_parameters}), we confirmed
that results from the literature are overestimated by 0.08~dex at
most, when compared to our results.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics*{FIGURE1.EPS}} \caption{V
vs. [Fe/H] diagram for the sample stars. The diagonal line is a
cut off between the halo and the disk populations (below and
above, respectively). Velocity components V were calculated by us.
The metallicities initially employed, represented as open squares,
were taken from the literature (Table~\ref{tab:photometry}); small
filled circles represent metallicities derived in this work.}
\label{fig:kinematics}
\end{figure}
According to~\citet{buseretal98}, the local density of thin disk,
thick disk, and halo stars in the Galaxy conforms to the
proportion 2000:108:1, respectively. If we take a completely
random sample of 20 stars, there is a 34.9~\% probability that it
will be totally free of thick disk stars. The probability that it
will contain 3 or more thick disk stars is low (8.0~\%).
Consequently, if there are thick disk stars in our sample, they
can be regarded as a contamination without significant bearing on
our results.
\subsection{ESO 1.52~m telescope spectra}
Observations with the ESO 1.52~m telescope were performed by the
authors in two runs, in March and August, 2001. All obtained FEROS
spectra have high nominal resolving power (R~=~48\,000),
signal-to-noise ratio ($\mbox{S/N}\ge300$ in the visible) and
coverage (3500~\AA\ to 9200~\AA\ spread over 39~\'echelle orders).
Spectra were acquired for all sample stars and for the Sun
(\object{Ganymede}), and were used for atmospheric parameter and
chemical abundance determination. Order \#9 of the FEROS spectra,
which extends from 4039~\AA\ to 4192~\AA, was used for the
determination of Eu abundances by synthesis of the \ion{Eu}{ii}
line at 4129.72~\AA.
FEROS spectra are reduced automatically by a MIDAS script during
observation, immediately after the CCD read out. Reduction is
carried out in the conventional way, applying the following steps:
bias, scattered light, and flat field corrections, extraction,
wavelength calibration (using ThAr calibration spectra), and
baricentric radial velocity correction. As a last step, the
reduction script merges all \'echelle orders to form one
continuous spectrum with a 5200~\AA\ coverage. However, the merge
process is sometimes faulty, and strong curvatures and
discontinuities are introduced in the final spectrum as result. It
becomes difficult to distinguish between the ``natural'' curvature
of the spectrum and that introduced by the merge, and the
continuum normalization procedure is rendered very insecure. We
chose to re-reduce all spectra, employing the same script used for
the automatic reduction, but eliminating the merge, so that we
could work with each order independently. Unfortunately, after the
re-reduction we found out that each individual order is plagued by
a strong discontinuity itself, which demonstrates that some
reduction steps prior to the merge are also faulty. We corrected
this by simply trimming each order and retaining only the section
beyond the discontinuity. As there are intersections in wavelength
coverage between adjacent orders, total loss of coverage caused by
the trimmings was only $\sim500$~\AA. After trimming the orders,
we corrected them for the stellar radial velocity. Stars that were
observed more than once had their spectra averaged, using their
S/N ratio as weight. Finally, the spectra were normalised by
fitting selected continuum windows with order 2 to 5 Legendre
polynomials. Figure~\ref{fig:ESO_spectrum} is an example of a
solar (\object{Ganymede}) normalised ESO spectrum.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics*{FIGURE2.EPS}}
\caption{Example of a solar (\object{Ganymede}) normalised FEROS
spectrum (order~\#20), obtained with the ESO 1.52~m telescope.}
\label{fig:ESO_spectrum}
\end{figure}
\subsection{OPD spectra}
Observations at OPD were performed by the authors in five runs, in
May and October, 2000, and in May, August and October, 2002. The
instrumental set-up was the following:
1800~lines~$\mathrm{mm}^{-1}$ grating,
250~{\usefont{U}{psy}{m}{n}{m}}m slit, 1024
24{\usefont{U}{psy}{m}{n}{m}}m-pixel back-illuminated CCD (\#101
or \#106). All obtained spectra have moderate nominal resolving
power (R~$\sim$~22\,000) and high signal-to-noise ratio
($\mbox{S/N}\ge200$). The spectra are centered at the H$\alpha$
line (6562.8~\AA) and cover $\sim$150~\AA. Spectra were obtained
for all sample objects, with the exception of \object{HD~9562},
and for the Sun (\object{Ganymede}), and were used for
$T_{\mathrm{eff}}$ determination through profile fitting
(Sect.~\ref{sec:h_alfa_fitting}). FEROS spectra cannot be used for
this purpose because the echelle order~\#29, which contains
H$\alpha$, has a useful range of 6430~\AA\ to 6579~\AA, leaving
only 16~\AA\ at the red wing of the line. Since H$\alpha$ is very
broad, it does not reach continuum levels this close to its
center, precluding normalisation. On the hand, OPD spectra reach
75~\AA\ at each side of H$\alpha$, allowing the selection of
reliable continuum windows.
Reduction was performed using the Image Reduction and Analysis
Facility (IRAF\footnote{IRAF is distributed by the National
Optical Astronomy Observatories, which are operated by the
Association of Universities for Research in Astronomy, Inc., under
cooperative agreement with the National Science Foundation.}),
following the usual steps of bias, scattered light and flat field
corrections, and extraction. Wavelength calibration was performed
individually for each star by comparing the central wavelength of
selected absorption lines with their laboratory values. This way,
Doppler corrections were not necessary, since the wavelength
calibration corrects the spectra to the rest frame. Finally,
spectra were averaged and normalised in the same way the ESO
spectra were. Figure~\ref{fig:LNA_spectrum} is an example of a
solar (\object{Ganymede}) normalised OPD spectrum.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics*{FIGURE3.EPS}}
\caption{Example of a solar (\object{Ganymede}) normalised OPD
spectrum.} \label{fig:LNA_spectrum}
\end{figure}
\subsection{ESO 3.60~m telescope and CAT spectra}
Observational spectra for the Th abundance determinations were
obtained with the CES fed by the ESO 3.60~m telescope, for 16 of
the 20 sample stars and for the Sun. Observations were carried out
by the authors in January, 2002. Spectra were obtained with the
blue optical path and the high resolution image slicer (nominal
resolving power $\mbox{R}\sim235\,000$). However, unknown problems
during the observations degraded the effective resolving power
down to $\mbox{R}\sim130\,000$. The spectra were centered at the
\ion{Th}{ii} line (4019.13~\AA) and have a 27~\AA\ coverage. Due
to the low efficiency of the spectrograph at 4000~\AA, we limited
the signal-to-noise ratio to $\mbox{S/N}\sim200$ for stars with
visual magnitude $V\ge6$ and $\mbox{S/N}\sim300$ for brighter
objects.
CES spectra were also obtained with the ESO's CAT, with the aim of
determining the abundances of both Eu and Th. Observations were
carried out by the authors in August, 1998. Not all sample objects
have been observed: we obtained Eu spectra for 8 stars and the
Sun, and Th spectra for 9 stars and the Sun. All spectra have high
resolving power ($\mbox{R}=100\,000$ for Th, and
$\mbox{R}=50\,000$ for Eu), and high signal-to-noise ratio
($\mbox{S/N}\sim300$); coverage is 18~\AA.
Table~\ref{tab:available_observations} details which objects were
observed with which instruments and telescopes.
\begin{table}
\caption[]{Available observations.}
\label{tab:available_observations}
\begin{tabular}{ l @{\hspace{0em}} c @{\hspace{0.7em}} c @{\hspace{0.7em}}
c @{\hspace{0.1em}} c c @{\hspace{0em}} c @{\hspace{0.7em}} c
@{\hspace{0.7em}} c @{\hspace{0.7em}}}
\hline \hline HD & FEROS & OPD & CES + 3.60~m & & \multicolumn{3}{c}{CES + CAT}&\\
\cline{6-8} & & & & & Europium && Thorium&\\
\hline
2151 & $\surd$ & $\surd$ & $\surd$ && $\surd$ && $\surd$&\\
9562 & $\surd$& & $\surd$ && &&&\\
16\,417 & $\surd$ & $\surd$ & $\surd$ && &&&\\
20\,766 & $\surd$ & $\surd$ & $\surd$ && &&$\surd$&\\
20\,807 & $\surd$ & $\surd$ & $\surd$ && $\surd$ && $\surd$&\\
22\,484 & $\surd$ & $\surd$ & $\surd$ && $\surd$ && $\surd$&\\
22\,879 & $\surd$ & $\surd$ & $\surd$ && &&&\\
30\,562 & $\surd$ & $\surd$ & $\surd$ && &&&\\
43\,947 & $\surd$ & $\surd$ & $\surd$ && &&&\\
52\,298 & $\surd$ & $\surd$ & $\surd$ && &&&\\
59\,984 & $\surd$ & $\surd$ & $\surd$ && &&&\\
63\,077 & $\surd$ & $\surd$ & $\surd$ && $\surd$ && &\\
76\,932 & $\surd$ & $\surd$ & $\surd$ && &&&\\
102\,365 & $\surd$ & $\surd$ & $\surd$ && &&&\\
128\,620 & $\surd$ & $\surd$ & $\surd$ && $\surd$ && $\surd$&\\
131\,117 & $\surd$ & $\surd$ & $\surd$ && &&&\\
160\,691 & $\surd$ & $\surd$ & && $\surd$ && $\surd$&\\
196\,378 & $\surd$ & $\surd$ & && $\surd$ && $\surd$&\\
199\,288 & $\surd$ & $\surd$ & && && $\surd$&\\
203\,608 & $\surd$ & $\surd$ & && $\surd$ && $\surd$&\\
\hline
\end{tabular}
\end{table}
Reduction followed the usual steps of bias and flat field
corrections, extraction and wavelength calibration (using ThAr
calibration spectra). Correction for the stellar radial velocities
was implemented by comparing the wavelengths of a number of
absorption lines with their rest frame wavelengths. Finally, the
spectra were normalised by fitting selected continuum windows with
order 2 to 5 Legendre polynomials. Stars that were observed in
more than one night did \emph{not} have spectra from different
nights averaged. Figure~\ref{fig:ces_360_spectrum} shows as an
example a solar (sky) normalised CES spectrum, observed with the
3.60~m. Examples of solar (sky) normalised CES spectra, observed
with the CAT, are presented in Fig.~\ref{fig:ces_cat_eu_spectrum}
(for the Eu region), and in Fig.~\ref{fig:ces_cat_th_spectrum}
(for the Th region).
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics*{FIGURE4.EPS}}
\caption{Example of a solar (sky) normalised CES spectrum,
observed with the ESO 3.60~m telescope, in the \ion{Th}{ii} line
region at 4019.13~\AA. The inset shows the \ion{Th}{ii} line in
greater detail.} \label{fig:ces_360_spectrum}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics*{FIGURE5.EPS}}
\caption{Example of a solar (sky) normalised CES spectrum,
observed with the ESO CAT, in the \ion{Eu}{ii} line region at
4129.72~\AA. The inset shows the \ion{Eu}{ii} line in greater
detail.} \label{fig:ces_cat_eu_spectrum}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics*{FIGURE6.EPS}}
\caption{Example of a solar (sky) normalised CES spectrum,
observed with the ESO CAT, in the \ion{Th}{ii} line region at
4019.13~\AA. The inset shows the \ion{Th}{ii} line in greater
detail.}\label{fig:ces_cat_th_spectrum}
\end{figure}
\section{Atmospheric parameters}
\label{sec:atmospheric_parameters}
Atmospheric parameters (effective temperatures, surface gravities,
microturbulence velocities ($\xi$), and metallicities) have been
obtained through an iterative and totally self-consistent
procedure, composed of the following steps:
\begin{enumerate}
\item Determine photometric $T_{\mathrm{eff}}$ ([Fe/H] from literature).
\item Determine $\log g$.
\item Determine $T_{\mathrm{eff}}$ from H$\alpha$ fitting.
\item Set $T_{\mathrm{eff}}=(\mbox{photometric} +
\mbox{H}\alpha)/2$.
\item Determine {[Fe/H]} and $\xi$.
\item Determine photometric $T_{\mathrm{eff}}$ ([Fe/H] from step~5).
\item Set $T_{\mathrm{eff}}=(\mbox{photometric} +
\mbox{H}\alpha)/2$.
\item Have the parameters changed since step~8 of last iteration?\\ If
yes, go back to step~2.
\item END.
\end{enumerate}
Notice that a minimum of 2 iterations is required, since in step~8
we must compare the results obtained with those from the preceding
iteration. Input parameters for each step are simply the output
parameters of the preceding step. At all times, the adopted
effective temperature is the average of the two determinations
(photometric and from H$\alpha$ fitting). Owing to this, a new
average is performed each time a new determination is made
(steps~4 and 7, following $T_{\mathrm{eff}}$ determinations by
H$\alpha$ fitting and photometry, respectively). The techniques
used at each step are described in the next sections, and the
final results are presented in
Table~\ref{tab:adopted_atmospheric_parameters}.
\subsection{Model atmospheres and partition functions}
\label{sec:model_atm_part_funct}
The determination of effective temperatures through H$\alpha$
profile fitting, and of microturbulence velocities and chemical
abundances through detailed spectroscopic analysis, demands the
use of model atmospheres. Those employed by us were the plane
parallel, flux constant, LTE model atmospheres described and
discussed in detail by \citet{edvardssonetal93}. The $T(\tau)$
laws were interpolated using a code gently made available by
Monique Spite (Observatoire de Paris-Meudon, France). Throughout
this work, we have used [Fe/H] as the metallicity of the model
atmosphere. Partition functions were calculated by us using the
polynomial approximations of \citet{irwin81}.
\subsection{Effective temperature}
\label{sec:effective_temperature}
\subsubsection{Photometric calibrations}
\label{sec:photometric_calibrations}
We determined effective temperatures using the photometric
calibrations from \citet{artigo_tese_Gustavo}. These
metallicity-dependent calibrations were constructed using highly
accurate effective temperatures obtained with the infrared flux
method \citep{blackwelletal80,blackwelletal91}, and make use of
the $(B-V)$ and $(V-K)$ Johnson, $(b-y)$ and $\beta$ Str\"omgren,
and (B$_{\rm{T}}-$V$_{\rm{T}}$) Hipparcos colour indices:
\begin{displaymath}
\begin{array}{l r}
T_{\mathrm{eff}}(\mathrm{K})=7747-3016\,(\mathrm{B}-\mathrm{V})\,\{1-0.15\,\mathrm{[Fe/H]}\},
& \sigma=65\,\mathrm{K}\\
T_{\mathrm{eff}}(\mathrm{K})=8974-2880\,(\mathrm{V}-\mathrm{K})+440\,(\mathrm{V}-\mathrm{K})^2,
&\sigma=50\,\mathrm{K}\\
T_{\mathrm{eff}}(\mathrm{K})=8481-6516\,(\mathrm{b}-\mathrm{y})\,\{1-0.09\,\mathrm{[Fe/H]}\},
& \sigma=55\,\mathrm{K}\\
T_{\mathrm{eff}}(\mathrm{K}) =11\,654\,\sqrt{\beta-2.349}, & \sigma=70\,\mathrm{K}\\
T_{\mathrm{eff}}(\mathrm{K})=7551-2406\,(\mathrm{B}_{\mathrm{T}}-\mathrm{V}_{\mathrm{T}})\,\{1-0.2\,\mbox{[Fe/H]}\},
& \sigma=64\,\mathrm{K}
\end{array}
\end{displaymath}
A complete description of the derivation of these calibrations is
given in \citet{artigo_tese_Gustavo}.
Table~\ref{tab:photometry} contains photometric data from the
literature. The only sample star absent on the table is
\object{HD~128\,620} ($\alpha$~Cen~A), whose photometric data is
unreliable due to its high apparent magnitude. The metallicities
presented in the table were taken from papers individually chosen
from the \citet{cayreldestrobeletal01} catalogue, with the
exception of \object{HD~160\,691} \citep{artigo_tese_Gustavo}, and
were only used on step~1 of the atmospheric parameter
determination for each star. The $(b-y)$ colour indices taken from
\citet{gronbech&olsen76,gronbech&olsen77} and
\citet{crawfordetal70} were originally in the \citet{olsen83}
photometric system. Since the calibration used by us was based on
the \citet{olsen93} photometric system, we converted the
afore-mentioned data to this system through the relation
$(b-y)_{\mathrm{Olsen~1993}}=0.8858\,(b-y)_{\mathrm{Olsen~1983}}+0.0532$,
which is Equation~(1) from \citet{olsen93}.
Photometric temperatures have been averaged using the maximum
likelihood method, assuming that each value follows a Gaussian
probability distribution, which results in
$$\overline{T_{\rm{eff}}}=\sum_{i=1}^N
\frac{T_{\rm{eff}\,\emph{i}}}{\sigma_i^2}\left/\sum_{i=1}^N
\frac{1}{\sigma_i^2}\right.,$$ where $N$ is the number of
calibrations effectively used. We took the standard deviation of
the averages as their uncertainties. These vary from 27~K to 37~K,
and such a short range prompted us to adopt a single value
$\sigma=32~\mbox{K}$ for the average photometric
$T_{\mathrm{eff}}$ of each sample star. Take heed that these are
internal uncertainties only; for a detailed discussion, see
\citet{artigo_tese_Gustavo}. Final photometric effective
temperatures are presented in
Table~\ref{tab:adopted_atmospheric_parameters}.
\begin{table*}
\caption[]{Photometric indices and metallicities for all sample
stars. Metallicities were taken from the literature for use in the
kinematic characterization of the sample
(Sect.~\ref{sec:kinematic}) and as input for the first step in the
iterative determination of atmospheric parameters
(Sect.~\ref{sec:effective_temperature}).} \label{tab:photometry}
\begin{tabular}{l @{\hspace{2em}} c c @{\hspace{2em}} c c @{\hspace{2em}}
c c @{\hspace{2em}} c c @{\hspace{2em}} c c @{\hspace{2em}} c c}
\hline \hline HD & (B$-$V) & Ref. & (b$-$y) & Ref. & (V$-$K) &
Ref. & $\beta$ &
Ref. & (B$_{\rm{T}}-$V$_{\rm{T}}$)& Ref. & $[$Fe/H$]$ & Ref.\\
\hline
2151 & 0.62 & 1 & 0.379 & 3 & -- & --& 2.597 & 11 & -- & --& $-$0.11 & 15\\
9562 & 0.64 & 1 & 0.408 & 3 & 1.422 & 13 & 2.585 & 10 & 0.709 & 14 & +0.13 & 16\\
16\,417 & 0.66 & 1 & 0.412 & 3 & -- & --& -- & --& 0.730 & 14 & +0.00 & 17\\
20\,766 & 0.64 & 1 & 0.404 & 4 & 1.537 & 13 & 2.586 & 5 & -- & --& $-$0.22 & 18\\
20\,807 & 0.60 & 1 & 0.383 & 4 & -- & --& 2.592 & 5 & -- & --& $-$0.22 & 18\\
22\,484 & 0.58 & 1 & 0.363 & 3 & 1.363 & 13 & 2.608 & 5 & 0.626 & 14 & $-$0.13 & 19\\
22\,879 & 0.54 & 2 & 0.369 & 3 & -- & --& -- & --& 0.581 & 14 & $-$0.84 & 20\\
30\,562 & 0.62 & 1 & 0.403 & 3 & 1.410 & 13 & 2.610 & 5 & 0.709 & 14 & +0.27 & 21\\
43\,947 &-- &--& 0.377 & 6 & -- & --& 2.598 & 6 & 0.604 & 14 & $-$0.33 & 19\\
52\,298 & 0.46 & 2 & 0.320 & 3 & -- & --& -- & --& 0.500 & 14 & $-$0.84 & 22\\
59\,984 & 0.54 & 1 & 0.354 & 7 & -- & --& 2.599 & 5 & 0.566 & 14 & $-$0.75 & 23\\
63\,077 & 0.60 & 1 & 0.387 & 5 & -- & --& 2.590 & 5 & -- & --& $-$0.78 & 23\\
76\,932 & 0.53 & 1 & 0.368 & 5 & 1.410 & 13 & 2.595 & 5 & 0.556 & 14 & $-$0.76 & 16\\
102\,365 & 0.66 & 1 & 0.411 & 3 & -- & --& 2.588 & 6 & -- & --& $-$0.28 & 24\\
131\,117 & 0.60 & 1 & 0.389 & 4 & -- & --& 2.621 & 9 & 0.662 & 14 & +0.09 & 16\\
160\,691 & 0.70 & 1 & 0.433 & 3 & -- & --& -- & --& 0.786 & 14 & +0.27 & 24\\
196\,378 & 0.53 & 1 & 0.369 & 5 & -- & --& 2.609 & 5 & 0.579 & 14 & $-$0.44 & 23\\
199\,288 & 0.59 & 2 & 0.393 & 6 & -- & --& 2.588 & 6 & 0.638 & 14 & $-$0.66 & 25\\
203\,608 & 0.49 & 1 & 0.338 & 8 & 1.310 & 12 & 2.611 & 8 & 0.522 & 14 & $-$0.67 & 23\\
\hline
\end{tabular}
References: 1~-~\citet{brightstar}; 2~-~\citet{mermilliod87};
3~-~\citet{olsen94b}; 4~-~\citet{olsen93};
5~-~\citet{gronbech&olsen76,gronbech&olsen77};
6~-~\citet{olsen83}; 7~-~\citet{manfroid&sterken87};
8~-~\citet{crawfordetal70}; 9~-~\citet{olsen&perry84};
10~-~\citet{crawfordetal66}; 11~-~\citet{hauck&mermilliod98};
12~-~\citet{koornneef83}; 13~-~\citet{dibenedetto98};
14~-~\citet{hipparcos}; 15~-~\citet{castroetal99};
16~-~\citet{grattonetal96}; 17~-~\citet{gehren81};
18~-~\citet{delpelosoetal00}; 19~-~\citet{chenetal00};
20~-~\citet{fuhrmann98}; 21~-~\citet{dasilva&portodemello00};
22~-~\citet{hartmann&gehren88}; 23~-~\citet{edvardssonetal93};
24~-~\citet{artigo_tese_Gustavo}; 25~-~\citet{axeretal94}.
\end{table*}
\subsubsection{H$\alpha$ profile fitting}
\label{sec:h_alfa_fitting}
The H$\alpha$ profiles have been studied in detail (by
\citealt{dasilva75}, \citealt{gehren81},
\citealt{fuhrmannetal93,fuhrmannetal94}, \citealt{grattonetal96},
and \citealt{cowley&castelli02}, among others), and have been
shown, in the case of cool stars, to be rather insensitive to
surface gravity, microturbulence velocity, metallicity, and mixing
length parameter variations. They are, notwithstanding, very
sensitive to the effective temperature of the atmosphere. By
comparing the observations with theoretical profiles calculated
for various effective temperatures, we can estimate this
atmospheric parameter.
Theoretical profiles were calculated with a code developed by us
from the original routines of \citet{praderie67}. This code takes
into account the convolution of radiative, Stark
\citep{vidaletal71}, Doppler and self-resonance
\citep{cayrel&traving60} broadenings.
In order to check the accuracy of the determinations, we analysed
the solar spectrum (Fig.~\ref{fig:h_alpha_fit_sun}). The adopted
solar parameters were $T_{\mathrm{eff}}=5777$~K \citep{neckel86},
$\log g=4.44$, $[\mbox{Fe/H}]=0.00$, microturbulence velocity
$\xi$ = 1.15~km~s$^{-1}$ \citep{edvardssonetal93}, and
$n(\mbox{He})/n(\mbox{H})=0.10$. The obtained temperature was
5767~K, 10~K lower than the solar value adopted by us. This small
discrepancy was corrected by adding +10~K to all stellar effective
temperatures determined by H$\alpha$ profile fitting.
Uncertainties in the determinations were estimated using the Sun
as a standard (Table~\ref{tab:h_alpha_uncertainties}). The
influence of atmospheric parameter uncertainties were evaluated
changing these parameters (one at a time) by an amount deemed
representative of their uncertainties, and then redetermining the
effective temperature. We tested the influence of the continuum
placement uncertainty, which is $\sim$0.2\%, by multiplying the
flux by 1.002, and redetermining the temperature. Since the
fitting of theoretical profile to the observation is done via a
\emph{fit-by-eye} technique, the personal judgement of who is
making the analysis introduces uncertainty to the obtained
temperature; we estimated this uncertainty to be $\sim$20~K.
Finally, some uncertainty comes from the difference between blue
and red wings, which are best fit by profiles of effective
temperatures differing by $\sim$20~K. The total uncertainty was
estimated at 43~K by the root-mean-square (RMS) of the individual
sources cited above. Final H$\alpha$ effective temperatures are
presented in Table~\ref{tab:adopted_atmospheric_parameters}.
\begin{table}
\caption[]{Uncertainty of the effective temperatures determined by
H$\alpha$ profile fitting, using the Sun as a standard. For
detailed explanation, see text.} \label{tab:h_alpha_uncertainties}
\begin{tabular}{@{} l r @{.} l c @{}}
\hline \hline Parameter & \multicolumn{2}{c}{Parameter} & T$_{\mathrm{eff}}$\\
& \multicolumn{2}{c}{uncertainty} & uncertainty\\
\hline
Continuum & 0 & 20~\% & 25~K\\
$\log g$ & 0 & 20~dex & 20~K\\
$[\mbox{Fe/H}]$ & 0 & 08~dex & \ \ 5~K\\
$\xi$ & 0 & 10~km~s$^{-1}$ & \ \ 5~K\\
Personal judgement & \multicolumn{2}{c}{--} & 20~K\\
Blue/red wings & \multicolumn{2}{c}{--} & 20~K\\
\hline TOTAL & \multicolumn{2}{c}{--} & 43~K\\
\hline
\end{tabular}
\end{table}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics*{FIGURE7.EPS}}
\caption{H$\alpha$ profile of the Sun. Dashed line is the best fit
(5767~K). Dotted lines show profiles computed when the effective
temperature was changed by $\pm$50~K and $\pm$100~K.}
\label{fig:h_alpha_fit_sun}
\end{figure}
\subsubsection{Adopted effective temperatures}
The adopted affective temperature for each star was the arithmetic
mean of the photometric and H$\alpha$ determinations, after
convergence was achieved in the iterative procedure. The
uncertainty of the adopted $T_{\mathrm{eff}}$ was taken as the
standard deviation of the mean, resulting in 27~K.
\subsection{Surface gravity}
Surface gravities were estimated from effective temperatures,
stellar masses and luminosities, employing the known equation
\begin{equation}
\label{eqn:surface_gravity}
\log\left(\frac{g}{g_{\mbox{\scriptsize\sun}}}\right)=
\log\left(\frac{m}{m_{\mbox{\scriptsize\sun}}}\right)\
+4\log\left(\frac{T_{\mathrm{eff}}}{T_{\mathrm{eff}\mbox{\scriptsize\sun}}}\right)
+ 0.4\,(M_{\mathrm{bol}}-M_{\mathrm{bol}\mbox{\scriptsize\sun}}),
\end{equation}
in which we adopted $M_{\mathrm{bol}\mbox{\scriptsize\sun}}=4.75$
\citep{neckel86}.
Luminosities were calculated by us from the visual magnitudes of
Table~\ref{tab:sample} and bolometric corrections from
\citet{habets&heintze81}. Stellar masses were estimated using HR
diagrams with the Gen92/96 sets of evolutionary tracks. Each
diagram was calculated by Gen92/96 for 1 of 5 different
metallicities ($Z$ = 0.0010, 0.0040, 0.0080, 0.0200, and 0.0400,
with $Z_{\mbox{\scriptsize\sun}}=0.0188)$, with tracks for 5
different masses (0.80, 0.90, 1.00, 1.25, and
1.50~$m_{\mbox{\scriptsize\sun}}$). Masses were estimated for each
star, interpolating among the tracks calculated for the
metallicities above. Then, we interpolated among metallicities to
obtain the correct mass for the star. Figure~\ref{fig:masses}
presents one such diagram, calculated for $Z=0.0200$, in which we
used the final, adopted effective temperatures and metallicities
(Table~\ref{tab:adopted_atmospheric_parameters}).
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics*{FIGURE8.EPS}}
\caption{Example of a Gen92/96 set of evolutionary tracks, for
$Z=0.0200$. Filled squares are the sample stars, after convergence
of the atmospheric parameter determination procedure. The Sun is
represented by the symbol~$\sun$. The effective temperatures and
metallicities are the final, adopted values
(Table~\ref{tab:adopted_atmospheric_parameters}).}
\label{fig:masses}
\end{figure}
In order to check the accuracy of the determinations, we estimated
the solar mass itself in the Gen92/96 diagrams. The obtained value
was $0.97\,m_{\mbox{\scriptsize\sun}}$. This discrepancy was
corrected by adding $+0.03\,m_{\mbox{\scriptsize\sun}}$ to the
masses obtained for the sample stars.
The uncertainty of the surface gravities was obtained by error
propagation from Equation~(\ref{eqn:surface_gravity}), resulting
in the expression
$$\sigma_{\log~g}=\sqrt{\sigma_{\log(m/m_{\mbox{\scriptsize\sun}})}^2+16\,\sigma_{\log~T_{\rm{eff}}}^2};$$
analysis showed that the influence of the bolometric magnitude
uncertainty is negligible, and as such it was not taken into
consideration.
Stellar mass uncertainty is a function of the position of the star
in the HR diagram. Stars located in regions where the evolutionary
tracks are closer together will have larger errors than those
located between tracks further apart. Nevertheless, we have chosen
to adopt a single average uncertainty of
0.03~$m_{\mbox{\scriptsize\sun}}$, so as to simplify the
calculations. Taking stellar masses and effective temperatures
typical of our sample (1.00~$m_{\mbox{\scriptsize\sun}}$ and
6000~K, respectively), we have
$\sigma_{\log~T_{\rm{eff}}}=1.9\,10^{-3}$ and
$\sigma_{\log(m/m_{\mbox{\scriptsize\sun}})}=1.3\,10^{-2}$, and
consequently $\sigma_{\log~g}=0.02$~dex. This value does not take
into consideration the uncertainties intrinsic to the calculation
of evolutionary tracks, which are very difficult to assess.
\subsection{Microturbulence velocity and metallicity}
Microturbulence velocities and metallicities have been determined
through detailed, differential spectroscopic analysis, relative to
the Sun, using EWs of \ion{Fe}{i} and \ion{Fe}{ii} lines. In such
differential analysis, the Sun is treated like any other sample
star: its spectra are obtained under the same conditions, with the
same resolving power, signal-to-noise ratio, and equipment, and
are reduced by the same procedures; the same model atmospheres and
analysis codes are utilised. This way, systematic errors and
eventual non-LTE effects are reduced, because they are partially
cancelled out.
\subsubsection{Equivalent width measurements}
\label{sec:EW_measurement}
We were not interested solely in Fe lines, but also in the lines
of other elements that contaminate the spectral regions of the
\ion{Th}{ii} line in 4019.13~\AA\ (V, Cr, Mn, Co, Ni, Ce, and Nd),
and of the \ion{Eu}{ii} line in 4129.72~\AA\ (Ti, V, Cr, Co, Ce,
Nd, and Sm), whose EWs were then used for the abundance
determinations of Sect.~\ref{sec:chemical_abundances}.
An initial list of lines was constructed from
\citet{linhas_solares}, \citet{steffen85},
\citet{cayreldestrobel&bentolila89}, \citet{brown&wallerstein92},
\citet{furenlid&meylan90}, and \citet{meylanetal93}. Then, a
series of selection criteria were applied: lines with wavelengths
lower than 4000~\AA\ were removed, since the line density in this
region is very high, lowering the continuum and rendering its
normalization insecure, besides making it too difficult to find
acceptably isolated lines; lines with wavelengths greater than
7000~\AA\ were discarded, due to excessive telluric line
contamination in this region; lines with
$\mbox{EW}>110\,\mbox{m\AA}$ in the Sun, which are more sensitive
to uncertainties in microturbulence velocity due to saturation,
were eliminated; only isolated lines were kept, eliminating those
with known contaminations; lines whose EWs could not have been,
for any reason, accurately measured in the solar spectra, were
rejected.
EWs have been measured by Gaussian profile fitting. In order to
check the measurement accuracy, we compared our solar EWs with
those from \citet{meylanetal93}. The EWs from
\citeauthor{meylanetal93} were measured by Voigt profile fitting,
using the \citet{atlas_solar} solar atlas, which has very high
resolving power and signal-to-noise ratio ($R=522\,000$ and
$S/N=3000$, between 4500~\AA\ and 6450~\AA). A linear fit between
the two data sets,
$$\mbox{EW}_{\mathrm{Meylan~et~al.~1993}}=(1.03463\pm0.00649)\,\mbox{EW}_{\mathrm{this~work}},$$
shown in panel~(a) of Fig.~\ref{fig:EW_atlas}, presents low
dispersion ($\sigma=2.9$~m\AA) and high correlation coefficient
($R=0.994$). This relation is used to correct all our EWs. The
mean percent difference between the measurements is $-2.5\%$,
presenting no dependence on the EW (Fig.~\ref{fig:EW_atlas}, panel
(b)). The larger percent differences exhibited by the weakest
lines are expected, since these are more prone to uncertainties.
More than 86\% of the lines agree at a 10\% level.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics*{FIGURE9.EPS}}
\caption{Panel (a): Comparison between our solar EWs and those
from \protect\citet{meylanetal93}. Solid line corresponds to a
$y=x$ relation, and the dotted one is a linear regression.
Panel~(b): Percent differences between the same EWs. Solid lines
correspond to mean differences, and the dotted ones represent the
mean differences modified by their standard deviations
($\pm1\,\sigma$).} \label{fig:EW_atlas}
\end{figure}
A last selection criterium was applied to the line list: given the
2.9~m\AA\ dispersion of the linear relation between our EWs and
those from \citeauthor{meylanetal93}, we rejected all Fe lines
with $\mbox{EW}<6$~m\AA\ (i.e., $\sim2\,\sigma$). This criterium
was not implemented for the lines of the other elements because
they have fewer interesting lines available, and a cut off at
6~m\AA\ would be too restrictive. The final list of lines is
composed of 65~\ion{Fe}{i}, 10~\ion{Fe}{ii}, 31~Ti, 6~V, 21~Cr,
6~Mn, 8~Co, 13~Ni, 5~Ce, 2~Nd, and 1~Sm lines, in the Sun. Not all
these lines have been used for all sample stars, since random
events, like excessive profile deformation by noise, prevented
sometimes an accurate measurement.
Table~\ref{tab:sample_ews} presents a sample of the EW data. Its
complete content, composed of the EWs of all measured lines, for
the Sun and all sample stars, is only available in electronic form
at the CDS.\footnote{Via anonymous ftp to {\tt
cdsarc.u-strasbg.fr} or via {\tt http://} {\tt
cdsweb.u-strasbg.fr/cgi-bin/qcat?}} Column~1 lists the central
wavelength (in angstroms), Column~2 gives the element symbol and
degree of ionization, Column~3 gives the excitation potential of
the lower level of the electronic transition (in eV), Column~4
presents the solar $\log gf$ derived by us, and the subsequent
Columns present the EWs, in m\AA, for the Sun and the other stars,
from \object{HD~2151} to \object{HD~203\,608} (in order of
increasing HD number).
\begin{table}
\caption[]{A sample of the EW data. The complete content of this
table is only available in electronic form at the CDS. For a
description of the columns, see text
(Sect.~\ref{sec:EW_measurement}).} \label{tab:sample_ews}
\begin{tabular}{@{} c c c c c @{\hspace{1em}} c @{\hspace{1em}}
c @{}} \hline \hline $\lambda$~(\AA) & Element &
$\chi$~(eV) & $\log gf$ &
Sun & $\cdots$ & \object{HD~203\,608}\\
\hline
5657.436 & \ \ion{V}{i} & 1.06 & $-$0.883 & \ \ 9.5 & $\cdots$ & \ \ 0.0\\
5668.362 & \ \ion{V}{i} & 1.08 & $-$0.920 & \ \ 8.5 & $\cdots$ & \ \ 0.0\\
5670.851 & \ \ion{V}{i} & 1.08 & $-$0.452 & 21.6 & $\cdots$ & \ \ 0.0\\
$\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ \\
5427.826 & \ion{Fe}{ii} & 6.72 & $-$1.371 & \ \ 6.4 & $\cdots$ & \ \ 0.0\\
6149.249 & \ion{Fe}{ii} & 3.89 & $-$2.711 & 40.9 & $\cdots$ & 24.7\\
\hline
\end{tabular}
\end{table}
\subsubsection{Spectroscopic analysis}
\label{sec:fe_analysis}
The first step of the spectroscopic analysis consisted of
determining \emph{solar} $\log gf$ values for all measured lines.
For such determination, we employed a code written by us, based on
routines originally developed by Monique Spite. This code
calculates the $\log gf$ that an absorption line must have for its
abundance to be equal to the standard solar value from
\citet{grevesse&sauval98}, given the solar EW and lower excitation
potential of the line. For all stars, chemical abundances were
determined using these solar $\log gf$ values.
Microturbulence velocity and metallicity have been determined
through a procedure which is iterative, and which can be
summarised by the following sequence of steps:
\begin{enumerate}
\item Determine [Fe/H] for all lines, using\\
$[$Fe/H$]_{\mathrm{model~atmosphere}}$ = $[$Fe/H$]_{\mathrm{literature}}$.
\item Set $[$Fe/H$]_{\mathrm{model~atmosphere}}$~=~average $[$Fe/H$]$ from lines.
\item Determine [Fe/H] for all lines.
\item Is $[$Fe/H$]_{\mathrm{model~atmosphere}}$~=~average $[$Fe/H$]$ from
lines?\\ If no, go back to step 2.
\item Draw a [Fe/H] vs. EW graph for all lines, and fit a line.
\item Is the angular coefficient of the line negligible?\\ If
no, modify $\xi$ and go back to step 3.
\item END.
\end{enumerate}
In steps 1 and 3, iron abundances are calculated line-by-line
using a code very similar to the one used for the determination of
the solar $\log gf$ values. The only difference is that, in this
case, the $\log gf$ are fixed at their solar values, and an
abundance is calculated for each line. In step 6, the angular
coefficient is considered negligible when it is less than 25\% of
its own uncertainty (obtained by the fitting calculation). It is
important to note that effective temperature and surface gravity
remain constant at all times. In practice, the procedure is
executed automatically by a code, developed by \citet{dasilva00}.
This code iterates the parameters, calculating abundances and
linear fits, and making the appropriate comparisons, without human
interference.
Iron abundances obtained using \ion{Fe}{i} and \ion{Fe}{ii} lines
agree very well, their differences being always lower than
dispersion among the lines (as can be seen in
Table~\ref{tab:abundances_1}). The differences show no dependence
on any atmospheric parameter. Ionization equilibrium was not
enforced by the analysis procedure, but it was, nonetheless,
achieved.
Microturbulence velocity uncertainties have been estimated by
varying this parameter until the uncertainty of the angular
coefficient of the linear fit to the [Fe/H] vs. EW graph became
equal to the coefficient itself. We found out that the
uncertainties are metallicity-dependent, and can be divided into
three ranges:
$$ \sigma_{\xi} = \left\{ \begin{array}{l @{\mbox{,\space\space if\space}} c c c c c}
0.23~\mbox{km s}^{-1} & & & \mbox{[Fe/H]} & < & -0.7\\
0.13~\mbox{km s}^{-1} & -0.7 & \le & \mbox{[Fe/H]} & < & -0.3\\
0.05~\mbox{km s}^{-1} & -0.3 & \le & \mbox{[Fe/H]} & &
\end{array} \right.$$ This behaviour is expected, as the
dispersion between Fe lines increases for metal-poor stars.
Metallicity uncertainties were estimated in the same way that
uncertainties for the other elements were
(Sect.~\ref{sec:errors_assessment}), and are presented in
Table~\ref{tab:adopted_atmospheric_parameters}, along with the
final, adopted values of all atmospheric parameters.
\begin{table}
\caption[]{Adopted atmospheric parameters, including the
photometric and H$\alpha$ effective temperatures used to obtain
the adopted mean values, and the stellar masses used to obtain the
surface gravities. Uncertainties of photometric, H$\alpha$, and
mean effective temperatures, and of stellar masses and surface
gravities, are the same for all stars: 32~K, 43~K, 27~K,
$0.03~m_{\mbox{\scriptsize\sun}}$, and 0.02~dex, respectively.}
\label{tab:adopted_atmospheric_parameters}
\begin{tabular}{ l @{\hspace{0.47em}} c @{\hspace{0.47em}} c @{\hspace{0.3em}}
c @{\hspace{0.3em}} c @{\hspace{0.45em}} c @{\hspace{0.47em}} c
@{\hspace{0.47em}} c @{\hspace{0.47em}} c @{\hspace{0.47em}} c }
\hline \hline HD & \multicolumn{3}{c}{$T_{\mathrm{eff}}$ (K)} & &
$m_{\mbox{\scriptsize\sun}}$ & $\log g$
& [Fe/H] & $\xi$ (km~s$^{-1}$)\\
\cline{2-4} & Phot. & H$\alpha$ & MEAN & & & & & \\
\hline
2151 & 5909 & 5799 & 5854 & & 1.19 & 3.98 & $-$0.03 $\pm$0.09 & 1.32 $\pm$0.05 \\
9562 & 5794 & -- & 5794 && 1.23 & 3.95 & +0.16 $\pm$0.09 & 1.45 $\pm$0.05\\
16\,417 & 5821 & 5817 & 5819 && 1.15 & 4.07 & +0.13 $\pm$0.09 & 1.38 $\pm$0.05\\
20\,766 & 5696 & 5715 & 5706 && 0.95 & 4.50 & $-$0.21 $\pm$0.09 & 1.01 $\pm$0.05\\
20\,807 & 5866 & 5863 & 5865 && 0.99 & 4.48 & $-$0.23 $\pm$0.10 & 1.18 $\pm$0.05\\
22\,484 & 5983 & 6063 & 6023 && 1.16 & 4.11 & $-$0.03 $\pm$0.09 & 1.44 $\pm$0.05\\
22\,879 & 5928 & 5770 & 5849 && 0.73 & 4.34 & $-$0.76 $\pm$0.12 & 0.69 $\pm$0.23\\
30\,562 & 5883 & 5855 & 5869 && 1.18 & 4.09 & +0.19 $\pm$0.09 & 1.51 $\pm$0.05\\
43\,947 & 5940 & 5937 & 5889 && 0.95 & 4.32 & $-$0.27 $\pm$0.10 & 1.02 $\pm$0.05\\
52\,298 & 6305 & 6200 & 6253 && 1.09 & 4.41 & $-$0.31 $\pm$0.10 & 1.44 $\pm$0.13\\
59\,984 & 5968 & 5848 & 5908 && 0.97 & 3.96 & $-$0.67 $\pm$0.11 & 1.07 $\pm$0.13\\
63\,077 & 5752 & 5713 & 5733 && 0.77 & 4.15 & $-$0.76 $\pm$0.11 & 0.78 $\pm$0.23\\
76\,932 & 5874 & 5825 & 5850 && 0.82 & 4.14 & $-$0.84 $\pm$0.11 & 0.84 $\pm$0.23\\
102\,365 & 5705 & 5620 & 5663 && 0.88 & 4.43 & $-$0.29 $\pm$0.09 & 1.05 $\pm$0.05\\
128\,620 & -- & 5813 & 5813 && 1.09 & 4.30 & +0.26 $\pm$0.09 & 1.23 $\pm$0.05\\
131\,117 & 5994 & 5813 & 5904 && 1.25 & 3.96 & +0.10 $\pm$0.09 & 1.49 $\pm$0.05\\
160\,691 & 5736 & 5676 & 5706 && 1.06 & 4.19 & +0.28 $\pm$0.09 & 1.27 $\pm$0.05\\
196\,378 & 6014 & 6044 & 6029 && 1.14 & 3.97 & $-$0.37 $\pm$0.10 & 1.64 $\pm$0.13\\
199\,288 & 5785 & 5734 & 5760 && 0.82 & 4.35 & $-$0.59 $\pm$0.10 & 0.84 $\pm$0.13\\
203\,608 & 6057 & 5987 & 6022 && 0.86 & 4.31 & $-$0.67 $\pm$0.11 & 1.18 $\pm$0.23\\
\hline
\end{tabular}
\end{table}
\section{Abundances of contaminating elements}
\label{sec:chemical_abundances}
\subsection{Abundance determination}
The abundance of the elements other than iron have been determined
using the same code as in Sect.~\ref{sec:fe_analysis}, with the
adopted atmospheric parameters and the solar $\log gf$ values.
Adopted abundances have been obtained taking the arithmetic mean
of the abundances from all lines of each element, irrespective of
ionization state. When dispersion between lines was less than
0.10~dex, we eliminated those that deviated by more than
$2\,\sigma$ from the mean, and when it was more than 0.10~dex, we
eliminated those that deviated by more than $1\,\sigma$. The
number of lines kept is presented in Tables~\ref{tab:abundances_1}
and \ref{tab:abundances_2}, along with adopted abundances,
dispersions between the lines, and uncertainties, calculated
according to the procedure described in
Sect.~\ref{sec:errors_assessment}.
The lines of V, Mn, and Co exhibit hyperfine structures (HFSs)
strong enough that they must be taken into account, or else severe
inaccuracies would be introduced. At the resolving power achieved
by FEROS, line profiles remain Gaussian even for elements with
important HFSs, so that we can still use the EWs we measured by
Gaussian profile fitting. For V and Mn lines, we took the
wavelengths of the HFS components from the line lists of
\citet{kurucz_homepage}, whereas for the Co lines, we calculated
the wavelengths ourselves \citep{delpelosoetal05c}. These
calculations were realized using Casimir's equation
\citep{casimir_63}, and HFS laboratory data from
\citet{guthohrlein&keller90}, \citet{pickering96}, and
\citet{pickeringþe96}. The relative intensities of the
components were obtained from the calculations of White \&
Eliason, tabulated in \citet{condon&shortley67}. A modified
version of the code, which takes HFS into consideration, was used
for the determination of the $\log gf$ values and abundances.
\begin{sidewaystable*}\begin{minipage}[t][180mm]{\textwidth}
\caption[]{Fe, Ti, V, Cr, and Mn abundances, relative to H. N is
the number of absorption lines effectively used for each abundance
determination, and $\sigma_{\mathrm{[X/H]}}$ is the standard
deviation of the lines of element~X. The uncertainties of
abundances relative to H and to Fe, calculated as described in
Sect.~\ref{sec:errors_assessment}, are also presented (as
uncert.$_{\mathrm{[X/H]}}$ and uncert.$_{\mathrm{[X/Fe]}}$,
respectively).} \label{tab:abundances_1}
\begin{tabular}{ l @{\hspace{1em}}
c @{\hspace{1em}} c @{\hspace{1em}} c @{\hspace{1em}} c
@{\hspace{1em}} c @{\hspace{1em}} c @{\hspace{1em}} c
@{\hspace{1em}} c @{\hspace{1em}} c @{\hspace{1em}} c
@{\hspace{1em}} c @{\hspace{1em}} c @{\hspace{1em}} c
@{\hspace{1em}} c @{\hspace{1em}} c } \hline \hline HD &
[\ion{Fe}{i}/H] & N & $\sigma_{\mathrm{[\ion{Fe}{i}/H]}}$ &
[\ion{Fe}{ii}/H] & N & $\sigma_{\mathrm{[\ion{Fe}{ii}/H]}}$ &
[Fe/H] & N & $\sigma_{\mathrm{[Fe/H]}}$ &
uncert.$_{\mathrm{[Fe/H]}}$ & [Ti/H] & N &
$\sigma_{\mathrm{[Ti/H]}}$
& uncert.$_{\mathrm{[Ti/H]}}$ & uncert.$_{\mathrm{[Ti/Fe]}}$\\
\hline 2151 & $-$0.03 & 54 & 0.08 & $-$0.03 & 8 & 0.05 &
-0.03 & 62 & 0.08 & 0.09 & +0.00 & 29 & 0.12 & 0.09 & 0.02\\
9562 & +0.16 & 55 & 0.06 & +0.19 & 9 & 0.04 & +0.16 & 64
& 0.06 & 0.09 & +0.15 & 29 & 0.09 & 0.09 & 0.02\\
16\,417 & +0.13 & 55 & 0.04 & +0.12 & 8 & 0.03 & +0.13 & 63 & 0.04
& 0.09 & +0.14 & 28 & 0.06 & 0.09 & 0.02\\
20\,766 & $-$0.21 & 51 & 0.05 & $-$0.21 & 8 & 0.05 & $-$0.21 & 59
& 0.05 & 0.09 & $-$0.16 & 29 & 0.06 & 0.09 & 0.02\\
20\,807 & $-$0.23 & 48 & 0.04 & $-$0.25 & 8 & 0.04 &
$-$0.23 & 56 & 0.04 & 0.10 & $-$0.18 & 28 & 0.07 & 0.09 & 0.02\\
22\,484 & $-$0.03 & 51 & 0.07 & $-$0.03 & 8 & 0.05 & $-$0.03 & 59
& 0.07 & 0.09 & +0.01 & 28 & 0.09 & 0.10 & 0.01\\
22\,879 & $-$0.75 & 38 & 0.12 & $-$0.80 & 8 & 0.08 & $-$0.76 & 46
& 0.12 & 0.10 & $-$0.46 & 27 & 0.10 & 0.09 & 0.02\\
30\,562 & +0.19 & 58 & 0.05 & +0.19 & 7 & 0.04 & +0.19 & 65 & 0.05
& 0.09 & +0.22 & 28 & 0.06 & 0.09 & 0.02\\
43\,947 & $-$0.28 & 49 & 0.06 & $-$0.22 & 8 & 0.06 & $-$0.27 & 57
& 0.07 & 0.10 & $-$0.23 & 27 & 0.10 & 0.09 & 0.02\\
52\,298 & $-$0.31 & 38 & 0.06 & $-$0.28 & 8 & 0.06 & $-$0.31 & 46
& 0.06 & 0.10 & $-$0.23 & 25 & 0.12 & 0.09 & 0.02\\
59\,984 & $-$0.67 & 43 & 0.10 & $-$0.68 & 8 & 0.11 & $-$0.67 & 51
& 0.01 & 0.11 & $-$0.48 & 28 & 0.12 & 0.08 & 0.01\\
63\,077 & $-$0.76 & 43 & 0.12 & $-$0.78 & 8 & 0.10 & $-$0.76 & 51
& 0.11 & 0.09 & $-$0.48 & 25 & 0.09 & 0.09 & 0.03\\
76\,932 & $-$0.84 & 36 & 0.11 & $-$0.84 & 8 & 0.10 & $-$0.84 & 44
& 0.11 & 0.10 & $-$0.57 & 27 & 0.08 & 0.09 & 0.02\\
102\,365 & $-$0.28 & 52 & 0.05 & $-$0.33 & 8 & 0.05 & $-$0.29 & 60
& 0.05 & 0.09 & $-$0.17 & 28 & 0.05 & 0.09 & 0.02\\
128\,620 & +0.26 & 56 & 0.06 & +0.26 & 9 & 0.04 & +0.26 & 65 &
0.06 & 0.09 & +0.29 & 27 & 0.08 & 0.09 & 0.02\\
131\,117 & +0.09 & 53 & 0.05 & +0.14 & 9 & 0.05 & +0.10 & 62 &
0.05 & 0.09 & +0.09 & 29 & 0.06 & 0.09 & 0.02\\
160\,691 & +0.27 & 52 & 0.05 & +0.33 & 8 & 0.03 & +0.28 & 60 &
0.05 & 0.09 & +0.30 & 29 & 0.07 & 0.09 & 0.02\\
196\,378 & $-$0.37 & 32 & 0.08 & $-$0.33 & 9 & 0.06 & $-$0.37 & 41
& 0.07 & 0.10 & $-$0.30 & 20 & 0.08 & 0.09 & 0.01\\
199\,288 & $-$0.59 & 37 & 0.07 & $-$0.59 & 8 & 0.06 & $-$0.59 & 45
& 0.06 & 0.10 & $-$0.38 & 22 & 0.09 & 0.09 & 0.02\\
203\,608 & $-$0.68 & 36 & 0.10 & $-$0.60 & 9 & 0.11 & $-$0.67 & 45
& 0.11 & 0.10 & $-$0.62 & 20 & 0.12 & 0.09 & 0.02\\
\hline
\end{tabular}
\vspace{0.3cm}
\begin{tabular}{ l @{\hspace{1em}} c @{\hspace{1em}} c @{\hspace{1em}} c @{\hspace{1em}} c @{\hspace{1em}}
c @{\hspace{1em}} c @{\hspace{1em}} r @{\hspace{1em}} c
@{\hspace{1em}} c @{\hspace{1em}} c @{\hspace{1em}} c
@{\hspace{1em}} c @{\hspace{1em}} c @{\hspace{1em}} c
@{\hspace{1em}} c } \hline \hline HD & [V/H] & N &
$\sigma_{\mathrm{[V/H]}}$ & uncert.$_{\mathrm{[V/H]}}$ &
uncert.$_{\mathrm{[V/Fe]}}$ & [Cr/H] & N &
$\sigma_{\mathrm{[Cr/H]}}$ & uncert.$_{\mathrm{[Cr/H]}}$ &
uncert.$_{\mathrm{[Cr/Fe]}}$ & [Mn/H] & N &
$\sigma_{\mathrm{[Mn/H]}}$ & uncert.$_{\mathrm{[Mn/H]}}$ &
uncert.$_{\mathrm{[Mn/Fe]}}$\\ \hline 2151 & +0.02 & 4 & 0.01 &
0.10 & 0.04 & $-$0.04 & 19 & 0.06 & 0.10
& 0.02 & $-$0.09 & 6 & 0.04 & 0.08 & 0.03\\
9562 & +0.16 & 6 & 0.07 & 0.11 & 0.04 & +0.18 & 20 & 0.07 & 0.10 &
0.02 & +0.16 & 6 & 0.05 & 0.07 & 0.02\\
16\,417 & +0.18 & 6 & 0.07 & 0.11 & 0.04 & +0.14 & 19 & 0.04 &
0.10
& 0.02 & +0.14 & 6 & 0.03 & 0.07 & 0.03\\
20\,766 & $-$0.25 & 5 & 0.07 & 0.11 & 0.04 & $-$0.21 & 19 & 0.06 &
0.10 & 0.02 & $-$0.30 & 5 & 0.03 & 0.08 & 0.03\\
20\,807 & $-$0.23 & 4 & 0.03 & 0.11 & 0.04 & $-$0.23 & 19 & 0.08 &
0.11 & 0.02 & $-$0.29 & 6 & 0.03 & 0.08 & 0.03\\
22\,484 & +0.06 & 4 & 0.08 & 0.06 & 0.03 & $-$0.03 & 20 & 0.07 &
0.10 & 0.02 & $-$0.07 & 6 & 0.03 & 0.09 & 0.04\\
22\,879 & $-$0.50 & 4 & 0.15 & 0.13 & 0.05 & $-$0.69 & 13 & 0.10 &
0.11 & 0.01 & $-$0.96 & 3 & 0.10 & 0.08 & 0.03\\
30\,562 & +0.23 & 6 & 0.10 & 0.10 & 0.04 & +0.20 & 19 & 0.08 &
0.10 & 0.02 & +0.18 & 6 & 0.02 & 0.07 & 0.03\\
43\,947 & $-$0.12 & 4 & 0.08 & 0.11 & 0.04 & $-$0.27 & 20 & 0.08 &
0.11 & 0.02 & $-$0.41 & 6 & 0.09 & 0.08 & 0.03\\
52\,298 & $-$0.26 & 2 & 0.09 & 0.10 & 0.04 & $-$0.30 & 17 & 0.13 &
0.11 & 0.02 & $-$0.39 & 6 & 0.09 & 0.08 & 0.03\\
59\,984 & $-$0.44 & 4 & 0.08 & 0.13 & 0.06 & $-$0.67 & 17 & 0.13 &
0.12 & 0.01 & $-$0.83 & 6 & 0.09 & 0.09 & 0.04\\
63\,077 & $-$0.27 & 2 & 0.07 & 0.14 & 0.04 & $-$0.67 & 11 & 0.07 &
0.10 & 0.02 & $-$0.82 & 5 & 0.08 & 0.08 & 0.03\\
76\,932 & $-$0.51 & 3 & 0.17 & 0.13 & 0.05 & $-$0.76 & 10 & 0.07 &
0.11 & 0.02 & $-$1.01 & 2 & 0.06 & 0.08 & 0.03\\
102\,365 & $-$0.28 & 4 & 0.08 & 0.12 & 0.04 & $-$0.28 & 19 & 0.06
& 0.10 & 0.02 & $-$0.37 & 6 & 0.02 & 0.08 & 0.03\\
128\,620 & +0.29 & 6 & 0.10 & 0.11 & 0.04 & +0.26 & 20 & 0.06 &
0.10 & 0.02 & +0.29 & 6 & 0.04 & 0.07 & 0.02\\
131\,117 & +0.06 & 6 & 0.08 & 0.10 & 0.04 & +0.11 & 19 & 0.06 &
0.10 & 0.02 & +0.06 & 6 & 0.04 & 0.08 & 0.03\\
160\,691 & +0.25 & 4 & 0.06 & 0.12 & 0.04 & +0.28 & 19 & 0.04 &
0.10 & 0.02 & +0.32 & 6 & 0.05 & 0.05 & 0.01\\
196\,378 & -- & 0 & -- & -- & 0.04 & $-$0.38 & 14 & 0.08 & 0.11 &
0.02 & $-$0.44 & 3 & 0.01 & 0.08 & 0.03\\
199\,288 & $-$0.33 & 3 & 0.02 & 0.13 & 0.04 & $-$0.59 & 14 & 0.06
& 0.11 & 0.02 & $-$0.77 & 3 & 0.02 & 0.08 & 0.03\\
203\,608 & -- & 0 & -- & -- & 0.05 & $-$0.69 & 9 & 0.09 & 0.11 &
0.01 & $-$0.70 & 3 & 0.06 & 0.08 & 0.03\\
\hline
\end{tabular}
\vfill \end{minipage}
\end{sidewaystable*}
\begin{sidewaystable*}\begin{minipage}[t][180mm]{\textwidth}
\caption[]{The same as Table~\ref{tab:abundances_1}, but for Co,
Ni, Ce, Nd, and Sm.} \label{tab:abundances_2}
\begin{tabular} { l @{\hspace{1em}} c @{\hspace{1em}} c @{\hspace{1em}} c @{\hspace{1em}} c @{\hspace{1em}} c @{\hspace{1em}} c
@{\hspace{1em}} r @{\hspace{1em}} c @{\hspace{1em}} c
@{\hspace{1em}} c @{\hspace{1em}} c @{\hspace{1em}} c
@{\hspace{1em}} c @{\hspace{1em}} c @{\hspace{1em}} c }
\hline\hline HD & [Co/H] & N & $\sigma_{\mathrm{[Co/H]}}$ &
uncert.$_{\mathrm{[Co/H]}}$ & uncert.$_{\mathrm{[Co/Fe]}}$ &
[Ni/H] & N & $\sigma_{\mathrm{[Ni/H]}}$ &
uncert.$_{\mathrm{[Ni/H]}}$ & uncert.$_{\mathrm{[Ni/Fe]}}$ &
[Ce/H] & N & $\sigma_{\mathrm{[Ce/H]}}$ &
uncert.$_{\mathrm{[Ce/H]}}$ & uncert.$_{\mathrm{[Ce/Fe]}}$\\
\hline 2151 & +0.04 & 7 & 0.03 & 0.09 & 0.03 & $-$0.04 & 11 & 0.05
& 0.11 & 0.02 & $-$0.12 & 5 & 0.10 & 0.10 & 0.04\\
9562 & +0.18 & 7 & 0.07 & 0.09 & 0.04 & +0.20 & 11 & 0.09 & 0.10 &
0.02 & $-$0.04 & 5 & 0.05 & 0.10 & 0.04\\
16\,417 & +0.16 & 8 & 0.04 & 0.09 & 0.04 & +0.14 & 11 & 0.03 &
0.10 & 0.02 & +0.03 & 5 & 0.06 & 0.10 & 0.04\\
20\,766 & $-$0.22 & 8 & 0.06 & 0.10 & 0.04 & $-$0.22 & 11 & 0.07 &
0.11 & 0.03 & $-$0.22 & 4 & 0.06 & 0.11 & 0.04\\
20\,807 & $-$0.20 & 8 & 0.06 & 0.09 & 0.04 & $-$0.23 & 10 & 0.07 &
0.11 & 0.03 & $-$0.31 & 4 & 0.07 & 0.10 & 0.04\\
22\,484 & $-$0.01 & 8 & 0.08 & 0.05 & 0.02 & $-$0.07 & 9 & 0.05 &
0.10 & 0.01 & $-$0.14 & 5 & 0.07 & 0.07 & 0.0\\
22\,879 & $-$0.55 & 6 & 0.09 & 0.12 & 0.04 & $-$0.79 & 9 & 0.14 &
0.13 & 0.04 & $-$0.72 & 5 & 0.05 & 0.13 & 0.05\\
30\,562 & +0.24 & 8 & 0.06 & 0.09 & 0.03 & +0.23 & 11 & 0.07 &
0.10 & 0.02 & +0.14 & 5 & 0.05 & 0.10 & 0.04\\
43\,947 & $-$0.29 & 8 & 0.06 & 0.09 & 0.04 & $-$0.31 & 12 & 0.10 &
0.11 & 0.03 & $-$0.29 & 5 & 0.07 & 0.11 & 0.04\\
52\,298 & $-$0.23 & 5 & 0.08 & 0.09 & 0.03 & $-$0.38 & 5 & 0.05 &
0.11 & 0.03 & $-$0.27 & 4 & 0.09 & 0.10 & 0.04\\
59\,984 & $-$0.54 & 6 & 0.10 & 0.13 & 0.05 & $-$0.71 & 6 & 0.10 &
0.14 & 0.04 &
$-$0.81 & 5 & 0.04 & 0.14 & 0.05\\
63\,077 & $-$0.53 & 6 & 0.07 & 0.12 & 0.04 & $-$0.62 & 7 & 0.09 &
0.12 & 0.05 & $-$0.80 & 5 & 0.03 & 0.13 & 0.05\\
76\,932 & $-$0.61 & 4 & 0.02 & 0.11 & 0.04 & $-$0.80 & 7 & 0.09 &
0.12 & 0.04 & $-$0.81 & 5 & 0.08 & 0.13 & 0.05\\
102\,365 & $-$0.23 & 8 & 0.05 & 0.10 & 0.04 & $-$0.29 & 10 & 0.05
& 0.11 & 0.03 & $-$0.38 & 4 & 0.08 & 0.11 & 0.04\\
128\,620 & +0.30 & 8 & 0.06 & 0.09 & 0.04 & +0.28 & 9 & 0.04 &
0.10 & 0.02 & +0.15 & 5 & 0.08 & 0.10 & 0.04\\
131\,117 & +0.05 & 8 & 0.04 & 0.08 & 0.03 & +0.12 & 12 & 0.04 &
0.11 & 0.02 & $-$0.03 & 5 & 0.05 & 0.09 & 0.04\\
160\,691 & +0.29 & 8 & 0.04 & 0.09 & 0.04 & +0.34 & 11 & 0.04 &
0.09 & 0.02 & +0.18 & 5 & 0.06 & 0.09 & 0.04\\
196\,378 & $-$0.31 & 2 & 0.17 & 0.09 & 0.04 & $-$0.40 & 8 & 0.11 &
0.12 & 0.03 & $-$0.41 & 3 & 0.03 & 0.11 & 0.04\\
199\,288 & $-$0.39 & 4 & 0.05 & 0.11 & 0.04 & $-$0.63 & 6 & 0.05 &
0.12 & 0.04 & $-$0.66 & 3 & 0.07 & 0.12 & 0.05\\
203\,608 & $-$0.18 & 3 & 0.09 & 0.11 & 0.04 & $-$0.74 & 5 & 0.09 &
0.12 & 0.03 & $-$0.73 & 4 & 0.09 & 0.12 & 0.05\\
\hline
\end{tabular}
\vspace{0.3cm}
\begin{tabular}{ l @{\hspace{1em}}
c @{\hspace{1em}} c @{\hspace{1em}} c @{\hspace{1em}} c
@{\hspace{1em}} c @{\hspace{1em}} c @{\hspace{1em}} c
@{\hspace{1em}} c } \hline \hline HD & [Nd/H] & N &
uncert.$_{\mathrm{[Nd/H]}}$ & uncert.$_{\mathrm{[Nd/Fe]}}$ &
[Sm/H] & N &
uncert.$_{\mathrm{[Sm/H]}}$ & uncert.$_{\mathrm{[Sm/Fe]}}$\\
\hline 2151 & $-$0.03 & 2 & 0.11 & 0.04 & $-$0.24 & 1 & 0.11
& 0.04\\
9562 & +0.06 & 2 & 0.12 & 0.04 & +0.06 & 1 & 0.11 & 0.04\\
16\,417 & +0.06 & 2 & 0.12 & 0.04 & +0.06 & 1 & 0.11 & 0.04\\
20\,766 & $-$0.10 & 2 & 0.12 & 0.05 & $-$0.11 & 1 & 0.12 & 0.04\\
20\,807 & $-$0.12 & 2 & 0.12 & 0.05 & $-$0.19 & 1 & 0.11 & 0.04\\
22\,484 & +0.04 & 2 & 0.08 & 0.04 & $-$0.13 & 1 & 0.07 & 0.04\\
22\,879 & $-$0.29 & 2 & 0.14 & 0.06 & $-$0.49 & 1 & 0.13 & 0.05\\
30\,562 & +0.16 & 2 & 0.11 & 0.04 & +0.14 & 1 & 0.11 & 0.04\\
43\,947 & $-$0.17 & 2 & 0.12 & 0.05 & $-$0.28 & 1 & 0.11 & 0.04\\
52\,298 & $-$0.10 & 2 & 0.12 & 0.05 & $-$0.29 & 1 & 0.11 & 0.04\\
59\,984 & $-$0.47 & 2 & 0.15 & 0.06 & $-$0.74 & 1 & 0.14 & 0.05\\
63\,077 & $-$0.40 & 2 & 0.14 & 0.06 & $-$0.85 & 1 & 0.13 & 0.05\\
76\,932 & $-$0.44 & 2 & 0.14 & 0.06 & $-$0.93 & 1 & 0.13 & 0.05\\
102\,365 & $-$0.13 & 2 & 0.13 & 0.05 & $-$0.23 & 1 & 0.12 & 0.05\\
128\,620 & +0.28 & 2 & 0.12 & 0.04 & +0.26 & 1 & 0.11 & 0.04\\
131\,117 & +0.10 & 2 & 0.11 & 0.04 & $-$0.11 & 1 & 0.10 & 0.04\\
160\,691 & +0.14 & 2 & 0.12 & 0.03 & +0.28 & 1 & 0.12 & 0.04\\
196\,378 & -- & 0 & -- & -- & $-$0.24 & 1 & 0.11 & 0.05\\
199\,288 & $-$0.32 & 1 & 0.13 & 0.05 & $-$0.39 & 1 & 0.13 & 0.05\\
203\,608 & -- & 0 & -- & -- & $-$0.62 & 1 & 0.12 & 0.05\\
\hline
\end{tabular}
\vfill\end{minipage}
\end{sidewaystable*}
\subsection{Uncertainty assessment}
\label{sec:errors_assessment}
Chemical abundances determined using EWs are subject to three
sources of uncertainty: atmospheric parameters, EWs, and model
atmospheres. If these sources were independent from one another,
the total uncertainty could be estimated by their RMS. However,
there is no such independency, as can be clearly perceived when
the iterative character of the analysis is called to mind. In this
circumstance, RMS yields an estimate of the \emph{maximum}
uncertainty. Given the extreme complexity of the interdependency
between the sources of uncertainty, we did not attempt to
disentangle them, and used the RMS instead.
We chose four stars as standards for the assessment, with extreme
effective temperatures and metallicities: \object{HD~160\,691}
(cool and metal-rich), \object{HD~22\,484} (hot and metal-rich),
\object{HD~63\,077} (cool and metal-poor), and \object{HD~59\,984}
(hot and metal-poor). \object{HD~203\,608} is as metal-poor as,
but hotter than \object{HD~59\,984}, and would apparently be
better suited as a standard star; yet we did not use it, because
it lacks measured V and Nd lines, and thus could not be used to
estimate the uncertainties for these elements
(Fig.~\ref{fig:error_stars}).
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics*{FIGURE10.EPS}}
\caption{Stars used as uncertainty standards are labelled.
\object{HD~203\,608} was not chosen when assessing the
uncertainties of contaminating elements, because it lacks V and Nd
abundance determinations; however, it was chosen as the hot,
metal-poor standard when assessing Eu and Th uncertainties. The
effective temperatures and metallicities are the adopted ones.
Dotted lines are an example of the weights used for uncertainty
determination in stars not chosen as standards -- see text for
details.} \label{fig:error_stars}
\end{figure}
Initially, we determined the iron abundance uncertainty. For this
purpose, we evaluated the influence of all uncertainty sources,
independently. The influence of effective temperature, surface
gravity and microturbulence velocity was obtained by recalculating
the Fe abundances using the adopted atmospheric parameters
modified, one at a time, by their own uncertainties: +27~K,
+0.02~dex, and +0.05~km~s$^{-1}$ (metal-rich stars) and
+0.23~km~s$^{-1}$ (metal-poor stars), respectively. The difference
between the adopted Fe abundance and the one obtained with the
modified parameter is our estimate of the influence of the
parameter on the total uncertainty.
The influence of EWs was determined by recalculating the
abundances using EWs modified by percentages obtained when
comparing our measurements with those from \citet{meylanetal93}
(Fig.~\ref{fig:EW_atlas}). This comparison allows us to estimate
the influence of the EW measurement per se, as well as the
influence of continuum placement. The percent difference between
the two sets of measurements presents a 15\% dispersion for lines
with $\mbox{EW}<20$~m\AA, and 5\% for those with
$\mbox{EW}>20$~m\AA. The Fe EWs were, then, modified by these
amounts. However, this was only done for the two metal-rich
standard stars. For the metal-poor stars, the influence of the
continuum placement is stronger than that estimated by comparing
solar EWs, because EWs are considerably lower. On the other hand,
it is easier to find good continuum windows, because they are
wider and more numerous. So, we compared the EWs of
\object{HD~22\,879} and \object{HD~76\,932}, which are metal-poor
stars with very similar atmospheric parameters (specially their
effective temperatures, which are virtually identical). The high
similarity of atmospheric parameters mean that differences in EWs
are caused almost exclusively by the uncertainties in the EW
measurements and continuum placement. Only \ion{Fe}{i} lines were
compared, in order to minimize the effect of the stars having
slightly different surface gravities. The EW percent difference
between the two stars present 19\% dispersion for lines with
$\mbox{EW}<15$~m\AA, and 5\% for those with $\mbox{EW}>15$~m\AA\
(Fig.~\ref{fig:EW_poor_stars}); the Fe EWs of the metal-poor
standard stars were modified by these values.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics*{FIGURE11.EPS}}
\caption{Percent differences between the EWs of
\object{HD~22\,879} and \object{HD~76\,932}. Solid lines
correspond to mean differences, and the dotted ones represent the
mean differences modified by their standard deviations
($\pm1\,\sigma$).} \label{fig:EW_poor_stars}
\end{figure}
$\mbox{Log }gf$ are also a source of uncertainty. But, since they
are derived from solar EWs, their contribution is the same as that
from the EWs themselves. That is to say, in practice we take into
account the influence of EWs twice, so as to account for the solar
EWs per se, and for the $\log gf$ values.
The influence of metallicity in the Fe abundance uncertainty is
the uncertainty itself. This means that the Fe abundance
uncertainty determination is, rigorously speaking, an iterative
process. As a first step, Fe abundances were modified by the
average dispersion \emph{between lines} of all stars (0.07~dex).
As can be seen in Table~\ref{tab:errors}, the influence of
metallicity is very weak. The contribution of this source would
have to be 0.04~dex higher for the total uncertainty to be raised
just 0.01~dex. As a second step, we should have modified the Fe
abundances by the total Fe uncertainty just obtained, on the first
step, by the RMS of all sources (including the metallicity
itself). But, given the low contribution of this source, we
stopped at the first step.
The total Fe uncertainties for the standard stars were obtained by
the RMS of the six contributions: effective temperatures, surface
gravities, microturbulence velocities, EWs, $\log gf$, and
metallicities. Total uncertainties for the other stars were
obtained by weighted average of the standard stars values, using
as weight the reciprocal of the distance of the star to each
standard star, in the $T_{\mathrm{eff}}$ vs. [Fe/H] plane. In
Fig.~\ref{fig:error_stars} we see an example, in which the star at
the center uses the reciprocal of the lengths of the dotted lines
as weights.
Uncertainties for the abundances of the other elements were
determined the same way as for Fe. Although we determined
individual [Fe/H] uncertainties for each star, we used a sole
value of 0.10~dex when estimating the metallicity contribution to
the uncertainties of the other elements. This can be justified by
the low dispersion between [Fe/H] uncertainties, and by the very
low sensitivity of the abundances to the metallicity of the model
atmosphere. Table~\ref{tab:errors} presents contributions from the
individual sources, total uncertainties, and some averages for the
standard stars, relative to H. Values relative to Fe can be
obtained simply by subtracting the [Fe/H] value from the
[element/H] value (e.g.,
$\mbox{uncert.}_{\mathrm{[V/Fe]}}=\mbox{uncert.}_{\mathrm{[V/H]}}
-\mbox{uncert.}_{\mathrm{[Fe/H]}}$).
\begin{sidewaystable*}\begin{minipage}[t][180mm]{\textwidth}
\caption[]{[element/H] abundance uncertainties for the standard
stars. Last column contains the average uncertainty for each row,
and the last row contains\\ the average total uncertainty for the
four standard stars.} \label{tab:errors}
\begin{tabular} { c r @{\space} r c r @{.} l r @{.} l r @{.} l r @{.} l r @{.} l
r @{.} l r @{.} l r @{.} l r @{.} l r @{.} l r @{\space} l }
\hline\hline Parameter & \multicolumn{2}{c}{$\Delta$Parameter} &
HD & \multicolumn{2}{c}{[Fe/H]} & \multicolumn{2}{c}{[Ti/H]} &
\multicolumn{2}{c}{[V/H]} & \multicolumn{2}{c}{[Cr/H]} &
\multicolumn{2}{c}{[Mn/H]} & \multicolumn{2}{c}{[Co/H]} &
\multicolumn{2}{c}{[Ni/H]} & \multicolumn{2}{c}{[Ce/H]} &
\multicolumn{2}{c}{[Nd/H]} & \multicolumn{2}{c}{[Sm/H]} &
\multicolumn{2}{c}{Average}\\
\hline & & & 160\,691 & +0 & 01 & +0 & 02 & +0 & 03 & +0 & 02 & +0
& 02 & +0 & 02 & +0 & 02 &
+0 & 00 & +0 & 01 & +0 & 01 & +0.02 & $\pm$0.01\\
& & & 22\,484 & +0 & 02 & +0 & 02 & +0 & 03 & +0 & 02 & +0 & 03 &
+0 & 03 & +0 & 02 &
+0 & 01 & +0 & 01 & +0 & 01 & +0.02 & $\pm$0.01\\
\raisebox{1.5ex}[0pt]{$T_{\mathrm{eff}}$} &
\multicolumn{2}{c}{\raisebox{1.5ex}[0pt]{+27~K}} & 63\,077 & +0 &
02 & +0 & 02 & +0 & 03 & +0 & 01 & +0 & 02 & +0 & 02 & +0 & 02 &
+0 & 01 & +0 & 01 & +0 & 01 & +0.02 & $\pm$0.01\\
& & & 59\,984 & +0 & 02 & +0 & 02 & +0 & 03 & +0 & 02 & +0 & 03 &
+0 & 02 & +0 & 02 &
+0 & 01 & +0 & 02 & +0 & 01 & +0.02 & $\pm$0.01\\
\hline & & & 160\,691 & +0 & 00 & +0 & 00 & +0 & 00 & +0 & 00 & +0
& 00 & +0 & 00 & +0 & 00 &
+0 & 01 & +0 & 01 & +0 & 01 & +0.00 & $\pm$0.00\\
& & & 22\,484 & +0 & 00 & +0 & 00 & +0 & 01 & +0 & 00 & +0 & 00 &
+0 & 01 & +0 & 00 &
+0 & 01 & +0 & 01 & +0 & 01 & +0.01 & $\pm$0.01\\
\raisebox{1.5ex}[0pt]{$\log g$} &
\multicolumn{2}{c}{\raisebox{1.5ex}[0pt]{+0.02~dex}} & 63\,077 &
+0 & 00 & +0 & 00 & +0 & 00 & +0 & 01 & +0 & 00 & +0 & 00 & +0 &
00 &
+0 & 01 & +0 & 01 & +0 & 01 & +0.00 & $\pm$0.01\\
& & & 59\,984 & +0 & 00 & +0 & 01 & +0 & 00 & +0 & 00 & +0 & 00 &
+0 & 02 & +0 & 00 &
+0 & 01 & +0 & 01 & +0 & 01 & +0.01 & $\pm$0.01\\
\hline & & & 160\,691 & $-$0 & 01 & $-$0 & 02 & +0 & 00 & $-$0 &
01 & $-$0 & 01 & +0 & 00
& $-$0 & 02 & $-$0 & 01 & $-$0 & 01 & $-$0 & 01 & $-$0.01 & $\pm$0.01\\
& \multicolumn{2}{c}{\raisebox{1.5ex}[0pt]{+0.05 km s$^{-1}$}} &
22\,484 & $-$0 & 01 & $-$0 & 01 & $-$0 & 01 & $-$0 & 02 & +0 & 00
& $-$0 & 01
& $-$0 & 01 & $-$0 & 01 & $-$0 & 01 & $-$0 & 01 & $-$0.01 & $\pm$0.00\\
\cline{2-3} \raisebox{1.5ex}[0pt]{$\xi$} & & & 63\,077 & $-$0 & 03
& $-$0 & 03 & $-$0 & 01 & $-$0 & 04 & +0 & 00 & +0 & 00 &
+0 & 00 & $-$0 & 02 & $-$0 & 01 & $-$0 & 01 & $-$0.01 & $\pm$0.01\\
& \multicolumn{2}{c}{\raisebox{1.5ex}[0pt]{+0.23 km s$^{-1}$}} &
59\,984 & $-$0 & 03 & $-$0 & 03 & +0 & 00 & $-$0 & 04 & +0 & 00 &
+0 & 00
& $-$0 & 01 & $-$0 & 01 & $-$0 & 01 & $-$0 & 01 & $-$0.01 & $\pm$0.01\\
\hline & $\mbox{EW}<20\mbox{~m\AA}$: & +15\% & 160\,691 & +0 & 06
& +0 & 07 & +0 & 04 & +0 & 07 & +0 & 06 & +0 & 03 & +0 & 07 &
+0 & 04 & +0 & 05 & +0 & 04 & +0.05 & $\pm$0.01\\
\raisebox{1ex}[0pt]{EW} & $\mbox{EW}>20\mbox{~m\AA}$: & +5\% &
22\,484 & +0 & 06 & +0 & 06 & +0 & 08 & +0 & 07 & +0 & 03 & +0 &
06 & +0 & 06 &
+0 & 06 & +0 & 08 & +0 & 08 & +0.06 & $\pm$0.02\\
\cline{2-3} \raisebox{1.5ex}[0pt]{and} &
$\mbox{EW}<15\mbox{~m\AA}$: & +19\% & 63\,077 & +0 & 07 & +0 & 05
& +0 & 09 & +0 & 08 & +0 & 06 & +0 & 09 & +0 & 10 &
+0 & 10 & +0 & 10 & +0 & 10 & +0.08 & $\pm$0.02\\
\raisebox{2ex}[0pt]{$\log gf$} & $\mbox{EW}>15\mbox{~m\AA}$: &
+5\% & 59\,984 & +0 & 06 & +0 & 06 & +0 & 10 & +0 & 06 & +0 & 05 &
+0 & 08 & +0 & 08 &
+0 & 09 & +0 & 10 & +0 & 09 & +0.08 & $\pm$0.02\\
\hline & & & 160\,691 & +0 & 01 & +0 & 01 & +0 & 01 & +0 & 01 & +0
& 00 & +0 & 01 & +0 & 01 &
+0 & 03 & +0 & 04 & +0 & 04 & +0.02 & $\pm$0.01\\
& \multicolumn{2}{c}{\raisebox{1.5ex}[0pt]{Fe: +0.07~dex}} &
22\,484 & +0 & 00 & +0 & 01 & +0 & 00 & +0 & 01 & +0 & 00 & +0 &
01
& +0 & 01 &
+0 & 03 & +0 & 03 & +0 & 03 & +0.01 & $\pm$0.01\\
\raisebox{1.5ex}[0pt]{[Fe/H]} &
\multicolumn{2}{c}{\raisebox{0.5ex}[0pt]{Other elements:}}
& 63\,077 & +0
& 00 & +0 & 01 & +0 & 01 & +0 & 01 & +0 & 00 & +0 & 00 & +0 & 00 &
+0 & 02 & +0 & 03 & +0 & 01 & +0.01 & $\pm$0.01\\
& \multicolumn{2}{c}{\raisebox{0.5ex}[0pt]{+0.10~dex}} & 59\,984 &
+0 & 00 & +0 & 01 & +0 & 00 & +0 & 01 & +0 & 01 & +0 & 01 & +0 &
01 &
+0 & 02 & +0 & 02 & +0 & 01 & +0.01 & $\pm$0.01\\
\hline \hline & & & 160\,691 & 0 & 09 & 0 & 09 & 0 & 12 & 0 & 10 &
0 & 05 & 0 & 09 & 0 & 09 & 0 & 09 &
0 & 12 & 0 & 12 & 0.08 & $\pm$0.02\\
\raisebox{-0.3ex}[0pt]{Total} & & & 22\,484 & 0 & 09 & 0 & 10 & 0
& 06 & 0 & 10 & 0 & 09 & 0 & 05 & 0 & 10 & 0 & 07 &
0 & 08 & 0 & 07 & 0.10 & $\pm$0.02\\
\raisebox{0.3ex}[0pt]{uncertainty} & & & 63\,077 & 0 & 09 & 0 & 09
& 0 & 14 & 0 & 10 & 0 & 08 & 0 & 12 & 0 & 12 & 0 & 13 &
0 & 14 & 0 & 13 & 0.12 & $\pm$0.02\\
& & & 59\,984 & 0 & 11 & 0 & 08 & 0 & 13 & 0 & 12 & 0 & 09 & 0 &
13 & 0 & 14 & 0 & 14 &
0 & 15 & 0 & 14 & 0.11 & $\pm$0.02\\
\hline \hline \raisebox{-0.3ex}[0pt]{Average} & & & & 0 & 10 & 0 &
09 & 0 & 11 & 0 & 11 & 0 & 08
& 0 & 10 & 0 & 11 & 0 & 11 & 0 & 12 &
0 & 12 & & \\
\raisebox{0.3ex}[0pt]{total} & & & & $\pm$0 & 01 & $\pm$0 & 01 &
$\pm$0 & 04 & $\pm$0 & 01 & $\pm$0 & 02 &
$\pm$0 & 04 & $\pm$0 & 02 & $\pm$0 & 03 & $\pm$0 & 03 & $\pm$0 & 03 &
\raisebox{1.5ex}[0pt]{0.10} & \raisebox{1.5ex}[0pt]{$\pm$0.03}\\
\hline
\end{tabular}
\vspace{0.1cm}
Note: Contributions from EW and $\log gf$ are equal. The values
which are presented correspond to \emph{each} contribution, and
not to the total of these two sources.
Therefore, they are taken in consideration \emph{twice} when
calculating the total uncertainties. \vfill\end{minipage}
\end{sidewaystable*}
\subsection{Adopted abundances}
Figure~\ref{fig:abundances} presents the abundance patterns for
all contaminating elements. Error bars were taken as the
dispersion between the lines or the estimated [element/Fe]
uncertainty, whichever is the largest (see
Tables~\ref{tab:abundances_1} and \ref{tab:abundances_2}). The
presence of a few ($\sim4\%$) outliers can be noticed. For these
stars, these abundances will not be used for the Eu and Th
spectral syntheses. In these cases, the discrepant abundances will
be substituted by the value they would have if they agreed with an
exponential fit to the well behaved data points. One might wonder
if the discrepant abundances would be a hint of chemical
peculiarity. If this was true, one such star would exhibit
peculiar abundances for all elements produced by one specific
nucleosynthesis process, and this behaviour is not observed. Mark
that the three stars which present discrepant abundances
(\object{HD~63\,077}, \object{HD~76\,932}, and
\object{HD~203\,608}) are all metal-poor, with
$\mbox{[Fe/H]}<-0.67$, and therefore more sensitive to noise and
continuum placement errors. The Nd and V abundances of
\object{HD~196\,378} and \object{HD~203\,608}, which have no
measured lines, were established this same way.
\begin{figure*}
\centering
\includegraphics*[width=17cm]{FIGURE12.EPS}
\caption{Abundance patterns for all contaminating elements.
Outliers are indicated by arrows labelled with their respective HD
numbers. Solid lines, when present, represent exponential fits to
the well behaved data points.} \label{fig:abundances}
\end{figure*}
\section{Eu and Th abundance determination}
The Eu and Th abundance determination was carried out, for our
sample stars, using a single absorption line for each element. The
only acceptably strong Th line, which allows an accurate abundance
determination, is the \ion{Th}{ii} line located at 4019.13~\AA.
This line is, nevertheless, blended with many others, rendering
imperative the use of spectral synthesis. One other line is often
cited in the literature: a \ion{Th}{ii} line located at
4086.52~\AA. Unfortunately, although less contaminated than the
line at 4019.13~\AA, it is too weak to be used.
Eu has only one adequately strong and uncontaminated line, located
at 4129.72~\AA, available for abundance determinations. However,
this line has a significantly non-Gaussian profile by virtue of
its HFS and isotope shift. Therefore, spectral synthesis is also
required for this element. In stars of other spectral types and
other luminosity classes (e.g. giants), it is possible to use
other Th and Eu lines. This is, nonetheless, not true for our
sample stars (late-F and G dwarfs/subgiants).
The code used to calculate the synthetic spectra was kindly made
available to us by its developer, Monique Spite. This code
calculates a synthetic spectrum based on a list of lines with
their respective atomic parameters ($\log gf$, and excitation
potential of the lower level of the transition $\chi$, in eV). We
employed the same model atmospheres and partition functions used
in the determination of atmospheric parameters and abundances of
contaminating elements (Section~\ref{sec:model_atm_part_funct}).
The only exception was for Th, whose partition functions were
calculated by us, based on data from H. Holweger that was
published as a \emph{private communication} in MKB92. Irwin's data
are too inaccurate, because for the singly and doubly ionised
states of Th, he scales the partition functions along
iso-electronic sequences from a lower mass element, due to lack of
specific Th data. This resulted in values much lower than those of
Holweger, which were calculated by summing a large number of Th
atomic levels.
\subsection{Europium}
\subsubsection{Spectral synthesis}
\label{sec:eu_abundance_determination}
The list of lines used for calculation of the synthetic spectrum
of the Eu region was based on the line lists of the Vienna Atomic
Line Database -- VALD \citep{kupkaetal99}. Initially, we took all
lines found between 4128.4~\AA\ and 4130.4~\AA. The first cut
eliminated all lines of elements ionised two or more times, and
all lines with lower-level excitation potential greater than
10~eV, since these lines are unobservable in the photospheres of
stars with effective temperatures as low as those in our sample.
As a next step, we calculated synthetic spectra for the four stars
with the most extreme effective temperatures and metallicities in
our sample (\object{HD~22\,484}, \object{HD~59\,984},
\object{HD~63\,077}, and \object{HD~203\,608}) using laboratory
$\log gf$ values, and the atmospheric parameters and abundances
previously obtained. Lines with EWs lower than 0.01~m\AA\ in these
four stars were removed. The \ion{Sc}{i} line located at
4129.750~\AA, cited as important by \citet{mashonkina&gehren00},
presents negligible EW in all four standard stars. 23 Ti, V, Cr,
Fe, Co, Nb, Ce, Pr, Nd, Sm, and Dy lines were kept, besides the Eu
line itself. Adopted wavelengths were taken from the VALD list
\citep{bard&kock94,kurucz93,kurucz94,whaling83,wickliffeetal94}.
To improve the fits further, three artificial \ion{Fe}{i} lines
were added as substitutes for unknown blends
\citep{lawleretal01,mashonkina&gehren00}; the influence of these
artificial lines in the obtained Eu abundances is nevertheless
small, as they are located relatively far from the center of the
\ion{Eu}{ii} line. The final list is presented in
Table~\ref{tab:eu_line_list}.
In the spectral synthesis calculations, the \ion{Eu}{ii} line was
substituted by its HFS components, calculated by us using
Casimir's equation \citep{casimir_63}, and HFS laboratory data
from \citet{beckeretal93}, \citet{brostrometal95},
\citet{villemoes&wang94}, and \citet{molleretal93}. Isotope shift
was taken into account, using data from \citet{brostrometal95} and
the solar abundance isotopic ratio
$\varepsilon(\mbox{\element[][151]{Eu}})/\varepsilon(\mbox{\element[][153]{Eu}})=1.00\pm0.29$
\citep{lawleretal01}. The solar value was used for all stars
because there are strong indications that it is not
metallicity-dependent, since several works dealing with r-process
nucleosynthesis in very-metal-poor halo stars find similar results
\citep{pfeifferetal97,cowanetal99,snedenetal02,aokietal03}.
\begin{table*}
\caption[]{Line list used in the spectral synthesis of the
\ion{Eu}{ii} line at 4129.72~\AA.} \label{tab:eu_line_list}
\begin{tabular}{ c c c c c c c }
\hline \hline $\lambda$ (\AA) & Element & $\chi$ (eV) & $\log
gf_{\mathrm{CES}}$
& $\log gf_{\mathrm{FEROS}}$ & $\log gf$ source & Obs.\\
\hline
4128.742 & \ion{Fe}{ii} & 2.580 & $-$3.832 & $-$3.554 & solar fit & \\
4129.000 & \ion{Cr}{i} & 4.210 & $-$2.603 & $-$2.603 & VALD & \\
4129.040 & \ion{Ru}{i} & 1.730 & $-$1.030 & $-$1.030 & VALD & \\
4129.147 & \ion{Pr}{ii} & 1.040 & $-$0.100 & $-$0.100 & VALD & \\
4129.159 & \ion{Ti}{ii} & 1.890 & $-$2.330 & $-$2.210 & solar fit & \\
4129.166 & \ion{Ti}{i} & 2.320 & +0.131 & +0.251 & solar fit & \\
4129.165 & \ion{Cr}{i} & 3.010 & $-$1.948 & $-$1.948 & VALD & \\
4129.174 & \ion{Ce}{ii} & 0.740 & $-$0.901 & $-$0.901 & VALD & \\
4129.196 & \ion{Cr}{i} & 2.910 & $-$1.374 & $-$1.254 & solar fit & \\
4129.220 & \ion{Fe}{i} & 3.420 & $-$2.280 & $-$2.160 & solar fit & \\
4129.220 & \ion{Sm}{ii} & 0.250 & $-$1.123 & $-$1.123 & VALD & \\
4129.425 & \ion{Dy}{ii} & 0.540 & $-$0.522 & $-$0.522 & VALD & \\
4129.426 & \ion{Nb}{i} & 0.090 & $-$0.780 & $-$0.780 & VALD & \\
4129.461 & \ion{Fe}{i} & 3.400 & $-$2.180 & $-$1.920 & solar fit & \\
4129.530 & \ion{Fe}{i} & 3.140 & $-$3.425 & $-$3.455 & solar fit & artificial\\
4129.610 & \ion{Fe}{i} & 3.500 & $-$3.700 & $-$3.730 & solar fit & artificial\\
4129.643 & \ion{Ti}{i} & 2.240 & $-$1.987 & $-$1.987 & VALD & \\
4129.657 & \ion{Ti}{i} & 2.780 & $-$2.297 & $-$2.297 & VALD & \\
4129.721 & \ion{Eu}{ii} & 0.000 & +0.173 & +0.173 & \citet{komarovskii91} & HFS\\
4129.817 & \ion{Co}{i} & 3.810 & $-$1.808 & $-$1.808 & VALD & \\
4129.837 & \ion{Nd}{ii} & 2.020 & $-$0.543 & $-$0.543 & VALD & \\
4129.965 & \ion{Fe}{i} & 2.670 & $-$3.390 & $-$3.290 & solar fit & artificial\\
4129.994 & \ion{V}{i} & 2.260 & $-$1.769 & $-$1.769 & VALD & \\
4130.037 & \ion{Fe}{i} & 1.560 & $-$4.195 & $-$4.030 & solar fit & \\
4130.038 & \ion{Fe}{i} & 3.110 & $-$2.385 & $-$2.225 & solar fit & \\
4130.068 & \ion{Fe}{i} & 3.700 & $-$3.763 & $-$3.763 & VALD & \\
4130.073 & \ion{Cr}{i} & 2.910 & $-$1.971 & $-$1.971 & VALD & \\
\hline
\end{tabular}
References: see text.
\end{table*}
We adopted a projected rotation velocity for the integrated solar
disk $v\,\sin i=1.8~\mbox{km s}^{-1}$
\citep{mashonkina00,mashonkina&gehren00}. The width of the
Gaussian profile convolved to the synthetic spectrum to take
macroturbulence and instrumental broadenings into account was
obtained by fitting the \ion{Fe}{ii} located at 4128.742~\AA,
which is sufficiently isolated for this purpose
\citep{woolfetal95,koch&edvardsson02}. Solar $\log gf$ were
obtained by keeping abundances fixed at their solar values
\citep{grevesse&sauval98}, and fitting the observed solar
spectrum. Only the stronger Ti, Cr, and Fe lines had their $\log
gf$ adjusted. For the \ion{Eu}{ii} line, we adopted a fixed $\log
gf$ from \citet{komarovskii91}; this value was distributed among
the HFS components according to relative intensities calculated by
White \& Eliason, tabulated in \citet{condon&shortley67}. The
other, weaker lines had their $\log gf$ kept at their laboratory
values adopted from the VALD list
\citep{kurucz93,kurucz94,wickliffeetal94}. As we kept the Eu $\log
gf$ fixed, we allowed its abundance to vary. The complete solar
spectrum fit procedure was carried out iteratively because it
requires alternate Gaussian profile adjustments (which modify the
\emph{shape} of the lines) and $\log gf$ adjustments (which modify
the \emph{EWs} of the lines). The solar $\log gf$ and Gaussian
profile determination was accomplished independently for the two
sources of Eu spectra used, leading to different Eu solar
abundances ($\log\varepsilon(\mbox{Eu})_{\sun}=+0.37$ and $+0.47$
for the CES and FEROS, respectively). The stellar [Eu/H] abundance
ratios, used in this work in lieu of the absolute abundances, were
obtained by subtracting the appropriate solar value from the
stellar absolute abundances
($\mbox{[Eu/H]}=\log\varepsilon(\mbox{Eu})_{\mathrm{star}}-
\log\varepsilon(\mbox{Eu})_{\sun}$). Figure~\ref{fig:eu_sun}
presents the observed and synthetic solar spectrum in the Eu
region. The strongest lines that compose the total synthetic
spectrum are presented independently.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics*{FIGURE13.EPS}}
\caption{Synthesized spectral region for the \ion{Eu}{ii} line at
4129.72~\AA, for the Sun. The two thick lines represent the total
synthetic spectrum and the \ion{Eu}{ii} line. The thin lines
represent the most dominant lines in the region, including two
artificial \ion{Fe}{i} lines. Points are the observed FEROS solar
spectrum.} \label{fig:eu_sun}
\end{figure}
For the Eu synthesis in the sample stars, we kept the abundances
of all elements other than Eu fixed at the values determined using
EWs, in Sect.~\ref{sec:chemical_abundances}. Small adjustments
were allowed only to the abundances of Ti, Cr, and Fe, in order to
improve the fit. The adjustments were kept within the
uncertainties of the abundances of these elements -- see
Sect.~\ref{sec:chemical_abundances}. The abundances of elements
for which we did not measure any EW were determined scaling the
solar abundances, following elements produced by the same
nucleosynthetic processes. We chose to scale the Sm abundances as
well, even though they were determined by us, because our results
for this element exhibit high scatter and absence of well-defined
behaviour (see Fig.~\ref{fig:abundances}). Sm and Dy abundances
were obtained following Eu (produced mainly by r-process,
according to \citealt{burrisetal00}); Nb, Rb, and Pr were obtained
following Nd (produced by both r-process and s-process, also
according to \citealt{burrisetal00}). Mark that the influence of
these lines is marginal, since they are weak and are not close to
the Eu line. Projected rotation velocities $v\,\sin i$ and
macroturbulence and instrumental broadening Gaussian profiles were
determined by fitting the \ion{Fe}{ii} line at 4128.742~\AA, as
done with the solar spectrum.
Stars observed more than once had their spectra analysed
independently, and the results were averaged. Abundances obtained
with the different spectra of one object presented a maximum
variation of 0.02~dex. Examples of spectral syntheses can be seen
in Fig.~\ref{fig:examples_eu} for the two stars with extreme Eu
abundances that were observed with both FEROS and CES
(\object{HD~63\,077} and \object{HD~160\,691}). Note the effect of
CES spectra higher sampling, which better puts the asymmetric
profile of the \ion{Eu}{ii} line in evidence.
\begin{figure*}
\centering
\includegraphics*[width=17cm]{FIGURE14.EPS}
\caption{Examples of spectral syntheses of the \ion{Eu}{ii} line
at 4129.72~\AA. FEROS and CES spectra are presented for two stars
with extreme Eu abundances (\object{HD~63\,077} and
\object{HD~160\,691}). Points are the observed spectra. Thick
lines are the best fitting synthetic spectra, calculated with the
shown Eu abundances. Thin lines represent variations in the Eu
abundance $\Delta\log\varepsilon(\mbox{Eu})=\pm0.05~\mbox{dex}$.
The shown abundances are the ones that best fit the presented
spectra, and not the average values that are presented in
Table~\ref{tab:th_eu_abundances}.} \label{fig:examples_eu}
\end{figure*}
Eu abundances obtained using CES and FEROS spectra are compared in
Fig.~\ref{fig:eu_ces_eu_feros}. A linear fit was calculated, and
the resulting relation was used to convert FEROS results into the
CES system. This way, we ended up with two sets of Eu abundances:
one totally homogeneous, obtained with FEROS spectra, and one
obtained partially with CES spectra and partially with
corrected-into-CES-system FEROS spectra. We plotted a [Eu/Fe] vs.
[Fe/H] diagram for both sets of abundances, presented in
Fig.~\ref{fig:eu_fe_fe_h_ces_feros}. Linear fits calculated for
each set show that the CES abundances present a lower
scatter~$\sigma$, and for this reason the CES results were adopted
for all subsequent analyses.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics*{FIGURE15.EPS}}
\caption{Comparison between Eu abundances determined using CES and
FEROS spectra. Solid line is the identity. Dotted line is a linear
fit, whose parameters are shown in the figure.}
\label{fig:eu_ces_eu_feros}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics*{FIGURE16.EPS}}
\caption{[Eu/Fe] vs. [Fe/H] diagram for all sample stars.
Abundances obtained with CES and FEROS spectra are presented. The
Sun is not included in the comparison, since its abundance is
always equal to zero by definition. Solid and dotted lines are
linear fits to these two data sets, whose scatters are shown in
the figure. Mark that data from the CES are slightly better than
those from FEROS.} \label{fig:eu_fe_fe_h_ces_feros}
\end{figure}
\subsubsection{Uncertainty assessment}
[Eu/H] abundance uncertainties were estimated with a procedure
identical to the one used for the abundances of contaminating
elements determined with EWs, thoroughly described in
Sect.~\ref{sec:errors_assessment}. It consists basically of
varying the atmospheric parameters and continuum position
independently, by an amount equal to their uncertainties, and
recalculating the synthetic spectra. Total uncertainties are
obtained by RMS of the individual sources of uncertainty. We chose
four stars as standards for the assessment, with extreme effective
temperatures and metallicities: \object{HD~160\,691} (cool and
metal-rich), \object{HD~22\,484} (hot and metal-rich),
\object{HD~63\,077} (cool and metal-poor), and
\object{HD~203\,608} (hot and metal-poor) --
Fig.~\ref{fig:error_stars}.
Continuum placement is the preponderant source of Eu abundance
uncertainty. The placement uncertainty itself is higher for
metal-rich stars, as these present stronger absorption lines,
lowering the apparent continuum which restricts the choice of good
normalisation windows. To estimate the influence of continuum
placement uncertainties, we multiplied the spectral flux of the
metal-rich standard stars by 0.98, and that of the metal-poor
standard stars by 0.99, and recalculated the synthetic spectra.
Table~\ref{tab:th_eu_uncertainties} presents the [Eu/H], [Eu/Fe],
[Th/H], [Th/Fe], and [Th/Eu] uncertainties for each standard star.
Metallicity and continuum placement clearly dominate.
Microturbulence velocity variations have no effect in the [Eu/H]
abundances. This happens because the HFS broadens the line so much
that it becomes nearly unsaturated, as found by \citet[\space
WTL95]{woolfetal95}, \citet{lawleretal01} and \citet[\space
KE02]{koch&edvardsson02}. [Element/Fe] uncertainties were obtained
simply by subtracting the [Fe/H] uncertainty from the [element/H]
value (e.g.,
$\mbox{uncert.}_{\mathrm{[Eu/Fe]}}=\mbox{uncert.}_{\mathrm{[Eu/H]}}-\mbox{uncert.}_{\mathrm{[Fe/H]}}$).
Total uncertainties for the other sample stars were obtained by
weighted average of the standard stars values, using as weight the
reciprocal of the distance of the star to each standard star, in
the $T_{\mathrm{eff}}$ vs. [Fe/H] plane; this procedure is
identical to the one used for the uncertainties of contaminating
elements (Sect.~\ref{sec:errors_assessment}).
Table~\ref{tab:th_eu_abundances} presents the final [Eu/H],
[Th/H], and [Th/Eu] abundance ratios, for all sample stars, along
with their respective uncertainties relative to H and Fe.
Non-LTE effects in the abundances obtained with the \ion{Eu}{ii}
line at 4129.72~\AA\ have been calculated by
\citet{mashonkina&gehren00}. In the line formation layers, the
ground state was found to be slightly underpopulated, and the
excited level overpopulated. As a consequence, our results would
show a small difference if our method employed absolute
abundances. But since we employ a differential analysis, these
differences cancel each other partially, becoming negligible
(WTL95 and KE02).
\subsubsection{Comparison with literature
results}
We conducted a painstaking search of the literature for works with
Eu abundances, and selected WTL95 and KE02 as the ones with the
most careful determinations of disk stars, as well as a sizable
sample of high statistical significance. Their sample is composed
of a subset from \citet{edvardssonetal93}, and they use the same
atmospheric parameters (obtained from Str\"omgren photometric
calibrations). The Eu abundances were determined by spectral
synthesis using the same line we used, with a procedure
fundamentally identical to ours. KE02 merged their database with
that from WTL95 by means of a simple linear conversion, obtained
by intercomparison. We used this merged set of abundances to
compare to our results.
Figure~\ref{fig:eu_h_fe_h_wtl95_ke02} presents a [Eu/H] vs. [Fe/H]
diagram with our results and those from WTL95 and KE02. Both data
sets exhibit the same behaviour, but our results present scatter
36\%~lower than those of WTL95 and KE02, even though our sample is
6~times smaller than theirs (linear fits result in
$\sigma_{\mathrm{our~work}}=0.050$ and
$\sigma_{\mathrm{WTL95/KE02}}=0.078$). The lower scatter of our
results is a consequence of improvements introduced in our
analysis, like the use of atmospheric parameters obtained by us
through the detailed and totally self-consistent spectroscopic
analysis carried out in Sect.~\ref{sec:atmospheric_parameters}.
WTL95 and KE02 demonstrated, using Monte Carlo simulations of
observational errors, that the observed scatter is not real, but
mainly a result of observational, analytical, and systematic
errors.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics*{FIGURE17.EPS}}
\caption{[Eu/H] vs. [Fe/H] diagram for our sample stars and those
from WTL95 and KE02. Average error bars for the two data sets are
provided in the lower right corner. Note that the behaviour of
both data sets is similar, but that our abundances present
considerably lower scatter.} \label{fig:eu_h_fe_h_wtl95_ke02}
\end{figure}
\subsection{Thorium}
\subsubsection{Spectral synthesis}
\label{sec:th_abundance_determination}
The list of lines used for calculation of the synthetic spectrum
of the Th region was constructed following the same procedure used
for the Eu analysis. 14 V, Mn, Fe, Co, Ni, W, Ce, and U lines were
kept, besides the Th line itself. Adopted wavelengths were taken
from the VALD list
\citep{bard&kock94,kurucz93,kurucz94,whaling83}, with the
exception of the \ion{Fe}{i} line at 4019.043~\AA, \ion{Ni}{i} at
4019.067~\AA, and \ion{Th}{ii} at 4019.130~\AA, determined in
laboratory by \citet{learneretal91} with a precision higher than
0.0005~\AA. The \ion{Co}{i} lines were substituted by their HFSs
components, calculated by us using Casimir's equation, and HFS
laboratory data from \citet{childs&goodman68},
\citet{guthohrlein&keller90}, \citet{pickeringþe96}, and
\citet{pickering96}. To improve the fits further, one artificial
\ion{Fe}{ii} line was added as substitute for unknown blends
(MKB92; \citealt{francoisetal93}); the influence of this
artificial line in the obtained Th abundances is nevertheless
small, as it is located relatively far from the center of the
\ion{Th}{ii} line. The final list is presented in
Table~\ref{tab:th_line_list}.
\begin{table*}
\caption[]{Line list used in the spectral synthesis of the
\ion{Th}{ii} line at 4019.13~\AA.} \label{tab:th_line_list}
\begin{tabular}{ c c c c c c c }
\hline \hline $\lambda$ (\AA) & Element & $\chi$ (eV) & $\log
gf_{\mathrm{3.60~m}}$
& $\log gf_{\mathrm{CAT}}$ & $\log gf$ source & Obs.\\
\hline
4018.986 & \ion{U}{ii} & 0.04 & $-$1.391 & $-$1.391 & VALD & \\
4018.999 & \ion{Mn}{i} & 4.35 & $-$1.497 & $-$1.497 & VALD & \\
4019.036 & \ion{V}{ii} & 3.75 & $-$2.704 & $-$2.704 & VALD & \\
4019.042 & \ion{Mn}{i} & 4.67 & $-$1.031 & $-$1.026 & solar fit & \\
4019.043 & \ion{Fe}{i} & 2.61 & $-$3.150 & $-$3.145 & solar fit & \\
4019.057 & \ion{Ce}{ii} & 1.01 & $-$0.470 & $-$0.445 & solar fit & \\
4019.067 & \ion{Ni}{i} & 1.94 & $-$3.404 & $-$3.329 & solar fit & \\
4019.130 & \ion{Th}{ii} & 0.00 & $-$0.228 & $-$0.228 & \citet{nilssonetal02} & \\
4019.132 & \ion{Co}{i} & 2.28 & $-$2.270 & $-$2.270 & \citet{lawleretal90} & HFS\\
4019.134 & \ion{V}{i} & 1.80 & $-$1.300 & $-$1.300 & VALD & \\
4019.206 & \ion{Fe}{ii} & 3.00 & $-$5.380 & $-$5.425 & solar fit & artificial\\
4019.228 & \ion{W}{i} & 0.41 & $-$2.200 & $-$2.200 & VALD & \\
4019.293 & \ion{Co}{i} & 0.58 & $-$3.232 & $-$3.232 & VALD & HFS\\
4019.297 & \ion{Co}{i} & 0.63 & $-$3.769 & $-$3.769 & VALD & HFS\\
\hline
\end{tabular}
References: see text.
\end{table*}
The width of the Gaussian profile used to take macroturbulence and
instrumental broadenings into account was obtained by fitting the
\ion{Co}{i} lines located at 4019.293~\AA\ and 4019.297~\AA. Solar
$\log gf$ were obtained by keeping abundances fixed at their solar
values \citep{grevesse&sauval98}, and fitting the observed solar
spectrum. Only the stronger Mn, Fe, Ce, and Ni lines had their
$\log gf$ adjusted. For the \ion{Th}{ii} line, we adopted a fixed
$\log gf$ from \citet{nilssonetal02}; for the \ion{Co}{i} lines we
adopted fixed $\log gf$ from \citealt{lawleretal90} (4019.132~\AA)
and VALD \citep[4019.293~\AA\ and 4019.297~\AA,\ ][]{kurucz94}.
The other, weaker lines had their $\log gf$ kept at their
laboratory values adopted from the VALD list
\citep{kurucz93,kurucz94}. As we kept the Th $\log gf$ fixed, we
allowed its abundance to vary. The solar $\log gf$ and Gaussian
profile determination was accomplished independently for each
telescope used, leading to slightly different Th solar abundances
($\log\varepsilon(\mbox{Th})_{\sun}=+0.03$ and $+0.04$ for the
3.60~m and CAT, respectively). As with Eu, the stellar [Th/H]
abundance ratios were obtained by subtracting the appropriate
solar value from the stellar absolute abundances
($\mbox{[Th/H]}=\log\varepsilon(\mbox{Th})_{\mathrm{star}}-\log\varepsilon(\mbox{Th})_{\sun}$).
Figure~\ref{fig:th_sun} presents the observed and synthetic solar
spectrum in the Th region. The strongest lines that compose the
total synthetic spectrum are presented independently.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics*{FIGURE18.EPS}}
\caption{Synthesized spectral region for the \ion{Th}{ii} line at
4019.13~\AA, for the Sun. The two thick lines represent the total
synthetic spectrum and the \ion{Th}{ii} line. The thin lines
represent the most dominant lines in the region, including an
artificial \ion{Fe}{ii} line. Points are the observed CES solar
spectrum, obtained with the 3.60~m telescope.} \label{fig:th_sun}
\end{figure}
For the Th synthesis in the sample stars, we kept the abundances
of all elements other than Th fixed at the values determined using
EWs, in Sect.~\ref{sec:chemical_abundances}. Small adjustments
were allowed only to the abundances of Mn, Fe, Ce, and Ni, in
order to improve the fit. The adjustments were kept within the
uncertainties of the abundances of these elements -- see
Sect.~\ref{sec:chemical_abundances}. The W abundances, for which
we did not measure any EW, were determined scaling the solar
abundances, following the Fe abundances. The U abundances were
estimated from a simplifying hypothesis that the [U/H] stellar
abundance at the time of formation is independent of metallicity.
We allied this hypothesis to a simple linear age-metallicity
relation, in which a star with $\mbox{[Fe/H]}=-1.00$ is 10~Gyr
old, and one with $\mbox{[Fe/H]}=+0.00$ is 4.57~Gyr old, i.e.,
$\mbox{age(Gyr)}=4.57-5.43\,\mbox{[Fe/H]}$. Given the known
\element[][238]{U} half-life $t_{1/2}=4.46~\mbox{Gyr}$, we reach a
relation $\mbox{[U/H]}=-0.50+0.37\,\mbox{[Fe/H]}$. Mind that the W
and U lines have only a small influence on the final Th derived by
the analysis, since they are weak (with
$\mbox{EW}<0.2~\mbox{m\AA}$ for the metal-richest stars) and are
not close to the Th line. We employed the projected rotation
velocities $v\,\sin i$ determined in the Eu analysis, and the
macroturbulence and instrumental broadening Gaussian profiles were
obtained by fitting the \ion{Co}{i} lines at 4019.293~\AA\ and
4019.297~\AA, like done with the solar spectrum.
Stars observed more than once had their spectra analysed
individually, and the results were averaged. Different spectra for
each star presented a 0.02~dex maximum variation. Examples of
spectral syntheses can be seen in Fig.~\ref{fig:examples_th} for
the two stars with extreme Th abundances that were observed with
both 3.60~m and CAT (\object{HD~20\,766} and
\object{HD~128\,620}). Mark that, due to problems that degraded
the resolving power of spectra obtained with the 3.60~m telescope,
these are almost identical to the ones obtained with the CAT.
\begin{figure*}
\centering
\includegraphics*[width=17cm]{FIGURE19.EPS}
\caption{Examples of spectral syntheses of the \ion{Th}{ii} line
at 4019.13~\AA. 3.60~m and CAT spectra are presented for two stars
with extreme Th abundances (\object{HD~20\,766} and
\object{HD~128\,620}). Points are the observed spectra. Thick
lines are the best fitting synthetic spectra, calculated with the
shown Th abundances. Thin lines represent variations in the Th
abundance $\Delta\log\varepsilon(\mbox{Th})=\pm0.20~\mbox{dex}$.
The shown abundances are the ones that best fit the presented
spectra, and not the average values that are presented in
Table~\ref{tab:th_eu_abundances}.} \label{fig:examples_th}
\end{figure*}
Th abundances determined using spectra obtained with the 3.60~m
and CAT are compared in Fig.~\ref{fig:th_360_th_cat}. A linear fit
was calculated, and the resulting relation was used to convert
3.60~m results into the CAT system, and vice versa. This way, we
ended up with two sets of Th abundances: one obtained partially
with 3.60~m spectra and partially with
corrected-into-3.60-m-system CAT spectra, and one that is just the
opposite, obtained partially with CAT spectra and partially with
corrected-into-CAT-system 3.60~m spectra. We plotted a [Th/Fe] vs.
[Fe/H] diagram for both sets of abundances, presented in
Fig.~\ref{fig:th_fe_fe_h_360_cat}. Linear fits calculated for each
set show that the CAT abundances present a lower scatter~$\sigma$,
and for this reason the CAT results were adopted for all
subsequent analyses. Thus, our adopted Eu and Th were obtained in
one and the same system, related to one single instrument (CES)
and telescope (CAT), reinforcing the homogeneity of the analysis.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics*{FIGURE20.EPS}}
\caption{Comparison between Th abundances determined using CES
spectra obtained with the CAT and with the 3.60~m. Solid line is
the identity. Dotted line is a linear fit, whose parameters are
shown in the figure.} \label{fig:th_360_th_cat}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics*{FIGURE21.EPS}}
\caption{[Th/Fe] vs. [Fe/H] diagram for all sample stars.
Abundances determined using CES spectra obtained with the CAT and
the 3.60~m are presented. The Sun is not included in the
comparison, since its abundance is always equal to zero by
definition. Solid and dotted lines are linear fits to these two
data sets, whose scatters are shown in the figure. Mark that data
from the CAT are better than those from 3.60~m.}
\label{fig:th_fe_fe_h_360_cat}
\end{figure}
\subsubsection{Uncertainty assessment}
[Th/H] abundance uncertainties were obtained with a procedure
identical to the one used for Eu.
Table~\ref{tab:th_eu_uncertainties} contains the results of the Th
uncertainty analysis for the four standard stars, and
Table~\ref{tab:th_eu_abundances} contains the abundances for all
sample stars, with their respective uncertainties. Dependence of
Th abundances on atmospheric parameters is similar to Eu, but
exhibit 3 to 5 times higher sensitivity to continuum placement
variations. This results from the much lower EW of the Th line,
which ranges from 1~m\AA\ to 8~m\AA\ for our sample stars, whereas
Eu ranges from 20~m\AA\ to 90~m\AA. Furthermore, the Eu line is
practically isolated from contaminations, while the Th line is
located in the red wing of a feature composed of strong Mn, Fe,
Ce, and Ni lines, and is blended with one \ion{V}{i} and one
\ion{Co}{i} line that, depending on the metallicity of the star,
has an EW comparable to Th itself.
\begin{table*}
\caption[]{[Eu/H], [Eu/Fe], [Th/H], [Th/Eu], and [Th/Eu] abundance
uncertainties for the standard stars.}
\label{tab:th_eu_uncertainties}
\begin{tabular} { c c c c c c c c }
\hline\hline Parameter & $\Delta$Parameter & HD &
[Eu/H] & [Eu/Fe] & [Th/H] & [Th/Fe] & [Th/Eu]\\
\hline & & 160\,691 & +0.01 & +0.00 & +0.00 & $-$0.01 & $-$0.01\\
& & 22\,484 & +0.01 & $-$0.01 & +0.01 & $-$0.01 & +0.00\\
\raisebox{1.5ex}[0pt]{$T_{\mathrm{eff}}$} &
\raisebox{1.5ex}[0pt]{+27~K} & 63\,077 & +0.01 & $-$0.01 &
+0.01 & $-$0.01 & +0.00\\
& & 203\,608 & +0.02 & +0.00 & +0.01 & $-$0.01 & $-$0.01\\
\hline & & 160\,691 & +0.01 & +0.01 & +0.01 & +0.01 & +0.00\\
& & 22\,484 & +0.01 & +0.01 & +0.01 & +0.01 & +0.00\\
\raisebox{1.5ex}[0pt]{$\log g$} & \raisebox{1.5ex}[0pt]{+0.02~dex}
& 63\,077 &
+0.01 & +0.01 & +0.01 & +0.01 & +0.00\\
& & 203\,608 & +0.01 & +0.01 & +0.01 & +0.01 & +0.00\\
\hline & & 160\,691 & +0.00 & +0.01 & +0.00 & +0.01 & +0.00\\
& \raisebox{1.5ex}[0pt]{+0.05 km s$^{-1}$} &
22\,484 & +0.00 & +0.01 & +0.00 & +0.01 & +0.00\\
\cline{2-2} \raisebox{1.5ex}[0pt]{$\xi$} & & 63\,077 & +0.00 &
+0.03 & +0.00
& +0.03 & +0.00\\
& \raisebox{1.5ex}[0pt]{+0.23 km s$^{-1}$} &
203\,608 & +0.00 & +0.03 & +0.00 & +0.03 & +0.00\\
\hline
& & 160\,691 & +0.03 & +0.02 & +0.04 & +0.03 & +0.01\\
& & 22\,484 & +0.02 & +0.02 & +0.03 & +0.03 & +0.01\\
\raisebox{1.5ex}[0pt]{[Fe/H]} & \raisebox{1.5ex}[0pt]{+0.10~dex} &
63\,077 & +0.02 & +0.02 &
+0.02 & +0.02 & +0.00\\
& & 203\,608 & +0.01 & +0.01 & +0.02 & +0.02 & +0.01\\
\hline & & 160\,691 & +0.02 & $-$0.04 & +0.09 & +0.03 & +0.07\\
& \raisebox{1.5ex}[0pt]{$+2\%$} &
22\,484 & +0.03 & $-$0.03 & +0.10 & +0.04 & +0.07\\
\cline{2-2} \raisebox{1.5ex}[0pt]{Continuum} & & 63\,077 & +0.02 &
$-$0.05 & +0.11
& +0.04 & +0.09\\
& \raisebox{1.5ex}[0pt]{$+1\%$} &
203\,608 & +0.03 & $-$0.03 & +0.14 & +0.08 & +0.11\\
\hline \hline
& & 160\,691 & 0.04 & 0.05 & 0.10 & 0.05 & 0.07\\
\raisebox{-0.3ex}[0pt]{Total} & & 22\,484 & 0.04 & 0.04 & 0.11 & 0.05 & 0.07\\
\raisebox{0.3ex}[0pt]{uncertainty} & & 63\,077 & 0.03 & 0.06 & 0.11 & 0.06 & 0.09\\
& & 203\,608 & 0.04 & 0.04 & 0.14 & 0.09 & 0.11\\
\hline
\end{tabular}
\end{table*}
\begin{table*}
\caption[]{[Eu/H], [Th/H], and [Th/Eu] abundance ratios for all
sample stars, with their respective uncertainties relative to H
and Fe.} \label{tab:th_eu_abundances}
\begin{tabular} { l c c c c c c c c }
\hline\hline HD & [Eu/H] & uncert.$_{\mathrm{[Eu/H]}}$ &
uncert.$_{\mathrm{[Eu/Fe]}}$ & [Th/H] &
uncert.$_{\mathrm{[Th/H]}}$
& uncert.$_{\mathrm{[Th/Fe]}}$ & [Th/Eu] & uncert.$_{\mathrm{[Th/Eu]}}$\\
\hline
2151 & $-$0.04 & 0.04 & 0.05 & +0.00 & 0.11 & 0.06 & +0.04 & 0.08\\
9562 & +0.05 & 0.04 & 0.05 & +0.04 & 0.11 & 0.06 & $-$0.01 & 0.08\\
16\,417 & +0.12 & 0.04 & 0.05 & +0.06 & 0.11 & 0.06 & $-$0.06 & 0.08\\
20\,766 & $-$0.15 & 0.04 & 0.05 & $-$0.25 & 0.11 & 0.06 & $-$0.10 & 0.08\\
20\,807 & $-$0.12 & 0.04 & 0.05 & $-$0.17 & 0.11 & 0.06 & $-$0.05 & 0.08\\
22\,484 & $-$0.01 & 0.04 & 0.04 & +0.04 & 0.11 & 0.05 & +0.05 & 0.07\\
22\,879 & $-$0.42 & 0.04 & 0.05 & $-$0.58 & 0.12 & 0.07 & $-$0.16 & 0.09\\
30\,562 & +0.23 & 0.04 & 0.05 & +0.19 & 0.11 & 0.06 & $-$0.04 & 0.08\\
43\,947 & $-$0.18 & 0.04 & 0.05 & $-$0.11 & 0.12 & 0.06 & +0.07 & 0.08\\
52\,298 & $-$0.13 & 0.04 & 0.04 & $-$0.09 & 0.12 & 0.06 & +0.04 & 0.09\\
59\,984 & $-$0.48 & 0.04 & 0.05 & $-$0.42 & 0.12 & 0.07 & +0.06 & 0.09\\
63\,077 & $-$0.48 & 0.03 & 0.06 & $-$0.48 & 0.11 & 0.06 & +0.00 & 0.09\\
76\,932 & $-$0.48 & 0.04 & 0.05 & $-$0.54 & 0.12 & 0.07 & $-$0.06 & 0.09\\
102\,365 & $-$0.11 & 0.04 & 0.05 & $-$0.23 & 0.11 & 0.06 & $-$0.12 & 0.08\\
128\,620 & +0.16 & 0.04 & 0.05 & +0.09 & 0.11 & 0.06 & $-$0.07 & 0.08\\
131\,117 & +0.07 & 0.04 & 0.05 & +0.10 & 0.11 & 0.06 & +0.03 & 0.08\\
160\,691 & +0.16 & 0.04 & 0.05 & +0.09 & 0.10 & 0.05 & $-$0.06 & 0.07\\
196\,378 & $-$0.22 & 0.04 & 0.04 & $-$0.31 & 0.12 & 0.07 & $-$0.10 & 0.09\\
199\,288 & $-$0.30 & 0.03 & 0.05 & $-$0.32 & 0.11 & 0.06 & $-$0.02 & 0.09\\
203\,608 & $-$0.50 & 0.04 & 0.04 & $-$0.46 & 0.14 & 0.09 & +0.04 & 0.11\\
\hline
\end{tabular}
\end{table*}
\subsubsection{Comparison with literature results}
There are only two works available in the literature in which Th
abundances have been determined for Galactic disk stars:
\citet{dasilvaetal90} and MKB92. In the classic work of
\citet{butcher87}, which was the first to propose the use of
stellar Th abundances as chronometers, the ratio between the EWs
of Th and Nd is adopted as an estimate of [Th/Nd], but no
independent Th abundances are derived. We did not compare our
results to those from \citeauthor{dasilvaetal90} because only
preliminary data is presented for a small sample of four stars.
MKB92 is a re-analysis of the \citet{butcher87} data, with the
same stellar sample and spectra, but making use of more up-to-date
model atmospheres and more detailed spectral syntheses.
Unfortunately, MKB92 do not present a detailed Th abundance
uncertainty analysis. The authors investigated the influence of
effective temperature, surface gravity and continuum placement
variations, but overlooked metallicity and microturbulence
velocity. No estimate of total uncertainty is presented.
Therefore, we decided not to include error bars in
Fig.~\ref{fig:th_h_fe_h_mkb92}, which depicts [Th/H] vs. [Fe/H]
for our results and those from MKB92. The two data sets exhibit a
similar behaviour, but our results present scatter 61\% lower than
those of MKB92 (from linear fits we get
$\sigma_{\mathrm{our~work}}=0.064$ and
$\sigma_{\mathrm{MKB92}}=0.165$). The lower scatter of our results
is a consequence of many enhancements introduced in our analysis,
like the use of homogeneous and precise atmospheric parameters
obtained by us, while MKB92 used multiple literature sources, and
refinements in the spectral synthesis.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics*{FIGURE22.EPS}}
\caption{[Th/H] vs. [Fe/H] diagram for our sample stars and those
from MKB92. Note that the behaviour of both data sets is similar,
but that our abundances present considerably lower scatter.}
\label{fig:th_h_fe_h_mkb92}
\end{figure}
\section{Conclusions}
Homogenous, fully self-consistent atmospheric parameters, and
their respective uncertainties, have been obtained for all sample
stars. The use of two different and internally homogeneous
criteria for effective temperature determination allowed us to
achieve a very low uncertainty (27~K) for this important
parameter. Uncertainties of the other parameters were also found
to be adequate for our needs.
Abundances of the elements that contaminate the \ion{Th}{ii} and
\ion{Eu}{ii} spectral regions -- Ti, V, Cr, Mn, Co, Ni, Ce, Nd and
Sm -- have been determined by detailed spectroscopic analysis,
relative to the Sun, using EWs. For the elements with meaningful
HFSs (V, Mn, and Co), this has been taken into account. A thorough
estimation of the uncertainties has been carried out. The average
[element/H] uncertainty -- $(0.10\pm0.02)$~dex -- was found to be
satisfactorily low.
Eu and Th abundances have been determined for all sample stars
using spectral synthesis of the \ion{Eu}{ii} line at 4129.72~\AA\
and the \ion{Th}{ii} line at 4019.13~\AA. Comparison of our
results with the literature shows that our analysis yielded a
similar behaviour, but with considerably lower scatters (36\%
lower for Eu, and 61\% lower for Th). The [Th/Eu] abundance ratios
thus obtained were used to determine the age of the Galactic disk
in Paper~II.
\begin{acknowledgements}
This paper is based on the PhD thesis of one of the authors
\citep{delpeloso03}. The authors wish to thank the staff of the
Observat\'orio do Pico dos Dias, LNA/MCT, Brazil and of the
European Southern Observatory, La Silla, Chile. The support of
Martin K\"urster (Th\"uringer Landessternwarte Tautenburg,
Germany) during the observations with the ESO's 3.60~m telescope
was greatly appreciated. We thank R. de la Reza, C. Quireza and
S.D. Magalh\~aes for their contributions to this work. EFP
acknowledges financial support from CAPES/PROAP and FAPERJ/FP
(grant E-26/150.567/2003). LS thanks the CNPq, Brazilian Agency,
for the financial support 453529.0.1 and for the grants
301376/86-7 and 304134-2003.1. GFPM acknowledges financial support
from CNPq/Conte\'udos Digitais, CNPq/Institutos do
Mil\^enio/MEGALIT, FINEP/PRONEX (grant 41.96.0908.00), and
FAPESP/Tem\'aticos (grant 00/06769-4). Finally, we are grateful to
the anonymous referee for a thorough revision of the manuscript,
and for comments that helped to greatly enhance the final version
of the work.
\end{acknowledgements}
\bibliographystyle{aa}
|
{
"timestamp": "2005-02-21T21:56:29",
"yymm": "0411",
"arxiv_id": "astro-ph/0411698",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411698"
}
|
\section{Introduction}
A crucial prediction of the Minimal Supersymmetric Standard Model
(MSSM)~\cite{susy} is the existence of at least one light Higgs
boson. The search for this particle
is one of the main goals at the present and the next generation
of colliders. Direct searches at LEP have already ruled out a
considerable fraction of the MSSM parameter
space~\cite{LEPHiggsSM,LEPHiggsMSSM}, and the forthcoming
high-energy experiments at the Tevatron, the LHC, and the International
Linear Collider (ILC) will either
discover a light Higgs boson or rule out Supersymmetry (SUSY)
as a viable theory
for physics at the weak scale. Furthermore, if one or more Higgs
bosons are discovered, bounds on their masses and couplings will be
set at the LHC~\cite{LHCHiggs,HcoupLHCSMearly,HcoupLHCSM}.
Eventually the masses and couplings will be determined
with high accuracy at the ILC~\cite{tesla,orangebook,acfarep}.
Thus, a
precise knowledge of the dependence of masses and mixing angles in
the MSSM Higgs sector on the relevant supersymmetric parameters is of
utmost importance to reliably compare the predictions of the MSSM with
the (present and future) experimental results.
The status of the available results for the higher-order contributions to
the neutral ${\cal CP}$-even MSSM Higgs boson masses can be summarised as follows.
For the
one-loop part, the complete result within the MSSM is
known~\cite{ERZ,mhiggsf1lA,mhiggsf1lB,mhiggsf1lC}. The
dominant one-loop contribution is the \order{\alpha_t} term due to top and stop
loops ($\alpha_t \equiv h_t^2 / (4 \pi)$, $h_t$ being the
superpotential top coupling).
Corrections from the bottom/sbottom sector can also give large effects,
in particular for
large values of $\tan \beta$, the ratio of the two vacuum expectation values,
$\tan \beta = v_2/v_1$. The computation of two-loop corrections is also
quite advanced. It has now reached a
stage such that all the presumably dominant
contributions are known. They include the strong corrections, usually
indicated as \order{\alpha_t\alpha_s}, and Yukawa corrections, \order{\alpha_t^2},
to the dominant one-loop \order{\alpha_t} term, as well as the strong
corrections to the bottom/sbottom one-loop \order{\alpha_b} term
($\alpha_b \equiv h_b^2 / (4\pi)$), i.e.\ the \order{\alpha_b\alpha_s} contribution,
derived in the limit $\tan \beta \to \infty$.
Presently, the
\order{\alpha_t\alpha_s}~\cite{mhiggsEP1b,ahoang,mhiggsRG1,mhiggsRG2,mhiggsletter,mhiggslong,mhiggsEP0,mhiggsEP1,bse,reconc},
\order{\alpha_t^2}~\cite{mhiggsEP1b,ahoang,mhiggsEP3,mhiggsEP2} and the
\order{\alpha_b\alpha_s}~\cite{mhiggsEP4} contributions to the self-energies
are known for vanishing external momenta. Most recently also the corrections
\order{\alpha_t\alpha_b} and \order{\alpha_b^2}~\cite{mhiggsEP5}, a ``full'' two-loop\
effective potential calculation~\cite{effpotfull} and an evaluation of
the leading two-loop momentum dependent effects~\cite{mhiggsp2} have
become available. In the (s)bottom
corrections the all-order resummation of the $\tan \beta$-enhanced terms,
\order{\alpha_b(\alpha_s\tan \beta)^n}, is also performed \cite{deltamb1,deltamb}.
Reviews with further references can be found in
\citeres{mhiggsAEC,habilSH,Allanach:2004rh}.
The $b/\tilde{b}$ sector has attracted considerable attention in the last years,
since its corrections to the MSSM Higgs boson sector have been found to
be large in certain parts of the MSSM parameter space, possibly even
exceeding the size of the top/stop corrections.
This can happen especially for large values of $\tan \beta$ and
the supersymmetric Higgs mass parameter $\mu$.
For illustration, we show in \reffi{fig:deltamh}
the shift in the lightest ${\cal CP}$-even Higgs boson mass, $\DeM_h$, arising from
the $b/\tilde{b}$ sector at the one-loop\ level (all two-loop\
corrections are omitted here) as a function of the bottom-quark mass for
large $\tan \beta$ and $|\mu|$. The bottom-quark mass in this plot is
understood to be an effective mass that includes higher-order effects
(see the discussion in \refse{sec:ren}). The figure demonstrates that
corrections from the $b/\tilde{b}$ sector can get large if the effective
bottom mass is bigger than about 3~GeV.
\begin{figure}[htb!]
\begin{center}
\epsfig{figure=plots/mbmh01.bw.eps, width=14cm,height=11cm}
\caption{
The shift in the lightest ${\cal CP}$-even Higgs-boson mass from the
one-loop\ corrections in the $b/\tilde{b}$~sector is shown as a function of
the (effective) bottom-quark mass for $\mu = \pm 1000 \,\, {\rm GeV}$,
$\tan\beta = 50$, $M_{\text{SUSY}} = 600 \,\, {\rm GeV}$, $A_t = A_b = 500 \,\, {\rm GeV}$,
$M_A = 700 \,\, {\rm GeV}$.
}
\label{fig:deltamh}
\end{center}
\end{figure}
The possibly large size of the corrections from the $b/\tilde{b}$ sector
makes it desirable to investigate the
corresponding two-loop corrections and thus to analyse
the renormalisation in this sector. An
inconvenient choice could give rise to artificially large corrections,
whereas a convenient scheme absorbs the dominant contributions into the
one-loop result such that higher-order corrections remain small. The
comparison of different schemes (where no artificially enhanced
corrections appear) gives an indication of the possible size of missing
higher-order terms of \order{\alpha_b\alpha_s^2}.
In this paper we derive the result for the \order{\alpha_b\alpha_s}
corrections in various renormalisation schemes. The relations between the
different parameters in these schemes are worked out in detail. The
absorption of leading higher-order contributions into an effective
bottom-quark mass is discussed. We perform a numerical
analysis of the various schemes and compare our results
with a previous evaluation of the \order{\alpha_b\alpha_s} corrections carried
out in the limit where $\tan \beta$ is infinitely large~\cite{mhiggsEP4}.
We discuss the dependence of our result on the renormalisation scale and
provide an estimate of the remaining theoretical uncertainties in this
sector.%
\footnote{
This kind of issues have not been addressed in
\citeres{effpotfull,mhiggsp2}.
}
The paper is organised as follows: in \refse{sec:higgssector} we
briefly review the MSSM Higgs boson sector, outline the corresponding
renormalisation at the two-loop\ level, and describe the evaluation of the
diagrams of \order{\alpha_b\alpha_s}. \refse{sec:ren} contains a detailed
description of the renormalisation of the scalar top and scalar bottom
sector, which is explicitly carried out in four different renormalisation
schemes for the latter. The numerical analysis of the \order{\alpha_b\alpha_s}
corrections, the comparison of the different schemes, the investigation
of the renormalisation scale, and the
comparison with the previous result are
performed in \refse{sec:numres}. The conclusions can be found in
\refse{sec:conclusions}.
\section{The Higgs sector at higher orders}
\label{sec:higgssector}
We recall that the Higgs sector of the MSSM~\cite{hhg} comprises two
neutral ${\cal CP}$-even Higgs bosons, $h$ and $H$ ($m_h < m_H$), the
${\cal CP}$-odd $A$~boson,%
\footnote{
Throughout this paper we assume that ${\cal CP}$ is conserved.
}%
~and two charged Higgs bosons, $H^\pm$.
At the tree-level, the masses $m_{h,{\rm tree}}$ and $m_{H,{\rm tree}}$
can be calculated in terms of $M_Z$, $M_A$ and $\tan \beta$ from
the mass matrix for the
neutral ${\cal CP}$-even Higgs components (denoted by~$\phi$)
\begin{eqnarray}
\label{higgsmassmatrixtree}
{\cal M}_\phi &=& \left( \begin{array}{cc} M_A^2 \sin^2\beta\hspace{1mm} +
M_Z^2 \cos^2\beta\hspace{1mm} & -(M_A^2 + M_Z^2) \sin \beta \cos \beta\hspace{1mm} \\ -(M_A^2 + M_Z^2) \sin \beta \cos \beta\hspace{1mm} &
M_A^2 \cos^2\beta\hspace{1mm} + M_Z^2 \sin^2\beta\hspace{1mm} \end{array} \right)
\end{eqnarray}
by diagonalization,
\begin{eqnarray}
\label{higgsmassmatrixtreediag}
\left( \begin{array}{cc} m_{H,{\rm tree}}^2 & 0 \\ 0 & m_{h,{\rm tree}}^2 \end{array} \right) &=&
{\cal U}_\phi \, {\cal M}_\phi \, {\cal U}_\phi^\dagger~, \qquad
{\cal U}_\phi = \left( \begin{array}{cc} \cos \alpha\hspace{1mm} & \sin \alpha\hspace{1mm} \\ -\sin \alpha\hspace{1mm} & \cos \alpha\hspace{1mm} \end{array} \right) \, ,
\end{eqnarray}
with the angle $\alpha$ determined by
\begin{equation}
\tan 2\alpha = \tan 2\beta\, \frac{M_A^2 + M_Z^2}{M_A^2 - M_Z^2},
\quad - \frac{\pi}{2} < \alpha < 0 .
\label{alpha}
\end{equation}
\bigskip
In the Feynman-diagrammatic (FD) approach, the higher-order corrected
Higgs boson masses, $M_h$ and $M_H$, are derived as the poles of the
$h,H$-propagator matrix,
i.e.\ by solving the equation
\begin{equation}
\left[p^2 - m_{h,{\rm tree}}^2 + \hat{\Sigma}_{hh}(p^2) \right]
\left[p^2 - m_{H,{\rm tree}}^2 + \hat{\Sigma}_{HH}(p^2) \right] -
\left[\hat{\Sigma}_{hH}(p^2)\right]^2 = 0\,.
\label{eq:proppole}
\end{equation}
The renormalised self-energies
\begin{equation}
\label{SEmatrix}
\hat{\Sigma}(p^2) = \left( \begin{array}{cc} \hat{\Sigma}_{HH}(p^2) & \hat{\Sigma}_{hH}(p^2) \\[.2em]
\hat{\Sigma}_{hH}(p^2) & \hat{\Sigma}_{hh}(p^2) \end{array} \right)
\end{equation}
can be expanded according to
the one-, two-, \ldots loop-order contributions,
\begin{equation}
\label{renSE}
\hat{\Sigma}(p^2) = \hat{\Sigma}^{(1)}(p^2)
+ \hat{\Sigma}^{(2)}(p^2) + \cdots ~.
\end{equation}
The dominant one-loop\ contributions to the Higgs boson self-energies
(and thus to the Higgs boson masses) from the $b/\tilde{b}$~sector are
of \order{\alpha_b} and arise from the Yukawa part of the theory (neglecting the
gauge couplings) evaluated at $p^2=0$. This has been verified by comparison
with the full one-loop result from the $b/\tilde{b}$~sector.
Hence, the leading two-loop\ corrections from the $b/\tilde{b}$~sector are the
\order{\alpha_s} corrections to those dominant one-loop\ contributions; they
are obtained in the same limit, i.e.\ for zero external momentum and
neglecting the gauge couplings (the same approximations have been made
in \citere{mhiggsEP4}). This approach is analogous to the way the
leading one- and two-loop\ contributions in the top/stop sector have been
obtained, see e.g.~\citere{mhiggslong}.
The renormalisation of the Higgs-boson mass matrix for
the \order{\alpha_b\alpha_s} corrections under consideration follows the
description for the \order{\alpha_t\alpha_s} terms given in
\citere{mhiggslong}.
Renormalisation can be performed by adding the appropriate
counterterms,
\begin{align}\label{massenmatrixrenormierung}
{\cal M}_\phi &\to {\cal M}_\phi + \delta{\cal M}^{(1)}_\phi + \delta{\cal M}^{(2)}_\phi + \cdots \, ,
\end{align}
where $\delta{\cal M}^{(i)}_\phi$ denotes the {\it i}th-loop
counterterm matrix consisting of the
counterterms to the parameters in the tree-level
mass matrix \refeq{higgsmassmatrixtree}.
Field renormalisation is not
needed for the leading \order{\alpha_b\alpha_s} corrections.
The renormalised two-loop\ Higgs boson self-energies with the leading
contributions of \order{\alpha_b\alpha_s} are thus given by
\begin{align}
\label{loopHh}
\hat{\Sigma}^{(2)}(0) &=
\Sigma^{(2)} (0) - {\cal U}_\phi \delta {\cal M}_\phi^{(2)} {\cal U}^\dagger_\phi~.
\end{align}
\bigskip
The counterterm matrix in \refeq{loopHh}
is composed of the counterterms for
the $A$-boson mass and for the tadpoles $t_{h,H}$
(with $s_\mathrm{w} \equiv \sin\theta_W$, $c_\mathrm{w} = \cos\theta_W$),
\begin{align}
\delta
{\cal M}_{\phi}^{(2)} &=
\begin{pmatrix} \sin^2 \beta & - \sin \beta \cos
\beta \\ - \sin \beta \cos \beta & \cos^2 \beta \end{pmatrix} \delta M_A^{2\, (2)}
\nonumber
\\[1mm]&
\nonumber
\quad\ + \frac{e}{2 M_Z c_\mathrm{w} s_\mathrm{w}}
\begin{pmatrix}- \cos \beta (1 + \sin^2 \beta) &- \sin^3 \beta
\\ -\sin^3 \beta &\cos \beta \sin^2 \beta \end{pmatrix}
(\cos \alpha \, \delta t_H^{(2)} - \sin \alpha \, \delta t_h^{(2)})\\[1mm]&
\quad\ + \frac{e}{2 M_Z c_\mathrm{w} s_\mathrm{w}}
\begin{pmatrix} \cos^2 \beta \sin \beta&- \cos^3 \beta
\\-\cos^3 \beta & - (1+ \cos^2 \beta) \sin \beta \end{pmatrix}
(\sin \alpha \, \delta t_H^{(2)} + \cos \alpha \, \delta t_h^{(2)}) \, .
\end{align}
\noindent
The counterterms are determined by the following conditions:
\begin{itemize}
\item[(i)] On-shell renormalisation of the $A$-boson mass,
formulated in the
approximation of vanishing external momentum, determines the two-loop
$A$-mass counterterm $\deM_A^{2\,(2)}$ according to
\begin{align}
\deM_A^{2\,(2)} = \Sigma^{(2)}_{AA} (0) \, .
\end{align}
\item[(ii)]
Tadpole renormalisation determines
the tadpole counterterms
by the requirements
\begin{eqnarray}
\delta t_H^{(2)} &= - t_H^{(2)}\, , \quad
\delta t_h^{(2)} &= - t_h^{(2)} \, ,
\end{eqnarray}
which means that the minimum of the Higgs potential is not shifted.
\end{itemize}
\begin{figure}[htb!]
\vspace{3em}
\begin{center}
\includegraphics[width=0.9\linewidth]{plots/selfhh2lbotbildpub}
\vspace{0.5em}
\caption{Generic two-loop\ diagrams for the Higgs-boson self-energies
($\phi = h, H, A$;
$\;i,j,k,l = 1,2$).}
\label{fig:FD2L}
\end{center}
\end{figure}
\begin{figure}[htb!]
\vspace{3em}
\begin{center}
\includegraphics[width=0.8\linewidth]{plots/2lhiggsbildpub}
\caption{Generic two-loop\ diagrams for the Higgs tadpoles ($\phi= h,H$;
$\;i,j,k = 1,2$).}
\label{fig:Tad2L}
\end{center}
\end{figure}
\begin{figure}[htb!]
\vspace{3em}
\begin{center}
\includegraphics[width=0.9\linewidth]{plots/selfhh2lctbotbildpub}
\vspace{0.5em}
\caption{Generic one-loop\ diagrams with counterterm insertion for the
Higgs-boson self-energies ($\phi = h, H, A$),
$\;i,j,k = 1,2$).}
\label{fig:FD1LCT}
\end{center}
\end{figure}
\begin{figure}[htb!]
\begin{center}
\includegraphics[width=0.9\linewidth]{plots/2lcthiggsbildpub}
\caption{Generic one-loop diagrams with counterterm insertion for the
Higgs tadpoles ($\phi=h,H$, $\;i,j = 1,2$).}
\label{fig:Tad2Lct}
\end{center}
\end{figure}
The genuine two-loop\ Feynman diagrams to be
evaluated for the Higgs boson self-energies and the tadpoles
are shown \reffi{fig:FD2L} and \reffi{fig:Tad2L}.
The diagrams with subloop renormalisation are depicted in
\reffi{fig:FD1LCT} and \reffi{fig:Tad2Lct}.
The counterterms for the insertions, where different renormalisation
schemes will be investigated, are specified in the next section.
The diagrams and the corresponding amplitudes have been generated with
the package {\em FeynArts}~\cite{feynarts,famssm}. The further evaluation has
been done using the program {\em TwoCalc}~\cite{twocalc}. The resulting expressions
are given in terms of the one-loop functions $A_0$ and
$B_0$~\cite{oneloop}, and the two-loop vacuum integrals~\cite{twoloop}.
\section{Renormalisation of the quark/squark sector}
\label{sec:ren}
Since the two-loop self-energy is considered at
$\mathcal O(\alpha_{\{t,\,b\}} \alpha_s)$ it is sufficient to
determine the counterterms
induced by the strong interaction only.
The squark-mass terms of the Lagrangian, for a given species of
squarks $\tilde{q}$,
can be written as the bilinear expression
\begin{align}
{\cal L}_{\tilde{q}\text{-mass}} &= - \begin{pmatrix} \tilde{q}_L^\dagger, \tilde{q}_R^\dagger
\end{pmatrix}
{\cal M}_{\tilde{q}} \begin{pmatrix} \tilde{q}_L \\ \tilde{q}_R \end{pmatrix} ,
\end{align}
with ${\cal M}_{\tilde{q}}$ as the squark-mass matrix squared,
\begin{align}\label{Sfermionmassenmatrix}
{\cal M}_{\tilde{q}} = \begin{pmatrix}
M_L^2 + m_q^2 + M_Z^2 c_{2 \beta} (T_q^3 - Q_q s_\mathrm{w}^2) &
m_q (A_q - \mu \kappa) \\[.2em]
m_q (A_q - \mu \kappa) &
M_{\tilde{q}_R}^2 + m_q^2 +M_Z^2 c_{2 \beta} Q_q s_\mathrm{w}^2
\end{pmatrix},
\end{align}
where the quantities $M_L^2$, $M_{\tilde{q}_R}^2$, $A_q$ are
soft-breaking para\-meters, and $\mu$ is the supersymmetric Higgs mass
parameter.
Since we are dealing in this paper with a ${\cal CP}$-conserving
Higgs sector, these parameters are treated as real.
As an abbreviation, $c_{2\beta} \equiv \cos(2\beta)$ is introduced;
$\kappa$ is defined as $\kappa = \cot\beta$ for {\it up}-type
squarks and $\kappa = \tan \beta$ for {\it down}-type squarks.
$m_q$, $Q_q$, and $T_q^3$ are mass, charge, and
isospin of the quark $q$.
The mass matrix \eqref{Sfermionmassenmatrix}
can be diagonalised by a unitary transformation,
which in our case of real parameters
involves a mixing angle $\theta\kern-.15em_{\tilde{q}}$,
\begin{align}
\label{transformation}
\begin{pmatrix} \tilde{q}_1 \\ \tilde{q}_2 \end{pmatrix} =
{\cal U}_{\tilde{q}} \begin{pmatrix} \tilde{q}_L \\ \tilde{q}_R \end{pmatrix} \, ,
\qquad
{\cal U}_{\tilde{q}} = \begin{pmatrix}U_{\tilde{q}_{11}} & U_{\tilde{q}_{12}} \\
U_{\tilde{q}_{21}} & U_{\tilde{q}_{22}} \end{pmatrix} =
\begin{pmatrix} \cos \theta\kern-.15em_{\tilde{q}} & \sin \theta\kern-.15em_{\tilde{q}} \\
- \sin \theta\kern-.15em_{\tilde{q}} & \cos \theta\kern-.15em_{\tilde{q}} \end{pmatrix}\,.
\end{align}
In the $(\tilde{q}_1, \tilde{q}_2)$-basis, the squared-mass matrix is diagonal,
\begin{align}
{\cal D}_{\tilde{q}} &= {\cal U}_{\tilde{q}} {{\cal M}}_{\tilde{q}}{{\cal U}}_{\tilde{q}}^\dagger =
\begin{pmatrix} m_{\tilde{q}_1}^2 & 0 \\ 0 & m_{\tilde{q}_2}^2 \end{pmatrix}\,,
\end{align}
with the eigenvalues $m_{\tilde{q}_1}^2$ and $m_{\tilde{q}_2}^2$ given by
\begin{align} \nonumber
&m_{\tilde{q}_{1,2}}^2 = \frac{1}{2} (M_L^2 +M_{\tilde{q}_R}^2) + m_q^2 +
\frac{1}{2} T_q^3 M_Z^2 c_{2 \beta}
\pm \frac{1}{2} \frac{M_L^2 - M_{\tilde{q}_R}^2+ M_Z^2 c_{2 \beta} (T_q^3 - 2 Q_q
s_\mathrm{w}^2)}{|M_L^2 - M_{\tilde{q}_R}^2+ M_Z^2 c_{2 \beta} (T_q^3 - 2 Q_qs_\mathrm{w}^2)|}
\\& \qquad\ \quad
\times \sqrt{\bigl[ M_L^2 - M_{\tilde{q}_R}^2
+ M_Z^2 c_{2 \beta} (T_q^3 - 2 Q_q s_\mathrm{w}^2)\bigr]^2
+ 4 m_q^2 (A_q - {\mu} \kappa)^2} \; .
\label{Sfermionmassenmatrixeigenwerte}\end{align}
The squark-mass matrix can now be expressed in terms of the two mass
eigenvalues and the mixing angle, yielding
\begin{equation}
\label{stopmassenmatrix}
{\cal M}_{\tilde{q}} = \left( \begin{array}{cc} \cos^2\tsq \; m_{\tilde{q}_1}^2 + \sin^2\tsq \; m_{\tilde{q}_2}^2 &
\sin\tsq \cos\tsq \, (m_{\tilde{q}_1}^2 - m_{\tilde{q}_2}^2) \\[.4em]
\sin\tsq \cos\tsq \, (m_{\tilde{q}_1}^2 - m_{\tilde{q}_2}^2) &
\sin^2\tsq \;m_{\tilde{q}_1}^2 + \cos^2\tsq \;m_{\tilde{q}_2}^2 \end{array} \right)~.
\end{equation}
\subsection{Renormalisation of the top and scalar top sector }
\label{subsec:stoprenorm}
The $(t,\tilde{t})$ sector contains four independent parameters:
the top-quark mass $m_{t}$, the stop masses $m_{\tilde{t}_1}$ and $m_{\tilde{t}_2}$, and
either the squark mixing angle $\theta_{\tilde{t}}$ or, equivalently, the
trilinear coupling $A_t$. Accordingly, the renormalisation of this sector
is performed by introducing four counterterms that are determined by
four independent renormalisation conditions.
The following renormalisation conditions are imposed
(the procedure is equivalent to that of \citere{hr}, although
there no reference is made to the mixing angle).
\begin{itemize}
\item[(i)]
On-shell renormalisation of the top-quark mass yields the top
mass counterterm,
\begin{align}\label{dmt}
\dem_{t} = \frac{1}{2} m_{t} \bigl [\mathop{\rm Re} \Sigma_{{t}_L} (m_{t}^2) +
\mathop{\rm Re} \Sigma_{{t}_R} (m_{t}^2) +
2\mathop{\rm Re} \Sigma_{{t}_S} (m_{t}^2)
\bigr]\; ,
\end{align}
with the scalar coefficients of the
unrenormalised top-quark self-energy, $\Sigma_t (p)$,
in the Lorentz decomposition
\begin{align}
\label{Fermionselbstenergiezerlegung}
\Sigma_t (p) &= \SLASH{p}{.2} \omega_-
\Sigma_{{t}_L} (p^2) + \SLASH{p}{.2} \omega_+
\Sigma_{{t}_R} (p^2) + m_{t} \Sigma_{{t}_S} (p^2)\; .
\end{align}
\item[(ii)]
On-shell renormalisation of the stop masses
determines the mass counterterms
\begin{equation}
\label{dmst}
\delta m_{\tilde{t}_1}^2 =
\mathop{\rm Re} \Sigma_{\tilde{t}_{11}}(m_{{\tilde{t}}_{1}}^2) \, , \quad
\delta m_{\tilde{t}_2}^2 =
\mathop{\rm Re} \Sigma_{\tilde{t}_{22}}(m_{{\tilde{t}}_{2}}^2)\; ,
\end{equation}
in terms of the diagonal squark self-energies.
\item[(iii)]
The counterterm for the mixing angle, $\theta_{\tilde{t}}$, (entering
\refeq{stopmassenmatrix}) is fixed in the following way,
\begin{align}
\label{ZusammenhangdeltathetadeltaM}
\delta \theta_{\tilde{t}} =
\frac{\mathop{\rm Re} \Sigma_{\tilde{t}_{12}}(m_{{\tilde{t}}_{1}}^2)+\mathop{\rm Re}
\Sigma_{\tilde{t}_{12}}(m_{{\tilde{t}}_{2}}^2)}{2(m_{\tilde{t}_1}^2-m_{\tilde{t}_2}^2)}\; ,
\end{align}
involving the non-diagonal squark self-energy. (This is a convenient
choice for the treatment of \order{\alpha_s} corrections. If
electroweak contributions were included, a manifestly gauge-independent
definition would be more appropriate.)
\end{itemize}
In the renormalised vertices with squark and Higgs fields,
the counterterm of the trilinear
coupling $A_t$ appears.
Having already specified $\delta \theta_{\tilde{t}}$,
the $A_t$ counterterm cannot be defined
independently but follows from the relation
\begin{align}
\sin 2 \theta_{\tilde{t}} = \frac{ 2 m_{t} ( A_t - \mu \cot \beta)}{m_{\tilde{t}_1}^2 -m_{\tilde{t}_2}^2} \, ,
\end{align}
yielding
\begin{align}
\delta A_t &= \frac{1}{m_{t}}
\Bigl[\frac{1}{2} \sin 2 \theta_{\tilde{t}}
\bigl(\delta m_{\tilde{t}_1}^2 - \delta m_{\tilde{t}_2}^2\bigr)
+ \cos 2 \theta_{\tilde{t}}
(m_{\tilde{t}_1}^2 - m_{\tilde{t}_2}^2)\, \delta \theta_{\tilde{t}}
\nonumber \\[1.5mm] & \quad\ \quad\ \label{deltaAt}
- \frac{1}{2 m_{t}} \sin 2 \theta_{\tilde{t}} (m_{\tilde{t}_1}^2 -m_{\tilde{t}_2}^2) \, \delta m_{t} \Bigr]~.
\end{align}
This relation is valid at \order{\alpha_s} since both $\mu$ and $\tan\beta$
do not receive one-loop contributions from the strong interaction.
\subsection{Renormalisation of the bottom and scalar bottom sector}
\label{subsec:sbotrenorm}
Because of SU(2)-invariance
the soft-breaking parameters for the left-handed {\em up}- and
{\em down}-type squarks are identical, and thus
the squark masses of a given generation are not
independent. The stop and sbottom masses are connected via the relation
\begin{align}
\cos^2\tsb m_{\tilde{b}_1}^2 + \sin^2\tsb m_{\tilde{b}_2}^2 =
\cos^2\tst m_{\tilde{t}_1}^2 + \sin^2\tst m_{\tilde{t}_2}^2 +
m_{b}^2 - m_{t}^2 - M_W^2 \cos (2 \beta)~,
\label{MSb1gen}
\end{align}
with the entries of the rotation matrix in \refeq{transformation}.
Since the stop masses have already been renormalised
on-shell, only one of the sbottom mass
counterterms can be determined independently. In the following, the
$\tilde{b}_2$~mass is chosen%
\footnote{
This choice is possible since
\eqref{transformation}--\eqref{Sfermionmassenmatrixeigenwerte}
ensure that
the $\tilde{b}_2$-field and the $\tilde{b}_L$-field do not coincide.
}%
~as the pole mass yielding the counterterm from an on-shell
renormalisation condition, i.e.\
\begin{align}
\delta m_{\tilde{b}_2}^2 &= \mathop{\rm Re} \Sigma_{\tilde{b}_{22}}(m_{\tilde{b}_2}^2)\; ,
\label{eq:msbz}
\end{align}
whereas the counterterm for $m_{\tilde{b}_1}$ is determined
as a combination of other counterterms,
according to
\begin{align}
\delta m_{\tilde{b}_1}^2 &=
\frac{1}{\cos^2 \theta_{\tilde{b}}} \Bigl(
\cos^2 \theta_{\tilde{t}} \delta m_{\tilde{t}_1}^2
+ \sin^2 \theta_{\tilde{t}} \delta m_{\tilde{t}_2}^2
- \sin^2 \theta_{\tilde{b}} \delta m_{\tilde{b}_2}^2
- \sin 2 \theta_{\tilde{t}} (m_{\tilde{t}_1}^2 -m_{\tilde{t}_2}^2)\delta \theta_{\tilde{t}} \nonumber
\\[1.5mm]
& \quad
+ \sin 2 \theta_{\tilde{b}} (m_{\tilde{b}_1}^2 -m_{\tilde{b}_2}^2)\delta \theta_{\tilde{b}}
- 2 m_{t}\, \dem_{t} + 2 m_{b}\, \delta m_{b} \Bigr)~.
\label{ms1CT}
\end{align}
Accordingly, the numerical value of $m_{\tilde{b}_1}$ does not
correspond to the pole mass. The pole mass can be obtained
from $m_{\tilde{b}_1}$ via a finite shift of $\mathcal O(\alpha_s)$
(see e.g.~\citere{delrhosusy2loop}).
There are three more parameters with counterterms to be determined:
the $b$-quark mass $m_{b}$, the mixing angle $\theta_{\tilde{b}}$,
and the trilinear coupling~$A_b$. They are connected via
\begin{align}
\label{mixingangleAparametermbrelation}
\sin 2 \theta_{\tilde{b}} = \frac{ 2 m_{b} ( A_b - \mu\tan \beta)}{m_{\tilde{b}_1}^2 -m_{\tilde{b}_2}^2} \, ,
\end{align}
which reads in terms of counterterms
\begin{align}
\label{deltaSbot}
2 \cos 2\tsb\; \delta\theta_{\tilde{b}} &= \sin 2\tsb \frac{\dem_{b}}{m_{b}}
+ \frac{2m_{b}\,\deA_b}{m_{\tilde{b}_1}^2 - m_{\tilde{b}_2}^2}
- \sin 2\tsb \frac{\dem_{\tilde{b}_1}^2 - \dem_{\tilde{b}_2}^2}
{m_{\tilde{b}_1}^2 - m_{\tilde{b}_2}^2}~.
\end{align}
Only two of the three counterterms, $\dem_{b}$, $\delta\theta_{\tilde{b}}$, $\deA_b$ can
be treated as independent, which offers a variety of choices.
In the following,
four different renormalisation schemes, see \refta{tab:sbotren},
will be investigated.
Two of them are on-shell schemes
in the sense that the Higgs self-energies do not depend on the
renormalisation scale $\mu^{\overline{\rm{DR}}}$.
\begin{table}[!htb]
\renewcommand{\arraystretch}{1.7}
\begin{tabularx}{16.5cm}{|c||X|X|X|X|}
\hline
scheme & $m_{\tilde{b}_2}^2$ & $m_{b}$ & $A_b$ & $\theta_{\tilde{b}}$ \\ \hline\hline
analogous to $t/\tilde{t}$ sector (``$m_{b}$ OS'') & on-shell &
on-shell & & on-shell\\\hline
$\overline{\rm{DR}}$\ bottom-quark mass (``$m_{b}$ $\overline{\rm{DR}}$'') & on-shell &
$\overline{\rm{DR}}$ & $\overline{\rm{DR}}$ & \\\hline
$\overline{\rm{DR}}$\ mixing angle and $A_b$ (``$A_b$, $\theta_{\tilde{b}}$ $\overline{\rm{DR}}$'') & on-shell & &
$\overline{\rm{DR}}$ & $\overline{\rm{DR}}$ \\\hline
on-shell mixing angle and $A_b$ (``$A_b$, $\theta_{\tilde{b}}$ OS'') & on-shell & &
on-shell & on-shell \\\hline
\end{tabularx}
\caption {\small Summary of the four renormalisation schemes for the bottom
quark/squark sector investigated below. Blank entries indicate
dependent quantities.}
\label{tab:sbotren}
\end{table}
The schemes are described in the following subsections, prior to the
discussion of their quantitative numerical features in~\refse{sec:numres}.
\subsubsection{Analogous to the top quark/squark sector}
A straight-forward
possibility is to impose renormalisation conditions in analogy to those
of the top quark/squark sector in \refse{subsec:stoprenorm}.
\begin{itemize}
\item[(i)]
On-shell renormalisation of the bottom quark mass $m_{b}$
determines the corresponding counterterm as follows,
\begin{align}
\delta m_{b} = \frac{1}{2} m_{b} \bigl[\mathop{\rm Re} {\Sigma}_{{b}_L} (m_{b}^2) +
\mathop{\rm Re} {\Sigma}_{{b}_R} (m_{b}^2) + 2\mathop{\rm Re} {\Sigma}_{{b}_S} (m_{b}^2) \bigr] \, .
\label{eq:dembos}
\end{align}
\item[(ii)]
The counterterm for the sbottom mixing angle
$\theta_{\tilde{b}}$ is determined in the following way,
\begin{align} \label{dthetab}
\delta \theta_{\tilde{b}} =
\frac{\mathop{\rm Re} \Sigma_{\tilde{b}_{12}}(m_{\tilde{b}_1}^2)
+\mathop{\rm Re} \Sigma_{\tilde{b}_{12}}(m_{\tilde{b}_2}^2)}
{2(m_{\tilde{b}_1}^2 - m_{\tilde{b}_2}^2)}\; .
\end{align}
\end{itemize}
The dependent counterterm $\delta m_{\tilde{b}_1}^2$ for the $\tilde{b}_1$ mass
is then fully specified by~\refeq{ms1CT}. Moreover,
$A_b$ is treated here as a dependent quantity;
the corresponding counterterm
$\deA_b$ follows from the relation \refeq{deltaSbot},
yielding in combination with (\ref{ms1CT}) the expression
\begin{align} \nonumber
\delta A_b &= \frac{1}{m_{b}}\Bigl[
-\tan \theta_{\tilde{b}} \delta m_{\tilde{b}_2}^2 + (m_{\tilde{b}_1}^2-m_{\tilde{b}_2}^2) \delta \theta_{\tilde{b}}
- \delta m_{b} \Bigl( \frac{1}{2 m_{b}}(m_{\tilde{b}_1}^2-m_{\tilde{b}_2}^2) \sin 2 \theta_{\tilde{b}}
- 2 \tan \theta_{\tilde{b}} m_{b} \Bigr) \\[1.5mm]
\label{Abparameter}
& \quad\ \quad\
+ \tan \theta_{\tilde{b}} \Bigl(\cos^2 \theta_{\tilde{t}} \delta m_{\tilde{t}_1}^2
+ \sin^2 \theta_{\tilde{t}} \delta m_{\tilde{t}_2}^2
- \sin 2 \theta_{\tilde{t}}
(m_{\tilde{t}_1}^2-m_{\tilde{t}_2}^2) \delta \theta_{\tilde{t}}
-2 m_{t} \delta m_{t}\Bigr) \Bigr]~.
\end{align}
While formally the renormalisation described in this section is the same as
in the top/stop sector, there are nevertheless important differences.
The top-quark pole mass can be directly extracted from experiment and,
due to its large numerical value as compared to other quark masses
and the fact that
the present experimental error is much larger than the QCD scale,
it can be used as input for theory predictions in a well-defined way.
For the mass of the bottom quark, on the other hand, problems related to
non-perturbative effects are much more severe.
Therefore the parameter extracted from the comparison of theory and
experiment~\cite{pdg}
is not the bottom pole mass. Usually the value of the bottom mass
is given in the $\overline{\rm{MS}}$\ renormalisation scheme,
with the renormalisation scale $\mu^{\overline{\rm{MS}}}$ chosen as the
bottom-quark mass, i.e.\ $m_{b}^{\overline{\rm{MS}}}(m_{b}^{\overline{\rm{MS}}})$~\cite{pdg}.
Another important difference to the top/stop sector is the replacement
of $\cot\beta \rightarrow \tan\beta$. As will be discussed in more detail below,
very large effects can occur in this scheme for large values of $\mu$
and $\tan\beta$.
\subsubsection{\boldmath{$\overline{\rm{DR}}$} bottom-quark mass}
Potential problems with the bottom pole mass can be avoided by adopting
a renormalisation scheme with a running bottom-quark mass. In the
context of the MSSM it seems appropriate to use the $\overline{\rm{DR}}$\
scheme~\cite{dred} and to include the SUSY contributions at
$\mathcal O(\alpha_s)$ into the running. We therefore choose a scheme where
$m_{b}$ and $A_b$ are both renormalised in the $\overline{\rm{DR}}$\ scheme.
The following renormalisation conditions are imposed
for the independent quantities.
\begin{itemize}
\item[(i)]
The $b$-quark mass is defined in the $\overline{\rm{DR}}$\ scheme, which determines
the mass counterterm by the expression
\begin{align}
\delta m_{b} = \frac{1}{2} m_{b} \bigl[
\mathop{\rm Re} {\Sigma}_{{b}_L}^{\rm div} (m_{b}^2) +
\mathop{\rm Re} {\Sigma}_{{b}_R}^{\rm div} (m_{b}^2) +
2\mathop{\rm Re} {\Sigma}_{{b}_S}^{\rm div} (m_{b}^2)
\bigr]\; ,
\label{eq:dembdrbar}
\end{align}
where ${\Sigma}^{\rm div}$ means replacing the one- and two-point integrals
$A$ and $B_0$ in the quark self-energies
by their divergent parts in the following way,
\begin{align}
\nonumber
A(m)|_{\rm div} &= m^2 \Delta \; , \\
\label{Ersetzung}
B_0(p^2, m_1, m_2)|_{\rm div} &= \Delta\;,
\end{align}
with $\Delta = 2/\epsilon-\gamma+\log 4\pi$, and $D = 4 - \epsilon$.
\item[(ii)]
Besides $m_{b}$, also the trilinear coupling $A_b$ is
defined within the $\overline{\rm{DR}}$\ scheme. Using \refeq{Abparameter} and
inserting the self-energies yields the counterterm
\begin{align} \nonumber
\delta A_b &=
\frac{1}{m_{b}}\Bigl[
-\tan \theta_{\tilde{b}}
\mathop{\rm Re} \Sigma_{\tilde{b}_{22}}^{\rm div}(m_{\tilde{b}_2}^2)
+ \frac{1}{2}
(\mathop{\rm Re} \Sigma_{\tilde{b}_{12}}^{\rm div}(m_{\tilde{b}_1}^2)+
\mathop{\rm Re} \Sigma_{\tilde{b}_{12}}^{\rm div}(m_{\tilde{b}_2}^2))\\[1.5mm]
& \nonumber \quad\ \quad\
+ \tan \theta_{\tilde{b}} \Bigl(\cos^2 \theta_{\tilde{t}}
\mathop{\rm Re} \Sigma_{\tilde{t}_{11}}^{\rm div}(m_{\tilde{t}_1}^2)
+ \sin^2 \theta_{\tilde{t}}
\mathop{\rm Re} \Sigma_{\tilde{t}_{22}}^{\rm div}(m_{\tilde{t}_2}^2) \\[1.5mm]
& \nonumber \quad\ \quad\
- \frac{1}{2} \sin 2 \theta_{\tilde{t}}
(\mathop{\rm Re} \Sigma_{\tilde{t}_{12}}^{\rm div}(m_{\tilde{t}_1}^2)+
\mathop{\rm Re} \Sigma_{\tilde{t}_{12}}^{\rm div}(m_{\tilde{t}_2}^2)) \Bigr)
\\[1.5mm]
& \nonumber \quad\ \quad\
- m_{t}^2
\Bigl(\mathop{\rm Re} {\Sigma}_{{t}_L}^{\rm div}(m_{t}^2) +
\mathop{\rm Re} {\Sigma}_{{t}_R}^{\rm div}(m_{t}^2) +
2\mathop{\rm Re} {\Sigma}_{{t}_S}^{\rm div}(m_{t}^2)\Bigr)
\Bigr] \\[1.5mm]
& \nonumber \quad\
+ \frac{1}{2} \bigl(2 \tan \theta_{\tilde{b}} m_{b} - \frac{1}{2 m_{b}}(m_{\tilde{b}_1}^2-m_{\tilde{b}_2}^2) \sin 2
\theta_{\tilde{b}} \bigr)\\[1.5mm]
& \quad \quad
\Bigl
(\mathop{\rm Re} {\Sigma}_{{b}_L}^{\rm div} (m_{b}^2)
+ \mathop{\rm Re} {\Sigma}_{{b}_R}^{\rm div} (m_{b}^2)
+ 2\mathop{\rm Re} {\Sigma}_{{b}_S}^{\rm div} (m_{b}^2) \Bigr)~.
\label{AbcountertermDR}
\end{align}
\end{itemize}
The counterterms for the mixing angle, $\delta \theta_{\tilde{b}}$, and
the $\tilde{b}_1$ mass, $\delta m_{\tilde{b}_1}^2$,
are dependent quantities and can be determined
as combinations of the independent counterterms,
invoking (\ref{ms1CT})
and~(\ref{deltaSbot}),
\begin{align}\nonumber
\delta \theta_{\tilde{b}} &= \frac{1}{m_{\tilde{b}_1}^2 -m_{\tilde{b}_2}^2} \Bigl[
m_{b} \delta A_b
+\tan \theta_{\tilde{b}} \delta m_{\tilde{b}_2}^2
+ \dem_{b}
\Bigl(\frac{1}{2 m_{b}}(m_{\tilde{b}_1}^2 -m_{\tilde{b}_2}^2) \sin 2 \theta_{\tilde{b}} - 2 \tan \theta_{\tilde{b}}
m_{b}\Bigl)
\\[1.5mm] & \quad\ \label{thetabinAbMBdrbar}
- \tan \theta_{\tilde{b}} \Bigl(\cos^2 \theta_{\tilde{t}} \dem_{\tilde{t}_1}^2
+ \sin^2 \theta_{\tilde{t}} \dem_{\tilde{t}_2}^2
- \sin 2 \theta_{\tilde{t}} (m_{\tilde{t}_1}^2 -m_{\tilde{t}_2}^2) \delta \theta_{\tilde{t}}
- 2 m_{t} \delta m_{t} \Bigr) \Bigr] ~,
\\[2mm]
\delta m_{\tilde{b}_1}^2 &= \tan^2 \theta_{\tilde{b}} \delta m_{\tilde{b}_2}^2
+ 2 \tan \theta_{\tilde{b}} m_{b}\deA_b
+ 2 \Bigl( \frac{1}{m_{b}} \sin^2 \theta_{\tilde{b}} (m_{\tilde{b}_1}^2-m_{\tilde{b}_2}^2)
+ (1 - \tan^2 \theta_{\tilde{b}}) m_{b} \Bigr) \delta m_{b} \nonumber
\\[1mm]
& \quad \label{msb1inAbMBdrbar}
+(1 - \tan^2 \theta_{\tilde{b}})
\Bigl(\cos^2 \theta_{\tilde{t}} \delta m_{\tilde{t}_1}^2
+ \sin^2 \theta_{\tilde{t}} \delta m_{\tilde{t}_2}^2
- \sin 2 \theta_{\tilde{t}} (m_{\tilde{t}_1}^2-m_{\tilde{t}_2}^2) \delta \theta_{\tilde{t}} - 2 m_{t} \delta m_{t} \Bigr)~.
\end{align}
The renormalised quantities in this scheme depend on the $\overline{\rm{DR}}$\
renormalisation scale $\mu^{\overline{\rm{DR}}}$.
If not stated differently, in all numerical results given in this paper the
$\overline{\rm{DR}}$\ scale refers to the top-quark mass, i.e.\
$\mu^{\overline{\rm{DR}}} = m_t$.
\bigskip
In order to determine the value of
$m_{b}^{\overline{\rm{DR}}, \text{MSSM}}(\mu^{\overline{\rm{DR}}})$ from the
value $m_{b}^{\overline{\rm{MS}}}(\mu^{\overline{\rm{MS}}})$ that is extracted from the experimental
data one has to note that by definition $m_{b}^{\overline{\rm{DR}}, \text{MSSM}}$ contains
all MSSM contributions at \order{\alpha_s}, while $m_{b}^{\overline{\rm{MS}}}$ contains
only the \order{\alpha_s} SM correction, i.e.\ the gluon-exchange
contribution. Furthermore, a finite shift arises from the transition
between the $\overline{\rm{MS}}$\ and the $\overline{\rm{DR}}$\ scheme. As input value for
$m_{b}^{\overline{\rm{MS}}}(M_Z)$ we use in this paper $m_{b}^{\overline{\rm{MS}}}(M_Z) =
2.94$~GeV\cite{mbmsbarmz}.
The expression for $m_{b}^{\overline{\rm{DR}}, \text{MSSM}}(\mu^{\overline{\rm{DR}}})$
is most easily derived
by formally relating $m_{b}^{\overline{\rm{DR}}, \text{MSSM}}$ to the bottom pole mass
first and
then expressing the bottom pole mass in terms of the $\overline{\rm{MS}}$\ mass (the
large non-perturbative contributions affecting the bottom pole mass drop
out in the relation of $m_{b}^{\overline{\rm{DR}}, \text{MSSM}}$ to $m_{b}^{\overline{\rm{MS}}}$).
Using the equality $m_{b}^{\rm OS} + \delta m_{b}^{\rm OS} = m_{b}^{\overline{\rm{DR}},
\text{MSSM}} + \delta m_{b}^{\overline{\rm{DR}}, \text{MSSM}}$ and the expressions for
the on-shell counterterm and
the $\overline{\rm{DR}}$\ counterterm given in \refeq{eq:dembos} and \refeq{eq:dembdrbar},
respectively, one finds
\begin{equation}
\label{eq:mbdrbar1}
m_{b}^{\overline{\rm{DR}}, \text{MSSM}}(\mu^{\overline{\rm{DR}}}) = m_{b}^{\text{OS}} + \frac{1}{2} m_{b}
\bigl(\Sigma^{\rm fin}_{b_L}({m_{b}}^2) + \Sigma^{\rm fin}_{{b}_R} ({m_{b}}^2) \bigr)
+ m_{b}\, \Sigma^{\rm fin}_{b_S}(m_{b}^2)~.
\end{equation}
Here the $\Sigma^{\rm fin}$ are the UV-finite parts of the self-energy
coefficients in \refeq{eq:dembos}.
They depend on the $\overline{\rm{DR}}$\ scale
$\mu^{\overline{\rm{DR}}}$ and are evaluated for
on-shell momenta, $p^2 = m_{b}^2$.
Inserting $m_{b}^{\text{OS}} =
m_{b}^{\overline{\rm{MS}}}(M_Z) b^{\text{shift}}$, where
\begin{align}\label{dregdredbmassshift}
b^{\text{shift}} \equiv \Bigl[1 +
\frac{\alpha_s}{\pi} \Bigl(\frac{4}{3} - \ln
\frac{(m_{b}^{\overline{\rm{MS}}})^2}{M_Z^2} \Bigr)\Bigr]~,
\end{align}
one finds the desired expression for $m_{b}^{\overline{\rm{DR}}}$,
\begin{align}
\label{eq:mbdrbar2}
m_{b}^{\overline{\rm{DR}}, \text{MSSM}}(\mu^{\overline{\rm{DR}}}) &= m_{b}^{\overline{\rm{MS}}}(M_Z)
b^{\text{shift}} + \frac{1}{2} m_{b}
\Bigl(\Sigma^{\rm fin}_{b_L}({m_{b}}^2) + \Sigma^{\rm fin}_{{b}_R} ({m_{b}}^2)
\Bigr)
+ m_{b}\, \Sigma^{\rm fin}_{b_S}(m_{b}^2)~.
\end{align}
\subsubsection{\boldmath{$\overline{\rm{DR}}$} mixing angle and \boldmath{$A_b$}}
A further possibility is to impose renormalisation conditions for the
mixing angle $\theta_{\tilde{b}}$ and for $A_b$, and to treat the
counterterm of the $b$-quark mass as a dependent quantity determined
as a combination of the other counterterms using the relation
\refeq{deltaSbot}.
The renormalisation conditions in this case read explicitly:
\begin{itemize}
\item[(i)]
$\deA_b$ is determined in the $\overline{\rm{DR}}$\ scheme as in the previous case
by the expression~\refeq{AbcountertermDR}.
\item[(ii)]
The mixing angle $\theta_{\tilde{b}}$, defined in the $\overline{\rm{DR}}$\ scheme,
is renormalised by the counterterm
\begin{align}\label{thetadrbar}
\delta \theta_{\tilde{b}} =
\frac{\mathop{\rm Re} \Sigma_{\tilde{b}_{12}}^{\rm div}(m_{\tilde{b}_1}^2)+
\mathop{\rm Re} \Sigma_{\tilde{b}_{12}}^{\rm div}(m_{\tilde{b}_2}^2)}
{2(m_{\tilde{b}_1}^2-m_{\tilde{b}_2}^2)} \; .
\end{align}
\end{itemize}
The counterterm for the $b$-quark mass, $\delta m_{b}$, can be obtained
using \refeq{deltaSbot}
and the constraint~\refeq{ms1CT}.
It is given by the following
quantity (which is well-behaved for $\theta_{\tilde{b}} \to 0$),
\begin{align}
\delta m_{b} &= \Bigl[\tan \theta_{\tilde{b}} \Bigl(- \delta m_{\tilde{b}_2}^2
+ \cos^2 \theta_{\tilde{t}} \delta m_{\tilde{t}_1}^2 + \sin^2 \theta_{\tilde{t}} \delta m_{\tilde{t}_2}^2 - \sin 2 \theta_{\tilde{t}}
(m_{\tilde{t}_1}^2 -m_{\tilde{t}_2}^2)\delta \theta_{\tilde{t}} - 2 m_{t}\, \delta m_{t}\Bigr)
\nonumber \\[1.5mm]\label{deltambabh}
& \quad\ - m_{b}\, \delta A_b +
(m_{\tilde{b}_1}^2 - m_{\tilde{b}_2}^2) \delta \theta_{\tilde{b}}
\Bigr]
\Bigl[\frac{m_{\tilde{b}_1}^2 -m_{\tilde{b}_2}^2}{ 2 m_{b}} \sin 2 \theta_{\tilde{b}}
- 2 m_{b} \tan \theta_{\tilde{b}} \Bigr]^{-1} ~.
\end{align}
The numerical value of $m_{b}$ in this scheme is obtained from
\refeq{eq:mbdrbar2} and the (finite) difference of the counterterms
given in \refeq{deltambabh} and \refeq{eq:dembdrbar}.
Finally, \eqref{ms1CT} yields also
the counterterm for the dependent squark mass, $\delta m_{\tilde{b}_1}^2$,
with the specification~\refeq{deltambabh}
for the $b$-mass counterterm.
\subsubsection{On-shell mixing angle and \boldmath{$A_b$} }
\label{subsubsec:AbtsbOS}
In \citere{mhiggsEP4} a renormalisation condition was imposed on the
$A\tilde{b}_1\tilde{b}_2$ vertex in order to avoid an explicit dependence on the
renormalisation scale $\mu^{\overline{\rm{DR}}}$. For the purpose of comparing our
results with those of \citere{mhiggsEP4} we include such a renormalisation
scheme in our discussion. While in \citere{mhiggsEP4}
the limit $\tan \beta \to \infty$ has been used to
derive all the renormalisation conditions and counterterms, we have
derived the relevant quantities for arbitrary values of $\tan \beta$.
We call this scheme ``on-shell'' (as in \citere{mhiggsEP4}), although
the vertex is taken at an off-shell value of the $A$-boson momentum.
Similarly to the previous scheme,
the counterterm for the $b$-quark mass is derived as
a linear combination of other counterterms by means
of~\refeq{deltaSbot}.
The independent renormalisation conditions can be formulated as follows.
\begin{itemize}
\item[(i)]
The counterterm for the mixing angle $\theta_{\tilde{b}}$ is defined by
\begin{align}\label{mixangleUIF}
\delta \theta_{\tilde{b}} =
\frac{\mathop{\rm Re} \Sigma_{\tilde{b}_{12}}(m_{\tilde{b}_1}^2)+
\mathop{\rm Re} \Sigma_{\tilde{b}_{12}}(m_{\tilde{b}_2}^2)}{2(m_{\tilde{b}_1}^2-m_{\tilde{b}_2}^2)}\; ,
\end{align}
as in the scheme ``analogous to the top quark/squark sector''.
\item[(ii)]
$A_b$ is determinded by imposing the condition
\begin{align}\label{lambdahutbed}
\hat{\Lambda}(0, m_{\tilde{b}_1}^2, m_{\tilde{b}_1}^2) + \hat{\Lambda}(0, m_{\tilde{b}_2}^2, m_{\tilde{b}_2}^2) = 0\; ,
\end{align}
with $\hat{\Lambda}(p_A^2, p_{\tilde{b}_1}^2, p_{\tilde{b}_2}^2)$ as the
renormalised three-point $A\tilde{b}_1\tilde{b}_2$
vertex function, \\
\begin{center}
\vspace{-3em}
\setlength{\unitlength}{1pt}
\begin{picture}(260, 180)
\DashArrowLine(140,090)(220,130){5}
\DashArrowLine(220,050)(140,090){5}
\DashLine(50,90)(140,90){5}
\put(35,85){$A$}
\put(225,45){$\tilde{b}_2$}
\put(225,130){$\tilde{b}_1$}
\put(240,85){$\Hat{=} \; \hat{\Lambda}(p_A^2, p_{\tilde{b}_1}^2, p_{\tilde{b}_2}^2)$~,}\,
\GCirc(140,90){10}{.5}
\end{picture}
\vspace{-3em}
\end{center}
\begin{align}\nonumber
\hat{\Lambda}(p_A^2, p_{\tilde{b}_1}^2, p_{\tilde{b}_2}^2) &= \Lambda (p_A^2,
p_{\tilde{b}_1}^2, p_{\tilde{b}_2}^2) + \frac{i e}{2 M_W \sin \theta_W}\Bigl[
m_{b} \tan \beta\, \deA_b \\ & \quad + (\mu + \tan \beta A_b)\Bigl( \delta m_{b} +
\frac{1}{2} m_{b} (\delta
Z_{\tilde{b}_1 \tilde{b}_1} + \delta Z_{\tilde{b}_2 \tilde{b}_2})\Bigr)\Bigr] .
\label{lambdahut}
\end{align}
In the large-$\tan \beta$ limit,
this requirement reproduces the
condition applied in \citere{mhiggsEP4}.
\noindent
Condition (ii) can be formulated as an equation determining the counterterm for
$A_b$ in the following way,
\begin{align}
\nonumber
\delta A_b &=
i \frac{M_W \sin \theta_W}{e\, m_{b} \tan \beta}
\Bigl( \Lambda(0, m_{\tilde{b}_1}^2, m_{\tilde{b}_1}^2) + \Lambda(0, m_{\tilde{b}_2}^2, m_{\tilde{b}_2}^2) \Bigr)
- \frac{\mu + A_b \tan \beta}{2 \tan \beta}
(\delta Z_{\tilde{b}_1 \tilde{b}_1} +
\delta Z_{\tilde{b}_2 \tilde{b}_2}) \\
& \quad \nonumber
+ \frac{\mu + A_b \tan \beta}{m_{b} \tan \beta}\Bigg[
-i \frac{M_W \sin \theta_W}{e \tan \beta}
\Bigl(\Lambda(0, m_{\tilde{b}_1}^2,m_{\tilde{b}_1}^2) + \Lambda(0, m_{\tilde{b}_2}^2, m_{\tilde{b}_2}^2)\Bigr) \\[1.5mm]
& \qquad \nonumber
+ \frac{m_{b}(\mu + A_b \tan \beta)}{2 \tan \beta}
(\delta Z_{\tilde{b}_1 \tilde{b}_1} + \delta Z_{\tilde{b}_2 \tilde{b}_2})
+ (m_{\tilde{b}_1}^2 - m_{\tilde{b}_2}^2) \delta \theta_{\tilde{b}} \\[1.5mm]
& \qquad \nonumber
- \tan \theta_{\tilde{b}} \Bigl( \delta m_{\tilde{b}_2}^2 -\cos^2 \theta_{\tilde{t}} \delta m_{\tilde{t}_1}^2
- \sin^2 \theta_{\tilde{t}} \delta m_{\tilde{t}_2}^2 + \sin 2 \theta_{\tilde{t}} (m_{\tilde{t}_1}^2 - m_{\tilde{t}_2}^2)\delta \theta_{\tilde{t}}
\\[1.5mm]& \qquad
+ 2 m_{t} \delta m_{t}
\Bigr)
\Bigg]
\left[ \mu \bigl(\tan \beta +
\frac{1}{\tan \beta}\bigr)
+2 m_{b} \tan \theta_{\tilde{b}} \right]^{-1} ,
\label{dAbUIFartig}
\end{align}
where the $Z$~factors are defined as
\begin{align}
\delta Z_{\tilde{b}_i \tilde{b}_i} &=
-\frac{\Sigma_{\tilde{b}_{ii}}(m_{\tilde{b}_1}^2) -
\Sigma_{\tilde{b}_{ii}}(m_{\tilde{b}_2}^2)}
{m_{\tilde{b}_1}^2 -m_{\tilde{b}_2}^2} \;.
\end{align}
\end{itemize}
Again,
the dependent counterterm for the $b$-quark mass is determined by
\refeq{deltaSbot} and
the constraint~\refeq{ms1CT}, but now inserting the above
specification~\refeq{dAbUIFartig} for $\delta A_b$, yielding
\begin{align} \nonumber
\delta m_{b} &= - \Bigl[
- i \frac{M_W \sin \theta_W}{e\,\tan \beta}
\Bigl(\Lambda(0, m_{\tilde{b}_1}^2, m_{\tilde{b}_1}^2) + \Lambda(0, m_{\tilde{b}_2}^2, m_{\tilde{b}_2}^2) \Bigr)
+ \frac{m_{b}(\mu + A_b \tan \beta)}{2 \tan \beta}
(\delta Z_{\tilde{b}_1 \tilde{b}_1} + \delta Z_{\tilde{b}_2 \tilde{b}_2})
\\[1.5mm]
& \quad\ \nonumber
+ (m_{\tilde{b}_1}^2 - m_{\tilde{b}_2}^2) \delta \theta_{\tilde{b}}
+ \tan \theta_{\tilde{b}}
\Bigl(
- \sin 2 \theta_{\tilde{t}} (m_{\tilde{t}_1}^2 -m_{\tilde{t}_2}^2)\delta \theta_{\tilde{t}}
- \delta m_{\tilde{b}_2}^2
+ \cos^2 \theta_{\tilde{t}} \delta m_{\tilde{t}_1}^2
\\[1.5mm]
& \quad\
+ \sin^2 \theta_{\tilde{t}} \delta m_{\tilde{t}_2}^2
- 2 m_{t} \delta m_{t}
\Bigr)
\Bigr]
\Bigl[
\mu\bigl(\tan \beta + \frac{1}{\tan \beta}\bigr)
+ 2 m_{b} \tan \theta_{\tilde{b}} \Bigr]^{-1} ~.
\label{dmbmitthetauAbonshell}
\end{align}
The numerical value of $m_{b}$ in this scheme is obtained from
\refeq{eq:mbdrbar2} and the (finite) difference of
the counterterms
given in \refeq{dmbmitthetauAbonshell} and \refeq{eq:dembdrbar}.
With the specification of $\delta m_{b}$ in \refeq{dmbmitthetauAbonshell},
also the $\tilde{b}_1$-mass counterterm $\delta m_{\tilde{b}_1}^2$ in the general
relation~\refeq{ms1CT} is fully determined.
\subsection{Resummation in the \boldmath{$b/\tilde{b}$} sector}
\label{subsec:botresum}
The relation between the bottom-quark mass and the Yukawa coupling $h_b$,
which in lowest order reads $m_{b} = h_b v_1/\sqrt{2}$, receives radiative
corrections proportional to $h_b v_2 = h_b \tan \beta \, v_1$. Thus, large
$\tan \beta$-enhanced contributions can occur, which need to be properly taken
into account. As shown in \citeres{deltamb1,deltamb} the leading terms of
\order{\alpha_b(\alpha_s\tan \beta)^n} can be resummed by using an appropriate
effective bottom Yukawa coupling.
Accordingly, an effective bottom-quark mass is obtained by extracting
the UV-finite $\tan \beta$-enhanced term $\Delta m_{b}$ from
\refeq{eq:mbdrbar2} (which enters through $\Sigma_{b_S}$) and writing it
as $1/(1 + \Delta m_{b})$ into the
denominator. In this way the leading powers of $(\alpha_s\tan \beta)^n$ are
correctly resummed~\cite{deltamb1,deltamb}. This yields
\begin{equation}
\label{eq:mbdrbarresum}
m_{b}^{\overline{\rm{DR}}, \text{MSSM}}(\mu^{\overline{\rm{DR}}}) =
\frac{m_{b}^{\overline{\rm{MS}}}(M_Z) b^{\text{shift}} + \frac{1}{2} m_{b}
\Bigl(\Sigma^{\rm fin}_{b_L}({m_{b}}^2) + \Sigma^{\rm fin}_{{b}_R} ({m_{b}}^2)
\Bigr)
+ m_{b}\, \widetilde{\Sigma}^{\rm fin}_{b_S}(m_{b}^2)}{1 + \Delta m_{b}}~,
\end{equation}
where $\widetilde{\Sigma}_{b_S} \equiv \Sigma_{b_S} + \Delta m_{b}$ denotes the
non-enhanced remainder of the scalar $b$-quark self-energy at
\order{\alpha_s}, and $b^{\text{shift}}$ is given in \eqref{dregdredbmassshift}.
The $\tan \beta$-enhanced scalar part of the
$b$-quark self-energy, $\Delta m_{b}$, is given at \order{\alpha_s} by%
\footnote{
There are also corrections of \order{\alpha_t} to $\Dem_{b}$ that can be
resummed~\cite{deltamb}. These effects usually amount up to 5--10\% of
the \order{\alpha_s} corrections. Since in this paper we are interested only in the
\order{\alpha_b\alpha_s} contributions to the MSSM Higgs sector, these
corrections have been neglected. Further corrections from subleading
resummation terms can be found in \citere{deltamb3}.
}%
\begin{align}
\label{eq:deltamb}
\Delta m_{b} = \frac{2}{3 \pi} \alpha_s\tan \beta\, \mu\, m_{\tilde{g}}\,
I (m_{\tilde{b}_1}^2, m_{\tilde{b}_2}^2, m_{\tilde{g}}^2) ,
\end{align}
with
\begin{align}
I (m_{\tilde{b}_1}^2, m_{\tilde{b}_2}^2, m_{\tilde{g}}^2) =
- \frac{m_{\tilde{b}_1}^2 m_{\tilde{b}_2}^2 \log(m_{\tilde{b}_2}^2/m_{\tilde{b}_1}^2) +
m_{\tilde{b}_1}^2 m_{\tilde{g}}^2 \log(m_{\tilde{b}_1}^2/m_{\tilde{g}}^2) +
m_{\tilde{g}}^2 m_{\tilde{b}_2}^2 \log(m_{\tilde{g}}^2/m_{\tilde{b}_2}^2)}
{(m_{\tilde{b}_1}^2 - m_{\tilde{g}}^2) (m_{\tilde{g}}^2 - m_{\tilde{b}_2}^2) (m_{\tilde{b}_2}^2 - m_{\tilde{b}_1}^2)}~,
\label{eq:I}
\end{align}
and $\Delta m_{b} > 0$ for $\mu > 0$.
In the ``$m_{b}$ $\overline{\rm{DR}}$'' scheme we
use the effective bottom-quark mass as given in \refeq{eq:mbdrbarresum}
everywhere instead of the $\overline{\rm{DR}}$\ bottom quark mass
(in particular, we use this bottom mass in the
sbottom-mass matrix squared, \refeq{Sfermionmassenmatrix}, from which
the sbottom mass eigenvalues are determined).
The numerical values of the bottom-quark mass in the other
renormalisation schemes can be obtained from \refeq{eq:mbdrbarresum} as
explained above, and from \refeq{mbdef} below.
We incorporate the effective bottom-quark mass of \refeq{eq:mbdrbarresum}
(or the correspondingly shifted value in the other renormalisation
schemes) into our one-loop results for the
renormalised Higgs boson self-energies, which determine the Higgs boson masses
at one-loop order according to
\refeq{eq:proppole}--\refeq{renSE}.
In this way the leading effects
of \order{\alpha_b\alpha_s} are absorbed into the one-loop result.
We refer to the genuine two-loop contributions,
which go beyond this improved one-loop result, as ``subleading
\order{\alpha_b\alpha_s} corrections'' in the following.
\section{Numerical results}
\label{sec:numres}
\subsection{Evaluation}
If not mentioned explicitly in the text the default set of parameters shown in
\refta{tab:inputparameter} is used.
Large values of $\tan \beta$ and $|\mu|$ are chosen in order to
illustrate possibly large effects in the $b/\tilde{b}$ sector.
\begin{table}[!htb]
\renewcommand{\arraystretch}{1.7}
\begin{tabularx}{15.5cm}{|X|X|}\hline
\multicolumn{2}{|l|}{SM parameters:}\\
\multicolumn{2}{|l|}{$m_{t} = 174.3 \,\, {\rm GeV}$,
$m_{b}^{\overline{\rm{MS}}}(M_Z) = 2.94 \,\, {\rm GeV}$,}\\
\multicolumn{2}{|l|}{$M_Z = 91.1875 \,\, {\rm GeV}$,
$M_W = 80.426 \,\, {\rm GeV}$, $G_F = 1.16639 \; 10^{-5}$}\\
\hline\hline
\multicolumn{2}{|l|}{parameters of the Higgs sector:}\\
\multicolumn{2}{|c|}{$M_A = 120 \,\, {\rm GeV}$ \hspace{2.5cm} $\tan \beta = 50$
\hspace{2.5cm} $\mu = -1000 \,\, {\rm GeV}$}\\ \hline\hline
\multicolumn{2}{|l|}{soft-breaking parameters:}\\\hline
for the gauginos: & for the
sfermions:
\\
\multicolumn{1}{|c|}{$M_1 = \displaystyle{\frac{5}{3} \frac{\sin^2
\theta_W}{\cos^2 \theta_W}} M_2$} & \multicolumn{1}{c|}{$M_L =
M_{L_{\{\tilde{q}_i,\,\tilde{l}_i\}}} = 1000 \,\, {\rm GeV}$ with $i = 1,\, 2,\,3$}
\\
\multicolumn{1}{|c|}{$M_2 = 100 \,\, {\rm GeV}$} &
\multicolumn{1}{c|}{$M_{\tilde{f}_R} = 1000 \,\, {\rm GeV}$ with $f =
u,\,c,\,t,\,d,\,s,\,b,\,e,\,\mu,\,\tau$}
\\
\multicolumn{1}{|c|}{$M_3 = 1000 \,\, {\rm GeV}$} & \multicolumn{1}{c|}{$A_{\{u,\,c,\,t\}} =
A_{\{d,\,s,\,b\}} = A_{\{e,\,\mu,\,\tau\}} = 2000 \,\, {\rm GeV}$} \\\hline
\end{tabularx}
\caption {\small Set of default input parameters.}
\label{tab:inputparameter}
\end{table}
We will mostly discuss the case of negative $\mu$, since
according to \refeqs{eq:mbdrbarresum}--(\ref{eq:I}) this sign of $\mu$
leads to a negative $\Dem_{b}$ and therefore to an increase of the effective
bottom-quark mass. This gives rise to an
enhancement of the corrections from the $b/\tilde{b}$ sector, see
\reffi{fig:deltamh}. While the negative sign of $\mu$ is disfavoured
from the comparison of the MSSM prediction~\cite{g-2MSSMf1l,g-2review}
with the experimental data on
the anomalous magnetic moment of the muon~\cite{g-2exp},
it would seem premature at this
stage to completely disregard this possibility. For $\mu > 0$, on the
other hand, the corrections to the Higgs-boson masses from the $b/\tilde{b}$
sector will normally
not exceed the GeV level if the result is expressed in terms of an
appropriately chosen running bottom-quark mass (see \reffi{fig:deltamh}).
It should be noted, however, that the prospective experimental accuracy
on $M_h$ at the LHC and the ILC will be significantly below the GeV
level, so that the inclusion of non-enhanced two-loop corrections will
be necessary in order to achieve the same level of precision for the
theoretical prediction (see the discussion below).
For the calculation of the Higgs boson masses presented below
the complete one-loop\ self-energies have been used, with
$\tan \beta$ renormalised in the $\overline{\rm{DR}}$\
scheme~\cite{dissMF,renormA,renormB}
and with the $Z$~boson mass on-shell.
At the two-loop\ level, besides the \order{\alpha_b\alpha_s}
corrections also the contributions \order{\alpha_t\alpha_s} using
the one-loop sub-renormalisation
of \refse{subsec:stoprenorm} have been included.
For simplicity we have neglected the \order{\alpha_t^2} terms~\cite{mhiggsEP2}.
For the \order{\alpha_t\alpha_s} corrections the top pole mass,
$m_{t} = 174.3 \,\, {\rm GeV}$, has been used.
The inclusion of all known corrections and the new experimental top
quark mass value of $m_{t} = 178.0 \,\, {\rm GeV}$~\cite{mtopexpnew} in our analysis
would yield an
increase in $M_h$ of \order{8 \,\, {\rm GeV}}~\cite{mhiggsAEC}.
Therefore the mass values given
in our numerical analysis should not be viewed as predictions of
$M_h$; they are rather illustrations of the $\alpha_s$-corrections
to the bottom Yukawa contributions at the two-loop level.
(It should be noted that the chosen parameters are such that they are
not in conflict with the experimental lower bounds on
$M_h$~\cite{LEPHiggsSM,LEPHiggsMSSM}.)
\subsection{Comparison of the different renormalisation schemes}
\label{subsec:numanal}
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In order to compare the different renormalisation schemes, the
parameters entering the one-loop result have to be transformed according
to the different renormalisation prescriptions. As our default for which
the input parameters are fixed we have chosen the ``$m_{b}$ $\overline{\rm{DR}}$'' scheme,
where $m_{b}$ and $A_b$ are defined as $\overline{\rm{DR}}$\ parameters.
As explained in \refse{sec:ren}, the parameters are converted to a
different renormalisation scheme RS
(with counterterms \nolinebreak $\delta x^{\text{RS}}$)
with the help of the following transformations,
\begin{align}
\label{mbdef}
m_{b}^{\text{RS}} &=
m_{b}^{\overline{\rm{DR}}} -\dem_{b}^{\text{RS}}|_{\text{finite}}\, , \\
\label{Abdef}
A_b^{\text{RS}} &=
A_b^{\overline{\rm{DR}}} -\delta A_b^{\text{RS}}|_{\text{finite}}\;,
\end{align}
and analogously for the other parameters.
If not stated differently, the
$\overline{\rm{DR}}$\ scale has always been chosen as $\mu^{\overline{\rm{DR}}} = m_{t}$.
As an example,
in \refta{wertetanbeta30} and \refta{wertetanbeta50} numerical values
for the bottom quark mass, $A_b$
and the sbottom masses in the different schemes
(see \refta{tab:sbotren}), are
shown for $\tan \beta = 30$ and $\tan \beta = 50$ and using the default
values given in \refta{tab:inputparameter} otherwise.
\begin{table}[htb!]
\renewcommand{\arraystretch}{1.4}
\begin{center}
\begin{tabular}{|c||r|r|r|}
\hline
scheme & $m_{b}$ [GeV] & $A_b$ [GeV] & $m_{\tilde{b}_1}$ [GeV] \\\hline\hline
$m_{b}$ $\overline{\rm{DR}}$ & 3.79 & 2000.00 & 1059.95 \\ \hline
$A_b$, $\theta_{\tilde{b}}$ $\overline{\rm{DR}}$ & 3.04 & 2000.00 & 1039.50 \\ \hline
$A_b$, $\theta_{\tilde{b}}$ OS & 2.99 & 2332.81 & 1039.04 \\ \hline
$m_{b}$ OS & 3.77 & -4284.56 & 1039.25 \\ \hline
\end{tabular}
\end{center}
\caption{Values of the bottom quark mass, $A_b$ and $m_{\tilde{b}_1}$
in the different schemes for $\tan \beta = 30$ and $\mu = -1000 \,\, {\rm GeV}$. The
value of $m_{\tilde{b}_2}$, which is
renormalised on-shell (see \refeq{eq:msbz}), is the same in all four
schemes, $m_{\tilde{b}_2} = 938.44$~GeV.
\label{wertetanbeta30}}
\end{table}
\begin{table}[htb!]
\renewcommand{\arraystretch}{1.4}
\begin{center}
\begin{tabular}{|c||r|r|r|}
\hline
scheme & $m_{b}$ [GeV] & $A_b$ [GeV]& $m_{\tilde{b}_1}$ [GeV] \\\hline\hline
$m_{b}$ $\overline{\rm{DR}}$ & 5.82 & 2000.00 & 1142.16 \\\hline
$A_b$, $\theta_{\tilde{b}}$ $\overline{\rm{DR}}$ & 5.26 & 2000.00 & 1117.93 \\\hline
$A_b$, $\theta_{\tilde{b}}$ OS & 5.24 & 2219.40 & 1118.02 \\\hline
$m_{b}$ OS & 4.93 & 6508.12 & 1122.04 \\\hline
\end{tabular}
\end{center}
\caption{Values of the bottom quark mass, $A_b$ and $m_{\tilde{b}_1}$
in the different schemes for $\tan \beta = 50$ and $\mu = -1000 \,\, {\rm GeV}$. The
value of $m_{\tilde{b}_2}$
is the same in all four schemes, $m_{\tilde{b}_2} = 836.48$~GeV.
\label{wertetanbeta50}}
\end{table}
The values given in \refta{wertetanbeta30} and \refta{wertetanbeta50}
indicate that the ``$m_{b}$ OS'' scheme leads to huge corrections in $A_b$
that invalidate the applicability of this scheme. The other schemes give
rise to only moderate shifts in the parameters.
The reason for the problematic behaviour of the ``$m_{b}$ OS'' scheme is
easy to understand. The renormalisation condition in the ``$m_{b}$ OS''
scheme is a condition on the sbottom mixing angle $\theta_{\tilde{b}}$ and thus on the
combination $(A_b - \mu \tan \beta)$ (see
\refeq{deltaSbot}). In parameter regions where
$\mu \tan \beta$ is much larger than $A_b$, the counterterm $\deA_b$ receives
a very large finite shift when calculated from the counterterm
$\delta\theta_{\tilde{b}}$.
More specifically, $\deA_b$ as given in \refeq{Abparameter} contains
the contribution
\begin{eqnarray}
\label{largedAb}
\deA_b &=& \frac{1}{m_{b}} \left[- \frac{\dem_{b}}{2\,m_{b}} \,
(m_{\tilde{b}_1}^2 - m_{\tilde{b}_2}^2) \sin 2\theta_{\tilde{b}} + \ldots \right] \nonumber \\
&=& \frac{1}{m_{b}} \left[- \dem_{b} (A_b - \mu \, \tan \beta) + \ldots
\right] ,
\end{eqnarray}
that can give rise to very large corrections to $A_b$. This problem is
avoided in the other renormalisation schemes introduced in
\refta{tab:sbotren}, where the renormalisation condition is applied
directly to $A_b$, rather than deriving $\deA_b$ from the
renormalisation of the mixing angle.
\smallskip
We now turn to the numerical comparison of the different
renormalisation schemes.
As discussed above, the $\tan \beta$-enhanced terms of \order{\alpha_b\alpha_s} entering
via $\Delta m_{b}$ have been absorbed into the one-loop result.
The meaning of the various curves in the following figures
is specified as (see also \refta{tab:sbotren}):
\begin{itemize}
\item
dashes with dots (black): \order{\alpha_t\alpha_s} with $m_{b}^{\overline{\rm{DR}},\text{MSSM}}$,
results without subleading two-loop \order{\alpha_b\alpha_s}\ terms
\item
dot-dash (light blue):
``$m_{b}$ OS'' scheme for
subleading two-loop\ \order{\alpha_b\alpha_s} terms
\item
solid (red): ``$m_{b}$ $\overline{\rm{DR}}$'' scheme for
subleading two-loop\ \order{\alpha_b\alpha_s} terms
\item
dotted (dark blue): ``$A_b$, $\theta_{\tilde{b}}$ $\overline{\rm{DR}}$'' scheme for
subleading two-loop\ \order{\alpha_b\alpha_s} terms
\item
dashes with stars (green): ``$A_b$, $\theta_{\tilde{b}}$ OS'' scheme for
subleading two-loop\ \order{\alpha_b\alpha_s} terms
\end{itemize}
\begin{figure}[htb!]
\begin{center}
\epsfig{figure=plots/plot1a1neuer.eps, width=14cm,height=10cm}\\[1.5cm]
\epsfig{figure=plots/plot1a2neuer.eps, width=14cm,height=10cm}
\caption{
$\tan \beta$-dependence of $M_h$ and $M_H$ for $M_A = 120 \,\, {\rm GeV}$ and $\mu$ negative.
}
\label{fig:mhtb_MAsmall_MUneg}
\end{center}
\end{figure}
We start our analysis of the different renormalisation schemes by
comparing the results for $M_h$ and $M_H$ as a function of $\tan \beta$
in \reffi{fig:mhtb_MAsmall_MUneg}.
The other parameters are as given in \refta{tab:inputparameter}.
As expected from the discussion of
\refta{wertetanbeta30} and \refta{wertetanbeta50}, the
``$m_{b}$ OS'' scheme gives rise to artificially large corrections and
shows very large deviations from the other
schemes for intermediate and large values of $\tan \beta$. This
behaviour is even more pronounced for $M_H$ than for $M_h$, as
can be seen in the lower plot of \reffi{fig:mhtb_MAsmall_MUneg}.
These extremely large corrections are a consequence of the large
contributions to the counterterm of
the parameter~$A_b$ (see \refeq{largedAb}).
The Higgs self-energy contribution from virtual
sbottoms contains a term proportional to $A_b^2$.
Using as input a value for $A_b$ according to
\eqref{Abdef}, very large contributions proportional to
$(\deA_b)^2$ are introduced.
These corrections are more pronounced
in $\Sigma_{HH}$, where they enter like $(\cos \alpha\hspace{1mm} A_b)^2$, than in
$\Sigma_{hh}$, where they enter like $(\sin \alpha\hspace{1mm} A_b)^2$ ($|\alpha| \ll 1$ in our
analysis). The unacceptably large contributions to $\deA_b$ in the
``$m_{b}$~OS'' scheme invalidate a perturbative treatment in this scheme.
We therefore discard this scheme in the following and focus our
discussion on the other three schemes defined in
\refta{tab:sbotren}.
The other schemes all give similar and numerically well-behaved
results, where $M_h$ starts to decrease
rapidly with $\tan \beta$ for $\tan \beta \gsim 40$. Negative mass squares
are reached at $\tan \beta \simeq 53$.
The main effect comes from the leading contributions of
\order{\alpha_b\alpha_s} that enter via the resummation of $\Dem_{b}$, see
\refeq{eq:mbdrbarresum}.
The decrease with increasing $\tan \beta$ is mainly due to the dependence of
$\Dem_{b} \sim \mu\tan \beta$ in \refeq{eq:deltamb}. The subleading
\order{\alpha_b\alpha_s} corrections, which arise from the genuine two-loop
diagrams, are of \order{1\,\, {\rm GeV}}. The differences between the three
renormalisation schemes are of similar size.
For this particular parameter choice the ``$A_b$, $\theta_{\tilde{b}}$ $\overline{\rm{DR}}$'' scheme
enhances $M_h$, whereas the other two schemes decrease~$M_h$ compared to
the case where the genuine two-loop corrections are omitted.
\begin{figure}[htb!]
\begin{center}
\epsfig{figure=plots/plot1b1ii.eps, width=14cm,height=10cm}
\caption{
$\tan \beta$-dependence of $M_h$ for $M_A = 700 \,\, {\rm GeV}$ and $\mu$ negative.
}
\label{fig:mhtb_MAlarge_MUneg}
\end{center}
\end{figure}
In \reffi{fig:mhtb_MAlarge_MUneg} we show $M_h$ as a function of $\tan \beta$
for the same parameters as in \reffi{fig:mhtb_MAsmall_MUneg}, but with
$M_A = 700 \,\, {\rm GeV}$. This results in general in larger $M_h$ values, but
the general behaviour as a function of $\tan \beta$ is the same as for
$M_A = 120 \,\, {\rm GeV}$; $M_h$ drops steeply for large $\tan \beta$ values.
In all three schemes the subleading terms increase $M_h$ by a few GeV,
depending on $\tan \beta$.
\begin{figure}[t!]
\begin{center}
\epsfig{figure=plots/plot2a1.eps, width=14cm,height=10cm}\\[1cm]
\epsfig{figure=plots/plot2b1.eps, width=14cm,height=10cm}
\caption{
$\tan \beta$-dependence of $M_h$ for $M_A = 120 \,\, {\rm GeV}$ (upper plot)
and $M_A = 700 \,\, {\rm GeV}$ (lower plot) for positive $\mu$.
}
\label{fig:mhtb_MUpos}
\end{center}
\end{figure}
As discussed above, large corrections from the $b/\tilde{b}$ sector are
only expected for negative values of $\mu$.
In \reffi{fig:mhtb_MUpos}
we show the results for $M_h$ as a function of $\tan \beta$ with positive $\mu$
and $M_A = 120$ and $700 \,\, {\rm GeV}$, respectively. The other parameters are
given in \refta{tab:inputparameter}.
The positive sign of $\mu$ results in a positive $\Dem_{b}$ and thus a
smaller numerical value of $m_{b}^{\overline{\rm{DR}},\text{MSSM}}$.
As expected%
\footnote{
See also the discussion in \citere{mhiggsEP4}, where the
opposite sign convention for $\mu$ is used.
}%
, the variation of
$M_h$ with $\tan \beta$ is much smaller than for negative $\mu$. Both, the
leading corrections, i.e.\ the $\tan \beta$ enhanced terms of
\order{\alpha_b\alpha_s}, as well as
the subleading corrections are at the level of \order{100 \,\, {\rm MeV}}.
The ``$m_{b}$ $\overline{\rm{DR}}$'' scheme does not show any visible corrections
beyond the resummed contributions.
This leads to the conclusion that for positive $\mu$ the corrections
beyond the one-loop\ level coming from the $b/\tilde{b}$~sector are
sufficiently well under control. However, in view of the fact that the
anticipated ILC accuracy on $M_h$~\cite{tesla,orangebook,acfarep} and
the parametric uncertainty of the theory prediction from the ILC
measurement of the top-quark mass~\cite{tbexcl,deltamt}
will both be about 100 MeV, ultimately the aim will be to
reduce the theoretical uncertainties from unknown higher-order
corrections to at least this level. This will require the inclusion of
all two-loop corrections (and a significant part of
corrections beyond two-loop order).
For the further analysis in this paper we focus on negative values of
$\mu$.
\begin{figure}[t!]
\begin{center}
\epsfig{figure=plots/plot3a1i.eps, width=14cm,height=10cm}\\[1cm]
\epsfig{figure=plots/plot3b1i.eps, width=14cm,height=10cm}
\caption{
$\mu$-dependence of $M_h$ for $M_A = 120 \,\, {\rm GeV}$ (upper plot)
and $M_A = 700 \,\, {\rm GeV}$ (lower plot) for $\tan \beta = 50$.
}
\label{fig:mhmu}
\end{center}
\end{figure}
The variation of $M_h$ with $\mu$ (for $\mu < 0$) for $\tan \beta = 50$
is shown in \reffi{fig:mhmu}.
As can be expected from
\refeq{eq:deltamb} the corrections at \order{\alpha_b\alpha_s} increase with
increasing~$|\mu|$. Typically the genuine two-loop contributions are
of \order{1 \,\, {\rm GeV}}.
For large $M_A$ all the
schemes lead to an increase of $M_h$,
whereas for small $M_A$ both negative and positive shifts can occur.
Differences in the $M_h$ predictions induced by the different
renormalisation schemes are below the GeV level for large $M_A$.
In \reffi{fig:mhMA} the dependence of $M_h$ on $M_A$ is shown for the
different renormalisation schemes, with the other
default parameters from \refta{tab:inputparameter}.
For $M_A \gsim 200 \,\, {\rm GeV}$ the subleading terms of
all three schemes enhance $M_h$ by \order{1 \,\, {\rm GeV}}. A decrease only
occurs for small values of $M_A$, depending on the scheme.
The differences
in the $M_h$ prediction resulting from the use of different
renormalisation schemes decrease for $M_A \gsim 200 \,\, {\rm GeV}$ to
\order{0.1 \,\, {\rm GeV}}.
\begin{figure}[t!]
\begin{center}
\epsfig{figure=plots/plot4a1.eps, width=14cm,height=10cm}
\caption{
$M_A$-dependence of $M_h$ for $\tan \beta = 50$ and $\mu$ negative.
}
\label{fig:mhMA}
\end{center}
\end{figure}
\begin{figure}[htb!]
\begin{center}
\epsfig{figure=plots/plot5b1.eps, width=14cm,height=10cm}
\caption{
$m_{\tilde{g}}$-dependence of $M_h$ for $M_A = 700 \,\, {\rm GeV}$, $\tan \beta = 50$ and $\mu$
negative.
}
\label{fig:mhmgl}
\end{center}
\end{figure}
In \reffi{fig:mhmgl} it can be seen that the behaviour of the corrections
strongly depends on the choice of $m_{\tilde{g}}$. The figure shows $M_h$ as a
function of $m_{\tilde{g}}$ for $\mu = -1000 \,\, {\rm GeV}$, $\tan \beta = 50$ and $M_A = 700 \,\, {\rm GeV}$.
For $m_{\tilde{g}} \lsim 1000 \,\, {\rm GeV}$ all
schemes lead to an increase of $M_h$ from the subleading
\order{\alpha_b\alpha_s} corrections. For $m_{\tilde{g}} \gsim 1500 \,\, {\rm GeV}$, on the other
hand, all schemes lead to a decrease, where the size of the individual
corrections also strongly varies with $m_{\tilde{g}}$. Accordingly, the relative
size of the corrections in the different schemes also
varies with $m_{\tilde{g}}$. Corrections up to about $3 \,\, {\rm GeV}$ are possible.
The differences between the three schemes
are of \order{2 \,\, {\rm GeV}} for large $m_{\tilde{g}}$.
It should be noted that the effects of the higher-order corrections to
$M_h$ do not decouple with large $m_{\tilde{g}}$. The corrections at
\order{\alpha_t\alpha_s}~\cite{mhiggslong} as well as \order{\alpha_b\alpha_s} grow
logarithmically in the renormalisation schemes that we have adopted.
The above analysis of the three schemes
``$m_{b}$ $\overline{\rm{DR}}$'', ``$A_b$, $\theta_{\tilde{b}}$ $\overline{\rm{DR}}$'', and ``$A_b$, $\theta_{\tilde{b}}$ OS'' in
various parameter regions yields numerically well-behaved and physically
meaningful results. As there is no clear preference for one of the
schemes on physical grounds, the difference between the results obtained
in the three schemes can be interpreted as an indication of the possible
size of missing higher-order corrections.
The
size of the individual corrections and also the differences
between the renormalisation schemes sensitively depend on the input
parameters. Typically we find that the genuine two-loop corrections
in the $b/\tilde{b}$~sector yield a shift in $M_h$ of \order{1 \,\, {\rm GeV}}.
\mbox{The differences between the three schemes are usually somewhat smaller.}
\subsection{Numerical analysis of the renormalisation scale dependence}
\label{subsec:numanalmudim}
While in the previous section we compared the results of different
renormalisation scheme, we now focus on the ``$m_{b}$~$\overline{\rm{DR}}$'' scheme
and investigate the effect of varying the renormalisation scale of the
\order{\alpha_b\alpha_s} result obtained in this scheme. We vary the scale
within the interval $m_{t}/2 \le \mu^{\overline{\rm{DR}}} \le 2\,m_{t}$, resulting in a shift
which is formally of \order{\alpha_b\alpha_s^2}.
The results are shown as a function of $m_{\tilde{g}}$ for $\tan \beta = 50$ in
\reffi{fig:mhmudim} for $M_A = 120 \,\, {\rm GeV}$ and $M_A = 700 \,\, {\rm GeV}$.
\begin{figure}[hb!]
\begin{center}
\epsfig{figure=plots/MhmudimMA120tb50mun03.cl.eps,
width=10cm,height=7.9cm}\\[1cm]
\epsfig{figure=plots/MhmudimMA700tb50mun03.cl.eps, width=10cm,height=7.9cm}
\caption{
$\mu^{\overline{\rm{DR}}}$ dependence of $M_h$ as a function of $m_{\tilde{g}}$ for
$M_A = 120 \,\, {\rm GeV}$ (upper plot) and $M_A = 700 \,\, {\rm GeV}$ (lower plot) for
$\mu = -1000 \,\, {\rm GeV}$, $\tan \beta = 50$. The black area
corresponds to the \order{\alpha_t\alpha_s} result including resummation,
i.e.\ the result without the subleading two-loop \order{\alpha_b\alpha_s} terms.
}
\label{fig:mhmudim}
\end{center}
\end{figure}
The $\mu^{\overline{\rm{DR}}}$ variation of the leading contribution (the \order{\alpha_t\alpha_s}
result including resummation) is shown as the
dark shaded (black) band. The results including the subleading
corrections in the ``$m_{b}$ $\overline{\rm{DR}}$'' scheme are shown as a light shaded
(red) band. It can be seen that the variation with $\mu^{\overline{\rm{DR}}}$ is
strongly reduced by the inclusion of the subleading contributions.
The variation with $\mu^{\overline{\rm{DR}}}$ within the ``$m_{b}$ $\overline{\rm{DR}}$'' scheme is tiny
for $m_{\tilde{g}} \lsim 500 \,\, {\rm GeV}$, and reaches $\pm 2 \,\, {\rm GeV}$ for large
$m_{\tilde{g}}$ values.
Thus, the $\mu^{\overline{\rm{DR}}}$ variation causes a similar shift in $M_h$ as the
comparison between the three renormalisation schemes discussed above.
We have also analysed the variation with $\mu^{\overline{\rm{DR}}}$ in the case
$\mu > 0$, which is not shown here. As for negative $\mu$, the
variation with $\mu^{\overline{\rm{DR}}}$ is of the same order as the differences
between the three renormalisation schemes, see
\reffi{fig:mhtb_MUpos}.
Therefore, for $\mu > 0$ the unknown higher-order corrections to $M_h$
from the $b/\tilde{b}$~sector can be estimated to be of \order{100 \,\, {\rm MeV}}.
\subsection{Comparison with existing calculations}
\label{subsec:UIFcomp}
Finally we compare our result with the existing calculation of the
\order{\alpha_b\alpha_s} corrections presented in \citere{mhiggsEP4}.
The renormalisation employed there consists of an on-shell
renormalisation of the two scalar bottom masses and the on-shell
condition for $A_b$ shown in \refse{subsubsec:AbtsbOS}.
We denote it as ``$m_{\tilde{b}},\;A_b$~OS'' renormalisation.
Thus, the differences between our ``$A_b,\;\theta_{\tilde{b}}$~OS'' and
the ``$m_{\tilde{b}},\;A_b$~OS'' renormalisation are the different treatment of the
$m_{\tilde{b}_1}$ renormalisation,
as well as the treatment of $\tan \beta$. We kept
$\tan \beta$ as a free parameter, whereas in \citere{mhiggsEP4} it was set to
infinity in the subleading \order{\alpha_b\alpha_s} corrections.
In \citere{mhiggsEP4} the shift of the sbottom masses due to the
SU(2)-invariance was taken into
account in the numerical evaluation of the sbottom masses following the
prescription in \citere{bartl} (see also \citere{delrhosusy2loop}).
Our result for $M_h$ in the ``$A_b,\;\theta_{\tilde{b}}$~OS'' scheme is compared with
the result of \citere{mhiggsEP4} in \reffi{fig:UIFcomp}. For the
implementation of the latter (``$m_{\tilde{b}},\;A_b$~OS'' scheme)
the Fortran code of \citere{mhiggsEP4}
for the numerical evaluation of the $\mathcal O(\alpha_s \alpha_b)$ corrections
to the Higgs-boson self-energies has been used~\cite{pietro}.
Thereby the input values were determined according to
\eqref{mbdef} and \eqref{Abdef}. Using these input values for $A_b$ and
$m_{b}$ the sbottom masses were calculated taking the sbottom mass shift
into account~\cite{bartl}.
$M_h$ is shown as function of $m_{\tilde{g}}$ for
$\mu < 0$, $\tan \beta = 50$, and $M_A = 700 \,\, {\rm GeV}$. Our result in the
``$A_b,\;\theta_{\tilde{b}}$~OS'' scheme is shown as the dash-star (green) curve, while
the result of \citere{mhiggsEP4} (``$m_{\tilde{b}},\;A_b$~OS'' scheme) is given by the
fine-dotted (pink) curve. The leading contribution in the two schemes,
i.e.\ the \order{\alpha_t\alpha_s} result including resummation, is also shown:
the light-dot-dashed (orange)
curve shows the \order{\alpha_t\alpha_s} result using the
``$A_b,\;\theta_{\tilde{b}}$~OS'' renormalised
parameters; the corresponding result for the ``$m_{\tilde{b}},\;A_b$~OS''
renormalised parameters is shown as the light-dotted (gray) curve.
\reffi{fig:UIFcomp} shows that the \order{\alpha_t\alpha_s} results in the two
schemes differ from each other by up to $2 \,\, {\rm GeV}$ for large $m_{\tilde{g}}$. The
inclusion of the subleading two-loop corrections reduces this difference
significantly. Our result in the ``$A_b,\;\theta_{\tilde{b}}$~OS'' scheme agrees with the
result of \citere{mhiggsEP4} to better than $0.5 \,\, {\rm GeV}$.
\begin{figure}[th!]
\begin{center}
\epsfig{figure=plots/plot5b1vglbneu.eps, width=14cm,height=9.0cm}
\caption{
Comparison of our \order{\alpha_b\alpha_s} result for $M_h$
in the ``$A_b,\;\theta_{\tilde{b}}$~OS'' scheme and the
result of \citere{mhiggsEP4} (``$m_{\tilde{b}},\;A_b$~OS'' scheme) as a function
of $m_{\tilde{g}}$.
The \order{\alpha_t\alpha_s} results in the two schemes, where the subleading
\order{\alpha_b\alpha_s} corrections are omitted (using the appropriate renormalised
parameters), are also shown.
}
\vspace{-2em}
\label{fig:UIFcomp}
\end{center}
\end{figure}
\begin{figure}[hb!]
\begin{center}
\epsfig{figure=plots/plot92c.eps, width=14cm,height=9.0cm}
\caption{
Comparison of our \order{\alpha_b\alpha_s} result in three different
renormalisation schemes and the result of \citere{mhiggsEP4}.
The three curves for $\DeM_h$ show the difference between our result in each
of the three schemes and the result of \citere{mhiggsEP4}
as a function of $\tan \beta$.
}
\label{fig:UIFcomp2}
\end{center}
\end{figure}
In \reffi{fig:UIFcomp2} we compare our result in each of the three
schemes discussed above, i.e.\ the ``$A_b,\;\theta_{\tilde{b}}$~OS'', the
``$m_{b}$~$\overline{\rm{DR}}$'' and the ``$A_b$, $\theta_{\tilde{b}}$ $\overline{\rm{DR}}$'' schemes, with the result of
\citere{mhiggsEP4}. The difference $\DeM_h$ between our result and the
result of \citere{mhiggsEP4} is shown for each of the three schemes as a
function of $\tan \beta$ for $m_{\tilde{g}} = 1500 \,\, {\rm GeV}$, $\mu = -1000 \,\, {\rm GeV}$, and
$M_A = 700 \,\, {\rm GeV}$.
The differences stay below $1 \,\, {\rm GeV}$ for $\tan \beta \lsim 50$, where our result
in the ``$A_b$, $\theta_{\tilde{b}}$ $\overline{\rm{DR}}$'' scheme shows the biggest deviation from the
result of \citere{mhiggsEP4}, while as expected, the difference is
smallest for the ``$A_b$, $\theta_{\tilde{b}}$ OS'' scheme. For $\tan \beta > 50$ large
deviations occur because of the sharp decrease of $M_h$ in this region
(see e.g.\ \reffi{fig:mhtb_MAlarge_MUneg}).
\section{Conclusions}
\label{sec:conclusions}
We have obtained results for the two-loop \order{\alpha_b\alpha_s} corrections
to the neutral ${\cal CP}$-even Higgs-boson masses in the MSSM within
different renormalisation schemes. The leading $\tan \beta$-enhanced
contributions of the $b/\tilde{b}$~sector can be incorporated into an
appropriately chosen bottom
Yukawa coupling, for which we use the bottom-quark mass in the $\overline{\rm{DR}}$\
scheme with a resummation of the leading contributions. We have
analysed in detail the impact of the genuine (subleading)
\order{\alpha_b\alpha_s}
two-loop corrections in different parts of the MSSM parameter space and
we have compared the results obtained in the different schemes.
We have shown that an on-shell scheme that is
frequently used in the $t/\tilde{t}$~sector leads to numerically unstable
results if it is applied in the $b/\tilde{b}$~sector. The origin of the huge
corrections in this scheme was traced to the fact that it involves a
renormalisation condition
for the sbottom mixing angle, $\theta_{\tilde{b}}$, rather than for
the trilinear coupling, $A_b$.
The other three schemes that we have analysed yield numerically
well-behaved and physically meaningful results. For $\mu > 0$ the effect
of the genuine \order{\alpha_b\alpha_s} two-loop corrections is rather small,
typically of \order{100 \,\, {\rm MeV}}. Corrections at this level
will nevertheless be relevant in view of the prospective accuracy of
measurements in the Higgs sector and of the top-quark mass at the ILC.
For $\mu < 0$ the effective bottom Yukawa coupling increases, leading to an
enhancement of the effects from the $b/\tilde{b}$~sector. While the
constraints arising from the measurement of the anomalous magnetic
moment of the muon favour the positive sign of $\mu$, it seems premature
at the present stage to discard the parameter region with $\mu < 0$.
For large values
of $\tan \beta$ and $m_{\tilde{g}}$ and large negative values of $\mu$ we find that the
genuine \order{\alpha_b\alpha_s} corrections can amount up to $3 \,\, {\rm GeV}$.
We have compared our result for the \order{\alpha_b\alpha_s} corrections
with the existing result in the literature, which was
obtained in the limit of $\tan \beta \to \infty$, and found good agreement.
The comparison of the results in the different schemes that we have
analysed and the investigation of the renormalisation scale dependence
give an indication of the possible size of missing higher-order
corrections in the $b/\tilde{b}$~sector. For $\mu > 0$ the higher-order
corrections from the $b/\tilde{b}$~sector (beyond \order{\alpha_b\alpha_s}) appear
to be sufficiently well
under control even in view of the prospective ILC accuracy.
This applies especially to the ``$m_{b}$ $\overline{\rm{DR}}$'' scheme, where
the corrections beyond the improved one-loop result have been found to
be particularly small. For $\mu < 0$, on the other hand, sizable
higher-order corrections from the $b/\tilde{b}$~sector are possible. The
size of the individual corrections and also the difference between the
analysed schemes varies significantly with the relevant parameters,
$\mu$, $\tan \beta$, $m_{\tilde{g}}$ and $M_A$. We estimate the uncertainty from missing
higher-order corrections in the $b/\tilde{b}$~sector to be about $2 \,\, {\rm GeV}$ in
this region of parameter space.
The results obtained will be implemented into the Fortran code
{\em FeynHiggs}~\cite{feynhiggs,feynhiggs2}.
The evaluation of the results within the three schemes will allow to
obtain
an estimate of the size of the missing higher-order corrections
as a function of the chosen input parameters.
\subsection*{Acknowledgements}
We thank A.~Hoang, U.~Nierste, P.~Slavich and D.~St\"ockinger
for helpful discussions. We are grateful to P.~Slavich for providing
the Fortran code for the ``$m_{\tilde{b}}$, $A_b$ OS'' renormalisation scheme.
S.H.\ and G.W.\ thank the Max Planck Institut f\"ur
Physik, M\"unchen, for kind hospitality during part of
this work. G.W.\ thanks the CERN theory group for kind hospitality
during the final stage of this paper.
This work has been supported by the European Community's Human
Potential Programme under contract HPRN-CT-2000-00149 ``Physics at
Colliders''.
\section*{Appendix: Counterterms of the quark/squark sector}
In section \ref{sec:ren} the counterterms have been given using the
definitions \eqref{transformation} and
\eqref{Sfermionmassenmatrixeigenwerte} for the sfermion masses and mixing
angles. In this appendix the counterterms are given in a more
general way allowing to use also other definitions for the sfermion masses and
mixing angles. Introducing a counterterm for the mixing angle needs a certain
choice of definitions of the sfermion masses and mixing angles. Instead of
using an explicit mixing angle counterterm the counterterm $\delta
Y_{\tilde{q}}$ is introduced as
\begin{align}
\delta Y_{\tilde{q}} = ({\cal U}_{\tilde{q}} \delta {\cal M}_{\tilde{q}} {\cal U}^{\dagger}_{\tilde{q}})_{12}=
({\cal U}_{\tilde{q}} \delta {\cal M}_{\tilde{q}} {\cal U}^{\dagger}_{\tilde{q}})_{21} ~,
\end{align}
where the counterterm mass matrix $\delta {\cal M}_{\tilde{q}}$ contains the
counterterms of the parameters appearing in
\eqref{Sfermionmassenmatrix}. With the definitions \eqref{transformation} and
\eqref{Sfermionmassenmatrixeigenwerte} $\delta Y_{\tilde{q}}$ is
related to the mixing angle counterterm as follows
\begin{align}
\delta Y_{\tilde{q}} &= (m_{\tilde{q}_1}^2 - m_{\tilde{q}_2}^2)\, \delta \theta\kern-.15em_{\tilde{q}} \,.
\end{align}
\noindent
$\bullet$ {\bf Top quark/squark sector:}
The counterterms for the top-quark mass \eqref{dmt} and the stop masses
\eqref{dmst} are already in a general form. The counterterm for the
mixing angle \eqref{ZusammenhangdeltathetadeltaM} is replaced by
\begin{align}
\delta Y_{\tilde{t}} =
\frac{1}{2}\Bigl(\mathop{\rm Re} \Sigma_{\tilde{t}_{12}}(m_{{\tilde{t}}_{1}}^2)+\mathop{\rm Re}
\Sigma_{\tilde{t}_{12}}(m_{{\tilde{t}}_{2}}^2)\Bigr) ,
\end{align}
and the counterterm of the A-parameter \eqref{deltaAt} is rewritten as
\begin{align}
\delta A_t &= \frac{1}{m_{t}}
\Bigl[U_{\tilde{t}_{11}} U_{\tilde{t}_{12}}
\bigl(\delta m_{\tilde{t}_1}^2 - \delta m_{\tilde{t}_2}^2\bigr)
+(U_{\tilde{t}_{11}} U_{\tilde{t}_{22}} + U_{\tilde{t}_{12}} U_{\tilde{t}_{21}})\,
\delta Y_{\tilde{t}}
- \delta m_{t} \, ( A_t - \mu \cot \beta) \Bigr]~.
\end{align}
\noindent
$\bullet$ {\bf Analogous to the top quark/squark sector:}
As in the top quark/squark sector the counterterm for the mixing angle
\eqref{dthetab} is replaced by
\begin{align}\label{dYbanalogtop}
\delta Y_{\tilde{b}} =
\frac{1}{2} \Bigl(\mathop{\rm Re} \Sigma_{\tilde{b}_{12}}(m_{\tilde{b}_1}^2)
+\mathop{\rm Re} \Sigma_{\tilde{b}_{12}}(m_{\tilde{b}_2}^2) \Bigr)\; .
\end{align}
The dependent counterterms of the $\tilde{b}_1$-mass \eqref{ms1CT}
and of the A-parameter \eqref{Abparameter} are rewritten as follows:
\begin{align} \nonumber
\delta m_{\tilde{b}_1}^2 &= \frac{1}{U_{\tilde{b}_{11}}^2}\bigl[
- U_{\tilde{b}_{12}}^2 \dem_{\tilde{b}_2}^2
+ 2 U_{\tilde{b}_{12}} U_{\tilde{b}_{22}}
\delta Y_{\tilde{b}}
+ U_{\tilde{t}_{11}}^2 \delta m_{\tilde{t}_1}^2
+ U_{\tilde{t}_{12}}^2 \delta m_{\tilde{t}_2}^2
\\[1.5mm] & \quad\
- 2 U_{\tilde{t}_{12}} U_{\tilde{t}_{22}} \delta Y_{\tilde{t}}
+ ( 2 m_{b} \delta m_{b} - 2 m_{t} \delta m_{t})\bigr]~,\label{dmsb1allg} \\[1.5mm]\nonumber
\delta A_b &= \frac{1}{m_{b}}\Bigl[
-\frac{U_{\tilde{b}_{12}}}{U_{\tilde{b}_{11}}} \delta m_{\tilde{b}_2}^2 +
\frac{U_{\tilde{b}_{22}}}{U_{\tilde{b}_{11}}} \delta Y_{\tilde{b}}
- \delta m_{b} ( A_{b} - \mu \tan \beta
- 2 \frac{U_{\tilde{b}_{12}}}{U_{\tilde{b}_{11}}} m_{b}) \\[1.5mm]
& \quad\ \quad\
+ \frac{U_{\tilde{b}_{12}}}{U_{\tilde{b}_{11}}} \bigl( U_{\tilde{t}_{11}}^2 \delta m_{\tilde{t}_1}^2
+ U_{\tilde{t}_{12}}^2 \delta m_{\tilde{t}_2}^2
- 2 U_{\tilde{t}_{12}} U_{\tilde{t}_{22}} \delta Y_{\tilde{t}}
-2 m_{t} \delta m_{t} \bigr) \Bigr]~.
\end{align}
\noindent
$\bullet$ {\bf \boldmath{$\overline{\rm{DR}}$} bottom-quark mass}
The A-parameter counterterm \eqref{AbcountertermDR} is written in the
following way
\begin{align} \nonumber
\delta A_b &=
\frac{1}{m_{b}}\Bigl[
-\frac{U_{\tilde{b}_{12}}}{U_{\tilde{b}_{11}}}
\mathop{\rm Re} \Sigma_{\tilde{b}_{22}}^{\rm div}(m_{\tilde{b}_2}^2)
+ \frac{U_{\tilde{b}_{22}}}{2 U_{\tilde{b}_{11}}}
\bigl(\mathop{\rm Re} \Sigma_{\tilde{b}_{12}}^{\rm div}(m_{\tilde{b}_1}^2)+
\mathop{\rm Re} \Sigma_{\tilde{b}_{12}}^{\rm div}(m_{\tilde{b}_2}^2)\bigr)\\[1.5mm]
& \nonumber \quad \quad\
+ \frac{U_{\tilde{b}_{12}}}{U_{\tilde{b}_{11}}}\Bigl( U_{\tilde{t}_{11}}^2
\mathop{\rm Re} \Sigma_{\tilde{t}_{11}}^{\rm div}(m_{\tilde{t}_1}^2)
+ U_{\tilde{t}_{12}}^2
\mathop{\rm Re} \Sigma_{\tilde{t}_{22}}^{\rm div}(m_{\tilde{t}_2}^2) \\[1.5mm]
& \nonumber \quad \quad\
- U_{\tilde{t}_{12}} U_{\tilde{t}_{22}}
\bigl(\mathop{\rm Re} \Sigma_{\tilde{t}_{12}}^{\rm div}(m_{\tilde{t}_1}^2)+
\mathop{\rm Re} \Sigma_{\tilde{t}_{12}}^{\rm div}(m_{\tilde{t}_2}^2)\bigr)
\\[1.5mm] & \nonumber \quad \quad\
- m_{t}^2 (\mathop{\rm Re} {\Sigma}_{{t}_L}^{\rm div}(m_{t}^2) +
\mathop{\rm Re} {\Sigma}_{{t}_R}^{\rm div}(m_{t}^2) +
2\mathop{\rm Re} {\Sigma}_{{t}_S}^{\rm div}(m_{t}^2))
\Bigr) \Bigr]
+ \frac{1}{2} (2
\frac{U_{\tilde{b}_{12}}}{U_{\tilde{b}_{11}}} m_{b}
\\[1.5mm] & \quad \quad
- A_{b} + \mu\tan \beta) \Bigl
(\mathop{\rm Re} {\Sigma}_{{b}_L}^{\rm div} (m_{b}^2)
+ \mathop{\rm Re} {\Sigma}_{{b}_R}^{\rm div} (m_{b}^2)
+ 2\mathop{\rm Re} {\Sigma}_{{b}_S}^{\rm div} (m_{b}^2) \Bigr)~,
\end{align}
avoiding an explicit definition of the mixing angles.
The dependent counterterm for the mixing angle \eqref{thetabinAbMBdrbar}
is replaced by
\begin{align}\nonumber
\delta Y_{\tilde{b}} &=
\frac{U_{\tilde{b}_{11}}}{U_{\tilde{b}_{22}}}m_{b} \delta A_b
+\frac{U_{\tilde{b}_{11}}}{U_{\tilde{b}_{22}}} \dem_{b}
( A_b - \mu \tan \beta - 2 \frac{U_{\tilde{b}_{12}}}{U_{\tilde{b}_{11}}} m_{b})
+\frac{U_{\tilde{b}_{12}}}{U_{\tilde{b}_{22}}} \bigl[\delta m_{\tilde{b}_2}^2\\[1.5mm]
& \quad\
- U_{\tilde{t}_{11}}^2 \dem_{\tilde{t}_1}^2
- U_{\tilde{t}_{12}}^2 \dem_{\tilde{t}_2}^2
+ 2 U_{\tilde{t}_{12}} U_{\tilde{t}_{22}} \delta Y_{\tilde{t}}
+ 2 m_{t} \delta m_{t} \bigr]~,
\end{align}
and the counterterm for the $\tilde{b}_1$-mass \eqref{msb1inAbMBdrbar} by
\begin{align}\nonumber
\delta m_{\tilde{b}_1}^2 &=
\frac{1}{U_{\tilde{b}_{11}}^2} \Bigl[
(1 - 2 U_{\tilde{b}_{12}}^2)
\Bigl(U_{\tilde{t}_{11}}^2 \delta m_{\tilde{t}_1}^2
+U_{\tilde{t}_{12}}^2 \delta m_{\tilde{t}_2}^2
-2 U_{\tilde{t}_{12}} U_{\tilde{t}_{22}} \delta Y_{\tilde{t}}
- 2 m_{t} \delta m_{t} \Bigr)
+ U_{\tilde{b}_{12}}^2 \delta m_{\tilde{b}_2}^2
\\[1mm] & \quad
+ 2 U_{\tilde{b}_{11}} U_{\tilde{b}_{12}} m_{b} \delta A_b
+ \Bigl( 2 U_{\tilde{b}_{12}} U_{\tilde{b}_{11}}
( A_b - \mu \tan \beta) + 2 (1 - 2
U_{\tilde{b}_{12}}^2) m_{b} \Bigr) \delta m_{b} \Bigr]~.
\end{align}
\noindent
$\bullet$ {\bf \boldmath{$\overline{\rm{DR}}$} mixing angle and \boldmath{$A_b$}}
The counterterm for the mixing angle \eqref{thetadrbar} is replaced by
\begin{align}
\delta Y_{\tilde{b}} =
\frac{1}{2}\bigl(\mathop{\rm Re} \Sigma_{\tilde{b}_{12}}^{\rm div}(m_{\tilde{b}_1}^2)+
\mathop{\rm Re} \Sigma_{\tilde{b}_{12}}^{\rm div}(m_{\tilde{b}_2}^2)\bigr) \; .
\end{align}
The dependent counterterm for the bottom quark mass \eqref{deltambabh}
is rewritten as the following combination of counterterms:
\begin{align}\nonumber
\delta m_{b} &= - \Bigl[m_{b}\, \delta A_b -
\frac{U_{\tilde{b}_{22}}}{U_{\tilde{b}_{11}}} \delta Y_{\tilde{b}}
+ \frac{U_{\tilde{b}_{12}}}{U_{\tilde{b}_{11}}} \delta m_{\tilde{b}_2}^2
- \frac{U_{\tilde{b}_{12}}}{U_{\tilde{b}_{11}}} \Bigl(
U_{\tilde{t}_{11}}^2 \delta m_{\tilde{t}_1}^2
+ U_{\tilde{t}_{12}}^2 \delta m_{\tilde{t}_2}^2
- 2 U_{\tilde{t}_{12}} U_{\tilde{t}_{22}} \delta Y_{\tilde{t}}
\\[1.5mm] & \quad\
- 2 m_{t}\, \delta m_{t}\Bigr) \Bigr]
\Bigl[A_b -\mu \tan \beta - 2 m_{b} \frac{U_{\tilde{b}_{12}}}{U_{\tilde{b}_{11}}} \Bigr]^{-1}
~. \label{mbinAbthetabdrbarallg}
\end{align}
The counterterm for the $\tilde{b}_1$-mass is obtained by inserting the
expression \eqref{mbinAbthetabdrbarallg} for the bottom quark mass into
the expression \eqref{dmsb1allg}.\\[1cm]
\noindent
$\bullet$ {\bf On-shell mixing angle and \boldmath{$A_b$} }
The renormalised vertex \eqref{lambdahut} has the following general form
\begin{align}\nonumber
\hat{\Lambda}(p_A^2, p_{\tilde{b}_1}^2, p_{\tilde{b}_2}^2) &= \Lambda (p_A^2,
p_{\tilde{b}_1}^2, p_{\tilde{b}_2}^2) + \frac{i e}{2 M_W \sin
\theta_W}(U_{\tilde{b}_{11}}U_{\tilde{b}_{22}}- U_{\tilde{b}_{12}}U_{\tilde{b}_{21}}) \Bigl[
m_{b} \tan \beta \delta A_b \\ & \quad + (\mu + \tan \beta A_b)\Bigl( \delta m_{b} +
\frac{1}{2} m_{b} (\delta
Z_{\tilde{b}_1 \tilde{b}_1} + \delta Z_{\tilde{b}_2 \tilde{b}_2})\Bigr)\Bigr]~.
\end{align}
Using the renormalisation condition \eqref{lambdahutbed} the
counterterms of the A-parameter and the bottom quark mass can be derived
as
\begin{align}
\nonumber
\delta A_b &=
i \frac{M_W \sin \theta_W}{e\, m_{b} \tan \beta (U_{\tilde{b}_{11}} U_{\tilde{b}_{22}}
- U_{\tilde{b}_{12}} U_{\tilde{b}_{21}})}
\Bigl( \Lambda(0, m_{\tilde{b}_1}^2, m_{\tilde{b}_1}^2) + \Lambda(0, m_{\tilde{b}_2}^2, m_{\tilde{b}_2}^2) \Bigr)
\\[1.5mm]& \quad \nonumber
- \frac{\mu + A_b \tan \beta}{2 \tan \beta}
(\delta Z_{\tilde{b}_1 \tilde{b}_1} + \delta Z_{\tilde{b}_2 \tilde{b}_2})
\\ & \quad \nonumber
- \frac{\mu + A_b \tan \beta}{m_{b} \tan \beta}\Bigg[
i \frac{M_W \sin \theta_W}{e \tan \beta (U_{\tilde{b}_{11}} U_{\tilde{b}_{22}}-
U_{\tilde{b}_{12}} U_{\tilde{b}_{21}})}
\Bigl(\Lambda(0, m_{\tilde{b}_1}^2,m_{\tilde{b}_1}^2) + \Lambda(0, m_{\tilde{b}_2}^2, m_{\tilde{b}_2}^2)\Bigr) \\[1.5mm]
& \qquad \nonumber
- \frac{m_{b}(\mu + A_b \tan \beta)}{2 \tan \beta}
(\delta Z_{\tilde{b}_1 \tilde{b}_1} + \delta Z_{\tilde{b}_2 \tilde{b}_2})
+\frac{U_{\tilde{b}_{12}}}{U_{\tilde{b}_{11}}} \delta m_{\tilde{b}_2}^2
- \frac{U_{\tilde{b}_{22}}}{U_{\tilde{b}_{11}}} \delta Y_{\tilde{b}}
-\frac{U_{\tilde{b}_{12}}}{U_{\tilde{b}_{11}}} \Bigl(
U_{\tilde{t}_{11}}^2 \delta m_{\tilde{t}_1}^2
\\[1.5mm]& \qquad
+ U_{\tilde{t}_{12}}^2 \delta m_{\tilde{t}_2}^2
-2 U_{\tilde{t}_{12}} U_{\tilde{t}_{22}} \delta Y_{\tilde{t}}
- 2 m_{t} \delta m_{t} \Bigr)
\Bigg]
\left[ \mu\Bigl(\tan \beta +
\frac{1}{\tan \beta}\Bigr)
+ 2 m_{b} \frac{U_{\tilde{b}_{12}}}{U_{\tilde{b}_{11}}}
\right]^{-1}~,
\end{align}
and
\begin{align} \nonumber
\delta m_{b} &= \Bigl[
i \frac{M_W \sin \theta_W}{e\,\tan \beta (U_{\tilde{b}_{11}} U_{\tilde{b}_{22}}-
U_{\tilde{b}_{12}} U_{\tilde{b}_{21}})}
\Bigl(\Lambda(0, m_{\tilde{b}_1}^2, m_{\tilde{b}_1}^2) + \Lambda(0, m_{\tilde{b}_2}^2, m_{\tilde{b}_2}^2) \Bigr)
\\[1.5mm]& \quad\ \nonumber
- \frac{m_{b}(\mu + A_b \tan \beta)}{2 \tan \beta}
(\delta Z_{\tilde{b}_1 \tilde{b}_1} + \delta Z_{\tilde{b}_2 \tilde{b}_2})
-\frac{U_{\tilde{b}_{22}}}{U_{\tilde{b}_{11}}} \delta Y_{\tilde{b}} +
\frac{U_{\tilde{b}_{12}}}{U_{\tilde{b}_{11}}} \delta m_{\tilde{b}_2}^2
- \frac{U_{\tilde{b}_{12}}}{U_{\tilde{b}_{11}}}
\Bigl( U_{\tilde{t}_{11}}^2 \delta m_{\tilde{t}_1}^2
\\[1.5mm] & \qquad\
+ U_{\tilde{t}_{12}}^2 \delta m_{\tilde{t}_2}^2
- 2 U_{\tilde{t}_{12}} U_{\tilde{t}_{22}}\delta Y_{\tilde{t}}
- 2 m_{t} \delta m_{t}
\Bigr)
\Bigr]
\Bigl[
\mu\Bigl(\tan \beta + \frac{1}{\tan \beta}\Bigr)
+2 m_{b} \frac{U_{\tilde{b}_{12}}}{U_{\tilde{b}_{11}}}
\Bigr]^{-1} ~,
\end{align}
replacing \eqref{dAbUIFartig} and \eqref{dmbmitthetauAbonshell}.
The counterterm of the mixing angle \eqref{mixangleUIF} is replaced by
\eqref{dYbanalogtop}.
\newpage
|
{
"timestamp": "2005-01-06T22:24:12",
"yymm": "0411",
"arxiv_id": "hep-ph/0411114",
"language": "en",
"url": "https://arxiv.org/abs/hep-ph/0411114"
}
|
\section{The Cosmological Paradigm}
The last couple of decades have been the golden age for cosmology, in much the same way as the mid-1900's
were a golden age for particle physics. Data of exquisite quality confirmed the broad paradigm of standard
cosmology and helped us to determine the composition of the universe. As a direct consequence, the cosmological observations have thrusted upon us a rather preposterous
composition for the universe which defies any simple explanation, thereby posing the greatest challenge
theoretical physics has ever faced.
To understand these exciting developments, it is best to begin by reminding ourselves of the standard
paradigm for cosmology. Observations show that the universe is fairly homogeneous and isotropic at scales
larger than about $150h^{-1}$ Mpc, where 1 Mpc$\approx 3\times 10^{24}$ cm is a convenient unit for extragalactic astronomy and $h\approx 0.7$ characterizes \cite{h} the current rate of expansion of the universe in dimensionless form. (The mean distance between galaxies is about 1 Mpc while the size of the visible universe is about $3000 h^{-1}$ Mpc.) The conventional --- and highly successful --- approach to cosmology
separates the study of large scale ($l\gtrsim 150h^{-1}$ Mpc) dynamics of the universe from the issue of structure formation at smaller scales. The former is modeled by a homogeneous and isotropic distribution of energy density; the latter issue is
addressed in terms of gravitational instability which will amplify the small perturbations in the energy density, leading to the formation of structures like galaxies.
In such an approach, the expansion of the background universe is described by a single function of time $a(t)$
which is governed by the equations (with $c=1$):
\begin{equation}
\frac{\dot a^2+k}{a^2} =\frac{8\pi G\rho}{3};\qquad d(\rho a^3)=-pda^3
\label{frw}
\end{equation}
The first one relates expansion rate to the energy density $\rho$ and $k=0,\pm 1$ is a parameter which characterizes the spatial curvature of the universe.
The second equation, when coupled with the equation of state
$p=p(\rho)$, determines the evolution of energy density $\rho=\rho(a)$ in terms of the expansion factor of the universe.
In particular if $p=w\rho$ with (at least, approximately) constant $w$ then, $ \rho \propto a^{-3(1+w)}$ and (if we assume $k=0$),
$ a \propto t^{2/[3(1+w)]}$.
It is convenient to measure
the energy densities of different components in terms of a \textit{critical energy density} ($\rho_c$) required to make $k=0$ at the present epoch. (Of course, since $k$ is a constant,
it will remain zero at all epochs if it is zero at any given moment of time.) From Eq.(\ref{frw}), it is clear that $\rho_c=3H^2_0/8\pi G$ where $H_0=(\dot a/a)_0$
is the rate of expansion of the universe at present. The variables $\Omega_i=\rho_i/\rho_c$
will give the fractional contribution of different components of the universe ($i$ denoting baryons, dark matter, radiation, etc.) to the critical density. Observations then lead to the following results:
\begin{itemize}
\item
Our universe has $0.98\lesssim\Omega_{tot}\lesssim1.08$. The value of $\Omega_{tot}$ can be determined from the angular anisotropy spectrum of the cosmic microwave background radiation (CMBR) (with the reasonable assumption that $h>0.5$) and these observations now show that we live in a universe
with critical density \cite{cmbr,kanduhere}.
\item
Observations of primordial deuterium produced in big bang nucleosynthesis (which took place when the universe
was about 1 minute in age) as well as the CMBR observations show that \cite{baryon} the {\it total} amount of baryons in the
universe contributes about $\Omega_B=(0.024\pm 0.0012)h^{-2}$. Given the independent observations on the Hubble constant \cite{h} which fix $h=0.72\pm 0.07$, we conclude that $\Omega_B\cong 0.04-0.06$. These observations take into account all baryons which exist in the universe today irrespective of whether they are luminous or not. Combined with previous item we conclude that
most of the universe is non-baryonic.
\item
Host of observations related to large scale structure and dynamics (rotation curves of galaxies, estimate of cluster masses, gravitational lensing, galaxy surveys ..) all suggest \cite {dm} that the universe is populated by a non-luminous component of matter (dark matter; DM hereafter) made of weakly interacting massive particles which \textit{does} cluster at galactic scales. This component contributes about $\Omega_{DM}\cong 0.20-0.35$.
\item
Combining the last observation with the first we conclude that there must be (at least) one more component
to the energy density of the universe contributing about 70\% of critical density. Early analysis of several observations
\cite{earlyde} indicated that this component is unclustered and has negative pressure. This is confirmed dramatically by the supernova observations (see \cite{sn}; for a critical look at the data, see \cite{tptirthsn1,tptirthsn2}). The observations suggest that the missing component has
$w=p/\rho\lesssim-0.78$
and contributes $\Omega_{DE}\cong 0.60-0.75$.
\item
The universe also contains radiation contributing an energy density $\Omega_Rh^2=2.56\times 10^{-5}$ today most of which is due to
photons in the CMBR. This is dynamically irrelevant today but would have been the dominant component in the universe at redshifts
larger that $z_{eq}\simeq \Omega_{DM}/\Omega_R\simeq 4\times 10^4\Omega_{DM}h^2$.
\item
Together we conclude that our universe has (approximately) $\Omega_{DE}\simeq 0.7,\Omega_{DM}\simeq 0.26,\Omega_B\simeq 0.04,\Omega_R\simeq 5\times 10^{-5}$. All known observations
are consistent with such an --- admittedly weird --- composition for the universe.
\end{itemize}
Before discussing the composition of the universe in greater detail, let us briefly consider the issue of structure formation. The key idea is that if there existed small fluctuations in the energy density in the early universe, then gravitational instability can amplify them in a well-understood manner (see e.g., \cite{tpsfuv3}), leading to structures like galaxies etc. today. The most popular theoretical model for these fluctuations is based on the idea that if the very early universe went through an inflationary phase \cite{inflation}, then the quantum fluctuations of the field driving the inflation can lead to energy density fluctuations\cite{genofpert,tplp}. It is possible to construct models of inflation such that these fluctuations are described by a Gaussian random field and are characterized by a power spectrum of the form $P(k)=A k^n$ with $n\simeq 1$. The models cannot predict the value of the amplitude $A$ in an unambiguous manner but it can be determined from CMBR observations. The CMBR observations are consistent with the inflationary model for the generation of perturbations and gives $A\simeq (28.3 h^{-1} Mpc)^4$ and $n=0.97\pm0.023$ (The first results were from COBE \cite{cobeanaly} and
WMAP \cite{kanduhere} has reconfirmed them with far greater accuracy).
So, to the zeroth order, the universe is characterized by just seven numbers: $h\approx 0.7$ describing the current rate of expansion; $\Omega_{DE}\simeq 0.7,\Omega_{DM}\simeq 0.26,\Omega_B\simeq 0.04,\Omega_R\simeq 5\times 10^{-5}$ giving the composition of the universe; the amplitude $A\simeq (28.3 h^{-1} Mpc)^4$ and the index $n\simeq 1$ of the initial perturbations.
The challenge is to make some sense out of these numbers from a more fundamental point of view.
\section{The Dark Energy}
It is rather frustrating that we have no direct laboratory evidence for nearly 96\% of matter in the universe.
(Actually, since we do not quite understand the process of baryogenesis, we do not understand $\Omega_B$ either;
all we can \textit{theoretically} understand now is a universe filled entirely with radiation!). Assuming that particle physics models will eventually come of age and (i) explain $\Omega_B$ and $\Omega_{DM}$ (probably as the lightest
supersymmetric partner) as well as (ii) provide a viable model for inflation predicting correct value for $A$,
one is left with the problem of understanding $\Omega_{DE}$. While the issues (i) and (ii) are by no means trivial or satisfactorily addressed, it is probably correct to say that the issue of dark energy is lot more perplexing,
thereby justifying the attention it has received recently.
The key observational feature of dark energy is that --- treated as a fluid with a stress tensor $T^a_b={\rm dia} (\rho, -p, -p,-p)$
--- it has an equation state $p=w\rho$ with $w \lesssim -0.8$ at the present epoch.
The spatial part ${\bf g}$ of the geodesic acceleration (which measures the
relative acceleration of two geodesics in the spacetime) satisfies an \textit{exact} equation
in general relativity (see e.g., page 332 of \cite{probbook}) given by:
\begin{equation}
\nabla \cdot {\bf g} = - 4\pi G (\rho + 3p)
\label{nextnine}
\end{equation}
This shows that the source of geodesic acceleration is $(\rho + 3p)$ and not $\rho$.
As long as $(\rho + 3p) > 0$, gravity remains attractive while $(\rho + 3p) <0$ can
lead to repulsive gravitational effects. In other words, dark energy with sufficiently negative pressure will
accelerate the expansion of the universe, once it starts dominating over the normal matter. This is precisely what is established from the study of high redshift supernova, which can be used to determine the expansion
rate of the universe in the past \cite{sn}.
Figure \ref{fig:tptrc} presents the supernova data as a phase portrait \cite{tptirthsn1,tptirthsn2} of the universe (plotting the `velocity' $\dot a$ against 'position' $a$). It is clear that the universe was decelerating at high redshifts and started accelerating when it was about two-third of the present size.
\begin{figure}[ht]
\begin{center}
\includegraphics[angle=-90,scale=.5]{Fig2c}
\end{center}
\caption{The ``velocity'' $\dot a$ of the universe is plotted against the ``position'' $a$ in the form of a phase portrait. The different curves are for models parameterized by the value of $\Omega_{DM}(=\Omega_m)$ keeping $\Omega_{tot}=1$. The top-most curve has $\Omega_m=1$ with no dark energy and the universe is decelerating at all epochs. The bottom-most curve has $\Omega_m=0$
and $\Omega_{DE}=1$ and the universe is accelerating at all epochs. The in-between curves show universes which were decelerating in the past and began to accelerate when the dark energy started dominating. The supernova data clearly favours such a model. (For a more detailed discussion of the figure, see
\cite{tptirthsn1,tptirthsn2}.)}
\label{fig:tptrc}
\end{figure}
The simplest model for a fluid with negative pressure is the
cosmological constant (for a review, see \cite{cc}) with $w=-1,\rho =-p=$ constant (which is the model used in Figure \ref{fig:tptrc}).
If the dark energy is indeed a cosmological constant, then it introduces a fundamental length scale in the theory $L_\Lambda\equiv H_\Lambda^{-1}$, related to the constant dark energy density $\rho_{DE}$ by
$H_\Lambda^2\equiv (8\pi G\rho_{DE}/3)$.
In classical general relativity,
based on the constants $G, c $ and $L_\Lambda$, it
is not possible to construct any dimensionless combination from these constants. But when one introduces the Planck constant, $\hbar$, it is possible
to form the dimensionless combination $H^2_\Lambda(G\hbar/c^3) \equiv (L_P^2/L_\Lambda^2)$.
Observations then require $(L_P^2/L_\Lambda^2) \lesssim 10^{-123}$.
As has been mentioned several times in literature, this will require enormous fine tuning. What is more,
in the past, the energy density of
normal matter and radiation would have been higher while the energy density contributed by the cosmological constant
does not change. Hence we need to adjust the energy densities
of normal matter and cosmological constant in the early epoch very carefully so that
$\rho_\Lambda\gtrsim \rho_{\rm NR}$ around the current epoch.
This raises the second of the two cosmological constant problems:
Why is it that $(\rho_\Lambda/ \rho_{\rm NR}) = \mathcal{O} (1)$ at the
{\it current} phase of the universe ?
Because of these conceptual problems associated with the cosmological constant, people have explored a large variety of alternative possibilities. The most popular among them uses a scalar field $\phi$ with a suitably chosen potential $V(\phi)$ so as to make the vacuum energy vary with time. The hope then is that, one can find a model in which the current value can be explained naturally without any fine tuning.
A simple form of the source with variable $w$ are scalar fields with
Lagrangians of different forms, of which we will discuss two possibilities:
\begin{equation}
L_{\rm quin} = \frac{1}{2} \partial_a \phi \partial^a \phi - V(\phi); \quad L_{\rm tach}
= -V(\phi) [1-\partial_a\phi\partial^a\phi]^{1/2}
\label{lquineq}
\end{equation}
Both these Lagrangians involve one arbitrary function $V(\phi)$. The first one,
$L_{\rm quin}$, which is a natural generalization of the Lagrangian for
a non-relativistic particle, $L=(1/2)\dot q^2 -V(q)$, is usually called quintessence (for
a sample of models, see \cite{phiindustry}).
When it acts as a source in Friedman universe,
it is characterized by a time dependent
$w(t)$ with
\begin{equation}
\rho_q(t) = \frac{1}{2} \dot\phi^2 + V; \quad p_q(t) = \frac{1}{2} \dot\phi^2 - V; \quad w_q
= \frac{1-(2V/\dot\phi^2)}{1+ (2V/\dot\phi^2)}
\label{quintdetail}
\end{equation}
The structure of the second Lagrangian in Eq.~(\ref{lquineq}) can be understood by a simple analogy from
special relativity (see the first reference in \cite{tptirth}). A relativistic particle with (one dimensional) position
$q(t)$ and mass $m$ is described by the Lagrangian $L = -m \sqrt{1-\dot q^2}$.
It has the energy $E = m/ \sqrt{1-\dot q^2}$ and momentum $k = m \dot
q/\sqrt{1-\dot q^2} $ which are related by $E^2 = k^2 + m^2$. As is well
known, this allows the possibility of having \textit{massless} particles with finite
energy for which $E^2=k^2$. This is achieved by taking the limit of $m \to 0$
and $\dot q \to 1$, while keeping the ratio in $E = m/ \sqrt{1-\dot q^2}$
finite. The momentum acquires a life of its own, unconnected with the
velocity $\dot q$, and the energy is expressed in terms of the momentum
(rather than in terms of $\dot q$) in the Hamiltonian formulation. We can now
construct a field theory by upgrading $q(t)$ to a field $\phi$. Relativistic
invariance now requires $\phi $ to depend on both space and time [$\phi =
\phi(t, {\bf x})$] and $\dot q^2$ to be replaced by $\partial_i \phi \partial^i
\phi$. It is also possible now to treat the mass parameter $m$ as a function of
$\phi$, say, $V(\phi)$ thereby obtaining a field theoretic Lagrangian $L =-
V(\phi) \sqrt{1 - \partial^i \phi \partial_i \phi}$. The Hamiltonian structure of this
theory is algebraically very similar to the special relativistic example we
started with. In particular, the theory allows solutions in which $V\to 0$,
$\partial_i \phi \partial^i \phi \to 1$ simultaneously, keeping the energy (density) finite. Such
solutions will have finite momentum density (analogous to a massless particle
with finite momentum $k$) and energy density. Since the solutions can now
depend on both space and time (unlike the special relativistic example in which
$q$ depended only on time), the momentum density can be an arbitrary function
of the spatial coordinate. This provides a rich gamut of possibilities in the
context of cosmology.
\cite{tptachyon,tptirth,bjp,tachyon},
This form of scalar field arises in string theories \cite{asen} and --- for technical reasons ---
is called a tachyonic scalar field.
(The structure of this Lagrangian is similar to those analyzed in a wide class of models
called {\it K-essence}; see for example, \cite{kessence})
The stress tensor for the tachyonic scalar field can be written
as the sum of a pressure less dust component and a cosmological constant (see the first reference in \cite{tptirth}).
To show this explicitly,
we break up the density $\rho$ and the pressure $p$
and write them in a more suggestive form as
$\rho = \rho_\Lambda + \rho_{\rm DM}; ~~
p = p_V + p_{\rm DM}$
where
\begin{equation}
\rho_{\rm DM} = \frac{V(\phi) \partial^i \phi \partial_i \phi}
{\sqrt{1 - \partial^i \phi \partial_i \phi}};\ \ p_{\rm DM} = 0;\ \
\rho_\Lambda = V(\phi) \sqrt{1 - \partial^i \phi \partial_i \phi};\ \
p_V = -\rho_\Lambda
\end{equation}
This means that the stress tensor can be thought of as made up of two components
-- one behaving like a pressure-less fluid, while the other having a negative
pressure. This suggests a possibility of providing a unified description of both dark matter
and dark energy using the same scalar field \cite{tptirth}.
When $\dot\phi$ is small (compared to $V$ in the case of quintessence or
compared to unity in the case of tachyonic field), both these sources have $w\to -1$ and
mimic a cosmological constant. When $\dot \phi \gg V$, the quintessence has $w\approx 1$ leading to
$\rho_q\propto (1+z)^6$; the tachyonic field, on the other hand, has $w\approx 0$ for $\dot\phi\to 1$
and behaves like non-relativistic matter. In both the cases, $-1<w<1$, though it is possible to construct more complicated scalar field Lagrangians \cite{phantom} with even $w<-1$ describing
what is called {\it phantom} matter.
(For some alternatives to scalar field models, based on brane world scenarios, see,
for example, \cite{branes}.)
Since the quintessence field (or the tachyonic field) has
an undetermined free function $V(\phi)$, it is possible to choose this function
in order to produce a given $H(a)$.
To see this explicitly, let
us assume that the universe has two forms of energy density with $\rho(a) =\rho_{\rm known}
(a) + \rho_\phi(a)$ where $\rho_{\rm known}(a)$ arises from any known forms of source
(matter, radiation, ...) and
$\rho_\phi(a) $ is due to a scalar field.
Let us first consider quintessence. Here, the potential is given implicitly by the form
\cite{ellis,tptachyon}
\begin{equation}
V(a) = \frac{1}{16\pi G} H (1-Q)\left[6H + 2aH' - \frac{aH Q'}{1-Q}\right]
\label{voft}
\end{equation}
\begin{equation}
\phi (a) = \left[ \frac{1}{8\pi G}\right]^{1/2} \int \frac{da}{a}
\left[ aQ' - (1-Q)\frac{d \ln H^2}{d\ln a}\right]^{1/2}
\label{phioft}
\end{equation}
where $Q (a) \equiv [8\pi G \rho_{\rm known}(a) / 3H^2(a)]$ and prime denotes differentiation with respect to $a$.
Given any
$H(a),Q(a)$, these equations determine $V(a)$ and $\phi(a)$ and thus the potential $V(\phi)$.
Every quintessence model studied in the literature can be obtained from these equations.
We shall now briefly mention some examples:
\begin{itemize}
\item
Power law expansion of the universe can be generated by
a quintessence model with $V(\phi)=\phi^{-\alpha}$. In this case, the energy
density of the scalar field varies as $\rho_\phi \propto t^{-2\alpha/(2+\alpha)} $;
if the background density $\rho_{\rm bg}$ varies as $\rho_{\rm bg} \propto t^{-2}$,
the ratio of the two energy densities changes as $(\rho_\phi/\rho_{\rm bg} =
t^{4/(2+\alpha)}$).
Obviously, the scalar field density can dominate over the background at late
times for $\alpha >0$.
\item
A different class of models arise if the potential is taken
to be exponential with, say, $V(\phi) \propto \exp(-\lambda \phi/M_{\rm Pl})$.
When $k=0$, both $\rho_\phi$ and $\rho_{\rm bg}$ scale in the same manner
leading to
\begin{equation}
\frac{\rho_\phi}{\rho_{\rm bg} +\rho_\phi} = \frac{3(1+w_{\rm bg})}{\lambda^2}
\end{equation}
where $w_{\rm bg}$ refers to the background parameter value. In this
case, the dark energy density is said to ``track'' the background energy
density. While this could be a model for dark matter, there are
strong constraints on the total energy density of the universe
at the epoch of nucleosynthesis. This requires $\Omega_\phi \lesssim 0.2$
requiring dark energy to be sub dominant at all epochs.
\item
Many other forms of $H(a)$ can be reproduced by a combination of non-relativistic matter and a suitable form of scalar field with a potential $V(\phi)$. In fact, one can make the dark energy to vary with $a$ in an unspecified manner \cite{coop98} as $a^{-n}$. In this case we need $H^2(a)=H_0^2[ \Omega_{\rm NR} a^{-3}+(1-\Omega_{\rm NR})a^{-n}]$ which can arise if the universe is populated with non-relativistic matter with density parameter $\Omega_{\rm NR}$ and a scalar field with the potential, determined using equations (\ref{voft}), (\ref{phioft}). We get
\begin{equation}
V(\phi)=V_0 \sinh^{2n/(n-3)}[\alpha(\phi -\psi)]
\end{equation}
where
\begin{equation}
V_0={(6-n)H_0^2\over 16\pi G}
\left[ \frac{\Omega_{\rm NR}^n}{(1- \Omega_{\rm NR})^3}\right]^{\frac{1}{n-3}};\quad
\alpha=(3-n)(2\pi G/n)^{1/2}
\end{equation}
and $\psi$ is a constant.
\end{itemize}
Similar results exists for the tachyonic scalar field as well \cite{tptachyon}. For example, given
any $H(a)$, one can construct a tachyonic potential $V(\phi)$ so that the scalar field is the
source for the cosmology. The equations determining $V(\phi)$ are now given by:
\begin{equation}
\phi(a) = \int \frac{da}{aH} \left(\frac{aQ'}{3(1-Q)}
-{2\over 3}{a H'\over H}\right)^{1/2}
\label{finalone}
\end{equation}
\begin{equation}
V = {3H^2 \over 8\pi G}(1-Q) \left( 1 + {2\over 3}{a H'\over H}-\frac{aQ'}{3(1-Q)}\right)^{1/2}
\label{finaltwo}
\end{equation}
Equations (\ref{finalone}) and (\ref{finaltwo}) completely solve the problem. Given any
$H(a)$, these equations determine $V(a)$ and $\phi(a)$ and thus the potential $V(\phi)$.
As an example, consider a universe with power law expansion
$a= t^n$. If it is populated only by a tachyonic scalar field, then $Q=0$; further,
$(a H'/H)$ in equation (\ref{finalone}) is a constant
making $\dot \phi $ a constant. The complete solution
is then given by
\begin{equation}
\phi(t) = \left({2\over 3n}\right)^{1/2} t + \phi_0; \quad
V(t) = {3n^2\over 8\pi G}\left( 1- {2\over 3n}\right)^{1/2} {1\over t^2}
\end{equation}
where $n>(2/3)$.
Combining the two, we find the potential to be
\begin{equation}
V(\phi) = {n\over 4\pi G}\left( 1- {2\over 3n}\right)^{1/2}
(\phi - \phi_0)^{-2}
\label{tachpot}
\end{equation}
For such a potential, it is possible to have arbitrarily rapid expansion with large $n$.
(For the cosmological model, based on this potential, see \cite{bjp}.)
A wide variety of phenomenological models with time dependent
cosmological constant\ have been considered in the literature all of which can be
mapped to a
scalar field model with a suitable $V(\phi)$.
While the scalar field models enjoy considerable popularity (one reason being they are easy to construct!)
it is very doubtful whether they have helped us to understand the nature of the dark energy
at any deeper level. These
models, viewed objectively, suffer from several shortcomings:
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.5]{rho_p1.ps}
\end{center}
\caption{Constraints on the possible variation of the dark energy density with redshift. The darker shaded region (magenta) is excluded by SN observations while the lighter shaded region (green) is excluded by WMAP observations. It is obvious that WMAP puts stronger constraints on the possible
variations of dark energy density. The cosmological constant corresponds to the horizontal line
at unity. The region between the dotted lines has $w>-1$ at all epochs. (For more details, see
\cite{jbp}.) }
\label{fig:bjp2ps}
\end{figure}
\begin{itemize}
\item
They completely lack predictive power. As explicitly demonstrated above, virtually every form of $a(t)$ can be modeled by a suitable ``designer" $V(\phi)$.
\item
These models are degenerate in another sense. The previous discussion illustrates that even when $w(a)$ is known/specified, it is not possible to proceed further and determine
the nature of the scalar field Lagrangian. The explicit examples given above show that there
are {\em at least} two different forms of scalar field Lagrangians (corresponding to
the quintessence or the tachyonic field) which could lead to
the same $w(a)$. (See ref.\cite{tptirthsn1} for an explicit example of such a construction.)
\item
All the scalar field potentials require fine tuning of the parameters in order to be viable. This is obvious in the quintessence models in which adding a constant to the potential is the same as invoking a cosmological constant. So to make the quintessence models work, \textit{we first need to assume the cosmological constant\ is zero.} These models, therefore, merely push the cosmological constant problem to another level, making it somebody else's problem!.
\item
By and large, the potentials used in the literature have no natural field theoretical justification. All of them are non-renormalisable in the conventional sense and have to be interpreted as a low energy effective potential in an adhoc manner.
\item
One key difference between cosmological constant\ and scalar field models is that the latter lead to a $w(a)$ which varies with time. If observations have demanded this, or even if observations have ruled out $w=-1$ at the present epoch,
then one would have been forced to take alternative models seriously. However, all available observations are consistent with cosmological constant\ ($w=-1$) and --- in fact --- the possible variation of $w$ is strongly constrained \cite{jbp} as shown in Figure \ref{fig:bjp2ps}.
(Also see \cite{wconstraint}).
\end{itemize}
Given this situation, we shall now take a more serious look at the cosmological constant\ as the source of dark energy in the universe.
\section{...For the Snark was a Boojam, you see }
If we assume that the dark energy in the universe is due to a cosmological constant\, then we are introducing a second length scale, $L_\Lambda=H_\Lambda^{-1}$, into the theory (in addition to the Planck length $L_P$) such that
$ (L_P/L_\Lambda)\approx
10^{-60}$. Such a universe will be asymptotically deSitter with $a(t)\propto \exp (t/L_\Lambda) $ at late times.
We will now explore several peculiar features of such a universe.
\begin{figure}
\begin{center}
\includegraphics[angle=-90,scale=0.6]{plumianA}
\end{center}
\caption{The geometrical structure of a universe with two length scales $L_P$ and $L_\Lambda$ corresponding to the Planck length and the cosmological constant \cite{plumian,bjorken}. Such a universe spends most of its time in two De Sitter phases which are (approximately) time translation invariant. The first De Sitter phase corresponds to the inflation and the second corresponds to the accelerated expansion arising from the cosmological constant. Most of the perturbations generated during the inflation will leave the Hubble radius (at some A, say) and re-enter (at B). However, perturbations which exit the Hubble radius
earlier than C will never re-enter the Hubble radius, thereby introducing a specific dynamic range CE during the inflationary phase. The epoch F is characterized by the redshifted CMB temperature becoming equal to the De Sitter temperature $(H_\Lambda / 2\pi)$ which introduces another dynamic range DF in the accelerated expansion after which the universe is dominated by vacuum noise
of the De Sitter spacetime.}
\label{fig:tpplumian}
\end{figure}
Figure \ref{fig:tpplumian} summarizes these features \cite{plumian,bjorken}. Using the characteristic length scale of expansion,
the Hubble radius $d_H\equiv (\dot a/a)^{-1}$, we can distinguish between three different phases of such a universe. The first phase is when the universe went through a inflationary expansion with $d_H=$ constant; the second phase is the radiation/matter dominated phase in which most of the standard cosmology operates and $d_H$ increases monotonically; the third phase is that of re-inflation (or accelerated expansion) governed by the cosmological constant in which $d_H$ is again a constant. The first and last phases are time translation invariant;
that is, $t\to t+$ constant is an (approximate) invariance for the universe in these two phases. The universe satisfies the perfect cosmological principle and is in steady state during these phases!
In fact, one can easily imagine a scenario in which the two deSitter phases (first and last) are of arbitrarily long duration \cite{plumian}. If $\Omega_\Lambda\approx 0.7, \Omega_{DM}\approx 0.3$ the final deSitter phase \textit{does} last forever; as regards the inflationary phase, nothing prevents it from lasting for arbitrarily long duration. Viewed from this perspective, the in between phase --- in which most of the `interesting' cosmological phenomena occur --- is of negligible measure in the span of time. It merely connects two steady state phases of the universe.
(In a way, this scenario provides the ultimate generalisation of the Copernican principle. It was
well known that we are not in a special position in {\it space} in our universe. The composition of the universe also shows that we are not made of the most dominant constituent of the universe. Finally, in this picture, we are not even existing at a generic moment of {\it time} in the evolution of the universe!)
Given the two length scales $L_P$ and $L_\Lambda$, one can construct two energy scales
$\rho_P=1/L_P^4$ and $\rho_\Lambda=1/L_\Lambda^4$ in natural units ($c=\hbar=1$). The first is, of course, the Planck energy density while the second one also has a natural interpretation. The universe which is asymptotically deSitter has a horizon and associated thermodynamics \cite{ghds} with a temperature
$T=H_\Lambda/2\pi$ and the corresponding thermal energy density $\rho_{thermal}\propto T^4\propto 1/L_\Lambda^4=
\rho_\Lambda$. Thus $L_P$ determines the \textit{highest} possible energy density in the universe while $L_\Lambda$
determines the {\it lowest} possible energy density in this universe. As the energy density of normal matter drops below this value, the thermal ambience of the deSitter phase will remain constant and provide the irreducible `vacuum noise'. Note that the dark energy density is the the geometric mean $\rho_{DE}=\sqrt{\rho_\Lambda\rho_P}$ between the two energy densities. If we define a dark energy length scale $L_{DE}$ such that $\rho_{DE}=1/L_{DE}^4$ then $L_{DE}=\sqrt{L_PL_\Lambda}$ is the geometric mean of the two length scales in the universe. The figure \ref{fig:tpplumian} also shows the variation of $L_{DE}$ by broken horizontal lines.
While the two deSitter phases can last forever in principle, there is a natural cut off length scale in both of them
which makes the region of physical relevance to be finite \cite{plumian}. Let us first discuss the case of re-inflation in the late universe.
As the universe grows exponentially in the phase 3, the wavelength of CMBR photons are being redshifted rapidly. When the temperature of the CMBR radiation drops below the deSitter temperature (which happens when the wavelength of the typical CMBR photon is stretched to the $L_\Lambda$.)
the universe will be essentially dominated by the vacuum thermal noise of the deSitter phase.
This happens at the point marked F when the expansion factor is $a=a_F$ determined by the
equation $T_0 (a_0/a_{F}) = (1/2\pi L_\Lambda)$. Let $a=a_\Lambda$ be the epoch at which
cosmological constant started dominating over matter, so that $(a_\Lambda/a_0)^3=
(\Omega_{DM}/\Omega_\Lambda)$. Then we find that the dynamic range of
DF is
\begin{equation}
\frac{a_F}{a_\Lambda} = 2\pi T_0 L_\Lambda \left( \frac{\Omega_\Lambda}{\Omega_{DM}}\right)^{1/3}
\approx 3\times 10^{30}
\end{equation}
Interestingly enough, one can also impose a similar bound on the physically relevant duration of inflation.
We know that the quantum fluctuations generated during this inflationary phase could act as seeds of structure formation in the universe \cite{genofpert}. Consider a perturbation at some given wavelength scale which is stretched with the expansion of the universe as $\lambda\propto a(t)$.
(See the line marked AB in Figure \ref{fig:tpplumian}.)
During the inflationary phase, the Hubble radius remains constant while the wavelength increases, so that the perturbation will `exit' the Hubble radius at some time (the point A in Figure \ref{fig:tpplumian}). In the radiation dominated phase, the Hubble radius $d_H\propto t\propto a^2$ grows faster than the wavelength $ \lambda\propto a(t)$. Hence, normally, the perturbation will `re-enter' the Hubble radius at some time (the point B in Figure \ref{fig:tpplumian}).
If there was no re-inflation, this will make {\it all} wavelengths re-enter the Hubble radius sooner or later.
But if the universe undergoes re-inflation, then the Hubble radius `flattens out' at late times and some of the perturbations will {\it never} reenter the Hubble radius ! The limiting perturbation which just `grazes' the Hubble radius as the universe enters the re-inflationary phase is shown by the line marked CD in Figure \ref{fig:tpplumian}. If we use the criterion that we need the perturbation to reenter the Hubble radius, we get a natural bound on the duration of inflation which is of direct astrophysical relevance. This portion of the inflationary regime is marked by CE
and can be calculated as follows: Consider a perturbation which leaves the Hubble radius ($H_{in}^{-1}$) during the inflationary epoch at $a= a_i$. It will grow to the size $H_{in}^{-1}(a/a_i)$ at a later epoch.
We want to determine $a_i$ such that this length scale grows to
$L_\Lambda$ just when the dark energy starts dominating over matter; that is at
the epoch $a=a_\Lambda = a_0(\Omega_{DM}/\Omega_{\Lambda})^{1/3}$.
This gives
$H_{in}^{-1}(a_\Lambda/a_i)=L_\Lambda$ so that $a_i=(H_{in}^{-1}/L_\Lambda)(\Omega_{DM}/\Omega_{\Lambda})^{1/3}a_0$. On the other hand, the inflation ends at
$a=a_{end}$ where $a_{end}/a_0 = T_0/T_{\rm reheat}$ where $T_{\rm reheat} $ is the temperature to which the universe has been reheated at the end of inflation. Using these two results we can determine the dynamic range of CE to be
\begin{equation}
\frac{a_{\rm end} }{a_i} = \left( \frac{T_0 L_\Lambda}{T_{\rm reheat} H_{in}^{-1}}\right)
\left( \frac{\Omega_\Lambda}{\Omega_{DM}}\right)^{1/3}=\frac{(a_F/a_\Lambda)}{2\pi T_{\rm reheat} H_{in}^{-1}} \cong 10^{25}
\end{equation}
where we have used the fact that, for a GUTs scale inflation with $E_{GUT}=10^{14} GeV,T_{\rm reheat}=E_{GUT},\rho_{in}=E_{GUT}^4$
we have $2\pi H^{-1}_{in}T_{\rm reheat}=(3\pi/2)^{1/2}(E_P/E_{GUT})\approx 10^5$.
For a Planck scale inflation with $2\pi H_{in}^{-1} T_{\rm reheat} = \mathcal{O} (1)$, the phases CE and DF are approximately equal. The region in the quadrilateral CEDF is the most relevant part of standard cosmology, though the evolution of the universe can extend to arbitrarily large stretches in both directions in time. This figure is definitely telling us something regarding the time translation invariance of the universe (`the perfect cosmological principle') and --- more importantly ---
\textit{about the breaking of this symmetry}, but it is not easy to translate this concept into a workable theory.
Let us now turn our attention to few of the many attempts to understand the cosmological constant. This is, of course, a non-representative sample (dictated by personal bias!) and a host of other approaches exist in literature, some of which can be found in \cite{catchall}.
\subsection{Dark energy from a nonlinear correction term}
One of the \textit{least} esoteric ideas regarding the dark energy
is that the cosmological constant term in the FRW equations arises because we have not calculated the energy density driving the expansion of the universe correctly. The motivation for such a suggestion arises from the following fact: The energy momentum tensor of the real universe, $T_{ab}(t,{\bf x})$ is inhomogeneous and anisotropic and will lead to a very complex metric $g_{ab}$ if only we could solve the exact Einstein's equations
$G_{ab}[g]=\kappa T_{ab}$.
The metric describing the large scale structure of the universe should be obtained by averaging this exact solution over a large enough scale, to get $\langle g_{ab}\rangle $. But what we actually do is to average the stress tensor {\it first} to get $\langle T_{ab}\rangle $ and {\it then} solve Einstein's equations. But since $G_{ab}[g]$ is nonlinear function of the metric, $\langle G_{ab}[g]\rangle \neq G_{ab}[\langle g\rangle ]$ and there is a discrepancy. This is most easily seen by writing
\begin{equation}
G_{ab}[\langle g\rangle ]=\kappa [\langle T_{ab}\rangle + \kappa^{-1}(G_{ab}[\langle g\rangle ]-\langle G_{ab}[g]\rangle )]\equiv \kappa [\langle T_{ab}\rangle + T_{ab}^{corr}]
\end{equation}
If --- based on observations --- we take the $\langle g_{ab}\rangle $ to be the standard Friedman metric, this equation shows that it has, as its source, \textit{two} terms:
The first is the standard average stress tensor and the second is a purely geometrical correction term
$T_{ab}^{corr}=\kappa^{-1}(G_{ab}[\langle g\rangle ]-\langle G_{ab}[g]\rangle )$ which arises because of nonlinearities in the Einstein's theory that leads to $\langle G_{ab}[g]\rangle \neq G_{ab}[\langle g\rangle ]$. If this term can mimic the cosmological constant\ at large scales there will be no need for dark energy! Unfortunately, it is not easy to settle this question to complete satisfaction \cite{avgg}. One possibility is to use some analytic approximations to nonlinear perturbations (usually called non-linear scaling relations, see e.g. \cite{nsr}) to estimate this term.
This does not lead to a stress tensor that mimics dark energy (Padmanabhan, unpublished) but this is not a conclusive proof either way. We mention this mainly because this issue deserves more attention than it has received.
\subsection{Unimodular gravity}
Another possible way of addressing this issue is to simply eliminate from the gravitational theory those modes which couple to cosmological constant. If, for example, we have a theory in which the source of gravity is
$(\rho +p)$ rather than $(\rho +3p)$ in Eq. (\ref{nextnine}), then cosmological constant\ will not couple to gravity at all. (The non linear coupling of matter with gravity has several subtleties; see eg. \cite{gravitonmyth}.) Unfortunately
it is not possible to develop a covariant theory of gravity using $(\rho +p)$ as the source. But we can achieve the same objective in different manner. Any metric $g_{ab}$ can be expressed in the form $g_{ab}=f^2(x)q_{ab}$ such that
${\rm det}\, q=1$ so that ${\rm det}\, g=f^4$. From the action functional for gravity
\begin{equation}
A=\frac{1}{16\pi G}\int d^4x (R -2\Lambda)\sqrt{-g}
=\frac{1}{16\pi G}\int d^4x R \sqrt{-g}-\frac{\Lambda}{8\pi G}\int d^4x f^4(x)
\end{equation}
it is obvious that the cosmological constant\ couples {\it only} to the conformal factor $f$. So if we consider a theory of gravity in which $f^4=\sqrt{-g}$ is kept constant and only $q_{ab}$ is varied, then such a model will be oblivious of
direct coupling to cosmological constant. If the action (without the $\Lambda$ term) is varied, keeping ${\rm det}\, g=-1$, say, then one is lead to a {\it unimodular theory of gravity} with the equations of motion
$R_{ab}-(1/4)g_{ab}R=\kappa(T_{ab}-(1/4)g_{ab}T)$ with zero trace on both sides. Using the Bianchi identity, it is now easy to show that this is equivalent to a theory with an {\it arbitrary} cosmological constant. That is, cosmological constant\ arises as an undetermined integration constant in this model \cite{unimod}.
While this is interesting, we need an extra physical principle to fix its value.
One possible way of doing this is to interpret the $\Lambda$ term in the action as a Lagrange multiplier for the proper volume of the spacetime. Then it is reasonable to choose the cosmological constant\ such that the total proper volume of the universe is equal to a specified number. While this will lead to a cosmological constant\ which has the correct order of magnitude, it has several obvious problems. First, the proper four volume of the universe is infinite unless we make the spatial sections compact and restrict the range of time integration. Second, this will lead to a dark energy density which varies as $t^{-2}$ (corresponding to $w= -1/3$ ) which is ruled out by observations.
\subsection{Scale dependence of the vacuum energy}
The conventional discussion of the relation between cosmological constant and vacuum energy density is based on
evaluating the zero point energy of quantum fields with an ultraviolet cutoff and using the result as a
source of gravity.
Any reasonable cutoff will lead to a vacuum energy density $\rho_{\rm vac}$ which is unacceptably high.
This argument,
however, is too simplistic since the zero point energy --- obtained by summing over the
$(1/2)\hbar \omega_k$ --- has no observable consequence in any other phenomena and can be subtracted out by redefining the Hamiltonian. The observed non trivial features of the vacuum state of QED, for example, arise from the {\it fluctuations} (or modifications) of this vacuum energy rather than the vacuum energy itself.
This was, in fact, known fairly early in the history of cosmological constant problem and, in fact, is stressed by Zeldovich \cite{zeldo} who explicitly calculated one possible contribution to {\it fluctuations} after subtracting away the mean value.
This
suggests that we should consider the fluctuations in the vacuum energy density in addressing the
cosmological constant problem.
If the vacuum probed by the gravity can readjust to take away the bulk energy density $\rho_P\simeq L_P^{-4}$, quantum \textit{fluctuations} can generate
the observed value $\rho_{\rm DE}$. One of the simplest models \cite{tpcqglamda} which achieves this uses the fact that, in the semiclassical limit, the wave function describing the universe of proper four-volume ${\cal V}$ will vary as
$\Psi\propto \exp(-iA_0) \propto
\exp[ -i(\Lambda_{\rm eff}\mathcal V/ L_P^2)]$. If we treat
$(\Lambda/L_P^2,{\cal V})$ as conjugate variables then uncertainty principle suggests $\Delta\Lambda\approx L_P^2/\Delta{\cal V}$. If
the four volume is built out of Planck scale substructures, giving $ {\cal V}=NL_P^4$, then the Poisson fluctuations will lead to $\Delta{\cal V}\approx \sqrt{\cal V} L_P^2$ giving
$ \Delta\Lambda=L_P^2/ \Delta{\mathcal V}\approx1/\sqrt{{\mathcal V}}\approx H_0^2
$. (This idea can be a more quantitative; see \cite{tpcqglamda}).
Similar viewpoint arises, more formally, when we study the question of \emph{detecting} the energy
density using gravitational field as a probe.
Recall that an Unruh-DeWitt detector with a local coupling $L_I=M(\tau)\phi[x(\tau)]$ to the {\it field} $\phi$
actually responds to $\langle 0|\phi(x)\phi(y)|0\rangle$ rather than to the field itself \cite{probe}. Similarly, one can use the gravitational field as a natural ``detector" of energy momentum tensor $T_{ab}$ with the standard coupling $L=\kappa h_{ab}T^{ab}$. Such a model was analysed in detail in ref.~\cite{tptptmunu} and it was shown that the gravitational field responds to the two point function $\langle 0|T_{ab}(x)T_{cd}(y)|0\rangle $. In fact, it is essentially this fluctuations in the energy density which is computed in the inflationary models \cite{inflation} as the seed {\it source} for gravitational field, as stressed in
ref.~\cite{tplp}. All these suggest treating the energy fluctuations as the physical quantity ``detected" by gravity, when
one needs to incorporate quantum effects.
If the cosmological constant\ arises due to the energy density of the vacuum, then one needs to understand the structure of the quantum vacuum at cosmological scales. Quantum theory, especially the paradigm of renormalization group has taught us that the energy density --- and even the concept of the vacuum
state --- depends on the scale at which it is probed. The vacuum state which we use to study the
lattice vibrations in a solid, say, is not the same as vacuum state of the QED. Using this feature, it is possible to construct systems in condensed matter physics \cite{volovikilya} wherein the quantity analogous to
vacuum energy density has to vanish on the average because of dynamical reasons.
In fact, it seems \textit{inevitable} that in a universe with two length scale $L_\Lambda,L_P$, the vacuum
fluctuations will contribute an energy density of the correct order of magnitude $\rho_{DE}=\sqrt{\rho_\Lambda\rho_P}$. The hierarchy of energy scales in such a universe has \cite{plumian,tpvacfluc}
the pattern
\begin{equation}
\rho_{\rm vac}={\frac{1}{ L^4_P}}
+{\frac{1}{L_P^4}\left(\frac{L_P}{L_\Lambda}\right)^2}
+{\frac{1}{L_P^4}\left(\frac{L_P}{L_\Lambda}\right)^4}
+ \cdots
\end{equation}
The first term is the bulk energy density which needs to be renormalised away (by a process which we do not understand at present); the third term is just the thermal energy density of the deSitter vacuum state; what is interesting is that quantum fluctuations in the matter fields \textit{inevitably generate} the second term.
The key new ingredient arises from the fact that the properties of the vacuum state depends on the scale at which it is probed and it is not appropriate to ask questions without specifying this scale.
(These ideas have been developed more generally in ref. \cite{holo}.)
If the spacetime has a cosmological horizon which blocks information, the natural scale is provided by the size of the horizon, $L_\Lambda$, and we should use observables defined within the accessible region.
The operator $H(<L_\Lambda)$, corresponding to the total energy inside
a region bounded by a cosmological horizon, will exhibit fluctuations $\Delta E$ since vacuum state is not an eigenstate of
{\it this} operator. The corresponding fluctuations in the energy density, $\Delta\rho\propto (\Delta E)/L_\Lambda^3=f(L_P,L_\Lambda)$ will now depend on both the ultraviolet cutoff $L_P$ as well as $L_\Lambda$.
To obtain
$\Delta \rho_{\rm vac} \propto \Delta E/L_\Lambda^3$ which scales as $(L_P L_\Lambda)^{-2}$
we need to have $(\Delta E)^2\propto L_P^{-4} L_\Lambda^2$; that is, the square of the energy fluctuations
should scale as the surface area of the bounding surface which is provided by the cosmic horizon.
Remarkably enough, a rigorous calculation \cite{tpvacfluc} of the dispersion in the energy shows that
for $L_\Lambda \gg L_P$, the final result indeed has the scaling
\begin{equation}
(\Delta E )^2 = c_1 \frac{L_\Lambda^2}{L_P^4}
\label{deltae}
\end{equation}
where the constant $c_1$ depends on the manner in which ultra violet cutoff is imposed.
Similar calculations have been done (with a completely different motivation, in the context of
entanglement entropy)
by several people and it is known that the area scaling found in Eq.~(\ref{deltae}), proportional to $
L_\Lambda^2$, is a generic feature \cite{area}.
For a simple exponential UV-cutoff, $c_1 = (1/30\pi^2)$ but cannot be computed
reliably without knowing the full theory.
We thus find that the fluctuations in the energy density of the vacuum in a sphere of radius $L_\Lambda$
is given by
\begin{equation}
\Delta \rho_{\rm vac} = \frac{\Delta E}{L_\Lambda^3} \propto L_P^{-2}L_\Lambda^{-2} \propto \frac{H_\Lambda^2}{G}
\label{final}
\end{equation}
The numerical coefficient will depend on $c_1$ as well as the precise nature of infrared cutoff
radius (like whether it is $L_\Lambda$ or $L_\Lambda/2\pi$ etc.). It would be pretentious to cook up the factors
to obtain the observed value for dark energy density.
But it is a fact of life that a fluctuation of magnitude $\Delta\rho_{vac}\simeq H_\Lambda^2/G$ will exist in the
energy density inside a sphere of radius $H_\Lambda^{-1}$ if Planck length is the UV cut off. {\it One cannot get away from it.}
On the other hand, observations suggest that there is a $\rho_{vac}$ of similar magnitude in the universe. It seems
natural to identify the two, after subtracting out the mean value by hand. Our approach explains why there is a \textit{surviving} cosmological constant which satisfies
$\rho_{DE}=\sqrt{\rho_\Lambda\rho_P}$
which --- in our opinion --- is {\it the} problem.
There is a completely different way of interpreting this result based on some imaginative ideas suggested by Bjorken
\cite{bjorken} recently. The key idea is to parametrise the universes by the value of $L_\Lambda$ which they have. It is a fixed, pure number for each universe in an ensemble of universes but all the other parameters of the physics are assumed to be correlated with
$L_\Lambda$. This is motivated by a series of arguments in ref. \cite{bjorken} and, in this approach, $\rho_{vac}\propto L_\Lambda^{-2}$ almost by definition; the hard work was in determining how other parameters scale with $L_\Lambda$. In the approach suggested here, a dynamical interpretation of the scaling $\rho_{vac}\propto L_\Lambda^{-2}$ is given as due to vacuum fluctuations of fields. We now reinterpret each member of of the ensemble of universes as having zero energy density for vacuum (as any decent vacuum should have) but the effective $\rho_{vac}$ arises from the quantum fluctuations {\it with the correct scaling}. One can then invoke standard anthropic-like arguments (but with very significant differences as stressed in ref. \cite{bjorken} ) to choose a range for the size of our universe. This appears to be much more attractive way of interpreting the result.
Finally, to be fair, this attempt should be
judged in the backdrop of other suggested solutions almost all of which require introducing
extra degrees of freedom in the form of scalar fields, modifying gravity or introducing higher dimensions etc. {\it and} fine tuning the potentials. At a fundamental level such approaches are unlikely to provide the final solution.
\section*{Acknowledgement}
I thank K. Subramanian for two decades of discussion about various aspects of cosmological constant and for sharing and reinforcing the view that any quick-fix solution to this problem will be futile. I also thank
Apoorva Patel for useful discussions.
|
{
"timestamp": "2004-11-02T05:48:09",
"yymm": "0411",
"arxiv_id": "astro-ph/0411044",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411044"
}
|
\section{Introduction}
The quest for a satisfactory theory of quantum dots is driven not only
by their obvious importance as mesoscopic devices revealed by a series
of groundbreaking experiments\cite{recent-expts}, but also by their
challenge as a unique confluence of disorder, interactions and
finite-size effects\cite{reviews}. For weak interactions, the
Universal Hamiltonian\cite{H_U,univ-ham} (UH) provides a satisfactory
description. For ballistic/chaotic quantum dots, we have
espoused\cite{mm,qd-ms1,longpaper} an approach based on the fermionic
Renormalization Group\cite{rg-shankar} (RG), $1/N$ expansions and the
fact that energy eigenstates around the Fermi energy in disordered
systems ought to be described by Random Matrix Theory
(RMT)\cite{alt1,RMT}. Our approach not only explains the UH as a fixed
point of the RG but also describes the physics outside its basin of
attraction. It predicts a phase transition at strong coupling and
allows a fairly detailed study\cite{longpaper} of the new phase and
the quantum critical region\cite{critical-fan} separating it from that
governed by the UH.
Our results, however, were predicated on a variety of RMT and RG
assumptions. To test our assumptions and the conclusions deduced from
them, we performed a detailed numerical study on a ballistic but
chaotic billiard (the Robnik-Berry billiard\cite{robnik-berry}) and we
report our findings here. Our expectations are borne out, with one
notable exception.
We recall our strategy and assumptions briefly so that the reader may
see in advance what sort of ideas are put to test in our study. In the
primordial problem of interest to us one has electrons confined to a
ballistic dot of size $L$, with no impurities inside, and edges so
irregular that classical motion is chaotic. The electrons experience
the Coulomb interaction. In momentum space, all momenta within the
bandwidth (of order $k_F$, the Fermi momentum) exist. The
semiclassical ergodicization time for an electron within the dot is a
few bounces, or $\tau_{erg}\simeq L/v_F$. By the uncertainty principle
this leads to an important energy scale, the Thouless energy
$E_T\simeq\hbar v_F/L$ which has a dual significance. First, it
controls the dimensionless conductance of a dot strongly coupled to
leads, as follows. Since the transport through the dot takes place in
a time such that energy is uncertain by an amount $E_T$, all single
particle states that fit into this band will each contribute a unit of
dimensionless conductance. If the average single-particle level
spacing is $\delta$, then the dimensionless conductance is
$g=E_T/\delta$. Second, in the other limit of dots very weakly coupled
to leads (which we focus on in this work), the Thouless band of width
$E_T$ centered on $E_F$, marks the scale deep within which RMT should
apply to the energies and eigenfunctions\cite{alt1}. In this context
$g$ is better denoted the Thouless number.
Since we are only interested in a narrow band of energies of width
$E_T <<E_F$, the first step in the program\cite{mm} is to use the RG
for fermions\cite{rg-shankar} to get an effective low energy theory by
eliminating all momentum states outside $E_T$. Should we worry that
we are not eliminating exact single particle eigenstates (labelled
here by ${\alpha}$)? No, because the disorder due to the boundaries will mix
momentum states at roughly the same energy, and it does not matter
whether we eliminate momentum states within any annulus of energy
thickness $E_T$ or the single particle states they evolve
into. Indeed, even the mixing within $E_T$ is due to the fact that
momentum itself not well defined in a finite dot, a point we will
elaborate on shortly. However, once we come down to within $E_T$
of $E_F$, we cannot eliminate the remaining states in one shot since
it is the flow of couplings {\em within} this band that is all
important in the RG.
Now it is
known\cite{rg-shankar} that the clean system RG (justified above)
leads to Landau's Fermi liquid interaction\cite{agd}
\begin{equation}
V=\sum_{{\mathbf k} {\mathbf k}'} F(\theta_{{\mathbf k}}-\theta_{{\mathbf k}'})\delta n({\mathbf k}
)\delta n({\mathbf k}' )
\end{equation}
at an energy scale $E_L$ which is small compared
to $E_F$. But since $E_L$ is a bulk scale it can always be made larger
than $E_T$ which vanishes as $L\to
\infty$. Thus Murthy and Mathur\cite{mm} perform their RG on the
hamiltonian (focussing on the spinless case for simplicity)
\begin{equation} H=\sum_{{\alpha}}c^{\dag}_{{\alpha}}c_{{\alpha}}\varepsilon_{{\alpha}}+
\sum\limits_{{\alpha} {\beta} {\gamma} {\delta}}V_{{\alpha} {\beta} {\gamma}
{\delta}}c^{\dag}_{{\alpha}}c^{\dag}_{{\beta}}c_{{\gamma}}c_{{\delta}}
\end{equation}
where
\begin{eqnarray}
V_{{\alpha} {\beta}
{\gamma} {\delta}} =&{1\over4} \sum\limits_{ {\mathbf k} {\mathbf k}'}F(\theta_{{\mathbf k}}-\theta_{{\mathbf k}'})(\phi^{*}_{{\alpha}}({\mathbf k} )\phi^*_{{\beta}}({\mathbf k}')-\phi^*_{{\beta}}({\mathbf k})\phi^*_{{\alpha}}({\mathbf k}'))\nonumber\\
&\times(\phi_{\gamma}({\mathbf k}')\phi_{\delta}({\mathbf k})-\phi_{\delta}({\mathbf k}')\phi_{\gamma}({\mathbf k})) \label{wof1}
\end{eqnarray}
is simply the Landau interaction written in the basis of exact
eigenstates, a statement that needs some elaboration. In usual RMT
treatments, $\phi_{{\alpha}}({\mathbf k} )$ is the exact eigenstate ${\alpha}$ written
in the infinite dimensional basis of all momentum states. In our
version which uses the RG to reduce the Hilbert space, the states
labeled by ${\mathbf k}$ are approximate momentum states with an uncertainty
$\delta k\simeq 1/L$ in both directions. The number of such wave
packets that fit into an annulus of radius $k_F$ and thickness
$E_T/v_F$ is ${\mathcal{O} } (k_F L)=g$. We call them the
Wheel-of-Fortune (WOF) states, see Figure (\ref{wof}). One way to
construct such packets is to pick $g$ plane waves of equally spaced
momenta on the Fermi circle and to chop them off at the edges of the
dot to respect the boundary conditions. This is what we mean by
${\mathbf k}$ in $\phi_{{\alpha}}({\mathbf k} )=\langle {\mathbf k} |{\alpha}\rangle$.
\begin{figure}[ht]
\includegraphics*[width=2.4in,angle=0]{fig1.eps}
\caption{The $g$ wheel-of-fortune states in the Thouless band.
They are packets in ${\mathbf k}$ space with average momenta equally
spaced on the Fermi circle.} \label{wof}
\end{figure}
We are now ready to state our two assumptions.\\
{\em {\bf Assumption 1}: We assume that the $g$ approximate
momentum states generated as above form a complete basis for the
$g$ exact eigenstates in the Thouless shell}.
{\bf Assumption II}: The energy eigenvalues $\varepsilon_{{\alpha}}$
obey RMT statistics as do the wavefunctions. For example we assume
that the ensemble averages (denoted by $\langle \rangle$) obey
\begin{equation} \langle \phi^{*}_{\mu}({\mathbf k}_1 )\phi^{}_{\mu}({\mathbf k}_2
)\phi^{*}_{\nu}({\mathbf k}_3 )\phi^{}_{\nu}({\mathbf k}_4) \rangle
={\delta_{12}\delta_{34}\over g^2} +O(1/g^3) \label{4pt}\end{equation}
To some, Assumption I seems remarkable -- How can we furnish {\em in
advance, independent of the dot shape} a basis of $g$ states for
expanding the $g$ exact eigenstates within $E_T$? After all, these
eigenstates are supposed to resemble those of a random matrix. The
point is that no matter how chaotic the dot, it can only mix states at
the same energy. While this sounds like Berry's
ansatz\cite{berry-ansatz} it is somewhat different in both scope and
content: Berry's ansatz states that every exact eigenstate ${\alpha}$ can be
expanded in terms of an {\it infinite number} of ${\mathbf k}$ states (with
the same energy $\epsilon_{{\alpha}}={\mathbf k}^2/2m$)in the bulk, while we claim
that only $g$ of WOF states are needed. Secondly, we claim that the
{\it same} $g$ WOF states can be used to describe all the states
within the Thouless band.
In this work we will show that these states are nearly orthonormal and
that the exact state right in the middle of the band has 99.9\%
overlap with the WOF states. The success of this extension of Berry's
conjecture to a finite dot exceeds our expectations in this
regard. However, we find that the $g$ WOF states become less effective
at describing states as we move away from $E_F$: the overlap drops to
50\% at $E_{{\alpha}}=E_F\pm E_T/2$. This
prepares us for the possibility that nonuniversal quantities may be
quantitatively inaccurate in our approach.
As for Assumption II, we have verified RMT behavior for the
eigenvalues (as have others before
us\cite{stone-bruus,alhassid-lewenkopf}) but not the eigenfunctions.
What we did instead was to see what extent the solution of a specific
dot resembled the picture we drew based on these two assumptions. We
begin by describing how one starts from Eqn. (\ref{wof1}), which
describes the effective hamiltonian, and use our two assumptions with
large-$N$ ideas to make our predictions\cite{qd-ms1,longpaper}. These
predictions are asymptotically exact as $g\to\infty$.
First one expands the Landau function as
\begin{equation} F(\theta ) = \sum_m u_m
e^{im\theta}.
\end{equation}
Barring accidents, the phase transition occurs in one channel with
some $m$ (recall superconductivity). This allows us to focus on a
single $u_m\ne 0$, ignoring all others. Then we carry out a
Hubbard-Stratovich transformation on the interaction using a
collective field $\mbox{\boldmath $\sigma $}$. We then formally integrate out the
fermions and get an effective action $S(\mbox{\boldmath $\sigma $} )$ for $\mbox{\boldmath $\sigma $}$. In
this process we make use of assumption II. The action in terms of
$\mbox{\boldmath $\sigma $}$ is obtained by summing one loop Feynman diagrams with
varying numbers of external $\mbox{\boldmath $\sigma $}$'s connected to a single fermion
line running around the loop. Each diagram is a sum over fermion
energy denominators multiplying products of a string of $\phi_{{\alpha}}({\mathbf k}
)$'s. We are able to show that these products may be replaced by
their ensemble averages in the large $g$ limit. In other words the sum
over so many terms in each diagrams leads to self-averaging. For the
averages we use relations like Eqn. (\ref{4pt}). When this is done,
the effective action can be cast into a form which has a $g^2$ in
front of it\cite{qd-ms1,longpaper}, so that the saddle point gives
exact answers as $g \to \infty$.
At this point let us collect all the results and predictions of the
RMT + large-$N$ theory\cite{longpaper} with a view to comparing them
with similar results without using Assumptions I and II on the
Robnik-Berry billiard.
\begin{itemize}
\item{} In the large-$g$ limit there is a sharp transition to a
phase in which $\mbox{\boldmath $\sigma $}$ acquires a vacuum expectation value. The
critical value of $u_m$ in our approximation turns out to be
$-1/\log{2}$ in the spinless case and $-1/2\log{2}$ in the spinful
case. The true critical value is most likely to be the bulk value
$-2$ (spinless) or $-1$ (spinful), as has been found in an
explicitly solvable model by Adam, Brouwer, and Sharma
recently\cite{nowindow}. This is an example of the nonuniversal
quantity alluded to earlier, that we cannot predict exactly in our
approach even as $g\to \infty$.
\item{} For finite $g$, instead of a sharp phase transition, there is
a crossover from the weak-coupling regime through a quantum critical
regime to a strong-coupling regime. Due to the explicit
symmetry-breaking at order $1/g$, there is always some nonzero order
parameter, which increases to a number of order $g$ (in the
normalization we use here, which is different from that of
ref.\cite{longpaper}) in the strong-coupling regime.
\item{} For symmetry-breaking in odd angular momentum channels there are
two exactly degenerate minima for every sample arising from
time-reversal invariance.
\item{} The ground state energy at the minimum in the strong-coupling
regime is lower than that in the weak-coupling regime by a number of
order $g^2{\delta}$.
\item{} The effective potential landscape in the strong-coupling
regime is in the approximate shape of a Mexican Hat, with the ripples
at the bottom of the hat being of order $g{\delta}$.
\item{} In the quantum-critical and strong-coupling regimes,
even low-energy quasiparticles acquire large widths given on average by
\begin{equation}
\Gamma(\varepsilon)\approx {{\delta}\over\pi}\log(\varepsilon/{\delta})
\end{equation}
\end{itemize}
We found that most of these predictions are verified by our
numerical results on the Robnik-Berry billiard, except that the
ripples at the bottom of the Mexican Hat turn out to be much
larger than expected for the $m=2$ Landau interaction channel.
\section{The Robnik-Berry billiard}
In this section we will describe how the dot is chosen and how the
single particle energy levels $\varepsilon_{{\alpha}}$ and eigenfunctions
$\phi_{{\alpha}}({\mathbf r} )$ are determined. We use a trick invented by Robnik
and Berry\cite{robnik-berry} and elaborated upon by
Stone and Bruus\cite{stone-bruus}. Consider a unit circle $|z|=1$ in the complex
plane of $z=x+iy$. The analytic function
\begin{equation}
w(z) = {z+bz^2+cz^3e^{i\chi}\over \sqrt{1+2b^2+3c^2}}
\label{conformal-map}\end{equation}
defines a map
under which the unit circle in $z$ gets mapped into a new shape in
$w$, which will be our dot. The shape of the dot can be varied by
varying the parameters $b,\ c, \ \mbox{and}\ \chi$. The denominator
ensures that the billiard has the same area ($\pi$) as the unit disc.
The wavefunction $\phi_{{\alpha}}(w,\bar{w}) =\phi_{{\alpha}}(u, v) $ is required
to vanish at the boundary and obey
\begin{eqnarray}
&-&\left( {{\partial}^2\over {\partial}
u^2}+{{\partial}^2\over {\partial} v^2}\right) \phi_{{\alpha}}(u,v)\nonumber \\
&=&-4{{\partial} \over {\partial} w} {{\partial} \over {\partial} \bar{w}}\phi_{{\alpha}}(w,\bar{w})
=\varepsilon_{{\alpha}} \phi_{{\alpha}}(w,\bar{w}).
\end{eqnarray}
(We have
chosen $\hbar = 2m =1$). If we now go the $z$ plane where the
wavefunction is $\phi_{{\alpha}}(w(z),\bar{w}(\bar{z}))$, the
Schr\"{o}dinger equation and boundary condition are
\begin{equation}
-4{{\partial}
\over {\partial} z} {{\partial} \over {\partial}
\bar{z}}\phi_{{\alpha}}(z,\bar{z})=\varepsilon_{{\alpha}}|w'(z)|^2
\phi_{{\alpha}}(z,\bar{z}) \label{se} \ \ \ \ \phi_{{\alpha}}(|z=1|)=0
\end{equation}
where $w'(z)=dw/dz$. This differential equation in the continuum
is next converted to a discrete matrix equation by writing
\begin{equation}
\phi_{{\alpha}}(z,\bar{z})\equiv \phi_{{\alpha}}(r,\theta)=\sum_j {1\over
{\gamma}_j}C^{{\alpha}}_{j}\psi_j(r,\theta )\label{exp}
\end{equation}
where
$\psi_j(r,\theta ) $ is the solution to the free Schr\"odinger
equation in the unit disk vanishing on the boundary:
\begin{equation}
-\nabla^2 \psi_j(r,\theta )={\gamma}^{2}_{j}\psi_j(r,\theta ).
\end{equation}
(That is, these are Bessel functions in $r$ times angular momentum
eigenfunctions in $\theta$. ) Feeding this expansion into Eqn.
(\ref{se}) one obtains the matrix equation \begin{equation} \sum_j
M_{ij}C^{{\alpha}}_{j}={1 \over \varepsilon_{{\alpha}}}C^{{\alpha}}_{i}\end{equation} where
\begin{equation} M_{ij}={1\over {\gamma}_i}\langle i||w'|^2|j\rangle {1\over {\gamma}_j}.
\end{equation} (Without the $1/{\gamma}_j$ in the expansion Eqn. (\ref{exp}), $M$
would not have been Hermitian. )
In practice one truncates $M$ to a finite size (we used 585
states) and expects the lower energy levels and wavefunctions to
be unaffected.
The parameters $b,c,\chi$ are chosen to lie in the range where
classical behavior is chaotic, and where quantum chaos as reflected in
the eigenvalue distribution has been established\cite{stone-bruus}. A
value we used repeatedly was $b=c=.2,\chi =.85$. A nonzero $\chi$ ensures
that the billiard has no reflection symmetry. This shape is often
called the Africa Billiard based on its resemblance to that continent,
as seen in Fig \ref{fig1} for our chosen values of parameters.
\begin{figure}[ht]
\includegraphics*[width=2.4in,angle=0]{fig2.eps}
\caption{The shape of the Robnik-Berry billiard for $b=c=0.2$, and $\chi=0.85$.}
\label{fig1}
\end{figure}
We shall refer to these eigenfunctions and eigenvalues as exact even
though they come from solving a truncated problem because we can
easily increase the accuracy by increasing the size of the truncated
Hilbert space.
\subsection{ Testing Assumptions I and II}
Our ability to solve the Schr\"odinger equation (to high accuracy)
implies in principle that we can test our two assumptions.
In the next subsection we will test Assumption I, i.e., see in
detail how well the WOF states serve a basis within the Thouless
band.
As for Assumption II, we and our predecessors
\cite{stone-bruus,alhassid-lewenkopf} have shown that the
eigenvalues and single eigenfunctions obey the distribution
expected by RMT for a GOE. (The ensemble is generated by varying
the parameters in $w(z)$.)
Similar information about wavefunction correlations is not known
in the ballistic problem (despite some recent progress using
supersymmetry methods\cite{supersymm}). We did not try to do this
here since our computing capabilities did not allow us to
generate an ensemble.
Instead we computed the fate of the interacting system without
recourse to Assumptions I and II and compare to our predictions
based on these assumptions.
\subsection{Completeness of the WOF basis}
Let $E_F$ be the Fermi energy. Then $g\simeq \sqrt{4\pi
N}\simeq\sqrt{\pi E_F}$, which we arrive at as follows. The Fermi
circle has a circumference $2\pi K_F$ and into this will fit
$g=2\pi K_F/(2\pi /L)$ WOF states each of width $2\pi /L$ in the
tangential direction. Finally $E_F=K_{F}^{2}/2m = K_{F}^{2}$,
$N=k_F^2L^2/4\pi$ and $L=\sqrt{\mbox{Area}}=\sqrt{\pi}$.
As a test case when we picked the Fermi energy to be the 100-th
level, we found $g=37$. How well is this state $|F>$ at $E_F$
spanned by the $g$ WOF states at the Fermi energy?
First we first take $g$ equally spaced points ${\mathbf k}_n$ on the Fermi
circle and form the WOF states \begin{equation} \psi_{WOF-n}({\mathbf r} ) = {1\over
\sqrt{\pi}}e^{i{\mathbf k} \cdot {\mathbf r}} \Theta (\mbox{dot})\end{equation} where $\Theta
(\mbox{dot})$ is unity inside the dot and zero outside. These states
are very close to being orthonormal. For example the overlaps of $n=1$
state (with ${\mathbf k} $ along the $y$-axis) with the others as we go around
the circle is shown in Fig. (\ref{fig2a}).
\begin{figure}[ht]
\includegraphics*[width=2.4in,angle=0]{fig3.eps}
\caption{The absolute value of $\langle n|1\rangle$, the inner
product of WOF state number 1 with the other$g-1=36$ states. }
\label{fig2a}
\end{figure}
Next we ask how much of the state $|F>$ at the Fermi energy is
contained in the WOF states. We find $\sum_{n=1}^{g}|\langle
n-WOF|F\rangle |^2 = .9993$. This is a rather remarkable result. It
says that $|F>$, which is a vector with 580 components (which was
the size of our truncated problem) can be expanded almost completely
in terms of $g=37 $ WOF states which are given in advance. In other
words as one changes the shape of the dot and works at fixed Fermi
energy, the state $|F>$ changes in a random way, but that randomness
is only in which particular combination of WOF states describes it,
not in the completeness of the WOF basis.
While this is very satisfactory we need more to implement our scheme:
we need to be expand all $g$ states in the WOF basis. Here we find
that as we move off the center of the Thouless band, the fractional
norm captured by the WOF basis drops. In a typical case, with $g=37$,
there are roughly 12 states (one third of $g$) where the number lies
above 95\%. At band edge this drops to 50\%, as shown in Fig
(\ref{fig2b}). Thus there is inevitably some error in transcribing
the Landau interaction written in terms of the WOF states labelled by
${\mathbf k}$ into the basis of $g$ exact eigenstates labelled by ${\alpha}$. This
just means that the location of the critical point will not be
correctly predicted by our RMT based analysis, as pointed out recently
by Adam, Brouwer, and Sharma\cite{nowindow}.
\begin{figure}[ht]
\includegraphics*[width=2.4in,angle=0]{fig4.eps}
\caption{The norm of the projection of the $n^{th}$ exact eigenstate from $E_F$
on to the subspace spanned by the WOF states. It can be seen that more
and more of the exact eigenstate lies outside this subspace as one
moves away from $E_F$. } \label{fig2b}
\end{figure}
This concludes our (partial) test of Assumptions I and II. We turn
to a comparison of our results based on these assumptions with a
direct solution of the problem with no recourse to the
assumptions.
\subsection{Hartree-Fock solution of interacting problem}
How can the knowledge of the "exact" eigenfunctions and
eigenvalues in the billiard help in the solution of the problem
with interactions? The tactic will be illustrated in schematic
form first. Suppose we have a four-Fermi interaction added to a
free hamiltonian which in first-quantization is given by some
differential operator $H_0$. Then the path integral becomes
\begin{equation} Z=\int d\psi d\bar{\psi} e^{S}
\end{equation}
where
\begin{equation} S= \int
d\tau \left[ \bar{\psi} \left( i\partial_{\tau} - H_0 \right) \psi
+{u\over 2} (\bar{\psi}\psi )^2\right] .
\end{equation}
Using a Hubbard-Stratanovich transformation we can write
\begin{equation} Z=\int d\psi d\bar{\psi} d\sigma
e^{S}
\end{equation}
where
\begin{equation} S= \int d\tau \left[ \bar{\psi} \left[
i\partial_{\tau} - H_0-\sigma \right] \psi -{\sigma^2\over 2u}
\right] .
\end{equation}
If the fermions are integrated out we will get an effective action
$S_{eff}(\sigma )$. {\em To find the minimum we need just the
action for static $\sigma$. } In this case it is clear that
\begin{equation}
\int d\psi
d\bar{\psi}\exp \left[ \int d\tau \bar{\psi} \left[
i\partial_{\tau} - H_0+\sigma \right] \psi \right]=
e^{-E_0(\sigma) T}
\end{equation}
where $T\to \infty$ is the length of the imaginary time $\tau$- axis
and $E_0(\sigma ) $ is the ground state energy of $\psi^{\dag}
(H_0+\sigma )\psi$. To find $E_0(\sigma )$ one simply solves for the
single particle levels of $ (H_0+\sigma )$ and fills up the ones with
negative energy. The effective action for static configurations, which
is also the effective potential, is
\begin{equation}
V_{eff}=E_0(\sigma )+{\sigma^2 \over 2u}.
\label{effpot}\end{equation}
At this point we have a mean-field theory. We still need to
justify its use by showing that fluctuations of the collective
field $\mbox{\boldmath $\sigma $}$ around its minimum are small. In our previous
work, based on Assumptions I and II we showed that the
fluctuations were indeed small in the limit of large $g$, since
the $g^2$ in front of the actions limits fluctuations. In the
billiard we will justify the mean field similarly, based on the
depth and curvature of the minimum.
When the Landau interaction is factorized, the hamiltonian
whose ground state gives us $E_0(\sigma )$ is
\begin{equation}
\sum_{{\alpha} {\beta}} \psi^{\dag}_{{\alpha}} (\delta_{{\alpha} {\beta}}{\large
\varepsilon}_{{\beta}}+\mbox{\boldmath $\sigma $}\cdot {\mathbf M}_{{\alpha} {\beta}})\psi_{{\beta}}
\label{ham-to-diag}\end{equation}
where, for the case $m=1$, for example,
\begin{equation} {\mathbf M}_{{\alpha} {\beta}}=\sum_{{\mathbf k}} \phi^{*}_{{\alpha}}({\mathbf k} )\phi_{{\beta}}({\mathbf k} )
{{\mathbf k} \over k}
\end{equation}
and ${\alpha} , {\beta}, {\mathbf k}$ are not restricted to the Thouless band. This is
because we want to solve the problem without any of the assumptions
that led to the effective low energy theory within the Thouless
band. Note that $\mbox{\boldmath $\sigma $}$ has two components, because the Landau
interaction associated with $u_m$ has two parts:
\begin{equation}
V_{L}= {u_m\over2}\sum_{{\mathbf k} {\mathbf k}'} \delta n_{{\mathbf k}}
\delta n_{{\mathbf k}'}(\cos m \theta_{{\mathbf k}} \cos m \theta_{{\mathbf k}'}+ \sin m
\theta_{{\mathbf k}} \sin m \theta_{{\mathbf k}'}).
\end{equation}
Once $S_{eff}$ is known (on a grid of points in the $\mbox{\boldmath $\sigma $}$
plane) one can ask if and when the minimum moves off the origin.
So far our considerations have been fairly generic, and the Landau
interaction has been written in momentum space. However, in testing
our approach in the billiard, we will find it more convenient to
represent the Landau interaction in real space, since the
eigenfunctions are known as linear combinations of Bessel functions
whose integrals are best carried out in real space. We have carried
out calculations for two Laudau parameters corresponding to $m=1$ and
$m=2$. The $m=1$ Landau interaction is chosen to be (in
second-quantized notation)
\begin{eqnarray}
{1\over2} &\int d^2r \Psi^{\dag}({\vec{r}}){1\over(2mH_0)^{1/4}}(-i{\vec\nabla}){1\over(2mH_0)^{1/4}}\Psi({\vec{r}}) \cdot\nonumber\\
&\times\int d^2r' \Psi^{\dag}({\vec{r}}'){1\over(2mH_0)^{1/4}}(-i{\vec\nabla}'){1\over(2mH_0)^{1/4}}\Psi({\vec{r}}')
\end{eqnarray}
The factors of ${1\over(2mH_0)^{1/4}}$ on each side of the $\nabla$
have the effect of $1/|{\mathbf k}|$ in momentum space. Since momentum does
not commute with the free Hamiltonian $H_0$, the factors have to be
placed symmetrically. Note that this corresponds only to the ${\vec{q}}=0$
part of the Landau interaction. In reality, all values of ${\vec{q}}$ up to
the scale $E_L/v_F$ exist in the Hamiltonian. Depending on the shape
of the dot a particular combination of them may break symmetry to give
the best energy. Still, we expect that since at large $g$ we are close
to the zero-dimensional limit, the best combination will consist
largely of very small ${\vec{q}}$ parts of the Landau interaction. In any
case, the energy of the true symmetry-broken state can only be
lower than what we calculate, so what we have here is a conservative
estimate of symmetry-breaking. Similarly the $m=2$ interaction (also
at ${\vec{q}}=0$) is
\begin{eqnarray}
{1\over2}&\int d^2r \Psi^{\dag}({\vec{r}}){1\over(2mH_0)^{1/2}}(\nabla_x^2-\nabla_y^2){1\over(2mH_0)^{1/2}}\Psi({\vec{r}}) \cdot\nonumber\\
&\times\int d^2r'
\Psi^{\dag}({\vec{r}}'){1\over(2mH_0)^{1/2}}((\nabla')_x^2-(\nabla')_y^2){1\over(2mH_0)^{1/2}}\Psi({\vec{r}}')\nonumber\\
&+{1\over2}\int d^2r
\Psi^{\dag}({\vec{r}}){1\over(2mH_0)^{1/2}}2\nabla_x\nabla_y{1\over(2mH_0)^{1/2}}\Psi({\vec{r}})
\nonumber\\
&\times\int d^2r'
\Psi^{\dag}({\vec{r}}'){1\over(2mH_0)^{1/2}}2(\nabla')_x(\nabla')_y{1\over(2mH_0)^{1/2}}\Psi({\vec{r}}')
\end{eqnarray}
The integrals are over $(w,\bar{w})$, but can be converted to
integrals over the disk by using the conformal mapping of Eq.
(\ref{conformal-map}). Of course, the derivative operators must
also be transformed in the process. In order to find the matrix
elements of ${\mathbf M}_{{\alpha} {\beta}}$ we had to take the matrix elements of
the above operators in the basis of exact billiard states. We
carried out the angular part of the integrals analytically, but
had to turn to numerical integration to evaluate the radial
integrals. This is a computationally intensive calculation, but
once the matrix ${\mathbf M}$ has been constructed, one simply
diagonalizes the Hamiltonian of Eq. (\ref{ham-to-diag}) for a mesh
of $\mbox{\boldmath $\sigma $}$ in the plane, adds up the energies of the lowest $N$
particles to obtain the fermionic ground state energy, and obtains
the effective potential landscape from Eq. (\ref{effpot}) for
various values of the coupling strength $u$. After this, it is a
simple matter to identify the global minimum, which gives us the
lowering of ground state energy and the value of the order
parameter as a function of $u$.
Let us proceed to the results, displayed in pictorial form. In
Fig. \ref{fig3} we show the absolute value of the order parameter,
normalized by the nominal value of $g=\sqrt{4\pi N}$, for three
values of the number of particles $N$. The bulk transition happens
at $u^*_{bulk}=2$. As can be seen, there is a nonzero order
parameter for any nonzero $u$, and it grows smoothly and
continuously as $u$ increases. Nothing discontinuous happens at
$u=2$ or even beyond, indicating that the instability does not
suddenly become first-order at the bulk value of $u^*$. Of course,
in these finite systems, the Thouless and bulk scales are related
by a factor $g/4\pi$, which is not that large (4.4 for the largest
system we considered, with $N=245$). So somewhere between $u=2.25$
and $u=2.5$ the instability seems to reach the bulk scale.
However, note that the size of systems we have considered
correspond quite closely to actual ballistic
samples\cite{recent-expts}, which typically have a few hundred
electrons. Further, the three curves seem to track each other
fairly closely, indicating that the expectation value of
$|\mbox{\boldmath $\sigma $}|$ indeed scales with $g$, as predicted by our earlier
work based on RMT assumptions.
\begin{figure}[ht]
\includegraphics*[width=2.4in,angle=0]{fig5.eps}
\caption{The absolute value of the order parameter normalized by $g$
as a function of coupling strength $u$ for three values of the number
of particles $N$. The fact that the curves track each other closely
indicates that the order parameter does indeed scale like
$g$. Furthermore, nothing discontinuous happens at the bulk critical
coupling strength $u^*_{bulk}=2$.}
\label{fig3}
\end{figure}
In Fig. \ref{fig4} we show the corresponding reduction in ground state
energy normalized by $g^2$. Once again, the curves track each other
fairly closely, indicating that the energy reduction due to
interactions is indeed of order $g^2$, as predicted by our earlier
work.
\begin{figure}[ht]
\includegraphics*[width=2.4in,angle=0]{fig6.eps}
\caption{Reduction in ground state energy normalized by $g^2$ for three values of the number of particles $N$.}
\label{fig4}
\end{figure}
In Fig. \ref{fig5} we show the effective potential landscape for
$m=2$, with $N=245$, at a value of $u=2.15$, at which the minimum is
well-developed, but the order parameter is still within the nominal
Thouless scale and has not reached the bulk scale. The RMT analysis
predicted a Mexican Hat landscape with ``small'' ripples (down by
$1/g$) in the circle of minima of the Mexican Hat. The landscape we
see bears no resemblance to this. Instead, it appears to be an
isolated minimum at a nonzero $\mbox{\boldmath $\sigma $}$. Upon close inspection it can
be seen that the minimum is shallower in the transverse direction than
in the radial direction, but this is the only indication we could find
of a (perhaps) incipient Mexican Hat structure.
\begin{figure}[ht]
\includegraphics*[width=2.4in,angle=0]{fig7.eps}
\caption{Effective potential landscape for symmetry-breaking in the
$m=2$ channel. Instead of a Mexican Hat minimum structure with small
ripples we see an isolated minimum. The minimum does seem shallower in
the transverse direction. }
\label{fig5}
\end{figure}
Fig. \ref{fig6} shows a similar effective potential landscape for
symmetry breaking in the $m=1$, channel, where the two exactly
degenerate minima expected from time-reversal invariance
considerations can be seen. The landscape also appears more
Mexican-Hat-like than in the $m=2$ case.
\begin{figure}[ht]
\includegraphics*[width=2.4in,angle=0]{fig8.eps}
\caption{Effective potential landscape for symmetry-breaking in the
$m=1$ channel for $N=200$ and $u=2.15$. The two exactly degenerate
minima required by time-reversal invariance can be seen, as can a
Mexican-Hat-like structure.}
\label{fig6}
\end{figure}
To trace the origin of this difference in behavior, we investigated
the average absolute value $\langle |M^i_{{\alpha}{\beta}}|\rangle$ and the rms
deviation of the matrix elements from the mean absolute value,
$\sqrt{\langle |M^i_{{\alpha}{\beta}}|^2\rangle-\langle |M^i_{{\alpha}{\beta}}|\rangle^2}$
for the two cases $m=1,2$. The results for the $i=1$ (corresponding to
$\nabla_x$ for $m=1$ and $\nabla_x^2-\nabla_y^2$ for $m=2$) shown in
Fig. \ref{fig7} are an energy average for a particular billiard, with
the parameters $b=c=0.20, {\delta}=0.85$. (We have confirmed similar
behavior of the matrix elements for other parameter values as well.)
Fig.
\ref{fig7} shows these quantities as a function of the energy
difference between the two states ${\alpha}$ and ${\beta}$. There are two
features that are particularly noteworthy.
\begin{itemize}
\item There is a ``hole'' in the $m=1$ matrix element near zero
energy difference.
\item The rms deviation of the $m=2$ matrix elements from their
mean absolute value is huge. As a rough estimate, if the matrix
elements were Gaussian distributed complex numbers, the rms
deviation should be roughly half the mean modulus.
\end{itemize}
\begin{figure}[ht]
\includegraphics*[width=2.4in,angle=0]{fig9.eps}
\caption{A plot of the absolute value and the rms deviation of the matrix elements
$M^x_{{\alpha}{\beta}}$ from their mean absolute value as a function of
${\varepsilon}_{{\alpha}}-{\varepsilon}_{{\beta}}$ for the two cases $m=1,2$.}
\label{fig7}
\end{figure}
However, the $i=2$ component (corresponding to $\nabla_y$ for $m=1$
and $2\nabla_x\nabla_y$ for $m=2$) shows very different behavior in
Fig. \ref{fig8}. While the $m=1$ case looks similar to the $i=1$
component, the fluctuations of the $m=2$ $i=2$ component are strongly
suppressed by almost an order of magnitude below the mean.
\begin{figure}[ht]
\includegraphics*[width=2.4in,angle=0]{fig10.eps}
\caption{A plot of the absolute value and the rms deviation of the matrix elements
$M^y_{{\alpha}{\beta}}$ from their mean absolute value as a function of
${\varepsilon}_{{\alpha}}-{\varepsilon}_{{\beta}}$ for the two cases $m=1,2$.}
\label{fig8}
\end{figure}
Consider first the "hole" at $E_F$ for $m=1$. By symmetry
considerations alone one can understand that the diagonal matrix
element $M_{{\alpha}\a}$ for $m=1$ has to be zero sample by sample in the
absence of an external magnetic field. Focusing on the $x$ component
of the order parameter
\begin{equation}
M^{x}_{{\alpha}\a}=\sum\limits_{{\mathbf k}} \cos(\theta_{{\mathbf k}}) \phi^*_{{\alpha}}({\mathbf k})\phi_{{\alpha}}({\mathbf k})
\end{equation}
By time-reversal invariance $\phi^*_{{\alpha}}({\mathbf k})=\phi_{{\alpha}}(-{\mathbf k})$. Noting
that $\theta_{-{\mathbf k}}=\theta_{{\mathbf k}}+\pi$, and that the $\cos$ term
changes sign, one concludes that $M_{{\alpha}\a}=-M_{{\alpha}\a}=0$. The reason
the "hole'' persists for finite energy differences for the operator
${\vec p}=-i{\vec\nabla}$ can be explained by noting
that\cite{barnett-pvt} for a billiard
\begin{eqnarray}
{\vec p}=&im[{\vec{r}},H]\nonumber\\
\Rightarrow (-i{\vec\nabla})_{{\alpha}{\beta}}=&-im({\vec{r}})_{{\alpha}{\beta}} ({\varepsilon}_{\alpha}-{\varepsilon}_{\beta})
\end{eqnarray}
which means that the matrix element must vanish at least linearly
with the energy difference. In fact, such ``banded'' matrix
elements have been found for many operators in ballistic
dots\cite{barnett}.
Consider next the fact that the distribution of the matrix
elements of $M^x_{{\alpha}{\beta}}$ for the $m=2$ case is much broader than
for the $m=1$ case, while the $M^y_{{\alpha}{\beta}}$ matrix elements have a
very narrow distribution. The RMT answer would have the rms
deviation of $M_{{\alpha}{\beta}}$ from the mean to be of the same order as
the mean absolute value. This seems to be roughly true for both
components of $m=1$ but grossly untrue for the $i=1$ component of
$m=2$. Since it is these mesoscopic fluctuations in $M_{{\alpha}\a}$
which determine the size of the ripples at the bottom of the
Mexican Hat in the RMT scenario, this broad distribution of
$M_{{\alpha}{\beta}}$ seems to be the cause of the failure of the RMT
prediction that the ripples should be subdominant by $1/g$. While
it is tempting to try to explain this in relation to the shape of
the billiard (Fig. \ref{fig1}), which certainly appears to favor
an $x^2-y^2$ type of symmetry, a satisfactory explanation of the
broad distribution of the $m=2$, $i=1$ matrix elements eludes us.
Our knowledge of the eigenfunctions at the global minimum allow us
to compute the effective action for time-dependent $\mbox{\boldmath $\sigma $}$ at
that minimum. Since the quasiparticles couple to this collective
field, the interaction induces a decay width for the
quasiparticles (details can be found in ref.\cite{longpaper}). In
Fig. \ref{fig9} we compare the numerically calculated values of
the width to the parameter-free theoretical prediction (solid
line) based on RMT\cite{longpaper}. On average, the RMT based
prediction seems consistent with the numerics, though there is a
lot of variation in the widths driven by large variations in the
matrix elements coupling the quasiparticle levels to the
collective mode.
\begin{figure}[ht]
\includegraphics*[width=2.4in,angle=0]{fig11.eps}
\caption{A plot of the decay width of quasiparticles induced by their coupling to
fluctuations of the collective field $\mbox{\boldmath $\sigma $}$, for $N=245$, $m=2$,
and $u=2.15$. The solid line is the theoretical prediction from our
previous RMT-based analysis\cite{longpaper}. While the prediction does
well on average there are huge variations in the widths due to large
variations in how strongly each level couples to the collective mode.}
\label{fig9}
\end{figure}
\section{Conclusions}
\medskip
In our earlier work, we used a global RG assumption to reduce the
problem on the scale of the Thouless energy to that of a disordered
noninteracting problem with Fermi-liquid interactions. This is quite
plausible for ballistic dots on very general grounds. To proceed
further we had to make two further assumptions; (i) That the $g$
approximate momentum states at the Fermi energy were a good basis in
which to expand the exact disorder eigenstates, and (ii) That the wave
functions of the exact eigenstates in the momentum basis obeyed all
the statistical properties of RMT. Based on these two assumptions we
were able to construct a solution to the problem which was
asymptotically exact in the limit $g\to\infty$. This solution led to
specific predictions for various physical quantities, including the
size of the order parameter, the reduction in energy due to
interactions, the shape of the energy landscape, and the size of the
quasiparticle decay widths.
In this paper we have carried out a calculation which is still
predicated on the validity of the Fermi-liquid form of the
interactions on a scale $E_L$ much larger than the Thouless energy
$E_T$. In retaining this assumption we are on firm ground, since after
all, the Thouless energy can be made as small as one wishes merely by
increasing the size of the system. We also assumed that the mean-field
description of the Landau Fermi-liquid interactions is valid, which is
justified by the fact that the minima in the effective potential
landscape are indeed of order $g^2{\delta}$. However, we explicitly eschewed
the other two assumptions that we made in previous work, with a view
to independently testing their validity. We found that our first
assumption, that the approximate momentum states were a good basis in
which to expand the exact disorder eigenstates, was extremely good
near the Fermi energy, but became increasingly inaccurate as one went
to the edge of the Thouless shell. We did not test the second
assumption about wavefunction correlations explicitly, but indirectly
through its effects on the predictions of our earlier work. We found
that most of the predictions held up, with the exception of the shape
of the effective potential landscape in the case of symmetry-breaking
in the $m=2$ channel. Even here, the minimum is shaped more like a
crescent, indicating the possible emergence of the Mexican Hat
structure at larger values of $g$ (we went to the largest value of $g$
that we could given that we kept only 585 states and had to keep at
least half the states empty). We traced the discrepancy back to the
anomalously broad distribution (compared to estimates based on a
complex gaussian distribution) of the matrix elements
$M^x_{{\alpha}{\beta}}$. However, we were unable to pin down a physical reason
for this broad distribution for the $m=2$ case.
In conclusion, much of the physics we uncovered using our RMT
assumptions seems to be valid in the Robnik-Berry billiard. The
second-order transition that we uncovered in the $g\to\infty$ limit
seems to indeed be broadened into a smooth croossover as expected,
belying fears that it may be overtaken by a first-order bulk
transition. The question of how large $g$ has to be before RMT becomes
fully applicable remains open; another way to phrase the question is
to ask what the nonuniversal corrections to RMT are in ballistic
systems. Finally, an important open question is whether the broad
distributions of matrix elements of interaction operators is a generic
feature of ballistic systems, rather than being a special feature of
the Robnik-Berry billiard, and if so, what physics determines the
width of those distributions. However, our results here give us
encouragement that the RMT assumptions can indeed be used with
confidence in making predictions in ballistic systems, at a
qualitative and semi-quantitative level.
\section{Acknowledgements}
We would like to thank Yoram Alhassid, Alex Barnett, Piet Brouwer,
and Doug Stone for illuminating conversations, and the Aspen
Center for Physics where part of this work was carried out. We are
also grateful to the NSF for partial support under grants DMR
0311761 (GM), DMR 0354517(RS).
|
{
"timestamp": "2004-11-10T22:20:27",
"yymm": "0411",
"arxiv_id": "cond-mat/0411280",
"language": "en",
"url": "https://arxiv.org/abs/cond-mat/0411280"
}
|
\section{Introduction}
\subsection{LBVs and their variations \label{lbvs}}
The most massive stars, above roughly 50\,\hbox{M$_{\sun}$}, pass an unstable phase
as they turn into {\it Luminous Blue Variable\/} (LBVs).
In the LBV phase, a transitional phase between the main-sequence and
Wolf-Rayet state, the stars lose large amount of mass
($> 10^{-5}$\hbox{M$_{\sun}$}\,yr$^{-1}$). As a consequence
the evolution of these stars towards cooler temperature
is stalled and reversed \citep{1994A&A...290..819L}. The most
massive stars seem not to enter the red supergiant phase
(see \citet{2001A&A...366..508V} for a discussion).
The coolest location of these stars in the
Hertzsprung-Russell Diagram (HRD) marks the empirical {\it Humphreys-Davidson
limit} \citep[e.g.][]{1994PASP..106.1025H}.
\begin{figure}
{\resizebox{\hsize}{!}{\includegraphics{Gi241f1.eps}}}
\caption{A g-band image of NGC 2403 taken with the Isaac Newton telescope.
A box indicates the area in which SN\,2002kg\ was found, and is shown as
enlargement in Fig. \ref{fig:astrometrie}.}
\label{fig:ngc2403}
\end{figure}
\begin{figure*}
{\resizebox{\hsize}{!}{\includegraphics{Gi241f2.eps}}}
\caption{The optical lightcurve of V37 (upper
panel) and B-V color (lower panel).
Symbols are coded as:
circles are B magnitudes,
squares are V magnitudes, triangles are R magnitudes.
The detection of SN\,2002kg\ is indicated with a star, and
is roughly an R magnitude. Filled symbols are data taken from \citet{1968ApJ...151..825T},
open symbols from this work, half-filled from
\citet{1992hdspiller}. The SN\,2002kg\ detection as well as
one R magnitude limit (hashed triangle) from \citet{2003IAUC.8051....1S}.
Possible (S)-SD periods are indicated.
}
\label{fig:light}
\end{figure*}
LBVs show variabilities on different timescales and
with different strength \citep[e.g.][]{1994A&AS..106..141S},
the most classical being the S~Dor variability
\citep[e.g.][]{1997A&AS..124..517V,
1997A&A...318...81V, 2001A&A...366..508V}. During an S Dor
phase, which is a cyclic phenomenon of expansion and contraction at constant
luminosity, the temperature decreases and rises, respectively.
As a consequence the spectrum changes from a hot O or B type to a cooler
mid A or early F type \citep{1992RvMA....5....1W}, so
the colors become red and than blue again.
It has been shown
that the temperature variation in such a cycle is larger for more luminous
LBVs (amplitude-luminosity-relation, \citep{1989A&A...217...87W}).
Now, \citet{2001A&A...366..508V} subdivides the S Dor variability
further in the {\it short S Dor phase, (S)-SD\/} and the
{\it long S Dor phase, (L)-SD\/}.
The (S)-SD is shorter than about 10 years, and
the (L)-SD is larger than 20 years.
More dramatic are the so called giant eruptions.
These are spontaneous outbursts in which an LBV increases
its luminosity by several magnitudes. It stays bright for a short time
before rapidly declining back to lower luminosities, sometimes
even lower as before the outburst. While becoming fainter their
appearance becomes redder, presumably due to the
formation of dust \citep[e.g.][]{1999PASP..111.1124H}.
The best known example was $\eta$~Carinae's
outburst around 1843, when it was with $-1^{\rm m}$ the second
brightest star in the southern
hemisphere \citep{1903AnCap...9....75,1997ARA&A..35....1D}.
Other giant eruption LBVs are P~Cygni
($\sim$1600; \citet{1988IrAJ...18..163D}),
SN1961V in NGC 1058 (1961; \citet{1989ApJ...342..908G}) and
SN1954J (=V12) in NGC 2403 (1954; \citet{1968ApJ...151..825T} ).
These LBVs are called 'giant eruption LBVs' or '$\eta$ Car Variables',
or more recently 'supernova impostors'.
The high mass loss of LBVs and the ejecta of mass during the giant
eruption leads to the formation of nebulae around LBVs
\citep{1995ApJ...448..788N,2001RvMA...14..261W}.
These nebula are generally small ($<$ 2\,pc) but their
expansion velocities show a large range.
Several expand slowly ($\sim$ 30\,\hbox{km\,s$^{-1}$}) others can be as fast
as 100\,\hbox{km\,s$^{-1}$} \citep{2003A&A...408..205W}. The fastest expansion velocities
detected are in $\eta$ Carinae with 600\,\hbox{km\,s$^{-1}$} (Homunculus) to
at least 2500\,\hbox{km\,s$^{-1}$} (outer ejecta)
\citep[e.g.][]{2001AGM....18S0211W,2004A&A...415..595W}.
LBV nebulae do show strong nitrogen emission due to
the CNO processed material \citep{1996A&A...305..229G}.
\subsection{SN2002kg and V37 in NGC\,2403 }\label{2002kg}
SN\,2002kg\ was detected \citep{2003IAUC.8051....1S} in an unfiltered
image from October 26, 2002 with the
{\it Katzman Automatic Imaging Telescope at
Lick observatory (KAIT)\/}.
SN\,2002kg\ is situated in NGC\,2403 (see Fig. \ref{fig:ngc2403})
a SBc spiral in the M81 group (distance modulus = 28.14
\citep{1988ngc..book.....T}).
At the time of discovery SN\,2002kg\ had a brightness of 19$^{\rm m}$ in the
unfiltered image.
A spectrum of the SN\,2002kg\ taken on January 6, 2003 taken with the Keck
telescope identified SN\,2002kg\ as type IIn, since
it showed for a supernova quite narrow Balmer lines ($<$ 500\,\hbox{km\,s$^{-1}$}),
casting already some doubt on being a classical supernova.
Additionally, broader components (FWHM $\sim$ 2500\,\hbox{km\,s$^{-1}$}) were also present.
Unusual was the detection of [N\,{\sc ii}] emission
at 6548\,\AA\ and 6584\,\AA.
V37 has first been identified as a bright blue irregular variable
by \citet{1968ApJ...151..825T},
together with V12 (alias SN1954J), V22,V35, V38,
today confirmed LBVs \citep{1994PASP..106.1025H}. The lightcurve
given in Fig.\ \ref{fig:light}
shows the irregular behavior of V37, the original data of
\citet{1968ApJ...151..825T} are included (filled symbols).
From the brightness and colors \citet{1968ApJ...151..825T} estimated
that these stars are F supergiants.
A spectrum of V37 obtained in 1985 shows according
to \citet{1987AJ.....94.1156H}
absorption-lines, a blue continuum, and no emission lines.
\begin{figure}
{\resizebox{\hsize}{!}{\includegraphics{Gi241f3.eps}}}
\caption{This enlargement of Fig \ref{fig:ngc2403} shows the
closer vicinity and position (error circle) of SN\,2002kg\ in NGC 2403.
The image was taken in with the INT in 2001, before the detection as SN\,2002kg.
}
\label{fig:astrometrie}
\end{figure}
\section{Observations and analysis}
\subsection{Astrometry}
The position of SN\,2002kg\ as given at discovery
\citep{2003IAUC.8051....1S} is $\alpha =$ 7h 37m 1.83s and
$\delta = +65\degr\,34\arcmin\,29\farcs3$ (2000.0).
We used a deep INT archival image in the g-band
to investigate the surroundings of SN\,2002kg. Using astrometry routines
in IRAF/STSDAS and KARMA we transferred the coordinate system of
the DSS image onto the CCD frame.
Given an uncertainty of the process and the not corrected higher
order distortions of the CCD we reach an absolute positioning
of slightly better than 1\arcsec. Assuming a similar accuracy for the
position of SN\,2002kg\ we generated Fig. \ref{fig:astrometrie},
with the g-band image in gray-scale and the error circle overlayed.
It shows clearly, that SN2002kg is coincident with the
stellar source identified as V37 by \citet{1968ApJ...151..825T}.
\begin{figure}
{\resizebox{\hsize}{!}{\includegraphics{Gi241f4.eps}}}
\caption{HST ACS/HRC image in the F435W ($\sim$ B) image
shown the same section as in Fig. \ref{fig:astrometrie}.
The area of V37 is shown as enlargement in the lower left
corner again.
}
\label{fig:acs}
\end{figure}
\subsection{Ground based photometry}
For the construction of a lightcurve (Fig. \ref{fig:light}) we used the
measurements by \citet{1968ApJ...151..825T} as starting point.
The B,V photometry
also provided us with a decent number of secondary photometric
standard stars in the field of NGC 2403. We uses these stars
to tie our photometry made on a large variety of images to
Tammann-Sandage's system. We ignore color terms between the photographic
photometry and our CCD images in most cases. For the few very good
data we verified that the effect is generally smaller than the
photometric uncertainties due to crowding, seeing, and centering
of the aperture. All measurements where done with IRAF/DAOPHOT
through Tammann-Sandage's 6\farcs7 aperture.
We measured the DSS, DSS2-blue, and DSS2-red (1955, 1997, 1989),
as well as two blue plates
from the Tautenburg 2\,m Schmidt (from 1970 and 1972), several
CCD images from the Isaac Newton 2.5m (1992, 1996, 2001),
Jacobus Kapteyn 1m telescopes (1996, 1998)
retrieved from the ING archive, and CCD images
from the Tautenburg 2\,m (2003, 2004). Whenever the object is near or below
detection limit, we also estimated the magnitude (or magnitude limit)
in comparison to
the stars of secondary standard sequence in the traditional way by eye.
For comparison we also added the measurements reported in the
discovery IAUC of SN\,2002kg\ \citep{2003IAUC.8051....1S}.
These measurements where taken
with an unfiltered CCD mounted at the KAIT.
We converted them into R band measurements following the recipe
described in \citet{2003PASP..115..844L}.
\subsection{HST ACS images}
Very recently HST ACS/WFC and ACS/HRC images of NGC 2403 were taken,
which have the region V37 in the field of view.
We retrieved WFC images in the F475W, F606W, F814W, and F658N filters,
and HRC images in the
F435W, and F625W filters from the HST archive.
V37 is present on the images as a blue point source at exactly the
position predicted based on the 2001 INT images. This clearly shows
that SN\,2002kg\ was not a SN event, but a brightening of V37.
We measured the brightness in the ACS B and V bands and added the
points to our lightcurve. These data points show that V37 is
fading again and of quite blue color.
The HRC images (Fig. \ref{fig:acs})
also reveal that next brightest (fainter by 1.6$^{\rm m}$
in F475W) object in the positional error
circle is not a star but a blue diffuse object of uncertain nature
(see enlarged box).
\section{Discussion and conclusions}
SN\,2002kg\ was from the detection on a very strange SN IIn.
Being quite faint at detection (about -9$^{\rm m}$, absolute)
and with an unusual
spectrum it was suspected that SN\,2002kg\ may not be a real
supernova. Unfortunately, its lightcurve was not monitored.
A comparison of ground based images (Fig. \ref{fig:astrometrie}) taken
before the
supernova event with the position of SN\,2002kg\ shows two sources, a diffuse object
and a bright one, the latter being the LBV V37.
Astrometry on the ground based and HST ACS images showed that
the bright star is indeed coincident with SN\,2002kg.
The HST ACS images, as well as images from Tautenburg show
both sources present after 2002, which clearly proves that SN\,2002kg\ was
rather the brightening of a luminous star, historically known as V37.
The diffuse source with a size of 0\farcs18 FWHM (3.7\,pc) is too
large for being a young SNR (created 2002). The source
is also present in our best archival images before 2002.
It is also too faint for
a tight cluster of massive stars, which could be progenitors of a
type IIn supernova. The SN\,2002kg\ could be a supernova in a more
distant background galaxy, but than the velocity
of the emission lines detected \citep{2003IAUC.8051....1S}
would be different, and this is not the case.
The lightcurve of V37 in Fig. \ref{fig:light} shows several possible
cycles of 2000 -- 3000 days period during the last 80 yrs as indicated
in Fig. \ref{fig:light}. This is consistent with (S)-SD type variations.
However the brightening around MJD 35000 shows a
change towards bluer colors. This is unusual as
S Dor variabilities normally show a redward color trend.
Such an increase in brightness with bluer colors is rarely seen
in LBVs, with the exceptions being presently $\eta$ Carinae
\citep{2004MNRAS.352..447W},
and NGC 2366 V1, which became UV brighter during its V-band fading
\citep{2001ApJ...546..484D}.
There is no indication for a giant eruption in the lightcurve of
V37,
but there are still large time gaps which in theory could
accommodate such an event in the past.
With our data sets we could generate two continuum corrected
H$_{\alpha}$\ images of the region around V37.
In 2001 V37 is a bright
H$_{\alpha}$\ emitter (see Fig.\ \ref{fig:ha}).
The HST ACS image from 2004 taken in the
F658N filter also shows bright emission at the position of V37 after
correction for the continuum emission.
The spectrum of SN\,2002kg\ (section \ref{2002kg})
indicates that H$_{\alpha}$\ and [N\,{\sc ii}]\ emission
is present at that time. One possible explanation is the creation
of an LBV nebula around V37 coinciding with the recent bright phase.
Such a nebula would show strong nitrogen emission
and could expand with velocities
as high as detected in the SN\,2002kg\ spectrum, see section \ref{lbvs}.
The H\,{\sc ii} region [H83]\,177 identified earlier
by \citet{1983AJ.....88..296H} on a deep H$_{\alpha}$+[N\,{\sc ii}]\ plate
seems to coincide within the errors
with the position of V37, too. That could indicate that the
star has been emitting H$_{\alpha}$\ and maybe [N\,{\sc ii}]\ as early
as the 1980s, which would point at an earlier nebula formation.
Still, missing emission lines in the spectrum of \citet{1987AJ.....94.1156H}
could mean, that the H$_{\alpha}$\
emission would just indicate a strong stellar and variable
H$_{\alpha}$\ line. The missing [N\,{\sc ii}]\ lines in that spectrum may imply that the
nebula has not been formed in 1985.
Whether there is nebula formation during earlier times
or not, may only be answered with the detection (or high quality
non-detection) of [N\,{\sc ii}]\ lines
in other historic spectra. In any case, it seems, that
V37 has a nitrogen enhanced nebula now.
In V37 we may witness currently the creation of an LBV nebula, but it
is not yet clear whether this nebula was created during
a giant eruption, as shell ejection during several SD periods
as in P Cyg \citep[e.g.][]{2001A&A...376..898M}, or recently
in connection with the brightening in 2002.
Determining the exact evolutionary state of V37 is therefore of
importance for
our understanding of the LBV phenomenon and very massive stars in general.
\begin{figure}
{\resizebox{\hsize}{!}{\includegraphics{Gi241f5.eps}}}
\caption{This H$_{\alpha}$\ emission line image generated from INT images
taken in 2001, indicates that V37 is an H$_{\alpha}$\ bright source.
}
\label{fig:ha}
\end{figure}
\begin{acknowledgements}
KW is supported by the state of North Rhine-Westphalia
(Lise-Meitner fellowship).
We thank S. Klose for kindly providing his Tautenburg 2m images, and
H. Meusinger for scanning the historic Tautenburg plates.
We thank the referee A.M. van Genderen for his comments that helped to
significantly improve the paper.
This research is partially based on data from the ING Archive.
Based partly on observations made with the NASA/ESA Hubble Space Telescope,
obtained from the data archive at the Space Telescope Institute.
This research has made use of NASA's Astrophysics Data System.
\end{acknowledgements}
|
{
"timestamp": "2004-11-17T18:00:47",
"yymm": "0411",
"arxiv_id": "astro-ph/0411504",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411504"
}
|
\section{Introduction}
The {\sc Planck} satellite will carry our high sensitivity all sky
surveys at 9 frequencies in the poorly explored range 30--860 GHz
(see the contribution by J. Tauber, this volume). At low
frequencies, {\sc Planck} will go several times deeper (and will
detect about ten times more sources) than WMAP, that has provided
the {f}{i}rst all-sky surveys at frequencies of tens of GHz,
comprising about 200 objects (Bennett et al. 2003).
Above 100 GHz, {\sc Planck} surveys will be the {f}{i}rst and will
remain the only all sky surveys available for many years to come.
They will {f}{i}ll an order of magnitude gap in our knowledge of
the spectrum of bright extragalactic sources and may discover new
populations, not represented, or not recognized, in lower or
higher frequencies surveys.
Rather than presenting a comprehensive review of the expected
scienti{f}{i}c results from {\sc Planck} measurements of
extragalactic sources (see, e.g., De Zotti et al. 1999) we will
focus on a couple of frequencies, one of the Low Frequency
Instrument, namely 30 GHz, and one of the High Frequency
Instrument, namely 350 GHz. The relatively shallow but all-sky
{\sc Planck} surveys will be ideal to study populations which are
both very powerful at mm/sub-mm wavelengths, and very rare, such
as radio sources with inverted spectra up to $\ge 30\,$GHz
[extreme GHz Peaked Spectrum (GPS) sources or High Frequency
Peakers (HFP)], thought to be the most recently formed and among
the most luminous radio sources, and ultra-luminous dusty
proto-spheroidal galaxies, undergoing their main and huge episode
of star formation at typical redshifts $\ge 2$ (Granato et al.
2001, 2004). And {\sc Planck} will observe such sources with an
unprecedented frequency coverage.
To estimate the detection limit, and the number of detectable
sources, we need to take into account, in addition to the
instrument noise, the {f\-l}uctuations due to Galactic emissions,
to the Cosmic Microwave Background (CMB), and to extragalactic
sources themselves. These {f\-l}uctuations will be brie{f}{l}y
reviewed in Section 2, while in Sections 3 and 4 we will discuss
the expected impact of {\sc Planck} data on our understanding of
HFPs and of ultra-luminous proto-spheroidal galaxies,
respectively. Our main conclusions are summarized in Sect. 5.
\begin{figure*}[t]
\centerline{
\protect\includegraphics[width=12cm,height=10cm]{dezotti_fig1.ps}}
\vskip-5cm \caption{Galactic (dot-dashed) and extragalactic
(dashed) contributions to the power spectrum of foreground
{f\-l}uctuations, compared with the CMB (dotted horizontal line)
for three values of the multipole number $\ell$. The solid lines
show the sum, in quadrature, of the two contributions. At
$\ell=100$ the dot-dashed line essentially coincides with the
solid line; the two lines largely overlap at high frequencies also
for higher $\ell$'s. The Galactic contributions are averages for
$|b|\ge 20^\circ$, after having applied the Kp0 mask which include
the point source removal, and comprise synchrotron, free-free and
thermal dust emissions, whose power spectra are normalized to the
K-band (22.8 GHz), V-band (60.8 GHz), and W-band (93.5 GHz) WMAP
data, respectively (where each component is best measured). The
extrapolation in frequency has been done adopting, for free-free,
the antenna temperature spectral index ($T_A\propto \nu^\beta$)
$\beta_{\rm ff}=-2.15$, and for synchrotron the expression
proposed by Jackson \& Wall (2002) for low-luminosity radio
sources ($\log S_\nu = \hbox{const} - 0.6424\log(\nu) -
0.0692(\log(\nu)^2)$, with $\nu$ in GHz); this formula, which
allows for the high-frequency steepening of the synchrotron
spectrum due to electron energy losses, is consistent with the
steepening observed in WMAP data (Fig. 9 of Bennett et al. 2003).
As for thermal dust we have considered two cases: $\beta_d=2.2$,
the best {f}{i}t value of Bennett et al. (2003), and the more
usual value $\beta_d=2$. With these spectra, an additional
component (spinning dust?) is necessary to account for the
foreground signal detected by WMAP particularly in the Q-band
(40.7 GHz); the solid lines include this component. Power spectra
at $\ell =100$ were derived directly from WMAP data. At higher
$\ell$'s we assume $C_\ell = C_{100} (\ell/100)^{-\gamma}$ with
$\gamma = 2$ or $\gamma = 3$. The upper dot-dashed curve
corresponds to $\gamma = 2$ and $\beta_d=2.2$, the lower one to
$\gamma = 3$ and $\beta_d=2$. The dashed curve includes, summed in
quadrature, the contributions of all classes of extragalactic
sources, based on models by De Zotti et al. (2004), including
canonical radio sources, starburst galaxies, proto-spheroidal
galaxies and Sunyaev-Zeldovich effects. The effect of clustering
of proto-spheroidal galaxies has been taken into account as in
Negrello et al. (2004a). }\label{foreground}
\end{figure*}
\section{Power spectra of foreground emissions}
For a very high sensitivity experiment, like {\sc Planck}, the
main limitation to the capability of mapping the CMB is set by
contamination by astrophysical sources (``foregrounds''), while
CMB {f\-l}uctuations may be the highest ``noise'' source for the
study of astrophysical emissions at mm wavelengths. The most
intense foreground source is our own Galaxy. Because of the
different power spectra of the various emission components, the
frequency of minimum foreground {f\-l}uctuations depends to some
extent on the angular scale (see Fig.~\ref{foreground}, where
$\delta T$ are {f\-l}uctuations of the CMB thermodynamic
temperature, in $\mu$K, related to the power spectrum $C_\ell$ by
$\delta T = [\ell(\ell+1)C_\ell/(2\pi)]^{0.5}(e^x-1)^2/(x^2 e^x)$,
with $x=h\nu/kT_{\rm CMB}$). So long as diffuse Galactic emissions
dominate the {f\-l}uctuations ($\theta \,\lower2truept\hbox{${> \atop\hbox{\raise4truept\hbox{$\sim$}}}$}\, 30'$; see De Zotti et
al. 1999), they have a minimum in the 60--80 GHz range (depending
also on Galactic latitude; cf. Bennett et al. 2003).
But the power spectra of diffuse Galactic emissions decline rather
steeply with increasing multipole number (or decreasing angular
scale). Thus, on small scales, {f\-l}uctuations due to
extragalactic sources, whose Poisson contribution has a
white-noise power spectrum (on top of which we may have a,
sometimes large, clustering contribution) take over, even though
their integrated emission is below the Galactic one. At high
frequencies, however, Galactic dust may dominate {f\-l}uctuations
up to $\ell$ values of several thousands. Unlike the relatively
quiescent Milky Way, the relevant classes of extragalactic sources
have strong nuclear radio activity or very intense star formation,
or both. Thus, although in many cases their SEDs are qualitatively
similar to that of the Milky Way, there are important quantitative
differences. In particular, dust in active star forming galaxies
is signi{f\-i}cantly hotter and the radio to far-IR intensity
ratio of the extragalactic background is much higher than that of
the Milky Way. Both factors, but primarily the effect of radio
sources, cooperate to move the minimum of the SED to 100--150 GHz.
\begin{figure}[t]
\centerline{\includegraphics[width=5.5cm,height=5cm]{dezotti_fig2a.ps}
\includegraphics[width=5.5cm,height=5cm]{dezotti_fig2b.ps}}
\caption{Predicted 30 GHz differential counts. The left-hand panel
shows the counts of all the main populations (see De Zotti et al.
2004 for details). The right-hand panel details the contributions
of three sub-classes of canonical radio sources: FSRQs, BL Lac
objects, and steep-spectrum sources } \label{30GHzcounts}
\end{figure}
\begin{figure*}
\centerline{\includegraphics[width=5.5cm,height=5cm]{dezotti_fig3a.ps}
\includegraphics[width=5.5cm,height=5cm]{dezotti_fig3b.ps}}
\caption{Redshift distributions of WMAP FSRQs (left-hand panel)
and BL Lacs (right-hand panel) compared with the model by De Zotti
et al. (2004, solid line). }\label{WMAP_FSRQ_BLLacs}
\end{figure*}
\section{30 GHz counts}
Figure~\ref{30GHzcounts} provides a synoptic view of the
contributions of different source classes to the global counts of
extragalactic sources. Shallow surveys, such as those by WMAP and
{\sc Planck}, mostly detect canonical radio sources. As shown by
the right-hand panel of Fig.~\ref{30GHzcounts}, detected sources
will be mostly {f}{l}at-spectrum radio quasars (FSRQs), while the
second more numerous population are BL Lac objects. {\sc Planck}
will detect about ten times more sources than WMAP, thus allowing
a substantial leap forward in the understanding of evolutionary
properties of both populations at high frequencies, only weakly
constrained by WMAP data (Fig.~\ref{WMAP_FSRQ_BLLacs}).
{\sc Planck} will also provide substantial complete samples of
sources not (yet) represented in the WMAP catalog, such as
Sunyaev-Zeldovich (1972) signals and extreme GPS sources or HFPs
(Dallacasa et al. 2000).
GPS sources are powerful ($\log P_{\rm 1.4\, GHz} \,\lower2truept\hbox{${> \atop\hbox{\raise4truept\hbox{$\sim$}}}$}\,
25\,\hbox{W}\,\hbox{Hz}^{-1}$), compact ($\,\lower2truept\hbox{${<\atop\hbox{\raise4truept\hbox{$\sim$}}}$}\, 1\,$kpc) radio
sources with a convex spectrum peaking at GHz frequencies. It is
now widely agreed that they correspond to the early stages of the
evolution of powerful radio sources, when the radio emitting
region grows and expands within the interstellar medium of the
host galaxy, before plunging in the intergalactic medium and
becoming an extended radio source (Fanti et al. 1995; Readhead et
al. 1996; Begelman 1996; Snellen et al. 2000). Conclusive
evidence that these sources are young came from measurements of
propagation velocities. Velocities of up to $\simeq 0.4c$ were
measured, implying dynamical ages $\sim 10^3$ years (Polatidis et
al. 1999; Taylor et al. 2000; Tschager et al. 2000). The
identi{f}{i}cation and investigation of these sources is therefore
a key element in the study of the early evolution of radio-loud
AGNs.
There is a clear anti-correlation between the peak (turnover)
frequency and the projected linear size of GPS sources. Although
this anti-correlation does not necessarily de{f}{i}ne the
evolutionary track, a decrease of the peak frequency as the
emitting blob expands is indicated. Thus high-frequency surveys
may be able to detect these sources very close to the moment when
they turn on. The self-similar evolution models by Fanti et al.
(1995) and Begelman (1996) imply that the radio power drops as the
source expands, so that GPS's evolve into lower luminosity radio
sources, while their luminosities are expected to be very high
during the earliest evolutionary phases, when they peak at high
frequencies. De Zotti et al. (2000) showed that, with a suitable
choice of the parameters, this kind of models may account for the
observed counts, redshift and peak frequency distributions of the
samples then available. The models by De Zotti et al. (2000)
imply, for a maximum rest-frame peak frequency $\nu_{p,i}
=200\,$GHz, about 10 GPS quasars with $S_{30{\rm GHz}} > 2\,$Jy
peaking at $\geq 30\,$GHz over the 10.4 sr at $|b| >10^\circ$.
Although the number of {\it candidate} GPS quasars (based on the
spectral shape) in the WMAP survey is consistent with such
expectation, when data at additional frequencies (Trushkin 2003)
are taken into account the GPS candidates look more blazars caught
during a {f}{l}are optically thick up to high frequencies.
Furthermore, Tinti et al. (2004) have shown that most, perhaps two
thirds, of the quasars in the sample of HFP candidates selected by
Dallacasa et al. (2000) are likely blazars.
Thus, WMAP data are already providing strong constraints on the
evolution of HFPs. {\sc Planck} will substantially tighten such
constraints and may allow us to directly probe the earliest phases
(ages $\sim 100\,$yr) of the radio galaxy evolution, hopefully
providing hints on the still mysterious mechanisms that trigger
the radio activity.
We note, in passing, that contrary to some claims, we do not
expect that {\sc Planck} can detect the late phases of the AGN
evolution, characterized by low accretion/radiative ef{f}{i}ciency
(ADAF/ADIOS sources).
At faint {f\-l}ux densities, other populations come out and are
expected to dominate the counts. In addition to SZ effects, we
have active star-forming galaxies, seen either through their radio
emission, or through their dust emission, if they are at
substantial redshift. The latter is the case for the sub-mm
sources detected by the SCUBA surveys if they are indeed at high
redshifts (see below).
Such sources may be relevant in connection with the interpretation
of the excess signal on arc-minute scales detected by CBI (Mason
et al. 2003; Readhead et al. 2004) and BIMA (Dawson et al. 2002)
experiments at 30 GHz, particularly if, as discussed below, they
are highly clustered, so that their contribution to
{f\-l}uctuations is strongly super-Poissonian (Toffolatti et al.
2004). In fact, to abate the point source contamination of the
measured signals, the CBI and BIMA groups could only resort to
existing or new low frequency surveys. But the dust emission is
undetectable at low frequencies. Although our reference model
(Granato et al. 2004), with its relatively warm dust temperatures
yielded by the code GRASIL (Silva et al. 1998), imply dusty galaxy
contributions to small scale {f\-l}uctuations well below the
reported signals, the (rest-frame) mm emission of such galaxies is
essentially unknown and may be higher than predicted, e.g. in the
presence of the extended distribution of cold dust advocated by
Kaviani et al. (2003) or of a widespread mm excess such as that
detected in several Galactic clouds (Dupac et al. 2003) and in
NGC1569 (Galliano et al. 2003). This is another instance of the
importance of a multifrequency approach, like {\sc Planck}'s,
capable of keeping under control all the relevant emission
components, with their different emission spectra.
\begin{figure}
\centerline{\includegraphics[width=11.5cm,height=10.5cm]{dezotti_fig4.ps}
} \vskip-5cm \caption{Left-hand panel: contributions of different
populations to the 350 GHz counts. Central panel: effect of
lensing on counts of proto-spheroidal galaxies. Right-hand panel:
estimated counts of ``clumps'' of proto-spheroids observed with
{\sc Planck} resolution. } \label{lensplusclust}
\end{figure}
\section{350 GHz counts}
The 350 GHz counts of extragalactic sources have been determined
in the range from $\simeq 10\,$mJy to $\simeq 0.25\,$mJy by
surveys with the SCUBA camera, covering small areas of the sky
(overall, a few tenths of a square degree). These surveys have led
to the discovery of very luminous high-$z$ galaxies, with
star-formation rates $\sim 10^3\,M_\odot$/yr, a result
con{f}{i}rmed by 1.2mm surveys with MAMBO on the IRAM 30m
telescope. These data proved to be extremely challenging for
semi-analytic galaxy formation models, and have indeed forced to
reconsider the evolution of baryons in dark matter halos.
The bright portion of observed counts appears to be declining
steeply with increasing {f\-l}ux density, probably re{f\-l}ecting
the exponential decline of the dark-halo mass function at large
masses implied by the Press \& Schechter formula, so that one
would conclude that {\sc Planck} cannot do much about these
objects, but rather detect brighter sources such as blazars and
relatively local star-forming galaxies, or SZ signals. There are,
however, two important effects to be taken into account, that may
change this conclusion: gravitational lensing and clustering.
We refer here to the model by Granato et al. (2004) according to
which SCUBA sources are large spheroidal galaxies in the process
of forming most of their stars. Forming spheroidal galaxies, being
located at relatively high redshift, have a substantial optical
depth for gravitational lensing, and the effect of lensing on
their counts is strongly ampli{f\-i}ed by the steepness of the
counts. This is illustrated by the left-hand panel of
Fig.~\ref{lensplusclust}, based on calculations by Perrotta et al.
(2003). Strong lensing is thus expected to bring a signi{f\-i}cant
number of high-$z$ forming spheroids above the estimated {\sc
Planck} $5\sigma$ detection limit.
If indeed SCUBA galaxies are massive spheroidal galaxies at high
$z$, they must be highly biased tracers of the matter
distribution, and must therefore be highly clustered. There are in
fact several, although tentative, observational indications of
strong clustering with comoving radius $r_0 \simeq
8\hbox{h}^{-1}\,$Mpc (Smail et al. 2004; Blain et al. 2004;
Peacock et al. 2000), consistent with theoretical expectations.
But if massive spheroidal proto-galaxies live in strongly
over-dense regions, low resolution experiments like {\sc Planck}
unavoidably measure not the {f\-l}ux of individual objects but the
sum of {f\-l}uxes of all physically related sources in a
resolution element.
This is an aspect of the ``source confusion" problem, whereby the
observed {f\-l}uxes are affected by unresolved sources in each
beam. The problem was extensively investigated in the case of a
Poisson distribution, particularly by radio astronomers (Scheuer
1957, Murdoch et al. 1973, Condon 1974, Hogg \& Turner 1998). The
general conclusion is that unbiased {f\-l}ux measurements require
a $S/N \ge 5$.
Not much has been done yet on confusion in the presence of
clustering (see however Hughes \& Gaztanaga 2000). The key
difference is that, for a Poisson distribution, a bright source is
observed on top of a background of unresolved sources that may be
either above or below the all-sky average, while in the case of
clustering, sources are preferentially found in over-dense
regions.
Clearly, the excess signal (over the {f\-l}ux of the brightest
source in the beam) depends on the angular resolution. For a
standard $\xi(r) = (r/r_0)^{-1.8}$ the mean clustering
contribution is $\propto r_0^{1.8} r_{\rm beam}^{1.2}$. The {\sc
Planck} beam at this frequency corresponds to a substantial
portion of the typical clustering radius at $z\simeq 2$--3, so
that {\sc Planck} will actually measure a signi{f\-i}cant fraction
of the {f\-l}ux of the clump, which may be substantially larger
than the {f\-l}ux of any member source. The effect on counts
depends on the joint distribution of over-densities and of $M/L$
ratios. The former depends on both the two- and the three-point
correlation function, while the latter depends on the luminosity
function.
Preliminary estimates of the distribution of excess luminosities
due to clustering around bright sources have been obtained by
Negrello et al. (2004b). The right-hand panel of
Fig.~\ref{lensplusclust} shows the estimated counts of clumps
observed with {\sc Planck} resolution for three models for the
evolution of the coef{f}{i}cient $Q$ of the three-point
correlation function. Obviously {\sc Planck} can provide
information only on the brightest clumps, and, except in the
extreme case of $Q=1$ at all cosmic times, the clumps will only
show up as $< 5\sigma$ {f\-l}uctuations. On the other hand, such
{f\-l}uctuations will provide a rich catalogue of candidate
proto-clusters at substantial redshifts (typically at $z\simeq
2$--3), very important to investigate the formation of large scale
structure and, particularly, to constrain the evolution of the
dark energy thought to control the dynamics of the present day
universe.
\section{Conclusions}
Although extragalactic surveys are not the primary goal of the
mission, {\sc Planck} will provide unique data for several
particularly interesting classes of sources. Examples are the
FSRQs, BL Lac objects, but especially extreme GPS sources that may
correspond to the earliest phases of the life of radio sources,
and proto-spheroidal galaxies. Thus {\sc Planck} will investigate
not only the origin of the universe but also the origin of radio
activity and of galaxies. Sub-mm surveys will provide large
samples of candidate proto-clusters, at $z\simeq 2$--3, shedding
light on the evolution of the large scale structure (and in
particular providing information on the elusive three-point
correlation function) and of the dark energy, across the cosmic
epoch when it is expected to start dominating the cosmic dynamics.
\begin{acknowledgments}
Work supported in part by MIUR through a PRIN grant and by ASI.
\end{acknowledgments}
\begin{chapthebibliography}{1}
\bibitem{}
Begelman, M.C. (1996). In ``Cygnus A -- Study of a Radio Galaxy'',
C.L. Carilli \& D.E. Harris eds., Cambridge University Press, p.
209
\bibitem{}
Bennett, C.L., et al. (2003). ApJS, 148, 97.
\bibitem{}
Blain, A. W., S.C. Chapman, I. Smail, and R. Ivison (2004). ApJ,
611, 725
\bibitem{}
Condon, J.J. (1974). ApJ, 188, 279.
\bibitem{}
Dallacasa, D., C. Stanghellini, M. Centonza, and R. Fanti (2000).
A\&A, 363, 887.
\bibitem{}
Dawson, K.S., et al. (2002). ApJ, 581, 86.
\bibitem{}
De Zotti, G., et al. (1999). In L. Maiani, F. Melchiorri and N.
Vittorio, editors, {\em Proceedings of the EC-TMR Conference ``3K
Cosmology'', AIP CP}, 476, 204.
\bibitem{}
De Zotti, G., G.L. Granato, L. Silva, D. Maino, and L. Danese
(2000). A\&A, 354, 467.
\bibitem{}
De Zotti, G., R. Ricci, D. Mesa, L. Silva, P. Mazzotta, L.
Toffolatti, and J. Gonz\'alez-Nuevo (2004). A\&A, in press
\bibitem{}
Dupac, X., et al. (2003). A\&A, 404, L11.
\bibitem{}
Fanti, C., et al. (1995). A\&A, 302, 317.
\bibitem{}
Galliano, F., et al. (2003). A\&A, 407, 159.
\bibitem{}
Granato, G.L., L. Silva, P. Monaco, P. Panuzzo, P. Salucci, G. De
Zotti, and L. Danese (2001). MNRAS, 324, 757.
\bibitem{}
Granato, G.L., G. De Zotti, L. Silva, A. Bressan, and L. Danese
(2004). ApJ, 600, 580.
\bibitem{}
Hogg, D.W., and E.L. Turner (1998). PASP, 110, 727.
\bibitem{}
Hughes, D.H., and E. Gaztanaga (2000). In {\em Star formation from
the small to the large scale}, proc. 33rd ESLAB symp., F. Favata,
A. Kaas, and A. Wilson eds., ESA SP 445, p. 29.
\bibitem{}
Jackson, C.A., and J.V. Wall (2001). In {\em Particles and Fields
in Radio Galaxies}, ASP Conf. Proc. 250, R.A. Laing and K.M.
Blundell eds., p. 400.
\bibitem{}
Kaviani, A., M.G. Haehnelt, and G. Kauffmann (2003). MNRAS, 340,
739.
\bibitem{}
Mason, B.S., et al. (2003). ApJ, 591, 540.
\bibitem{}
Murdoch, H.S., D.F. Crawford, and D.L. Jauncey (1973). ApJ, 183,
1.
\bibitem{} Negrello, M., M. Magliocchetti, L. Moscardini, G. De Zotti,
G.L. Granato, and L. Silva (2004a). MNRAS, 352, 493.
\bibitem{}
Negrello, M., M. Magliocchetti, L. Moscardini, G. De Zotti, and L.
Danese (2004b). MNRAS, submitted
\bibitem{}
Peacock, J.A., et al. (2000). MNRAS, 318, 535.
\bibitem{}
Perrotta, F., M. Magliocchetti, C. Baccigalupi, M. Bartelmann, G.
De Zotti, G.L. Granato, L. Silva, and L. Danese (2003). MNRAS,
338, 623.
\bibitem{}
Polatidis, A., et al. (1999). NewAR, 43, 657.
\bibitem{}
Readhead, A.C.S., G.B. Taylor, T.J. Pearson, and P.N. Wilkinson
(1996). ApJ, 460, 634.
\bibitem{}
Readhead, A.C.S., et al. (2004). ApJ, 609, 498.
\bibitem{}
Scheuer, P.A.G. (1957). Proc. Cambridge Phil. Soc., 53, 764.
\bibitem{}
Silva, L., G.L. Granato, A. Bressan, and L. Danese (1998). ApJ,
509, 103.
\bibitem{}
Smail, I., S.C. Chapman, A.W. Blain, and R.J. Ivison (2004). In
``The Dusty and Molecular Universe: A Prelude to Herschel and
ALMA'', A. Wilson ed., ESA Conf. Ser., p. 16.
\bibitem{}
Snellen, I.A.G., et al. (2000). MNRAS, 319, 445.
\bibitem{}
Sunyaev, R. A., and Ya.B. Zeldovich (1972). Comm. Ap. Space Phys.,
4, 173.
\bibitem{}
Taylor, G.B., J.M. Marr, T.J. Pearson, and A.C.S. Readhead (2000).
ApJ, 541, 112.
\bibitem{}
Tinti, S., D. Dallacasa, G. De Zotti, A. Celotti, and C.
Stanghellini (2004). A\&A, in press.
\bibitem{}
Toffolatti, L., J. Gonz\'alez-Nuevo, M. Negrello, G. De Zotti, L.
Silva, G.L. Granato, and F. Arg\"ueso (2004). A\&A, submitted
\bibitem{}
Trushkin, S.A. (2003). Bull. Spec. Astrophys. Obs. N. Caucasus,
55, 90, astro-ph/0307205.
\bibitem{}
Tschager, W., et al. (2000). A\&A, 360, 887.
\end{chapthebibliography}
\end{document}
|
{
"timestamp": "2004-11-08T10:36:44",
"yymm": "0411",
"arxiv_id": "astro-ph/0411182",
"language": "en",
"url": "https://arxiv.org/abs/astro-ph/0411182"
}
|
\section{Introduction}
Twistor methods in gauge theory have a long history (summarized in ref. [P90]). A common feature of these methods is that spacetime is replaced by a twistor (or ambitwistor) analytic manifold $\mathbf{T}$ . Equations of motion "emerge" (in terminology of Penrose) from complex geometry of $\mathbf{T}$.
The twistor approach turns to be a very useful technical innovation. For example difficult questions of classical gauge theory, e.g. the ones that appear in the theory of instantons, admit a translation into a considerably more simple questions of analytic geometry of space $\mathbf{T}$. On this way classifications theorems in the theory of instantons has been obtained (see ref. [AHDM]).
The quantum theory did not have a simple reformulation in the language of geometry of the space $\mathbf{T}$ so far. One of the reasons is that the quantum theory formulated formally in terms of a path integral requires a Lagrangian. Classical theory, as it was mentioned earlier, provides only equations of motion whose definition needs no metric. In contrast a typical Lagrangian requires a metric in order to be defined. Thus a task of finding the Lagrangian in (ambi)twistor setup is not straightforward. In this paper we present a Lagrangian for N=3 D=4 Yang-Mills (YM) theory formulated in terms of ambitwistors.
Recall that N=3 YM theory coincides in components with N=4 YM theory. The easiest way to obtain N=4 theory is from N=1 D=10 YM theory by dimensional reduction. The Lagrangian of this ten dimensional theory is equal to
\begin{equation}
(<F_{ij},F_{ij}>+<D\!\!\!\!/ \chi,\chi>)dvol
\end{equation}
In the last formula $F_{ij}$ is a curvature of connection $\nabla$ in a principal U$(n)$-bundle over $\mathbb{R}^{10}$. An odd field $\chi$ is a section of $S\otimes Ad$, where $S$ is a complex sixteen dimensional spinor bundle, $Ad$ is the adjoint bundle, $D\!\!\!\!/$ is the Dirac operator, $<.,.>$ is a Killing pairing on $\mathfrak{u}(n)$. The measure $dvol$ is associated with a flat Riemannian metric on $\mathbb{R}^{10}$, $F_{ij}$ are coefficients of the curvature in global orthonormal coordinates. The N=4 theory is obtained from this by considering fields invariant with respect to translations in six independent directions. The theory is conformally invariant and can be defined on any conformally flat manifold,e.g. $S^4$ with a round metric.
E. Witten in 1978 in ref. [W78] discovered that it is possible to encode solutions N=3 supersymmetric YM-equation by holomorphic structures on a vector bundle defined over an open subset $U$ in a superquadric $\mathcal{Q}$. We shall call the latter a complex ambitwistor superspace. In this description the action of N=3 superconformal symmetry on the space of solutions is manifest. Symbol $n|m$ denotes dimension of a supermanifold. More precisely the quadric $\mathcal{Q} \subset \mathbf{P}^{3|3}\times \mathbf{P}^{*3|3}$ is defined by equation
\begin{equation}\label{E:kdsj}
\sum_{i=0}^3 x_ix^i+\sum_{i=1}^3\psi_i\psi^i=0,
\end{equation}
in bihomogeneous coordinates
\begin{equation}
x_0,x_1,x_2,x_3,\psi_1,\psi_2,\psi_3;\ x^0, x^1, x^2, x^3,\psi^1,\psi^2,\psi^3
\end{equation}
in $\mathbf{P}^{3|3}\times \mathbf{P}^{*3|3}$($x_i,x^j$-even, $\psi_i,\psi^j$-odd, a symbol $*$ in the superscript stands for the dual space). The quadric a is complex supermanifold. It makes sense therefore to talk about differential $(p,q)$-forms $\Omega^{p,q}(\mathcal{Q})$.
Let ${\cal G}$ be a holomorphic vector bundle on $U$. Denote by
\begin{equation}\label{E:hjdur}
\Omega^{0\bullet}End{\cal G},
\end{equation}
a differential graded algebra of smooth sections of $End{\cal G}$ with coefficients in $0,p$-forms.
Let $\bar \partial$ and $\bar \partial'$ be two operators corresponding to two holomorphic structures in ${\cal G}$. It is easy to see that $(\bar \partial'-\bar \partial)b=ab$, where $a\in \Omega^{0,1}End{\cal G}$. The integrability condition $\bar \partial^{'^2}=0$ in terms of $\bar \partial$ and $a$ becomes a Maurer-Cartan (MC) equation:
\begin{equation}\label{E:fdsaf}
\bar \partial a+\frac{1}{2}\{a,a\}=0
\end{equation}
The first guess would be that the space of fields of ambitwistor version of N=3 YM would be $\Omega^{0,1}(U)End{\cal G}$, where ${\cal G}$ is a vector bundle on $U$ of some topological type. Witten suggested in ref. [W03] that the Lagrangian in question should be similar to a Lagrangian of holomorphic Chern-Simons theory.
\begin{equation}\label{E:gdfsguqd}
CS(a)=\int tr(\tilde(\frac{1}{2}a\bar \partial a +\frac{1}{6}a^3))Vol
\end{equation}
where $Vol$ is some integral form. The action (\ref{E:gdfsguqd}) reproduces equations of motion (\ref{E:fdsaf}). The hope is that perturbative analysis of this quantum theory will give some insights on the structure of N=3 YM.
The main result of the present note is that we
give a precise meaning to this conjecture.
Introduce a real supermanifold $\mathcal{R} \subset \mathcal{Q}$ of real superdimension $8|12$. It is defined by equation
\begin{equation}\label{E:dfsad}
x_1\bar{x}^2-x_2\bar{x}^1+x_3\bar{x}^4-x_4\bar{x}^3+\sum_{i=1}^3\psi_i\bar{\psi}^i=0
\end{equation}
In Section \ref{S:llfddfgg} we discuss the meaning of reality in superalgebra and geometry.
\begin{definition}
Let $M$ be a $C^{\infty}$ supermanifold, equipped with a subbundle $H$ of the tangent bundle $T$. We say that $M$ is equipped with CR-structure if $H$ carries a complex structure defined by fiberwise transformation $J$.
The operator $J$ defines a decomposition of the complexification $H^{\mathbb{C}}$ into a direct sum of eigensubbundles ${\cal F}+\overline{{\cal F}}$.
We say that the CR-structure $J$ is integrable if the sections of ${\cal F}$ form a Lie subalgebra of $T^{\mathbb{C}}$ under the bracket of vector fields.
\end{definition}
A tautological embedding of $\mathcal{R} $ into the complex manifold $\mathcal{Q}$ induces a CR structure specified by distribution ${\cal F}$. Properties of this CR structure are discussed in Section \ref{S:qjscmmz}. A global holomorphic supervolume form $vol$ on $\mathcal{Q}$ is constructed in Proposition \ref{P:jsdhd}. When restricted on $\mathcal{R} $ it defines a section of $\ ^{int}\Omega^{-3}_{{\cal F}}$-a CR integral form. Functorial properties of this form are discussed in Section \ref{S:gufjasklly}. For any CR-holomorphic vector bundle ${\cal G}$ we define a differential graded algebra $\Omega^{\bullet}_{{\cal F}}End({\cal G})$. It is the tangential CR complex. We equip it with a trace
\begin{equation}
\int :a \rightarrow \int_{\mathcal{R} }tr(a)\ vol
\end{equation}
We define a CS-action of the form (\ref{E:gdfsguqd}), where we replace an element of $\Omega^{0\bullet}(U)End({\cal G})$ by an element of $\Omega^{\bullet}_{{\cal F}}(\mathcal{R} )End({\cal G})$. The integral is taken with respect to the measure $vol$. We make some assumptions about topology of ${\cal G}$ as it is done in classical twistor theory. The space $\S$ is a superextension of $S^4$ (see Section \ref{S:hfdgjpo} for details). There is a projection
\begin{equation}\label{E:sdsfsew}
p:\mathcal{R} \rightarrow \S
\end{equation}
We require that ${\cal G}$ is topologically trivial along the fibers of $p$. It is an easy exercise in algebraic topology to see that topologically all such bundles are pullbacks from $S^4$. On $S^4$ unitary vector bundles are classified by their second Chern classes.
\begin{conjecture}\label{C:gdfsgjj}
Suppose ${\cal G}$ is a CR-holomorphic vector bundle on $\mathcal{R} $ of rank $n$.
Under the above assumptions a CS theory defined by the algebra $\Omega^{\bullet}_{{\cal F}}End({\cal G})$ is equivalent to N=3 YM theory on $S^4$ in a principal U$(n)$ bundle with the second Chern class equal to $c_2(End({\cal G}))$.
\end{conjecture}
For perturbative computations in YM theory it is convenient to work in BV formalism. See ref. [Sch00] for mathematical introduction and ref. [MSch06] for applications to YM.
\begin{conjecture}\label{C:ttc2}
In assumptions of Conjecture \ref{C:gdfsgjj} we believe that N=3 YM theory in BV formulation is equivalent to a CS theory defined by the algebra $\Omega^{\bullet}_{{\cal F}}End({\cal G})$, where the field $a\in \Omega^{\bullet}_{{\cal F}}End({\cal G})$ has a mixed degree.
\end{conjecture}
The following abstract definition will be useful
\begin{definition}\label{D:rtje}
Suppose we are given a differential graded algebra $(A,d)$ with a $d$-closed trace functional $\int$. We can consider $A$ as a space of fields in some field theory with Lagrangian defined by the formula
\begin{equation}
CS(a)=\int(\frac{1}{2}ad(a)+\frac{1}{6}a^3)
\end{equation}
We call it a Chern-Simons $CS$ theory associated with a triple $(A,d,\int)$
We say that two theories $(A,d,\int)$ and $(A',d',\int')$ are classically equivalent if there is a quasiisomorphism of algebras with trace $f:(A,d,\int)\rightarrow (A',d',\int')$
See Appendix of [MSch05] for extension of this definition on A$_{\infty}$ algebras with a trace.
\end{definition}
Thus the matrix-valued Dolbeault complex $(\Omega_{{\cal F}}^{\bullet}(\mathcal{R})\otimes Mat_n,\bar \partial)$ with a trace defined by the formula $\int(a)=tr\int_ravol$ would give an example of such algebra.
We shall indicate existence of classical equivalence of N=3 YM defined over $\Sigma=\mathbb{R}^4\subset S^4$ and a CS theory defined over $p^{-1}(U)$,
where $U$ is an open submanifold of $\S$ with $U_{red}=\Sigma$.
Here is the idea of the proof.
We produce a supermanifold $Z$ and an integral form $Vol$ on it. We show that a CS theory constructed using differential graded algebra with a trace $A(Z)$, associated with manifold $Z$ is classically equivalent to N=3 YM theory. We interpret algebra $A(Z)$ as a tangential CR-complex on $Z$.
We shall construct a manifold $Z$ and algebra $A$ in two steps .
Here is a description of the steps in more details:
{\bf Step 1.} We define a compact analytic supermanifold $\widetilde {\Pi F}$ and construct an integral form $\mu$ on it in a spirit of [MSch05].
Let $A_{pt}$ be the Dolbeault complex of $\widetilde {\Pi F}$. Integration of an element $a \in A_{pt}$ against $\mu$ over $\widetilde {\Pi F}$ defines a $\bar \partial$-closed trace functional on $A_{pt}$. We show that the Chern-Simons theory associated with dga $(A_{pt}\otimes Mat_n,\bar \partial,\int\otimes tr_{Mat_n})$ is classically equivalent to N=3 Yang-Mills theory with gauge group U$(n)$ reduced to a point.
{\bf Step 2.} From the algebra $A_{pt}$ we reconstruct a differential algebra $A$. The algebra $A\otimes Mat_n$ conjecturally encodes full N=3 Yang-Mills theory with gauge group U$(n)$ in a sense of Definition \ref{D:rtje}. If we put aside the differential $d$, $A$ is equal to $A_{pt}\otimes C^{\infty}(\Sigma)$. The integral form we are looking for is equal to $Vol=\mu dx_1dx_2dx_3dx_4$, where $\Sigma$ is equipped with global coordinates $x_1,x_2,x_3,x_4$.
The manifold $Z$ is a CR submanifold. We identify it with an open subset of $\mathcal{R} $.
Finally we would like to formulate an unresolved question. Restriction of a holomorphic vector bundle ${\cal G}$ over $U$ on $U\cap \mathcal{R}$ defines a CR-vector bundle over the intersection. Is it true that any CR-holomorphic vector bundles can be obtained this way? The answer would be affirmative if we impose some analyticity conditions on the CR structure on ${\cal G}$. Presumably super Levi form will play a role in a solution of this problem.
It is tempting to speculate that there is a string theory on $\mathcal{Q}$ and $\mathcal{R} $ defines a D-brane in it.
We need to say few words about the structure of this note.
In Section \ref{S:dfdrf} we make definitions and provide some constructions used in formulation of conjectures (\ref{C:gdfsgjj}) and (\ref{C:ttc2}).
In Section \ref{S:main} we give a geometric twistor-like description of N=3 YM theory reduced to a point( Step 1). In Section \ref{S:ffkewx} we do the Step 2.
Appendix contains some useful definitions concerning reality in superalgebra and CR-structures.
\section{Infinitesimal constructions}\label{S:dfdrf}
In this section we shall show that the space $\mathcal{R} $ is homogeneous with respect to the action of a real form of N=3 superconformal algebra $\mathfrak{gl}(4|3)$ . Here we also collected facts that are needed for coordinate-free description of space $\mathcal{R}$ in terms of Lie algebras of symmetry group and isotropy subgroup.
\subsection{Real structure on the Lie algebra $\mathfrak{gl}(4|3)$}\label{S:qjscmmz}
In this section we describe a graded real structure on $\mathfrak{gl}(4|3)$. It will be used later in construction of CR-structure on real super-ambitwistor space.
The reader might wish to consult Section \ref{S:llfddfgg} for the definition of a graded real structure. There the reader will find explanation of some of our notations. By definition $\mathfrak{gl}(4|3)$ is a super Lie algebra of endomorphisms of $\mathbb{C}^{4|3}=\mathbb{C}^{4}+\Pi\mathbb{C}^{3}$. Symbol $\Pi$ stands for the parity change. This Lie algebra consists of matrices of a block form $\mat{A}{B}{C}{D}$ with $A \in Mat(4 \times 4,\mathbb{C}),D \in Mat(3\times 3,\mathbb{C}), C \in Mat(3\times 4,\mathbb{C}),B\in Mat(4 \times 3,\mathbb{C})$. Elements $\mat{A}{0}{0}{D}$ belong the even part $\mathfrak{gl}_{0}(4|3)$, elements $\mat{0}{B}{C}{0}$ to the odd $\mathfrak{gl}_{1}(4|3)$.
In the following a symbol $\mathfrak{g}(\mathbb{K})$ will stand for a Lie algebra defined over a field $\mathbb{K}$. If the field is not present it means that the algebra is defined over $\mathbb{C}$. The same applies to Lie groups.
Let $\mathfrak{g}$ be a complex super Lie algebra. By definition a map $\rho$ that defines a graded real structure on on super Lie algebra $\mathfrak{g}$ if $\rho$ is a homomorphism: $\rho[a,b]=[\rho(a),\rho(b)]$. In ref. [Man] Yu. I. Manin suggested several definitions of a real structure on a (Lie) superalgebra. In notations of [Man] these definitions are parametrized by a triple $(\epsilon_1,\epsilon_2,\epsilon_3), \epsilon_i=\pm$. Our real structure correspond to the choice $\epsilon_1=-,\epsilon_2=\epsilon_3=+$.
The reader will find a complete classification of graded real structures on simple Lie algebras in the work [Serg].
Define a matrix $J$ as :
\begin{equation}
J=\mat{0}{id}{-id}{0}
\end{equation}
where $id$ is a $2\times 2$ identity matrix. A map $\rho$ is defined as
\begin{equation}\label{E:gfjeiuq}
\rho\mat{A}{B}{C}{D}=\mat{J}{0}{0}{id}\mat{\overline{A}}{\overline{B}}{\overline{C}}{\overline{D}}\mat{-J}{0}{0}{id}=\mat{-J\overline{A}J}{J\overline{B}}{-\overline{C}J}{\overline{D}}
\end{equation}
The identity $\rho^2=sid$\footnote{The operator $sid$ is defined in the Appendix in Definition \ref{D:gdfjgdf}.} is a corollary of equation $J^2=-id$.
It is useful to analyze the Lie subalgebra $\mathfrak{gl}_{0}(4|3)^{\rho}$ of real points in $\mathfrak{gl}_0(4|3)=\mathfrak{gl}(4,\mathbb{C})\times \mathfrak{gl}(3,\mathbb{C})$. Due to (\ref{E:gfjeiuq}) we have $\mathfrak{gl}(3)^{\rho}=\mathfrak{gl}(3,\mathbb{R})$. To identify $\mathfrak{gl}^{\rho}(4)$ we interpret $\mathbb{C}^{4}=\mathbb{C}^{2}+ \mathbb{C}^{2}$(whose algebra of endomorphisms is $\mathfrak{gl}(4)$ ) as a two-dimensional quaternionic space $\mathbb{H}+\mathbb{H}$. Let $1,i,j,k$ be the standard $\mathbb{R}$-basis in quaternions, $<e_1,e_2>$ be an $\mathbb{H}$-basis in $\mathbb{H}+\mathbb{H}$. The space $\mathbb{H}+\mathbb{H}=\mathbb{C}^{2}+ \mathbb{C}^{2}$ has a complex structure defined by the right multiplication on $i$. The right multiplication on $j$ defines an $i$-antilinear map. In a $\mathbb{C}$-basis $e_1,e_2,e_1j,e_2j$ a matrix of right multiplication on $j$ is equal to $J$. From this it is straightforward to deduce that $\mathfrak{gl}^{\rho}(4)=\mathfrak{gl}(2,\mathbb{H})$.
\begin{definition}
Let $M$ be a $C^{\infty}$ supermanifold with a tangent bundle $T$. Let $H\subset T$ be a subbundle equipped with a complex structure $J$. This data defines a (nonintegrable) CR-structure on $M$. There is a decomposition \footnote{In the following a letter $\mathbb{C}$ in superscript denotes complexification.} $H^{\mathbb{C}}={\cal F}+\overline{{\cal F}}$. A CR-structure $(H,J)$ is integrable if a space of sections of ${\cal F}$ is closed under commutator. In this case we also say that ${\cal F}$ is integrable.
\end{definition}
\begin{definition}
Let $M_{red}$ denote the underlying manifold of supermanifold $M$.
\end{definition}
If $M$ is a real submanifold of a complex supermanifold $N$ then at any $x\in M$ the tangent space $T_x$ to $M$ contains a maximal complex subspace $H_x$. If $rank H_x$ is constant along $M$ then a family of spaces $H$ defines an integrable CR-structure. In our case the manifold $\mathcal{R} \subset \mathcal{Q}$ is defined by equation (\ref{E:dfsad}).
Denote by $GL(4|3)$ an affine supergroup with Lie algebra $Lie(GL(4|3))$ equal to $\mathfrak{gl}(4|3)$\footnote{For global description of $(GL(4|3),\rho)$ see ref. [Pel].}. We will show later that $\mathcal{R}$ is a homogeneous space of a real form of $GL(4|3)$ described above. The induced CR-structure is real-analytic and homogeneous with respect to the group action .
A CR-structure on a supermanifold enables us to define an analog of Dolbeault complex . Suppose a supermanifold $M$ carries a CR structure ${\cal F}\subset T^{\mathbb{C}}$. A space of complex 1-forms $\Omega^{1}_M$ contains a subspace $I$ of forms pointwise orthogonal to ${\cal F}$. It is easy to see that ${\cal F}$ is integrable iff the ideal $(I)$ is closed under $d$. Define a tangential CR-complex $(\Omega^{\bullet}_{{\cal F}},\bar \partial)$ to be $(\Omega^{\bullet}/(I),d)$ .
A vector bundle ${\cal G}$ is CR-holomorphic if the gluing cocycle $g_{ij}$ satisfies $\bar \partial g_{ij}=0$. In such case we can define a ${\cal G}$-twisted CR-complex $\Omega_{{\cal F}}^{\bullet}{\cal G}$.
\begin{remark}
Denote by $\sigma$ an operation of complex conjugation. Define an antilinear map
\begin{equation}\label{E:gjhsirpqqa}
s=\sigma\circ\mat{J}{0}{0}{id}:\mathbb{C}^{4|3}\rightarrow \mathbb{C}^{4|3} .
\end{equation}
A map $a\rightarrow sas^{-1}, a\in \mathfrak{gl}(4|3)$ coincides with the real structure $\rho$. Let us think about LHS of equation (\ref{E:kdsj}) as a quadratic function associated with an even bilinear form $(a,b)$. It is easy to see that LHS of equation (\ref{E:dfsad}) is equal to $(a,s(a))=0$. Naively thinking the centralizer of operator $s$ would precisely be the real form of $(\mathfrak{gl}(4|3), \rho)$ and it would preserve equations (\ref{E:kdsj}) and (\ref{E:dfsad}). The problem is that we cannot work pointwise in supergeometry. Instead we consider equations (\ref{E:kdsj}, \ref{E:dfsad}) as a system of real algebraic equations. We interpret them as a system of sections of some line bundles on ${\cal C}_{\mathbb{H}}$ manifold $M=\mathbf{P}^{3|3}\times \mathbf{P}^{*3|3}\times \overline{\mathbf{P}}^{3|3}\times \overline{\mathbf{P}}^{*3|3}$ (see Section \ref{S:llfddfgg} for discussion of reality in supergeometry). The space $M$ carries a canonical graded real structure $\rho$, that leaves the space of equations invariant. The $\rho$-twisted diagonal action of $\mathfrak{gl}(4|3)$ also leaves the equations invariant.
The graded real structure induces a graded real structure $\rho$ on $\mathfrak{gl}(4|3)^{\rho}$ and makes a supermanifold $\mathcal{R} $ an algebraic graded real supermanifold.
\end{remark}
\subsection{Symmetries of the ambitwistor space}\label{S:hfdgjpo}
We define a space $\mathcal{R}_{GL(4|3)} $ as a homogeneous space of a real supergroup $(GL(4|3),\rho)$. In this section we establish an isomorphism $\mathcal{R}_{GL(4|3)}\cong \mathcal{R}$.
In fig. (\ref{F:gdfgdjh}) the reader can see a graphical presentation of some matrix $\mat{A}{B}{C}{D}\in \mathfrak{gl}(4|3)$.
The isotropy subalgebra $\mathfrak{a}\subset \mathfrak{gl}(4|3)$ of a base point in the space $\mathcal{R}_{GL(4|3)} $ is defined as a linear space of matrices whose nonzero entries are in the darkest shaded area of a matrix in fig. (\ref{F:gdfgdjh}).
\begin{figure} [ht]
\centering
\includegraphics[width=.3\textwidth,height=.3\textwidth]{subalgebrapict.eps}
\caption{}
\label{F:gdfgdjh}
\end{figure}
\begin{lemma}
Subspace $\mathfrak{a}\subset \mathfrak{gl}(4|3)$ is a $\rho$-invariant subalgebra.
\end{lemma}
\begin{proof}
Direct inspection.
\end{proof}
Let $A$ be an algebraic subgroup of $GL(4|3)$ with a Lie algebra $\mathfrak{a}$.
The space $\mathcal{R}_{GL(4|3)} $ carries a homogeneous CR structure (see Section \ref{S:gfgjheq} for related discussion). Define a subspace $\mathfrak{p}\subset \mathfrak{gl}(4|3)$ as a set of matrices with nonzero entries in gray and dark gray areas in fig. (\ref{F:gdfgdjh}).
\begin{lemma}
Subspace $\mathfrak{p}\subset \mathfrak{gl}(4|3)$ is a subalgebra. It satisfies $\mathfrak{p}\cap \rho(\mathfrak{p})=\mathfrak{a}$
\end{lemma}
\begin{proof}
Direct inspection.
\end{proof}
Let $P$ denote an algebraic subgroup with Lie algebra $\mathfrak{p}$ .
A complex supermanifold $X=GL(4|3)/P$ has an explicit description.
Equation (\ref{E:kdsj}) is preserved by the action of $GL(4|3)$.
\begin{proposition}
There is a $GL(4|3)$-equivariant isomorphism $X=\mathcal{Q}$.
\end{proposition}
\begin{proof}
We can identify the quadric $\mathcal{Q}$ with the space of partial flags ${\mathbb C}^{4|3}$ in as it is done in purely even case (see [GH] for example). A spaces $\mathcal{Q}$ is a connected component of the flag space containing the flag
\begin{equation}\label{E:yhggb}
F_1\subset F_2\subset {\mathbb C}^{4|3}
\end{equation}
with $F_1\cong {\mathbb C}^{1|0}$ and $F_2\cong {\mathbb C}^{3|3}$. This flag can be interpreted as a pair of points $F_1\in \mathbf{ P}^{3|3}$, $F_2\in \mathbf{P}^{*3|3}$. The condition (\ref{E:yhggb}) is equivalent to (\ref{E:kdsj}).
Let us choose a standard basis $e_1,\dots,e_7$ of ${\mathbb C}^{4|3}$ such that the parities of elements are $\varepsilon(e_1)=\varepsilon(e_2)=\varepsilon(e_3)=\varepsilon(e_7)=1$, $\varepsilon(e_4)=\varepsilon(e_5)=\varepsilon(e_6)=-1$. In this notations the standard flag $F$ has the following description:
\begin{equation}\label{E:ksk}
\begin{split}
&F_1=\mbox{span}<e_7>\\
&F_2=\mbox{span}<e_2,\dots e_7>
\end{split}
\end{equation}
The flag defines a point in the space $\mathcal{Q}$.
It is easy to compute a shape of the matrix of an element from the stabilizer $P_{F}$ of $F$. The following picture is useful
\begin{equation}\label{E:fgojkxcz}
\mbox{
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\put(1276,-436){\makebox(0,0)[lb]{$\mathfrak {gl}(2|3)$}}%
\put(1276,-1336){\makebox(0,0)[lb]{$\mathbb{C}^{*2|3}$}}%
\put(2476,-1336){\makebox(0,0)[lb]{$\mathbb {C}$}}%
\put(451,539){\makebox(0,0)[lb]{$\mathbb{C}$}}%
\put(2401,539){\makebox(0,0)[lb]{$\mathfrak{rad}$}}%
\put(451,-1336){\makebox(0,0)[lb]{$\mathbb{C}^{1|0}$}}%
\put(451,-436){\makebox(0,0)[lb]{$\mathbb{C}^{2|3}$}}%
\end{picture}
}
\end{equation}
The Lie algebra $\mathfrak{p}_F$ of stabilizer $P_F$ is formed by matrices with zero entries below thick solid line in picture (\ref{E:fgojkxcz}).
Conjugating with a suitable permutation of coordinates $t$ we see that $\mathfrak{p}^t_F=\mathfrak{p}$.
\end{proof}
\begin{remark}
A transformation $s$ defined in equation (\ref{E:gjhsirpqqa}) acts on the space of flags.
By definition an $s$-invariant flag belongs to the subvariety $\mathcal{R} $ .
A direct inspection shows that the nonzero entries of the matrix of an element of stabilizer are located in the darkest shaded area on fig. (\ref{F:gdfgdjh}). The manifold $\mathcal{R} _{red}$ fibers over $\mathbf{P}^3$ with connected fibers. Thus $\mathcal{R} $ is connected. From this and a simple dimension count we conclude that subvariety $\mathcal{R} $ coincides with $\mathcal{R}_{GL(4|3)} $. Identification of CR structure also follows from this.
\end{remark}
The Lie algebra $\mathfrak{gl}(4|3)$ contains a subalgebra $\mathfrak{l}$. The elements of this subalgebra have nonzero entries in the darkest area on fig (\ref{F:gdfgdjh}) and also spots marked by $*$. This algebra is invariant with respect to the real structure $\rho$. Denote by $L$ an algebraic subgroup of $GL(4|3)$ with Lie algebra $\mathfrak{l}$.
The quotient $(\mathcal{S}_{GL(4|3)},\rho)=((GL(4|3)/L),\rho)$ is a supermanifold with $(\mathcal{S}_{GL(4|3)})^{\rho}_{red}=S^4$. Indeed the real points $L_{red}^{\rho}$ of the group $L_{red}$ are conjugated to quaternionic matrices of the form $\mat{a}{b}{0}{d}\in GL(2,\mathbb{H})$. Thus the quotient space $GL(2,\mathbb{H})/L^{\rho}_{red}=\mathbb{H}\mathbf{P}^1$ is isomorphic to $S^4$. Denote by $p$ projection
\begin{equation}\label{E:sdjhwuwq}
\mathcal{R}_{GL(4|3)} \rightarrow \mathcal{S}_{GL(4|3)}
\end{equation}
An easy local exercise with Lie algebras reveals that the fibers of projection $p$ are CR-holomorphic and are isomorphic to $\mathbf{P}^1\times\mathbf{P}^1$.
The following direct geometric description of ambitwistor space will be useful. Let $M$ be a $C^{\infty}$ 4-dimensional Riemannian manifold. A metric $g$ defines a relative quadric (the ambitwistor space) in the projectivisation of a complexified tangent bundle $A(M)\subset \mathbf{P}(T^{\mathbb{C}})$. By construction there is a projection $p:A(M)\rightarrow M$. The space $A(M)$ carries a CR structure (it could be nonintegrable). Indeed a fiber of the distribution ${\cal F}$ at a point $x\in A(M)$ is a direct sum of the holomorphic tangent space to the fiber through $x$ and a complex line in $T^{\mathbb{C}}(M)$ spanned by $x$. From the point of view of topology the space $A(M)$ coincides with a relative Grassmannian of oriented 2-planes in $T_M$. A constructed complex distribution depends only on conformal class of the metric. From this we conclude that $A(S^4)$ ($S^4$ has a round metric induced by the standard embedding into $\mathbb{R}^5$) is a homogeneous space of $Conf(S^4)=PGL(2,\mathbb{H})$, $A(S^4)=PGL(2,\mathbb{H})/A_{red}^{\rho}$ and the CR structure is integrable.
An appropriate super generalization of this construction is as follows . We have an isomorphism
\begin{equation}\label{E:aadfjs}
{\cal W}_l\otimes {\cal W}_r\overset{\Gamma}{\cong}T^{\mathbb{C}}_M
\end{equation}
In the last formula ${\cal W}_l,{\cal W}_r$ are complex two-dimensional spinor bundles on $M$ (we assume that $M$ has a spinor structure). The isomorphism $\Gamma$ is defined by Clifford multiplication. Let $T$ be a 3-dimensional linear space. This vector space will enable us to implement N$=dim(T)=3$ supersymmetry. To simplify notations we keep ${\cal W}_l\otimes T+{\cal W}_r\otimes T^*$ for the pullback $p^*({\cal W}_l\otimes T+{\cal W}_r\otimes T^*)$. Define a split, holomorphic in odd directions\footnote{The reader might wish to consult Section \ref{S:llfddfgg} on this.} supermanifold $\widetilde{{\cal A}}(M)$ associated with a vector bundle $\Pi({\cal W}_l\otimes T+{\cal W}_r\otimes T^*)$ over $A(M)$. To complete the construction we define a superextension of the CR structure. Introduce odd local coordinates $\theta^i_{\alpha},\tilde{\theta}^{j\beta}$ ($1\leq i,j\leq 2,1\leq \alpha,\beta\leq 3$) on fibers of $\Pi({\cal W}_l\otimes T+{\cal W}_r\otimes T^*)$. We decompose local complex vector fields $\Gamma(\pr{\theta^i_{\alpha}}\otimes \pr{\tilde{\theta}^{j\beta}})$ in a local real basis $\pr{x^s}$ as $\delta_{\alpha}^{\beta}\Gamma^{ s}_{ij}\pr{x^s}, 1\leq s\leq 4$. The odd part of the CR distribution ${\cal F}$ is locally spanned by vector fields
\begin{equation}\label{E:gfdjfur}
\begin{split}
&\pr{\theta_{i\alpha}}+\tilde{\theta}^{j}_{\alpha}\Gamma_{ij}^{s}\pr{x^{s}}\\
& \pr{\tilde{\theta}^{j}_{\alpha}}+\theta^{j\alpha}\Gamma^{s}_{ij}\pr{x^{s}}
\end{split}
\end{equation}
This construction of a superextension of ordinary CR structure depends only on the conformal class of the metric. It is convenient to formally add the complex conjugate odd coordinates. This way we get ${\cal A}(M)=\Pi({\cal W}_l\otimes T+{\cal W}_r\otimes T^*+\overline{{\cal W}_l\otimes T}+\overline{{\cal W}_r\otimes T^*})$, equipped with a graded real structure. As in the even case the symmetry analysis allows to identify CR-space ${\cal A}(S^4)$ with $\mathcal{R} $.
A tangent space $\mathfrak{ m}$ to $\mathcal{Q}$ at a point fixed by $\mathfrak{p}$ is formed by elements with nonzero entries below thick solid line on the picture (\ref{E:fgojkxcz}) . It decomposes into a sum ${\mathbb C}^{2|3}+{\mathbb C}^{*2|3}+{\mathbb C}^{1|0}$ of irreducible $GL(2|3)$ representations.
The elements $a_{21},a_{31},\alpha_{41},\alpha_{51},\alpha_{61},a_{71},a_{72},a_{73},\alpha_{74},\alpha_{75},\alpha_{76}$ stand for matrix coordinate functions on the linear space $\mathfrak{ m}$( coordinates $a$ are even, $\alpha$ are odd).
\begin{proposition}\label{P:jsdhd}
An element
\begin{equation}\label{e:jksdss}
vol=da_{21}\wedge da_{31}\wedge d\alpha_{41}\wedge d\alpha_{51}\wedge d\alpha_{61}\wedge da_{71}\wedge da_{72}\wedge da_{73}\wedge d\alpha_{74}\wedge d\alpha_{75}\wedge d\alpha_{76}
\end{equation}
belongs to $Ber(\mathfrak{ m}^*)$. It is invariant with respect to the action of $P$.
\end{proposition}
\begin{proof}
Simple weight count.
\end{proof}
We spread a generator of $Ber(\mathfrak{ m}^*)$ by the action of $GL(4|3)$ over $\mathcal{Q}$ and form a $GL(4|3)$-invariant section $vol$ of $Ber_{\mathbb{C}}(\mathcal{Q})$
\section{Reduced theory}\label{S:main}
As a preliminary step in construction of the superspace $Z$ we introduce a "holomorphic" manifold $\widetilde{\Pi F} $ and an integral form on it (see Section \ref{E:gfgdfgjr} for the explanation of quotation marks ). The form defines a functional $\int$ on Dolbeault complex of this manifold. We prove that the CS theory constructed by the triple $(\Omega^{0\bullet}(\widetilde{\Pi F})\otimes Mat_n,\bar \partial,\int\otimes tr_{Mat_n})$ is classically equivalent to N=3 YM theory with gauge group U$(n)$ reduced to a point.
\subsection{ A Manifold $\Pi F$}\label{S:ncjsjs}
A manifold $\widetilde{\Pi F}$ is a deformation of a more simple manifold $\Pi F$. In this section we give relevant definitions concerning $\Pi F$.
Denote a product $\mathbf{P}^1\times \mathbf{P}^1$ by $X$. It has two projections $p_i:X\rightarrow \mathbf{P}^1 , i=l,r$. Let ${\cal O}(1)$ denote the dual to the Hopf line bundle over $\mathbf{P}^1$. The Picard group of $X$ is $\mathbb{Z}+\mathbb{Z}$. It is generated by the classes of line bundles $\pi ^*_l{\cal O}(1)={\cal L}_l$, \quad$\pi ^*_r {\cal O}(1)={\cal L}_r$ that can serve as coordinates in $Pic(X)$. Let ${\cal O}(a,b)$ denote a line bundle ${\cal L}^{\otimes a}_l\otimes {\cal L}^{\otimes b}_r$.
{\bf Convention} We denote by $H^{\bullet}(Y,{\cal G})$ the cohomology of (super)manifold $Y$ with coefficients in a vector bundle ${\cal G}$. It can be computed as cohomology of Dolbeault complex $\Omega^{0\bullet}(Y){\cal G}$. It is tacitly assumed that in the Section \ref{S:main} the omitted argument $Y$ in $\Omega^{0\bullet}(Y){\cal G}$ implies $Y=X$ . If ${\cal G}$-argument is missing we assume that ${\cal G}={\cal O}$.
Denote by $\mathrm{Sym} V,\Lambda V$ symmetric and exterior algebras of a vector space (bundle).
Denote by $\Theta$ a line bundle isomorphic to ${\cal O}(1,1)$.
We construct a vector bundle $F$ over $X$ as a direct sum:
\begin{equation}\label{E:sjs}
\begin{split}
&F=T\otimes {\cal L}_l+T^*\otimes {\cal L}_r+\Theta^* \\
&H=T\otimes{\cal L}_l+T^*\otimes {\cal L}_r.
\end{split}
\end{equation}
As before $T$ is a three dimensional vector space.
The reader may have noticed that the manifold $X$ has also appeared as a fiber of projection (\ref{E:sdsfsew}). We shall see this is not accidental.
\subsection{Properties of the manifold $\Pi F$}\label{E:gfgdfgjr}
In this section we devise an infinitesimal deformation of a complex structure on $\Pi F$. This deformation will be promoted to the actual deformation which we denote by $\widetilde{\Pi F}$. The algebra $A_{pt}$ from the introduction is equal to $\Omega^{0\bullet}(\widetilde{\Pi F})$. We construct on $\widetilde{\Pi F}$ we construct an integral form that will enable us to define a functional $\int_D:\Omega^{0\bullet}(\widetilde{\Pi F})\rightarrow \mathbb{C}$.
The manifold $\Pi F$ is a complex split supermanifold. The Dolbeault complex $(\Omega ^{0\bullet}(\Pi F),\bar \partial)$ is defined on supermanifold $\Pi F$, considered as a graded real supermanifold. The complex $(\Omega ^{0\bullet}(\Pi F),\bar \partial)$ contains as a differential subalgebra the Dolbeault complex $\Omega ^{0\bullet}\Lambda F^*$.
\begin{proposition}
Differential algebras $\Omega ^{0\bullet}(\Pi F)$ and $\Omega ^{0\bullet}\Lambda F^*$ are quasiisomorphic.
\end{proposition}
\begin{proof}
The same as a proof of Proposition \ref{E:fdsafjh}.
\end{proof}
The canonical line bundle ${\cal K}_X$ is equal to ${\cal O}(-2,-2)$. There is a nontrivial cohomology class- the "fundamental" class: $\alpha \in H^2(X,{\cal O}(-2,-2))\overset {id}{\subset } H^2(X,{\cal O}(-2,-2)\otimes T\otimes T^*)\subset H^2(X,\Lambda^2 (H^*)\otimes \Theta ^*)\subset H^2(X,\Lambda F^*\otimes \Theta^*)$.
We interpret $\Lambda (F^*)\otimes \Theta^* $ as a sheaf of local holomorphic differentiations of $\Pi F$ in direction of $\Theta^*$.
A representative $\alpha =f d\bar {z}_ld\bar{z}_r\pr{\theta}$ ($z_l,z_r$ are local coordinates on $X$) of the class $[\alpha ]$ can be extended to a differentiation of $\Omega ^{0\bullet}(\Pi F)$.
The main properties of $D=\bar \partial+\alpha$ are:
\\
$1)$ it is a differentiation of $\Omega ^{0\bullet}(\Pi F)$,
\\
$2)$ equation $ D^2=0$ holds.
\\
All of them are corollaries of $\bar \partial $-cocycle equation for $\alpha$.
The operator $D$ defines a new "holomorphic" structure on $\Pi F$ \footnote{This definition is not standard, because usually a deformation cocycle $\alpha$ is an element of $ \Omega^{0,1}{\cal T}$ (${\cal T}$ is a holomorphic tangent bundle), whereas in our case $\alpha \in \Omega^{0,2}{\cal T}$. We however continue to use a traditional wording and call it a deformation of a complex structure, though a more precise term would be deformation of the Dolbeault algebra $(\Omega^{0\bullet},\bar)$. This algebra in our approach becomes a substitute for the underlying manifold.
}.
This new complex manifold will be denoted by $\widetilde{ \Pi F}$.
The manifold $\Pi F$ is Calabi-Yau. By this we mean that $Ber_{\mathbb{C}}$ is trivial. Indeed the determinant line bundle of $F$ is equal to $det(T\otimes {\cal L}_l)\otimes det (T^*\otimes {\cal L}_r)\otimes det ({\cal O}(-1,-1))={\cal O}(3,0)\otimes {\cal O}(0,3)\otimes {\cal O}(-1,-1)={\cal O}(2,2);\quad Ber_{\mathbb C}\Pi F={\cal K}_X\otimes det F={\cal O}$-is trivial.
It implies that the bundle $Ber_{\mathbb C}\Pi F$ admits a nonvanishing section $vol_{\Pi F}$.
This section is $SO(4)$-invariant. The action of $\mathfrak{u}=\mathbb{C}+\mathbb{C}$-the unipotent subgroup of Borel subgroup $B\subset SO(4)$ on the large Schubert cell of $X$ is transitive and free.
Hence the section of $Ber_{\mathbb{C}} \Pi F$ in $\mathfrak{u}$-coordinates is
$$vol_{\Pi F}=dz_l\wedge dz_r\wedge d\alpha _1\wedge d\alpha _2\wedge d\alpha _3\wedge d\tilde {\alpha} _1\wedge d\tilde {\alpha }_2\wedge d\tilde {\alpha} _3\wedge d\theta,$$ where $\alpha_1,\dots,\tilde {\alpha} _3,\theta$ are $\mathfrak{u}$-invariant coordinates on the odd fiber. The section $vol_{\Pi F}$ is in the kernel of $D$ by construction.
We can construct on manifold $\Pi F$ a global holomorphic integral $-2$-form the way explained in remark (\ref{E:hhghssa}).
In our case it is equal to $c_{\Pi F}=\alpha_1\dots\tilde{\alpha}_3\theta d\alpha_1\wedge \dots \wedge d\tilde{\alpha}_3\wedge d\theta$.
\begin{proposition}
The form $\mu=vol_{\Pi F} \otimes \bar {c}_{\Pi F}$ is $D$ closed nontrivial integral $(0,-2)$-form on the underlying real graded $\Pi F$.
\end{proposition}
\begin{proof}
Direct inspection in local coordinates.
\end{proof}
An integral form $\mu$ defines a $\bar \partial+\alpha$-closed trace on $\Omega^{0\bullet}(\Pi F)\otimes Mat_n$
$$\int a=tr\int_{\Pi F}a\mu$$
\begin{definition}
By definition an A$_{\infty}$ algebra algebra is a graded linear space , equipped with a series of maps $\mu_n:A^{\otimes n}\to A, n\ge 1$ of degree $2-n$ that satisfy quadratic relation:
\begin{equation}
\begin{split}
&\sum_{i+j=n+1}\sum_{0\le l\le i}\epsilon(l,j)\times\\
&\mu_i(a_0,...,a_{l-1},\mu_j(a_l,...,a_{l+j-1}),a_{l+j},...,a_n)=0
\end{split}
\end{equation}
where $a_m\in A$, and
$\epsilon(l,j)=(-1)^{j\sum_{0\le s\le l-1}deg(a_s)+l(j-1)+j(i-1)}$.
In particular, $\mu_1^2=0$.
\end{definition}
\begin{remark}\label{E:gfdadfjj}
Suppose we have an A$_{\infty}$ algebra $A$ equipped with a projector $\pi$. A homotopy $H$ such that $\{d,H\}=id-\pi$ can be used as an input data for construction of a new A$_{\infty}$ structure on $\mathrm{Im} \pi$ (see [Kad],[Markl] for details). The homotopy $H$ is not unique. The resulting A$_{\infty}$ algebras will have different multiplications, depending on $H$. All of them will be A$_{\infty}$ equivalent. An additional structure on $A$ helps to fix an ambiguity in a choice of $H$. In our case algebra $A$ is a Dolbeault complex of a manifold with an operator $\pi$ being an orthogonal projection on cohomology. If the manifold is compact, K\"{a}hler and a $G$-homogeneous there is natural choice of $H$: $H=\bar \partial^*/\Delta'$. The operator $\bar \partial^*$ $\Delta'$ are build by a $G$-invariant metric. The operator $\Delta'$ is equal to $\Delta$ on $\mathrm{Ker} \Delta^{\perp}$ and equal to identity on $\mathrm{Ker} \Delta$.
\end{remark}
\begin{remark}\label{E:gfdadfjjdsfd}
A construction described in remark (\ref{E:gfdadfjj}) admits a generalization. Suppose an A$_{\infty}$ algebra $A$ has a differential $d$ that is a sum of two anticommuting differentials $d_1$ and $d_2$. Assume that $\{d_1,H\}=id-\pi$ and a composition $d_2H$ is a nilpotent operator . Then $\mathrm{Im} \pi$ carries a structure an A$_{\infty}$ algebra quasiisomorphic to $A$. The same statement is true for A$_{\infty}$ algebras with a trace.
The proof goes along the same lines as in ref. [Markl], but we allow two-valent vertices.
\end{remark}
Technically it is more convenient to work not with algebra $\Omega ^{0\bullet}(\widetilde{\Pi F})$ but with a quasiisomorphic subalgebra $(\Omega ^{0\bullet}\Lambda F,D)$ .
In application of the constructions from the remarks (\ref{E:gfdadfjj}, \ref{E:gfdadfjjdsfd}) we choose $\pi$ to be an orthogonal projector from the Hodge theory, corresponding to $SO(4,\mathbb{R})$-invariant metric on $X$. We also use a decomposition $D=d_1+d_2=\bar \partial+\alpha$.
The algebra of cohomology of $(\Omega ^{0\bullet}\Lambda F,\bar \partial)$ carries an A$_{\infty}$-algebra structure. We denote it by $C=H^{\bullet}(X,\Lambda(H^*)\otimes \Lambda(\Theta))$. Denote by $\psi$ a quasiisomorphism $(\Omega ^{0\bullet}\Lambda F,\bar \partial)\rightarrow C$. We shall describe some properties of $C$.
Let $W_l$, $W_r$ be spinor representations of $SO(4)$. The vector representation $V$ is equal to $W_l\otimes W_r$.
The differential $\alpha$ induces a differential $[\alpha]$ on $C$. The ghost grading of the group $H^i(X,\Lambda^k(H^*)\otimes \Lambda^s(\Theta))$ is equal to $i+s$, the additional grading is equal to $k+2s$(preserved by $\bar \partial$ and $\alpha$). We used a nonstandard ghost grading that differs from the one used in physics on shift by one. In particular the ghost grading of the gauge (labeled by $V$) and spinor (labeled by spinors $W_l$, $W_r$) and matter ($SO(4)$ action is trivial ) fields is equal to one. In the table below you will find the field content (representation theoretic description ) of $C$ :
\begin{equation}\label{E:dkns}
\mbox{
\scriptsize{
$\begin{array}{lll|lll}
gh°&&gh°&\\
0&0&H^0(X,\Lambda^{0}(H^*))=\mathbb{C}&1&2&H^0(X,\Lambda^{0}(H^*)\otimes \Theta)=V\\
1&2&H^1(X,\Lambda^{2}(H^*))=\Lambda^2(T)+\Lambda^2(T^*)&1&3&H^0(X,\Lambda^{1}(H^*)\otimes \Theta)=W_l\otimes T+W_r\otimes T^*\\
1&3&H^1(X,\Lambda^{3}(H^*))=W_l+ W_r&1&4&H^0(X,\Lambda^{2}(H^*)\otimes \Theta)=T\otimes T^*\\
2&4&H^2(X,\Lambda^{4}(H^*))=T\otimes T^*&2&5&H^1(X,\Lambda^{3}(H^*)\otimes \Theta)=W_l+ W_r\\
2&5&H^2(X,\Lambda^{5}(H^*))=W_l\otimes\Lambda^2(T)+ W_r\otimes\Lambda^2(T^*)&2&6&H^1(X,\Lambda^{4}(H^*)\otimes \Theta)=T+T^*\\
2&6&H^2(X,\Lambda^{6}(H^*))=V&3&8&H^2(X,\Lambda^{6}(H^*)\otimes \Theta)=\mathbb{C}
\end{array}$
}
}
\end{equation}
The groups $H^0(X,\Lambda^{2}(H^*)\otimes \Theta)$ and $H^2(X,\Lambda^{4}(H^*))$ are contracting pairs, they are killed by the differential $[\alpha]$ and should be considered as auxiliary fields in the related CS theory.
An A$_{\infty}$ algebra $C$ besides differential $[\alpha]$ and multiplication has higher multiplication on three arguments (corresponding to cubic nonlinearity of YM equation). However operations in more then three arguments are not present.
This can be deduced from homogeneity of $\bar \partial$ and $\alpha$ with respect to the additional grading. Finally representation theory fixes structure maps up to a finite number of parameters. The integral $\int$ defines a nonzero map $tr:H^2(X,\Lambda^{6}(H^*)\otimes \Theta)\rightarrow \mathbb{C}$.
Presumably it is possible to complete this line of arguments to a full description of multiplications in $C$. We prefer do it indirectly through the relation to Berkovits construction [Berk].
\begin{remark}
Let $\mathbb{R}^{10}$ be a linear space, equipped with a positive-definite dot-product. Denote by $S$ an irreducible complex spinor representation of orthogonal group $SO(10)$. Denote by $\Gamma_{\alpha\beta}^i$ coefficients of the nontrivial intertwiner $\mathrm{Sym}^2(S)\rightarrow \mathbb{C}^{10}$ in some basis of $S$ and an orthonormal basis of $\mathbb{C}^{10}$.
We assume that $\mathbb{C}^{10}$-basis is real.
On a superspace $(\mathbb{R}^{10}+ \Pi S)\otimes \mathfrak{u}(n)$ we define a superfunction (Lagrangian)
\begin{equation}
S(A,\chi)=\sum_{i<j}tr([A_i,A_j][A_i,A_j])+\sum_{\alpha\beta i}tr(\Gamma_{\alpha\beta}^i[A_i,\chi^{\alpha}]\chi^{\beta})
\end{equation}
$A_1,\dots, A_{10}$ is a collection of antihermitian matrices labeled by the basis of $\mathbb{C}^{10}$. Similarly odd matrices $\chi^1,\dots,\chi^{16}$ are labeled by the basis of $S$.
This can be considered as a field theory, obtained from D=10, N=1 YM theory by reduction to zero dimensions. We call it IKKT after the paper [IKKT] where it has been studied. IKKT theory has a gauge invariance - invariance with respect to conjugation. A BV version of IKKT coincides with a CS theory associated with an A$_{\infty}$ algebra $\mathcal{A}_{IKKT}$ that we shall introduce presently.
\end{remark}
\begin{definition}
An A$_{\infty}$ algebra $\mathcal{A}_{IKKT}$ can be considered as vector space spanned
by symbols $x_{k}, \xi ^{\alpha }, c, x^{\ast k}, \xi _{\alpha }^{\ast }, c^{\ast }, 1\leq k\leq 10, 1\leq \alpha \leq 16$
with operations $\mu_{2}$ (multiplication), $\mu_{3}$(Massey product) defined by the following formulas:
\begin{align}
&\mu_2(\xi ^{\alpha }, \xi ^{\beta })=\Gamma^{\alpha\beta}_k x^{\ast k}\\
&\mu_2(\xi ^{\alpha }, x_{k})=\mu_2(x_{k}, \xi ^{\alpha })=
\Gamma^{\alpha\beta}_k\xi _{\beta }^{\ast }\\
&\mu_2(\xi ^{\alpha }, \xi _{\beta }^{\ast })=
\mu_2(\xi _{\beta }^{\ast }, \xi ^{\alpha })=c^{\ast }\\
&\mu_2(x_{k}, x^{\ast k})=\mu_2(x^{\ast k}, x_{k})=c^{\ast }\\
&\mu_3(x_{k}, x_{l}, x_{m})=\delta_{kl}x^{\ast m}-\delta_{km}x^{\ast l}\\
&\mu_2(c, \bullet)=\mu_2(\bullet, c)= \bullet \\
\end{align}
All other products are equal to zero.
An element $c$ is a unit.
All operations $\mu_{k}$ with $k\neq 2, 3$ vanish.
The algebra carries a trace functional $tr$ equal to one on $c^*$ and zero on the rest of the generators. It induces a dot-product by the formula $(a,b)=tr(\mu_2(a,b))$, compatible with $\mu_k$.
\end{definition}
By definition an A$_{\infty}$ algebra has a grading (we call it a ghost grading ) such that operation $\mu_n$ has degree $2-n$. An A$_{\infty}$ algebra might also have an additional grading such that all operations have degree zero with respect to it. See [MSch05] for details on gradings of ${\cal A}_{IKKT}$.
\begin{proposition}\label{P:bbncnsq}
A differential graded algebra with a trace $(\Omega ^{0\bullet}(\widetilde{\Pi F}),\bar \partial+\alpha,tr_{\mu})$
is quasiisomorphic to $\mathcal{A}_{IKKT}$.
\end{proposition}
\begin{proof}
We shall employ methods developed in [MSch05]. Recall that the manifold of pure spinors in dimension ten is equal to $\mathcal{P}=SO(10,\mathbb{R})/U(5)$. As a complex manifold it is defined as the space of solutions of homogeneous equations $\Gamma_{\alpha\beta}^i\lambda^{\alpha}\lambda^{\beta}=0$, where $\lambda^{\alpha}$ are homogeneous coordinates on $\mathbf{P}^{15}$. A space $\mathbf{P}^{15}$ is the projectivisation of irreducible complex spinor representation of $Spin(10,\mathbb{R})$.
We denote by $R$ the restriction on $\mathcal{P}$ of the twisted tangent bundle $T_{\mathbf{P}^{15}}(-1)$. Denote by $A$ the coordinate algebra $\mathbb{C}[\lambda^{1},\dots,\lambda^{16}]/\Gamma_{\alpha\beta}^i\lambda^{\alpha}\lambda^{\beta}$ of $\mathcal{P}$. Denote by $B$ the Koszul complex $A\otimes \Lambda[\theta^{1},\dots\theta^{16}]$ with differential $\lambda^{\alpha}\pr{\theta^{\alpha}}$.
The algebra $B$ can be equipped with various gradings. The cohomological grading is defined on generators as $|\lambda^{\alpha}|=2$, $|\theta^{\alpha}|=1$; the grading by the degree (or homogeneous grading) is $deg\lambda^{\alpha}=deg\theta^{\alpha}=1$. The differential has degree one with respect to $||$ and zero with respect to $deg$. As a result the cohomology groups are bigraded: $H(B)=H^{ij}$, where $i$ corresponds to $||$-grading, $j$ corresponds to $deg$.
In [MSch05] we proved that $\bigoplus_{i-j=k} H^{ij}(B)=H^k(\Omega^{0\bullet}(\mathcal{P})\Lambda(R^*))$. In fact we proved that the identification map is a quasiisomorphism of differential graded algebras with a trace.
In the language of supermathematics we may say that cohomology $H^k(\Omega^{0\bullet}(\mathcal{P})\Lambda(R^*))$ is Dolbeault cohomology of a split supermanifold $\Pi R$.
In the course of the proof of quasiisomorphism we have identified the algebra of functions on the fiber of projection
\begin{equation}
\Pi R \rightarrow \mathcal{P}
\end{equation}
over a point $pt=(\lambda_0^{\alpha})\in \mathcal{P}$ with cohomology of algebra $B^{\bullet}_{pt}=(\Lambda[\theta^{1},\dots\theta^{16}],d)$ where $d=\lambda_0^{\alpha}\pr{\theta^{\alpha}}$. An analog of the complex $B^{\bullet}_{pt}$ can be defined for any subscheme $U$ of $\mathcal{P}$ as a tensor product $B^{\bullet}_{U}=A_{U}\otimes \Lambda[\theta^{1},\dots\theta^{16}]$. The algebra $A_U$ is equal to $\bigoplus_{i\geq 0} H^0(U,\O(i))$. The cohomology of $B_U$ is equal to $ H^{\bullet}B_{pt}\otimes \O(U)$ if $\O(1)$ is trivial on $U$ .
We proved in [MSch06] that the algebra $B$ with Berkovits trace $tr$ is quasiisomorphic to algebra ${\cal A}_{IKKT}$.
We need to present a useful observation from [MSch05]. Let us decompose a set $\{\lambda^{1},\dots,\lambda^{16}\}$ into a union $\{\lambda^{\alpha_1},\dots,\lambda^{\alpha_s}\}\cup\{\lambda^{\beta_1},\dots,\lambda^{\beta_k}\}$ such that $\{\lambda^{\alpha_1},\dots,\lambda^{\alpha_s}\}$ is a regular sequence. Then the algebras $(B,d)$ and $(B',d)=(A/(\lambda^{\alpha_1},\dots,\lambda^{\alpha_s})\otimes \Lambda[\theta^{\beta_1},\dots,\theta^{\beta_k}],d)$ are quasiisomorphic.
The following construction has been described in [MSch05].The spin representation $S$ of $\mathfrak{so}(10)$ splits after restriction on $\mathfrak{gl}(3)\times \mathfrak{sl}_l(2)\times \mathfrak{sl}_r(2)$ into $T\otimes W_l+T^*\otimes W_r+W_l+W_r$. We choose coordinates on $W_l+W_r$- equal to $(\lambda^{\alpha_i})=(\tilde{w}_l^+,\tilde{w}_l^-,\tilde{w}_r^+,\tilde{w}_r^-)$. They form a regular subsequence of $\lambda^{\alpha}$ . The manifold corresponding to $A/(\lambda^{\alpha_i})$ is equal to $Q\cap \mathbf{P}(T\otimes W_l+T^*\otimes W_r)$. The intersection is isomorphic to $F(1,2)\times X$. The algebra of homogeneous functions $A'=A(F(1,2)\times X)$ on $F(1,2)\times X$ is generated by $s^{i\alpha},t^j_{\alpha }(1\leq \alpha\beta \leq 3, 1\leq ij\leq 2)$. The relations are
\begin{equation}
\sum_{\alpha}s^{i\alpha}t^j_{\alpha}=0, det(s^{i\alpha})=0, det(t^j_{\alpha})=0
\end{equation}
In the formula $det$ stands for a row of $2\times 2$ minors of $2\times 3$ matrix.
We plan to follow almost the same method of construction of supermanifold as for the Koszul complex of pure spinors. Denote by $g$ a projection $$g:F(1,2)\times X\rightarrow X$$ We fix a point $x\in X$. The algebra $A'_{g^{-1}(x)}$ is isomorphic to \\$\mathbb{C}[p_1,\dots p_3,u^1\dots,u^3]/(p_iu^i)$. The algebra $B'_{p^{-1}(x)}$ is isomorphic to
\begin{equation}
A'_{g^{-1}(x)}\otimes\Lambda[\pi_1,\dots,\pi_3,\nu^1,\dots,\nu^3,\tilde{\pi}_1,\dots,\tilde{\pi}_3,\tilde{\nu}^1,\dots,\tilde{\nu}^3],d
\end{equation}
with a differential
\begin{equation}\label{E:asdfhhjdjd}
d=p_i\pr{\tilde{\pi}_i}+u^i\pr{\tilde{\nu}^i}
\end{equation}
The cohomology of this differential is equal to $$\Lambda[E_x]=\Lambda[\pi_1,\dots,\pi_3,\nu^1,\dots,\nu^3,\theta]$$ The induced A$_{\infty}$ algebra structure on cohomology has no higher multiplications.
The element $\theta$ is represented by a cocycle $\tilde{\pi_i}u^i$. The linear space $E_x$ coincides with the fiber $F_x$ of vector bundle $F$ (\ref{E:sjs}).
The main distinction between this computation and a computation with pure spinors is that we encountered a noncanonical A$_{\infty}$ morphism $\iota:B'_{p^{-1}(x)}\rightarrow \Lambda[E_x]$, which could be not a homomorphism of associative algebras.
Recall that we viewed the cohomology of $B_{pt}$ as functions of the fiber of projections $p:\Pi R \rightarrow \mathcal{P}$. We have a natural identification of fibers over different patches of $\mathcal{P}$. It gives us a consistent construction of a split manifold $\Pi R$.
In case of the manifold $X$ if we ignore the issues related to ambiguities of choice of morphism $\iota$ tt is not hard to see that an isomorphism of fibers $F_x\cong H_x$ can be extended to a $Spin(4)$ equivariant isomorphism of the vector bundles $F_x\cong H_x$ (use homogeneity of both vector bundles with respect to $\mathfrak{sl}^l(2)\times \mathfrak{sl}^r(2)$ action).
This way we recover the manifold $\Pi F$. In reality when we try to glue rings of functions on different patches the structure isomorphisms will be A$_{\infty}$-morphisms. We may claim on general grounds that we get an A$_{\infty}$ structure on a space of \v{C}ech chains of $\Pi H\cong \Pi F$. This structure can be trivialized by a twist on a local A$_{\infty}$-morphism (reduced to the standard multiplication in Grassmann algebra) on every double intersection $U_{ij}$ of patches (if $U_{ij}\subset X$ is sufficiently small). An ambiguity in a choice of such twist leads to appearance of \v{C}ech 2-cocycle $\beta_{ijk}$ with values in infinitesimal (not A$_{\infty}$ ) transformations of the fiber $\Pi F$.
In the Dolbeault picture this cocycle corresponds to $\alpha$. Finally we use \v{C}ech-Dolbeault equivalence. This proves the claim.
\end{proof}
\begin{remark}
The algebra $C$ carries a differential $d=[\alpha]$. The minimal model of $C$ (by definition it is a quasiisomorphic A$_{\infty}$ algebra without a differential) constructed for an obvious homotopy of differential $d=[\alpha]$ is quasiisomorphic to ${\cal A}_{IKKT}$. If we ask for a quasiisomorphism to be compatible with all gradings that exist on both algebras this quasiisomorphism is an isomorphism.
From this is is quite easy to recover all multiplications in the algebra $C$.
\end{remark}
\section{Nonreduced theory}\label{S:ffkewx}
A manifold $Z$ with an integral form is constructed in this section.
\subsection{Construction of the algebra $A(Z)$}\label{S:wfdke}
In this section we construct an algebra $A(Z)$. It will be a linking chain between YM theory and ambitwistors.
We construct a manifold $Z$ as a direct product $\Pi F\times \Sigma$. Intuitively speaking the algebra $\Omega^{0\bullet}(\widetilde{\Pi F})$ carries all information about YM theory reduced to a point, whereas algebra $C^{\infty}(\Sigma)$ contains similar information about the space $\Sigma$. The idea is that a tensor product $\Omega^{0\bullet}(\widetilde{\Pi F})\otimes C^{\infty}(\Sigma)$ with a suitably twisted differential will contain all information about 4-D YM theory.
Later we will interpret the same complex as tangential CR complex on the manifold $Z$.
The linear space
\begin{equation}\label{E:gdfhd}
\Sigma^{\mathbb{C}}=V=W_l\otimes W_r
\end{equation}
has coordinates $ x^{ij}, 1\leq i,j \leq 2$. The vector space $V$ has an $SO(4,\mathbb{C})$ action compatible with decomposition (\ref{E:gdfhd}). It is induced from the $SO(4,\mathbb{R})$ action on $\Sigma$ .
Define a differentiation of $\Omega ^{0\bullet}(\widetilde{\Pi F})\otimes C^{\infty}(\Sigma)$ as follows.
There is an $SO(4)$ equivariant isomorphism $V\cong H^0({\cal O}(1,1))$.
The line bundle ${\cal O}(1,1)$ is generated by its global sections.
We have a short exact sequence:
\begin{equation}\label{Q:wert}
0\rightarrow M \rightarrow V\overset{m}{\rightarrow}\Theta={\cal O}(1,1)\rightarrow 0
\end{equation}
where $V$ is considered as a trivial vector bundle with fiber $V$. The differentiation $\delta$ of $\Omega ^{0\bullet}(\widetilde{\Pi F}) \otimes C^{\infty}(\Sigma)$ is equal $\delta=m(x^{ij})\pr{x^{ij}}$. We interpret $\pr{x^{ij}}$ as global sections of $T^{\mathbb{C}}\Sigma$.
Recall that the differential $D$ in $\Omega ^{0\bullet}(\widetilde{\Pi F})$ is equal to $\bar \partial+\alpha$.
It becomes clear from explicit computation of cohomology that coefficients $\beta^{ij}$ in $\{\alpha, \delta\}=\beta^{ij}\pr{x^{ij}}$ are $\bar \partial$-exact.
Choose $\gamma^{ij}$ such that $\bar \partial \gamma^{ij}=-\beta^{ij}$. Define a differentiation $\gamma$ of $\Omega ^{0\bullet}(\widetilde{\Pi F})\otimes C^{\infty}(\Sigma)$ as $\gamma^{ij}\pr{x^{ij}}$ and zero on the rest of the generators.
It follows from our construction that $D_{ext}=D+\delta+\gamma$ satisfies $D_{ext}^2=0$ . Denote $D'=\delta+\gamma$.
By definition the integral form on manifold $\widetilde{\Pi F}\times \Sigma$ is :
\begin{equation}
Vol=\mu\otimes \bigotimes^2_{i,j=1} dx^{ij}
\end{equation}
\begin{proposition}
$Vol$ is invariant with respect to $D$, $D'$ and therefore with respect to $D_{ext}$.
\end{proposition}
\begin{proof}
Direct inspection.
\end{proof}
\subsection{Proof of the equivalence}
\begin{proposition}\label{P:jfjhsg}
Denote $Z=\Pi F\times \Sigma$. We equip algebra $A(Z)=\Omega ^{0\bullet}(\widetilde{\Pi F})\otimes C^{\infty}(\Sigma)\otimes Mat_n$ with a trace
\begin{equation}
\int a=tr\int_{Z}a Vol
\end{equation}
The CS theory constructed by the triple $(A(Z),D_{ext},\int)$ is classically equivalent to N=3 euclidean YM with a gauge group U$(n)$.
\end{proposition}
The equivalence should hold also on a quantum level.
\begin{proof}
We shall only outline the basic ideas.
A precise mathematical statement is about quasiisomorphism of certain A$_{\infty}$ algebras. One of them is $A(Z)$. The reader should consult [MSch06] for information about A$_{\infty}$ algebra with a trace corresponding to N=3 YM theory.
Suppose $\psi:A\rightarrow B$ is a quasiisomorphism of two A$_{\infty}$ algebras. Let $m$ be an associative algebra. Then we have a quasiisomorphism of tensor products $\psi:A\otimes m\rightarrow B\otimes m$. Let $a$ be a solution of MC equation in $A\otimes m$ (the reader may consult [MSch05] for the definition). The map $\psi$ transports solution $a$ to a solution $\psi(a)$ of MC equation in $B\otimes m$.
In a formal interpretation of structure maps of A$_{\infty}$ algebra $A$ as the Taylor coefficients of a noncommutative vector field on noncommutative space $\mathbb{A}$ solution $a$ of MC equation corresponds to a zero of the vector field. We can expand the vector field into series at $a$ and get some new A$_{\infty}$ algebra. This construction is particularly transparent in case of a dga. A solution of MC equation defines a new differential $\tilde{d}x=dx+[a,x]$. It corresponds to a shift of a vacuum in a physics jargon. Denote by $A_a$ an A$_{\infty}$ algebra constructed by the element $a$ .
It is easy to see that the map $\psi$ defines (under some mild assumptions in $a$) a quasiisomorphism $$\psi:A\otimes m_a\rightarrow B\otimes m_{\psi(a)}$$
Denote by $Diff(\Sigma)$ an algebra of differential operators on $\Sigma$.
We would like to apply construction from the previous paragraph to the tensor products $\Omega^{0\bullet}(\widetilde{\Pi F})\otimes Diff(\Sigma)$ and $C\otimes Diff(\Sigma)$. We can interpret $\delta+\gamma$ as a solution of MC equation for algebra $\Omega^{0\bullet}(\widetilde{\Pi F})$ with coefficients in $Diff(\Sigma)$. The quasiisomorphism $\psi$ maps $\delta+\gamma$ into an element $y^{ij}\pr{x^{ij}}$, where $y^{ij}$ is a basis of $H^0(X,\Lambda^{0}(H^*)\otimes \Theta)\subset C$.
The rest is a matter of formal manipulations. It is straightforward to see that $C\otimes Diff(\Sigma)_{\psi(a)}$ is an A$_{\infty}$ mathematics counterpart of YM equation where the gauge potential, spinors, matter fields are having their coefficients not in functions on $\Sigma$ but in $Diff(\Sigma)$.
This is not precisely what we have hoped to obtain. We shall address this issue presently.
Suppose an associative algebra $m$ contains a subalgebra $m'$. The previous construction has a refinement. The noncommutative vector field on a space $\mathbb{A}_m$ corresponding to A$_{\infty}$ algebra $A\otimes m$ is tangential to a noncommutative subspace $\mathbb{A}_{m'}$ (because $m'$ is closed under multiplication). We say that a solution of MC $a\in A\otimes m$ is compatible with $m'$ if the vector field defined by the algebra $ A\otimes m$ is tangential to the space $a+\mathbb{A}_{m'} \subset \mathbb{A}_m$. This is merely another way to say that a linear space $A\otimes m'$ is a subalgebra of $(A\otimes m)_a$. If $a$ is compatible with $m'$ the map $\psi$ (under some mild assumptions on $a$) induces a quasiisomorphism $\psi:A\otimes m'\rightarrow B\otimes m'_{\psi(a)}$.
We apply this construction to subalgebra $C^{\infty}(\Sigma)\subset Diff(\Sigma)$ for which mentioned above condition on $\delta+\gamma$ is met.
Suppose in addition that algebras $A,B,m'$ have a traces and a morphism $\psi:A\rightarrow B$ is compatible with the traces. Assume moreover that the induced A$_{\infty}$ structure $A\otimes m'_{a}$ is compatible with a trace. Then the induced morphism $\psi:A\otimes m'_a\rightarrow B\otimes m'_{\psi(a)}$ is compatible with traces.
In our case the operator $D'$ preserves the integral form $Vol$ and the above conditions are met.
The CS theory associated with the algebra $C\otimes C^{\infty}(\Sigma)$ has the following even part of the Lagrangian
$$<F_{ij},F_{ij}>+<\nabla_i\phi^{\alpha},\nabla_i\phi_{\alpha}>+<[\phi^{\alpha},\phi_{\beta}],K_{\alpha}^{\beta}>+<K_{\alpha}^{\beta},K^{\alpha}_{\beta}>$$
The theory besides of the gauge field corresponding to connection $\nabla_i$ in a principle U$(n)$ bundle , matter fields $\phi^{\alpha}\in Ad\otimes T,\phi^{\beta}\in Ad\otimes T^*$ contains an auxiliary field $K_{\alpha}^{\beta}\in Ad\otimes T\otimes T^*$. This theory is equivalent to N=3 YM( the odd parts of the Lagrangians coincide ).
\end{proof}
\subsection{Relation between a CR structure on $Z$ and an algebra $A(Z)$}\label{S:gsfre}
In this section we give a geometric interpretation of the algebra $A=\Omega^{0\bullet}(\widetilde{\Pi F})\otimes C^{\infty}(\Sigma)$.
Fibers of projection
\begin{equation}\label{E:cjsgwu}
p:X\times \Sigma\rightarrow \Sigma
\end{equation}
have holomorphic structure. Denote $T_{vert}$ a bundle of $p$-vertical vector fields. We define a distribution ${\cal G}$ as $T_{vert}^{1,0}\subset T_{vert}^{\mathbb{C}}$
Choose a linear basis $e_1,\dots,e_4\in \Sigma$. Define $\pr{x^s}$ the differentiations in the direction of $e_s$.
Restrict a map $m$ from short exact sequence (\ref{Q:wert}) on $\Sigma \subset V$, then $m(e_s)$ is a set of holomorphic sections of $\Theta$.
For any point $x\in X$ we have a subspace ${\cal H}_x\subset T^{\mathbb{C}}_{\Sigma}$ spanned by
\begin{equation}
\sum_{i=1}^4m(e_s)_x\pr{x^d}
\end{equation}
A union of such subspaces defines a complex distribution ${\cal H}$ on $X\times \Sigma$ .
Define an integrable distribution ${\cal F}={\cal G}+{\cal H}\subset T^{\mathbb{C}}_{X\times \Sigma}$.
The reader can see that the CR structure on the space $X\times \Sigma$ literally coincides with the CR structure on ambitwistor space $A(\Sigma)$ defined in Section \ref{S:hfdgjpo}.
In light of this identification an element $\theta$ (a local coordinate on $\Theta^*$) can be interpreted as a local CR-form with nonzero values on $\overline{{\cal H}}$.
Restriction of the vector bundle $H^*$ (used in the construction of supermanifold $F$ in equation (\ref{E:sjs})) on $X\times \Sigma$ is holomorphic along ${\cal F}$. Additionally we can interpret component $\alpha$ in the differential $D$ on $\Omega ^{0\bullet}(\widetilde{\Pi F})$ as a contribution from a superextension of the CR structure defined in (\ref{E:gfdjfur}).
From this we deduce that the algebra $A(Z)=(\Omega ^{0\bullet}(\widetilde{\Pi F})\otimes C^{\infty}(\Sigma),D_{ext})$ we have constructed coincides with the CR tangential complex on an open subset of the manifold $\mathcal{R}_{GL(4|3)} $.
The integral form $Vol$ is the only CR-holomorphic form invariant with respect to $SU(2)\times SU(2)\ltimes \mathbb{R}^4$.
From this we deduce that it coincides (up to a multiplicative constant) with the integral form defined by $vol$
It is possible reconstruct an action of the super Poincare group SP on $A$. The reader will find explanations of why the measure is not invariant with respect to the full superconformal group in Section \ref{S:gufjasklly}.
\section{Appendix}
\subsection{On the definition of a graded real superspace}\label{S:llfddfgg}
A tensor category of complex superspaces ${\cal C}_{\mathbb{C}}$ (see ref.[DMiln] for introduction to tensor categories) has two real forms. The first is a category of real superspaces ${\cal C}_{\mathbb{R}}$. It is more convenient to think about objects of this category as of complex superspaces, equipped with an antiholomorphic involution $\sigma$.
Another tensor category related to ${\cal C}_{\mathbb{C}}$ is formed by complex superspaces, equipped with an antilinear map $\rho$
\begin{definition}\label{D:gdfjgdf}
Suppose $V$ is a $\mathbb{Z}_2$-graded vector space over complex numbers. An antilinear map $\rho:v\rightarrow \bar{v}$ is a graded real structure if
\begin{equation}
\begin{split}
&\rho^{2}=sid\\
& sid(v)=(-1)^{|v|}v
\end{split}
\end{equation}
we denote by $|v|$ a parity of $v$.
An element $v$ is real iff $\bar v=v$. Only even elements can be real with respect to a graded real structure. A graded real superspace is a pair $(V,\rho)$. Grader real superspaces form a tensor category ${\cal C}_{\mathbb{H}}$
\end{definition}
We shall be mostly interested in categories ${\cal C}_{\mathbb{C}}$ and ${\cal C}_{\mathbb{H}}$.
The categories ${\cal C}_{\mathbb{C}}$ ${\cal C}_{\mathbb{H}}$ are related by tensor functors.
The first functor is complexification ${\cal C}_{\mathbb{H}}\Rightarrow {\cal C}_{\mathbb{C}} $, $V\Rightarrow V^{\mathbb{C}}$. It forgets about the map $\rho$.
The second functor is ${\cal C}_{\mathbb{C}} \Rightarrow {\cal C}_{\mathbb{H}}$, $V\Rightarrow V^{\mathbb{H}}$. The object $V^{\mathbb{H}}$ is a direct sum $V+\overline{V}$. There is an antilinear isomorphism $\sigma:V\rightarrow \overline{V}$. For $v=a+b+c+d\in V^0+V^1+\overline{V}^0+\overline{V}^1$ define
\begin{equation}\label{E:qdsfdfbshe}
\rho(a+b+c+d)=\sigma^{-1}(c)-\sigma^{-1}(d)+\sigma(a)+\sigma(b)
\end{equation}
By construction $\rho^2=sid$.
A language of tensor categories can be used as a foundation for developing Commutative Algebra and Algebraic Geometry. If we start off in this direction with a category ${\cal C}_{\mathbb{C}}$ the result will turn to be Algebraic Supergeometry.
A category ${\cal C}_{\mathbb{H}}$ provides us with some real form of this geometry.
First of all a ${\cal C}_{\mathbb{H}}$ or a real graded manifold is an algebraic supermanifold $M$ defined over $\mathbb{C}$. The manifold $M$ carries some additional structure. The manifold $M_{red}$ is equipped with an antiholomorphic involution $\rho_{red}$. There is also an antilinear isomorphism of sheaves of algebras
\begin{equation}\label{E:jjshdga}
\rho:\rho_{red}^*\O\rightarrow \O,
\end{equation}
such that $\rho^2=sid$.
There is a $C^{\infty}$ version of a ${\cal C}_{\mathbb{H}}$ manifold. It basically mimics a stricture of $C^{\infty}$ completion of algebraic ${\cal C}_{\mathbb{H}}$ manifold at locus of $\rho_{red}$ fixed points. Any $C^{\infty}$ manifold admits a noncanonical splitting: $M\cong \Pi E$, where $E$ is some complex vector bundle over $M_{red}$. A ${\cal C}_{\mathbb{H}}$-structure manifests in an antilinear automorphism $\rho$ of $E$, that satisfies $\rho^2=-id$. Observe that $\rho$, together with multiplication on $i$ defines a quaternionic structure on $E$.
For any complex algebraic supermanifold $M$ there is the underlying ${\cal C}_{\mathbb{H}}$ manifold. As an algebraic supermanifold it is equal to $M\times \overline M$. There is a canonical antiinvolution on $(M\times \overline M)_{red}$. The morphism of sheaves (\ref{E:jjshdga}) in local charts is defined by formulas similar to (\ref{E:qdsfdfbshe}).
This construction manifests itself in $C^{\infty}$-setting as follows. Any complex supermanifold $M$ defines a $C^{\infty}$ supermanifold $\tilde M$ that is holomorphic in odd directions. It is a $C^{\infty}$-completion of $M\times \overline{M}_{red}$ near diagonal of $M_{red}\times \overline{M}_{red}$. Suppose $\Pi E$ is a splitting of $\tilde M$, where $E$ is a complex vector bundle on $\tilde{M}_{red}$. The vector bundle $E+\overline{E}$ has a natural quaternionic structure and defines a $C^{\infty}$ ${\cal C}_{\mathbb{H}}$-manifold $\Pi(E+\overline{E})$. This manifold is isomorphic to completion of $M\times \overline{M}$. Sometimes it is more convenient to work with the manifold $\tilde{M}$.
\subsection{On homogeneous CR-structures}\label{S:gfgjheq}
Suppose we are given an ordinary real Lie group and a closed subgroup $A\subset G$ with Lie algebras $\mathfrak{a}\subset \mathfrak{g}$. Additionally we have a complex subgroup $P\subset G^{\mathbb{C}}$ in complexification of $G$, with Lie algebras $\mathfrak{p}\subset\mathfrak{g}^{\mathbb{C}}$. We assume that the map
\begin{equation}\label{E:frtw}
p:G/A\rightarrow G^{\mathbb{C}}/P
\end{equation}
is a local embedding. By construction $G^{\mathbb{C}}/P$ is a holomorphic homogeneous space. It tangent space $T_x, x\in G/A$ contain a subspace $H_x=T_x\cap JT_x$. The operator $J$ is an operator of complex structure on $G^{\mathbb{C}}/P$. Due to $G$-homogeneity spaces $H_x$ have constant rank and form a subbundle $H\subset T$. We can decompose $H\otimes \mathbb{C}={\cal F}+\overline{{\cal F}}$. It follows from the fact that $\mathfrak{p}$ is a subalgebra that the constructed distribution ${\cal F}$ is integrable and defines a CR-structure.
A condition that the map $p$ (\ref{E:frtw}) is a local embedding is equivalent to $\mathfrak{g}\cap \mathfrak{p}=\mathfrak{a}$. In other words
\begin{equation}
\mathfrak{p}\cap\bar{\mathfrak{p}}=\mathfrak{a}^{\mathbb{C}}
\end{equation}
It is easy to see that a fiber ${\cal F}_x$ at a point $x$ is isomorphic to $\mathfrak{p}/\mathfrak{a}^{\mathbb{C}}$
This construction of CR-structure can be extended to a supercase. A consistent way to derive such extension is to use a functorial language of ref. [DM].
However since this exercise, which we leave to the interested reader, is quite straightforward we provide only the upshot.
We start off with description of a data that defines a homogeneous space of a supergroup.
A complex homogeneous space $X$ of a complex supergroup $G$ with Lie algebra $\mathfrak{g}$ is encoded by :
{\bf 1.Global data:} An isotropy subgroup $H\subset G_{red}$ (closed, analytic, possibly nonconnected ). This data defines an ordinary homogeneous space $X_{red}=G_{red}/H$;
{\bf 2.Local data:} a pair of complex super Lie algebras $\mathfrak{p}\subset \mathfrak{g}$ such that $\mathfrak{p}_0=Lie(H), Lie(G_{red})=\mathfrak{g}_0$ .
In the cases when we specify only Lie algebra of isotropy subgroup is clear from the context.
A real graded structure on a homogeneous space $X=G/A$ is encoded by an antiholomorphic involution on $G_{red}$ that leaves subgroup $A_{red}$ invariant; a graded real structure $\rho$ on $\mathfrak{g}$,such that $\rho(\mathfrak{a})\subset\mathfrak{a}$.
If we are given a real subalgebra $(\mathfrak{a},\rho)\subset (\mathfrak{g},\rho)$ and a complex subalgebra $\mathfrak{p}\subset\mathfrak{g}$ such that $a=\mathfrak{p}\cap \rho(\mathfrak{p})$ we claim that a supermanifold $G/A$ carries a $(G,\rho)$-homogeneous CR-structure.
\subsection{General facts about CR-structures on supermanifolds}\label{S:gufjasklly}
In this section we will discuss mostly general facts about CR structures specific to supergeometry.
Suppose $M$ is a supermanifold equipped with an integrable CR distribution ${\cal F}$.
We present some basic examples of ${\cal F}$-holomorphic vector bundles on $M$.
{\bf Example} Sections of vector bundle $T^{\mathbb{C}}/\overline{{\cal F}}$ is a module over Lie algebra of sections of $\overline{{\cal F}}$. Thus the gluing cocycle of this bundle is CR-holomorphic. It implies that the bundle $Ber((T^{\mathbb{C}}/\overline{\F})^*)$ is also CR-holomorphic.
Suppose we have a trivial CR-structure on $\mathbb{R}^{n_1|n_2}\times\mathbb{C}^{m_1|m_2}$. We assume that the space is equipped with global coordinates $x_i,\eta_j,z_k,\theta_l$. The algebra of tangential CR complex $\Omega_{{\cal F}}^{\bullet}(\mathbb{R}^{n_1|n_2}\times\mathbb{C}^{m_1|m_2})$ has topological generators $x_i,\eta_j,z_k,\theta_l,\bar{z}_k,\bar{\theta}_l,d\bar{z}_k,d\bar{\theta}_l$. Denote by $A$ a subalgebra generated by $x_i,\eta_j,z_k,\theta_l,\bar{z}_k,,d\bar{z}_k$ and by $K$ a subalgebra generated by $\bar{\theta_l},d\bar{\theta}_l$. We have $\Omega^{0\bullet}=A\otimes K$. The algebra $K$ has trivial cohomology. As a result projection $\Omega^{0\bullet}\rightarrow A$ is a quasiisomorphism.
It turns out that this construction exists in a more general context of an arbitrary CR-manifold. Informally we may say think that a super-CR manifold is affine in holomorphic odd directions. It parallels with the complex case.
The construction requires a choice of $C^{\infty}$-splitting of CR-manifold $M$.
Suppose $Y$ is an ordinary manifold, $E$ is a vector bundle. Let $\Pi E$ denote the supermanifold whose sheaf of functions coincides with a sheaf of sections of the Grassmann algebra of the bundle $E^*$. Such supermanifold is called split. By construction it admits projection $p:\Pi E\rightarrow Y$. Any $C^{\infty}$ manifold is split, but the splitting is not unique. A space of global functions on $\Pi E$ is isomorphic to a space of sections of the Grassmann algebra $\Lambda E^*$ of vector bundle $E$.
To make a connection with our considerations we identify $Y=M_{red}$.
Denote $\overline{{\cal F}}_{red}=\overline{{\cal F}}_{red}^0+\overline{{\cal F}}_{red}^1$ restriction of $\overline{{\cal F}}$ on $M_{red}$. In terms of the splitting operator $\bar \partial$ can be encoded by a pair of operators of the first order $\bar \partial_{ev}:C^{\infty}(M_{red})\rightarrow \Lambda E^*\otimes \overline{{\cal F}}_{red}, \bar \partial_{odd}:E^*\rightarrow \Lambda E^*\otimes \overline{{\cal F}}_{red}$.
The lowest degree component in powers $\Lambda^i E^*$ of the operator $\bar \partial_{odd}$ is $\bar \partial^0_{odd} E^*\rightarrow \overline{{\cal F}}_{red}^1$. It is a $C^{\infty}(Y)$ linear map, with a locally free image. The image $S_0$ of a splitting $\overline{{\cal F}}^1_{red}\rightarrow E^*$ can be used to generate a differential ideal $(S)$ of $\Omega_{{\cal F}}$. Denote the quotient $\Omega_{{\cal F}}/S$ by $\Omega_{s{\cal F}}$.
The complex $\Omega_{s{\cal F}}$ is a differential graded algebra. We can interpret it as a space of functions on some superspace $L$. A possibility to split $\bar \partial^0_{odd}$ implies smoothness of $L$.
\begin{proposition}\label{E:fdsafjh}
The map $\Omega_{{\cal F}}\rightarrow \Omega_{s{\cal F}}$ is a quasiisomorphism.
\end{proposition}
\begin{proof}
Follows from consideration of a spectral sequence associated with filtration $F^i\Omega^p_{{\cal F}}=(S_0)^{i-p}\Omega^p_{{\cal F}}$(we denote by $(S_0)^k$ the k-th power of the ideal generated by $S_0$ ).
\end{proof}
A CR-manifold $M$ is locally embeddable to $\mathbb{C}^{m|n}$ if in a neighborhood of a point there is a collection of $z_1,\dots,z_m$ even and $\theta_1,\dots,\theta_n$ odd function that are annihilated by $\bar \partial$ and whose Jacobian is nondegenerate.
\begin{definition}\label{D:wpprnfj}
Let us assume that ${\cal F}^1|_{M_{red}}+\overline{{\cal F}}^1|_{M_{red}}=T^{\mathbb{C}1}$(superscript $1$ denotes the odd part),i.e. dimension of the odd part of CR distribution is maximal possible.
We can locally generate ideal $S_0$ by elements $\overline{\theta}_1,\dots,\overline{\theta}_n$ and take $S$ as a $\bar \partial$ closure of $S_0$. It is not hard to check that under such assumptions Proposition \ref{E:fdsafjh} holds. Denote by $sM$ a submanifold specified by $S_0$
\end{definition}
\begin{remark}\label{E:fdfdsdfv}
The Lie algebra of infinitesimal automorphisms of CR-structure is equal to $Aut_{{\cal F}}=\{a\in T^{\mathbb{C}}|[a,b]\in \overline{{\cal F}}, \mbox{ for all }b\in \overline{{\cal F}}\}\}$ with $Out_{{\cal F}}=Aut_{{\cal F}}/\overline{{\cal F}}$. In purely complex case the quotient construction can be replaced by $Out_{complex}=\{a\in {\cal F}|[a,b]\in \overline{{\cal F}}, \mbox{ for all }b\in \overline{{\cal F}}\}\}$ and the extension
\begin{equation}
0\rightarrow Inn_{{\cal F}} \rightarrow Aut_{{\cal F}}\rightarrow Out_{{\cal F}}\rightarrow 0
\end{equation} has a splitting. By construction elements $\overline{\theta}_1,\dots,\overline{\theta}_n$ are invariant along vector fields from distribution ${\cal F}$. We can guarantee that differential ideal generated by $\overline{\theta}_i$ is invariant with respect to elements of $Out_{complex}$. As a result we can push the action of $Out_{complex}$ to $\Omega^{\bullet}_{s{\cal F}}$-this is familiar fact from super complex geometry . This contrasts with absence of an action of $Aut_{{\cal F}}$ or $Out_{{\cal F}}$ on the ideal $S$ and on $\Omega^{\bullet}_{s{\cal F}}$ for a general CR structure. One can prove however that $\Omega^{\bullet}_{s{\cal F}}$ admits an A$_{\infty}$ action of $Aut_{{\cal F}}$ . A partial remedy is to consider subalgebra $\widetilde{Out}_{{\cal F}}=\{a\in {\cal F}|[a,b]\in \overline{{\cal F}}, \mbox{ for all }b\in \overline{{\cal F}}\}\}\subset Out_{{\cal F}}$. This subalgebra acts upon $\Omega^{\bullet}_{s{\cal F}}$ . However this algebra is trivial if the Levi form of ${\cal F}$ is not degenerate
\end{remark}
Supergeometry provides us with a complex of CR-integral forms. Let $\Lambda \overline{{\cal F}}$ be the super Grassmann algebra of $\overline{{\cal F}}$. Let $Ber$ be the Berezinian line bundle of a real manifold $M$. Denote $\ ^{int}\Omega^{-p}_{{\cal F}}$ the tensor product $Ber\otimes \Lambda \overline{{\cal F}}$. There is a pairing $(\omega,\nu)=\int_{M}<\omega,\nu>$ between sections $\omega\in \Omega^{p}_{{\cal F}}$ and $\nu\in \ ^{int}\Omega^{-p}_{{\cal F}}$. The symbol $<.,.>$ denotes contraction of a differential form with a polyvector field. The operation $<.,.>$ takes values in sections of $Ber$. The value $<\omega,\nu>$ can be used as an integrand for integration over $M$.
The orthogonal complement to the ideal $S\subset \Omega^{p}_{{\cal F}}$ is a subcomplex $\ ^{int}\Omega^{-p}_{s{\cal F}}\subset\ ^{int}\Omega^{-p}_{{\cal F}}$. It is a differential graded module over $\Omega^{p}_{s{\cal F}}$.
\begin{proposition}\label{P:fdfdhw}
Let $M$ be a supermanifold with a CR distribution ${\cal F}$ of dimension $(n|k)$.
There is an isomorphism $i: \ ^{int}\Omega^{p-n}_{s{\cal F}} \rightarrow \Omega^{p}_{s{\cal F}}Ber((T^{\mathbb{C}}/\overline{\F})^*)$ compatible with a stricture of $\Omega^{\bullet}_{s{\cal F}}$-module.
The isomorphism is unique.
\end{proposition}
\begin{proof}
Using $C^{\infty}$ splitting we can identify $\Omega^{\bullet}_{s{\cal F}}Ber((T^{\mathbb{C}}/\overline{\F})^*)$ and $\ ^{int}\Omega^{p-n}_{s{\cal F}}$ with sections of some vector bundles $A_p$ and $B_p$ over $M_{red}$. It is fairly straightforward to establish an isomorphism of $A_p$ and $B_p$ with the help of the splitting. In particular there is a $C^{\infty}$ isomorphism $Ber((T^{\mathbb{C}}/\overline{\F})^*)=\ ^{int}\Omega^{-n}_{s{\cal F}}$.
One can think about $\Omega^{0}_{s{\cal F}}$ as of a space of functions on a supermanifold $sM$. Then a space of sections $Ber(sM)$ coincide with $\Omega^{n}_{s{\cal F}}Ber((T^{\mathbb{C}}/\overline{\F})^*))$. It elements can be integrated over $sM$. The integral defines a pairing $(.,.)_s:\Omega^{\bullet}_{s{\cal F}}\otimes \Omega^{\bullet}_{s{\cal F}}Ber((T^{\mathbb{C}}/\overline{\F})^*))\overset{\int<.,.>_{s}}\rightarrow \mathbb{C}$, which is nondegenerate.
An element $a\in \ ^{int}\Omega^{0}_{s{\cal F}}$ defines a functional $f\rightarrow \int_{M} a f$ ($f\in C^{\infty}(sM)$). We may think about it as of an integral $\int_{sM} fi(a)$ , where $i(a)\in \Omega^{n}_{s{\cal F}}Ber((T^{\mathbb{C}}/\overline{\F})^*)$. Such interpretation of the integral uniquely specifies map $i$. Since $\Omega^{n}_{s{\cal F}}$ is invertible the induced isomorphism $i:Ber((T^{\mathbb{C}}/\overline{\F})^*)=\ ^{int}\Omega^{-n}_{s{\cal F}}$ is compatible with $\bar \partial$ (use pairings $(.,.),(.,.)_s$ to check this).
\end{proof}
Suppose that a super CR-manifold $M$ is CR embedded into a complex super manifold $N$. Denote by $J$ an operator of complex structure in tangent bundle $TN$. We assume
$TM+JTM=TN_{|M}$. Denote by $Ber_{\mathbb{C}}(N)$ the complex Berezinian of $N$. An easy local computation shows that $Ber_{\mathbb{C}}(N)_{|M}$ is isomorphic to $Ber((T^{\mathbb{C}}/\overline{\F})^*)$. Suppose $N$ is a Calabi-Yau manifold , i.e. it admits a global nonvanishing section $vol$ of $Ber_{\mathbb{C}}(N)$. A restriction of $vol$ on $M$ defined a global CR-holomorphic section of $Ber((T^{\mathbb{C}}/\overline{\F})^*)$. An isomorphism of Proposition \ref{P:fdfdhw} provides a $\bar \partial$ closed section of $\ ^{int}\Omega^{-n}_{s{\cal F}}\subset \ ^{int}\Omega^{-n}_{{\cal F}}$.
\begin{remark} The proof Proposition \ref{P:fdfdhw} parallels with the proof of Serre duality in super case given in ref. [HW]. Haske and Wells used sheaf-theoretic description of a complex supermanifold, which significant simplify the argument . The main simplification comes from the local Poincare lemma, which is absent is CR-case.
\end{remark}
It is worthwhile to mention that there is no canonical ($Aut_{{\cal F}}$ or $Out_{{\cal F}}$ equivariant) map $ \Omega^{\bullet}_{{\cal F}}Ber((T^{\mathbb{C}}/\overline{\F})^*)\rightarrow \ ^{int}\Omega^{\bullet}_{{\cal F}}[-n]$. This seems to be one of fundamental distinctions of purely even and super cases.
We think the reason is that the only known construction of this map is through the intermediate complex $\Omega^{\bullet}_{s{\cal F}}Ber((T^{\mathbb{C}}/\overline{\F})^*)$. As have have already mentioned in remark (\ref{E:fdfdsdfv}) the complex $\Omega^{\bullet}_{s{\cal F}}$ carries only A$_{\infty}$ action of $Aut_{{\cal F}}$.
A real line bundle $Ber$ on a complex super manifold is a tensor product $Ber_{\mathbb C}\otimes \overline{Ber}_{\mathbb C}$, where $Ber_{\mathbb C}$ is a holomorphic Berezinian.
A decomposition $T^{\mathbb{C}}={\cal T}=\overline{{\cal T}}$ implies that
\begin{equation}
\begin{split}
&^{int}\Omega^{-k}=Ber\otimes \\
&\Lambda^{k}({\cal T})=\bigoplus_{i+j=k} Ber_{\mathbb C}\otimes \Lambda^{i}({\cal T}_{\mathbb C})\otimes \overline{Ber}_{\mathbb C} \otimes \Lambda^{j}(\overline{{\cal T}}_{\mathbb C})=\bigoplus_{i+j=k}\ ^{int}\Omega^{-i,-j}
\end{split}
\end{equation}
\begin{remark}\label{E:hhghssa}
Suppose $M$ is a complex $n$-dimensional manifold, $E$ is an $k$-dimensional vector bundle.
On the total space $ \Pi E$ there is a canonical section of $c_{\Pi E} \in Ber_{\mathbb{C}}\otimes \Lambda^{n}({\cal T})$.
In local odd fiberwise coordinates $\theta_i$ it is equal to
\begin{equation}\label{E:jsdfdmdas}
c_{\Pi V}=\theta _1\dots \theta _k d\theta _1\wedge \dots \wedge d \theta _k.
\end{equation}
\end{remark}
\begin{proposition}\label{P:kahedy}
The forms $c_{\Pi V}$ and $\bar c_{\Pi V}$ are $\bar \partial$-closed.
\end{proposition}
\begin{proof}
Since the formula (\ref{E:jsdfdmdas}) does not depend on a choice of coordinates on $M$ one can do a local computation, which is trivial.
\end{proof}
\section{Acknowledgments}
The author would like to thank P. Candelas, P.Deligne, M.Kontsevich, L. Maison, A.S. Schwarz for useful comments and discussions. The note was written while the author was staying at Mittag-Leffler Institut, Max Plank Institute for Mathematics and Institute for Advanced Study. The author is grateful to these institutions for kind hospitality and support. After the preprint version of this paper [Mov] had been published, the author learned about the work by L.J.Mason, D.Skinner [MS] where they treat a similar problem.
|
{
"timestamp": "2008-10-20T01:42:01",
"yymm": "0411",
"arxiv_id": "hep-th/0411111",
"language": "en",
"url": "https://arxiv.org/abs/hep-th/0411111"
}
|
\section{Introduction}
The problem of dark matter (DM) of the Universe was revealed about
70 years ago. Several possible physical candidates were suggested
since that time and several approaches to probe these candidates
appeared. However, observed phenomena, which have unclear nature
yet, could not be decisively matched with expected effects
caused by existing candidates.
An important step in exploration of DM problem was a development
of direct searches for Weakly Interacting Massive Particles
(WIMP). Underground detectors were created in which effects of
nucleus recoil induced by interaction of cosmic WIMP with nucleus
were searched for. A positive result at 6.3 sigma C.L. was obtained
in the DAMA/NaI underground set-up at the Gran Sasso National Laboratory of
I.N.F.N.
by exploiting the distinctive WIMP annual modulation
signature \cite{DAMA-review}. Being model independent
this positive result cannot be directly compared with the single
model-dependent negative results
of other groups, which have also used different target-nuclei, different
experimental strategies, different set-ups
and all assumptions fixed at a single set \cite{DAMA-review}. Moreover, it
can be shown \cite{DAMA-review} that
these negative results are actually not incompatible with the positive
signal by DAMA/NaI.
On the other hand, an indirect probing of DM can be based on
cosmic ray (CR) data. The presence of DM in Galaxy in form of
WIMPs can cause an appearance of cosmic particles of high energy
due to an annihilation or decay of the WIMPs. An implication of
these annihilation (decay) sources of cosmic rays could remove
possible contradiction between observed cosmic ray fluxes and
their predictions on the base of standard cosmic ray model.
The result of DAMA/NaI is a challenge for DM studies, expected to shed
a light on the origin of DM. In fact, the existing physical
candidates of dominant DM can hardly or not at all provide an
explanation of the result of DAMA/NaI. WIMP candidates such as
neutralino, axion, gravitino, sterile neutrino, axino, mirror
(shadow) matter are able to compose all the required missing mass
of the Universe, however, all these candidates, except neutralino,
are virtually sterile particles in respect to their interactions
with an ordinary matter. Therefore, it looks like the measurements
of DAMA/NaI as well as anomalies observed in cosmic rays spectra
require a non-sterile DM which, in particular, could be a
non-dominant DM component in the form of Heavy neutrinos of the
4th generation \cite{Fargion}.
A possibility to explain the DAMA/NaI result within the framework of
Standard Model extended to the 4th generation of fermions,
revealed in \cite{Fargion}, is the subject of current
consideration.
The Heavy neutrino ($N$) is supposed to be a neutral fermion of
a new 4th generation possessing the standard weak interaction.
According to recent analysis of precision electroweak
data\cite{Okun}, where possible virtual contributions of 4th
generation particles were taken into account, a fit is compatible
with the 4th neutrino, being Dirac and (quasi-)stable, in a mass
range about 50 GeV (47-50 GeV is $1\sigma$ interval, 46.3-75 GeV
is $2\sigma$ interval) \cite{Okun} and other 4th generation
particles satisfying their direct experimental constraints (above
80-130 GeV). In the following we will assume that the 4th neutrino
mass is about 50 GeV.
If the fourth neutrino is sufficiently long living or absolutely
stable, its primordial gas from the early Universe can survive to
the present time and concentrate in the Galaxy. In the case of
charge symmetry of 4th generation particles the primordial 4th
neutrinos can not account for a bulk of missing mass in the
present Universe. In the mass range about $50-80$ GeV the 4th neutrinos
can make up $10^{-5}-10^{-2}$ of total density of the Universe,
being a non-dominant DM component. This leads to a scenario of
multi component dark matter consisting of a subdominant Heavy
neutrino component and a sterile dominant component. A complex
analysis of astrophysical effects induced by a presence of Heavy
neutrino (non-sterile) DM component is the purpose of present
paper.
It is worth noting that the 4th generation of quarks and leptons,
considered here and neutralino, which is widely considered as the
candidate for WIMPs, are naturally incorporated in the framework
of heterotic string phenomenology. \footnote{Mirror(shadow)
matter, also naturally follows from heterotic string
phenomenology. It is usually considered as sterile but its WIMP
effects may play a role\cite{Foot}}. It appeals to future
multi-component dark matter analysis of the results of direct and
indirect WIMP searches. However, the astrophysical uncertainties
revealed below even for the model of 4th generation neutrino,
which is the simplest physical model and implies the minimal
amount of parameters, demonstrate all the complications to be
expected in such multi-component dark matter approach.
We should remark that even if in principle the UHE
Neutrino of fourth family might produce a very exciting
scattering with relic Heavy neutrino (a Heavy-Z-Boson Burst)
nevertheless the cosmic relic N are too diluted and poor in number to
be anyway competitive with the much more abundant (by $13$ order of
magnitude)
and effective light neutrino \cite{Z-Far}
scattering. So there is little consequence of any Heavy-Z-Boson Burst
model interactions.
The eventuality for a UHE Neutrino (produced, for
instance in Top-Down models as \cite{Khlopov})
to be a source of amplified resonant interaction with electron or
quark (as for SUSY UHE neutralino scattering into s-electron and s-quark
channels \cite{Datta}) is absent:
there are not s-channel interaction able to overcome the electro-weak
cross-sections but only much less effective t-channel processes.
Finally, the UHE N created by Top-Down models may nevertheless
induce charged and neutral current interactions in Neutrino Detectors
(SK,UNO,$km^3$) almost un-distinguishable from lighter UHE neutrino
scattering in matter. This effect has just the ability to increase by a small
factor ($\sim 25 \%$) the event rate in $km^3$ or EUSO neutrino induced events
if Top-Down mechanism is the main source of UHECR.
\section{Estimation of local Heavy neutrino density from DAMA/NaI
experiment}
A contribution of Heavy neutrinos $\rho_{loc\,N}$ to the total
local density $\rho_{loc}$ is given by a ratio
\begin{equation}
\xi_{loc}=\frac{\rho_{loc\,N}}{\rho_{loc}}.
\end{equation}
Approximately this parameter can be estimated as
$\Omega_N/\Omega_{CDM}$ by assuming a dominance of Cold Dark
Matter (CDM) in the Galaxy and by choosing the local fraction of relic
Heavy neutrinos equal to their contribution to the cosmological density of
CDM.
The results of DAMA/NaI, based on measurements of an "active" DM
component, i.e. in our assumptions on cosmic Heavy neutrinos, give
the fraction $\xi_{loc}$ of Heavy neutrinos in the local galactic
density. Heavy neutrinos interact with nuclei
($^{23}Na,\,^{127}I$) of DAMA/NaI detector through the
spin-independent coherent vector weak coupling. A spin-dependent
axial weak coupling of neutrino and nuclei would contribute
significantly within the considered mass range only if
the corresponding WIMP-nucleon cross section exceeds the
spin-independent one by several orders of magnitude, what in
general is not the case for the 4th neutrino. The value
$\xi_{loc}$ for Heavy neutrinos is deduced from the result of DAMA/NaI
in term of $\xi_{loc}\sigma_{SI}$, where $\sigma_{SI}$ is the
effective spin-independent WIMP-nucleon cross-section
\begin{equation}
\sigma_{SI}=\frac{G_F^2\mu^2}{8\pi}\frac{\beta_{Na}+\beta_{I}}{V_{Na}^{-2}\beta_{Na}+V_{I}^{-2}\beta_{I}}.
\end{equation}
Here $G_F$ is the Fermi constant, $\mu=mm_{nucl}/(m+m_{nucl})$ is the
reduced mass of neutrino $m$ and nucleon ($m_{nucl}=0.94$ GeV),
$\beta_i=4mm_i/(m+m_i)^2$, $V_i=1-(2-4sin^2\theta_W)Z_i/A_i$ with
$Z_i$ and $A_i$ being numbers of protons and nucleons in nucleus
respectively, $\theta_W$ is the Weinberg angle.
Figure 1 shows a favorable region
for Heavy neutrinos of the 4th generation measured by DAMA and the
fraction corresponding to a Heavy neutrinos contribution to a
local galactic density. The result of DAMA/NaI takes into account
existing uncertainties in DM distribution parameters, in
form-factor of nuclei, and in the other experimental
parameters \cite{DAMA-review}. The fraction corresponding to a
Heavy neutrinos contribution was estimated by taking
$\Omega_{CDM}=0.3$.
\begin{figure}
\begin{center}
\centerline{\epsfxsize=7cm\epsfbox{DAMA.eps}} \caption{Plot of
DAMA favorable region (between upper and lower solid lines) for
Heavy neutrinos of the 4th generation. A dashed line shows the
fraction corresponding to a contribution of the Heavy neutrinos to
CDM of the Universe.} \label{DAMA}
\end{center}
\end{figure}
Note that the present work is based on the essentially updated
DAMA/NaI results in comparison with the previous works \cite{Fargion},
\cite{BK}.
\section{Shadows of Heavy neutrino annihilations in cosmic rays}
Due to a concentration in Galaxy Heavy neutrinos
can annihilate. The products of such annihilation contribute in cosmic
ray fluxes and cosmic gamma radiation. Observational data on cosmic
positrons, antiprotons
and gamma-radiation are sensitive to this contribution.
An analysis of cosmic rays is complicated because a description of
CR production and propagation contains significant uncertainties.
Observational data do not allow to choose parameters of physical
models of CR in a unique way for CR origin (injection spectra of
each CR species) and CR propagation (diffusion coefficients and
their energy dependence, parameters of convection,
re-acceleration, magnetic halo parameters, matter distribution in
Galaxy, model of solar modulation etc.). Recently a detailed study
of CR models was performed in \cite{SM-0},\cite{SM-1},\cite{SM-2}.
In order to study effects of possible DM annihilation
in Galaxy, it is reasonable to accept the most
conservative CR model. Eligible models should reproduce possible
CR data which are the least sensitive to effects of WIMP annihilation
(data on nuclear component of CR, its isotope composition).
"Conventional model (C)" in \cite{SM-1} and "diffusion
re-acceleration model (DR)" in \cite{SM-2} are the most suitable.
We will use in our consideration fluxes of secondary positrons,
antiprotons and gamma-radiation predicted in these models as a
"background". Dark matter annihilation sources will be included
in these models to reproduce DM effects.
An uncertainty in our analysis comes also from unknown
distribution of subdominant Heavy neutrino DM in the Galaxy. There
are many models of distribution of CDM in Galaxy. We will use
models for dominant DM, re-scaling a dark matter density in an
appropriate way for a non-dominant Heavy neutrino DM component .
By fixing the density distribution of DM component in Galaxy we
relate the result of DAMA, sensitive to local density of DM, with
results of CR analysis, sensitive to DM density distribution in
Solar neighborhood.
As a basic model for our estimations we select Evan's halo model,
which in \cite{DAMA-review} was named as C2. The values of
parameters in this model are $v_0=170$ km/sec, $\rho_{loc}=0.67$
GeV/cm$^3$. Density distribution of DM in Galaxy is given by
Eq(41) of \cite{DAMA-review} or Eq(34) of \cite{Belli} with the
parameters $q=1/\sqrt{2}$ and $R_c=5$ kpc
\begin{equation}
\rho(R, z)=const\frac{2R_c^2+R^2}{(R_c^2+R^2+2z^2)^2},
\end{equation}
where $R$ and $z$ are the radius in the galactic plane and
the cylindrical coordinate axis perpendicular to it, $const$ is defined
from a
condition $\rho(R=R_0=8.5 {\rm kpc},z=0)=\rho_{loc}$.
It is worth to note the chosen DM distribution is smooth and does
not have a sharp profile near the Galactic center (GC).
Also, to illustrate a dependence on a halo model choice, we will
consider an isothermal halo model with a sharp behavior of density
near GC. Density distribution of the isothermal halo is given by
\begin{equation}
\rho(R)=\rho_{loc}\frac{R_c^2+R_0^2}{R_c^2+R^2}
\end{equation}
with $R_c=1$ kpc, where $R$ is the distance from GC.
Note that the use of cuspy halo model, like Navarro-Frank-White
model \cite{Navarro}, leads to intermediate results.
\subsection{Signature of Heavy neutrinos annihilation in cosmic gamma
fluxes}
An excess of cosmic gamma-radiation observed by EGRET over
predicted galactic $\gamma$-emission, often called "extragalactic"
$\gamma$-background, can be considered as a possible effect of
dark matter sources. A flux of cosmic gamma-radiation near the
Earth from annihilation of relic Heavy neutrinos is defined by
\begin{equation}
I=\frac{dN_{\gamma}}{dtdSd\Omega dE}=\frac{1}{4\pi}\frac{1}{4}<\sigma
v>\int_0^{\infty}n^2dl\,\frac{dN_{\gamma E}}{dE}.
\end{equation}
Here $<\sigma v>$ is the product of cross section of $N\bar{N}$
annihilation and a relative velocity of neutrinos averaged over
velocity distribution, $n=\rho/m$ is the number density of
neutrinos in Galaxy. An integration is performed along the line of
sight with formally infinite upper limit, $dN_{\gamma E}/dE$ is the
mean multiplicity of photons created in an act of annihilation for
$E-(E+dE)$ energy interval. The factor $1/4$ comes from the fact
that the number densities of neutrinos and antineutrinos are equal
to a half of their total number density $n$. To obtain a
distribution $dN_{\gamma E}/dE$ a code PYTHIA 6.2 was used.
Figures 2 and 3 show $\gamma$-fluxes at the Earth from two
directions: from the Galactic center (Fig.2) and from a halo in
the direction of Galactic zenith (Fig.3). Corrected EGRET data for
the halo were taken from \cite{SM-3}, where EGRET data were
re-analyzed in an advanced approach in which the galactic
contribution was subtracted giving pure isotropic "extragalactic"
$\gamma$-radiation. Dashed/solid lines in these figures show
annihilation/annihilation plus background $\gamma$-fluxes for
neutrino mass ranging 47-80 GeV. The annihilation
$\gamma$-fluxes were obtained selecting density parameter
$\xi_{loc}$ at given (Evan's) density distribution to fit in the
best way the observation data for the accepted value of neutrino
mass. In a fitting procedure $\chi^2$ criterion was used by fixing
the Galactic contribution (background) and changing the
annihilation flux by the $\xi_{loc}$-parameter. Neutrino masses
were chosen as 47, 50, 55, 60, 65, 70, 75, 80 GeV.
The low energy part of $\gamma$-spectrum from halo (Fig.3) is not
reproduced by annihilation of Heavy neutrinos in the halo. However
this low energy part can be explained by extragalactic emission
based on a blazar population \cite{SM-3},\cite{neutralino-gamma}.
This emission is expected with a similar slope in spectra as data
points exhibit. Extraction of pure "extragalactic" $\gamma$-flux
in direction of Galactic center (GC) is more complicated problem.
We used observational data and a prediction of galactic
contribution for $\gamma$-flux from GC \cite{SM-1} in accordance
with model "C". The galactic contribution (refereed to as a
background) is shown in Fig.2 by dot-dashed line.
\begin{figure}
\begin{center}
\centerline{\epsfxsize=7.5cm\epsfbox{gamma-GC.eps}}
\caption{Cosmic gamma-radiation from galactic center
($0.5^o<l<30.0^o,330.0^o<l<359.0^o$): EGRET data, predicted
background (dot-dashed line), and the best-fit contribution from
47-80 GeV neutrino DM for Evan's halo model. The set of dashed
lines corresponds to pure annihilation gamma-fluxes, the set of
solid lines is the sum of background and annihilation
fluxes.\label{fig.2}}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\centerline{\epsfxsize=7.5cm\epsfbox{gamma-halo.eps}}
\caption{Cosmic gamma-radiation from zenith galactic direction:
EGRET data and the best-fit gamma-flux from 47-80 GeV neutrino
DM annihilation (the set of lines) for Evan's halo model.
\label{fig.3}}
\end{center}
\end{figure}
All $\xi_{loc}$-parameters, fitting in the best way the predicted
annihilation with background fluxes to observational data, will be
presented in a figure below.
\subsection{Signature of Heavy neutrinos annihilation in cosmic $e^+$ and
$\overline{p}$ fluxes}
The flux of charged particles from annihilation of Heavy neutrinos
in Galaxy near the Earth is defined by a diffusion of particles
from $N\bar{N}$ annihilation to the region around the Earth of the
size as large as the characteristic diffusion length. For antiprotons,
which do not experience significant energy loss, this region is
determined by the size of halo where they are trapped by magnetic
field. For positrons the size of region of dark matter
annihilation sources contributing to a flux near the Earth
depends on energy loss of positrons. This does not allow
positrons, created with an energy $E_0$, to come with energy $E$
from distance strongly exceeding
\begin{equation}
\lambda(E,E_0)=\left(\int_{E_0}^E\frac{D(E')}{b(E')}dE'\right)^{1/2}.
\end{equation}
Here $D$ is the diffusion coefficient which is energy (rigidity)
dependent, $b(E)$ is the rate of energy loss defined as
\begin{equation}
\frac{dE}{dt}=-b(E).
\end{equation}
Charged cosmic particles experience a solar modulation. To take
into account this effect we will use force-field model
\cite{solar modulation}. In this approximation the intensity
measured at the top of the Earth's atmosphere (inner heliosphere) at
the energy $E_{Earth}$ corresponds to a local interstellar (LIS)
intensity (outer heliosphere) through the relation
\begin{equation}
I_{LIS}(E=E_{Earth}+\Phi(t))=\frac{(E_{Earth}+\Phi(t))^2-m_p^2}{E_{Earth}^2-m_p^2}
\,I_{Earth}(E_{Earth},t).
\end{equation}
Here $m_p$ is the mass of cosmic particle, $\Phi(t)$ is the
energy lost by the cosmic particles during their travel in the
heliosphere. $\Phi(t)$ is the parameter of the model which can be
derived from an observation of appropriate period of Solar
activity. A dependence of solar modulation of CR on the sign of
particle charge appears at low energy (at LIS
energy, less than 1-2 GeV). It is shown in \cite{Casadei} this dependence
during
nineties (positive half-cycle of the Sun) broke the force-field
approximation for description of data on negatively charged
particles (electrons, antiprotons) in low energy range. Whereas
positively charged particles were well described by the
force-field model in that period. We will use data on cosmic
positrons and antiprotons transformed into LIS by Eq(8), what
will lead to some underestimation of LIS antiproton flux at low
energy.
Note that the energy scale of modulation is detemnined by $\Phi(t)$,
being around $1$GeV, and for $E>>1$ GeV effect of
modulation is negligible.
For an estimation of positron flux from $N\bar{N}$ annihilation
we adopted the diffusion approximation of positron propagation in
Galaxy without inclusion of the effect of diffusion zone
boundaries \cite{Ginzburg}. It is well-known that a strong energy
loss of high energy cosmic $e^{\pm}$ makes their spectra
dependent on the space distribution of density of
$e^{\pm}$-sources. It disfavors the use of "leaky-box" model for
a quantitative estimation of the effects of tangled, diffusion
propagation of $e^{\pm}$ in Galaxy. A diffusion coefficient and
energy loss parameter were chosen following "DR" model \cite{SM-2}
\begin{eqnarray}
D(E)=6.1\times10^{28}\left(\frac{E}{4\,{\rm GeV}}\right)^{0.33}\,{\rm cm^2s^{-1}},\\
b(E)=\beta E^2,\,\,\,\beta=1.52\times10^{-9}(0.5+0.5(H/3\,{\rm \mu G})^2)\,{\rm yr^{-1}GeV^{-1}},
\end{eqnarray}
Here in the expression for $\beta$ the dependence on the averaged galactic magnetic
field $H$ and its value $3\,{\rm \mu G}$ are taken in accordance with \cite{Turner}.
Such parameters (Eq(9-10)) allow positrons originated as far as
at GC to contribute to the flux near the Earth. A background
(secondary) positron flux was taken as predicted in "DR" model
keeping accordance with the choice of parameters (Eq(9-10)).
"DR" model takes into account effect of boundaries and effect of
re-acceleration, acceleration of cosmic particles (initially
accelerated in their sources) during their propagation in
interstellar medium. A disagreement between our estimation of
annihilation flux and the used prediction of background is not
significant. An effect of boundaries of diffusion zone is not
very important for Evan's DM halo model as it is seen from
\cite{SM-4}.
The decrease of halo size $z_h$ leads to the de-population of
distribution of positrons with $\lambda>z_h$ (Eq(6)) because they
escape the diffusion zone more intensively than at larger $z_h$
and also because a contribution from dark matter annihilation
sources situated outside the diffusion zone is not taken into
account. The escape from the diffusion zone leads to diminishing
relative contribution of annihilation positrons from GC with a
decreasing halo size, what is more marked effect for halo models
with a sharp density profile near GC. The use of Evan's halo model
with its smoothed density distribution provides better accuracy
for the used approximation than other models with sharper
profiles. Also to reduce a deviation of annihilation fluxes
obtained in our approximation from that one which would be
predicted in "DR" model, induced by the neglecting of the
boundaries, we excluded a contribution from dark matter
annihilation sources situated outside the diffusion zone of
$z_h=4$ kpc. Effect of re-acceleration appears below 5 GeV in
positron spectra \cite{SM-4} (or the right part of Fig.5 in the
ref. \cite{reacceleration}), where the role of secondary positrons
shades the possible dark matter annihilation sources contribution.
In Fig.4 the predicted LIS positron fluxes as compared to
observational data of HEAT \cite{HEAT} are presented. There are
the secondary positron flux (dot-dashed line), the sum of
secondary and the best-fit annihilation fluxes (the set of solid
lines) and separately the last ones (the set of dashed lines),
obtained for the range of neutrino mass 47-80 GeV. Note, that
the curves corresponding to the annihilation fluxes are extended
up to the energy equal to $m$ of neutrino, above this energy
the predicted total fluxes (secondary plus annihilation positrons)
are the same as the secondary flux at $E>m$.
HEAT data were
"demodulated" taking $\Phi=664$ MeV, derived from data of CAPRICE
\cite{Casadei}. A fitting was performed in the same way as
described above for gamma-radiation. First point of HEAT (slightly
below 2 GeV in Fig.4) was omitted in the procedure of fitting.
This point is apparently inconsistent even with the predicted
secondary positron flux and also inconsistent with the other
measurements of cosmic positrons \cite{Casadei}. Note that the low
energy range should be considered cautiously, because predictions
in this range depend on possible effects of re-acceleration.
Let's consider an effect of diffusion coefficient variation, or
correspondingly (see Eq(6)) a variation of positron diffusion
length. A decrease in $D$ by a factor 10 modifies a little a slope
(makes steeper) of spectra of annihilation positrons near the
Earth and changes the best-fit $\xi_{loc}$ parameter by less than 20\%.
The further decrease in $D$ causes no influence, because mean free path
length $\lambda$ becomes less than typical physical scales of the
problem (the distance to the nearest boundary of diffusion zone or
the length scale of variation in the DM density distribution).
\begin{figure}
\begin{center}
\centerline{\epsfxsize=7.5cm\epsfbox{positron.eps}}
\caption{Cosmic positrons (LIS): HEAT data, predicted background
(dot-dashed line), and the best-fit contribution from 47-80 GeV
neutrino DM (the set of dashed lines is pure annihilation positron
fluxes, the set of solid lines is the sum of background and
annihilation fluxes) for Evan's halo model. \label{fig.4}}
\end{center}
\end{figure}
To estimate the flux of antiprotons from $N\bar{N}$ annihilation
near the Earth we accept a leaky-box model. In this model the
intensity can be defined by a simple expression
\begin{equation}
I=\frac{v_{\bar{p}}}{4\pi}<\dot{n}_{\bar{p}}>\tau_{conf}.
\end{equation}
Here $v_{\bar{p}}$ is the velocity of antiproton,
$<\dot{n}_{\bar{p}}>=\frac{\int\frac{1}{4}n^2<\sigma
v>dV}{V_{Gal}}\frac{dm_{\bar{p}}}{dE}$ is the mean number of
antiprotons created in a unit volume per second per energy
interval $E-(E+dE)$, averaged over the volume of Galaxy,
$\tau_{conf}$ is the confinement time, the other notations are
analogous to those introduced in Eq(5). Time $\tau_{conf}$ is a
parameter of the model and it has the meaning of the time of
$\bar{p}$ confinement in Galaxy. We chose $\tau_{conf}$ to be
$10^7$ years as in the early works \cite{Fargion}. The volume of
Galaxy is supposed to be the volume of region where antiprotons
are confined and this region is chosen in form of a disk with
radius 25 kpc and semiheight $z_h=4$ kpc, typical for CR models.
Energy losses of antiprotons are neglected in Eq(11). Such
simplification is justified by small mean matter column
(5 g/cm$^2$) traveled
by cosmic nuclei, , which was deduced from CR analysis.
A small fraction of antiprotons, which lose their energy in an
inelastic scattering on protons of medium ("tertiary" component),
appears in antiproton spectra at very low energy \cite{SM-2}.
As a background the secondary antiprotons predicted in model "C" of
\cite{SM-1} were used. For a comparison with observations the
combined data of BESS'95 and '97 were used \cite{BESS}, which were
demodulated with parameter $\Phi=540$ MeV, derived from BESS'95
\cite{BESS-all}. Data BESS'98 belong to time of high solar
activity what is less suitable from point of view of detection of
possible dark matter annihilation sources \cite{BESS-all}. Figure
5 shows a spectrum of cosmic antiprotons. Unlike the cases of
gamma-radiation and positrons the spectra of antiprotons from
$N\bar{N}$ annihilation for different values of Heavy neutrino
mass coincide within an energy interval presented in Fig.5. A
charge-sign dependence of solar modulation and an effect of
re-acceleration are significant at low energy, making results in
this energy range less certain.
\begin{figure}
\begin{center}
\centerline{\epsfxsize=7.5cm\epsfbox{antiproton.eps}}
\caption{Cosmic antiprotons (LIS): BESS(95+97) data, predicted
background (dot-dashed line), and the best-fit contribution from
47-80 GeV neutrino DM (the set of dashed lines is pure
annihilation antiproton fluxes, the set of solid lines is the sum
of background and annihilation fluxes) for Evan's halo model.
Note, that for the considered interval of neutrino masses
the sets of dashed and solid lines are virtually reduced to
single lines.
\label{fig.5}}
\end{center}
\end{figure}
An increase of parameter $\tau_{conf}$ or/and a decrease of volume
comprising $\bar{p}$ propagation zone in Galaxy lead to a decrease
in the best-fit density parameter $\xi_{loc}$. Uncertainties in
$\tau_{conf}$ and $V_{Gal}$ lead to overall uncertainty of about a
factor 2.
Note, that the analysis of CR carried out in this work differs
from analogous analysis performed in the previous works
\cite{Fargion}, \cite{4thN-CR} by more refined consideration, in
particular more realistic CR models and models of distribution of
Heavy neutrinos in Galaxy.
\section{Heavy neutrino in underground versus cosmic rays signals}
As one can see from figures 2-5, the presence of dark matter
annihilation sources in the form of Heavy neutrinos improves
description of existing data on cosmic gamma-radiation, positrons,
antiprotons with corresponding $\xi_{loc}$ selected in the best
way from observational data. The annihilation fluxes in these
figures were obtained for Evan's halo model (Eq(3)) as described
above. In the same manner as in case of Evan's model the
parameters $\xi_{loc}$, allowing to fit in the best way CR data,
were obtained for isothermal halo model (Eq(4)). All these
parameters for different values of neutrino mass are shown in
Fig.6 in comparison with those preferable in measurements of DAMA.
There is the set of black lines in upper half of figure, starting
at $m=46$ GeV and ending at $m=80$ GeV, which corresponds to
$\xi_{loc}$ parameters, inferred from CR analysis. Pairs of solid
(dot-dashed), dotted and dashed lines of this set are related with
the best-fit $\xi_{loc}$ for gamma-radiation from halo (GC), for
cosmic positrons and antiprotons respectively. Upper and lower
lines of each pair correspond to Evan's and isothermal halo models
respectively. Solid and dashed grey lines, going across the
picture, enclose favorable region of DAMA. For consistent
comparison of $\xi_{loc}$, inferred from CR analysis using Evan's
halo model, the values of $\xi_{loc}$, derived from analysis of
DAMA/NaI measurements based on the same Evan halo model, are shown
by dashed grey lines.
\begin{figure}
\begin{center}
\centerline{\epsfxsize=8cm\epsfbox{DAMAgammapositantip.eps}}
\caption{DAMA favorable region (as in Fig.1) and
the best-fit density parameters deduced from cosmic
gamma-radiation (from halo and CG), positron and antiproton
analysis. Horizontal grey dashed and solid lines enclose DAMA
favorable region accepting Evan's halo model and other halo models,
respectively. The set of upper lines corresponds to the $\xi_{loc}$
parameters preferable for CR data. In this set of lines, upper and
lower lines of the same type correspond to Evan's halo model and
to isothermal halo model, respectively. Vertical grey dashed and
solid lines restrict $1\sigma$ and $2\sigma$ allowable range of
the 4th neutrino mass deduced from the particle physics data analysis.
\label{fig.6}}
\end{center}
\end{figure}
All $\xi_{loc}$ parameters, obtained from CR analysis, define
parameters favored by CR data as well as upper constraints
imposed by CR data. So, given results allow to make a conclusion
that CR data are consistent with measurements of DAMA/NaI treated in
framework of hypothesis about the 4th generation neutrino.
An additional
source of information about possible existence of WIMPs is data
from the search for light neutrino fluxes from annihilation of
WIMPs accumulated inside the Earth and Sun. But existing data of
underground measurements of neutrino fluxes exhibit, contrary to
CR data, a lack of neutrinos as compared to predicted background
(atmospheric neutrinos). New physics is possibly required here. An
interpretation based on 3-flavor neutrino oscillations fails to
reproduce all appropriate existing data without an
introduction of a new sterile
neutrino. An estimation of muon neutrino fluxes from annihilation
of Heavy neutrinos inside the Earth gives a result comparable with
the expected corresponding atmospheric neutrino flux in the energy
range $>3$ GeV for $m=50$ GeV for acceptable parameters of Evan's
halo model. In this analysis the result depends also on WIMP
velocity distribution which affects the capture rate of WIMPs by
the Earth. This ratio is reduced by a few times if the velocity
distribution given by Evan's halo model (Eq(A1-A4) \cite{Belli})
with velocity parameter $v_0=170$ km/sec is replaced by Maxwellian
distribution with r.m.s. velocity $v_0=220$ km/sec. An analysis of
underground measurements of upward-going muons (Super-Kamiokande,
MACRO, Baksan), induced by neutrino fluxes, requires its further
development.
A question about an agreement between all predicted parameters for
the 4th generation neutrino, helping to improve description data
of different species, is of greater interest. Striking is a
relative closeness of $\xi_{loc}$ parameters preferable for
different observations.
Figure 6 shows that "play" with the form of density distribution
of Heavy neutrinos in Galaxy is able to change significantly
$\xi_{loc}$ derived from different observations. An agreement
between the considered data is possible. For the chosen isothermal
model all $\xi_{loc}$ parameters (lower lines from pairs of lines
of upper (black) lines in Fig.6) favored by CR data, a neutrino
mass is close to its lower constraint, in the region of
corresponding magnitudes, deduced from DAMA experiment accounting
for different halo models.
In the case of isothermal model, the
value of $\xi_{loc}$ parameter inferred from analysis of cosmic
gamma-radiation from halo (solid line), differs from the other
ones. This discrepancy can be due to a possible extragalactic
$\gamma$-radiation (see Fig.3) the account for which can lead to
better agreement. But, of course, results of indirect WIMP
searches should be treated cautiously taking into account the low
precision of the corresponding experimental data.
Given Evan's halo model, predictions of $\xi_{loc}$ from the data
on different CR species are differed by a factor of three. In a
view of approximations in CR analysis described above it should
not be considered as the principal discrepancy. An agreement achieved
between predictions of parameters preferable for CR data and for
measurement of DAMA would require reduction of parameter
$\xi_{loc}$, preferred by CR data, by a factor a few - ten
in the allowed range of neutrino mass below 50 GeV. Such
a reduction corresponds to an amplification of the annihilation
flux proportional to the square of that factor.
In other words, the results of DAMA/NaI experiment are compatible with
indirect effects of 4th neutrino annihilation, but the observational
indications to WIMP annihilation effects in cosmic rays and gamma
radiation can be explained together with these results only for some models
and for a very narrow interval of neutrino masses (46-47 GeV). To increase
the range of neutrino masses, at which direct and indirect WIMP signals
can find simultaneous explanation,
the rate of 4th neutrino annihilation in Galaxy should be much larger.
\section{Three ways to extend neutrino models and mass range}
\subsection {Amplification of neutrino annihilation due to
clumpiness}
There is a possibility to amplify CR flux created by dark matter
annihilation sources maintaining local density and average density
distribution of DM in Galaxy. This possibility is related with a
clumpy DM distribution in Galaxy (\cite{Berezinsky}
and references therein). In particular we are considering local
clustering in our galactic center and halo with no peculiar
density enhancement for our neighborhood. The opposite situation
(higher Solar and lower global galactic density) is a
possibility much less probable and attractive.
CDM might form clumps on the stage of structure formation in the
Universe. As it was shown in \cite{Berezinsky} a small fraction of
total DM mass (a few$\times10^{-3}$) can survive to present time
in form of clumps and it is enough to provide strong enhancement
(up to a few orders of magnitude) of annihilation signal. Being
a non-dominating DM component, Heavy neutrinos most likely
do not form their own clumps. A formation of clumps should be governed
by the dominating CDM component. Heavy neutrinos should subserve in
such processes for values of the formed clump mass, $M_{clump}$,
exceeding some minimal one, $M_{N\,min}$. The last one is defined by
the size of a proto-clump equal to free-streaming length of
neutrinos, $\lambda_{fs}$, when inhomogeneities start to grow.
$\lambda_{fs}$ depends on the moment of Heavy neutrinos
decoupling from an ambient plasma. For 50 GeV neutrino the
temperature of decoupling is estimated as $T_d\sim
20$ MeV,
whereas the mass $M_{N\,min}\sim 0.6\times10^{-6}M_{\odot}$ (Eq(37)
of \cite{Berezinsky}). Provide a dominant CDM component has a less
free-streaming scale (that is, for instance, quite probable for
neutralinos and heavy gravitinos), there would exist the clumps with
masses in the range $M_{clump}>M_{min}$ so that
$M_{min}<M_{N\,min}$ (for neutralino $M_{min}$ is estimated as
$\sim10^{-8}M_{\odot}$ \cite{Berezinsky}).
For Heavy neutrinos the
ranges of clump masses are $M_{min}<M_{clump}<M_{N\,min}$ and
$M_{clump}>M_{N\,min}$. Creation of clumps only of the second mass
range is not expected to proceed with a separation of the dominant CDM
component and Heavy neutrinos. We will suppose a conservation of
proportionality (ratio) between densities of dominant CDM
component and Heavy neutrinos in a such clump creation. Clumps
lighter than $M_{N\,min}$, if they are, can be populated by Heavy
neutrinos in less degree in accordance with the mechanism of
adiabatic loss of energy by collisionless particles (neutrinos) in
an external variable gravitation field \cite{ZKKC}.
Estimations of \cite{Berezinsky} for the enhancement factor of
dominant DM annihilation flux due to the presence of clumps can be
applied to Heavy neutrinos. This factor is defined as
\begin{equation}
\eta=\frac{I_{clump}+I_{hom}}{I_{hom}}
\end{equation}
where $I_{clump}$ and $I_{hom}$ are the intensities of
annihilation fluxes from clumps of DM and homogeneously distributed
DM. It crucially depends on minimal mass of clumps,
$\eta(M_{min})$, the lightest clumps give the main contribution
into an annihilation rate. Under assumption on proportionality
between densities the predictions for enhancement factor can be
referred to non-dominating Heavy neutrinos DM assuming the minimal
clump mass to be $M_{N\,min}$
\begin{equation}
\eta_N=\eta(M_{N\,min}).
\end{equation}
Note, that Fig.1 implies some deviation from such proportionality
within a factor 1-100 at $m=50$ GeV for an average local density
(which is deduced from dynamical observation to be within 0.17-1.7
GeV/cm$^3$), if real local density (in vicinity of the Earth) does
not differ significantly from the average one.
Additional contribution into an enhancement of the neutrino
annihilation rate will result from the clumps with a mass in the
range $M_{min}-M_{N\,min}$. The enhancement from these clumps can
be roughly estimated as a corresponding estimation in
\cite{Berezinsky} for $M_{min}$ reduced with respect to smaller
relative compression of Heavy neutrino density inside the given
clumps as compared to that of dominant DM. In each such clump the
relative compression of neutrino density, i.e. the ratio of
densities inside a clump and outside it (of homogeneous component
near the clump), should be smaller than that of dominant component
of matter (CDM) in accordance with \cite{ZKKC}
\begin{equation}
\frac{\rho_{N\,clump}}{\rho_{N\,hom}}=\left(\frac{\rho_{CDM\,clump}}{\rho_{CDM\,hom}}\right)^{3/4}.
\end{equation}
As a first approximation we neglect details (differences) of
density distributions of dominant component and Heavy neutrinos
inside these clumps. So, the factor $\eta$ is determined by
squared ratio of densities above both for Heavy neutrinos and for
CDM. The enhancement factor for Heavy neutrinos for a clump mass
between $M_{min}$ and $M_{N\,min}$ can be estimated as
\begin{equation}
\eta_{N\,add}=\eta(M_{min})^{3/4},
\end{equation}
where $\eta(M_{min})$ is the enhancement factor as predicted in
\cite{Berezinsky} (for dominant DM component).
Factor $\eta$ increases with a decrease of clump mass
\cite{Berezinsky}, so a contribution from the clumps of mass
$M_{min}<M_{clump}<M_{N\,min}$, which are relatively less
populated by Heavy neutrinos, can be comparable with those of mass
$M_{clump}>M_{N\,min}$. For instance, for an index of primeval
perturbation spectrum $n_p=1.05$, index of power-like density
distribution inside the clump $\beta=1.8$ at
$M_{N\,min}=0.6\times10^{-6}M_{\odot}$ we have $\eta_N\approx20$
(see Fig.5 of \cite{Berezinsky}), and assuming
$M_{min}=10^{-8}M_{\odot}$ we obtain
$\eta_{N\,add}\approx(30)^{3/4}\approx13$. For essentially smaller
values of $M_{min}$ the contribution from clumps with
$M_{min}<M_{clump}<M_{N\,min}$ is turned out to prevail.
A density parameter $\xi_{loc}$ should decrease as a square root
of enhancement factor $\eta_N$, so in example above we get
$\xi_{loc}$ by a factor 5 less at 50 GeV than it is in Fig.6. For
other values of neutrino mass of interest the result is virtually
the same. An agreement between measurement of DAMA/NaI and CR
observation data is possible for the hypothesis of 4th neutrino
with Evan's halo model within allowed neutrino mass range below 50
GeV (see Fig.7).
\begin{figure}
\begin{center}
\centerline{\epsfxsize=8cm\epsfbox{DAMAgammapositantip-33.eps}}
\caption{Effect of neutrino clumpiness. All notations are analogous to those
of fig. 6.
\label{fig.7}}
\end{center}
\end{figure}
Note that in our approximation we did not pay attention to a
correction for given quantitative estimations of an effect of
clumps due to a different DM density distribution in Galaxy
(Evan's) than it was supposed in \cite{Berezinsky}. Also note,
that a destruction of clumps near GC \cite{Berezinsky} should
partially decrease the enhancement of annihilation $\gamma$-flux
from GC due to clumps. This may improve agreement between results
for photons from GC and the halo.
In the case if a minimal possible DM clump mass, $M_{min}$, is
greater than $M_{N\,min}$ (there is only a unique mass range),
then the enhancement factor would be given by Eq(13) with
$M_{min}$ instead of $M_{N\,min}$.
\subsection {Amplification of neutrino annihilation due to
new Coulomb-like interaction}
An annihilation signal as a signature of existence of DM particles
like Heavy neutrinos implies a condition of the presence of both
particles and antiparticles. The case considered in the present
article was based on
an assumption of charge symmetry of 4th generation particles,
i.e. an equality between the numbers of primordial 4th neutrinos and
antineutrinos. Such a statement can find physical foundation in
superstring models. New charge(s) is(are) predicted there which,
being strictly conserved, can be ascribed only to 4th generation
particles \cite{Shibaev}. It accounts for absolute stability of
the lightest particle bearing this charge (assumed to be the 4th
neutrino) and an equality between particles and antiparticles of a
new generation.
An important consequence of a new charge is an effect of new
interaction. In a wide class of models this charge is $U(1)$-gauge
charge which leads to existence of corresponding massless gauge bosons
($y$-photons) and to a Coulomb-like interaction of 4th neutrinos. It was
revealed in \cite{Sakharov
enhancement}, that this new
interaction does not influence significantly
the results of 4th neutrino freezing out in early Universe, but it can
increase their annihilation
signal in Galaxy by a few hundred
times. The matter is that in the considered range of neutrino masses
ordinary electroweak
$Z$-bozon resonance annihilation channel dominates over the new channel of
2$y$annihilation
and the main effect of new interaction is Coulomb-like factor in the cross
sections
of slow charged particles. Such factor, $C$, first deduced by A.Sakharov
\cite{Sakharov} for
Coulomb interaction of slow electrically charged particles, in the case
of Coulomb-like interaction with "fine structure constant" $\alpha_y \sim
1/137 - 1/14$
has the form \cite{Sakharov enhancement}
\begin{equation}
C=\frac{2 \pi \alpha_y c/v}{1 - \exp({-2 \pi \alpha_y c/v})},
\end{equation}
where $c$ is the speed of light and $v$ is the relative velocity of charged
particles.
At $v/c \ge 1/10$, what is the case for the period of 4th neutrino freezing
out in the early Unvierse,
this factor is close to 1, but for $v/c \ll 1$, being the case for
neutrinos in Galaxy, it increases
their annihilation rate by the factor of Sakharov's enhancement
\cite{Sakharov enhancement}
\begin{equation}
C(v)={2 \pi \alpha_y \frac{c}{v}} \simeq 10^2
\left(\frac{\alpha_y}{1/60}\right)\left(\frac{300\,{\rm km/s}}{v}\right).
\end{equation}
It would provide a decrease of predicted the best-fit
$\xi_{loc}$ parameters as square root of that enhancement
{\bf (being properly averaged over velocity distribution).}
The account for this new Coulomb-like
interaction, possessed by 4th neutrinos, extends the possibility
of unified explanation of CR and DAMA/NaI data in the framework
of 4th neutrino hypothesis for the most part of considered
interval of neutrino masses. Figure 8 corresponds to the case of
$\alpha_y=1/30$, which is the most natural value for this coupling constant.
\begin{figure}
\begin{center}
\centerline{\epsfxsize=8cm\epsfbox{DAMAgammapositantip-30.eps}}
\caption{Effect of neutrino Coulomb-like new
interaction. All notations are analogous to those
of fig. 6.
\label{fig.8}}
\end{center}
\end{figure}
The case of clumpiness of Heavy neutrinos with new interaction, while
being possible, is more complex and involves both factors ($\eta$ and
$\alpha_y$). Its combined role will offer more tunable scenarios;
however for sake of simplicity we wil not take it into account here.
On the first sight Sakharov's enhancement does not influence the annihilation
rate of
accumulated 4th neutrinos inside the Earth and Sun, which is
defined by their capture rate. However, the existence of
stable quark of 4th generation can make the picture more complicated.
Being compatible \cite{Shibaev} with the constraints on the abundance
of anomalous isotopes, the presence of small amount of anomalous hadrons,
containing this quark (or antiquark) and possessing
the $U(1)$-gauge charge can cause asymmetry in capture rates for 4th
neutrino
and antineutrino and thus influence their annihilation rate
in Earth and Sun. A role of this gauge interaction,
its charges and field in
effects of Heavy neutrinos as well as the account of possible
existence of stable 4th generation hadrons requires a separate discussion.
\subsection {The role of Heavy neutrino asymmetry and decays}
If neutrinos of 4th generation do not possess new $U(1)$-gauge
charge, there does not appear any fundamental reason for their
absolute stability as well as for their strict charge symmetry.
If charge symmetry is absent, which is the case for
baryons in the Universe, the 4th neutrinos would prevail over
their antineutrinos (or vise versa) and they could decay. A
magnitude of this asymmetry, being defined as the ratio of
difference of present number densities of 4th neutrinos and 4th
antineutrinos ($\delta n$) and present relic photon number
density, is an additional parameter of the problem.\footnote{In the case
of light neutrinos, which decouple in the conditions of thermodynamical
equilibium, such asymmetry would determine their chemical potential
(see review in \cite{Dolgov}).}
This parameter
should be much less than that of baryons in order not to exceed an
essentially relative contribution of 4th neutrino into the density
of CDM derived from measurements of DAMA (Fig.1). A difference
$\delta n\sim {\rm a\, few}\times n_{sym}$, where $n_{sym}$ is the
relic Heavy neutrino density in the case of charge symmetry,
leads to an increase of relic 4th neutrino density by a few times
and to an exponential decrease of density of relic 4th
antineutrinos by a few orders of magnitude. As seen in Fig.1 this
would be especially favored for neutrino mass around 50 GeV. If
relic neutrinos survive to present time, their annihilation
signals from Galaxy and from the Earth and Sun weaken by a few
orders of magnitude as compared to the symmetric case.
However, in the asymmetric case another signature of 4th neutrino DM in CR
is possible.
Neutrinos of 4th generation are unstable in this case
and their decays in the galactic halo can lead to effects similar to the
ones from
stable neutrino annihilation.
In this case CR data can be reproduced,
if 4th neutrino decay lifetime is, for Evan's
density distribution,
\begin{equation}
\tau\approx \xi_{loc}\cdot (0.2-2)\times10^{19}{\rm\, years}\frac{50\,{\rm
GeV}}{m}.
\label{unutime}
\end{equation}
Here $\xi_{loc}$ is the local density fraction of neutrinos in charge
asymmetric case.
The value of the decay rate preferable for CR data changes weakly with
variation of neutrino mass
(since it is defined by the observed CR fluxes), so uncertainty 0.2-2 in
estimation (\ref{unutime}) is mainly induced by uncertainty of best-fit
$\xi_{loc}$
parameters deduced from data of different CR species (for Evan's halo
model).
Note that there is a difference
by a factor 2 in energy release for annihilation reaction and
decay process. But in the numerical estimations for the effects of decay
the predictions for CR fluxes,
induced by neutrino annihilation, can be used without significant change
with only proper account for the change in the energy release,
provided that hadron modes are present in decay with sufficient
probability.
Parameter $\xi_{loc}\approx 0.01 - 0.001$ would provide for neutrino
of mass about 47-70 GeV with the lifetime given above
an agreement between the measurement of DAMA/NaI and CR data.
So, preferable lifetime of 4th neutrino is
$\tau\sim (2\times10^{15}-2\times 10^{17})$ years.
Note, that clumpiness does not affect the
fluxes of products of Heavy neutrino decay in the Galaxy. Also
note, that the annihilation rate of neutrinos accumulated inside
the Earth and Sun in the symmetric case corresponds to a timescale
less than the age of Solar system (much less in case of the Sun).
So the decay rate with the lifetime above (\ref{unutime}) is strongly
suppressed as compared to a corresponding annihilation rate for
the symmetric case.
\section{Conclusions}
In the present work it was shown that the positive result at 6.3
sigma C.L. obtained in DAMA/NaI and observed possible excesses in
cosmic gamma-radiation, positrons and antiprotons can be in
agreement within the framework of hypothesis of a 4th neutrino
mass hidden nearby half the Z-boson mass. The evident advantage of
this hypothesis is the minimal number of physical parameters. In
the simplest case it is only the mass of neutrino (to be compared
with minimally 5 parameters in the case of SUSY dark matter).
But even in this simplest case we have revealed the complex model
dependence
on the galactic mass distribution and cosmic ray diffusion. We
have shown its compatibility within realistic values of Heavy
neutrino mass. The model may be naturally extended in the case of
in-homogeneous (clumpy) galactic halo, new Heavy neutrino
Interactions related to its necessary stability, relic neutrino
asymmetry and it consequent unstability and decay.
In those models there are room for better agreement between
underground and cosmic rays signals.
The lightest neutrino masses ($\sim 50$ GeV) might be searched
inside the old LEP data regarding electron pair annihilations into
one photon with missing energy \cite{accelerators}; the largest ones
($\sim 57-75$ GeV) might be
discovered by near future LHC search of invisible Higgs boson
decay\cite{Higgs}.
Additional satellites and antimatter search in Space might
define the exact parameter range for this extension of the
lepton sector, whose existence might be tested also in the
search for a novel pair of 4th quark family\cite{4quark}.
\section{Acknowledgment}
We are grateful to R.Bernabei, P.Belli and D.Prosperi for important
discussions.
K.B. thanks Universita' di Roma "Tor Vergata", INFN section Roma2 and
Universita di Roma' "La Sapienza" for their support and hospitality and
K.I. Shibaev for the help. M.Kh. is grateful to Abdus Salam ICTP (Trieste)
for hospitality.
This work was supported in part
by grant RFBR 02-02-17490 and by the Federal
Program of the Russian Ministry of Industry,
Science and Technology 40.022.1.1.1106.
|
{
"timestamp": "2004-11-05T20:11:16",
"yymm": "0411",
"arxiv_id": "hep-ph/0411093",
"language": "en",
"url": "https://arxiv.org/abs/hep-ph/0411093"
}
|
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